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Complex paths and perturbations for slow-roll cosmologies Losic, Bojan 2001

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C O M P L E X P A T H S A N D P E R T U R B A T I O N S F O R S L O W - R O L L C O S M O L O G I E S B y Bojan Losic B. Sc . i University of Waterloo , 1999 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F PHYSICS A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A July 2001 © Bojan Losic, 2001 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be.granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of Br i t i sh Columbia 6224 Agricul tura l Road Vancouver, B . C . , Canada V 6 T 1Z1 Date: Abstract We extend the model orignally proposed by Unruh and Jheeta in [6] in two ways. Firs t we derive the complex action to thi rd order in the slope of the slow-roll potential and find that it depends on precisely which endpoint is chosen in complex r space, and that it also has phase contributions to the semi-classical wave function. Secondly, we derive the reduced Hamil tonian action to second order in classical, non-homogenous scalar metric and matter fluctuations about an arbitrary F R W background. We analyze the I = 1 mode for the closed F R W universe and prove it is a gauge mode. We find a contour in r space for which this reduced action has a simple character. We use this character to make an argument concerning the appropriateness of using Linde's or Hawking's sign in determining the amplitude contribution to the wave-function from these classical fluctuations. i i Table of Contents Abstract ii Table of Contents iii List of Figures vii Acknowledgements viii 1 Introduction 1 1.1 Wave-function of the universe 1 1.1.1 Semi-classical approximation 3 1.2 Outlook 4 2 Complex solutions for 'slow-roll' cosmologies 6 2.1 Decomposition of spacetime: the '3+1' formalism 6 2.1.1 Hamil tonian action 9 2.2 The slow-roll model 11 2.2.1 Init ial and final conditions: the N B P and the reality of aF and (j)Bf 13 2.2.2 Power series solution of complex equations of motion 14 2.2.3 Zeroth order solutions 16 2.2.4 Zeroth order action 18 2.2.5 First order action 20 2.2.6 Second order action 20 2.2.7 T h i r d order action 22 i i i 2.3 Summary and Outlook 24 3 Constrained Hamiltonian Formulation 26 3.1 Introduction 26 3.2 Example of a constrained Hamiltonian formulation 27 3.2.1 Geometric interpretation of constraints; first and second class con-straints 32 3.2.2 Gauge fixing to isolate physical degrees of freedom 34 3.3 Hamil tonian Formulation of General Relat ivi ty 35 3.3.1 Constraint algebra of vacuum G R in closed spaces 36 3.3.2 Extension to asymptotically flat and asymptotically A d S space-times 42 3.3.3 Incorporation of surface terms into the Hamil tonian formulation 44 3.4 Summary of chapter 47 4 Perturbations 49 4.1 Introduction 49 4.2 Linearization Stabili ty 50 4.2.1 Linearization stability in non-vacuum, symmetric, background spacetimes 52 4.3 Metr ic and Matter Perturbations 55 4.4 Formulation of the action 57 4.5 Hamil tonian action for metric and matter perturbations 58 4.6 Ac t ion for K = +1 60 4.7 Gauge fixation and 'gauge invariant' variables 64 4.8 Possible gauge choices 66 4.8.1 Setting E = 0 = $ 66 iv 4.8.2 Setting B = 0 = E 67 4.8.3 Setting A = ip,B = 0 68 4.8.4 Setting A = 0, B = 0 68 4.8.5 Setting n$ = 0 = E 69 4.8.6 Setting ^ = 0 = E 70 4.9 Reduction of the K = +1 Act ion 70 4.9.1 Reduction in different gauges: E = & = 0 72 4.9.2 The £ = 1 mode 74 4.10 l£ = V mode for K = -1, 0 77 4.11 Summary of Chapter 78 5 Complex Paths 80 5.1 Zeroth order contour: Hartle Hawking contour 81 5.2 Simplified background dependence: non-canonical second-order action . 82 5.3 Higher order contour in r-space 85 5.4 Matching conditions 89 5.5 Particular background solution 89 5.5.1 Semi-classical probabilties 92 5.6 Conclusion 93 Bibliography 94 A Spatial Harmonics 97 A . l General derivation of harmonics 97 A . l . l K = 1 98 A.1.2 K = - l 99 A . 1.3 K = 1 Eigenfunctions 101 v A.1.4 Convenient representations of K = 1 polynomials 102 A . 1.5 K = - 1 Eigenfunctions 103 A . 2 Summary of eigenvalues 103 B Transformation of Perturbations 104 C Decoupling of Scalar, Vector, and Tensor Modes 106 D Reduction of Hamiltonian Action in Spherical Symmetry 111 v i List of Figures 1.1 P a t h in tegra l v i s u a l i z e d 2 2.1 L a p s e N a n d shift Nl specify coord ina te sys tem 8 2.2 So lu t ions for T R , 77 whenever va,f > 1 18 3.1 B o u n d a r y in tegra l i n flat spacet ime 43 5.1 H a r t l e - H a w k i n g C o n t o u r : Z e r o t h - O r d e r C o n t o u r 81 5.2 F i r s t - o r d e r C o m p l e x C o n t o u r 88 5.3 E v o l u t i o n of b a c k g r o u n d scale factor 91 v i i Acknowledgements M y family, friends, and supervisor B i l l Unruh al l have my immense gratitude for their support in al l senses of that word. Special thanks are also due to Robert Brandenberger and Ignacio Olabarrieta for many helpful discussions and a crit ical reading of this thesis. More specifically, I would like to thank my parents for helping me with my algebra on the small Strawberry Shortcake table in Selma & Mik i ' s room before watching M r . E d . v i i i Chapter 1 Introduction Quantum cosmology is an attempt to obtain semi-classical predictions from general rel-at ivity in lieu of a full quantum theory of gravitation. Among al l of the approximations to such a full theory that we currently have, the Euclidean path-integral formulation [1] of quantum gravity has proven quite useful in describing the origin and evolution of spacetime. One important application of this formulation is the so-called No Boundary Proposal ( N B P ) due to Hartle and Hawking [2], in which they suggested a compelling theory of the ini t ia l conditions for a closed universe. 1.1 Wave-function of the universe The basic idea is to define a complex-valued function called the wave-function of the universe, denoted \T>, within the Euclidean path-integral formalism. This function \T> wi l l depend on the three-geometries hij induced by slices of constant time E t as well as any classical fields ^4> o n that three-geometry. Its modulus squared wi l l be interpreted as the relative probability of obtaining a particular h'^, ( 3 V from a given ^<f). The formal definition originally suggested involved a path integral over al l non-singular four-geometries gap and matter fields ^<j> which induced three-geometries h'^ plus matter fields ^4>' on the boundary DM, namely m ^ ^ m ^ h i ^ U l h ' i ^ U ' y = £ / e - ^ ^ W f l W ] . ( i . i ) M 1 Chapter 1. Introduction 2 Here, SE is the Euclidean action for the metric gap and matter fields ^4> which induce the hij, ( 3 V , and the sum over M is over some suitable class of manifolds. We can visualize the path integral in general by imagining a sum of spacetimes wi th fixed in i t ia l and final hypersurfaces, E j and E / , each possessing progressively more and more clasically forbidden three-geometries and matter configurations h^, ( i - e - those not allowed by any classical Cauchy development of the ini t ia l data) weighted by the number of these clasically forbidden configurations [3], as shown in Figure 1.1. The N B P is a restriction of the sum over the classes of M. The manifolds M are taken to have a lone boundary dM on which h'^, ^<f>' are specified and no other boundary. Thus the sum is over classes of compact manifolds which have only the boundary necessary to specify the arguments of the wave function \1>. It is essentially this topological specification which not only defines the N B P but provides hope that the idea may be generalizable to any full quantum theory. Classical Cauchy Development Figure 1.1: This figure represents the path integral (1.1) as a sum of probabilities asso-ciated wi th the evolution of a given spacetime. The leading term is the classical Cauchy development. The shaded slices in al l subsequent quantum terms are the classically for-bidden configurations hij, ^(f). The Cauchy development of the spacetime dominates the wave function, but there are non-zero contributions, weighted by the number of shaded slices, from more exotic quantum evolutions of the same ini t ia l data to the same final data. Whi le elegant, equation (1.1) is purely formal for a number of reasons. The most Chapter 1. Introduction 3 obvious is that it is impossible to evaluate even in simple circumstances, so its practical ut i l i ty without resorting to some approximation is very small at present. Another major reason is that the Euclidean action for classical general relativity is not bounded from below 1 , so that in order to use equation (1.1) at all we must admit complex-valued configurations simply to guarantee convergence [4]. 1.1.1 Semi-classical approximation Given that we admit complex metrics gap and matter fields ^4>, we can extract physical predictions from equation (1.1) by assuming that the extrema of the Euclidean action SE dominate the above path integral, so that $ is taken as i This is known as the semi-classical approximation. The index £ sums over the £ four-metric and four-matter configurations ( ( ^ i j ) * / 3 V * ) which minimize the action (SE)(, and the factors At are determined by integrals over fluctuations about the solution[5]. These £ configurations are the 'saddle points' of the action and represent solutions of the equations of motion with the N B P (together with any regularity conditions the N B P may imply via the equations of motion) imposed at the ini t ia l three-surface, and reality conditions at the final surface2 [5]. It would seem reasonable to conclude that the semi-classical approximation to the full path integral implies that the history of the universe is simply the classical solution of the Einstein field equations ( E F E ) which dominates the path integral. We can see that this 1This can be seen by conformally transforming the Einstein-Hilbert Lagrangian, the Ricci scalar R, by the transformation gap = ^29a0- ^ is strictly positive. In four dimensions, R = H ~ 2 [R - 6gaKVaWK lnH - 6<?aK(Va lnfi)V« lnfi], which can be made arbitrarily negative when we Wick rotate the time. 2The reality of the final three-metric hij and any matter fields ^4> is a reflection of the fact that we can only observe real quantities, and not the NBP, which demands initial regularity. Therefore the arguments of the wave function on £ / , (h'ij,^cj)'), must be real. Chapter 1. Introduction 4 cannot be true because, in general, complex solutions of the E F E wi l l contribute v ia the semi-classical approximation in addition to wholly real solutions. A s a complex solution cannot represent a history of the universe i t follows that the semi-classical approximation does not imply that the dominating real classical solution is the correct history [6],[5],[4]. 1.2 Outlook In order to obtain a more precise idea of what these complex solutions wi l l imply for the semi-classical wave function of the universe, we wi l l follow Unruh and Jheeta [6] and consider a simple model in which the geometry and matter fields are highly symmetric, namely possessing the symmetry of S3. Their scalar matter field has a non-trivial 'slow-ro l l ' potential, taken to be linear with very small slope e, and they solve the complex equations of motion in a power series in e. They found that there were an infinite number of complex solutions, of which many were analytically related, and each of which gave an action for the final state of the system which was the same for large classes of paths and endpoints (to the order of approximation they examined) i n complex time. Another result of their analysis was that in the absence of non-homogenous modes of the metric and matter fields the N B P principle does not, on its own, select the sign of the amplitude of the semi-classical wave function. This sign has been the subject of considerable controversy ([7] and refereces therein). In this thesis we achieve two things. First , in Chapter 2, we extend the order of approximation of the above work and find, that the action to higher order does indeed depend on which of the paths is selected. We then compute the physical (reduced) Hamil tonian action to second order in non-homogenous metric and matter fluctuations in Chapter 4, using the constrained Hamiltonian formalism developed in Chapter 3. The Dirac reduction of the action, the discussion of the £ = 1 mode, and the elucidation of Chapter 1. Introduction 5 the decoupling mode theorems in Appendix C are al l original. In Chapter 5 we link the results with the slow-roll model [6] and give our conclusions. Chapter 2 Complex solutions for 'slow-roll' cosmologies In this chapter we briefly review the main results of [6] and then present the higher order calculations. In reviewing their results and for further applications in Chapter 3, it wi l l be necessary to first briefly introduce the standard A D M formalism. 2.1 Decomposition of spacetime: the '3+1' formalism The standard four-dimensional spacetime of general relativity may be decomposed into a infinite series of spacelike hypersurfaces parametrized by some timelike parameter, in what is called the '3+1' formalism [8]. W i t h i n such a formalism each of these hyper-surfaces has a geometry described by an induced three-metric, and the manner in which successive hypersurfaces are related is described by additional functions (which turn out to be non-dynamical). In addition to providing an ini t ia l value formulation for the whole theory, which is very important for doing analytic and especially numerical calculations, such a decomposition provides considerable insight into not only the dynamics of gen-eral relativity but the constraints such dynamics must satisfy as enforced by the Bianchi identities 1 . Such a process is sometimes called the foliation of spacetime, which is clearly 1The Bianchi identities are the statement that the covariant four-divergence of the Einstein tensor is zero, and reflect the diffeomorphism (infinitesimal coordinate transformation) invariance of the theory. Because of the field equations, they also imply that stress-energy must be conserved. 6 Chapter 2. , Complex solutions for 'slow-roll' cosmologies 7 arbitrary in the choice of this global parameter. However, for globally hyperbolic 2 space-times there is a very useful foliation named after Arnowit t , Deser, and Misner ( A D M ) , who first proposed it in [9]. Following Wald [8], let us briefly describe the details of the A D M formulation. The spacetime (with coordinates x01) is foliated by hypersurfaces Et (with coordinates yl) defined by a constant value of the global time function t, which are called spacelike. The projection of the four dimensional metric gap onto these surfaces of constant time is a three-dimensional, positive-definite (Riemannian) 3 metric given by hije\e3p = gap + nanp, (2.1) where na is the unit normal to the hypersurface 4 and ela = (dyl/dxa)\t. Let t ing ta be a vector field satisfying the normalization taVat = 1, one can decompose ta into parts tangential and normal to the hypersurface S t by defining N = -tana, (2.2) Nt = hiitP. (2.3) These are the lapse scalar and shift vector respectively, and they describe how one may specify the correspondence between the ini t ia l hypersurface and the 'next' hypersurface, i.e. how to 'move forward in time'. Specifying a coordinate system within the context of the A D M foliation is equivalent to picking a cooridinate system yl covering (or at least partly covering) a hypersurface £ t , and then specifying how this is 'propagated' by specifying a point-identification map l inking T,t wi th T,t+st- Thus, geometrically, the 2 Global hyperbolicity is an existence statement for a Cauchy surface in the spacetime. This surface will be intersected by all causal (timelike or null) worldlines exactly once, i.e. it 'captures' the entire evolution of the space in the sense that the domain of dependence of the surface is the whole spacetime. It is a strong topological and causal condition as well. 3Unless otherwise noted, the signature will be taken as (-+++). Negative-definite metrics are also called Riemannian. 4 Greek indices run from 0 to 3 whilst latin indices are purely spatial and run from 1 to 3. Chapter 2. Complex solutions for 'slow-roll' cosmologies 8 lapse is the magnitude of the vector collinear with the normal to E 4 which joins Ylt+st (physically the magnitude of proper time between E t and E t + ( $ t ) , and the shift is the vector displacement along E t + < 5 t to reach the same coordinate point y%. Figure 2.1: The lapse N , shift Nl, and spatial coordinates yl are equivalent to the space-time coordinate system xa. The are basis vectors on T,t+st a n a " n 1 S the unit normal to E t . Since now we are interpreting the field equations as second-order evolution equations for the Riemannian three-metric with respect to the time parameter, then any ini t ia l data formulation must specify an ini t ia l three-metric and its first time derivative. A quantity closely analagous to this time derivative defines the extrinsic curvature -K^, a tensor proportional to the Lie derivative of the three-metric wi th respect to the normal vector na, Kij = \LK. (2.4) Covariant derivatives Di on the hypersurfaces are similarly defined by the projection of the regular four-dimensional covariant derivative onto the hypersurface E t : Di^hijV*. (2.5) Similarly, in decomposing al l other tensors into normal and tangential parts to the hy-persurfaces, one can express the three-Riemann tensor ^Rijki in terms of the (restricted) Chapter 2. Complex solutions for 'slow-roll' cosmologies 9 four-Riemann tensor Rijki and extrinsic curvatures. These are the Gauss-Codazzi rela-tions [8] ( 3 ) ^ - f c ' = hrh?h{h\R^*-K*Kil + KkiKi\ (2.6) 1D[aKab] = Rcanahcb. (2.7) For vacuum spacetimes, equations (2.7) hint at ini t ia l value constraints since we can write Rijrfh?k — Gijnlh\ and verify that there are no second time derivatives present for any of the metric components. Similarly it can be shown that Rijkihlkh?1 = 2GijnlnJ, which means (2.36) can be used to find the fully normal components of the Einstein tensor purely in terms of three-dimensional quantities: Gijrtn? = \ ( ( 3 ) i ? + K2 - KijKij) . (2.8) The first three constraints, equations (2.7) wi th the right-hand side set to zero, are the momentum constraints whilst the right-hand side of equation (2.8) set equal to zero is the Hamiltonian constraint. A l l of this extends easily to the case with source by decomposing the stress-energy in exactly the same way [8]. The constraints and the algebra they form wi l l be discussed in detail in Chapter 3. 2.1.1 Hamiltonian action We start by observing that the full spacetime metric as expressed in the A D M formalism is ds2 = -N2dt2 + hij(dxi + Nldt)(dxj + Njdt). (2.9) Raising or lowering of indices on a hypersurface E t is done by means of the three-metric hS* or its inverse hij. Using equation (2.9) in equation (2.4), we obtain the extrinsic Chapter 2. Complex solutions for 'slow-roll' cosmologies 10 curvature as K i j = 2N 2Nm - hij (2.10) The vertical stroke denotes spatial covariant derivative, as defined above, and the () notation denotes symmetrization 5 . Then the four-Ricci scalar becomes, by the previous section, R = WR + KijIC* -{ICi)2, (2.11) Thus, in terms of these variables the standard Einstein-Hilbert action S = f RyJ—\g\d4x with matter becomes S = J\J (MR + KvK* -(K\Y)Nj\h\G?y dt + Smatter (2-12) One should note that the shift dependence (as well as additional lapse dependence) of the action is hidden in the extrinsic curvature terms. A s equation (2.12) expresses the action in terms of the three-metric and its time derivative hij we mpay pass to phase space from configuration space by making the standard Legendre transformation by defining momenta in the usual way (n ~ dL/dq ): TTN w 0, (2.13) Try, ^ 0, (2.14) ohij IK . dL ' , = (2-16) The first four equations are weakly equal to zero, as explained in Chapter 3, and are the primary constraints of general relativity within the A D M foliation. They clearly indicate 5Q(a\b) = \{Qa\b + Qb\a) Chapter 2. Complex solutions for 'slow-roll' cosmologies 11 that the lapse and shift cannot be thought of as dynamical. Performing the Legendre transformation on equation (2.12), one arrives at the so-called 'canonical form' of the first-order act ion 6 wi th minimally coupled scalar matter 7 •S = Jdtj [T?% + *<t>B<£B - NU± - rSTHi] d3y, (2.17) where (K = 8irG) \h\ didj<f>B,. (2.18) ' 2 Hi = - 2 a f c ( / i i j 7 r J ' f c ) + 7 r ' m a i / i ; m - f 7 r 0 B ^ B . (2.19) Stationarity of the action under variations of the lapse and shift implies, respectively, the Hamil tonian and momentum constraints: /H± = 0 and Hi = 0. Much more wi l l be said about these constraints in Chapter 3. 2 . 2 The slow-roll model We assume a spacetime geometry such that the metric is ds2 = N(t)2dt2 -a2(t)(dr2 + sin2(r) (d92 + s i n 0 2 # 2 ) ) along with a homogenous scalar field <f>B{t)- The lapse scalar 7Y, scale factor a, and 4>B are in complex-valued functions, although as pointed out earlier the real ('final') universe wi l l have real scale factor a and scalar field 4>B- Whenever the lapse N is purely imaginary we wi l l have a Euclidean spacetime and whenever purely real a Lorentzian spacetime, so that 6 Thi s first order action is valid only for closed spacetimes, as we shall explain later. 7 Here denoted by 4>B both to avoid confusion with the constraint (f> notation used in later chapters. <f>B will later represent the unperturbed (background) scalar field. Chapter 2. Complex solutions for 'slow-roll' cosmologies 12 the transition to different signatures is done by modifying N, not t. The Hamil tonian action, equation (2.17), reduces to S = J (7rad + TT^JB - NH±) dt (2,20) under the symmetries we have assumed (namely that E 4 = S3). Here, the Hamil tonian constraint (equation (2.19)) can easily be calculated, resulting in H± = - 6 f l - ^ + ^ + fl3%)=0. (2.21) The equations of motion for the fields a and (f>B and their conjugate momenta, as implied by equation (2.17), are Sna = 0^d(r) = - i V ^ , (2.22) 6**B = 0-+fair) = (2.23) 5a = 0^-ka(T) = -N 24a 2 2a 4 (2.24) 5<J>B = 0^7T4b{T) = -Nc?VjB, (2.25) 5N = 0 ^ H ± = 0, (2.26) Not ing the form of the above equations, it is convenient to define ' = d/Ndt = d/d(r), where the proper time r is in turn defined by r(t) = j N(t)dt. (2.27) r ( i ) in general describes a parametrized path through the complex r plane. We shall define the zero of r to correspond with the value of t at which the scale factor a is zero. In terms of r , the metric and action are somewhat simpler: ds2 = dr2 - a2(t)(dr2 +sin2(r)dQ2), (2.28) Chapter 2. Complex solutions for 'slow-roll' cosmologies 13 and S = J (7raa' + 7r(j>B(f,Bl -H±)dT. (2.29) We can also write al l of the equations of motion in terms of r in a compact way by-re-expressing the Hamil tonian constraint using equations (2.25) and (2.26), so that if 2 - 6 a , 2 - 6 + a 2 ^ f - + a2V(</>B) = 0, (2.30) (a^B1)' = -a%BV(4>B), (2.31) forms a more convenient, completely equivalent, set of equations to deal wi th . 2.2.1 Initial and final conditions: the NBP and the reality of dp and The N B P demands that we close off the geometry at r = 0, i.e. that a(0) = 0. Since the r r component of the Einstein tensor goes as GTT = ^{a'2 + 1), we can see that a regularity condition forces a' to go as ±ir (i.e. the metric w i l l be purely Euclidean near the 'origin' a — 0). Furthermore, the boundary condition on -K^ is that it must go to zero at r = 0. Summarizing, we have 'o(O) = 0, (2.32) o'(0) = /Si 7ra « - 1 2 r for r « 0, (2.33) B^(0) = 0-tM0) = 0, (2.34) where ft = ± 1 . In order to ensure that the final universe is physically reasonable we impose two final conditions on the final value TJ at which we wi l l evaluate the semi-classical wave function, namely the reality of a/ and (j)Bf. Indeed, we demand that a(Tf) = ± | o / | e f t , ; . (2.35) <t>B(rf) = 4Bf eto, (2-36) Chapter 2. Complex solutions for 'slow-rolV cosmologies 14 where both signs of the scale factor are acceptable since since only a 2 terms appear in the metric. There are five conditions in total, four being for a and (f>Bf since they obey two second-order equations, and one final condition on 77 to impose the reality of the final scalar field a,f and final scalar factor <pBf-2.2.2 Power series solution of complex equations of motion The equations of motion (2.22)-(2.26) are rather difficult to solve in closed form. However, considering the potential V(<PB) in the inflationary 'slow-roll ' approximation V(<j>B) = Vo + ec^s, (2.37) we can solve the equations of motion in a power series in e (assuming that the solution is analytic in e). Denoting the dynamical variables a and (J)B by qi and calling the zeroth order solution q^°\ the full solution is taken to be Qi = ft(0) + E^^T- (2-38) We point out that at each order in e the value of TJ w i l l also change, and define r 0 to be Tf at zeroth order. Then, qi(Tf(e)) = qif (2.39) can be shown to imply Sqi(r0) = -q^HroWrf),: (2.40) S{2)qi(r0) - . - ( ^ / ) 9 / ( 0 ) ( r 0 ) - ( 5 r / ) V ( 0 ) ^ o ) - 2 5 r / ^ / ( r 0 ) , e t c . (2.41) We are now in a position to compute how the action changes order by order. So, starting wi th the fact that S = fF(e'\mi -n±)dT, (2.42) Jo Chapter 2. Complex solutions for 'slow-roll' cosmologies 15 one writes dS de 'dirt , dq{ . de de (2.43) 'd'H±dni d'H±dqi dH^ dr. v 37Tj de ' dqi de ' de Using Hamilton's equations of motion 7r/ = —dqi'H± and g / = d^M^ in equation (2.45), we recover the simpler result dS dr, r n . ( dqi\\Tf ndH± de _ rv dUL Jo de -dr, de -dr, (2.44) (2.45) where in the last equality we have used the chain rule on equation (2.4.1). Now, for our slow-roll potential we have d'H±/de = o?(j), but we also know by equation of motion (2.33) that a 3 = —\(a3(/>')'. This means equation (2.47) can be re-written in a more suggestive form, namely 1 rs dS_ de •= - / 1' (a3<f>B)'<(>BdT e Jo = \ \ [ S {4>Bf(a3<f>B')' - a3<j>B'2) dr - -<t>Bf j*' azdr - ^  jTJ a3(j)B'2dT B'2dT. Using exactly the same chain of arguments, one can show that dS n dUt f Jo dV0 Jo dV0 which we use to find the final desired expression 1 m •dr, (2.46) .(2.47) dS ± dS Te = ' dVn Jo B'2dr. (2.48) Chapter 2. Complex solutions for 'slow-roll' cosmologies 16 Thus, given solutions to the equations of motion order by order, we can use equation (2.50) to calculate the action order by order i f we know what the zeroth order action is. 2.2.3 Zeroth order solutions Inserting the slow-roll potential into the equations of motion (2.33) - (3.34), we find that to zeroth order _ 6 A (0 ) ' 2 _ 6 + A(0) 2^|! + A(0)2 (T/O) = Q ) ( 2 4 Q ) ( a ( 0 ) V f l ) ' = 0, (2.50) which immediately give that a?(f)' is forced to be a constant. If this constant is not zero, then since a ~ /3ir near r ~ 0 one would have <j)'B ~ 1/r 3 , so that 4>B itself would diverge at least as 1/r 2 . This violates the (initial) N B P conditions and so the constant must be precisely zero, implying by the final condition (2.38) that <fiB(r) = (j)Bf. Therefore the scalar field to zeroth order, being a constant, automatically satisfies the N B P and regularity in i t ia l conditions as well as the final condition (equations (2.37) and (2.39)). The zeroth order solution for the scale factor a which obeys equations (2.35) and (2.36) is fl(°)(r) = ^ s i n ( z ^ r ) , (2.51) where (2.52) is defined purely for convenience. In order for the solution to be complete we must specify al l of the solutions of the final reality condition a^(r0) = a/ . Since is partly periodic, this condition wi l l Chapter 2. Complex solutions for 'slow-roll' cosmologies 17 in fact have an infinite number of solutions. Defining Tr + ITi = r 0 (2.53) we see by an identity that — sm(iv(TR + ITI)) = ^ [i sinh(uTR) COS(UTI) — cosh(uTR)], (2.54) which implies that TR and 77 must obey the two constraints cos (1/77) = 0, (2.55) —/3 cosh(z/77j) sin(uTi) — vaf, (2.56) in order for a/ to be real. For vaf > 1, equations (2.57) - (2.58) have the solutions T l = ^(2n-Bsgn(af)^,VneM (2,57) cosh(^T#) = v\af\, (2.58) some of which are displayed in Figure 2.2. A l l of these points in the complex r plane represent possible endpoints for the zeroth order solution a^(r), and three have been included in Figure 2.2. as an example. The cases vaf < 1 are restricted to the imaginary axis and satisfy sin(i/7j) = -/3isaf. (2.59) We wi l l only consider the case where va; > 1 in this thesis. Chapter 2. Complex solutions for 'slow-roll' cosmologies 18 Im(T) Re(T) Figure 2.2: Solutions for T R , TJ whenever vaf > 1 are represented by two (one for each sign or TR) truncated vertical stacks of dots in the complex r plane. These solutions are endpoints for three possible paths, as shown, of a^ in the complex plane, is real at these endpoints. 2.2.4 Zeroth order action Using equation (2.44), the zeroth order action S^°\af,(j)f) is (since 4>W' — 0), S^(af,4>Bf) = jTJ~T\aa'dT = —12— f— j j sin(ivT) [iv cos(ivr)] dr [-Z3 +1 7 (V0af2 12/33 3v2i = 4/3 f 6 i V - -1 vQ V0 V 6 where have defined the sign 7 and the quantity Z by Z = cos(ivTQ) = 7 (^+i\^v2af2 - 1|^ , for vaf > 1. (2.60) (2.61) The two signs ft and 7 wi l l characterize the parametrized paths through complex r space, and the definition for Z proves convenient for later work. In the semi-classical approximation to the wave function, the real part of the action wi l l determine the phase Chapter 2. Complex solutions for 'slow-roll' cosmologies 19 of the wave function whilst the imaginary part wi l l determine its amplitude. Thus the two values of 7 wi l l correspond to complex conjugation of the wave function and the two values of B w i l l allow for two possible values of the amplitude. These two values of B are the subject of the current controversy concerning the appropriate choice of sign of the action [7], [6]. We can see that the action (2.62) is independent of the path taken through complex T space to obtain a given value for a/ or (J>BJ, since more than one path through r space can give a certain a/ value. Also, as there is only a Z dependence in the action and no explicit r 0 dependence, the particular choice of endpoint wi l l not affect the action. This property wi l l not always be true to higher order, as we shall see when we discuss the third order action in section (2.2.7). It should also be noted that the function 5 ^ ( a / , <f>Bf) obeys the constraint equation (2.51) cast in Hamilton-Jacobi (HJ) form for e = 0 (and so is a H J function at this order): The fact that for each order the action must satisfy the constraint equation in H J form proves to be a very useful check when verifying the accuracy of a given expression for the act ion 8 . A more important observation is that since in general the higher order actions w i l l have non-trivial imaginary parts (amplitude contributions) the H J equations w i l l not be satisfied by the purely real parts (phase contributions) of the higher order actions. This is the mathematical statement that in the semi-classical approximation the history of the universe wi l l not just be a real solution, as mentioned in Chapter 1. 8In fact, it is the only check available. Chapter 2. Complex solutions for 'slow-roll' cosmologies 20 2.2.5 First order action Given the zeroth order action S^, we use equation (2.50) to find the first order action by noting de v u 1 2! z 3! ° " v V ^ / / a y 0 2! ' 3! 1 /"T/ e [Tf a34>B'2dT, (2.63) which implies simply « = ( 0 B / ) f . (2 ;64) The first order action 5S obeys the first-order H J equation _ j _ ^ g ( . / . f a / ) a " ' g ( « / . f e / ) + a ? ) V R ( 2 . 6 6 ) 12a/ 5a / 5a / 7 7 The imaginary part of 55 wi l l not contribute to the H J equation at this order, so to first order the real part of the action is the H J function and it alone satisfies equation (2.66). 2.2.6 Second order action Given the first order action 5S, we use equation (2.50) to find9 (2e)S<2> = 26 (</>B/) ^ " 2c jH a'S^dr. (2.66) Thus, we need the first order solution 5<J>B to compute the second order action. To first order, the equations of motion (2.33)-(2.34) are -12a^'6a' + 2Voa^8a + a^2(j)Bf = 0, (2.67) (aW35<j>B) = - a ^ 3 , (2.68) 9 We use the fact that the endpoint ry has shifted at this order. This changes the numerical coefficient in front of the final term in equation (2.67). Chapter 2. Complex solutions for 'slow-roll' cosmologies 21 which can be shown to imply (using the boundary conditions (2.35) - (2.37)) i f cos3(iur) . . . 2 \ 5$ = , 3 ^ — ^ - c o s t i / r + - . v sin (iz/r) \ 3 3 / (2.69) We use equation (2.71) in equation (2.68) and the change of variable z = cos(ivr) to obtain (2e) ( 2 )5 = 2e</>, ,<92(°)S •/ o z ( cos3 (ivr) .. . 2 ' • 3 r \ o cos(«ivr).+ -z/sm (wr ) V 3 2e0: d 2( 0>g ,d2^S + 2e-^ / ( z 3 - 3z + 2) 9z/5 7i v y i / ( l - z2 ) 2 rdz i8 rcos(^ To) / ( z + 2)(z - l ) ' z + l dz. (2 One can evaluate the last integral exactly and also evaluate any contributions from the encircling of the pole. Indeed, cos(^ ro) n z + 2)(z-iy [ 7+1 dz = 1 3 V Z + l +2/c7ri&(2==_i)j 1 2 + 1 2 l n ( l ± l ) + i 7 - 1 2 Z + Z3 (2.71) The last term counts the q times the integrand encircles the pole at z — — 1 and ac-cumulates a residue of b(z=-i) = 4. Since the residue is real, there wi l l only be phase contributions from the poles to the semi-classical wave function at this order. Pu t t ing all of this together, the second order action 52S is * q ,2d2So .18 6b — (pr-^T + f dV<? 9v6 1 2 + 1 2 1 n f ^ ^ ) + 1 7 - 1 2 Z + Z 3 ) +8qm Z + l (2.72) where again 7Z \ \V0a2 - 1 (2.73) Chapter 2. Complex solutions for 'slow-roll' cosmologies 22 The second order action 52S satisfies the second order H J equation in the sense that 12a ; '85S' dat, + dS0 d52S ddf daf 1 a6 85S + dSn 852S d(pB f J d(j)B f d(j>B f 0, (2.74) and it is clear that the real part of the second order action wi l l not satisfy equation (2.76) because the imaginary part of the second order action is non-trivial in its dependence on (j)B and a, and wi l l therefore contribute in the H J equation. Also, the action to second order st i l l only has a Z dependence which means that it s t i l l makes no difference which particular endpoint 1 0 is selected. 2.2.7 Third order action To compute the third order action we need 8a and the second order solution 52(f)B' in addition to 54B • One can show, solving equation (2.69) for 5a under conditions (2.35)-(2.36), that 3 rr 52(j)B = — [T \aW25a + 2aW25a5<j>B']dT JO L J (2.75) • \ fcos3(WT) .. . 2 \ „ — sm3(ivr) + ( • 3 " cos(zi/r) + - J (2.76) 4i + 2i (iur cos(iur) — sin(zi/r)) [ 3 ' sin(iiAr) The third order action itself follows from a straightforward manipulation of equation (2.50), the result of which is . (e2)53S = e2{<j>Bf)0-^-l2e2 jf ' ( H H ^ ' B ? + a ^ S 2 ^ ) dr, (2.77) and which we may simplify by again introducing z = cos(ivr): 1 0Recall that the endpoint r/ will shift from order to order owing to equations (2.42) - (2.43) and their higher order analogues. It is to these shifted endpoints that we refer. Chapter 2. Complex solutions for 'slow-roll' cosmologies 23 s{iVT0) (j)Bf / iB 12< (z + 2)(z-l) z + l -368{z + 2) arccos(z) fz — 1 dz. (2.78) We notice immediately that the last term wi l l endow at least the integrand wi th an explicit dependence on r 0 , giving a hint of endpoint dependence in the final result. The first term in the integrand is of the same form as that of the second-order action's integrand, to which we can apply the above analysis. We therefore focus on solving the latter integral, and, perhaps surprisingly, find that it can be carried out in closed form. The result is r Z arccosz (z —1\2, _x . „ . ,„ ^ „,_i . ,„ , (1 + Z^ j[ (z + 2)dz = -Z + 4/3 (1 + Z ) - 1 - 4/3 In — (arcsin(Z)) 2 V I - Z 2 ( 3 Z 2 + 1 0 Z + 1 1 ) arccos(Z) (1 + Z ) 2 +6arcsin(Z)arccos(Z) + l / 3 + l / 4 7 r 2 , (2.79) which allows us to write the entire third order action (using the second order results from section (2.2.6)) S3S 4>. _d_ BfdVQ +Sqiri]} _<hj_ fiP Vo IK,3 d2S0 , iB o + 9 ^ 6 1 2 + l 2 1 n ( ^ ) + 1 7 - 1 2 Z + Z 3 Z + l 1 2 + 1 2 1 n ( ^ - h +17-12Z + Z 3 ) +8q7ri Z + l 2 36/3 {-Z + 4/3.(1 + Zy1 - 4/3 In ~ (arcsin(Z)) 5 + 6 arcsin(Z) arccos(Z) y/1 -Z2(3Z2 + 10Z + 11) (i + zy 9 16 +1/3 + 1/47T 2-— iqn (2.80) Chapter 2. Complex solutions for 'slow-roll' cosmologies 24 The thi rd order action must satisfy the H J equation to third order, namely 1 3 - : ^ + 1 + n3 ' assays d(j)Bfd(j)Bf 0. (2.81) 12a/ [~ daj daj ' daj daj We see further that there is an explicit dependence on arccos(Z) = IVTQ, meaning that in third order there is a dependence on precisely which endpoint is chosen to terminate the contour. The last term in the third order action is the residue contribution from each of the poles T = ™ , m an odd integer, since in the r variables (as opposed to the z variables) there is a regular pole and no branch cuts to worry about. Thus, the contribution from the poles to third order is st i l l purely to the phase of the semi-classical wave-function and not to the amplitude. There are indications that to fourth order the wave-function wi l l receive amplitude contributions from the poles. 2.3 Summary and Outlook In this Chapter we reviewed the A D M formalism in section 2.1, in whose language we then formulated, reviewed and extended the complex slow-roll model in section 2.2. We found that the higher order actions do depend on the particular choice of endpoint r/(e) in complex time r and that furthermore the semi-classical wave function does not receive amplitude contributions from poles in the complex plane. We are led to conclude from the above analysis that the sign 8 (which controls the amplitude of the semi-classical wave function) cannot be completely determined in the absence of non-homogenous perturbations to the metric and scalar fields, and so the generalization of the slow-roll model to include these perturbations wi l l be the task of the next two chapters. Because of the unique problems associated wi th isolating genuine perturbations from coordinate perturbations in general relativity, we shall first introduce, in Chapter 3, the well-established constrained Hamiltonian formalism to better keep track of physical degrees of freedom in the perturbations. In Chapter 4 we shall make use of Chapter 2. Complex solutions for 'slow-roll' cosmologies 25 this formalism to interpret the action to second order in these linear, non-homogeneous metric and matter (classical) fluctuations, and only in Chapter 5 shall we again make contact wi th the slow-roll model and interpret our results. Chapter 3 Constrained Hamiltonian Formulation 3.1 Introduction A theory which specifies a physical state in terms of variables which themselves contain a non-physical redundancy is called a gauge theory. In other words, wi thin such a theory, there exists an entire class of these variables which correspond to one and the same physical state. Transformations between variables within this class, i.e. transformations which do not affect the physical state, are called gauge transformations. Therefore, any solution to the equations of motion wi l l remain a solution under a gauge transformation because it w i l l be gauge-invariant. One manner in which to treat gauge theories is to cast them into a Hamil tonian formu-lation. In practice, however, a Lagrangian is written down first and one must consistently pass to the Hamil tonian v ia the standard Legendre transformation from configuration space to phase space. The problem is that, due to their non-physical redundancy, not a l l of the canonical variables w i l l be independent and they wi l l in general satisfy relations called constraints. Thus, to study a gauge theory is to study its constrained Hamiltonian formulation, i.e. every gauge theory is a constrained Hamil tonian system 1 . The function of this chapter is to provide the necessary background for computing a physical Hamil tonian for cosmological perturbations, as discussed in later chapters. In doing so, however, it is useful to note that the full non-linear theory of relativity cannot 1We note, however, that the converse statement (that every possible constraint in a Hamiltonian system results from a gauge invariance) is not true, as some constraints do not imply an underlying gauge transformation. 26 Chapter 3. Constrained Hamiltonian Formulation 27 be considered a gauge theory in the above sense. The essential difference stems from the fact that, while spacetimes related by a diffeomorphism are regarded as identical (which may in some sense be regarded as a different sort of redundancy [10]), there are no known variables or fields which are physical (i.e. gauge invariant) except constants in spacetime 2 . First an example of consistently passing from a Lagrangian to a constrained Hami l -tonian, following the method of Dirac [11], wi l l be treated in some detail. Then the constrained Hamil tonian and its constraint algebra for general relativity, as explained below, w i l l be presented and discussed so as to gain insight into what the equivalent formulation might look like in linear perturbation theory. More modern, but entirely equivalent, methods on the reduction of constrained and unconstrained Hamil tonian sys-tems (see [12] and [13]) which have recently been popularized in the literature wi l l not be discussed. Finally, the importance of surface integrals in the Hamil tonian reduction of the action wi l l be examined as it pertains to the cosmological perturbations this thesis examines. 3.2 Example of a constrained Hamiltonian formulation Let us first consider an example (taken from [14]) to illustrate the basic procedure of passing from a Lagrangian to a constrained Hamiltonian formulation, both to gain insight into the more complicated formulation in the case of gravity, which we shall consider in 2 However, in linearized gravity with sources there exist combinations of variables which are invariant under linear diffeomorphisms [40]. Thus, linearized gravity with sources is indeed a gauge theory in this sense, so we are justified in applying these Hamiltonian methods to that theory. Chapter 3. Constrained Hamiltonian Formulation 28 section (2.3), and to fix the notation we use. Consider the Lagrangian 3 9i2 HQUQX) = - y + 9 3 9 2 - 9 4 / ( 9 1 , 9 2 ) . (3-1) One passes from the configuration space Lagrangian L(q, q) to the phase space Hamil to-nian H(q,n) firstly, by defining the momenta in the usual way, | 4 = 7^, and secondly by performing the Legendre transformation (change of variables) to form the Hamil tonian H(q, ^'^Eimf-Lfaq). Thus, 7T! = ft, (3.2) 7T2 = 93, (3.3) 7T3 = 0, (3.4) 7T4 = 0, (3.5) are the canonical momenta associated with the variables q,. The last three relations form constraints which delimit the swath of accessible phase space for any consistent evolution of the canonical variables (g, 7r), and some of them may generate the gauge transformations of the theory. These constraints can be written as <f>i = 7T2 - qz « 0 (3.6) 4>2 = 7 T 3 « 0 (3.7) 03 = TT4 sa 0 (3.8) where the « notation denotes the so-called 'weak-equality', which is a regular ('strong') equality up to a linear combination of the constraints which themselves are numerically zero 4 . 3We note that part of this Lagrangian is already in first order form because of the term. So a total Legendre transformation is optional, not required, and amounts to a partial duplication of efforts in the analysis. 4I.e., AttB<-tA-B = cl(q, n)<j)i(q, IT), where the cl are arbitrary. Chapter 3. Constrained Hamiltonian Formulation 29 The Hamil tonian is arbitrary up to a linear combination of these constraints, and the coefficients of these linear combinations are Lagrange multipliers, i.e. arbitrary functions of the phase space variables q, TT plus time t. To obtain the most general evolution possible from this Hamiltonian one must add to it the constraints fa times the Lagrange multipliers Xl(q,-K,t). This defines the so called total Hamiltonian: Htatal = H0(q,7t) + Xi(q,7T,t)fa (3.9) = ntf - L(q,q) + \l(q,n,t)fa Equation (3.9) explicitly incorporates some of the invariances of the original Lagrangian, equation (3.1), by including some of the constraints which generate such invariances 5. The quantity H0 is sometimes called the 'Hamiltonian evaluated at the constraint surface', and is explicit ly found to be H0(q,ir) = 7riql-L(q,q) = M l + + 7T393 + 7T494 ~ + ^2 ~ Qij'(92, 9 3 ^ (fc=0) 7Ti 2 , , = - y H - 9 4 / ( 9 2 , 9 3 ) - (3-10) Given HQ, once Htotai is known it is possible to determine how the canonical variables evolve by using the Poisson brackets, which are determined by the symplectic structure of the phase space (see [14]). The usual definition of the Poisson bracket happens to suffice in this case: r m _ Y - (da dP d/3 da\ where a and B are arbitrary functions of the phase space variables. The total time derivative of a dynamical variable F(q, TT) is then F = {F,Htotal}, (3.12) 5Technically, we must add all independent constraints to Ho in forming Htotai, including any con-straints which result from demanding that the fa hold at all times. Chapter 3. Constrained Hamiltonian Formulation 30 which allows the standard Hamiltonian equations of motion to be compactly expressed as 4i — {qi,Htotal} = d^Htotal, (3.13) tit = {iri,Htotal} = -dqiHtotal. (3.14) Now, in order for the whole theory to be consistent the pr imary 6 constraints, equations (3.6)-(3.9), themselves must be preserved in time. These further consistency conditions, called secondary constraints, can either result in new constraints amongst the canonical variables, restrictions on the Lagrange multipliers Xl(q,7v,t)7, or a contradiction (as in , say, the case of L(q,q) — q) in which case the original Lagrangian is abandoned as inconsistent. Hence, we determine the secondary constraints by computing 01 = {0i, Htotal} = {01, Ho} + {01, K4>1} = {TT2 - 93, H0) + {TT2 - g 3 , Ai(7r2 - q3) + A27r3 + A37r4} = - 9 4 d 9 2 / - A 2 , (3.15) 02 = {n3, Htotal} = - 9 4 ^ / + A i , (3.16) 03 = Htotal} = - f(q2,q3)- (3-17) Thus there is one additional constraint (= 04) amongst the canonical variables as well as two constraints on the Lagrange multipliers: . _ 0 4 = / (92 ,9 3 )~0, (3.18) 6Primary constraints (eqns. (3.2) - (3.5)) follow simply by definition of the momenta in terms of the Lagrangian, whilst secondary constraints follow from the equations of motion. 7These restrictions go as {<j>j,H0} + \m{<f>j,</»m} w 0, where the index m sums over the primary constraints and j labels any one of the constraints. So there are j inhomogenous equations in the unknowns A m with coefficients which are functions of the phase variables. Thus, Xm can be expressed as a sum of a particular inhomogenous solution Vm of the restriction and the most general homogenous solution Um: Xm = Um + r)k(t)V™. The arbitrary functions of time r]k (t) endow the Lagrange multipliers with their time dependence. Chapter 3. Constrained Hamiltonian Formulation 31 A i - ~ qAdqJ, (3.19) A 2 . « - ? 4 3 w / . (3.20) The secondary constraints themselves are preserved because fa = {f(q2,q3),Htotal} = {f,H0} + \i{f,<f>i} = 0-X1(-dq2)f-X2(-dg3)f = q4l(dqJ)(dqJ)-(dqJ)(doJ)} = 0, (3.21) where in the last equality we have used equations (3.19) and (3.20). Equation (3.21) states that there can arise no further constraints and the tower of constraints (fa through fa) is said to close8. This is the mathematical statement that the constrained Hamil tonian formulation is internally consistent, i.e. that the canonical variables obey a consistently constrained evolution by evolving only on the constraint surface in the phase space. The Hamil tonian that generates this evolution wi l l contain all of the independent constraints (primary, secondary, tertiary, etc.), so that the final Hamil tonian wi l l be of the form equa-t ion (3.9) wi th the summation index i changed to /z, where fi runs over al l independent constraints. Do the constraints fa through fa have a geometric interpretation? One would cer-tainly expect them to generate the allowed gauge transformations for the canonical vari-ables, but is this true for each of the constraints and when is it not true? To answer these questions we wi l l , following [11], have to further classify the constraints in terms of their mutual Poisson brackets. 8Actually, only weak equality is required to obtain closure. I.e., if the time derivative of a constraint returns linear combinations of the other known constraints there is no further unique constraint. Chapter 3. Constrained Hamiltonian Formulation 32 3.2.1 Geometric interpretation of constraints; first and second class con-straints The distinction between primary and secondary (and teritary, etc.) constraints is, as suggested above, of little importance to the final form of the theory. However, a different classification of constraints does play a more central role. In this scheme, constraints are divided into first or second class. Members of the first class have Poisson brackets wi th every constraint which are at least weakly zero, and the second class have at least one bracket with some constraint which is non-zero. It can be shown that the first-class property is preserved under Poisson brackets, i.e. Poisson brackets of two first-class functions return first class functions. What is important, however, is the Dime conjecture, which states that all first class constraints generate gauge transformations [15],[11]. The precise conditions under which this conjecture applies (and fails) were finally treated over 20 years later in an important paper by Castellani [16]. The conjecture is true for the systems considered in this thesis. Let us precisely define what we mean by a gauge transformation. We can see that Htotai contains arbitrary functions of time nk(t) in its dependence on the Lagrange mult i -pliers A*(g, 7T, t) = U{ + ?7 f c(t)y i k , which means that although a physical state is uniquely defined once a set of IT'S and q's is given, there is more than one set of values of the canonical variables representing a given physical state. Indeed, suppose we specify an ini t ia l set of canonical variables at time t\ and thereby completely specify the physical state. The equations of motion wi l l determine the physical state at later times, so then any ambiguity in the value of the canonical variables themselves at later times, say at £2 > h, is physically irrelevant. That in mind, a natural question to ask is what change a canonical variable W w i l l suffer at time t2 resulting from two different choices r)k(t), fjk(t) Chapter 3. Constrained Hamiltonian Formulation 33 of arbitrary functions at time t\. The form of this change must be SW = 6rf{W,<f>i} (3.22) where Srf = (if — 77') (£2 — h) = (if — ff)St, which again cannot alter the physical state 9 . Therefore, the transformation W —> W + SW is a gauge transformation of the canonical variable W. Whenever the Dirac conjecture holds, the constraints fa which appear in equation (3.22) w i l l be exclusively first-class, i.e. the first-class constraints w i l l generate the gauge transformation. B y constrast to first-class constraints, it can be shown that second-class constraints do not generate gauge transformations of any sort 1 0 [16],[15]. Thus, one strategy com-monly employed is to solve as many of the second class constraints as is necessary to render a l l the remaining constraints first-class so that the only constraints which remain unambiguously represent gauge transformations (modulo the conditions laid out in [16], which are al l satisfied for the systems we consider). Returning to our example, we thus compute the mutual brackets of the constraints to see which are first-class: {fa, fa} ? 0 f o r j = 2,4 (3.23) {fa, fa} # 0, f o r i = 2,4 . (3.24) {fa, fa} = 0, V i (3.25) {fa, fa} ^ 0 f o r i = 1,2 (3.26) Thus fa, fa and fa are al l second-class constraints with the exception of fa, which is 9Note, though, that (3.22) will not give the only transformations which do not change the physical state [15]. In particular, the Poisson bracket of any two first-class primary constraints and the Poisson bracket of any first class constraint with the first-class Hamiltonian will also leave the physical state unchanged [15]. 1 0There is also the Faddeev-Jackiw method of reduction [12], which claims no distinction of any sort between the classes of the constraints. Chapter 3. Constrained Hamiltonian Formulation 34 first-class. Now solve (substitute in) two of the second-class constraints (<j>i « 0 « 02) to render al l the remaining constraints first-class: 2 #tota(U=o = + 94/(92, ft) + A 37r 4 , (3.27) 02=0 * /(92,93) « 0. (3.28) This Hamil tonian, with its purely first class constraints 7r4 « 0 « f(q2,Qs), is now easier to interpret even though we st i l l have not completely isolated the physical degrees of freedom it describes. 3.2.2 Gauge fixing to isolate physical degrees of freedom Even after solving al l of the constraints the equations of motion wi l l s t i l l have arbitrary functions of time corresponding to further gauge freedom, i.e. the physical sector of the theory has not been completely isolated by just using the constraints and their consistency conditions. This means that ad-hoc relations, called gauge conditions, must be specified in some consistent manner to properly extract the physical content of the equations of motion. In general, the number of independent gauge conditions will equal the number of independent first-class primary constraints. For example, in general relativity there are four gauge conditions and there are also four first-class primary constraints (as wi l l be explicitly proven in the next section), one arising from the lapse and three arising from the shift vector components. O f course, just which variables to restrict and in what combination, should be guided solely by the requirement that there remain no residual transformations of the canonical variables which maintain the gauge, i.e. that the gauge choice used should completely eliminate the remaining freedom after a l l the constraints have been exhausted (i.e. resubstituted into the Hamiltonian) [15]. After this combination of back-substitution of constraints and gauge-fixation the Hamil tonian is said to be reduced and the equations of motion that it generates wi l l only describe Chapter 3. Constrained Hamiltonian Formulation 35 the evolution of genuinely physical degrees of freedom. A way of checking i f a l l of the available gauge freedom has been exhausted is to verify that al l the Poisson brackets of the gauge conditions with the constraints are non-zero, i.e. that only second-class constraints remain after gauge fixation11. To continue with this particular example, we reduce the phase space by solving the constraint 4>3 = 7r4 « 0 and gauge-fixing </4. The only motivation in doing this is to interpret g 4 as a Lagrange multiplier associated with a first-class constraint and not as a dynamical variable. In particular, this step is not required by the standard Dirac method analysis [14],[11]. This reduced phase space is described by 2 Htotai = ^ + 9 4 / ( 9 2 , 0 3 ) , (3.29) where the only (first class) constraint is /(<fe,<fe)*0. (3.30) Subject to this constraint, the equations of motion this reduced Hamil tonian wi l l generate wi l l describe only physical degrees of freedom. 3.3 Hamiltonian Formulation of General Relativity Drawing on the above example, the following sections wi l l discuss the Hamil tonian for-mulation of general relativity. Since such a formulation intrinsically requires some type of special parameter akin to 'time' against which to evolve dynamics, we shall use the 3+1 A D M formalism introduced in section 2.1. We shall first discuss the constraint algebra of the Hamil tonian for closed spacetimes in section 3.3.1. Then, the role of surface integrals "Caution must be used, however, since these are local conditions. Gribov first pointed out that globally these conditions may not ensure that the surface denned by the gauge conditions will intersect the gauge orbits, which lie on the constraint surface denned by the constraints, once and only once. This is the Gribov obstruction [17]. Chapter 3. Constrained Hamiltonian Formulation 36 in the Hamil tonian formulation of non-closed spacetimes is briefly discussed in sections 3.3.4 and 3.3.5. 3.3.1 Constraint algebra of vacuum GR in closed spaces We start from the constrained Hamiltonian action for gravity on a closed spacetime wi th minimal ly coupled scalar matter, equation (2.18). It is important to realize from the beginning that this analysis wi l l only apply to a closed universe. The subtleties of extending the formalism to open and asymptotically flat spacetimes, for which equation (2.18) w i l l not apply due to surface terms contributing from spatial infinity, wi l l be discussed in sections (3.3.2) and (3.3.3). Equation (2.18) read ' S = j d t j [^kj + IT^JB - NH± - N'-Hi] d3y, (3.31) where (K = 8nG) \h\ V " * S 2K 2sJ\h\ + ^y-ditBdi<j>B (3.32) Hi = -2dk{hi^k) +TTlmdihlrn+ 71^8^3. (3.33) The form of equation (3.31) implies that the Hamil tonian is the linear combination Htotai = / s [Nn± + NiHl]d3y +j^X^d3y, (3.34) where the primary constraints are = 7rjv, • (3.35) (f>i = TTNi. (3.36) Chapter 3. Constrained Hamiltonian Formulation 37 The secondary constraints (commonly called consistency relations) follow immediately as TCn = {nN,Htotal} = - 6 - ^ ^ = 0 ^ H ± = 0, . (3.37) 7iNi = {7rNi,Htotal} = — ^ - = 0 ^ n i = 0. (3.38) Thus, the secondary constraints are the Hamiltonian and momentum constraints respec-tively, i.e. the primary constraints (ir^ — 0, irNi = 0) wi l l hold for al l time only i f the Hamil tonian and momentum constraints hold on all slices E t foliating the spacetime 1 2 . In keeping wi th the consistency algorithm implied by the earlier example, we must check to see that the Hamil tonian and momentum constraints are themselves preserved by computing their brackets with the Hamiltonian (i.e. computing their ' time derivatives') and seeing i f they at least weakly vanish. Wr i t ing the vacuum Hamiltonian and momentum constraints with the proper weak equalities[14], we have H±{ir,h) = -7== U^if- ^) - v f r t f « 0, (3.39) J\h\ \ 1 ) > l Wfah) = -2hijDkirjk&0. (3.40) In order to take the appropriate Poisson brackets we must construct, again following Romano [14], constraint functions which basically 'smear out' the constraints wi th test fields N and JV a on E : N 1 / ij 1*1 V - 1 3 \h\®R d3y, (3.41) C(N) = -2 f ^{hijD^afy. (3.42) 1 2The notation Hy_ for the Hamiltonian constraint is suggestive, and rightly so, since it will be shown to generate 'normal' ('time') reparametrization gauge transformations Chapter 3. Constrained Hamiltonian Formulation 38 Consider any functions /(hij,^), g(hij,ix%:>) of the canonical variables. Their Poisson bracket w i l l here be defined as ~8f 8g 5g 8f 8hij 8TT%i 8hij 8TT1J dsy. (3.43) The next step is to evaluate the functional derivatives of equations (3.39) and (3.40) since we want to compute their Poisson brackets with the Hamil tonian. Consider the momentum constraints C(N) first. Start by rewriting them via an integration by-parts: C(N) = - 2 I Ni[hijDkirjk]d3y (3.44) = 2J TTjkNJlkd3y = 2 / (C^hij)^d3y = -2 [ (Cfirfhijtfy, where in the last expressions we have used 2D^Nj) = Cfihij. B y inspection, the functional derivatives of equation (3.44) are then 8C(N) 8 hij •2Cf}irtJ, (3.45) ^ = + 2 £ A , (3.46) Consider now the equivalent calculation for the scalar constraint function C±(N), equation (3.40). The functional derivative of Cj_(N) wi th respect to the three-metric w i l l be difficult since there is a non-polynomial dependence entering through the three-Ricci term. Nevertheless, the result is [14] dhij 2 Vh \ 2 j v • ' -y/h^LPN -hijAN) . (3.47) A much simpler calculation gives the funcional derivative with respect to the momenta: K±.(N) 1 / nhab\ ^ - - 2 N T h V a b ~ ~ ) - ( 3 - 4 8 ) Chapter 3. Constrained Hamiltonian Formulation 39 Equations (3.43) - (3.46) allow us to evaluate the mutual Poisson brackets of the con-straints as well as their Poisson brackets with the total Hamil tonian. Let us first compute the Poisson brackets of the vector constraints C(N) wi th them-selves and wi th the scalar constraints C±(N). Now let us define a generic scalar function f(M) of the canonical variables, by splitt ing apart its dependence on the canonical vari-ables (characterized by / f c ' ; . . j ) from its dependence on spatial coordinates (characterized by M ' " \ , ) , by the integral . / ( M ) = / Mi\Jk±j(ir,h)d3y. (3.49) We compute its bracket (so as to emulate the bracket with any scalar constraint) wi th the vector constraint to be {C(N),f(M)} = 2jf = + 2 / J Lit = 2f{CjfM). (3.50) Therefore, it follows that al l Poisson brackets of the momentum constraints wi th them-selves and wi th the scalar constraint must be of the form {C(N),C(M)} = 2C(Cf}Ma)=C([N,M}), (3.51) {C(N),C±(M)} = 2C{CnM), (3.52) where we have defined the standard Lie bracket of M, N. Final ly, we now proceed to compute the bracket of two scalar constraints. Observe that, owing to the form of equation (3.45), the Poisson bracket w i l l only involve the subtraction of terms interchanged in N and M. Thus, only terms locking 7Y and M in d3y Chapter 3. Constrained Hamiltonian Formulation 40 derivatives can possibly survive in the bracket. Keeping this in mind, { C 1 ( i V ) , C x ( M ) } = / h ^ - W A N ) ^ ^ - ^ ) \h\ * where - (-y/\h\(M^ - h ^ A M ) ^ ^ - \hi3) d3y = j [-(AN)M+ (AM)N}(ir\)-2TTij\N\ijM-M^N] +(7Tii) [(AN)M — (AM)N] d3y = -2 f [M&N-N&M]^^)^ = C(K), (3.53) Ka = MdaN-NdaM = hab(MdbN-NdbM), (3.54) A = LVDj. So indeed the scalar constraint function is also of first class, but, as is well known, its Poisson bracket with another scalar constraint function returns a vector constraint function which indicates that i t is not closed under the Poisson bracket. The totali ty of scalar and vector constraints, however, is, so that means that together the Hamil tonian and momentum constraints form a first class set. Whenever the Dirac conjecture holds, this implies that al l four constraints wi l l imply coordinate transformations under which the theory remains invariant. Al though strictly speaking we have only proven closure for the totality of test functions C±(N) and C(N), it can be shown that this implies that the original constraints, H±_ = 0 and Hi = 0, are also first-class and lead to no further constraints [14]. Since the Hamil tonian is a linear combination of constraints, then Poisson brackets of the constraints wi th the Hamiltonian are merely mutual brackets which we have already Chapter 3. Constrained Hamiltonian Formulation 41 computed above in equations (3.49) - (3.51). This means that the time derivatives of the constraints are equal to linear combinations of those constraints, i.e. they are weakly zero and the constraint tower closes, thereby terminating the Dirac algorithm at this iteration. Let us review and interpret the results of this section. The total Hamil tonian for vacuum gravitation in a compact spacetime, internally consistent owing to (3.49)-(3.51), is Htotai = ^ [NU± + ITUi] d3x + £ X^^x, (3.55) where 0o = KN> (3-56) <f>i • = TTNi, ; (3.57) are the primary constraints. In this form it is now obvious that evolution of the lapse N and shift Nl are completely arbitrary (since their first time derivatives, variations of the action with respect to their momenta, are equal to A). The Hamil tonian (3.53) is numerically equal to zero, mathematically since it is a linear combination of constraints which themselves are numerically zero, and physically since we are dealing with closed universes in which there is no obvious notion of total energy and momentum. In general this wi l l not be true for open universes wi th boundary, since in that case there is a viable concept of total energy and total angular momentum as there are non-trivial surface terms induced from spatial infinity which contribute. Also useful to note, following Langlois [18], is the fact that the Hamil tonian (3.55) contains only constraints by contrast with electrodynamics, say, in the sense that the latter also contains an additional dynamical part. The situation wi l l also be different for the case of interest to this thesis, in which the Hamiltonian to second order in metric and matter Chapter 3. Constrained Hamiltonian Formulation 42 fluctuations wi l l be calculated. The second order Hamil tonian action wi l l possess a dynamical part in addition to a part associated with the constraints. Another important point to note concerns equations (3.51) and (3.52). Since the vector field Ka depends on the metric the Poisson bracket is said to involve structure functions which are non-trivial. The constraints (equations (3.39) and (3.40)) do not form a Lie algebra, which is one of the reasons why quantization of this theory has been so difficult. 3.3.2 Extension to asymptotically flat and asymptotically AdS spacetimes How must the above analysis be modified when the spacetime is asymptotically flat or asymptotically AdS? Essentially, surface terms 1 3 must be added to the standard Einstein-Hilbert action in order to reproduce the standard field equations of general relativity. In order to explicitly compute and interpret the additional terms that w i l l result in the final Hamiltonian, consider first the action for an asymptotically flat spacetime [19] Here, K is the trace of extrinsic curvature of the foliating slices as embedded in the spacetime in question and K0 is the extrinsic curvature of the slices as embedded in flat spacetime. In order to better understand the surface term, consider a simple example to illustrate the need for the K0 term [20]. For flat spacetime, the surface term J E t K\J\h\d3y is easy to evaluate. Choose the boundary to be a cylinder formed by two bounding hypersurfaces as shown in Figure (2.2). Since on the cylinder the metric is ds2 — —dt2 + R2dVt2 and the normal is na = dar, then the extrinsic curvature trace is K = na.a = - | . This implies that the surface integral is SnRfo — h); which diverges 1 3 The form of these surface terms is deduced by allowing the normal derivatives of the variations of the metric on dM (of a spacetime (M,gap)) to be non-zero when varying the Einstein-Hilbert action. S (3.58) Chapter 3. Constrained Hamiltonian Formulation 43 — — _ Figure 3.1: Boundary of cylinder used to evaluate the surface integral for R —> oo. Thus, one must subtract the term to make sure the difference K — K0 is well-defined in the l imit of R —> oo. For our purposes, though, we see that a flat F R W universe is spatially flat everywhere, so K = KQ and the surface term does not contribute, but the point is that surface terms of this form wi l l alter the Hamiltonian of asymptotically A d S 1 4 spacetimes and also open F R W universes. There have traditionally been two equivalent tracks to obtaining the Hamil tonian action appropriate to given asymptotic conditions. Namely one could vary a naive (com-pact) Hamil tonian action and then add correction terms by hand to make the variation, and hence the equations of motion, well-defined 1 5 . This was the original strategy of Regge and Teitelboim [21]. The second approach, as expounded by Hawking and Horowitz [19], starts with the correct action from the very beginning and deduces the additional terms for the Hamiltonian. For our purposes we shall adopt the former approach. 1 4Anti-DeSitter spacetimes are static solutions with negative cosmological constant. In the case of a FRW universe filled only with negative cosmological constant (negative energy density), the spatial curvature must be K = -1 in order to satisfy the Hamiltonian constraint (i.e. the time-time components of the field equations). I note in passing that the curvature of the spatial sections used to foliate a De Sitter spacetime is not similarly constrained. All choices are consistent with A > 0 but only the K = +1 foliations cover all of the spacetime. Indeed, one recalls that Minkowski spacetime can be foliated by negative curvature slices to obtain the Milne universe (a K = -1, p = 0 solution of the Friedmann equations). 1 5Taking H from equation (3.59), one would find that S J(TTJ^1 — H)dt = 0 has no solutions, essentially because the ordinary phase space for compact spacetimes is 'too small' to contain the necessary extremal trajectories without the 'introduction of boundary conditions as further canonical variables' [21]. It is in this sense that the equations of motion wouldn't be well-defined. Chapter 3. Constrained Hamiltonian Formulation 44 3.3.3 Incorporation of surface terms into the Hamiltonian formulation In an important paper in 1974 Regge and Teitelboim [21] analyzed the role of surface integral terms in the Hamiltonian formulation of general relativity. Here, we wi l l trace out the central point of their paper. Consider the total Hamiltonian of general relativity in vacuum, given by equation (3.53), H = [ \Nn± + Nini]d3y+ I X^dPy. (3.59) When one takes the variation of this Hamiltonian, in order to obtain the equations of motion, the form that is required is SH = I [Ai^y)Shij(y) + Bij(y)S^(y)]d3y. (3.60) The equations of motion are then defined in terms of the coefficients of the variations: A i j = iir> <3-61) B « = ( 3 - 6 2 > However, actually varying the above Hamiltonian results in SH = f [A^(y)Shij(y) + Bij(y)S^(y)}d3y J Sf -^Gijke(NShiJlk-NkShi:i)d^ . - / (2NkS-Kke + (2Nkirji - Ne7rjk)Shjk) dZ£, (3.63) JdT,t V ' l t where Gijki = /hikhji + huhjk _ 2hijhH) . (3.64) Therefore in order to recover Einstein's equations in Hamil tonian form these surface integrals must somehow cancel. This occurs naturally for closed spacetimes but does not Chapter 3. Constrained Hamiltonian Formulation 45 happen for spacetimes which are asymptotically flat or A d S . Specifically, the first surface integral w i l l not be zero in these latter cases. In order to determine which integrals cancel and which do not one has to have information about the asymptotic behaviour of the fields and their conjugate momenta. Let us treat the asymptotically flat case first. As [21] points out, any solution of the field equations representing a finite-mass, asymptotically flat spacetime can be cast in the (static) Schwarzschild form near spatial infinity, namely V 87rr/ V 87T r6 J Although the precise form of this metric may be altered by a change of coordinates there exists no coordinate system such that, for M ^ 0, all components of the metric can be made to decrease faster than r _ 1 . Hence we, at best, can choose metric functions such that h^ — 5ij w r _ 1 , (3.66) hijlk « r - 2 . (3.67) In order to specifically compute which integrals wi l l survive in the l imi t of spatial infinity, they go on to assume the more specific falloff conditions (3.68) (3.69) (3.70) The first equation comes from considering whether or not the above asymptotic expansion is invariant under Poincare transformations at infinity [21], whilst the restrictions on the lapse and shift are based on the asymptotic form of the metric above and aim to fix the TT r^t N » Chapter 3. Constrained Hamiltonian Formulation 46 coordinates to be Minkowskian at spatial infinity. Taking these asymptotic conditions, one sees that the only term which survives is - / dZt&WShiM, (3.71) which may be rewritten in the more convenient form - / dXeGijke8hiJlk = -5(f dZ^hikj-hiiri^-Eihij]. (3.72) Therefore one must add this surface integral to the total Hamil tonian in order for the variation to yield well-defined equations of motion: o~(Htotai + E[hij]) = f \Aij{x)6hij(x) + Bij(x)5irij(x)]d3y. . (3.73) This modified Hamil tonian wi l l now have a non-zero numerical value for solutions, equal to E[hij]. Since it does not depend on time, E[hij] must be a constant of motion, identified with the energy E. This wi l l be important when finding the reduced Hamil tonian for a open F R W background. One can also think of these additional terms in the following physical sense. In a closed universe there is no viable notion of total energy or momentum since there is no asymptotic measure.by which to define these quantities (such as 'time translations at spatial infinity'). This is not true for open universes and in that case one must have boundary terms corresponding to the total energy-momentum and total angular momentum. Another way of expressing this is that, assuming that the lapse and shift assume the form N» ~ of + B£xx , + / V = 0 (3.74) at spatial infinity, the addition to the Hamil tonian must be of the form Hasympt = -o^P^ + ^ M ^ (3.75) Chapter 3. Constrained Hamiltonian Formulation 47 in order for the requirement of l im^oo 5N^ = 0 to be true. The particular form of the component P ° has already been computed above (i.e., it is E ) , and by similar con-siderations (by insisting on asymptotic translation invariance and asymptotic rotation invariance 1 6 ) one can obtain the other components: Pk = -2<f dSVfc, (3.76) JdT,t Li = 2<f dZeeIJK7Tejxk = etjMij. (3.77) Thus finally one can see that the fully corrected Hamil tonian for asymptotically flat spacetimes is Hcorrected = Htotal - O^Pfj, + -B^M^, (3.78) wi th the corrections due to the additional surface integrals. The Hamil tonian action that would correspond to equation (3.78) could then be reduced via some gauge choice to find the physical degrees of freedom as expressed by the reduced equations of motion. Variat ion of such reduced actions is not always straightforward, however (see Appendix D ) . 3.4 Summary of chapter In this chapter we have studied Dirac's method of consistently passing from a Lagrangian formulation to a reduced Hamiltonian formulation of a field theory, first by considering an example in section 3.2 and then by extending to the case of general relativity in section (3.3). Using the A D M foliation of spacetime introduced in section 2.1 and the Hamil tonian form of the Einstein-Hilbert action derived there, analyzed the resultant constraint algebra in section 3.3.1. We then considered the effects of the additional surface 16See pages 295-296 of [21] for specific definitions. Chapter 3. Constrained Hamiltonian Formulation 48 terms, which are induced from spatial infinity, for open universes on the total Hamil tonian in sections 3.3.2 - 3.3.3. Appendix D presents an example [22] where the variation of a reduced Hamil tonian action also induces non-trivial (and entirely analagous) surface terms which have to be included in order to obtain the correct equations of motion. Chapter 4 Perturbations 4.1 Introduction The non-homogenous fluctuation modes that we require are first-order metric and scalar field matter fluctuations. Since the first order fluctuations obey the equations of motion the first correction term within the semi-classical approximation wi l l come from the second order action in the perturbation variables: $ ^ E I ( ( 0 ) S + ( 3 ) S + ( 3 > S + . . . ) (4.1) The zeroth order action that we wi l l be primarily interested in wi l l be for the compact K = +1 F R W ('spherical') universe, although to retain generality and bui ld physical intuit ion perturbations about all three F R W universes (K = 0, ± 1 ) w i l l be considered. Perturbations of the metric can be broken up into three different types: scalar, vec-torial, and tensorial [24],[25],[23]. Physically, the scalar perturbations represent per-turbations of the energy density of the universe, vectorial perturbations correspond to rotations, and tensorial perturbations correspond to gravitational waves. Since the back-ground spacetimes we are considering have a vanishing Weyl tensor, gravitational waves wi l l only arise from the perturbations and in some sense the Weyl tensor could be used to measure the tensorial perturbation. In this thesis we wi l l only consider scalar per-turbations. For this particular choice of background, due to its maximal symmetry, the vectorial perturbations are purely gauge transformations so only the tensorial perturba-tions are being omitted [27]. That the tensorial and scalar perturbations wi l l decouple 49 Chapter 4. Perturbations 50 and that one is therefore justified in examining only scalar perturbations is shown in A p -pendix C [28]. We wi l l also comment on the linearization stability of the field equations about a F R W background, in particular about a compact k = + 1 F R W universe. The purpose of this chapter is thus to derive and interpret the action to second order in the fluctuations in the background metric and scalar field. In the first section we outline the perturbations and establish notation, and in the next the Lagrangian action to second order of the perturbation variables is considered. Then this Lagrangian action is converted into Hamil tonian form and reduced according to the methods of Chapter 2. The end result is that the perturbations, in this order of approximation, behave as simple harmonic oscillators with time dependent frequencies which depend on the background scale factor a and scalar field 4>B-4.2 Linearization Stability The relationship between the linearization of a full non-linear solution to a non-linear field equation and a solution to the linearized version of that field equation is not always simple. A way of characterizing this relationship is by defining the concept of linearization stability [8] about a given background solution QB of the field equations. Indeed, i f A represents the size of the perturbation and £(#B) = 0 is the background field equation, then the existence of a one-parameter family g(X)1 of exact solutions to the full non-linear equation £(g(\)) = 0 implies a solution to d 0, (4.2) A=0 which is a linear equation for ^ [<?(A)]|A_0 = 0. Whenever all solutions to equation (4.2) imply the existence of a corresponding one-parameter family of solutions g(X) to the non-linear equation £(g(\)) = 0, i.e. whenever the converse statement holds, the background 1Such that g(0) — QB and the A dependence is smooth. Chapter 4. Perturbations 51 gs is said to be linearization stable. In other words, the linearization stability of a non-linear equation is an existence statement for a one-parameter family of full solutions g(X) corresponding to every solution of the linear equation (4.2). For backgrounds which are linearization unstable, solutions to the linearized equation which are not tangent to the one-parameter family of solutions at A = 0 exist, and so are spurious linearizations of a full non-linear solution. Tensorial modes of such spurious solutions in general relativity, for example, could not be interpreted as gravitational waves since they are not true modes of the gravitational field. The issue of linearization stability for the vacuum Einstein equations has been ex-tensively studied 2 . One of the results of these various analyses is that i f the background spacetime possesses a compact, spacelike Cauchy surface (i.e., is closed and globally hy-perbolic) then the field equations are linearization stable i f and only if the background does not possess any isometries [30],[31]. If it does have any isometries, an additional second-order integral constraint 2 involving those K i l l i n g fields must be satisfied by the linearized metric in order for there to exist a one-parameter family g(X) [8]. Unless this constraint is satisfied the linearized metric wi l l not be tangent to any one-parameter curve of exact solutions. This restriction is clearly relevant for perturbations about a k = +1 F R W spacetime, as discussed in [32]. Linearization stability is believed to hold for al l asymptotically flat perturbations on asymptotically flat spacetimes regardless of the presence or absence of K i l l i n g fields in the background spacetime [32],[8], and in particular Minkowski spacetime is linearization stable [29]. For cases with matter source the analysis is more complicated 3 . In the case of perfect 2See [8] and references therein. Also see Fischer and Marsden in [29], Moncrief in [30] and [31], and D'Eath in [32]. An essential point to note in all of these proofs is that only the linearization of the non-linear constraints is analyzed. This is because the linearization stability of a well-posed hyperbolic system of partial differential equations is equivalent to the linearization stability of any non-linear constraints present. 3 Although the linearization stability of the Einstein-Maxwell system has been proven in a straight-forward generalization of [30],[31] (and the work of Fischer and Marsden cited therein) in [36]. Chapter 4. Perturbations 52 fluid source, for example, one may always perturb the fluid density and three-velocity in such a way as to 'make up' for the metric perturbations of the closed F R W universe and thereby avoid possible linearization instabilities 4 [32],[35]. Wha t this means concretely is that the A parameter dependence of the stress-energy sources is defined v i a the Hami l -tonian and momentum constraints and in this way a one-parameter family of solutions wi th matter is explicitly constructed, thereby ensuring linearization stability [32]. 4.2.1 Linearization stability in non-vacuum, symmetric, background space-times Let us first briefly skim over the argument due to Fischer and Marsden [29], in which they treat the case of vacuum linearization stability for spacetimes with compact spa-t ia l sections E t . They define a mapping $ which essentially describes the Hamil tonian and momentum constraints' (denoted 7i±(^g,Tr) — 0 and 5l(^g,ir) = 0, respectively) restriction of the full phase space 5 (^<7,7r) to a physical subspace (^<?*,7r*) (4.3) where ^g is the three-metric on E 4 and TT is the gravitational momentum on E t , defined by equation (2.15). The linearization of this mapping, denoted D$(^g,7T), w i l l be evaluated at the kernel of the full non-linear constraints described by equation (4.3) so as to invoke the Implicit Function Theorem, which one can cast into the following form: If the linearized constraint equations are surjective 6 (ie. D$(^g, TT) is surjective) at 4Note, though, that there are dissenting opinions on this issue. [33] and [34] recently claimed that closed FRW universes are always linearization unstable with perfect fluid matter as a source (under some slightly modified definition of linearization stability). Their contention, which is significant, has not to my knowledge been analyzed in the literature. Here it will simply be noted that they appear to place additional restrictions on the matter perturbations, which destroys the usual linearization stability.; This suggests that these additional restrictions are invalid. See also [37]. technically, this is a mapping from the 'cotangent bundle' of the spatial E t to the space of scalar densities cross space of vector densities. 6Every element in the range is an image of an element in the domain. Chapter 4. Perturbations 53 the kernel of the full non-linear constraints described by equation (4.3), then absolutions to the linearized constraint equations are tangent to the full solutions of equation (4.3) at the kernel of the full non-linear constraints. Now, using some results in elliptic theory Fischer and Marsden prove that D$(^g, TT) is surjective iff its associated adjoint mapping D$*(( 3)#, 7r) is injective. Thus, using the Implicit Function Theorem as given above, the proof of linearization stability reduces to showing that this adjoint mapping is injective, i.e. that its kernel is zero-dimensional (trivial). Fischer and Marsden find that in order for D$*(^g,7r) to be injective the necessary and sufficient conditions are i) If 7r — 0, ( 3 ) # is not flat, (4.4) ii) If £ x ( 3 ) £ = 0 a n d £ X 7 r = 0, then Xa = 0, (4.5) iii) TrK = qy/\Wg\, Vg e N+. (4.6) Moncrief in [30],[31] relaxes this third condition, and also proves that the dimension of the kernel of this adjoint mapping is in fact equal to the number of K i l l i n g vectors in the background spacetime. He also proves that any non-trivial kernel wi l l be generated by a vector which satisfies Ki l l ing ' s equations, i.e. a K i l l i n g vector. In order to exclude spurious solutions to the linearized equations D$(^g,7r) (which must exist i f there are K i l l i n g vectors in the background spacetime), one must place certain restrictions on the perturbations themselves. W i t h these restrictions in place, one can go ahead and perturb linearization unstable spacetimes and st i l l obtain valid linearizations of the full field equations. Such restrictions have been derived for the vacuum case, and for the case of matter perturbation similar constraints exist [37] which amount to generalized Gauss's laws for general relat ivity 7 . These integral constraints (due 7 B y constrast to the background spacetime, a Killing vector will not lead to conservation of a matter perturbation <5TM", since dT^" is not even covariantly conserved. Chapter 4. Perturbations 54 t o Traschen) define a 'general ized K i l l i n g vector ' , or ' in tegra l cons t ra in t vec tor ' ( I C V ) , w h i c h is defined by ana logy w i t h Gauss ' s l aw i n spec ia l r e l a t i v i t y : F o r G some spacel ike v o l u m e w i t h n o r m a l n a n d b o u n d a r y dG, we require tha t the in tegra l o f ST^V^ria over G be equa l to a surface in tegra l over dG. F o r F R W spacet imes, shw shows there are t en such cons t ra in t vectors , o f w h i c h s ix are genuine ly K i l l i n g vectors i n G ( in G a u s s i a n coordinates) a n d the o ther four are n o t 8 ( a l though they reduce to s t anda rd t i m e and space t r ans la t ions i n flat space t ime) [38]. W h a t is in te res t ing a n d relevant to th is thesis, however, is tha t the presence o f these I C V ' s is l i n k e d to the l i n e a r i z a t i o n s t a b i l i t y of the n o n - v a c u u m equat ions i n tha t they also require tha t non- l inear cons t ra in ts o n the pe r tu rba t ions s i m i l a r to those i n the v a c u u m case. A first order r e s t r i c t ion (for compac t spaces) sets out a necessary c o n d i t i o n for existence of so lu t ions and a c o n d i t i o n quadra t i c i n the pe r tu rba t ions ensures t ha t the so lu t ions so o b t a i n e d are tangent to the fu l l f a m i l y o f solutions[38]. T h i s l a t t e r c o n d i t i o n relates te rms q u a d r a t i c i n the me t r i c and m o m e n t u m pe r tu rba t ions to the second order v a r i a t i o n i n the sources. In t e rms of a m i n i m a l l y coup led scalar field w i t h a n o n - t r i v i a l p o t e n t i a l as a source, i t is u n k n o w n whether or not the sys tem is l i n e a r i z a t i o n s table , a l t h o u g h i n p r i n c i p l e any m a t t e r p e r t u r b a t i o n o f the field w h i c h satisfies the non- l inear cons t ra in t s o f T ra schen w i l l be l i n e a r i z a t i o n stable. W e w i l l s i m p l y assume tha t the sys tem we consider be low is l i n e a r i z a t i o n s table . 8 I n vacuum, all of these constraint vectors are Kil l ing vectors. I.e. in vacuum, Ki l l ing vectors do generate conservation laws for the perturbations. Chapter 4. Perturbations 55 4.3 Metric and Matter Perturbations The most general metric perturbation about a F R W background spacetime, in conformal time, can be written as ds2 = a2(V)[-{l + 2A(V,y^}dV2 + 2{Bli(V,y^ + Vi(V,y)}dyidr} (4.7) + {(1 - 2i,(n, ft) jijiy) + 2Elij(n, y) + 2^m(rj, y) + t^n, y)} dfdf] , where 7;J is the background F R W spatial metric in regular isotropic coordinates (r, 9, (j>), rj is conformal time (i.e. a(n)drj = dt)9, and | is the spatial covariant derivative on the isotropic slices as defined earlier 1 0 . The scalar field (matter) perturbations are given simply by Hv,y) = 4>B(v) + Hv,y), (4.8) where <P(r], y) is the full scalar field and $(77, y) is the perturbation. Let us examine the metric perturbation, equation 4.7, more closely. The vectors V1 and P, along wi th the tensor Uj, are defined to satisfy constraints of the form DiP = 0 , DiV\= 0 , Djtij = 0 , fi = 0, (4-9) to ensure that they do not contain parts that transorm as scalars and/or vectors, i.e. that they are purely vectorial and tensorial perturbations. To better see the scalar, vectorial, and tensorial perturbations separately consider an equivalent way [24] to write equation (4.7): ' -2AM) +2DiB(ri)yi) K +2DiB{T], y*) +2DiDjE{ri, yl) - ^(77, y*)^ S(ds2) = a2(rj) (4.10) W(77) 0 - 2 ^ ( 7 7 , ^ ) ^ -2^(77,7/0 +2/J ( iF j )(77,y i) + a2(n) \ +2^(77,7 /0 J 9Primes denote differentiation with respect to n. 1 0 D will also be used in places where the notation may become cluttered. In such cases, Dakb = kb |Q. Chapter 4. Perturbations 56 The first part is the 'scalar' perturbation (even though it transforms as a tensor!), the second the 'vector' perturbation, and the last the 'tensorial' perturbation, even though, of course, a l l the perturbations are of the same valence. Al though the terminology is rather unfortunate, one should just remember that a 'scalar' perturbation is simply a tensorial perturbation made of the three-metric or covariant derivative acting on scalars, and similarly for vectorial perturbations. Thus, there are four metric scalar functions A,ip,E,B, two three-vectors Ti,Vi, and one symmetric three-tensor ty , giving 4(1) + 2(3) + 1(6) = 16 functions. But there are precisely six constraints as well from DiT1 = 0 , DiV* = 0 , Djtij = 0 , f j = 0, which leaves 1 6 - 6 — 10 independent components of the metric tensor, as required. The assumption that the background hypersurfaces are spatially isotropic11 means that.the spatial parts of the perturbations can be expanded in the eigenmodes of the Laplacian associated with the particular hypersurface under consideration (see Appendix The operator A is the spatial Laplacian (sometimes called the Laplace-Beltrami operator in this context), and is related to the induced metric by the relationship The eigenvalues k2 and eigenfunctions G w i l l be different for each background spacetime, forming a continuous spectrum for the flat (K = 0 ) 1 2 and hyperbolic (K = -1) universes (because the eigenfunctions do not satisfy periodic boundary conditions) and a discrete spectrum for the spherical (K = 1) case: nIsotropy implies homogeneity, but not the other way around. See [8] for precise definitions. 12Unless otherwise stated, K will always denote the constant curvature of the FRW background. A ) : tfDiGfax) = AG(r],x) = -k2G(n,x). (4.11) (4.12) Chapter 4. Perturbations 57 K = 0 , k2 = m 2 , m2 > 0 , V m € R (4.13) # = 1 , A;2 = £(£ + 2) , £e Z+ (4.14) I f = _ i ) f c 2 = m 2 + l , m 2 > 0 , V m € f t (4.15) The eigenfunctions are also discussed in Appendix A , but we note in passing that they are simply plane waves in the K = 0 case, generalized spherical harmonics in the case of K = +1, and the so-called Fock polynomials for K = -1. . 4.4 Formulation of the action The calculation of the Lagrangian (ignoring the surface terms for a moment) for the perturbations, i.e. expanding out the Ricc i tensor to second order in the perturbation variables, is a tedious task. However, appealing to the A D M formalism and expanding out the three-Ricci tensor that appears in equation (2.13) leads to considerable simplification in computing the gravitational part of the action [24]: So = ^ / < V = W * -^[(3)rf/y v/jfc[ + z > g ) a V (4.16) where DQ is a total derivative term which wi l l be suppressed. We now insert the scalar metric and matter perturbations, the first term of equation (4.10) and equation (4.8), into equation (4.16) by identifying N = a2(n)(l + 2A(rj,x)), (4.17) Nt = 2a2{r])DIB(ri,x), (4.18) Chapter 4. Perturbations 58 hij = a2(rl){{l-2^(rl,x))jij + 2Eij(n,x)}, (4.19) which, with equation (2.10) and keeping only terms at second order13, yields • = • £ - [ \a2(r))y/fr\ {-6t/>'2 - UHAip' + 2AiP{2A - </>) - 2 (H' + 2U2) A2 +K ( $ ' 2 + $ A $ - a2{r))VH§2) + 2K (3<f>'Bil>'$ - <f>'B$'A -a2{r})A^V^) +K[-6^2 + 2A2 + l2^A + 2{B-E')A(B-E')) + 4 A ( B - £ ' ) ( ^ - r M - / ) } d*x, (4.20) which matches the analagous result of [24],[13]. In obtaining the above result we have used the background (zeroth order) equations (in conformal time) for the Hubble parameter14 T-i and background scalar field (f)B : •H2-H' + K = (4.21) 2K' + n 2 + K = ^ ( - V 2 + 2 a V ( ^ ) ) , (4.22) and <f>B'' + 2H<l>B' + a%BV{<l>B) = 0. (4.23) These equations significantly simplify equation (4.16). 4.5 Hamiltonian action for metric and matter perturbations At this stage we define the canonical momenta in the usual way and obtain •' j * s l r , = 2 ^ ( _ 3 ^ + A £ ' + 3 ^ - A B - 3 ^ ) , ( 4 . 2 4 ) 1 3We recall that the first order terms are zero by the background equations of motion. 1 4 K = a{rj)'/a(r}) = a(t) (dta(t)) /a(t) — a(t), ' = dt, where t is proper time. Chapter 4. Perturbations 59 — =7T$ = a2ffi(&-<i>B'A), (4.25) § S „ B = ^ A ( ^ - ^ £ - K B + ^ ) , (4.26) a ^ - ^ = °' (4-2 7) 9B'=*B = ( 4 2 8 ) Inverting the above momenta for the 'velocities' E', and ip', noting the primary constraints TTA — 0 = irB, and performing the Legendre transformation, we find the action in Hamil tonian form to be 1 5 <2>S = J d4x (4.29) = j (TT^' + irEE' + 7 r $ $ ' ~^H)d4x = J dAX7r^1p' + 7T£.E' + 71-$$' Tfa) + 2 ^ ^ + + 2 ( A + 3 K ) y + ^ | ( A + 3 J ^ 2 - f ( ( A + 3tf) - H 2 - W + $ 2 } + K 0 W § + + ( - ( A + 3 i ^ ) ^ + | ( W B - </>£)$) J + 1*7^ where Tfa) = • (4.30) 4a 2(r7)^/H(A + 3 / 0 Equation (4.29) agrees exactly with equation (B6) of [13]. There are no terms quadratic in the lapse perturbation A in the Hamiltonian action since their coefficients vanish owing to equations (4.21) - (4.23). Equations (4.21)-(4.23) were also used to eliminate V((f)B) in favour of higher derivatives of <pB in equation (4.29) for purposes of comparison wi th [24],[13], even though it wi l l later prove useful to have the action exhibit an .explicit 1 5 A will be treated as an algebraic factor (equal to —k2) in what follows, owing to equation (4.7). Chapter 4. Perturbations 60 dependence on the potential and its derivatives. This explains the somewhat unnatural appearance of the (j)"B' term: it is due to the usage of a background equation on a term like d2BV(<t>B). 4 . 6 Action for K = + 1 Consider the case where the perturbations are about a closed F R W universe. In that case, the eigenvalues of the spatial Laplacian are k2 = £{£ + 2), £, £ Z+, and so the non-zero momenta become, from equations (4.20) - (4.24), 2 a 2 V ^ ( _ 3 V / - £(£ + 2)E> + * ^ L * + e { t + 2)B-SHA) , (4.31) (<*>' - <j>B'A), (4.32) K 2 7ty = 7r$ — a 7 T E = ^V^l , _ e , e + 2 ) ) (xE> + ^ £ -KB + HA^j . (4.33) It is clear that the momenta wi l l not be independent for £ = 1, in which case ir^ = TTE and there is thus a constraint <j>\ = — TTE ~ 0. Indeed, it wi l l be shown later that these modes have no physical degrees of freedom associated with them. However, for modes wi th £(£ + 2) > 2 the primary constraints are (f>i=7vA = 0, (4.34) (j)2 = 7rB = 0, (4.35) and their preservation in conformal time generates secondary constraints which are the (linear) Hamil tonian and momentum constraints for the scalar perturbations. The total Hamil tonian is (where the fa are the primary constraints), after integrating out the S3 spatial dependence, Htotal — j [H0 + X'fa] d 4x Chapter 4. Perturbations 61 3 T T B 2 rr2 j\\r(V) (-K^2 + 2 ^ E + _ ( ^ 2 ) ) + 2(-(*(* + 2)) + 3 ^ ) ^ +«*Wf + ^  {(" W + 2 ) ) + 3 ^ ) ^ - f «" W + 2)) + 3 f f ) -U2-n' + ^ j ^ + A {-U^ + 2a2JW\ + 2)) + + \(UfaB - + 57 r E (4.36) Expl ic i t ly , the secondary constraints are / n r TJ 1 ^Htotai _ ^ 7I\4 = U = {TTA, Htotai) = - = L>A, i n r t r 1 ^ Ht0tal _ n TTB = U = {7TB, Htotai) = ^ g - = *-B> (4.37) (4.38) where CA = C B = 7TE = 0. j-Tfcr, + #,TT. + (-(-*(* + 2) + 3 K > + (^H<f>B - <f/B)$) = 0, Equations (4.33) and (4.34) are the scalar analogues of the Hamil tonian and momentum constraints. We also note that equation (4.32) demonstrates that the total Hamil tonian for these perturbations contains not only constraints but also a 'dynamical part', unlike the background Hamil tonian which is only a linear combination of constraints. Next we proceed to verify what class these (linear) Hamil tonian and momentum constraints belong to so as to deduce any possible gauge transformations they may imply, but first we quickly check to see that they do not imply any further unique constraints: CJ! = {^ ,#0} + AM{C ,^<M = u,cA-dWcA-cB + \l{cA,iTA} + \2{cA^B} Chapter 4. Perturbations 62 CV = {7vE,H0} + Xm{TrE,<f>m} = 0 + X1{nE,nA} + \2{irE,TTB} = 0, (4.40) which agrees with [13] up to a sign on the operator. A t first glance, it appears that there is a new constraint implied by equation (4.35) owing to the presence of this strange derivative term, which only operates on the background fields. However, since the background fields depend on r\ this term is entirely expected as the time derivative defined by the Poisson bracket is the total time derivative [39]. Thus, equation (4.35) must be interpreted as •jt[CA] = [CA' + d^CA] = H,CA-CB + \L{CA,TrA} + \2{CA,7rB}. The left hand side is now the total derivative, which is equal to a linear combination of constraints. This means that there are actually no new constraints and so, by equation (4.35), the constraint tower closes. Now we finally proceed to verify which class they belong to by computing their mutual Poisson brackets: {CA,CB} = {CA,nE} = 0 (4.41) {CA,CA} = + ^ - + — ( - ( - ^ [ ^ + 2] + 3 ) ^ + - ^)a7 r 4 >J ^  ~ 0. (4.42) The other brackets (with the primary constraints) strongly vanish by inspection, which means that the above constraints are a l l of the first class and, i t turns out, generate linearized coordinate transformations under which the physical evolution of the dynam-ical variables is not affected. Let us explicitly demonstrate this by computing how the perturbations transform under an infinitesimal coordinate transformation (which is done the 'long way' in Appendix B ) . These transformations are extremely important since we Chapter 4. Perturbations 6 3 will use them to reveal to what extent a given choice of coordinate system exhausts the available coordinate freedom. For the most general 'scalar transformations' (i.e. those completely specifiable by supplying only scalar functions T and L which are later operated on by spatial covariant derivatives to obtain vectors) [24] f) ->• n + Tfax*), (4.43) yi ^ yi-r PL-fay*), (4.44) we can compute the transformations in the canonical variables generated by the con-straints. If for example a dynamical variable a is so transformed to a, then cu = a + Set (4.45) = a + 8va{a,Ca}, where we have defined 8a using the gauge functions 5ua (T and L here). B y the discussion in the previous chapter, the constraints generate the unique way in which the dynamical variables change under an arbitrary linear diffeomorphism. For the case of these scalar diffeomorphisms we find 8ij) = 8va{iP,Ca} = T{xp,CA} + L{^,CB} = T(d^CA)+L(p)=T{-H), . (4.46) 8$ = T{$,CA} + L{<!>,CB}=T{(j>B'), (4.47) 8E = T{E,CA} + L{E,CB} = T(0) + L{1), (4.48) 8^ = T(-d*CA) + L(0) = -T(a2^\{H(j)B'-fa")}, (4.49) Chapter 4. Perturbations 64 (A + SK) +1,(0), (4.50) 6irE = T(0) + L ( 0 ) = 0 . (4.51) To obtain the transformation properties of A and B from this method one uses the invari-ance of the action under these transformations, as enforced by the Bianchi identities, and deduces the unique way they must transform in order to satisfy this invariance. Al terna-tively, one could proceed by computing the brute force Lie derivative of the background metric along the vector field defined by this (scalar) infinitesimal coordinate transforma-tion and read off the transformation properties (Appendix B ) . Thus, in this section we have obtained the Hamil tonian action in second order in the scalar metric and matter perturbations and have analyzed its associated linear Hamil to-nian constraint and momentum constraints. It remains to reduce it by choosing a gauge and using the constraints. 4 . 7 Gauge fixation and 'gauge invariant' variables Unlike other physical theories, perturbation theory in general relativity is complicated by the fact there is no preferred notion of identifying the background about which one perturbs to the perturbed spacetime itself. The choice is completely arbitrary, and once made it amounts to a point-identification map between the two spacetimes called a gauge choice1^ in exactly the sense of chapter 3. This choice is equivalent to picking a coordi-nate system on the perturbed spacetime, having fixed the coordinates on the background spacetime. It is often convenient to make this choice in such a way that some of the per-turbations are actually zero. For example, i f one picks a coordinate system in which the 1 6 This is the so-called 'active' approach to gauge choices described in [24], and there are other formulations. Chapter 4. Perturbations 65 shift and off-diagonal spatial perturbations are zero, then one operates in so-called longi-tudinal gauge. Bu t , of course, there are other choices including the so-called synchronous gauge used in the first study of perturbation theory in general relativity [25], although this particular choice leaves non-trivial residual coordinate transformations whereas the longitudinal gauge does not. Now, in an important paper [40] (see also [23] and [24]) Bardeen pointed out that one could construct variables which were invariant under the transformations implied by the first-order equations (4.42)-(4.47), called gauge invariant variables. For example, one may verify that i> EE if} - Ti(B - E') (4.52) A = A + ^{{B - E')a)' (4.53) $ EE $ + 4>B'(B — E') (4.54) are invariant under the above transformations [24]. However, one can regard usage of these (or any other combination of so-called gauge invariant) variables as just using the perturbed metric in a certain gauge-fixed form for which the linearized equations of motion are identical to those using the new variables [41]. For example, the Bardeen variables ^ , A , $ are just the variables ip, A , $ with the gauge fixing B — E' = 0. Thus, physically and mathematically there is precisely no difference between using one particular set of gauge-invariant variables and fixing the gauge in some particular way [41]. However, whenever one can specify perturbations exclusively in terms of such variables then the gauge impl ic i t ly selected completely exhausts the coordinate freedom that one has. Chapter 4. Perturbations 66 4.8 P o s s i b l e gauge choices Let us consider some of the possible choices of coordinate systems, i.e. gauges. Some choices are more convenient than others because residual coordinate freedoms which are s t i l l consistent with the chosen gauge may exist, and also some choices are singular when cj)B' = 0 or a' = 0. In some cases one may choose to gauge-fix variables other than the lapse A and shift B, such as E or TT$. Since this choice must hold at al l times, then the equations of motion wi l l give the effective choice of lapse A and shift B, which are necessary to completely specify the coordinate system in the A D M foliation. In particular, we wi l l consider gauges which 'isolate the matter degree of freedom <&' of the perturbation (one of which is the ' longitudinal gauge') and those which similarly isolate the 'gravitational degree of freedom ip> of the perturbations. We wi l l consider the standard 'synchronous' gauge choice, as well as some others which fix the conjugate momenta to the perturbations rather than the perturbations themselves for the sake of completeness. 4.8.1 S e t t i n g E = 0 = <l The required coordinate transformation is (4.55) yi = yi-LyEfay'), (4.56) which is equivalent to using the gauge invariant variables $ = ^ + —j$, (4.57) (4.58) Chapter 4. Perturbations 67 The equations of motion derived from equation (4.29) imply that this choice of gauge is equivalent to specifying the scalar lapse and shift as A — (4.59) B = = J i : ( 2 ^ + ^ ) ' . (4.60) 4 a 2 ^ ( A + 3 K ) . A Selection of this gauge is often said to be tantamount to 'isolating the gravitational degree of freedom' of the scalar perturbation (as opposed to isolating the matter perturbation variable <3>) in the literature [42],[13], since it leaves the reduced action (assuming one uses the constraints judiciously) purely in terms of the metric perturbation ip in this gauge. 4.8.2. Setting B = 0 = E The required coordinate transformation, which casts the problem into ' longitudinal gauge', is fj = ri + iBfay^-E'fay*)), (4.61) f = f-D'Efay1), (4.62) which is equivalent to using the variables $ = ip - U{B - E'), (4.63) $ = ^ + (j)B'(B - E'). (4.64) The equations of motion derived from equation (4.29) imply, by restricting the off-diagonal and spatial momenta, that this choice of gauge is equivalent to specifying a lapse which is actually equal to the diagonal spatial perturbation [24],[42]: A , TTE = — = > A = ip.- (4.65) O Chapter 4. Perturbations 68 Selection of this gauge, for similar reasons as above, is said to 'isolate the scalar matter' degree of freedom $ . 4.8.3 Setting A = jj), B = 0 The required (nonlocal) coordinate transformation that this gauge represents is \ii.Li i\ A / „ „ .mj„ i Ci(y ) y *(»?) y' + D +C2(yi)] ![{+aWl a {nr^M)-AM))dr} + (4.66) dn (4.67) where C^x1) and C2{xi) are arbitrary spatial functions of integration. This choice admits residual coordinate transformations, = v ' + C M y * ) , (4.68) (4.69) because of which there exist no gauge-invariant variables to formulate this gauge choice in . The lapse and shift have been selected a-priori, by the gauge choice. The reason for mentioning this gauge is to emphasize that picking coordinates such that A — -0, B = 0 is not equivalent to picking longitudinal gauge as defined above. 4.8.4 Setting A = 0, B = 0 This requires the (non-local) coordinate transformation, called the 'synchronous gauge' since it picks a coordinate system comoving wi th the perturbation, »(»?) yi = y1 + Di (4.70) B(r},yi))drj Chapter 4. Perturbations 69 + (4.71) This choice also admits residual transformations because of the spatial constants of in-tegration that occur, so that the residual transformations are 1 77 = n + a(rj) yi = tf + D1 [ -Lc2(yi)dV + C^y1) J a{n) (4.72) (4.73) meaning there are no associated gauge invariant variables with this choice. Synchronous gauge is a popular choice for the analysis of perturbations in the literature, in particular being used in the first comprehensive study of the subject [25] because of its comparative simplicity to implement. 4.8.5 Setting ir* = 0 = E This requires the coordinate transformation r, = n 7T$ j? = y'-D'E, which is equivalent to using the variables ijj = + Tin® . _ 2 (A + 3X)7r$ KtfyfilWte' ~ 4B") This choice is equivalent to setting the lapse and shift to 1 A = B (H4B ~ 4B") K K ^+(n2 + n'-(A + sK)-^-rj^ 2oVl7 — ( 2 ^ + ^ ) (4.74) (4.75) (4.76) (4.77) (4.78) (4.79) Chapter 4. Perturbations 70 This choice of gauge is well-defined for a' = 0, which wi l l make it interesting for later applications. 4.8.6 Setting 7ty = 0 = E Fina l ly we consider the coordinate transformation V = V r = ^ , (4-80) xl = xi - D*E, (4.81) which is entirely equivalent to using the variables *• = **- 2(A + 3K) **' ( 4 " 8 2 ) * = * ^ T r , . (4.83) 2a 2 ^ ( A + 3K) The effective choices for the lapse and shift in this case are, by the equations of motion, A = tp (4.84) B = =J5 ( 2 ^ + ^ ) (4.85) It is interesting to observe that this gauge is only a rotated version of the longitudinal gauge. 4.9 Reduction of the K = +1 Action Now we must solve the constraints and select one of the above gauge choices to obtain the reduced action. The aim is to take the reduced action to second order in the per-turbation variable and deduce from it the time-dependent frequency of the oscillation of the perturbations about the background. Chapter 4. Perturbations 71 Let us select the longitudinal gauge as described above (i.e. B = E = 0). After inserting the choice of lapse that this gauge implies (A — ip), we solve the Hamil tonian constraint, equation (4.33), for ip, resubstitute this result back into the action, and triv-ial ly impose the momentum constraint 1 7 7ty — 0. The form of the reduced Hamil tonian ac t ion 1 8 is then (4.86) where a, 6,7 are complicated functions of the background fields. We then eliminate the cross-term 7r$$ in the Hamiltonian by defining the new variables 7T<j> = V ^ ) T . + — ± = ( -J [ ln(a ( i7 ) ) ] ' + 8(n)) 2 ^ ( 7 7 ) V 2 . ) so that the action becomes 1 9 (4.87) (4.88) 7f$$' -7rl , 2 / * 7 i (4.89) where u; 2^) 7 W « W : 3 — / + 7 4 4 2 v / o ^ ) 4 1 {y/<*(v))' (4-Equation (4.89) implies that the linear perturbations behave as oscillators, and that furthermore they oscilllate about the isotropic background with a time-dependent fre-quency whose square is given by equation (4.90). For our particular case, the coefficients 17Since TTE = — = 0 => ity = 0 if A ^ 0, which is true for non-homogenous perturbations only. 1 8 The detailed expression is not illuminating at this stage. 1 9 In arriving at this result we performed integrations by-parts. Chapter 4. Perturbations 72 tv(?7),/3(77),and 7(77) are 1 L , *<t> /2 2 ( A + 3 A T ) / ' ^ = ( 4 ' 9 2 ) 2 7(?7) = ~W(A + W) t(A + 3^ (4 '^0B' + 3K0B'3 " 8^20B' + 20B"') + 6kA</>b' AC {-ri2<t>B'2 + 20B'^B" - 0B"2) 0B'] - 2 ( A V | 3 X ) - (4.93) When these functions are substituted into equation (4.90) and the background equations (4.17)-(4.19) are used to make judicious simplifications, one can show that the frequency is actually given by the surprisingly simple expression - ( A + 3K) - - ^ - j ~ - 2 < t > B , (4.94) where Equation (4.94) differs by a term — §</>B'2 from the analagous expression in [42]. This is the same frequency one would obtain by using the rotated longitudinal gauge (ir^ — 0 = E), which has the advantage of being well-defined when a' = 0. 4.9.1 R e d u c t i o n i n different gauges: E = l» = 0 Instead of isolating the matter degrees of freedom, let us select the gauge choice E = $ = 0 and isolate the metric perturbation ip. Going through the exact same procedure as with longitudinal gauge, except this time solving the Hamil tonian constraint, equation Chapter 4. Perturbations 73 (4.33),. for 7r$ = 7r*(t/>,7ty) and imposing the momentum constraint irE = 0 directly, the reduced action is 5 | , ^ K ^ B M ) r 2 ' 2 2 • d4:r. B y using equation (4.81) and the background equations (4.17)-(4.19), the frequency can be compactly expressed as (4.96) 2 (A + 3K) K(j) ,2 B An2n' + K(n'(2n2 -H' + K) + WH) _ ; 2H2{A + 3K)- KK(f>B'2 where KK<J) I2 B (4.97) 2 o V l 7 l ^ ' 2 V 2 ( A + 3*0 In order to check the substantial amount of algebra that is required to obtain the above result, we check that at least that upon insertion of K — 0 we agree wi th the well-known expression found in [24]. Indeed, setting K = 0, we find 1 +2<j)B"H'2 -U(j}B"H"] - A u2(rj) n4(j)B' + u2^" - 2 u<j)B" n' + 2 n%4B" - n2v! <pB' Ufa) + A (4.98) where (4.99) This is indeed the expression obtained in [24],[13], adding confidence that equation (4.92) is in fact correct. Chapter 4. Perturbations 74 For homogenous perturbations (where A = 0 ) one can go through the same procedure as above (except this time there is only a Hamil tonian constraint to solve, and so only $ is gauge fixed) and, after using the background equations (4.17)-(4.19), obtain the frequency ^ ( l \ " m*fr + {AH2 - H' - 2K) cj2(a(V),Mv)) = -VZ[-7=) ~4K *" V —-2 - (4.100) WaJ 6H2 - K(j)B and o~(r}) is defined as 4.9.2 The 1=1 mode It was mentioned in section 3.5 that the £ = 1 mode of the perturbations in a closed ( K = 1) F R W universe wi l l have no physical degrees of freedom associated wi th them. In this section we provide the reasoning behind this claim. For the £ — 1 modes the second order Lagrangian in equation (4.16) reduces to (since E = 0 in an £ = 1 perturbation) ( 2 ) £ % i i/iU [-6ip'2-12HAip'-6ip (2A-ip)-2 (H' + 2H2)A2 (4.102) +K (V - 3 $ 2 + + 2H'-4H2^ +2 K (3 <j)B' $ $ - 4>B' A + A $ (<f>B" + 2 H<t>B')) +K (-6ip2+ 2A2+ 12ipA-6B2) - 12 B (1/2 K<J>B' $ - HA - 0')) , where U = (4.103) K Defining the canonical momenta TTjh = ^ r r r = 1/4(7 ( - 1 2 0 ' - 12HA + 6 K(f>B'$ +12 B), (4.104) dip' Chapter 4. Perturbations 75 7r«j> = ^ — - = 1/4U(2K$' -2K<f>B'A) (4.105) dip' we define the total Hamiltonian as usual { 2 ) H = 7 T ^ ' + 7 T $ $ ' - ( 2 ) £ (4.106) = -1/6^ + ^ ( - H A + 1/2K<I>B'$ + B) + J £ &UK (64B - 24B'" -3K4b'3 -in'4B'+ 8U24A +1/8 * — , >-4B +A ( - 1 / 2 K f /$ 4B" + 1/2 K 4B UH$ + 7T<p 4B') . The term proportional to A 2 again vanishes due to the background relations. The two secondary constraints of the theory (irA = 0 = irB') are CA = 1/2K4B'Um-nn^-1/2KU^4B"+ T^^ 4B'= 0, (4.107) CB = 7 ^ = 0. (4.108) They themselves are preserved because CA' = { C A T = 1 W H E = 1 } = H C A T = 1 - C B T = L * 0 , (4.109) CB' = { C B I = 1 , { 2 ) H I = 1 } = 0, (4.110) and furthermore {CA,CB} = 0, (4.111) so that the constraints are of first class. Since the Dirac conjecture holds for this system, this implies that CA and CB generate the scalar diffeomorphisms under which the theory should remain invariant. Indeed, for the case of I = 1 one has that the dynamical variables which are left (ip and $ ) transform as 5$ = x T{$,CA} + L{$,CB} = +4B'T, (4.112) Sip =L T{ip, CA} + L{xp, CB) — —HT + L. (4.113) Chapter 4. Perturbations 76 This means that by selecting a coordinate system in which $ = $ + 5$ = O = '0, we can completely specify the scalar gauge functions L and T, i.e. completely exhaust the available coordinate freedom. The coordinate transformation that fixes this is explicitly V '= V - ^ i , ( 4 . H 4 ) 9B ii-u—, 9B (4.115) Substitute the momentum constraint 7ty = 0 into equation (4.107) to obtain ^ 2 &UK ( 6 0 B ' - 2 0 B ' ' ' - 3 K 0 B ' 3 - 4 H ' 0 B ' + 8 ^ 2 0 B ' ) = + y f - + l / 8 ^ —7 L UK 9B +A(-1/2KU$9B" + l/2K</>B'UH$ + Tr*<f>B). (4.116) The gauge conditions -0 = 0 = $ must be preserved in conformal time, so ip' = 0 = These consistency conditions imply a particular choice for the scalar lapse A and scalar shift B v ia the equations of motion, i.e. <£' = 0 - > i = % , (4.117) ft = 0 - > B = 0. (4.118) However, the Hamil tonian constraint in this particular gauge implies that 7f$ = 0. Thus, by equation (4.117) this must imply that A = 0 as well. But that means that the coordinate transformation given by equations (4.114) and (4.115) effectively set all of the perturbations to zero, which means that the £ = 1 mode for closed F R W spacetimes is a coordinate artifact, i.e. a gauge mode, as first pointed out in 1946 [25]. Chapter 4. Perturbations 77 4.10 l£ = 1' mode for K = -1, 0 Final ly , let us consider perturbations for which E — 0 ab init io about the open and flat F R W backgrounds. Keeping A general but non-zero (since it is now continuous as opposed to discrete, the statement £ — 1 has no real meaning in this context), as well as K, we find that both the Hamiltonian and momentum constraints are now of second class: {CA,CB} = {CAL | ^ + UA(j + K)B} = - A ( A + 3 3 K ) U ? 0, (4.119) {CB,CB} = 0, (4.120) {CA,CA} = + Ud^ - fad* + ( - ( A + 3 # + \wB - 4>"B)d^) c A « 0. (4.121) This means that the scalar Hamiltonian and momentum constraints are no longer the generators of the (linear) scalar diffeomorphisms. Also, the preservation of the Hamil to-nian and momentum constraints is non-trivial in this case because C'A = {CAWH} = HCA-CBK0, (4.122) C'B = { C B ^ H } = - ^ ^ ^ - A ) . (4.123) Note that the slightly different form of the momentum constraint also formally modifies the transformation property of ip to Sib Kt+1 T{IP,CA} + L{TP,CB} = - H T - J L , (4.124) although technically this relation is to be treated with caution since the momentum and Hamil tonian constraints are second-class. Formally proceeding, however, equation (4.117) implies that in order for the momentum constraint to be consistently preserved Chapter 4. Perturbations 78 in conformal time we must pick a gauge such that the xp = A is t rue 2 0 . This effective re-striction to one possible gauge choice essentially forces a further gauge choice since wi thin the gauge choice ip = A both the rj reparametrization and the spatial reparametrization (because L = L(T) here) are arbitrary to spatial constants of integration. The only manner in which to completely fix the r\ reparametrization is to insist on $ = 0, which would imply a complicated relationship between A and $ in order to st i l l maintain the condition ip = A: ~ , = l(~ JvAdrj + d(y*)) . (4.125) Time preservation of this gauge ($ ' = 0) implies the relationship A = — f aAdn. (4.126) a J However, $ ' = 0 also implies, by an equation of motion, that A = ——P=—:, which wi l l in general be difficult to satisfy simultaneously with equation (4.131). This possible inconsistency indicates the breakdown of the reduction procedure for these modes, pr i -mari ly because the constraints are no longer first class and they no longer imply scalar diffeomor phisms. 4.11 Summary of Chapter In this chapter we have formulated, and justified as physical, linear metric and matter perturbations about a F R W background in sections 4.1 - 4.3. In section 4.4 we formulated the Lagrangian action to second order in these perturbations and then proceeded to construct the corresponding Hamiltonian action (using Chapter 2) in section 4.5, finally specializing to the special case of K = 1 in section 4.6. In section 4.7 we considered 2 0Which again we note is not the same as picking the longitudinal gauge B = 0 = E, as explained in section 4.7.3. Chapter 4. Perturbations 79 various gauge choices that we could make to try and isolate the physical fluctuations that the above actions implied, and then in sections 4.9 and 4.9.1 we reduced the Hamil tonian action twice using two such gauge choices. Finally, in sections 4.9.2 and 4.10 we analyzed £ = 1 modes for K = 1 perturbations and the analagous perturbations for K = —1,0 cases. Chapter 5 Complex Paths In this chapter we consider the behaviour of the background fields a and $ B to first order in e along a complex contour in r-space. After reviewing the contour that Hartle and Hawking (HH) originally used in their study [2] of the constant scalar field potential in Section 5.1, we shall recast the canonical Hamil tonian action (equation (4.89)) in the perturbations into a form where the background dependence is much more t r i v i a l 1 in Section 5.2. This wi l l motivate a contour in r-space appropriate to first order in e, as discussed in Section 5.3. That the junction conditions 2 are obeyed along this contour wi l l be discussed in Section 5.4. In Section 5.5 we wi l l briefly review one approximate solution first considered by Green and Unruh [47] which suggests that the matter (scalar field) part of the background (zeroth order) action flips sign, so as to to render Linde's choice of sign for the semi-classical wave function weighting favourable. In Section 5.6, we conclude by dicussing what this particular solution may imply for the second-order action in the metric and matter perturbations. 1In the sense of ensuring that the background parts of the second-order action computed in Chapter 4 are both well-defined and of a sufficiently simple character (for example, real) to be condusive to a compelling argument concerning the overall sign of the action of the perturbations. 2The Israel junction conditions (see [20]) are essentially that the three metric must be continuous across the join of two or more genuinely different spacetime geometries (i.e. the spacetimes are not related by a diffeomorphism), and that the extrinsic curvatures Kij may or may not be continuous across the join. Any finite discontinuity is physically interpreted as the stress-energy of a thin matter 'shell' at the join and is hence a tolerable discontinuity. However, no such discontinuity is allowed when matching across a spacelike hypersurface and in fact we do not have such a discontinuity along the contour we choose. 80 Chapter 5. Complex Paths 81 5.1 Zeroth order contour: Hartle Hawking contour The constant scalar field potential (cosmological constant) case is equivalent to a zeroth order analysis in our slow-roll model. In this case the contour in r space can be segregated into purely real and imaginary parts, and that is the contour H H originally used. Thei r contour starts off down the imaginary axis from 0 to —n/(2u), then continues parallel to the real axis unti l c o s h - 1 (z/a/)/V, as shown in the Figure 5.1. A long this contour is kept wholly real, also as shown in Figure 5.1, where a^(ro) = a; motivates the particular choice of the real part of the zeroth order endpoint r 0 . Along any other contour w i l l in general be complex before returning to r 0 = c o s h - 1 (va,f)/i/ — iirl(2v) and becoming wholly real. Im(T) ~ n 1 2 v 1 I 1 1 1 II Im( a°>) • arccosh( va .) / v Re (T) Re( <f ) Figure 5.1: The unique H H contour such that a ( 0 ) remains real throughout. A long any other contour wi l l acquire an imginary part before becoming real at the endpoint r 0 . Since the scalar field to zeroth order is just a constant, then the scalar field is equal to its final value 0B / , which is real by construction. Chapter 5. Complex Paths 82 5.2 Simplified background dependence: non-canonical second-order action In models wi th constant potential (cosmological constant) it is possible to segregate the complex contour in r into purely imaginary and real parts, as we just saw i n the previous section. But , as pointed out in [4] this is not possible in general and in particular it is not possible in this model [5]. In order to address the question of which of these generally complex contours is 'appropriate' for the analysis of the linear metric and matter perturbations, it is necessary to examine either the background dependence of the reduced second-order canonical act ion 3 or to examine that of a simpler, rescaled action (which is non-canonical). A contour such that this background dependence is simple (i.e. such that the background quantities at some order in the slope of the potential are either purely real or imaginary) wi l l be called 'appropriate'. The reason is that along such a contour the overall sign of the action of the perturbations wi l l be more apparent. Pursuing the first program, we note that Equation (4.89) reads (with the integral over S 3 contributing a factor of 7 r 2 ) '^2 dn (5.1) where the complex frequency (now in proper complex time r ) is w 2(a(r),fc,(r)) = -(-£(£+ 2) +3)-= - ( - £ ( £ + 2) + 4) 1 Q a<f> B '. d d2 aTr+adV2 a<j)i Q (2{£(£ + 2) + 3) + a2K(f)2By 2 4 d a 0 B V ^ H O ? + 2) + 3) 3 Which reduces to examining the background dependence of the frequency of oscillations of the linear modes (as given by Equation (4.89)). Chapter 5. Complex Paths 83 - a 2 ( - 1 2 K O 2 0 ! ( - ^ + 2) + 3) + 20(-£{£ + 2) + 3 ) 2 + « 2 a 4 0 B ) +aa ( 8 a 2 « 0 B ( - ^ +.2) + 3) + 12(-£(£ + 2) + 3 ) 2 + / c 2 a 4 $ , ) + 6 ( ^ j \ a 4 ( - £ ( £ + 2) + 3) ] , (5.2) and where Q(T) = 1 + a 2 (5.3) 2(£(£ + 2) + S) Here we have eliminated higher derivatives in (J>B in favour of terms involving the scalar field potential, and have set V^B<pB = 0 because of the slow-roll potential. We notice that the behaviour of the full physical frequency in proper time (equation (5.2)), in the l imi t of vanishing scale factor a, is along a Euclidean contour approach to r —>• 0. Notice that for homogenous modes the l imi t is negative whilst for £ > 2 (inhomogeneous, physical modes) the l imi t is positive, which is exactly what Tanaka finds in his 'Negative mode theorem' [42]. This adds further confidence that the frequencies obtained are correct. We can expand the background frequency in the slope of the potential to extract the leading order behaviour. Indeed, if we use the fact that the background scalar field is a constant at zeroth order, then the frequency can be expressed up to second order (for £ > 2) as l im U2(T) = £(£ + 2)-l (5.4) a(T)->0 (5.5) Chapter 5. Complex Paths 84 where B = [5a ao a0 5a (5.6) Thus, we can see that the physical frequency to first order wi l l only be purely real or purely imaginary i f 5a, ao and their higher derivatives are a l l simultaneously real or imaginary. Such a contour is difficult to find. This suggests that we should back up and try to rescale the action so that some of these complicated background dependencies drop out after an appropriate integration by-parts, and then to search for a contour for which the remaining background dependence is simplified. Indeed, we start by noticing that the term in ui2 (77) (from which the complicated background dependence arises) is of the form ^a". This allows us to rescale the action and use a by-parts integration to effectively eliminate that term. Defining the new variable $(77,x) = a(n)f(n,x), we find that 3 = T/^' 2 + (^")^ = y [+ V 2 ) \8 + l *dV - Ja' [2ff'a + fa'} dr, = y / [a2/'2 + a'2f + 2a'aff] dr, - J [a'2f + 2aa'ff] dr = y/« 2/%, (5.7) using f\d = 0. Thus, we obtain a simplified form for the quadratic action as ( 2 ) 5 fa l2 Q f'2 -{£{£ +2)-3-K(j) B dr,, (5.8) where Q = 1 + K(j) l2 B 2(-£(£ + 2) +3) (5.9) (5.10) Chapter 5. Complex Paths 85 Inserting the definition for Q, we can rewrite the action into the non-canonical form = JT(T,£) f2-[£{£ + 2)-3 + ^ f ) f 2 dV, (5.11) where - ' 2 T(T,£) = TT2{£(£ + 2 ) - 3 ) ^ - j . - (5.12) K ' V V '2{£{£ + 2)-3)-K<J>B'2 ' By. expanding T(T,£), we see that to lowest order in e, this non-canonical action is simply <2>S = TT2 j ( 6 M 2 [f2 - (£(£ + 2) - 3) f2} dr,, (5.13) so that along any contour which renders S(f)'B real, the entire action wi l l be real up to the properties of / and / ' . 5.3 Higher order contour in r-space Since the second order action wi l l vanish (which is non-sensical) unless (j)'B ^  0, and since to first order in e this quantity is simply S(p'B, we obtain the first requirement for an acceptable contour. Indeed, if <f>''B = 0 at any time, the matter and metric perturbations wi l l decouple and together represent no physical degrees of freedom, so this requirement must be satisfied for any contour to any order in this model. If we solve = a ^ S M r ) = —^[--3 C O S ^ T ) + - J = 0 , (5.14) the (non-trivial) roots are (provided ao ^ 0) f = [ 7 r - i l n ( 2 ± V3 ) ] = i - [ T T T ( i cosh _ 1 (2 ) ) ] (5.15) A l l possible contours must avoid these points, which are located in the lower half-plane. Chapter 5. Complex Paths 86 Furthermore, since the non-canonical action only depends on the first derivative of the background scalar field 0£, any contour along which 8faB is real wi l l also be the contour along which the background part of the second order action wi l l be wholly real. A long such a contour only the character of the perturbations themselves (as expressed by the variable / defined in equation (5.19)) wi l l determine if the total action is generally complex, real, or purely imaginary. Setting the imaginary part of the 5<p'B to zero, we find that along the complex path in proper time defined by TI{TR) = arccos Q ( - 3 c o s h ( i / r f l ) + i/9cosh(i>rR) 2 - 8)^ ^ , (5.16) which asymptotes to ix/2v for TR —>• oo and ir/v for TR —>• 0, S(j>'B w i l l remain wholly real provided a 0 ^ 0. We also note that along the imaginary r axis 84>'B w i l l be wholly imaginary, provided a0 ^ 0. Since o 0 w i l l be zero at (77 = TT/U,TR = 0), we wi l l have to avoid that point in any contour we choose. This means that the path defined by equation (5.15) w i l l have to be truncated near TR = 0 and the point 77 = TT/U encircled in some way. We may avoid that point by joining the imaginary r axis with the path defined by equation (5.15) v i a an arc of some radius R TI{TR) = - y J ( K 2 - T R * ) + ^ , VTR<R. (5.17) Also , the final endpoint of the contour to first order wi l l be slightly shifted (pro-portionally to the final value of the scalar field 4>Bf) wi th respect to the zeroth order endpoint, so that any contour we select must end at this shifted endpoint. Indeed, using equations (1.28) and (1.29), one obtains, for r 0 = + c o s l l " t / ^ Q / \ £ a (T 0 ) drF = - ^ ^ r ao(r0) Chapter 5. Complex Paths 87 2V0u - c o s h - 1 (vaf) + . V ( l f I + ^ • 0/TT (5.18) Thus, the first order endpoint is r> = r 0 + o Y F , or .(i) • j L [ - c 0 S h - i ( l / a / ) + . ^ + cosh ^ a / ) 0/7T 7T 4 V J 7 + 2l> = TO 1 + 2V0 af 2V, TOOL + (1 — a)af yfi^a 2 - 1 (5.19) (5.20) Final ly, since the contour defined by equation (5.15) asymptotes to TT/V then in order to end on the shifted endpoint it wi l l be necessary to introduce another contour, say a line, to jo in the shifted endpoint with whatever point the previous contour is truncated at. That line wi l l be described by the complex path TI(TR) arccos 3 u , , x / 9 c o s h ( ^ r R C ) 2 - 8N •- cosh(uTRC) + CV7T 2u TR - TRF + an 27' (TRC - TRF) (5.21) where TRC is the value of TR where we start the line (i.e. truncate the previous contour) and Tp is the final real (shifted) value of r as given by equation (5.19). We summarize al l of these considerations in Figure 5.2, which is not to scale. Chapter 5. Complex Paths 88 Im(T) = x i TC / V 71 / 2v -X- -X-(arccosh( va f ) / v, irc/2v) — T 0 'RC • - TC / V Re(x) = T R Figure 5.2: Complex contour along which 8<p!B ^ 0 (for r ^ 0). Along Contour I, SfaB is purely imaginary. A long Contour A R C (equation (5.16)), 5(j)'B develops a small and increasing real part whilst the imaginary part decreases to zero at the juncture with Contour II. Along Contour II (equation (5.15)) it is purely real, and along Contour III (equation (5.20)) it develops a small imaginary part. Note the shifted endpoint to first order and the (off-axis) forbidden points X at which 8(f)'B = 0. Chapter 5. Complex Paths 89 5.4 Matching conditions Given that we wish to match a DeSitter spacetime to a Euclidean four sphere at a partic-ular value of the 'time', we must consider the the junction conditions at the interface in order to ensure a smooth transition. In particular, the content of the junction conditions is that the three-metric across the joining surface is continuous, and, if possible, that also the extrinsic curvatures of the nearby hypersurfaces are continuous. Due to the simplici ty of the background metric, the extrinsic curvature to first order in e is (in proper time) simply, from equation (2.10), where ao = , Ho = <W°o are the zeroth order scale factor and Hubble parameter respectively. We see that for this particular case, precisely due to spatial isotropy, that continuity of the three-metric automatically impies continuity of its associated extrinsic curvature. Since the three-metric is continuous across the entire contour given by Figure 5.2 and the time dependent prefactor in equation (5.21) is also well-behaved along the contour, then the matching wi l l be smooth. 5.5 Particular background solution Let us examine a particular solution, first considered by Green and Unruh in [47], of the background equations of motion within the No Boundary Proposal, so as to specifically get an idea the behaviour of quadratic action in the metric and matter fluctuations for a given solution. In particular, we shall be interested in the behaviour of 5<fi'B because of the form of equation (5.12). - Ho H (5a- Ho5a) h^, (5.22) Chapter 5. Complex Paths 90 For a closed F R W universe, consider the Hamiltonian constraint, equation (2.21), in both Lorentzian and Euclidean t ime 4 : , 2 a 2 F(0B) ac = - 1 + ^ - + — g — . « s = I + - 1 2 - — • ( 5 - 2 3 ) The Euclidean equation immediately tells us (for real (f>B) that the potential cannot be zero whenever aE = 0. Using the 7^ equation of motion, one may also derive the second rates of change of the scale factor a and obtain a C f -4B2 + V{4B)\ , (5.24) (5.25) c 6 O-E = —~r 6 4B + V(4B) from which it is apparent that for real 4B only negative curvature of aE is possible in the Euclidean region. However, consider the following argument in the case where the potential has negli-gible slope. Then, an equation of motion in Lorentzian proper time dictates (fa'al)' = -a\V{4B),<t>B=V ^4B = 4> (5-26) where c ^ 0. Then the Lorentzian Hamiltonian constraint is, according to this approxi-mation ( and assuming that V(4B) ~ 0 ), ac'2 = - 1 + 4- (5.27) ac This implies, integrating both sides, that 2 [dr = [ a c dac, (5.28) J J 7c2-a4-4In this section, a dot represents differentiation with respect to imaginary proper time and a prime with respect to real, Lorentzian, proper time. Chapter 5. Complex Paths 91 which means that in order to make the transition to imaginary time the Lorentzian constraint equation demands that o?c > c. Since at precisely o?c = c we have a'c = 0 (i.e. the tunneling point), then we conclude that a must grow at least slightly in the Euclidean region. This implies that aE > 0 after the tunneling point, which contradicts equation (5.24) wi th wholly real fields. In order to allow for positive curvature we must in fact allow for 4>B to possess an imaginary component. In the case of Green and Unruh , the simplest case was studied: 4>B —>• i(j)B. The following plot summarizes the situation for the scale factor (which must remain real for purely Euclidean or Lorentzian metrics): Figure 5.3: The scale factor a evolves with one F R W maximum and one SA maximum, respectively. We can see that this universe is quite different from our own, in that it seems to be de-termined by the future. This should make the semi-classical probability of this geometry small. Chapter 5. Complex Paths 92 5.5.1 Semi-classical probabilties Green and Unruh then went on to look at what the apparently necessary rule 4>B ~+ i4>B implied in terms the semi-classical probability of that geometry. To compute this probability, we observe that the Lorentzian action is Sc = j (naa'c + n<t><p" - NH±) dt = j (-12aca'c2 + O?c(J)B2 - NU±) dt. (5.29) Let t ing r = ±.it, so that dt = +~idr, the corresponding Euclidean action is iSE = Ti fTmax (-l2aEdE\-l) + aE<j)B{-l) - NH±) dr = +2i fTmaX aE (6 - aEV(<j))) dr (5.30) ' t u r n Green and Unruh find that this integral can be explicitly evaluated using the Sing method to find the potential V(4>). They find that Psemiclassical = exp [ + 6 ( 2 0 ^ - a 2 „ r j ] , (5.31) and since the universe pictured in Figure 5.3 is determined by its future, one would like to pick the minus sign so as to damp the probability of such a universe occuring (since we live in a very different one). However, the substution rule 4>B —>• %<PB must be applied by the above reasoning so as to obtain a valid solution. Doing so renders the matter and geometric parts of the Euclidean action to be, surprisingly, of the same sign. More specifically, we see that the reduced Euclidean action transforms from the usual expression in the following fashion: / = ± | (-12aEdE2 + a 3 J 2 B ) d r E ^ B ±j (-\2aEdE2 - a\^B) drE. (5.32) Chapter 5. Complex Paths 93 Comput ing the semi-classical probability in this case, [47] finds that Psemicalssical = 6 X p [ i ( c i{ u r n £ ; 2fl m a a ; £;)] , (5.33) which clearly suggests selecting the positive sign i f we want the probability to be damped. This supports the contentions of Linde (see [7] and references therein), who has argued that Psemicalssical ~ e x p + 5 £ is the appropriate expression for the semi-classical probability, i.e. an anti Wick rotation is preferred. Wha t is also evident is that if we take 8(f>B —>• i8<pB we shall also obtain that the action to second order in the metric and matter perturbations, to lowest order in e (equation (5.13)), w i l l also flip sign because of the (84>B')2 prefactor. Since then the whole overall action wi l l be negative (because the Euclidean action of these perturbations has been proven positive [48]), then the choice of Linde's positive sign is also required from the standpoint of the perturbations. 5.6 Conclusion In this final Chapter we have discussed a contour in r-space which is appropriate to first order in e (in that it renders the background dependence of a rescaled second-order action in the perturbations simple). We considered the sign of the action for the case of a solution of the equations of motion which had a geometry very much unlike our own universe and found, following [47], that in order to damp its associated semi-classical probabili ty that we had to select Linde's positive sign, i.e. that ^ = e x p + 5 s . B y applying this solution to the background fields that appeared in the action to second-order in the metric and matter perturbations, we found the same rule applied. Bibliography [1] Peskin and Shroeder, Introduction to Quantum field theory, (1995). [2] J . B . Hartle and S. W . Hawking, Phys. Rev. D 28, 2960 (1983). [3] D . L . Wiltshire, " A n introduction to quantum cosmology," gr-qc/0101003. [4] J . J . Hal l iwell and J . B . Hartle, Phys. Rev. D 41, 1815 (1990). [5] G . W . Lyons, Phys. Rev. D 46, 1546 (1992). [6] W . G . Unruh and M . Jheeta, gr-qc/9812017. [7] J . Garr iga and A . Vi lenkin , Phys. Rev. D 56, 2464 (1997) [gr-qc/9609067]. [8] R . M . Wald , General Relativity,Chicago University Press (1984) [9] Arnowit t , R. , Deser,S., and Misner, C . W . 1962, The Dynamics of General Relativity, A n introduction to Current Research [10] W . G . Unruh, private communication [11] P. A . Dirac, Proc. Roy. Soc. Lond. A 246, 326 (1958). [12] L . Faddeev and R. Jackiw, Phys. Rev. Lett. 60, 1692 (1988). [13] J . Garriga, X . Montes, M . Sasaki and T . Tanaka, Nucl . Phys. B 513, 343 (1998) [astro-ph/9706229]. [14] J . D . Romano, Gen. Rel . Grav. 25, 759 (1993) [gr-qc/9303032]. [15] M . Henneaux and C . Teitelboim, Quantization of Gauge Systems, Princeton U n i -versity Press (1992) [16] L . Castellani, Annals Phys. 143, 357 (1982). [17] V . N . Gribov, Nucl . Phys. B 139, 1 (1978). [18] D . Langlois, Class. Quant. Grav. 11, 389 (1994). [19] S. W . Hawking and G . T . Horowitz, Class. Quant. Grav. 13, 1487 (1996) [gr-qc/9501014]. 94 Bibliography 95 [20] E . Poisson, Physics 789 Class Notes, University of Guelph (1998) [21] T . Regge and C . Teitelboim, Annals Phys. 88, 286 (1974). [22] W . G . Unruh, Phys. Rev. D 14, 870 (1976). [23] J . M . Stewart (1990), Class. Quant. Grav., 7, 1169 [24] V . F . Mukhanov, H . A . Feldman and R. H . Brandenberger, Phys. Rept. 215, 203 (1992). [25] E . Lifshitz (1946), Zh. Eksp. Teor. Fiz., 16, 587 [26] Stewart J M and Walker M (1974) Perturbations of spacetimes in general relativity Proc. R. Soc. A, 341 , 49-74 [27] S. W . Hawking, R . Laflamme and G . W . Lyons, Phys. Rev. D 47, 5342 (1993) [gr-qc/9301017]. [28] H . K o d a m a and M . Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984). [29] S. W . Hawking and W . Israel, Cambridge, United Kingdom: Univ.Pr. (1979) 919p. [30] V . Moncrief, J . Ma th . Phys. 16, 493 (1976). [31] V . Moncrief, J . Ma th . Phys. 17, 1893 (1976). [32] P. D . D ' E a t h , Annals of Physics, 98, 237-263 (1976) [33] L . Bruna and J . Girbau, Journal of Ma th . Phys. 40, 5117-5130 (1999) [34] L . Bruna and J . Girbau, Journal of Math . Phys. 40 , 5131-5137 (1999) [35] V . Moncrief and R . Brandenberger, private communication [36] J . M . Arms , Journal of Ma th . Phys. 18, 830-833 (1977) [37] J . Traschen, Phys. Rev. D 31 , 283 (1985). [38] D . Kastor and J . Traschen, Phys. Rev. D 47, 480 (1993). [39] D . M . Gi tman , I .V. Tyut in , Quantization of fields with constraints, Springer-Verlag (1990) [40] J . M . Bardeen, Phys. Rev. D 22, 1882 (1980). [41] W . G . Unruh, astro-ph/9802323. Bibliography 96 [42] A . Khvedelidze and G . Lavrelashvili, Phys. Rev. D 62, 083501 (2000) [gr-qc/0001041]. [43] E . R . Harrison (1967), Rev. M o d . Phys., 39, 862 [44] M . Abramowitz and I A Stegun, Handbook of Mathematical Functions (Dover, New York, 1964) [45] M . Sasaki, Prog. Theor. Phys. 70, 394 (1983). [46] R . Durrer and N . Straumann (1988), Helv. Phys. Ac ta , 61, 1027 [47] D . Green and W . G Unruh (2001), In preparation [48] G . Lavrelashvili , Phys. Rev. D 58, 063505 (1998) [arXiv:gr-qc/9804056]. Appendix A Spatial Harmonics The spatial dependence of perturbations about a spatially isotropic spacetime can be described in terms of scalar harmonics, which are plane waves in the flat K = 0 case and generalized harmonics in the spherical (K = 1) and hyperbolic (K = —1) cases. This Appendix goes on to supply more details of their derivation, following [43]. A . l General derivation of harmonics Consider the Helmholtz-type equation in spherical polar coordinates: Separating out the variables such that / = ty{r)Y™(9,0), one sees that only the radial function wi l l differ in the three cases of flat, compact and hyperbolic spatial sections because only the radial part of the Laplacian wi l l change with the sign of the curvature 1 . Since the eigenfunctions of the flat space radial operator are simply plane waves, which are well studied [43], we wi l l only consider the more complicated eigenfunctions associated wi th the compact and hyperbolic isotropic spatial sections of F R W . Indeed, ini t ia l ly writ ing the spatial F R W metric in the standard way, 1We don't write indices on the radial function because there is no way of knowing what kind of function it will be at the onset of the problem. In hindsight, of course, there should be indices. V2f + k2f = 0. (A.l) (A.2) 97 Appendix A. Spatial Harmonics 98 one may deduce the Laplacian of the background space from equation (4.8), namely 1 V 7 = (A.3) Vl - Kr2 dr (r2VT^Kr^drf) + V 2 ^ ) / , where V 2 e ^  is the (unchanged) angular part of the Laplacian. Since the eventual aim is to reduce the radial equation to an associated Legendre equation, it proves useful to transform the radial coordinate v ia the relation r (1 + T)' (A.4) so that the metric, equation (A.2), acquries an overall multiplicative factor: ds2= 1 , 2 (df2 + r2dn2). (A.5) Using equation (A.5) in equation (A.3) and imposing the separability ansatz, one finds the radial equation for spherical harmonics of degree n , q2T + 2 n(n +1) qrA * = 0, (A.6) where 1 2 A.1.1 K = 1 For the compact case, define a variable o such that r (A.7) sin a 1 + (A.8) so that Appendix A. Spatial Harmonics 99 Expressing equation (A.6) in terms of a, one obtains the more convenient representation n(n + 1)" ( s i n a ) ~ 2 d a (sin {ofda^) + k - (sincr) 2 * = 0. (A.9) Transforming to *T/ = HVs ino - in order to recover the standard Legendre equation, we finally obtain (sincr) ldc (sin (a)daU) + A(A + 1 ) - £ ! J £ (sin a ) 2 n = 0, (A.10) where A(A + 1) = ^ + - . ( A . l l ) For the compact case, the eigenvalues k2 w i l l have to be integers in order for the function \T/ appearing in equation (A.9) to be periodic and single-valued, so one may set k2 = £(£ + 2), W e Z+, because it implies a particularly simple relationship for A = A(^), namely A i > 2 = 4 ± ( ' + 1)-(A.12) (A.13) Thus the eigenvalues k2 = £{£ + 2) form a discrete spectrum of values for the compact K = 1 case. A.1.2 K = -1 For the hyperbolic spatial sections, we proceed similarly by defining a variable a such that sinh(o;) = I T* • 1 A (A.14) Appendix A. Spatial Harmonics 100 so that da dr (A.15) Equation (A.6) with the transformation ^ = n V s i n h a yields, after some algebra, (n + 1) 2 (sinh a) lda (sinh (a)dalT) + A(A + 1) (sinh a ) 2 J n = o, (A.16) wi th A(A + 1 ) ^ 2 - - . (A.17) Equat ion (A.16) is another associated Legendre equation. Equation (A.17) w i l l yield simple roots for A = A(m) provided we pick k2 = m 2 + l , V m € sft, so as to render (via equation (A.17)) A = A(m) in the form Ai o = - ± im. (A.18) (A.19) The eigenvalues for the open case form a continuous spectrum because the space is open and the eigenfunctions do not satisfy periodic boundary conditions. Following [43] again, we can unify the formalism for both curvatures, as derived in equations (A.16) and (A.10), by substituting f = \[Ka: (n + 1 ) 2 ' (sinO" 1^(sin(OW + A(A + 1) - ( s i n ^ * = 0, (A.20) where A(A + 1) = Kk2 + (A.21) Appendix A. Spatial Harmonics 101 The two linearly independent solutions to equation (A. 18) are the associated Legendre functions FJ,(aw(0),^(aw(0), (A.22) where ^ E Un (A.23) Z v EE A. (A.24) A . 1 . 3 K = 1 E i g e n f u n c t i o n s In order to find which linearly independent solution is regular at the origin and to tabulate the first couple of polynomials, let us observe a property of the associated Legendre polynomials [43]: P t = P t ^ M n = P%, (A.25) Since n ± v is an integer one may use P ^ and P - ^ as linearly independent solutions. These solutions, equations (A.22) expressed with normalizations, are n = \ / ^ ^ + \ + , c o s ( c v ) , (A.26) w 2 sin a 2 + t ^r-» = . / I Z Z p * " cos(a). (A.27) e V 2 s i n a 2+e K J v ' In order to determine which solution is regular at the origin, an identity is used to express the associated Legendre function in terms of the Gamma function and the associated hypergeometric function [44], resulting in P«%(coS(a)) = r f f j ^ ) n\-l.\+*.l*(\ + nY. * • ' ( § ))• (A.28) Appendix A. Spatial Harmonics 102 A s f —> 0 (<r —>• 0), equation (A.25) approaches P * * ? 1 (cos(a)) ~ ( C O t ^ 2 ) ) ± ( 1 / 2 + W ) (A 29) r i + A C 0 S W i r ( l T ( l / 2 + n ) ) ' [ ' so that near the origin only is regular because tan(a/2) —>• 0 for a - » 0. Hence, we choose equation (A.27) as the appropriate solution for the case where K = 1. A useful property to note about (A.27) is that \P~™(— cos(a)) = COS(TY(£ — n))^~nl(cos(a)), again showing that £ must be integer for \& to be single-valued. Wha t this also means is that the spatial polynomial is symmetric (antisymmetric) whenever £ — n is even (odd). A. 1.4 Convenient representations of K = 1 polynomials There are more concise representations of these polynomials [44]which wi l l s imply be quoted here for convenience of reference in future work. Indeed, for n = 0,1, ...£, the function \I> of equation (A.9) can be represented by s in"q d " + 1 [cos(l + £)a] e n ^ 0 [ ( l + ^ ) 2 - ^ 2 ] d(co8(a))»+i ' { A - M > Thus, for n = 0, one has that = (i+euin(a) a n c ^ similarly for n = £, one finds q-t = sin{ay S i r n i l a r l y find r 4 4 ] = ( c o s a ) * - ^ , (A.31) * i + 2 = ( l - ^ ^ ° ) {A.32) Final ly, we use equations (A.31)-(A.32) to tabulate the first few radial solutions to the Helmholtz equation for compact spatial sections: * ° 1 = cos a: , = ^ s i n a , (A.33) 4 1 # ° 2 = 1 - - s in 2 a : , \I> _ 1 2 = - s i n a cos a . (A.34) o z Appendix A. Spatial Harmonics 103 A.1.5 K = —1 E i g e n f u n c t i o n s Working through exactly the same arguments as with the K = 1 case, one can derive an entirely analagous regular solution (near the origin a —> 0) for the hyperbolic case [43]: » - -=\Z(2^K" - W c o s h Q ) - ( A - 3 5 ) We emphasize that in this expression 7 is continous now, but n remains an integer as in the above case. For these integral values of n, one can write down a more compact representation [44], namely: sinh"o; dn+1 (cosh ma) IT^o { - m 2 ( m 2 + 772)} d(cosha) ' Using equation (A.36), we may finally tabulate the first few eigenfunctions of equation ( A . l ) for K = - 1 [43], [44]: For n = 0, ^ rn. nn ( ™ 2 f,™ 2 _ i _ »n2 U J / „„„u ' 1 ^ and for n = —1, = (A.37) msinh(a) tf-^ = ( m 2 + l ) - 1 ( m c o t ( m a ) - c o t h ( a ) ) t f ° m , (A.38) etc. A . 2 S u m m a r y o f e igenvalues The eigenvalue spectra for equation ( A . l ) in an isotropic null , positively, or negatively curved spatial background (FRW) are: K = 0 , k2 = m2 , m2 > 0 , V m € 9ft . (A.39) K = 1 , k2 = £(£ + 2) , £e Z+ (A.40) K = - 1 , k2 = m 2 + l , m 2 > 0 , V m 6 » (A.41) Appendix B Transformation of Perturbations Under an infinitesimal coordinate change the metric and matter perturbation variables (A, tb, B, E, and $) themselves shall change. Consider the coordinate transformation fj = n + T and yl = yl + D%L, where T and L are gauge functions which are fully functions of time and space. Under such a change the metric ds2 = a2(ri)[-dr12 + jijdyidyj] ( B . l ) itself tranforms. Indeed, to first order, 7 ^ = 7 ^ + £ ^ 7 ^ , where £ Q = (T,DlL). So we essentially have to compute the Lie derivative of the background metric wi th respect to the vector £ a generated by the scalar diffeomorphisms which define T and L. Since drf = drf+ 2(T'dri2+ Tidr,dyi) + 0(T2,L2), (B.2) dy{dyj = dy{dyj + 2 (L' '{idyj)dr] +L'{iwdyj)dya)+0(T2,L2), (B.3) a2(f,) = a2(r))(l + 2HT) + 0(T2,L2), (BA) then after some manipulation one finds that the Lie derivative of the background metric wi th respect to £ a is fyy = a2 [-2(HT + T')dn2 + {-TT+dy* + 2 7 i iL''%^)efy + 2LlJ<Tdyjdy(T +2 (uTjij - jLJ dy'dy* (B.5) Equation (B.5) is then compared to the linear perturbations of the metric, which define the variables (A, ib, B,E), S^ds2 = a(n)2 [-2Adn2 + Zib^jdyW - 2B^dyidn + 2E\ijdyidyj] , (B.6) 104 Appendix B. Transformation of Perturbations 105 which immediately reveals how the first order perturbations transform to first order under the given tranformations. The matter perturbation is easily seen to transform as $ = $ + (j)'gT by virtue of the fact that the background scalar field can only depend on time (so in particular there wi l l be no L ' s in its transformation). Thus, A = A + HT + T', (B.7) B = B + L'-T, (B.8) 0 = ^ - U T - ^ L , (B.9) E = E + L, (B.10) <£ = $ + 0 B 'T, ( B . l l ) as deduced in equations (4.42) - (4.47) using different methods. Appendix C Decoupling of Scalar, Vector, and Tensor Modes In order to justify examining only scalar modes one must show they are decoupled from the tensor and vector modes to linear order. In this Appendix we consider a ' long' proof first due apparently to Sasaki [45], who prove the sufficiency of maximal symmetry in the background for decoupling. There is also a more elegant group-theoretic discussion given in Durrer [46]1. Recall that scalar modes physically represent perturbations in the energy density of the scalar field matter, whilst vectorial perturbations and tensorial perturbations respec-tively measure rotational perturbations and gravitational waves. Thus, at least roughly, it should 'make sense' that one is allowed to study energy density perturbations without considering gravitational waves. However, such a rough understanding does not permit a precise criterion for when such decoupling fails, and in particular one would suspect that linear perturbations about spacetimes with fewer symmetries might not decouple valence by valence 2 We first consider the proof that the assumption of a F R W background is "sufficient to guarantee that linear perturbations about it decouple valence by valence. The basic idea in this sufficiency proof [45] is to construct all possible terms which are linear and maximal ly of second differential order (because the linearized field equations are second *They present a proof which, however, only covers the cases of flat and closed FRW backgrounds. The open FRW case is left as an open question by contrast to the above 'algebraic' proof, which also covers the open case. 2 A n object with valence zero is a scalar, valence one is a vector or form, valence two is a covariant or contravariant tensor. 106 Appendix C. Decoupling of Scalar, Vector, and Tensor Modes 107 order equations), using only the spatial metric h^, the spatial covariant derivatives Da, and the Laplacian operator for that particular hypersurface (H3 or S3). Assume that we can decompose any vector va into a divergence-free part va and a scalar part Dav, and similarly decompose the tensor field £ u into a transverse, trace-free part £ l J', a divergence-free part t, and trace and scalar parts t and s. A n arbitrary scalar function 0 is already decomposed into a scalar. Thus, we have 0 = 0 va = Dav + va >, ( C . l ) fi = (D*£P' - ^ ) s + \ttii + + iij t where vl and il are divergenceless whilst f J is transverse-traceless, i.e. Dava = 0 , DaP = 0 , Djiij = 0 , t\ = 0, (C.2) and where we also define n EE dim (E„) , A EE hijDiDj , t EE h^tfj. (C.3) If we can show that the form of this decomposition, i.e. the form of equations ( C . l ) , remains for al l possible (linear) terms after they, have been operated on by combinations of h^, A , or Di, then any linear evolution equations which involve those combinations of h^, A , or Di w i l l also remain decoupled. For example, consider A T / - AD'v + Av* = [DlA - RdiDd) v + Au< EE D'v + v*. (CA) Clearly DiV1 = 0 must be true in order for the original decomposition oft; 1 to be preserved. Demanding DiV1 = 0 = DiAvi is the same as demanding that [A, D^v* = 0 owing to Appendix C. Decoupling of Scalar, Vector, and Tensor Modes 108 equations (C.2). Thus we see that a condition on the curvature wi l l arise: [Du A]v* = -hkl([Dk,Di]Dlvi + Dk[Dl)Di]vi)=0 = -hkl (^+RfkiDdvi + RakiDivd + Dk {+Rdlivd)) = -hPH^Dtf - 2RidkiDkvd - {DlRdll)vd = 2RdkDkvd + Dl{Rdlvd) = 0, (C.5) where we have defined the curvature by [8] [D^D^^-R^y™. (C.6) In an entirely analagous way we demand the equivalent statement for the tensorial decomposition, i.e. that [A,/};]?- 7 = 0 . Indeed, [ A , A]f^ = -2RdkiDkiid + 3RakDkid:> + idjDlRal - iid (DlRj) = 0, (C.7) which amounts to another condition on the curvature. Equations (C.5) and (C.7) are the only possible restrictions on the curvature arising from the demand that the form of the original vectors and tensors be preserved, because A * 8 J = 0. However, equation (C.5) can also be further simplified by considering the expression DaDbtab = —-A2s - Da(RdaDds) + A - + 2Dl{Rdltd), (C.8) n n whose last term clearly wi l l couple vectorial and scalar perturbations (since a scalar, DaD0tab, w i l l depend on f, a vector) unless the Ricc i tensor has a special form. Indeed, we notice that the last term of equation (C.5) is the same as the offending cross-coupling term in equation (C.8), which we must require to vanish. Therefore, with the demand of vector-scalar decoupling, the curvature restriction equations (C.5) and (C.7) simplify to RdkDkvd = 0, (C.9) -2RjdkiDktid + 3RdkDkidj + idjDlRdl - iid (plRjdl) = 0. (C.10) Appendix C. Decoupling of Scalar, Vector, and Tensor Modes 109 We immediately notice that equation (C.9) is t r ivial ly satisfied by an isotropic metric, since in that case Rij = 2Khij. Equation (CIO) is also satisfied for an isotropic metric because [Dt, A]fi = -2 (2 t f ) {Vftdi - PM) (Dl&) - tidDl (5\hdi - 5\hd!) + 6KhdkDktdJ + tdWl(2K)hdl = -AK (pH\ - Ddtid) - 0 + 6KDdlfr + 2KDdidi = 0. ( C . l l ) This is essentially what Kodama and Sasaki proved: sufficiency of an isotropic back-ground for valence-by-valence decoupling of linear perturbations about i t . Equation (C.9) is only satisfied for either an isotropic metric or any metric such that = 0, but equation (C.10) in general won't be satisfied by any metric such that Rij = 0. Thus, only the isotropic metric can be a simultaneous solution to equations (C.9) and (C.10). This proves the necessity of a maximally symmetric background for the decoupling of linear modes valence-by-valence. Let us explicitly display how the decoupling works by considering a l l possible terms linear in (j), va, and tlj for the F R W spacetime, following [45]: Scalar Terms A(f> = A</> Dava = Av DaDb (tab = 2=1 A (A - nK) s (C.12) Notice how only s, t, v, and (j) appear on the right hand sides of these equations. Appendix C. Decoupling of Scalar, Vector, and Tensor Modes 110 V e c t o r T e r m s Db<b = Db<f> vb = Dbv + vb Avb = Db ( A - 2K) v + Avb \ (C.13) DbDava = DbAv Dj (tv - ^f) = ^ D { ( A + nK) s + ( A + (ra - 1)K) & Notice how only vb, il, <f>, and s appear on the right hand sides of these equations. These vectors have divergence-free parts owing to [Di,Av] = 0. T e n s o r T e r m s hij(j) = ti'<j> hijA<p = hijA(j> ^Jjijjj _ fc|4) (j) = frijQ D(ivj) ... D(iyj) + 2A^v + 2^-Av n tij = Aijs + 2DH^ + ft + t*f \ (CU) D<-iDkt»k - = 2D« (A + (n- 1)K) 1?) + l&^l&D^A + nK)s Atij = Aiij + ^D^Atj) - AK(n - 1 )£>(»«,-)] + Atj [A - 3K(n - 1)] 5 + i [(ra - l)(ra - 3)KAs + At] hijDkDmt1™ = hi:i [^A(A + nK)s + f] The tensorial terms clearly preserve the original decomposition. It is apparent that for F R W backgrounds, all linear terms constructed from , Di, and respect the original decomposition of the tensor field, so any linear tensorial equation, which wi l l be a linear combination of the above terms, w i l l also have scalar,vector, and tensor parts separately equal to each other. This implies we obtain three separate sets of equations, which is the same as saying the equations are decoupled. Appendix D Reduction of Hamiltonian Action in Spherical Symmetry In a 1976 paper, Unruh [22] found the correct expression for a reduced Hamil tonian in a spherically symmetric geometry with minimally-coupled scalar field. It is instructive to go through the steps of the analysis to see exactly how surface terms arise when one varies the action after solving for a variable from the Hamiltonian (and in the case of a system wi th less symmetry, the momentum constraints) constraint. Choose spatial coordinates such that at constant t, the radial coordinate r is such that at that fixed r , 4nr2 represents the surface area of a constant-/; sphere, and also 9 and 0 have their usual meaning. The A D M action for asymptotically flat universes, equation 2.17 up to a boundary term, is then written as S ( D . l ) which under spherical symmetry reduces to S (D.2) Here, fj, is defined by requiring to be diagonal and given by h^ = diag(e 2 f \ r 2 , r 2 sin2 (6)), (D.3) wi th its correspondingly diagonal momenta 7T4-? given by K - d i a g ( 2 e > 4 r 2 ' 4 r 2 s i n 2 ( 0 ) 7TA = r ( e "9 r ( e -"7r / t ) + 7 r * 9 r $ ) ; (D.4) (D.5) 111 Appendix D. Reduction of Hamiltonian Action in Spherical Symmetry 112 One may also compute the Hamiltonian constraint and the form of the shift vector [22] as P ^ (7T ^ 7*17 H± = -^(dr^-TTfidrfJ,-^dr^) + 2r2(l-2rdrfi-e2^ rl \ 8 4 + ^ + r 4 ( d r * ) 2 ) , (D.6) • .Ni = (Nr,Ne,N(t>) = (^No,0,0)- (D.7) Now we must proceed with the actual reduction of the action. First we pick a convenient gauge and then solve the constraints (in this case only the Hamil tonian constraint), and we note the two processes are commutative in this particular example. We have the further freedom to reparametrize the parameter t only, i.e. we have no spatial freedom to adjust since the symmetry has restricted that already. Thus one may pick the parameter t such that 7TM = 0, (D.8) which effectively picks what iV is by the equations of motion. W i t h i n the A D M formal-ism a specification of the shift and lapse functions completely specifies the spacetime coordinate system for a given choice of coordinates on the slice E t , so we have now com-pletely exhausted the available coordinate freedom 1 To completely reduce the action we must go on and solve the Hamiltonian constraint for a canonical variable and subsitute it back into the action. Thus, solving the Hamiltonian constraint for // = / / ( 7 r $ , 7 r M , $ ) and resubstituting it back into the action, such that 7r M = 0, we find S = J [TT*$ - NU MTT*, TTM, $ ) | ^ = 0 , Tr*, $)] dtdr. (D.9) This is the reduced action for this particular gauge, which has completely exhausted the allowed coordinate freedom. However, in extracting the equation of motion from it 1We can also see this by computing the Poisson bracket of the gauge choice with the Hamiltonian constraint Wi.: {n^Hi.} ^ 0 implies the gauge has been completely fixed. Appendix D. Reduction of Hamiltonian Action in Spherical Symmetry 113 there is a subtlety due to the fact that the Hamiltonian constraint has been already been used to solve for fj,. Indeed, vary equation D.9 wi th respect to, say, ir$ to obtain where the primes indicate that the variation is taken with /J, as an independent function, but Sfj, is taken as the variation in // due to the variation in 7 r $ . The first term is what one would get varying the original action. The latter term, however, is the term one obtains by varying the original action with respect to fi, which now has a non-vanishing boundary contribution. The bulk contribution of the latter term is manifestly zero because of the gauge choice 7rM = 0 2 . Hence, the boundary contribution must be added to the Hamil tonian in order to obtain the correct equations of motion, just as we encountered in Chapter 3. 2Because S'H/S'n = -{•K^U} = - T T M = 0 (D.IO) 

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