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Dimensionless cosmology Narimani, Ali 2011

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Dimensionless cosmology by Ali Narimani M.Sc., Mechanical Engineering, Isfahan University of Technology, 2009 B.Sc., Physics, Isfahan University of Technology, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in THE FACULTY OF GRADUATE STUDIES (Astronomy) The University Of British Columbia (Vancouver) September 2011 c© Ali Narimani, 2011 Abstract The variability of fundamental physical constants has been a topic of interest both theoretically and experimentally for many years. Although it is interesting to in- vestigate the consequences of such a variation, it is important to realise that only the variation of dimensionless combination of constants can be meaningfully mea- sured and discussed. In this thesis, I try to justify this way of thinking and apply it to two basic cosmological observables, Big Bang Nucleosynthesis and Cosmic Microwave Background anisotropies. I will mention some related studies that are either wrong or not complete because of being dimensionful. Variation of constants could be considered on two different levels. On the first level one assumes that the constants are time invariant but they can assume different values in different Universes or patches of sky. A thought experiment describing a discussion with aliens having a different system of units with different coupling constants could be helpful, this idea will be returned to throughout the thesis. On the next level, the constants can be promoted to being smooth functions of time or space. It is good to have a firm understanding of what happens on the previous level before trying to consider genuinely variable constants. For variable constants we need to consider theories beyond the currently accepted ones, which are capable of consistently describing such a variation. I briefly review the scalar-tensor theory of gravity as a possible way to describe the variation of the gravitational coupling. I give a brief historical review on the subject and consider the theory in the two so called ‘frames’, discussing about the benefits of each frame mathematically and the physical meaning of these frames. Such theories could form the frame work in which further study of variable con- stants could be carried out. ii Preface The introduction is partly based on a published work. Moss, A. Narimani, A. and Scott, D. Int. J. Mod. Phys. D, 19:22892294. The main Body of the thesis, including chapters 2 to 5 are based on a submitted paper. Narimani, A. Moss, A. Scott, D. arXiv:1109.0492. Prof. D. Scott proposed the main idea of these works. I was responsible for performing calculations and Dr. A. Moss helped me through Cosmic Microwave Background codes and calculations. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Dimensionless Gravity . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Warm-Up Calculations . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Different Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Most General Case . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Some Challanging Issues . . . . . . . . . . . . . . . . . . . . . . 9 2 Analytic Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Dimensionless BBN . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Redshift of Recombination . . . . . . . . . . . . . . . . . . . . . 16 3 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 Varying αs and CMB Anisotropies . . . . . . . . . . . . . . . . . 19 iv 3.1.1 Recombination through RECFAST . . . . . . . . . . . . 19 3.1.2 Perturbations through CMBFAST . . . . . . . . . . . . . 21 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Appendix: Scalar-Tensor Theory of Gravity in a Nutshell . . . . . . . . 37 A Scalar-Tensor Theory of Gravity in a Nutshell . . . . . . . . . . . . 38 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 A.2 Scalar-Tensor Theory on Conceptual Grounds . . . . . . . . . . . 39 A.3 Brans-Dicke Theory . . . . . . . . . . . . . . . . . . . . . . . . . 40 A.4 Brans-Dicke Theory on Observational Grounds . . . . . . . . . . 41 A.5 More General Scalar-Tensor Theories . . . . . . . . . . . . . . . 42 A.6 More General Theories in Practice . . . . . . . . . . . . . . . . . 45 A.7 Scalar-Tensor Theories on Theoretical Grounds . . . . . . . . . . 47 A.8 Future Developements . . . . . . . . . . . . . . . . . . . . . . . 49 v List of Tables Table 5.1 Dimensionless cosmological model parameters. Calculated val- ues and uncertainties are from the Markov chains from the WMAP 6-parameter fits [69]. . . . . . . . . . . . . . . . . . . . . . . 30 vi List of Figures Figure 3.1 Effects on the ionization history of 1% increases in αem (left) and αg (right). . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Figure 3.2 Effects on the CMB anisotropies of a 1% increase in αg (left) and αem (right). The vertical axis is percentage. . . . . . . . . 22 Figure 3.3 Effects on the CMB anisotropies of time-varying constants. We have specifically used αg ∝ t−0.002 (left) and αem ∝ t−0.002 (right). The vertical axis is percentage. . . . . . . . . . . . . 23 Figure A.1 Inverse of the scalar field coupling, α , and the scalar field, Φ, in artbirary units. . . . . . . . . . . . . . . . . . . . . . . . . 48 vii Glossary BAO Baryon Acoustic Oscillations CMB Cosmic Microwave Background ISW Integrated Sachs-Wolfe viii Acknowledgments I would like to thank my supervisor Prof. Douglas Scott for his support and guid- ance throughout this research and for being generous with his time. Prof. Scott has aided and encouraged me during these years and I’m very grateful for that. I would also like to extend my appreciation to Dr. Adam Moss for his helpful comments and for all he taught me during this research. I owe my deepest gratitude to my dear parents and my sisters who have helped me through my life and made this world a lovely place for me to live in. ix Chapter 1 Introduction It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are or what your name is. If it doesn’t agree with experiment, it’s wrong. — Richard Feynman [ch 7, Seeking New Laws] For a long while, the ultimate aim of physics has been to discover the under- lying laws of nature. Through this process of discovery, physicists were expected to be able to explain the qualitative and quantitative aspects of some experiment in great detail. However, there are some practical difficulties associated with this way of thinking about physics which have been with us since the previous century. A large body of work in theoretical physics involves areas about which we can have no direct empirical information, e.g. the interior of a blackhole or energies so high that they are beyond the capacity of any accelerator made on Earth. The goal for researchers in these fields seems to be to construct internally consistent theories in a way which might one day lead to some specific predictive power, without any contact with relevant measurement at least the near future. [7, 87] Through the great improvements in theoretical physics over the past century, physicists are in a situation where they can try to find scientific answers to some old questions about the Universe. It is now possible to scientifically explore the Universe and see if it is infinite or finite in space, eternal or with a beginning, finely tuned or not. All of these issues date back to at least ancient Greece. Although these questions are seemed firmly in the realm of philosophers, modern cosmological 1 observations now make them amenable to impirical investigation. The possibility of variation in the laws of physics is an issue with some empiri- cal problems. On the one hand, despite of its promise for extending our theoretical knowledge and understanding of Universe, the most likely energy scales for find- ing such a variation seems well beyond the reach of any accelerator on Earth. On the other hand, it needs needs to be extremely careful to make meaningful state- ments on this topic before interpreting experiments. Attempts to clarify how one might test for changing laws of physics end up requiring careful thought about what one means by a measurement, the importance of units in a measurement, and other basic considerations that have all been present since the very first physical experiments. This thesis aims to try to find a consistent way of asking questions about the variation of fundamental constants in physics and, attempts to answer those ques- tions with some phenomenological inspections in cosmology as a relevant place to seek for such a variation. There has been a wide and varied literature on this topic, from at least the time of the numerological ponderings of Kelvin and Dirac [32, 33]. Dirac argues in these papers that the ratio of electrical forces to gravita- tional forces between an electron and a proton is of the same order of magnitude as the age of the Universe on an atomic time scale. The age of the Universe is an increasing number while the ratio of the two forces is constant so in order to keep this order of magnitude equality he suggested that Newton’s constant is changing with time. Dirac’s 1937 paper on ‘ The Cosmological Constants’ has been cited more than 500 times and there are numerous papers published since then claiming to constrain variations of the speed of light, c, electrical charge, e, Planck constant, h, or Newton’s constant, G, a number of which we will mention in this thesis. However, there is a long history of debate about whether one can measure the time-variation of a dimensionful constant. Although claims to the contrary continue to crop up, there is consensus among a long and illustrious line of physicists that only the variation of dimensionless combinations of constants can be meaningfully discussed. Among those pointing this out are Dicke [29], McCrea [82], Rees [105], Jeffreys [62], Duff et al. [36], Silk [116] and Wesson [133]. The basis for this view is that physical units are quite arbitrary, and that each individual constant can be removed through a suitable choice of units (see e.g. [35, 2 36, 108] for a comprehensive discussion). The simplest example is the speed of light, c, whose variation is now manifestly unmeasurable because of its designation as a fixed constant relating the definition of time and distance units. Dimensionless quantities, on the other hand, are independent of the choice of units and so should admit the possibility of genuinely observable changes. From a different perspective, by considering how physics works empirically, one finds realises that any measurement can be reduced to dimensionless ratios of quantities – in other words, a measurement of some quantity is always made relative to some other quantity of the same dimensions. Therefore, the outcome of any actual measurement is a dimensionless number. 1.1 Dimensionless Gravity Much has been written (see e.g. Uzan [129]) about variation in the fine structure constant, αem ≡ e 2 4piε0h̄c (1.1) (where we have retained SI units in order not to obscure arguments about dimen- sions and units). A great deal of the activity in recent years has been inspired by claims of measurements or tight constraints on the variability of this constant with time [65, 85, 91, 103, 121]. There is a very close analogy between the Coulomb force and classical gravity, and hence one can define a ‘gravitational fine structure constant’ : αg ≡ Gm2p h̄c . (1.2) This choice of notation goes back at least to Silk in 1977 [116]. It explicitly selects the proton mass as the ‘gravitational charge’, although any other particle mass can be used instead. Using current values for the fundamental constants one obtains αg = 5.9×10−39. The fact that this is so small underscores the weakness of gravity compared with electromagnetism. What the definition of αg means is that experimental observations should be sensitive only to G multiplied by the square of the gravitational charge (and nor- malized by Planck’s constant and the speed of light). Just as in the case of variation 3 of αem, one is not allowed to ask which of the parts of αg are varying with time. Any discussion of the variation of the strength of gravity should start from this point. In other words, constraints which appear to have been placed on the variation of G only make sense if they can be interpreted as constraints on Gm2p. It should also be noted that just because a quantity is dimensionless does not imply it is unit-independent. Ġ/G (or ∆G/G) suffers from exactly the same unit- dependence problems as Ġ. One could use a system of units in which G = 1 by definition, and hence there is no Ġ in that system of units at all! Having said all of this, it is perfectly reasonable to choose a system of units in which {c,h,mp} are fixed constants, and interpret any variation in αg as a variation in G. The main point that one should always bear in mind is that there is only one degree of freedom in αg. One can choose either G or mp to represent that degree of freedom, but not both of them at the same time. And one needs to make sure to do this consistantly throughout a set of calculations. 1.2 Warm-Up Calculations A well known example of the use of αg in astrophysics is in the determination of the minimum mass of a star from first principles [15]: M∗ ∼ α3/2em α−3/2g ( me mp )−3/4 mp, (1.3) or in other words the characteristic size of a star is ∼1057 protons. This is similar to the Chandrasekhar mass: MCh ∼ α−3/2g mp ∼ 1057mp. (1.4) This means that if one were to compare such a mass measured today with the same mass measured billions of years ago, then one could determine that there had been a change because a pure number (in this case the number of nucleons in a star) was different. An estimate developed later gives the mass of a galaxy from a cooling time 4 argument (e.g.Rees and Ostriker [106], Silk [116]): Mgal ∼ α5emα−2g ( mp me )1/2 mp ∼ 1069mp. (1.5) If we wish we can rewrite these in terms of the Planck mass by using mp =α 1/2 g mPl. We can also estimate (see e.g. Padmanabhan [100]) the characteristic lifetime of a Main Sequence star as: t∗ ∼ α−1emα−1g ( me mp )−1/2 h̄ mec2 . (1.6) We can rewrite this as a dimensionless number of Planck times t∗ ∼ α−1emα−3/2g ( me mp )−1 tPl. (1.7) where tPl ≡ ( h̄G c5 )1/2 . (1.8) 1.3 Different Units Comparing (1.6) with (1.7) we see that the same physical observable has different αg dependence based on the units used to measure that quantity. This might lead us think there should be a preferred system of units one can use, if we are going to make a prediction for the variation of fundamental constants. To clarify the issue, let us think of a physical observable with units of time, T , which based on a theoretical prediction has some αg dependence like T ∝α pg , when expressed in seconds. The same quantity has a different αg dependence in terms of the Planck time: T ∝ α p−1/2g . On the observational side, T is measured with some uncertainty δT which could be used to constrain a possible time variation of αg. Now, the question is: are these two different theoretical predictions the same when compared with observational data? The answer is: Yes. There is some G dependence in tPl that should be taken into account. Considering this dependency, one finds the same results in both cases. 5 Let us temporarily choose G to represent αg. The observed value of T in sec- onds, including measurement error, is T + δT . One can infer the following con- straint on G based on this observation: T +δT T = ( G+δG G )p . (1.9) The observed value of T in Planck times is (T + δT )/tPl. If one assumes a vari- ation in the strength of gravity, then tPl varies as well. Therefore, the inferred constraint on G would be: ( T +δT T ) . ( tPl t ′Pl ) = ( T +δT T ) . ( G G+∆G )1 2 = ( G+∆G G )p− 12 , (1.10) and tPl t ′Pl = ( G G+∆G ) 1 2 (1.11) therefore (1.10) is the same as (1.9). 1.4 Most General Case It is reasonable to see if one can express the results of any general physical obser- vation in terms of a few dimensionless parameters. In most of cosmology the relevant physical quantities for specific problems are formed from the set P = {c,h,ε0,e,G,mp,me, k,T}, plus extensive variables, such as distance, mass, rate, redshift, etc. Then, from dimensional analysis, the only dimensionless quantities which can be constructed are αem ≡ e 2 4piε0 h̄ c , αg ≡ Gm2p h̄ c , µ ≡ me mp , θ ≡ k T mp c2 , (1.12) and combinations of them. The importance of the dynamical variable temperature, T , in this set, as opposed to the other dimensional constants, will be highlighted in the following section. This analysis shows that if a specific quantity is dimensionless (which should be the case if one is talking about a physical measurement), and contains one of 6 the members of P, then the other elements have to appear in a way that leads to at least one of the dimensionless ratios introduced in (1.12). In this paper we will ex- plore consequences of adopting this dimensionless way of thinking for measuring observables in physical cosmology.1 There are many papers in the literature (e.g. [9, 25, 52, 53, 57, 76, 77, 83, 96, 101, 102, 112, 113, 117, 122]) which try to answer a question such as: “What would happen to observable X , if the speed of light or the gravitational constant or Planck’s constant had a different value”. Such questions are not well defined, since one can tune the other parameters of the dimensional set P, such that the di- mensionless ratios in (1.12) remain the same. Since any observable is essentially dimensionless, it can only depend on these dimensionless ratios, and one can de- duce that nothing happens to the observable X if, for example, G changes. On the other hand one can meaningfully consider how X might change if αg varies. The necessity of using dimensionless ratios is commonly illustrated through an experiment which tries to measure a variation in the speed of light (e.g. [58]). The basic idea is that any measurement of the speed of light will in the end reduce to clocks and rulers. Since the separation of the two ends of any metre stick is a func- tion of the spacing between its atoms, which is a function of the speed of light, the length of the metre stick depends on the speed of of light, and this makes any pu- tative variation of c unmeasurable. What we say is a bit different. We believe that any apparent variation in the speed of light is irrelevant in any measurable quantity, regardless of special relativity. A variation in the speed of light is spurious because it is dimensional, not because of its fundamental role in modern physics. Once one understands this connection to a choice of units, it becomes clear that the same principle applies to all dimensional constants – one can talk about the variation of αem, for example, but it is meaningless to ask whether this is due to a variation of one of e, ε0, h̄ or c. In this paper we will explore consequences of adopting this dimensionless thinking to measuring observables in physical cosmology. As a prelude to the rest of the paper, let us consider (BBN) and the final abun- dance ratio of helium atoms relative to baryons – a measurement which is clearly dimensionless. We will discuss BBN in a dimensionless manner in more detail 1In a somewhat different context the importance of using dimensionless numbers in cosmology was stressed earlier by Wesson [134]. 7 later on, but it may be helpful first to explain the logic and sketch the main idea. When we say “nothing happens to observable X if G changes”, one might naively think that changing G would, for example, change the expansion rate of the Uni- verse and hence lead to observable consequences. The point is that there is no observable which depends solely on the expansion rate. There also has to be some other rate (like the recombination rate, neutron decay rate, etc.) that one is con- sidering in the problem. The ratio of these two rates (the expansion rate and the other relevant rate) is dimensionless and, therefore, has to depend on the dimen- sionless ratios and not just G. Based on this, the general dependence of the primor- dial helium abundance Yp = Yp ( c,h,ε0,e,G,mp,me,mn ) can always be reduced to Yp = Yp ( αg,αem,µe,µn ) . µe and µn are respectively the electron and neutron mass ratios relative to the proton mass. Therefore, the dimensional constants should not be regarded as independent, and any variation in these constants should always be understood as a variation in the dimensionless ratios. In this thesis I will consider different aspects of cosmology in relation to fun- damental constants. The next section deals with some difficulties which arise when one tries to track a variation in the fundamental constants within the stan- dard framework of cosmology. In chapter 2 I try to show explicitly and in a com- pletely dimensionless manner how a change in αg will affect two different aspects of the early Universe, namely BBN and recombination. I keep everything explicit in that chapter, but also simplify some of the equations. Therefore they lack the full accuracy required for comparing with data, but nevertheless demonstrate the main physics. In chapter 3 I work through the publicly available codes for com- puting Cosmic Microwave Background (CMB) anisotropies, make them manifestly dimensionless, and present the results of a variation of αg or αem on the CMB power spectra. In chapter 4 I briefly discuss how this work could be extended to other areas of physical cosmology. In cosmology, a particularly important constant is Newton’s constant, G. This is because it enters the dynamics of the background, the growth of perturbations and large-scale geometric effects. There have been many studies of how one might constrain Ġ/G using cosmological or astrophysical data (e.g. [44]), as we will discuss further below. It seems odd that studies of αem and µ are usually careful to point out the importance of only considering dimensionless constants, while papers 8 on G (or even sections in review articles) tend to ignore this entirely. 1.5 Some Challanging Issues We can imagine two different scenarios when discussing variable constants in cos- mology, such as αg or αem. In the first, we assume our Universe is described by a fixed set of physical laws and we attempt to measure the values of the (dimension- less) free parameters in this model. In the second (and perhaps more conventional) approach we consider a possible time variation of the constants. These ideas are of course related, but it is simplest to first concentrate on the notion of constants which are constant in time, but could vary among different universes (or patches within the same universe). In Bridgman’s classic book on dimensional analysis [13], he uses the idea of a different country, with a widely different set of units, in order to help explain the relationship between units and measurements. Since cosmology involves param- eters which apply to the whole Universe, one can effectively extend Bridgman’s idea and consider some alien civilization in a separate universe to our own, or at least a volume in which the dimensionless parameters may be different from ours. An imaginary conversation with observers in the other universe is a useful thought experiment for understanding what is really measurable. These observers would likely have a completely different set of units (and dimensional constants), so any such dialogue should only focus on the dimensionless aspects. The goal of the discussion would be to communicate the value of the dimensionless constants in our Universe, and learn the aliens’ values. To do this we “simply” have to sort out concepts like “rest mass” and “charge” and so on, in order to be able to mean- ingfully say things like “the mass ratio of the two most common particles is 1836 for us, what about you?” Then we would describe the fundamental forces, making sure that we were talking about the same thing as the aliens, before progressing to defining the dimensionless couplings. Moving on to the second scenario, where the constants might change, one would attempt to constrain the space-time variation of the dimensionless constants in our Universe using cosmological observables. The standard models of cosmol- ogy and particle physics cannot accommodate a genuinely variable αg or αem. In 9 this case one has to promote these parameters to dynamical fields, such as in scalar- tensor theories of gravity. The detailed study of such models is beyond the scope of this thesis but I will have a brief review of the theory in the appendix. For full consistency one would then have to check that any additional terms in relevant equations (the Friedmann equation for example) are negligible at the times under consideration. Here I will be assuming that the variation of the constants is small enough that these conditions are likely to be satisfied. However, it is worth noting that this may not be true in every case. To do better one would need an explicit theory for the time variation, while I will try to keep everything general here. In cosmology one requires an additional set of parameters (as well as the di- mensionless physical constants) to specify the model. This is a little different from setting the values of (dimensionless) physical constants, since the cosmological parameters are (at least statistically speaking) chosen from among a set of possible universes, all with the same physics. Some parameters (or ratios) may be fully deterministic, but others may have stochastic values. Someday we might have a theory which tells us exactly the value for some of the parameters, but in the cur- rent state of cosmology, we consider them as free parameters, which are unknown a priori. The usual set consists of (at least): {ρB,ρM,ρR,ρΛ,H0,T0,A,n, . . .}. These are the energy densities in various components (baryons, matter, radiation and vac- uum), the expansion rate, CMB temperature, and amplitude and slope of the initial conditions. Usually the densities are expressed in terms of the critical density (which involves G) to give the set {ΩB,ΩM,ΩR,ΩΛ}. Parameters such as the am- plitude A and tilt n of the scalar power spectrum probably depend on fundamental constants, but in a way which is as yet unknown. The obvious question that arises here is “how do we define the cosmological model in a dimensionless way?” We can return to our alien-conversation thought experiment to help sort this out. To communicate information on the background cosmology, we would need to discuss what we mean by “baryons”, “photons”, “neutrinos”, etc., and then give their densities in some way. However, there is an additional issue within the realm of cosmology, which arises because many of the usual parameters (the Ωs, H, etc.) depend on time. Hence, even if we have established that we live in the same Friedmann model as our alien friends, we still have to establish whether we are observing that model at the same epoch or not. 10 The best way to do this would be to agree on a fiducial period in the evolution of the Universe and then discuss where we are relative to that. An obvious choice is the epoch of matter-radiation equality (discussed in a related context in [125]), but there are plenty of other possibilities: when the Hubble rate is equal to a particular reaction rate; when the Thomson optical depth is unity; when matter has the same energy density as the vacuum, etc. Assuming that the Universe has flat geometry (a clearly dimensionless statement), then we can give the values of the Ωs at the agreed fiducial epoch, together with one number to fix that epoch, say the value of θ at equality. Then we only need to give the value of one parameter today in order to fix the epoch at which we live (making sure this is a dimensionless parameter of course). This could be any one of the Ωs today, or the value of zeq, or the value of H0t0, or θ0 (≡ kT0/mpc2). Anything which is essentially monotonic (and changing in time) and dimensionless will do (so H0t0 is fine now, but useless billions of years ago, when it was hardly changing). In addition, we need a way to describe the amplitude of density perturbations, which is dimensionless – but this is relatively simple, by converting from A to the dimensionless power at some fiducial scale (e.g. ∆2R(k0) used by WMAP [69], σ8 derived from galaxy clustering or Martin Rees’ Q2 [124]). The slope n is already dimensionless and so offers no difficulty. The real problems come from the fact that the other main cosmological parameters are epoch-dependent quantities. This gives rise to two complications when describing the observable Universe, which are not there when one is discussing models of laboratary physics. Firstly, if one is not careful, then it is possible to effectively fix more than one of the parameters today, leading to inconsistent results. The second part is that different choices of the “what-time-is-it?” parameter are not entirely equivalent, because some contain a dependency on other physics parameters. Take for example the choice of either the CMB temperature T0, or Hubble constant, H0 (made suitably dimensionless). These two are related to each other via ΩR and αg using the Friedmann equation H20 = 8pi3 45 αg h2ΩR k4B T 4 0 m2p c4 . (1.13) Therefore, either H0 or T0 is enough to determine today’s epoch, and it is not con- 11 sistent to use the Friedmann equation and treat both of these as free parameters. However, it turns out that it matters whether we choose T0 or H0, since the pair (T0,H0) in our Universe do not satisfy the above equation in a universe with a different αg. The thermal energy to rest mass dimensionless ratio, θ , needs some explana- tion. Unlike the other dimensionless ratios, this one is not built through pure phys- ical constants but contains a “variable”, the temperature. Temperature is a property of many body systems, which describes the phase space occupation of the particles. The role of Boltzman constant in the set P is to make temperature dimensionless. Boltzman constant is a Universal constant which can not be replaced with other physical constants [88]. In an alternative approach, one can give reason for the presence of temperature among fundamental constants because he actually needs a variable to describe thermodynamical systems which are evolving with time. 12 Chapter 2 Analytic Calculations In order to illustrate how one ensures that calculation are dimensionless, we focus on two cosmological examples. 2.1 Dimensionless BBN BBN is an area of astrophysics which has been thoroughly investigated for signs of variation in the fundamental constants (see e.g. [56, 72, 97] for non-gravitational couplings and BBN). Among the constants which are effective during BBN, the gravitational constant plays a key role, and hence there have been many published studies of the effects of a varying G on the primordial abundance of the light el- ements (see e.g. [3, 5, 21, 23, 24, 28, 59, 61, 74, 136]). I explicitly calculate the abundance of helium synthesized during BBN in this section, focusing on the dom- inant parts of the physics and neglecting some of the finer details. The calculation, though crude, shows the role of αg in primordial nucleosynthesis and explicitly reveals the dimensionality in the relevant physics. The key parameter in BBN is the ratio of number densities of neutrons to pro- tons, which can be defined as: R≡ nn np = e−u. (2.1) 13 The quantity u is the ratio of mass difference to freeze-out temperature: u≡ ( mn−mp ) c2 k Tf . (2.2) Tf is explicitly the temperature at which the following reactions freeze out: e−+p→ νe +n; p+ ν̄e→ e++n. (2.3) According to Bernstein Bernstein [8], the rate for these reactions is: λ = 255 τn u5 ( 12+6u+u2 ) , (2.4) where τn is the lifetime of the neutron. In order to identify the effects of the gravi- tational coupling constant, we can write this lifetime as τn = f1(αw)h mp c2 , (2.5) where f1 is a dimensionless function of the weak coupling constant. The freeze-out temperature is set by the equality of this rate and the expansion rate of the Universe: λ = H ⇒ wu3−u2−6u−12 = 0, (2.6) where we have defined w through w≡ f1 255 √ 8pig∗ 3 ( 1− mp mn )2 α1/2g , and g∗ takes care of the number of relativistic species. This cubic equation can be solved to obtain the freeze-out temperature as a function of gravitational coupling: u = 1 3 (18w+1)2/3 +(18w+1)−1/3 +1 w . (2.7) After this time the most important remaining reaction is the β -decay of neu- trons, which continues until the formation of deuterium. Deuterium formation is delayed due to the large photon-to-baryon ratio. Its abundance is fixed at a temper- ature TD, which is roughly given by the equality of the number density of photons 14 with the number density of the deuterium which has been formed: exp ( BD k TD ) η = 1. (2.8) Here η is the ratio of baryons to photons and BD is the binding energy of deuterium, which can be written as a dimensionless function of the strong and electromagnetic fine structure constants times the proton mass, BD = f2(αs,αem)mp c2. The tem- perature TD could also be converted to an age through the Friedmann equation. By the time of deuterium formation neutrinos have already frozen out and electron- positron pairs have annihilated. Thus the ratio of the age of the Universe to the neutron lifetime becomes v≡ t τn = 1 f1 f 22 √ −3 ln(η) 54pi α−1/2g , (2.9) and the primordial helium fraction (by mass) is Yp = 2 R 1+R e−v. (2.10) We have aimed at an expression for the helium abundance which is manifestly dimensionless, i.e. it only depends on dimensionless ratios of physical quantities, including αg. This simple analysis leads to a primordial helium fraction of about Yp ∼ 0.22, which is within 10 percent of the value coming from more complicated numerical BBN codes. Now we can put all this together to track the effects of a possible variation of αg on Yp: δYp = e−(u+v) (1+R)2 α1/2G × [ u3 3wu2−2u−6 f1 255 √ 8pig∗ 3 ( 1− mp mn )2 + 1+R f1 f 22 √ −3 ln(η) 45pi α−1G ] ×δαG αG . (2.11) The first term in the square brackets comes from the change in the neutron’s freeze- out fraction and the second is due to a change in the age of the Universe. Both terms 15 have the same order of magnitude (∼ 10−2) and have the same sign. A higher αg leads to more neutrons at the freeze-out time and a lower age for the Universe, both of which enhance the primordial helium fraction. If one takes the measurement error on Yp to be ∼ 0.005, and assumes a power- law variation for αg ∝ t−x, this simple analysis shows that x should be less than 0.005, which is consistent with the results of other studies (see e.g. Yang et al. [136]). Although most of the papers on this subject could be considered to be valid when one interprets them in a dimensionless manner (i.e. translating a variation in G into a variation in αg), making the whole argument dimensionless reveals some assumptions which are obscured by dimensions. For example, sometimes statements are made like “whenever you see G, interpret it as G times some quark mass mx, or a combination of G and ΛQCD” (see e.g. Iocco et al. [58]). But actual measurements of G have always been made using normal atoms, and therefore when numbers are plugged into equations in order to derive some proposed time variation for G (or more properly, αg), then αg is effectively the one introduced in equation (1.12). In this sense, those papers which are assuming a simultaneous time variation for G and mp, are not valid (see e.g. Dent and Stern [28], Iocco et al. [58]). 2.2 Redshift of Recombination Cosmological recombination of hydrogen is mainly controlled by the population of the first excited state Zel’Dovich et al. [138]. This is because of the high optical depth for photons coming from transitions direct to the ground state. The rate of recombination to this first excited state is given as [34, 119] Γ= nBα(2) = 9.78nB ( E0 k T )1/2 ln ( E0 k T ) h̄2α2em m2e c , (2.12) here ‘B’ stands for baryons (i.e. protons and neutrons), ‘e’ for electrons, α(2) is the recombination rate to the second energy level of hydrogen and E0 is the binding energy of hydrogen, which is equal to α2emmec2/2. We have assumed that all of the atoms are ionized. It also simplifies things considerably (without qualitatively 16 changing the physics) if we ignore the mass difference of protons and neutrons, i.e. set mB = mp, then nB = ρB mB = ρ0B mp ( T T0 )3 . (2.13) This gives a rate Γ= 7.0 ( me c2 k T0 )1/2 h̄2α3e m2e c ρ0B mp ln ( E0 k T ) ( T T0 )5/2 . (2.14) Assuming for further simplicity that ΩM = 1 (this assumption could easily be re- laxed later), the Hubble constant is H = ( 8piGρ0 3 )1/2 ( T T0 )3/2 , (2.15) and one can also use the equalities ρ0B = ρ0R ΩB ΩR , ρ0 = ρ0R ΩM ΩR , (2.16) together with ρ0R = aT 40 = 8pi5k4 15c3h3 T 40 . (2.17) Putting all of these together, one can find an expression for the redshift of recom- bination: 1+ zr ∝ Tr T0 = 68α1/2g α−3em µ 3/2 θ−3/20 ( ΩM ΩR )1/2 (ΩB ΩR )−1 . (2.18) Again we can see that an observable, which is the redshift of recombination, depends only on the dimensionless ratios of physical quantities, together with some other dimensionless parameters. Although this expression contains much of the es- sential physics, unfortunately the numerical factor (which is neglected to be writ- ten down) is far from correct. That is because I have not taken account of the partial hydrogen ionization. It would be possible to take this further by using an approximation to the Saha equation to correct for the ionized fraction, changing the scalings with the dimensionless parameters. However, this rapidly gets compli- 17 cated, and accuracy really needs the numerical solution to the relevant differential equations (which is done in the next section). Nevertheless the main point is that one can see an important dependence on the dimensionless quantities of interest. The change in the redshift of recombination is one of the main cosmological effects of varying αg. There are also other effects on CMB anisotropies, as we will see in the next chapter. 18 Chapter 3 Numerical Calculations 3.1 Varying αs and CMB Anisotropies In the previous sections I wrote down expressions which were explicitly dimen- sionless, but at the expense of accuracy. Let us now numerically explore the effects of a varying αem or αg on the CMB anisotropies. I use RECFAST v1.5 [114] for the recombination history of the Universe and CMBFAST [115] for the calculation of CMB angular power spectra. It turns out that one would face some difficulties if one were to blindly dive into the codes and try to change “G” or add some factors of αem in the relevant parts. There are several issues which should be considered before one can promote these constants into dynamical parameters in the code. It turns out that this choice not only affects the feasibility of the problem, but also the outcome of the numerical calculation. As an example, CMBFAST is written so that it uses the total density contributions, {Ω}, both for the strength of inertia in acoustic modes, and as gravitational charge. Therefore, if one chooses these ratios as the free parameters, in place of densities, it is much easier to trace the effects of a variation in the gravitational constant, because there is an additional factor of “G” wherever the code needs the Ωs as seeds for gravitational collapse. 3.1.1 Recombination through RECFAST One can trace the effects of a varying fine structure constant or gravitational con- stant on the process of recombination through RECFAST (see e.g. [51, 66, 111]). In 19 order to convert the code to a form where it can run with a different αem, one has to track all of the relevant dependencies [89]. Most of the dependence shows up in the energy levels, since E ∝ α2em. The ‘Case-B’ recombination rate (see Rybiki and Lightman p. 282 [110]) has an α5em dependence and the complete analytic form of the 2s–1s two photon rate ([20, 120]) varies as α8em. Triplet transitions of he- lium (discussed in Lach and Pachucki [71]), case-B recombination and two photon transition rates for helium have the same scaling as for hydrogen. With this infor- mation in hand one can trace the effects of a different αem (or a time-dependent αem) on recombination. The G dependence for recombination lies entirely in the Hubble constant. Putting the rates together and defining the dimensionless ratios, A≡ npα H , B≡ KΛnp, C ≡ K β np, D≡ βH , (3.1) the Saha equation [114] will take the form: dxp dz = [ xe xp A− ( 1− xp ) D ] [1+B(1− xp)] (1+ z) [ 1+(B+C)(1− xp) ] . (3.2) This is explicitly for hydrogen, but there is a similar form for helium too. These dimensionless ratios have the following dependencies on the coupling constants: A ( αem,αg ) ∝ α5emα −0.5 g ; (3.3) B ( αem,αg ) ∝ α2emα −0.5 g ; (3.4) C ( αem,αg ) ∝ α−1em α −0.5 g ; (3.5) D ( αem,αg ) ∝ α5emα −0.5 g . (3.6) Figure 3.1 shows the results of a 1% increase in the fine structure constant and separately of the gravitational constant through the history of recombination. This is consistent in sign with equation Equation : recom – an increase in αem leads to a higher ionization fraction, which leads to a lower redshift for recombination. It is also noticeable that even a 1% increase in αem can lead to more than 1% variation in the recombination history, while such a variation in αg does not leave any significant trace, due to the weaker power-law scalings above and the relatively thin last scattering surface. 20  0  0.005  0.01  0.015  0.02  0.025  0.03  0.035  0.04  0.045  0  500  1000  1500  2000  2500  3000 ch an ge  in  io ni ze d fra ct io n x e redshift variation in αe -0.0006 -0.0005 -0.0004 -0.0003 -0.0002 -0.0001  0  0.0001  0  500  1000  1500  2000  2500  3000 ch an ge  in  io ni ze d fra ct io n x e redshift variation in αG Figure 3.1: Effects on the ionization history of 1% increases in αem (left) and αg (right). 3.1.2 Perturbations through CMBFAST The effects of an alteration in the recombination history could in principle be mea- sured through the CMB power spectra. This section complements sec : BBN in terms of comparing a hypothetical variation in the fundamental constants with known physics at a particular epoch – z ∼ 1000 in this case (rather than z ∼ 109 for BBN). As is discussed in Moss et al. [89], a different fine structure constant leaves its main imprint on the recombination history, and this shows up in CMB- FAST through two main effects: (1) the derivative of the opacity, which is propor- tional to Thomson scattering, with an α2em dependence; and (2) the number density of free electrons, which is basically the “freeze-out” value at the end of recombi- nation. Therefore, one can almost ignore the effects of a different fine structure constant after recombination. The case is different for αg. As is discussed in Ri- azuelo and Uzan [107], a different G will lead to a different sound horizon at the last scattering surface and a different distance from this surface to us, the combina- 21 -0.4 -0.3 -0.2 -0.1  0  0.1  0.2  0.3  0.4  0.5  0  300  600  900  1200  1500 re la tiv e va ria tio n in  C l l 1% increase in αG -3 -2 -1  0  1  2  3  4  0  300  600  900  1200  1500 re la tiv e va ria tio n in  C l l 1% increase in αe Figure 3.2: Effects on the CMB anisotropies of a 1% increase in αg (left) and αem (right). The vertical axis is percentage. tion of which will change the position of the peaks of the power spectrum. There is also an Integrated Sachs-Wolfe (ISW) effect which can enhance the amplitude of perturbations. This is examined in Zahn and Zaldarriaga [137] and, as explained in Chan and Chu [18] for the case of a constant but different G, the effects are much less important than they are for a time-varying G. The results of a 1% increase in αem and αg are shown in figure 3.2. In [137], the authors propose a scaling in G, (G→ λ 2 G), which is equivalent to the scaling of αg considered here. However, I cannot confirm their results, since they have assumed that {T0,H0,ΩR} have the same values as in the standard model. This is incorrect, because these three are not independent and a change in αg will result in a different value for at least one of them (depending on which one we have chosen as our dependent variable). The time varying αg or αem results are shown in figure 3.3. Our study is certainly not the first to explore the effects of variable fundamental 22 -5 -4 -3 -2 -1  0  1  2  3  4  0  300  600  900  1200  1500 re la tiv e va ria tio n in  C l l time varying αG -5 -4 -3 -2 -1  0  1  2  3  4  0  300  600  900  1200  1500 re la tiv e va ria tio n in  C l l time varying αe Figure 3.3: Effects on the CMB anisotropies of time-varying constants. We have specifically used αg ∝ t−0.002 (left) and αem ∝ t−0.002 (right). The vertical axis is percentage. constants on the CMB (see e.g. [18, 86, 137] for G and [4, 50, 80, 81, 109] for αem). The main point here is that we have insisted on doing everything in a dimensionless way to avoid finding any apparent effects which are simply a result of the choice of units. As an example, Boucher and et al. [11] suggest that one can search for a violation of the Strong Equivalence Principle using the CMB power spectra. The method, though at first sight quite elegant, is totally dimensional. When one starts to make it dimensionless, it becomes evident that the whole argument has to be reconsidered. A basic assumption is that a variation in G can make a difference in gravitational mass relative to the inertial mass of baryons. However, this is in general wrong, since one can always use the freedom of setting the gravitational charge and inertial mass of a given body (and only for one object) equal to each other. Several particular ideas for exploring the variation of G within cosmology have 23 been described elsewhere. We are suspicious of any of these studies in which the dimensionful quantity G is discussed along with other dimensionful constants, such as H0 (e.g. [42, 73, 75, 90, 128]). Specific ideas for time-varying constants may lead to specific forms for the function of time, e.g. αem(t). Following such detailed predictions are beyond the scope of the present paper. However, the methodology set out here should be useful in testing any explicit model that is proposed. 24 Chapter 4 Discussion I have already considered the effects of a variable αg or αem on BBN and CMB anisotropies, with the main emphasis on making things dimensionless. One can ex- tend this approach to other observables in cosmology such as weak lensing, Baryon Acoustic Oscillations (BAO), large-scale structure, ISW effect and the use of super- novae as standard candles. BAO follows the same basic physics as CMB anisotropies (see e.g. Eisenstein and Hu [38]), so there should in principle be the same kind of αg dependence. The important timescales for BAO are the equality epoch, zeq and the drag epoch, zd. The first, zeq, is purely set by the initial conditions of the Universe, and the time today, setting the epoch when ΩM = ΩR. The drag epoch is defined as the time when baryons are freed from the Compton scattering of photons, and therefore zd has the same αg or αem dependence as zr. These special redshifts set the scaling conditions for BAO. After the drag epoch, one should solve for the matter transfer function, where this function is basically the same as the CMB transfer function in terms of αg or αem dependence. It seems clear therefore, that if one wanted to use BAO to constrain the variation of αg, it would be straightforward to follow the same procedure discussed in chapter 3. In gravitational lensing (see [70] for an earlier study) there is a danger of be- coming confused by dimensions, since both the estimate of the lensing mass and the curvature depend on G, and they are mixed together in αg. The simple case of an Einstein ring is illuminating. Here the lens and the source are colinear and 25 in Euclidean space the radius of the Einstein ring is the geometric mean of the Schwarschild radius of the lens and the distance to the lens. This means that for a relatively nearby lens at distance d the lensing angle is ΘE ∼ √ GM/c2d, and if we estimate the mass through measuring a velocity dispersion v2 over a radius r, then ΘE ∼ √ Θv2/c2, where Θ≡ r/d is the apparent angular size of the object. Viewed this way we see that it is hard to use lensing to measure the strength of gravity, because the obvious dimensionless observables leave no G dependence! Let us look at this in a little more detail. In cosmology one can work out the lensing angle to be (e.g. [16]) ΘE = √ 4GMdLS c2 dL dS , (4.1) where dLS is the distance from the source to the lens, and dS and dL are, respec- tively, the distances from us to the source and to the lens. However, this seemingly innocent dimensionless equation is not useful for any experiment designed to mea- sure a change in the gravitational constant. This equation, or any equation for lensing, should be turned into an equation which only has αg or αem dependence before it can be used as a test of fundamental constant variation. One can work out all of the distances in the above equation and end up with the following relation: ΘE = Aα 3/4 G θ M mp , (4.2) with A ≡ √ 8pi3 45ΩR √ 1+ zL NL − 1+ zL NL . (4.3) Here zL and zS are, respectively, the redshift of the lens and the source and NL and NS are defined as: Nx ≡ ∫ 1 1 1+ zx dy y2 √ ΩΛ + ΩMy3ΩRy4 , (4.4) where x can be either S or L. Speaking in more general terms (and again thinking about communicating with another civilization), we can see two possibilities. The first is that one could imag- 26 ine choosing a lens which consists of a fixed number of particles, so that one knows the value of the pure number M/mp. That may be interesting from a philosophical point of view, but is hardly related to how we carry out observations in cosmology. A second idea is that perhaps one could carry out a statistical survey over many galaxy lenses, measuring statistics which depend on a characteristic galaxy mass Mgal. If that mass depends on fundamental constants through a cooling argument (e.g. [106]) then it may be that in a lensing survey the observables depend on the number α5emα−2g µ−1/2. Since this would be different in universes with different values of the constants, then this is potentially measurable, and hence it might be possible to constrain a redshift dependence of lensing observables. These ideas do not seem particlarly practical, and so we are left unclear about whether gravitational lensing could ever be used to constrain the time variation of αg. Things will presumably be unambiguous in explicit self-consistent models which contain a variable strength of gravity, and we leave this topic for future studies of different models. There are many other astrophysical phenomena which can in principle be used to test the variation of fundamental constants (see e.g. Garcı́a-Berro et al. [44], Uzan [129]). Large-scale structure and various measures of the power spectrum of galaxy and matter clustering (see e.g. Nesseris et al. [95]), will also have depen- dence on αg. It will again be important to ensure that this dependence is dimen- sionless, and that cosmological parameters are not over-constrained when doing so. In other words, one needs to realise that changing αg can effectively change the epoch at which we live. Supernovae have also been very useful as approximately standard candles in cosmology, and such studies can also be adapted to constrain a combination of fun- damental constant variations [37, 43, 46, 94]. Like other stellar sources, there is a dependence on α−1/2g in the evolutionary timescale (see e.g. [27, 45, 49, 79, 126]), and hence the use of supernovae, combining the standard candle property with lu- minosity distance, will involve a different combination of fundamental constant variations than for BAO and lensing studies. A combination of cosmological probes will therefore enable variations among different parameters to be distin- guished. Finally, it worths mentioning that among the set of different astrophysical phe- 27 nomena, cosmological studies have the potential of probing not only the time de- pendency of αg, but also any scale dependence. It is important to work out an appropriately consistent theory to describe these phenomena with a spatially vari- able fundamental constant. Although I have not considered such models here, I still suppose that it will be important to avoid dimensional quantities. 28 Chapter 5 Conclusions I am not advocating that all of cosmology henceforth should be represented in dimensionless forms. However, I would say that care has to be taken when dealing with variation of physical constants in cosmology. This is particularly because of the cosmic time ambiguity, which could lead to over-constraining the cosmological parameters. In terms of the standard cosmological picture, one could define dimensionless parameters at some fiducial epoch. If we choose matter-radiation equality as this epoch, then we can define the background cosmology by giving the values of the Ωs then, as shown in Tabel 5.1. (I have assumed massless neutrinos in the table, contributing to the “radiation” by the usual factor of 0.68 times the photon energy density. I have also assumed a flat background, and adopted values from the WMAP 6-parameter fits Komatsu et al. [69].) We then need to give one quantity (out of a wide range of possibilities) to define the time today. I have pointed out how it is possible to reach invalid conclusions about the variation of fundamental constants (particularly G) by fixing too many parameters in the cosmological model. Explicit physical mechanisms for the variation of fundamental constants will lead to specific forms for αg(t) etc. I believe that the proper place to start inves- tigating such theories is to make sure that there is a robust basis for comparing cosmologies with different constant values of the constants. Only then can one effectively deal with time-variable quantities. A time-dependent (or space-dependent) αg is not consistent with General Rel- 29 Table 5.1: Dimensionless cosmological model parameters. Calculated val- ues and uncertainties are from the Markov chains from the WMAP 6- parameter fits [69]. Parameter Value Uncertainty Coupling constants αem . . . . . . . . . . . . . . . . . . . . . 7.297×10−3 . . . αg . . . . . . . . . . . . . . . . . . . . . . 5.906× 10−39 . . . Quantities at equality Ωγ . . . . . . . . . . . . . . . . . . . . . . 0.296 . . . Ων . . . . . . . . . . . . . . . . . . . . . . 0.204 . . . ΩCDM . . . . . . . . . . . . . . . . . . . 0.415 0.004 ΩB . . . . . . . . . . . . . . . . . . . . . . 0.085 0.004 ΩΛ . . . . . . . . . . . . . . . . . . . . . . 4.4 × 10−11 1.2× 10−11 θ . . . . . . . . . . . . . . . . . . . . . . . 8.0 × 10−10 0.3× 10−10 Definitions of “now” zeq . . . . . . . . . . . . . . . . . . . . . . 3200 130 θ0 . . . . . . . . . . . . . . . . . . . . . . 2.503× 10−13 . . . ativity. One of the only explicit theories for accommodating such a variation is a scalar-tensor theory of gravity. Some studies have already investigated CMB anisotropies and other cosmological constraints in the simplest form of scalar- tensor model (e.g. [1, 19, 135]), the so-called Brans-Dicke theory [12]. While this version of a scalar-tensor theory seems to be already ruled out by solar sys- tem experiments [104], there are more general scalar-tensor theories which pass the solar system tests [123], and might be promising avenues of exploration (e.g. [92, 127]). 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However, Brans-Dicke theory lost its initial attraction for a while due to the following works of Dicke himself. He worked out BBN in their theory and suggested some tests using solar observations. All of these tests turned out to be compatible with General Relativity, without any need for further modification. It is hard to say that these tests ruled out the idea, since GR is just a special case in Brans-Dicke, but GR is simpler and has less free parameters, therefore, being compatible with GR is usually taken as disfavouring Brans-Dicke. In 1970, Wagoner [130] wrote the most general form of a scalar-tensor theory. Wagoner’s theory is dynamic in the sence that the theory can evolve with time; 38 gravity can start from a Brans-Dicke like theory in the early epochs of the Universe and evolve into GR at later times. It is also possible to look for scale-dependent gravity models which are the same as GR on small scales but are deviate from it on larger scales. There was a revival of interest in scalar-tensor theories after the type-Ia su- pernova observations showed acceleration in 1998 [2], which brought in the first observational support for a cosmological constant. Many theorists tried to look for theories of gravity that could lead to cosmological acceleration at late times or on large scales without the mysterious ad hoc cosmological constant fluid [17, 22, 39, 47, 60, 68]. Multidimensional inspired theories, f(R) theories and of course, scalar- tensor theories, gained lots of attention since then and there has been a large body of work discussing the internal relations among these seemingly different theories (see e.g. [41] and references therein). In this appendix I will go through the basics of Brans-Dicke theory and the more general forms of a scalar-tensor theory while trying to make conceptual, the- oretical, and observational arguments for why one should care about scalar-tensor theories in general. A.2 Scalar-Tensor Theory on Conceptual Grounds As is described in [12], one can think of two different approaches regarding space- time structure. In the first approach, space-time is an absolute physical structure with its own physical and geometrical properties. One is allowed to define inertial frames and therefore absolute motions in this space-time structure, without any reference to the rest of matter present in space. In the second approach, space-time does not have any physical or geometrical structure of its own. Space-time gains its physical meaning from the matter therein, and one can define a motion in space- time only relative to other physical objects. This second approach is called Mach’s principle [78]. While the second approach seems more attractive for physicists nowadays, there is not any accepted physical theory which is in complete accordance with Mach’s principle. Even in GR, that space-time geometry is affected by the matter distribution, it is not uniquely specified by the distribution of objects in space [12]. 39 There are two main thought experiments which led Brans and Dicke to their theory of gravity. The idea behind both of these is that if one assumes Mach’s principle, then any fictitious force on a test particle should be reinterpreted as the effect of the motion of other particles in the Universe relative to the test particle. In the first experiment, they imagine an observer with a gyroscope and a gun in a laboratory in an empty space-time. The observer can lean out of the window of his laboratory and shot a small bullet tangent to the circular walls of his lab which will make his laboratory start rotating. The gyroscope in the lab should point towards a nearly fixed direction relative to the direction of motion of the rapidly receding bullet. In the Mach’s principle approach to space-time, this means that the tiny bullet is more important in determining the motion of gyroscope than the massive walls of the laboratory. The second thought experiment deals with a particle falling in the gravitational potential of the Sun. In a coordinate system so chosen that the particle is at rest, the gravitational pull of the Sun should be balanced by the gravitational pull due to the rest of the mass in the Universe. The acceleration of the falling particle due to Sun’s gravity can be approximated through Newton’s law as a ∼ GM/r2. Based on dimensional analysis and the relevant parameters in this problem, one can approximate the acceleration due to the matter distribution in the Universe as a∼MRc2/M r2 , where R is the radius of the Universe and M is the total mass in the causally related Universe. By equating these two accelerations one gets: G∼ Rc 2 M , (A.1) which means that the gravitational constant should vary with time. A.3 Brans-Dicke Theory From the preceding general arguments, Brans and Dicke deduced that a Mach- compatible theory would presumably have a time-varying gravitational constant in an expanding Universe. Einstein’s theory is clearly inconsistent with such a hy- pothesis and should therefore be extended to a theory which admits a time varying 40 gravitational coupling. They started with the Einstein-Hilbert Lagrangian: LEH = √ g ( R+ 16piG c4 Sm ) . (A.2) Here R is the Ricci scalar and Sm is the Lagrangian of matter, including all non- gravitational interactions. The quantity g is the absolute value of the determinant of the space-time metric. They argued that it is not wise to replace G with a field in this Lagrangian, since that would lead to a non-conserved energy momentum tensor for matter defined via Tµν ≡ 2√g δSm δgµν , (A.3) with gµν representing the metric of the space-time. Furthermore, the particles will not move along the geodesics of the metric any more if Sm is coupled to a space- time dependent field. Based on these considerations, they divided the Einstein- Hilbert Lagrangian by G and replaced it with a scalar field (φ = 1/G): L ′ EH = √ g ( R G + 16pi c4 Sm ) =⇒ LBD =√g ( φR+ 16pi c4 Sm− ωφ φ,µφ ,µ ) . (A.4) The last term in the Brans-Dicke Lagrangian, LBD, is the usual kinetic term of a scalar field and the role of the field in the denominator is to makeω a dimensionless constant. The interesting point is that right after writing this Lagrangian the authors say: “In any sensible theory ω must be of the general order of magnitude of unity”. A.4 Brans-Dicke Theory on Observational Grounds Solar System tests.— Brans and Dicke immediately worked out the weak field limit of their theory and found the correction to general relativity due to adding a scalar field. They found that the amount of precession of Mercury in their theory is 4+3ω 6+3ω × (value of general relativity) , (A.5) which is always less than the value predicted by GR. The Brans-Dicke prediction approached GR assymptotically as ω → ∞ and this limit is often taken as the GR 41 limit. Dicke argued that the oblateness of the Sun can make a Newtonian 1/r3 potential [31]. This would resolve some of the famous 43′′ discrepancy between the observed prehelion of the orbit of Mercury and the prediction of Newtonian gravity. Therefore, the post-Newtonian corrections should be smaller than 43′′, in favour of Brans-Dicke’s theory. Following studies [48] showed that the oblateness of the Sun is smaller than what Dicke had proposed by about an order of magnitude. In 1970 Nordtvedt [98] parametrised the most general theory-independent form of a metric in the weak field limit, up to terms of order (1/c4) for g00 and (1/c3 and 1/c2), respectively, for g0k and gkk. The results of comparing observations with Brans-Dicke’s theory showed that the maximum fractional contribution of scalars to the Newtonian and post-Newtonian potential is less than one part in a thousand [99]. Big Bang Nucleosynthesis.— Dicke [30] worked out BBN in his own theory. He assigned an energy momentum tensor to the scalar field which would increase the expansion rate of the Universe. His conclusion was that his theory allows a Uni- verse without any primordial helium forming. Cosmic Microwave Background.— Chen and Kamionkowski worked out the per- turbation theory in detail for Brans-Dicke theory [19]. They also carried out the numerical calculations and checked for possible degeneracies of ω with other cos- mological parameters. Their conclusion is that PLANK should be able to identify values as large as ω ' 3000 with a standard deviation of about 0.00062. A.5 More General Scalar-Tensor Theories Tensor-MonoScalar Theories.— In 1970, Wagoner wrote down the most general tensor-mono scalar theory [130]. This is important, since adding a scalar seems the most simple and natural extension of general relativity in the field theory language. The list of his assumptions are: a) The theory should be convarient, leading to tensorial equations. b) The equations of motion are derived from a Lagrangian of the form L=√g(Lg+ LI)/16piG, with Lg the Lagrangian for gravitational fields alone and LI the La- grangian for the coupling of gravity with all other forms of interaction. These fields vanish at infinity. 42 c) Gravity is mediated with a tensor and a scalar. d) The field equations are at most second order differential equations, leading to the following form for Lg: Lg = h(φ)R+ l(φ)φ,µφ ,µ +2λ (φ). (A.6) e) Lastly, mutual coupling of gravity and matter of the following form is assumed: LI = LI(A2(φ)gµν , ...). (A.7) Here A is an arbitrary function of the scalar field. This form of coupling guaran- tees that a local inertial system can always be made in which the laws of special relativity and electromagnetism hold. It is also claimed by Nordtvedt [99] that the only consistent field theories without causality problems, negative energy modes, etc., are those which respect this form of coupling. By employing the following redefinitions g̃µν = hgµν , dφ dΦ = h { hl− 3 2 ( dh dφ )2}−1/2 (A.8) the Lagrangian is written in its final form (suppressing tilde signs): L = 1 16piG √ g ( R−Φ,µΦ,µ +2λ (Φ)+LI(A2(Φ)gµν , ...) ) . (A.9) This form of Lagrangian, which does not have any coupling of the scalar field to the Ricci scalar, but couples the scalar field with matter fields, has its own benefits which I will describe later. It is refered to as a Lagrangian written in the Einstein conformal frame. If one performs a conformal transformation on the metric of this Lagrangian, such that ḡµν = A2 gµν , then the new matter Lagrangian takes the form that we want, L̄I = L̄I(ḡµν , ...). Under such a transformation, the determinant of the metric transforms with a factor of A8, i.e. ḡ = A8 g, and the Ricci scalar transforms as [131]: R = A2 { R̄+6 lnA−6 ḡµν(Oµ lnA)(Oν lnA) } . (A.10) 43 The second term on the right hand side of this equation is a surface term which is zero by assumption. Putting all of these together one gets the Lagrangian written in the so called ‘Jordan-Fierz’ [63] frame (or simply Jordan frame), L = 1 16pi √ ḡ { φ R̄− ω(φ) φ φ,µφ ,µ +2 λ̄ (φ)+LI(ḡµν , ...) } , (A.11) with the following substitutions: GA2(Φ) = φ−1; (A.12) α2 = (2ω(φ)+3)−1; (A.13) GA8(Φ) λ (Φ) = λ̄−1. (A.14) Here α is defined as α ≡ d lnA(Φ) dΦ . (A.15) As we can see, the Brans-Dicke theory is a special case of (A.11), with λ̄ = 0 and a constant value for ω . Tensor-MultiScalar Theories.— As I mentioned before, writing the Lagrangian in the Einstein frame has some benefits. One of them is that it can straightforwardly be generalised to the tensor-multi scalar case, which seems to be the most general plausible theory for gravity mediate via a tensor and a number of scalar fields. Nordtvedt [99] writes the Lagrangian for the tensor-multi scalar case as L = 1 16piG √ g { R−gµν〈Φ,µΦ,ν〉+2λ (Φ)+LI(A2(Φ)gµν , ...) } , (A.16) where the angular brackets define an inner product in the space of scalar fields, 〈Φ,µΦ,ν〉= γabΦa,µΦb,ν , (A.17) and γab is the metric in the space of these fields. Field Equations.— The field equations of the Lagrangian in the Jordan-Fierz frame, 44 equation 11, are: R̄µν − 12 R̄ ḡµν − λ̄ ḡµν = 8piφ −1T̄µν +φ−2ω(φ){φ,µφ,ν − 12 ḡµνφ,ρφ ,ρ} +φ−1{φ,ν ;µ − ḡµνφ} φ̄ + dλ̄ dφ̄ = 1 2ω(φ)+3 ( 8piT̄ − dω dφ ḡµνφ,µφ,ν )  ≡ ∂µ∂ µ . (A.18) As can be seen from the above equation, the second derivatives of the scalar and tensor fields are related to each other in the Jordan-Fierz frame, and therefore this frame mixes the scalar and tensorial modes with each other. In fact the scalar field is acting like a source for the tensor field. One of the other benefits of the Einstein frame is that these modes are decoupled from each other. The field equations in the Einstein frame are: Rµν − 12Rgµν −λ gµν = Φ,µΦ,ν − 1 2 gµνΦ,ρΦ,ρ +8piGTµν ; Φ+ dλ dΦ = −4piGα(Φ)T. (A.19) A.6 More General Theories in Practice As was mentioned before, the Brans-Dicke theory is essnetially ruled out by solar system observations. However, the situation is different for more general scalar- tensor theories. Keeping in mind that general relativity is the limit of Brans-Dicke theory with ω → ∞, there is a natural mechanism in the tensor-mono scalar theory of Wagoner or general tensor-multi scalar theories which drives the coupling ω to infinity at late cosmological times. In this section, I will discribe this so called “attractor mechanism” which runs the scalar-tensor theories towards general rela- tivity at recent cosmological times. I will follow Nordtvedt [99], who has shown it in the Einstein frame. Barrow [6] has worked out the cosmological equations in the Jordan-Fierz frame for vacuum-and radiation-dominated cases. I will check the equations for a special situation the ω starts from a finite value in the the early Universe and tends to infinity at late cosmological times. In the following, I will 45 work in the Einstein frame, using equation (A.19), with λ = 0. As is beautifully derived in Weinberg [132], one obtains the following metric as soon as one assumes isotropy and homogeneity in space: ds2 =−dt2 +a(t)2 ( dr2 1− k r2 + r 2dΩ2 ) . (A.20) Here k is the spatial curvature indicator, which takes the values of {0,+1,−1}, respectively, for flat, spherical or hyperbolic spaces. I will assume k = 0 in the following. It is also argued in [132] that the perfect fluid energy momentum tensor, is the only choice under the assumptions of isotropy and homogeneity. Plugging this metric and energy momentum tensor into the field equations of (A.19), one obtains the following equations for the scale factor, a, and the scalar field: −3 ä a = 4piG(ρ+3P)+2(Φ̇)2; 3 ( ȧ a )2 = 8piGρ+2(Φ̇)2; Φ̈+3 ȧ a Φ̇ = −4piGα(ρ−3P). (A.21) Here p and ρ are the pressure and the density of matter in the Einstein frame. The energy momentum tensor is not conserved in this frame, but one obtains the following counterpart equation: d(ρa3) dt + p d(a3) dt = (ρ−3P)a3 d(A(Φ)) dt . (A.22) which considers the scalar field contribution to the total energy. One can decouple the scalar field, Φ, and the scale factor, a(t), by introducing the following variable, τ ≡ ln(a)+ constant, (A.23) such that dτ = ȧ a dt, (A.24) and writing all of the above differential equations in “τ-time”. The resultant equa- 46 tion for the scalar field is: 2 3− (Φ′)2Φ ′′ +(1−Γ)Φ′ =−(1−3Γ)α(Φ), (A.25) where Γ≡ p/ρ and a prime denotes d/dτ . The above equation looks like a damped harmonic oscillator with 2/(3− (Φ′)2) as its mass, (1−Γ) its damping factor and α(Φ) its driving force. Looking at (A.13), one finds that α ∼ 1/ω , and therefore, if (α = d lnA/dΦ) tends towards zero, the theory is being driven towards general relativity. (A.25) opens up such a possibility and is the “attractor mechanism”. In this equation, and in the whole scalar-tensor theory, α is a free God given function without any constraints on it! I took two possibilities for this function and solved (A.25) during the radiation and matter dominated eras for both of these cases. In one of them I took α to be the solution of an underdamped oscilla- tor, and in the other, I took α to be an exponentially decreasing function of time. During radiation domination, γ = 1/3, and the term containing α is negligible. Φ = constant is a plausible solution for this era. Using this solution as an initial condition, I solved the equation numerically for the matter domination era. The results are shown in Fig. A.1. A.7 Scalar-Tensor Theories on Theoretical Grounds Unifying Interactions.— Attempts to unify gravity with other interactions mostly end up with prediction of massless scalar fields [99]. This can be seen from the early Kaluza-Klein [64, 67] attempts of unifying general relativity with electro- magnetism and the compactification of the fifth dimension arising in that theory. This may indeed be a two way street, since in the standard model of elementary particles, particles gain mass through a scalar field, and this could be thought of as a manifestation of gravity. Looking at the current state of theoretical physics, the are three scalar fields in the two standard models for elementary particles and cosmology. A scalar field exists in elementary particle physics, namely the Higgs particle. And in cosmology, one scalar field is needed for inflation and probably another for dark energy. It seems that it would be nice to unify these three scalar fields or find a relation 47 -0.2  0  0.2  0.4  0.6  0.8  1  0  2  4  6  8  10 α (φ) time Underdamped α(φ)  0  1  0  2  4  6  8  10 φ time  0  0.2  0.4  0.6  0.8  1  0  2  4  6  8  10 α (φ) time Exp. decreasing α(φ)  0  1  0  2  4  6  8  10 φ time Figure A.1: Inverse of the scalar field coupling, α , and the scalar field, Φ, in artbirary units. between them. Different Theories of Gravity.— It has become clear over time that scalar-tensor theories are conformally equivalent to a wide range of gravitational theories, in- cluding any f(R) theory [118], or some multidimensionally insipred theories of gravity [41]. Despite this mathematical equivalence among different theories, there has been disagreement about which theory is physical. Mathematical equivalency means that the space of the solutions of these theories is the same, but lack of physical equivalency means that these theories predict different values for the same physical observables. Physical Frame.— As was shown before, there are two so called frames in the 48 context of scalar-tensor theories, the Einstein frame and the Jordan frame. While these two frames are related to each other through a conformal transformation, which is basically a redifinition of the fields, they are not nevertheless physically equivalent. As is discussed in [14, 40, 41], each frame has its own benefits and drawbacks, both mathematically and physically. The main drawback of the Jordan frame is that the scalar field energy is either negative definite or non-definite. This leads to stability problems at the classical level and the lack of any ground state at the quantum level. The positive feature of the Jordan frame is that the scalar field is minimally coupled to matter fields, and one can therefore use the usual standard model calculations in this frame. In the Einstein frame the scalar field has a positive definite energy, but it cou- ples to the matter fields. This coupling calls into question the weak equivalence principle and the soundness of using standard model predictions in this frame. Equivalence Principle.— There are two versions of the equivalence principle, the weak equivalence principle and the strong equivalence principle [12]. According to the first one, two test bodies will fall at the same rate in a gravitational field if they are released at the same time and from the same place. In this sense, the coupling strength of gravity to particles might vary with time or even from place to place, but this strength is the same for all particles in every single place at a certain moment of time. On the other hand, the strong version of the equivalence principle says that gravity has a constant coupling to matter,at every time and every place! It is obvious that scalar-tensor theories do not respect the equivalence principle in its strong sense, but they might respect the weak form of it, as long as they have the same coupling for every particle. It has been generally believed that the weak equivalence principle does not hold in the Einstein frame, since the matter particles do not move on geodesics of the metric anymore, but Hui and Nicolis [55] proved that the Einstein frame respects the weak equivalence principle as long as the scalar field self energy can be ignored. A.8 Future Developements Scalar-tensor theories form a set of consistent theories in which the strength of gravity can vary with time. By carrying out explicit candidates within these the- 49 ories, we should be able to find self-consistent predictions for how effectively αg can affect various cosmological observables. That will be the focus of future work. 50


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