A NEW COSMOLOGICAL MODEL by PO-YU SHEN B.S., National Taiwan University, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OP THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Physics We accept this thesis as conforming to the required standard ) THE UNIVERSITY OF BRITISH COLUMBIA July, 1970 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British 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 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 The University of British Columbia Vancouver 8, Canada Date July , 1970 - i i -ABSTRACT The present situation in cosmology i s discussed. We try to explain the observed distribution of quasars i n terms of an inhomogeneous model universe that consists of inner and outer Friedmann zones separated by a transition zone. The number - red-shift relation i s derived, and numerical calculations are performed on the assumption that the transition zone i s negligible. When the results are compared with observation, i t i s found that the dearth of quasars with red-shifts greater than 2 i s easily explained, but that one cannot account for their anisotropic distribution. Modifications of the model are suggested. - i i i -TABLE OF CONTENTS Abstract i i List of Tables • iv List of Figures • v Acknowledgment • v i Chapter I Introduction 1 1. Development of Theoretical Cosmology 1 2. Observational Cosmology • 2 3. Quasi-Stellar Objects 5 4. The Present Program • 9 Chapter II The Field Equations 13 Chapter III Friedmann Model 16 Chapter IV Red-Shift in the Friedmann Model • 20 Chapter V Number count in the Friedmann Model 24 Chapter VI A New Model Universe .26 Chapter VII Number - Red-Shift Relation in the New Model .——. 29 Chapter VIII Numerical Calculations 35 A. Procedures to Obtain the N — Z R e l a t i o n — 36 B. Results 40 Chapter IX Conclusion 68 Bibliography ?0 - i v -LIST OP TABLES Table I Possible Model Universes 28 Table II Numerical Results For Model I 43 Table III Numerical Results For Model II 45 Table IV Numerical Results For Model III 47 Table V Numerical Results For Model IV 49 Table VI Numerical Results For Model V 51 - V -LIST OF FIGURES Figure 1 . The Distribution Of Quasars With Known Red-Shifts • . • 11 Figure 2. N - Z Curves For Model I (0,W) • 53 Figure 3. N - Z Curves For Model I (V) • 54 Figure 4. N - Z Curves For Model II (0,W) 55 Figure 5. N — Z Curves For Model II (V) .56 Figure 6. N - Z Curves For Model III (0,W) 57 Figure 7. N - Z Curves For Model III (V) 58 Figure 8. N — Z Curves For Model IV (0,W) 59 Figure 9. N - Z Curves For Model IV (V) 60 Figure 10. N - Z Curves For Model V (0,W) 61 Figure 11 . N — Z Curves For Model V (V) • 62 - v i -ACKNOWLEDGMENT The author wishes to thank Dr. P. Rastall who suggested the investigation and gave invaluable guidance. Chapter I Introduction 1. Development of Theoretical Cosmology In applying Einstein's general theory of r e l a t i v i t y to cosmology, the' following assumptions are generally postulated in order to simplify the f i e l d equations. (i) Weyl's postulate^ The world lines of galaxies form a bundle of geodesies diverging from a point in the f i n i t e or inf i n i t e , past. ( i i ) The cosmological principle At a given cosmic time, the universe presents the same large-scale view to a l l fundamental observers. ( i i i ) Isotropy At a given cosmic time, the universe presents the same large-scale view in a l l directions. (iv) The perfect cosmological principle The universe presents the same large-scale view to a l l fundamental observers at a l l times. The postulates ( i ) , ( i i ) , and ( i i i ) lead to a line e l e m e n t 5 ' 4 , 5 , 6 where k = 0 , + 1, and - 1 . - 2 -This line element i s used to describe the Friedmann cosmology, which i s one of the 'big-bang' models. The postulates ( i ) , ( i i ) , ( i i i ) , and (iv) lead 7 to a line element and to the steady state cosmology. The big-bang models and the steady state models are the two major classes of cosmological models. In the former, the expansion of the universe began from a singu-l a r i t y in space and time, emerging from that state a f i n i t e time ago amidst conditions of extreme density and pressure. In the steady state models, on the other hand, the universe had no begining and w i l l have no end, but continuously remakes i t s e l f according to a fixed and immutable pattern. In the next section, we w i l l compare the theoretical predictions of the two classes of cosmological models with the observational data. 2. Observational Cosmology (A) Observations on the past light cone The relation between the apparent magnitude m of galaxies and the red-shift z for small values of z - 3 -gives information about q Q, the deceleration parameter at the present epoch. R e s u l t s ^ 5 ' s h o w that q Q i s positive and of value q Q = 1 .2 * 0.4, which differs from the steady state prediction of q Q = - 1 by five times the probable error. However, this value for q Q could be wrong because of possible errors arising from a selection effect in 11 12 favour of bright galaxies ' . The distribution of radio sources provides another test, which also contradicts the steady state theory. One defines N(S) to be the number of sources per steradian whose flux density measured in units of 10 w sterad Hz exceeds S at a given frequency. The results of the Cambridge group (Ryle and Clarke 1961) J in the 4C survey at 178Mc/s show that the log N - log S curve has a slope - 1 .8 down to about two flux units. The steady state model predicts a slope which starts at - 1.5 and get progressively less in magnitude. In the big-bang models, the slope i s always - 1.5 for small distances. At large distances i t can be greater or less than — 1 . 5 , depending on the model chosen. ( It i s less than — 1 . 5 for open universes. ) However, Veron^4(1965) found that the slope of the curve was — 1 .55 for radio galaxies and - 2.2 for quasars. The radio galaxies are' r e l a t i v e l y nearby objects with red-shifts not exceeding 0 . 5 - Hence the observed slope of - 1 .55 is not - 4 -inconsistent with either of the two classes of cosmological models discussed above. If the large red-shifts of quasars arise, entirely from the expansion of the universe (which at present i s a matter of controversy), then the slope, of — 2.2 could disprove the simple steady state theory. We w i l l talk about quasars l a t e r . ( B ) Observations inside the past lig h t cone The galaxy should be at least as old as the oldest stars in i t . Considerations of nucleosynthesis enable us to estimate the age of the stars in our galaxy as being between 1.0x10 1 0 and 1.5x10 1 0 y e a r s 1 5 , 1 6 . This i s i n agreement with the prediction of the big-bang models, which predict an age of the universe (defined to be the time that has elapsed since the big-bang) of HQ , the inverse 1 0 of the present value of the Hubble constant. Recent data — 1 9 9 show that HQ i s between 7.5x10 and 19.5x10 years. In . the steady state model, galaxies of a r b i t r a r i l y large ages can exist. However, according to this model, the average age of galaxies in the universe i s 1/3 HQ~ 1, with the young galaxies being much more numerous than the old ones. Present observations indicate that far more old galaxies exist than are predicted by this model. However, Hoyle and Narlikar (1962) have pointed out that we may by chance be l i v i n g i n a region of old galaxies, so that the evidence cannot be - 5 -considered as decisive. A .proper assessment must therefore wait u n t i l a good theory of galaxy formation i s available. The other important observations are the residual 3°K background radiation and the helium abundance. It i s 1 8 pointed out by the princeton group that i f the background radiation i s indeed degraded f i r e b a l l radiation, then a big-bang origin would seem possible. A similar conclusion follows i f the pristine helium abundance i s 30$ as stated 19 20 by Gamow "and Sandage because so much helium can appar-ently only be made in the f i r e b a l l immediately after the big-bang. 3. Quasi-Stellar Objects Since the discovery of the f i r s t quasi-stellar object in 1960, a considerable effort has been devoted both theoretically and observationally to investigating their 21 22 properties. It i s found that they are characterized by ' a) Large red-shifts b) The presence, in"some cases, of both emission and absorp-tion lines of almost the same red-shift c) Very large energy output, with an ultraviolet excess They are a hundred times more luminous than galax-ies on average. d) Variation of both radio and optical fluxes in times of the order of years, months, or even days - 6 -e) Anisotropic distribution in the sky f) A dearth of red-shifts larger than 2. 23 Several physical models have been proposed to account for the enormous energy output. None of them has found general favour, for each accounts only in a very schematic way for what i s seen, and i n i t i a l conditions from which the results are derived must be postulated. The important thing i s that, in a l l cases, i t i s necessary to postulate the existence 6f very large masses with sizes no bigger than about one li g h t year. This i s required to account for the great size and rapid variation of the energy output. The question arises whether the quasars are at cosmological distances or n o t ^ . If they are, i t i s hard to believe that objects a hundred times more luminous than galaxies could be so small. If the quasars are not at cosmological distances, their large red-shifts cannot simply be due to the expansion of the universe. There must be large red-shifts which are directly associated with the objects. The alternative explanations are that they are caused by the.recession of local objects at high speeds, by the presence of strong gravitational f i e l d s , or that the values of the atomic 'constants' in quasars are different from those in the rest of the universe. We w i l l discuss a l l these p o s s i b i l i t i e s . - 7 -i ) Local Doppler sh i f t In order to account for the large red-shift, recession at high speeds must be assumed. Setti and Woltjer (1966) assumed that the line widths are due to large random motions of gas in the shells of quasars and that the central mass must be large enough to stabilize the object gravitationally. In this way, they obtained masses 7 8 of quasars in the range 10' - 10 M@ on the local hypothesis. Bahcall, Peterson, and Schmidt (1966) estimated the mass of quasars on the local hypothesis by assuming that the absorp-tion lines are due to a continuous outflow of gas from the q surface. They obtained masses of order 10^ M . Thus they concluded that quasars are too massive to be local objects. However, i t i s clear that their conclusions are based on assumptions which may not be true and consequently, they do not disprove the local hypothesis. A more convincing argu-ment i s the absence of blue-shifts. If the quasars have been thrown out of many galaxies at high speeds, not only should red-shifts be seen, but also blue-shifts. Because objects moving toward the observer w i l l seem to be much brighter than similar objects moving away, blue-shifted objects should predominate in a survey down to a given ap-parent brightness. i i ) Gravitational red-shift - 8 -A model of this type has been suggested by Hoyle 27 and Fowler . The observed line spectrum arises from gas clouds at the center of a massive object. It i s possible to avoid the d i f f i c u l t i e s pointed out by Greenstein and Schmidt 28 with the energy requirements , but i t i s not clear at pre-sent whether models can be made with large enough red-shifts. 29 Bondi showed that a collapsed static object with plausible properties such as adiabatic s t a b i l i t y could not have a gravitational red-shift greater than 0.62. i i i ) The p o s s i b i l i t y that the masses or charges of nuclei or electrons are different in the quasars has not been taken seriously so f a r 5 ^ ' ^ . We now go back to the model in which the red-shift i s due to the expansion of the universe. There are several objections to this model which have not been resolved so far. One which has already been mentioned i s the small size required to account for the. variation in flux. The second i s the presence of multiple, absorption lines . Studies^ * show that for 3C 191, ZrtTB*w _ ~ 1.95; this indicates that Gm 3 . 0 S the absorption i s taking place in the quasar and not in the intervening medium. Studies of PKS 0237 - 23 show that there are at least five absorption red-shifts in addition to an emission r e d - s h i f t 5 ^ ' 5 ^ ' . Attempts have been made to explain the multiple absorptions as due to the intervening - 9 -matter^ »-;-7»^ t but the spectroscopic arguments suggest that they are more l i k e l y to arise in the objects themselves. If the l a t t e r i s true, then at least the differences between the absorption red-shifts must be i n t r i n s i c to the objects, and the simple explanation that the red-shifts are due to the expansion of the universe has to be modified. Despite these problems, most astronomers s t i l l take the view that the cosmological interpretation of quasars i s the most plausible one. In our work, we w i l l also take this view. 4. The Present Program A. The Aim Ih section 3, we described the characteristic fea-tures of quasars and discussed the nature of the red-shift, the physical models accounting for the energy output, the variation in flux, and the presence of multiple absorption red-shifts in addition to an emission red-shift. Despite the d i f f i c u l t i e s that we have mentioned, we shall interpret the red-shift of quasars as a cosmological effect — the expansion of the universe. It i s our aim to construct a cosmological model which w i l l account for the following two properties of quasars. i ) The dearth of red-shifts beyond Z = 2 (ref. 22). This effect could be attributed to a Lemaitre - 10 -type universe, or to a universe i n which the frequency of occurrence of quasars i s a strong function of epoch, so that there are no quasars of more than a certain age. However, we w i l l take the point of view that the effect i s due to a large-scale spatial inhomogeneity of the universe, i i ) The anisotropic distribution of quasars in the sky, particularly of those with large red-shifts. Strittmatter, Faulkner, and Walmsley41(1966) investigated the distribution of quasars over the sky. They divided the quasars into four groups: Group I with Z > 1.5, Group II with 1, .5 > Z > 1.0, Group III with 1.0 > Z > 0.5, and Group IV with 0.5 > Z > 0. When they carried out the analysis, 67 quasars with known red-shift were available. The complete distribution of 100 quasars at the end of March 1967 i s given by Burbidge and Burbidge 4 2. We reprint the figure (Fig. 8-1) given by them. It i s seen that a l l the objects in Group I divide into two very compact and almost antipodal groups. The objects i n Group II are more widely spread in the northern galactic hemisphere, though s t i l l i n about the same posi-tion as those of Group I, whereas the members of this group that are in the south are almost coincident with those of Group I. The members of Group III and Group IV are more widely spread over the sky. Possible selection - 11 -( From Burbidge and Burbidge ) o z < 0 . 5 FIG. 8.1 Redshifts of the 100 QSO's given in Table 3.1 plotted os a function of position in the sky in galactic coordinates. Different symbols are used for the four ranges of redshift, as indicated in the key. The direction of the Earth's north and south poles are denoted by N and S, and the broken line represents the projection of the Earth's equator on the sky. - 12 -effects that may affect this result; have been discussed and the conclusion i s that the anisotropy i s l i k e l y to be a real effect. B. A New Model We assume that i ) The universe i s composed of dust particles which exert negligible pressure. i i ) The dust particles are distributed isotropically i n space with respect to a particular origin. i i i ) The Einstein f i e l d equations with zero cosmological constant are v a l i d . iv) The distribution of dust particles i s such that the model consists of an inner Friedmann zone and an outer Friedmann zone, separated by a transition zone. We define N(Z) to be the number of galaxies and quasars per steradian whose red-shifts are smaller than Z. We w i l l calculate N(Z) for our model with the further assump-tion that the transition zone can be neglected. Chapter II The F i e l d Equations It i s simplest to use a set of comoving coordinat From the assumption of spherical symmetry and the absence pressure, the metric can be w r i t t e n 4 4 as is » di - c (de + sm f if ) - e dr J where oo, JL are functions of r and t. The components of the energy momentum tensor are T/ - Ta - 7/ - , , where i s the energy density. (We choose c = 1 . ) The Einstein f i e l d equations are 67T j f = 0 = e - c ^ 4 + * + ^ u KT , ' . - » r r/...^-^.V' i f, where the accents denote differentiation with respect to r and the dots with respect to t. For later convenience, we write - 14 -To solve the f i e l d equations, we f i r s t integrate (6), and obtain . where 1 + f i s an undetermined function of r with positive values. Substituting (7) into.(3) gives (7) Z%% + % - j ( r ) - * . (8) A f i r s t integral of (8) i s fcr) + V<% > (9) where F i s a second undetermined function of r. A f i r s t integral of (9) i s j(f<r) + FC'jli,] « t - &<r) ? ( 1 0 ) where G i s a third undetermined function of r. Evaluating the integral in ( 1 0 ) , we get f I ( 1 1 - D - 15 -\% j'*** = i - fCt f , ^ f - * , (11.3) From (5)» the energy density f can he expressed as -7'f!^*' • (12) The three unknown functions of r, f ( r ) , F ( r ) , and. G(r) w i l l be determined l a t e r from the i n i t i a l c o n d itions. -.16 -Chapter I I I Friedmann Model Consider the case when there exists a function S such that £ Cr,t) - r gd) ( 1 3 ) Combining ( 1 ) , ( 7 ) , and ( 1 3 ) , we can write the metric as Equations ( 8 ) , ( 9 ) , and ( 1 0 ) become 3 (14) for) - ( a X i r + i ' , V r , ( 1 5 _ 1 } / r r r ; = ~- £ *S r* f ( 1 5 . 2 ) CjCr) = t -H r(f + F r~0ct)J . (15.3) The three functions f, F, and G are completely determined by choosing a set of i n i t i a l values for S, S, and S. We write S± = S ( t i ) , S± = S(t±), and s \ = S ( t ± ) for the i n i t i a l time V . Equation ( 1 2 ) becomes - 17 -Since f and P are functions of r only, and S i s a function of t only, equations (15-1) and (15-2) imply that 7,$$ -r $* « constant = - CLK~2 ? (16) a. S & =• constant •' (17) where . ( ' r^ - - , ( 1 8 ) a* - ( / a w ^ i 4 ! / * , (19) and i s an arbitrary constant i f k = 0. Define q, the deceleration parameter, , the density parameter, and H, the Hubble parameter as follows: HCi) ~ Hay^c*) , ( 2 0 ) %C*) * -fWfrty&^V ^ ( 2 1 ) (TCt) = 4j[>^ ftD/H^Ct) % ( 2 2 ) - 18 -(15-1), (15-2), and (15-4) then become j*x f<v--yi.f - #*#\,-4f) , (23) y£ Fo; = ^ V / , ( 2 4 ) The metric can be rewritten as Making a coordinate transformation r = CLk <r^(<*>) (27) where Sink, cO we find that (26) becomes 4s = $(l)Utb +%c*>)C<U +s>»*dj)J^(28) (28) i s the metric for the well-known Friedmann model for a uniform distribution of dust. The universe i s called - . 1 9 -open i f k = - 1 , closed i f k = + 1 , and Euclidean i f k = 0. Now equations (11-1), (11-2), and (11-3) become, by the help of equations (15-1), (15-2), (15-3), (20). (21). and (22), ' (29-1) * 4 (29-3) where the subscripts ' i * refer to the i n i t i a l epoch t^. Remark: According to (23), / _ / ' ' , : r « = j + / if I \ > o.t • (30) - 20 -Chapter IV Red-Shift in the Friedmann Model We relate our theory to observation by assuming that there i s a comoving coordinate system with the properties previously discussed, in which galaxies are at rest, apart from small random motions which we w i l l ignore. This assumption i s consistent with the f a i r l y small scatter of o points in the red-shift - magnitude diagram . We consider light emitted by a source s and ob-served by an observer o, both s and o being stationary i n the comoving coordinate system. The wavelength measured by o i s }*» , and the wavelength measured by a stationary, observer near the source i s X$ . We assume that none of the atomic parameters i s time-dependent. The red-shift of a spectral line i s defined to be 2 = A 7 ~ • (3D Let us use the metric (28) i n the coordinate system 2 ( t,co,e , J> ). Light rays are described by setting ds =0 in (28). Considering only the radial incoming light rays, we have 7u7 =" - ^ • (32) - 21 -Integration gives co - co - <U ^ ( 3 3 ) where c O s and u>a are the radial coordinates of source and observer, and t and t are the times of emission and s o reception respectively. Differentiating (33) for constant cos - ^ 0 shows that i f two ligh t signals are emitted at times t and t - + dt , s s s and received at times t ..and t„ + dt , the time differences o o o satisfy dt0 oits ' ( 3 4 ) The number of waves emitted between t and t + dt i s s s s >4 <£ts , and the number of waves received between t and o t Q + dt Q i s . >4 cit0 , where ^ and are the frequency of emission and reception respectively. Since the number of waves must be conserved, i.e., , >> cLis = >> cL-k » w e h a v e x C x \ C S (35) i s (36) - 22 -From (31) and ( 3 6 ) , where we write S Q =-S(t ) and S g = S ( t g ) . For l a t e r use, we eliminate t between (33) and (37) to s obtain a r e l a t i o n between o)g - noa and Z: 5 » - , = -ML. (38) (39) (40) where In s e r t i n g (23),and (24) i n equation (9) gives We take p o s i t i v e values f o r S ( t ) , because a contracting universe i s inconsistent with the observations. From (38) and (40), / . . . (41) Integration, with the help of equation (23), we get ' Jt Sty, f*t- O+Z) Note that for an observer at the origin, - 24 -Chapter Y Number Count in the Friedmann Model Consider an observer at the origin. The mass seen. _ in a spherical shell of coordinate radius co and thickness o£<a i s iN=*+1CfCts) 0"(«>) « t o , ( 4 3 ) where t g i s the time of emission of ligh t signals by means of which the object i s observed at the present epoch t . o Note that f^ts) i s the proper mass density, and f^ts^ (&t: £ ) )* i s "the mass density per unit coor-dinate volume. Let us write (44) However, from equations (15-4) and (17), £ = - 4 ! & & - Consent . ( 4 5 ) Therefore we w i l l write fc instead of J*c (^sj Integration of (43) gives the mass within a coordinate radius cd (46) - 25 -Evaluating the i n t e g r a l i n (46) gives 2%ft [-^swi^ti -u>J -/ (47) 3 J c Now, by use of equation (42), we can express N as a func t i o n of Z, and we have a r e l a t i o n between the two d i r e c t l y observed q u a n t i t i e s , N and Z. - 26 -Chapter VI A New Model Universe Let us go back to the coordinate system ( "t, r, Q , J ), and consider a model in which the metric i s given by x o < r < r % (48-1) <£s = dt - & H)^ — - + TA(<Ld + sini dyjj. T h < T , ~0* T (48-2) where r b > r & and the subscripts "1" and. "2" stand for the regions. o< r < r & and r^ < r respectively. We w i l l say that the region 0 < r < r i s the a. inner Friedmann zone, .and. the region r^ < r, the outer Friedmann zone. In between the two Friedmann zones, there i s a transition zone with a metric such that the boundary conditions at r = r & and r = r^ are sat i s f i e d ( The metric and i t s f i r s t derivative must be continuous. ). This i s possible because we are free to choose the three functions f, F, and G in the transition zone. Moreover, according to equations (1), (7), (8), (9), and (10), motion of the - 27 -dust i n each Friedmann zone i s independent of the behavior of the other parts of the model. We w i l l assume, as the i n i t i a l conditions of our model, that £,L = & i (49) or e q u i v a l e n t l y (50-1 ) (50-2) X K > (50-3) where the subscript " i " r e f e r s to the value at the i n i t i a l epoch t ^ . The c o n d i t i o n (50-3) means that the i n i t i a l value of pro-per mass density of the outer Friedmann zone i s equal or l e s s than that of the inner Friedmann zone. Thus we have the possible model universes l i s t e d i n Table I. - 28 -TABLE I Possible Model Universes Model. k 1. Model type of the kp. Model type of the inner Friedmann zone. outer Friedmann zone I -1 open -1 open II 0 Euclidean -1 open 0 Euclidean III +1 closed -1 open IV +1 closed 0 Euclidean V +1 closed +1 closed We note that in the coordinate system ( t,to , 9 , f ), (48-1 ) and (48-2) become /OK O < ed, <COA J ( 5 1 , ± 4 s .i. _> for- c*h < (51 where - 29 -Chapter VII Number - Red-Shift Relation in the New Model. observed from the origin, (B) the relation observed from a point in the inner Friedmann zone half-way between the origin and the boundary of the inner Friedmann zone., Note that the relation applies to quasars under the assump-tion that the ratio of the number of quasars to the total number of galaxies i s a constant. (A) The N - Z relation observed from the origin true for each Friedmann zone, but we w i l l have to make some extra assumptions about the. transition zone. where Z i s the red-shift of lig h t signals emitted at r ci i and received at the present epoch at the origin, the subscript "o" refers to the present epoch and we have We w i l l consider two cases: (A) the relation The expressions derived in chapter IV and V are (i) For a source in the inner Friedmann zone, the a>— Z relations are Z < z a - 30 -( i i ) For a source in the outer Friedmann zone, the co ~ Z relations are f" 1 4 I (53) V «a fa fat where t^ i s the epoch such that li g h t emitted at t^ and at r = r^ w i l l be received at t at the origin. Z' and u>£ are the red-shift and radial coordinate of the source for an observer at r = r^ and at the epoch t^. Let Z^ be the red-shift of light signals emitted at r = r f e and received by an observer at the origin at the present epoch. The epoch at which these light signals are emitted i s then t^. Now for a source in the outer Fried-mann zone, we can imagine the situation that the signals are emitted with wavelength As and received by an observer at r = r b at the epoch t f e . According to this observer, the signals have a wavelength A. and a red-shift & — /% ~ ' • The light signals are then immediately re-emitted by the observer at r = r^» and reach the observer at the origin at the present epoch with wavelength X . . By definition, - 31 -^ = / , and the total red-shift i s ( 5 4 ) The coordinate distance of the source as calcu-lated by the observer at the origin i s ( 5 5 ) By means of ( 5 4 ) and ( 5 5 ) , equations ( 5 3 - 1 ) , ( 5 3 - 2 ) , and ( 5 3 - 3 ) are rewritten as zs;»~'/2ht±„ zafate-OO+za if KM+I ( 5 6 ) CO, - 6i, v o The N — eo relations are modified to » A 7 r ] * , e ()£ shim, -for r < r , where T - ^ f «a, ) , W « £ K j ^ ( t^) . - 32 -and t o (58) (59) and t a k e s a c c o u n t o f t h e e f f e c t o f t h e t r a n s i t i o n zone. As a f i r s t a p p r o x i m a t i o n , we assume t h a t t h e t r a n s i t i o n zone i s s m a l l and n e g l i g i b l e . Then we have t h e r e l a t i o n s : and N 3 = 0. (B) The N — Z r e l a t i o n o b s e r v e d from a p o i n t i n t h e i n n e r F r i e d m a n n zone h a l f - w a y between t h e o r i g i n and t h e boundary o f t h e i n n e r F r i e d m a n n zone As seen by t h i s o b s e r v e r , t h e u n i v e r s e i s a n i s o -t r o p i c as w e l l as inhomogeneous, and t h e number count i s a r « r . (60) - 33 -function of distance ( or red-shift ) and direction. For simplicity, we consider only the radial directions ( radial with respect to the origin ). The W-direction i s the one that passes through the origin and the V-direction i s the opposite one, as shown i n the diagram. * * M * ;> * X. observer o]*ewer A B In calculating the N — Z relation, we note that we are calculating the number of objects per unit solid angle as a function of Z in the W- and V-directions. Because of the difference i n density between the inner and outer Friedmann zones, lig h t rays that cross the boundary in other than radial directions w i l l be refracted. This means that a bundle of light rays that subtend a solid angle 'ii-as observed in the inner. Friedmann zone w i l l subtend a different solid angle in the outer Friedmann zone. However, this effect i s very small with the density that we may reasonably expect. As w i l l be seen in Tables II - VI, fio/ fd i s greater than 6.4x10 v i n a l l cases that we have considered. Since /% 2.0x10" g/cnr and J V ^ -Pi » - 34 -the density i s smaller than 3.1x10 -11 g/cm at a l l points of space-time. The equations (52) — (60) apply for both directions, but we must remember that the coordinate distance from the observer to the boundary of the inner Friedmann zone in the W-direction i s three times that i n the V-direction. This provides a relation between the boundary values of the red-shift Z„ in the two directions, which can be calculated from equation (52). In the f i r s t approximation, where we neglect the transition zone, we have the relations: H3 - « (61 ) where the superscripts (V) and (W) refer to the V- and W-directions respectively. - 35 -Chapter VIII Numerical Calculation As noted i n the introduction, we are trying to find values of the parameters H and q that f i t the observed distribution of quasars. The f i r s t approximation in which the transition zone i s neglected, w i l l be used in the f o l -lowing numerical work. Thus we have the relations given by (60) and (61), in addition to the basic assumptions of the model as given by equation (50). For an observer at the origin, the value Z, = Z b a w i l l be chosen as 2.0. ( As remarked earlier, very few quasars are observed with red-shift larger than 2.0. ) For an observer half-way between the origin and.the boundary of the inner Friedmann zone, the value Z, = Z i s also b a taken as 2.0 i n the W-direction. This observer therefore sees the same N — Z relation in this direction as does an observer at the origin in the other case. The value in the V-direction w i l l then be calculated from (61) and (50). A l l the five model universes given in Table I w i l l be considered. - 36 -A. Procedure to obtain the N — Z relation 1. Determine the value of q 1 o» "the deceleration parameter. q.j can be determined from the observed values of the Hubble constant H^ Q and the average energy density />,© at the present epoch according to the equation It can also be determined by other methods^. At present, there i s a large uncertainty in the values of 10 both q^ 0 and H^ Q . We may take the value of q^ as + 1 .2 * 0.4. However, as remarked in the introduction, this value i s very uncertain. In our calculation, we w i l l therefore take values of q^ Q ranging from + 0.025 to + 2.00. The value of H., i s not used in the calculation of the 1 o N — Z relation. ( The value of H. i s in fact between 1 o q q 7.5x10 and 19.5x10 years, as recently reported by 1 0 Sandage . ) 2. Calculate ///* ~?t from the equations I y 0 - 37 -where Zt i s "the time taken for light to travel from r to a the observer and also the time taken for light to travel from r^ to the observer in the f i r s t approximation-.- We have T"i ~ "^o " . Equation (62) i s derived as f o l -lows: , by equation (40), where v = 1 + Z. Integration, using (30), then gives (62). 3. Assume a value of ^. Then the value of q2^ i s determined from the equation which i s derived by combining equations (25) and (50). For the models I and III, we w i l l take ^/f^ - 0.1, together with = 1 ( single Friedmann model ) and ^/£ ) L = 0 ( no matter in the outer Friedmann zone ). For the model II, we have q^ = q ^ = 0.5. The model i s described by parameters H^ /.H^ and ^/j^c • For the model IV, we have qg^ - <i2± = 0.5. The only parameter i s ^/Ju = q2±/ q1 ± = 0.5/q1jL.; - 38 -For the model V, since 2.0 £ > ^ > 0.5 and ^ y >,i = q 2 i / q 1 i ' w e h a v e 1 / 4 * ^ L < 1 * 4. C a l c u l a t e h\j ( t - t± ) from equations (29-1,2,3). Then H.,Q ( t & - ^ ) i s given by H,o < < - t.) = H,o C*. -V - H l % r 4 _ 5. For models I, I I I , IV, and V, c a l c u l a t e H 0. ( t 0 - t. ) c i a i from This equation i s obtained by equating equations (23) and (24). For model I I , 6. For models I, I I , I I I , and V, c a l c u l a t e q 2 b from the equations (29-1,2). For model IV, c a l c u l a t e H 2 i / H 2 b from (29-3). " 7. C a l c u l a t e a^/a.^ f o r models I, I I I , and V from the equa-t i o n \s V which i s obtained from equation (23). - 39 -For models II and IV, since we are free to choose the value of a^ i n a Friedmann zone with k = 0, we can put a^ya^ = 1 . This i s equivalent only to a l i n e a r change of the scale of the r a d i a l coordinate. 8. For models I, I I I , IV, and V, c a l c u l a t e This formula i s obtained by equating equations (23) and (24) For model I I , 9. Calculate - % ^ c from the equation (44): 7?* " J?c ~ Ti c £ 7 > • 10. Calculate <£>W and /V(^>CZ)^ from equations (52) to (61). - 40 -B. R e s u l t s The n u m e r i c a l r e s u l t s f o r our.model w i t h n e g l e c t o f t h e t r a n s i t i o n zone a r e shown i n T a b l e s I I - V I and a r e s i m p l e F r i e d m a n n models and t h o s e w i t h J f/J\t = 0 a r e models h a v i n g no m a t t e r i n t h e o u t e r F r i e d m a n n zone. I n c o n s i d e r e d two u n i v e r s e s s uch t h a t an o b s e r v e r h a l f - w a y between t h e o r i g i n and t h e boundary o f t h e i n n e r F r i e d m a n n zone i n t h e second u n i v e r s e sees t h e same N — Z r e l a t i o n i n t h e W - d i r e c t i o n as does an o b s e r v e r a t t h e o r i g i n i n t h e f i r s t , we p l o t t h e N — Z r e l a t i o n s f o r them i n t h e same f i g u r e s ( t h e even numbered ones ) . I n t h e V - d i r e c t i o n , t h e b o u ndary v a l u e Z^ V^ i s d e t e r m i n e d by t h e c h o i c e o f z! W) = 2.00.. The v a l u e s o f z{Y^ depend on q 1 and a r e a a r Mo shown i n column 3 o f t h e t a b l e s . The N - Z r e l a t i o n s i n t h i s d i r e c t i o n a r e p l o t t e d i n t h e odd numbered f i g u r e s . We note t h a t t h e c u r v e s i n t h e f i g u r e s and t h e models i n t h e t a b l e s a r e l a b e l l e d w i t h t h e same model numbers. F i g u r e s 2 - 1 1 . I n t h e t a b l e s , t h e models w i t h M.c = 1 t h e f i g u r e s , S i n c e we have - 41 -The Quantities Listed On The' Tables &\o : The present value of the deceleration parameter of the inner Friedmann zone. Z : The maximum red-shift of the objects in the inner Friedmann zone. For an observer at the origin, 2 ^ = . For an observer half-way between the origin and the boundary of the inner Friedmann zone, the value i n (ur) the W-clirection i s also taken as 2.^ = 2.0 . Z& : The maximum value in the V-direction.is determined from c ? i L : The. i n i t i a l value of the deceleration parameter of the inner Friedmann zone. ? a j : The i n i t i a l value of the deceleration parameter of the outer Friedmann zone. The ratio of the mass density per unit coordinate volume of the outer Friedmann zone to that of.the inner Friedmann zone. ^yp^ > The ratio of the present proper density to the i n i t i a l proper density in the inner Friedmann zone. - 42 -This i s the ratio of the number of galaxies (qua-sars) seen by an observer at the origin, with red shifts between 2.00 and 3.00, to the number with red-shifts smaller than 2.00.~ This i s the ratio of the number of galaxies (qua-sars) seen by an observer half-way between the origin and the boundary of the inner Friedmann zone in the V-direction, with red-shifts between 1.50 and 2.00, to the same quantity in the W-direction. Table II Model I k1 = t,i 1 . .025 2.00 .435 .499 2. .025 2.00 .435 .450 3. .025 2.00 .435 .300 4. .025 2.00 5. .025 2.00 6. .100 2.00 .415 .499 7. .100 2.00 .415 " .450 8. .100 2.00 .41 5 .300 9. .100 2.00 10. .100 2.00 11 . .400 2.00 .366 .499 12. .400 2.00 .366 .490 13. .400 2.00 14. .400 2.00 43 -1 , k 2 = - 1 .0499 1.02x10 -5 1 .17x10 -12 .0450 3.63x10 -3 2.00x10 -7 .0300 2.77x10 -2 4.32x10 -5 1 .00 0.00 .0499 1 .02x10 -5 1.26x10 -10 .0450 3.63x10 -3 2.1 4x10 -5 .0300 2.77x10 -2 4.63x10 -3 1 .00 0.00 .0499 1 .02x10 -5 5.1 5x10 -7 .0490 3.31x10 -4 5.44x10 -4 1 .00 0.00 Table II (continued) X i 1. .0057 .0032 2. .043 .018 3. .086 .062 4. 5. 6. .0064 .0039 7. .049 .028 8. .11 .095 9. 10. 11 . .01 4 .0080 12. .048 .038 13. 45 -Table I I I Model I I k., = 0, k 2 = -trio 1. .500 2.00 2. .500 2.00 3. .500 2.00 4. .500 2.00 5. .500 2.00 6. .500 2.00 7. .500 2.00 8. .500 2.00 9. .500 2.00 10. .500 2.00 11 . ' .500 2.00 12. .500 2.00 13. .500 2.00 14. .500 2.00 15. .500 2.00 4" tax .360 .500 .499 .360 .500 .400 .360 .500 .010 .500 .500 .360 .500 .499 .360 .500 .400 .360 .500 .010 .500 .500 .360 .500 .499 .360 .500 .400 .360 .500 .010 .500 .500 -17 .998 6.40x10 .800 6 .40x10 -17 .020 6 .40x10 -17 1 .00 0.00 .998 6 .40x10 -11 .800 6 .40x10 -11 .020 6 .40x10 -11 1 .00 0.00 .998 6 .40x10 -5 .800 6 .40x10 -5 .020 6 .40x10 -5 1 .00 0.00 Table III (continued) 1. 2 .063 .035 2 . .0050 .0031 3. .0000 .0000 4. 5. 6. .65 .63 7. .050 .032 8. .0006 .0001 9. 1 0 . 11.- 4.3 1 90 12. .23 1 .8 13. .0026 .0020 14. - 47 -Table IV Model III k 2 = - 1 1. .600 2.00 .154 2. .600 2.00 .154 3. .600 2.00 4. .600 2.00 501 510 -6 .0501 1.02x10~5 1.72x10 .0510 3.31x10"4 1.63x10~3 1 .00 0.00 5. 1 .00 2.00 .138 .501 .0501 1.02x10 -5 6. 1 .00 2.00 .138 .550 .0550 3.76x10 -3 7. 1 .00 2.00 1 .00 8. 1 .00 2.00 0.00 9. 2.00 2.00 •:. 188 .501 .0501 1.02x10 -5 -2 10. 2.00 2.00 .118 .600 .0600 1.08x10 11 2.00 2 .00 1.00 12 2.00 2 .00 0.00 6.36x10 -8 1.10x10' Table IV (continued) * 3 1. .014 .012 2. .049 .055 3. 4. 5. .0067 .0075 6. .056 .084 7. 8. 9. .0043 .0070 10. .068 .11 11 . 12. - 49 -Table V Model IV ^ = + 1, k 2 = 0 L z^-zt & 'V* 1. .600 2.00 .154 .501 .500 .998 1 .72x10"b 2. .600 2.00 .154 .510 .500 .980 1 .63x10"3 3. .600 2.00 1 .00 4. .600 2.00 0.00 5. 1 .00 2.00 .138 .501 .500 .998 6.38x1 0'8 6. 1 .00 2.00 .1 38 .550 .500 .909 6.01x10"3 7. 1 .00 2.00 • 1 .00 8. 1 .00 2.00 0.00 9. 2 .00 2.00 .118 .501 .500 .998 . 1.88x10"8 10. 2.00 2.00 .118 .600 . .500 .833 1 .10x10"2 11 . 2.00 2.00 1 .00 12. 2.00 2.00 0.00 1 Table V (continued) x 1 1 . .60 1 .01 2. .60 1.01 3. 4. 5. .50 1.00 6. .50 1.00 7. 8. 9. .39 .99 10. .35 .98 11 . 12. - 51 -Table VI Model V 1. .600 2.00 2. .600 2.00 3. .600 2.00 4. 1.00 2.00 5. 1.00 2.00 6. 1.00 2.00 7. 2.00 2.00 8. 2.00 2.00 9. 2.00 2.00 k1 = + 1 , k 2 .154 .510 .501 .138 .510 .501 .118 .510 .501 = + 1 31.1 1.63x10" 1 .00 0.00 31.1 .6.03x10"' 1 .00 0.00 31.1 1.78x10~* 1 .00 0.00 Table VI (continued) 1. .53 1 .38 2. 3. 4. .50 1.30 5. 6. 7. .40 1.19 8. 9. - 56 -T" or 1 ' V . H 1 I s - 61 -- 63 -Column 8 of the tables gives the ratio of the number of quasars with red-shifts between 2.00 and 3.00_to the number with- red-shifts smaller than 2.00 as seen by an observer at the origin. It i s seen that the ratio depends largely on the values o f s i 0 / s - | i ' o r e ( l u i v a l e n t l y of for a fixed value of . This allows us in principle to f i x J\v for our model by comparison with observation.. However, at present, there are only 136 quasars with known . red-shifts, and i t i s hard to determine the proper value Z a of the red-shift at the boundary. If we had more quasars, we could perhaps choose Z in a non-arbitrary fashion as the value of Z for which the slope of the N — Z curve change suddenly. But we cannot draw any significant conclusions from only 136 quasars. If we assume that the presently observed distribution of quasars i s correct, then the above described ratio i s about 0.1, and we may exclude those mo-dels in which the ratio differs too much from 0.1 . In Chapter I, we discussed how the known quasars are distributed over the sky. It i s clear that i f the s i -tuation remains unchanged as more observational data are accumulated, then our simple model, consisting of an inner Friedmann zone and an outer Friedmann zone, must be wrong. In our model, an observer at the origin sees an - 6 4 -isotropic distribution; while the observed distribution of quasars with Z > 1.5 i s anisotropic. For an observer half-way between the origin and the boundary of the inner Friedmann zone, the distribution i s anisotropic but i n the wrong way. The quasars with the largest Z are concentrated in the W-direction instead of in two almost antipodal directions as i s observed. (See Fig. 1) Column 9 of the tables shows the calculated ratio y of the number of quasars with red-shifts between 1 .50 and 2.00 i n the V-direction to the number in the W-direction. It i s possible to divide the sky into sixteen similar regions, each subtending a solid angle of 3% » i n such a manner that a l l known quasars with red-shifts greater than 1 .50 l i e in just two of the regions. These two regions are antipodal. One of them contains 1 4 of the quasars with red-shifts greater than 1.50, and the other contains 7. We identify the direction of the center of the f i r s t region with the W-direction of our model, and that of the second with the Y-direction. Assume that at any epoch t and in any region at a distance r from the center of the universe, the probability of a galaxy being a quasar is p ( r , t ) . We have to determine p(r,t) on our observed past light-cones for r 1 c < r< r ? n , - 65 -where r^ ^ and Q correspond to distances for which Z = 1 .5 and Z = 2.0 respectively. As a rough approximation, we assume that p(r,t) i s constant in this region and equal to the ratio in this region of the total number of quasars to the total number of galaxies. We w i l l calculate the probability of the event TE that the ratio of the number of quasars with red-shifts 1.5< Z < 2.0 i n the V-direction to the number in the W-direction i s greater than 1/2. Consider f i r s t the case where the observer i s half-way between the origin and the boundary of the inner Friedmann zone. i ) From the assumptions of our model, and by comparing our model with observations, we find the N — Z relation in the inner Friedmann zone. Then Nw, the number of galaxies in a solid angle of in the W-direction at distances 9 r„ n < r < r 0 ~, i s calculated to be about 5x10 , and the 1.5 c. .U corresponding number N y in the V-direction i s yNw, where y i s given in column 9 of tables II — VI. •ii) To determine the constant value p.of p(r,t), we consider two cases. In the f i r s t case, we take account of the radio-quiet quasars and have p •~ 1 .0x10"4. In the second case, only the observed quasars with known red-shift are taken —9 into account and p/-v 4.0x10 . - 6 6 -i i i ) The "probability of the event TC i s However, i n the f i r s t case where p = 1.0x10"^, Nwp and Nvp are large, the Le Moivre Laplace approximation can be made. It i s found that for a l l the cases we have considered, fit, < ^ . _q In the second case where p = 4.0x10 , both Nwp and Nyp are small, the Poisson distribution can be used. The probability becomes s * . rt In the numerical calculation for Pg, we exclude the cases where y i s greater than 0.5, for they are accompanied by - 67 -large values of x and contradict the dearth of quasars with red-shift greater than 2.00. The probability i s again very small. For the case where y = 0.11 (the largest value of y 4 0.5 that we have considered), P 2 <>/ 1 .37x10 and P 2 decreases very rapidly as y decreases. For the universe with the observer at the origin, consider the event J that the ratio of the number of IT/ quasars in an arbitrary direction, in a solid angle of / i * , with red-shifts 1.5 < Z < 2.0, to the number i n any other . direction i s greater than 1/2. Since the number of galaxies i s the same in a l l the directions, the probability of the event ^ i - s "very small by the same argument as given above for the other universe. This means that the probability of finding that the quasars with red-shifts 1.5< Z < 2.0 are concentrated in two almost antipodal groups i s very small. - 6 8 -C h a p t e r IX C o n c l u s i o n I t i s seen from t h e n u m e r i c a l r e s u l t s t h a t o u r model c a n s u c c e s s f u l l y a c c o u n t f o r t h e d e a r t h o f q u a s a r s w i t h r e d - s h i f t l a r g e r t h a n 2.0 p r o v i d e d we choose a s u i t a b l e i n i t i a l v a l u e o f t h e d e c e l e r a t i o n p a r a m e t e r , but i t does n o t d e s c r i b e t h e way i n w h i c h q u a s a r s o f l a r g e s t r e d - s h i f t a r e d i s t r i b u t e d i n t h e s k y . w h i c h t h e i n n e r F r i e d m a n n zone i s deformed from a s p h e r e i n t o an e l l i p s o i d . A s c h e m a t i c diagram i s p l o t t e d as f o l l o w s : An o b v i o u s m o d i f i c a t i o n i s t o c o n s i d e r a model i n V z>—a,, o The o b s e r v e r i s somewhat o f f t h e c e n t e r o f t h e e l l i p s o i d and t h e dashed c u r v e c o r r e s p o n d s t o a r e d - s h i f t o f Z = 2.0. - 69 -An alternative i s to consider an universe in which there are inner Friedmann zones distributed randomly in an outer Friedmann zone as suggested by Rees and Sciama 4^' 4^. We may or may not be l i v i n g in one of the inner;Friedmann zones. A schematic diagram i s as follows: It i s assumed that by chance, there are two zones in the W- and V-directions with distances corresponding to a red-shift of Z = 2.0. The p o s s i b i l i t y of formation of such large-scale inhomogeneities i s discussed by Rees and Sciama. - 70 -BIBLIOGRAPHY 1. Weyl, H., 1923, Phys. Z., 24, 230. 2. Bondi, H., and Gold, T., 1948, Mon. Not. R. Astr. Soc, 1_08, 252 . 3. Friedmann, A., 1 924, Z. Phys., 21_, 326. 4. Robertson, H.P., 1935, Ap. J., 82, 284. 5. Robertson, H.P., 1936, Ap. J., 83, 187, 257. 6. Walker, A.G., 1936, Proc Lond. Math. Soc, (2), 42, 90. 7. De Sitter, W., 1917, Mon. Not. R. Astr. Soc, 78, 3. 8. Humason et a l , 1 956, Ap. J., 61_, 97. 9. Baum, W.A., 1957, Ap. J., 62, 6. 1961, Observatory, 81_, 114. 10. Sandage, R., 1970, Phys. Today, Feb. 1970, 34. 11. Davidson, V/., and Narlikar, J.V., 1966, Reports on Progr. in Phys., V. 29, pt. 2, 594. 12. Hoyle, F., 1968, Proc. Roy. Soc, Ser. A, 308, 1. 13. Ryle, M., and Clarke, R.W.,.1961, Mon. Not. R. Astr. Soc, 1_22, 349. 14. Veron, P., 1966, Nature, • 211_, 724. 15. Fowler, W.A., and Hoyle, P., 1960 Ann. Phys. N.Y., 1_0, 280. 16. Dicke, R.H., 1962, Rev. Mod. Phys., 34, 110. - 71 -17. Hoyle, F., and Narlikar, J.V., 1962, Observatory, 82, 13. 18. Partridge, R.B., 1969., American Scientist, 57, 37. 19. Gamow, G., 1948, Nature, lond., 1_62, 680. 20. Hoyle, F., and Taylor, R.J., 1964, Nature, 203, 1108. 21. Burbidge, G.R., and Burbidge, E.M., 1967, Quasi-Stellar Objects, Freeman and Company, P. 10. 22. Burbidge, G.R., and Burbidge, E.M.', 1969, Nature, 224, Oct. 4, 21. 2 3 . c.f. 21, P. 188. 24. Hoyle, F., Burbidge, G.R., and Sargent, W.L.W., 1966, Nature, 209, 751. 25. Setti, G., and Woltjer, I., 1966, Ap. J., 1_44, 838. 26. Bahcall, J.N., Petercon, B.A., and Schmidt, M., 1966, Ap. J., 145, 369. 27. Hoyle, F., and Fowler, W.A., 1967, Nature, 21_3, 373-28. Greenstein, J.L., and Schmidt, M., 1964, Ap. J., 140, 1. 29. Bondi, H., 1964, Proc. Roy. Soc, Ser. A, 282, 303. 30. c.f. 21, P. 206. 31. Huang, J.C., and Edwards, T.W., 1968, : •. Phys. Rev., 1_71_, 1331 . 32. c.f. 21, P. 32. 33. Burbidge, E.M., Loyds, C.R., and Stockton, A.N., 1966, Ap. J., 1_44, 447. - 72 -34. Stockton, A, N., and loyds, C.R., 1966, Ap. J., 1_44, 451 . 35. Burbidge, E.M., Loyds, C.R., and Stockton, A.N., 1968, Ap. J., 1_52, 1077. 36. B a h c a l l , J.N., Greenstein, J.L., and Sargent, W.L.W., 1968, Ap. J., 1_53, 689. 37. Burbidge, G.R., and Burbidge, E.M., 1969, Nature, 222, 735. 38. Roeder, R.C, and Verreault, R.T., 1969, Ap. J.,. 1 55, 1047. 39. B a h c a l l , J.N., and Peebles, P.J.E., 1969, Ap. J. Lett., 1 56, L. 7. 40. B a h c a l l , J.N., and Spitzer, L., 1969, Ap. J. L e t t . , 156, L. 63.. 41. S t r i t t m a t t e r , P.A., Faulkner, J., and Walmsley, M., 1966, Nature, 21_2, 1441. 42. c . f . 21, P. 90. 43. Penston, M.V., and Robinson, G.M.R., 1967, Nature, 21_3, 375. 44. Tolraan, R.C., 1934, Proc. N.A.S., 20, 169. 45. Rees, M.J., and Sciarna, L.W., 1967, Nature, 21_3, 374. 46. Rees, M.J., and Sciama, D.W.', 1968, Nature, 217, 511.
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New cosmological model Shen, Po-Yu 1970
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Title | New cosmological model |
Creator |
Shen, Po-Yu |
Publisher | University of British Columbia |
Date Issued | 1970 |
Description | The present situation in cosmology is discussed. We try to explain the observed distribution of quasars in terms of an inhomogeneous model universe that consists of inner and outer Friedmann zones separated by a transition zone. The number - red-shift relation is derived, and numerical calculations are performed on the assumption that the transition zone is negligible. When the results are compared with observation, it is found that the dearth of quasars with red-shifts greater than 2 is easily explained, but that one cannot account for their anisotropic distribution. Modifications of the model are suggested. |
Subject |
Cosmology |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0084789 |
URI | http://hdl.handle.net/2429/34930 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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