A NEW COSMOLOGICAL MODEL by PO-YU SHEN Taiwan U n i v e r s i t y , 1966 B.S., National A THESIS SUBMITTED IN PARTIAL FULFILMENT OP THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics We accept t h i s t h e s i s 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 s i t u a t i o n i n cosmology i s discussed. We t r y to explain the observed d i s t r i b u t i o n of quasars i n terms of an inhomogeneous model universe that c o n s i s t s of inner and outer Friedmann zones separated by a t r a n s i t i o n zone. The number - r e d - s h i f t r e l a t i o n i s derived, and numerical c a l c u l a t i o n s are performed the t r a n s i t i o n zone i s n e g l i g i b l e . on the assumption that When the r e s u l t s are compared with observation, i t i s found that the dearth of quasars with r e d - s h i f t s greater than 2 i s e a s i l y explained, but that one cannot account f o r t h e i r anisotropic distribution. M o d i f i c a t i o n s of the model are suggested. - i i i- TABLE OF CONTENTS Abstract i i L i s t of Tables L i s t of Figures • • Acknowledgment Chapter I iv v • vi Introduction 1 1. Development of T h e o r e t i c a l Cosmology 1 2. Observational Cosmology 2 3. Q u a s i - S t e l l a r Objects 4. The Present Program Chapter I I The F i e l d Equations • 5 • 9 13 Chapter I I I Friedmann Model 16 Chapter IV Red-Shift i n the Friedmann Model Chapter V Number count i n the Friedmann Model Chapter VI A New Model Universe • 20 24 .26 Chapter VII Number - Red-Shift R e l a t i o n i n the New Model Chapter V I I I Numerical C a l c u l a t i o n s .——. 2 9 35 A. Procedures to Obtain the N — Z R e l a t i o n — 36 B. Results 40 Conclusion 68 Chapter IX Bibliography ?0 - iv - LIST OP TABLES Table I Possible Model Universes 28 Table II Numerical Results For Model I 43 Table I I I Numerical Results For Model I I 45 Table IV Numerical Results For Model I I I 47 Table V Numerical Results For Model 49 Table VI Numerical Results For Model V IV 51 - V - LIST OF FIGURES Figure 1 . The D i s t r i b u t i o n Of Quasars With Known Red-Shifts • . • 11 Figure 2. N - Z Curves For Model I (0,W) Figure 3. N - Z Curves For Model I (V) 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 I I I (0,W) 57 Figure 7. N - Z Curves For Model I I I (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) • • 53 54 • 62 - vi - ACKNOWLEDGMENT The author wishes to thank Dr. P. R a s t a l l who suggested the i n v e s t i g a t i o n and gave invaluable guidance. Chapter I Introduction 1. Development of T h e o r e t i c a l Cosmology In applying E i n s t e i n ' s general theory of relativity to cosmology, the' f o l l o w i n g assumptions are generally postulated i n order to s i m p l i f y the f i e l d equations. ( i ) Weyl's postulate^ The world l i n e s of galaxies form a bundle of geodesies diverging from a point i n the f i n i t e or i n f i n i t e , past. ( i i ) The cosmological p r i n c i p l e At a given cosmic time, the universe presents the same l a r g e - s c a l e view to a l l fundamental observers. ( i i i ) Isotropy At a given cosmic time, the universe presents the same l a r g e - s c a l e view i n a l l d i r e c t i o n s . (iv) The perfect cosmological p r i n c i p l e The universe presents the same l a r g e - s c a l e 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 element ' 5 4 , 5 , 6 where k = 0 , + 1, and - 1 . - 2 - This l i n e 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 ( i v ) lead 7 to a l i n e element and to the steady state cosmology. The big-bang models and the steady state models are the two major c l a s s e s of cosmological models. former, the expansion In the of the universe began from a singu- l a r i t y i n 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, continuously remakes i t s e l f according to a f i x e d but and immutable pattern. In the next section, we w i l l compare the t h e o r e t i c a l p r e d i c t i o n s of the two classes of cosmological models with the o b s e r v a t i o n a l data. 2. Observational Cosmology (A) Observations on the past l i g h t cone The r e l a t i o n between the apparent magnitude m of galaxies and the r e d - s h i f t z f o r small values of z - 3 - gives information about q , Q the present epoch. and of value q Q R e s u l t s ^ ' s h o w that q 5 = 1 .2 * 0.4, state p r e d i c t i o n of q error. the deceleration parameter at Q Q i s positive which d i f f e r s from the steady = - 1 by f i v e times the probable However, t h i s value f o r q Q could be wrong because of possible errors a r i s i n g from a s e l e c t i o n e f f e c t i n 11 favour of bright galaxies The 12 ' . d i s t r i b u t i o n of radio sources provides another t e s t , which also c o n t r a d i c t s the steady state theory. defines N(S) to be the number of sources per whose f l u x density measured i n u n i t s of 10 exceeds S at a given frequency. group (Ryle and Clarke 1961) J The One steradian w sterad Hz r e s u l t s of the Cambridge i n the 4C survey at 178Mc/s show that the log N - l o g S curve has a slope - 1 .8 down to about two flux units. The slope which s t a r t s at - 1.5 magnitude. steady state model p r e d i c t s a and get progressively l e s s i n In the big-bang models, the slope i s always - f o r small distances. l e s s than — 1 . 5 , l e s s than — 1 . 5 At large distances i t can be greater or depending on the model chosen. ( It i s f o r open universes. ) found that the slope of the curve was galaxies and - 2.2 f o r quasars. The However, Veron^ (1965) 4 — 1 .55 f o r radio radio galaxies are' r e l a t i v e l y nearby objects with r e d - s h i f t s not 0.5- 1.5 Hence the observed slope of - 1.55 exceeding i s not - 4 - inconsistent with e i t h e r of the two c l a s s e s of cosmological models discussed above. I f the large r e d - s h i f t s of quasars arise, e n t i r e l y 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. w i l l t a l k about quasars We later. ( B ) Observations inside the past l i g h t cone The galaxy should be at l e a s t as o l d as the oldest s t a r s i n i t . Considerations of nucleosynthesis enable us to estimate the age of the stars i n our galaxy as being between 1.0x10 10 and 1.5x10 10 years 1 5 , 1 6 . This i s i n agreement with the p r e d i c t i o n of the big-bang models, which p r e d i c t an age of the universe (defined to be the time that has elapsed since the big-bang) of HQ , the inverse 10 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 e x i s t . However, according to t h i s model, the average age of galaxies i n the universe i s 1/3 H Q ~ , with the young 1 galaxies being much more numerous than the o l d ones. Present observations indicate that f a r more o l d galaxies e x i s t than are p r e d i c t e d by t h i s model. However, Hoyle and N a r l i k a r (1962) have pointed out that we may by chance be l i v i n g i n a region of o l d galaxies, so that the evidence cannot be - considered as d e c i s i v e . 5 - A .proper assessment must therefore wait u n t i l a good theory of galaxy formation i s a v a i l a b l e . The other important observations are the r e s i d u a l 3°K background r a d i a t i o n and the helium abundance. It i s 18 pointed out by the princeton group that i f the background r a d i a t i o n i s indeed degraded f i r e b a l l r a d i a t i o n , then a big-bang o r i g i n would seem p o s s i b l e . A s i m i l a r conclusion follows i f the p r i s t i n e helium abundance i s 30$ as stated 19 20 by Gamow "and Sandage because so much helium can appare n t l y only be made i n the f i r e b a l l immediately a f t e r the big-bang. 3. Q u a s i - S t e l l a r Objects Since the discovery of the f i r s t object i n 1960, quasi-stellar a considerable e f f o r t has been devoted both t h e o r e t i c a l l y and o b s e r v a t i o n a l l y to i n v e s t i g a t i n g t h e i r 21 properties. I t i s found that they are characterized by 22 ' a) Large r e d - s h i f t s b) The presence, in"some cases, of both emission and absorpt i o n l i n e s of almost the same r e d - s h i f t c) Very large energy output, with an u l t r a v i o l e t They are a hundred excess times more luminous than galax- i e s on average. d) V a r i a t i o n of both radio and o p t i c a l fluxes i n times of the order of years, months, or even days - 6 - e) Anisotropic d i s t r i b u t i o n i n the sky f ) A dearth of r e d - s h i f t s l a r g e r than 2. 23 Several p h y s i c a l models have been proposed to account f o r the enormous energy output. None of them has found general favour, f o r each accounts only i n a very schematic way f o r what i s seen, and i n i t i a l conditions from which the r e s u l t s are derived must be postulated. important The t h i n g i s that, i n a l l cases, i t i s necessary to postulate the existence 6f very large masses with s i z e s no bigger than about one l i g h t year. This i s required to account f o r the great size and r a p i d v a r i a t i o n of the energy output. The question a r i s e s whether the quasars are at cosmological distances or n o t ^ . I f they are, i t i s hard to believe that objects a hundred times more luminous than galaxies could be so small. I f the quasars are not at cosmological distances, t h e i r large r e d - s h i f t s cannot be due to the expansion of the universe. simply There must be large r e d - s h i f t s which are d i r e c t l y associated with the objects. The a l t e r n a t i v e explanations are that they are caused by the.recession of l o c a l objects at high speeds, by the presence of strong g r a v i t a t i o n a l f i e l d s , or that the values of the atomic 'constants' i n quasars are d i f f e r e n t from those these i n the rest of the universe. possibilities. We w i l l discuss a l l - 7 - i ) Local Doppler shift In order to account f o r the large r e d - s h i f t , recession at high speeds must be assumed. Woltjer S e t t i and (1966) assumed that the l i n e widths are due to large random motions of gas i n the s h e l l s of quasars and that the c e n t r a l mass must be large enough to s t a b i l i z e the object g r a v i t a t i o n a l l y . In t h i s way, 7 they obtained masses 8 of quasars i n the range 10' - 10 B a h c a l l , Peterson, and Schmidt M @ on the l o c a l hypothesis. (1966) estimated the mass of quasars on the l o c a l hypothesis by assuming that the absorpt i o n l i n e s 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 l o c a l o b j e c t s . However, i t i s c l e a r that t h e i r conclusions are based on assumptions which may not be true and consequently, they do not disprove the l o c a l hypothesis. ment i s the absence of b l u e - s h i f t s . A more convincing arguI f the quasars have been thrown out of many galaxies at high speeds, not only should r e d - s h i f t s be seen, but also b l u e - s h i f t s . Because objects moving toward the observer w i l l seem to be much b r i g h t e r than s i m i l a r objects moving away, b l u e - s h i f t e d objects should predominate parent brightness. i i ) Gravitational red-shift i n a survey down to a given ap- - 8 - A model of t h i s type has been suggested by Hoyle 27 and Fowler . The observed l i n e spectrum a r i s e s from gas clouds at the center of a massive object. I t i s possible to avoid the d i f f i c u l t i e s pointed out by Greenstein 28 with the energy requirements and Schmidt , but i t i s not c l e a r at pre- sent whether models can be made with large enough r e d - s h i f t s . 29 Bondi showed that a collapsed s t a t i c object with p l a u s i b l e properties such as adiabatic s t a b i l i t y could not have a g r a v i t a t i o n a l r e d - s h i f t 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 n u c l e i or electrons are d i f f e r e n t i n the quasars has not been taken s e r i o u s l y so f a r ^ ' ^ . 5 We now go back to the model i n which the r e d - s h i f t i s due to the expansion of the universe. There are several objections to t h i s model which have not been resolved so f a r . One which has already been mentioned i s the small size required to account f o r the. v a r i a t i o n i n f l u x . i s the presence of multiple, absorption l i n e s show that f o r 3C 191, Z *w rtTB Gm the absorption The second . Studies^ * _ ~ 1.95; t h i s i n d i c a t e s that 3.0S i s taking place i n the quasar and not i n the intervening medium. Studies of PKS 0237 - 23 show that there are at l e a s t f i v e absorption r e d - s h i f t s i n a d d i t i o n 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 a r i s e i n the objects themselves. If the l a t t e r i s true, then at l e a s t the differences between the absorption r e d - s h i f t s must be i n t r i n s i c to the objects, and the simple explanation that the r e d - s h i f t s are due to the expansion of the universe has to be modified. problems, most astronomers Despite these s t i l l take the view that the cosmological i n t e r p r e t a t i o n of quasars i s the most p l a u s i b l e one. In our work, we w i l l also take t h i s view. 4. The Present Program A. The Aim Ih section 3, we described the c h a r a c t e r i s t i c fea- tures of quasars and discussed the nature of the r e d - s h i f t , the p h y s i c a l models accounting f o r the energy output, the v a r i a t i o n i n f l u x , and the presence of m u l t i p l e absorption r e d - s h i f t s i n a d d i t i o n to an emission r e d - s h i f t . Despite the d i f f i c u l t i e s that we have mentioned, we s h a l l i n t e r p r e t the r e d - s h i f t of quasars as a cosmological e f f e c t — expansion of the universe. the I t i s our aim to construct a cosmological model which w i l l account f o r the following two p r o p e r t i e s of quasars. i ) The dearth of r e d - s h i f t s beyond Z = 2 ( r e f . 22). This e f f e c t could be a t t r i b u t e d 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 c e r t a i n age. However, we w i l l take the point of view that the e f f e c t i s due to a large-scale s p a t i a l inhomogeneity of the universe, i i ) The anisotropic d i s t r i b u t i o n of quasars i n the sky, p a r t i c u l a r l y of those with large r e d - s h i f t s . S t r i t t m a t t e r , Faulkner, and Walmsley (1966) 41 i n v e s t i g a t e d the d i s t r i b u t i o n 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 I I I with 1.0 > Z > 0.5, and Group IV with 0.5 > Z > 0. When they c a r r i e d out the a n a l y s i s , 67 quasars with known r e d - s h i f t were a v a i l a b l e . The complete d i s t r i b u t i o n of 100 quasars at the end of March 1967 i s given by Burbidge and Burbidge . 42 them. We r e p r i n t the f i g u r e ( F i g . 8-1) given by I t i s seen that a l l the objects i n Group I divide into two very compact and almost antipodal groups. The objects i n Group II are more widely spread i n the northern g a l a c t i c hemisphere, though s t i l l i n about the same p o s i t i o n as those of Group I, whereas the members of t h i s group that are i n the south are almost coincident with those of Group I. The members of Group I I I and Group IV are more widely spread over the sky. Possible s e l e c t i o n - 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 - e f f e c t s that may a f f e c t t h i s 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 p a r t i c l e s which exert negligible pressure. i i ) The dust p a r t i c l e s are d i s t r i b u t e d space with respect to a p a r t i c u l a r isotropically in origin. i i i ) The E i n s t e i n f i e l d equations with zero cosmological constant are v a l i d . i v ) The d i s t r i b u t i o n of dust p a r t i c l e s i s such that the model c o n s i s t s of an inner Friedmann zone and an outer Friedmann zone, separated by a t r a n s i t i o n zone. We define N(Z) to be the number of galaxies and quasars per steradian whose r e d - s h i f t s are smaller than Z. We w i l l c a l c u l a t e N(Z) f o r our model with the f u r t h e r assumpt i o n that the t r a n s i t i o n zone can be neglected. Chapter I I The F i e l d Equations It i s simplest to use a set of comoving coordinat From the assumption of s p h e r i c a l 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/ - T a - 7/ - , , i s the energy density. (We choose c = 1 . ) where The E i n s t e i n f i e l d equations are 67T j f = 0 = e - c ^4 + * +^ u KT,'.-»rr/...^-^.V' , if where the accents denote d i f f e r e n t i a t i o n with respect to r and the dots with respect to t . For l a t e r convenience, we write - 14 - To solve the f i e l d obtain equations, we f i r s t integrate ( 6 ) , and . (7) where 1 + f i s an undetermined function of r with p o s i t i v e values. S u b s t i t u t i n g (7) i n t o . ( 3 ) gives Z%% + % - j(r) - * . (8) A f i r s t i n t e g r a l of (8) i s f cr) + V<% (9) > where F i s a second undetermined function of r . A first i n t e g r a l of (9) i s j(f<r) + FC'jli,] « t - &< (10) r) ? where G i s a t h i r d undetermined function of r . Evaluating the i n t e g r a l i n ( 1 0 ) , we get f I (11-D - 15 - \% j'*** = i - fCtf , From (5)» t h e energy d e n s i t y f ^f - * , (11.3) c a n he e x p r e s s e d as -7' !^*' • (12) f The t h r e e unknown f u n c t i o n s o f r , f ( r ) , F ( r ) , and. G ( r ) w i l l be determined l a t e r from t h e i n i t i a l conditions. -.16 - Chapter I I I Friedmann Model Consider the case when there e x i s t s a f u n c t i o n S such that £ Cr,t) - r gd) (13) Combining ( 1 ) , ( 7 ) , and ( 1 3 ) , we can write the metric as (14) Equations ( 8 ) , ( 9 ) , and ( 1 0 ) become for) - ( a X i r + /rr ; r = ~- £ *S r(f CjCr) = t -H i',V r r* 3 , ( 1 5 f ( 1 5 + F r~0ct)J . _ . 1 } 2 ) ( .) 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 f o r S, S, and S. S ± = S(t ), i S ± = S(t ), ± V. Equation ( 1 2 ) becomes and s\ = We write S ( t ) f o r the i n i t i a l ± time - 17 - Since f and P are functions of r only, and S i s a f u n c t i o n ( 1 5 - 1 ) and (15-2) imply that of t only, equations 7,$$ S a. & -r $* « constant = - CL ~ 2 K ? =• constant •' (16) (17) where . a* - and ( ( / a w ^ i ' r^ - - , 4 ! / * i s an a r b i t r a r y constant , (18) (19) i f k = 0. Define q, the d e c e l e r a t i o n parameter, , the density parameter, and H, the Hubble parameter as f o l l o w s : HCi) ~ Hay^c*) , (20) %C*) * -fWfrty&^V (TCt) = 4j[>^ ftD/H^Ct) ^ % (21) (22) - 18 - (15-1), (15-2), and (15-4) then become j* f<v--yi.f - #*#\,-4 x y£ Fo; = f) , (23) ^ V / , (24) The metric can be rewritten as Making a coordinate transformation r = CL <r^(<*>) ) k (27 where Sink, cO we f i n d that (26) becomes 4s = $(l)Utb +%c*>)C<U +s>»*dj)J^(28) (28) i s the metric f o r the well-known Friedmann model f o r a uniform d i s t r i b u t i o n of dust. The universe i s c a l l e d -.19 open i f k = - 1 , - closed i f k = + 1 , and Euclidean i f k = 0. Now equations (11-1), (11-2), and (11-3) become, by the help of equations (15-1), (15-2), (15-3), (20). (21). and (22), ' * (29-3) 4 where the subscripts ' i * (29-1) r e f e r to the i n i t i a l epoch t ^ . Remark: According to (23), / « = j _ + / / ' ', if : r I \ > o. t • (30) - 20 - Chapter IV Red-Shift i n the Friedmann Model We r e l a t e our theory to observation by assuming that there i s a comoving coordinate system with the properties previously discussed, i n which galaxies are at r e s t , apart from small random motions which we w i l l ignore. This assumption i s consistent with the f a i r l y small s c a t t e r of o points i n the r e d - s h i f t - magnitude diagram . We consider l i g h t emitted by a source s and ob- served by an observer o, both s and o being s t a t i o n a r y i n the comoving coordinate system. o i s }*» The wavelength measured by , and the wavelength measured by a stationary, observer near the source i s X$ . We assume that none of the atomic parameters i s time-dependent. The r e d - s h i f t of a s p e c t r a l l i n e 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> ). i n (28). Light rays are described by s e t t i n g ds Considering only the r a d i a l incoming l i g h t rays, we have 7u7 =" - ^ • (32) =0 - 21 - Integration gives co - co - <U ^ (33) where c O and u> are the r a d i a l coordinates s a of source and observer, and t and t are the times of emission and s o reception r e s p e c t i v e l y . D i f f e r e n t i a t i n g (33) f o r constant two l i g h t s i g n a l s are emitted co - ^ s shows that i f 0 and t - + dt , s s s and received at times t ..and t„ + dt , the time d i f f e r e n c e s o o o satisfy dt oit 0 s at times t ' (34) The number of waves emitted between t and t s >4 <£t t Q s + dt i s s , and the number of waves received between t and o s + dt Q is . >4 cit 0 , where ^ and are the frequency of emission and reception r e s p e c t i v e l y . Since the number of waves must be conserved, i . e . , , >> cLi s = >> cL-k » w e h a v e (35) x C x \ C S is (36) - 22 - From (31) and (36), where we w r i t e S Q =-S(t ) and S = g S(t ). g F o r l a t e r use, we e l i m i n a t e t between (33) and (37) t o s o b t a i n a r e l a t i o n between o) - no and Z: g 5 - » , a -ML. = (38) where (39) Inserting (23),and (24) i n e q u a t i o n (9) g i v e s (40) We take p o s i t i v e v a l u e s f o r S ( t ) , because a c o n t r a c t i n g universe From i s i n c o n s i s t e n t w i t h the o b s e r v a t i o n s . (38) and (40), / . . I n t e g r a t i o n , with the h e l p o f e q u a t i o n ' Jt . ( 2 3 ) , we get Sty, *t- O+Z) f (41) Note that f o r an observer at the o r i g i n , - 24 - Chapter Y Number Count i n the Friedmann Model Consider an observer at the o r i g i n . The mass seen. _ i n a s p h e r i c a l s h e l l of coordinate radius co and thickness o£<a i s 0"(«>) « t o , iN=*+1CfCt ) s where t g ( 4 3 ) i s the time of emission of l i g h t signals by means of which the object i s observed at the present epoch t . o Note that f^t ) i s the proper mass density, and s f^ts^ (&t: £ ) )* i s "the mass density per unit coor- dinate volume. Let us write (44) However, from equations £ = - 4 (15-4) and (17), ! Therefore we w i l l write & f c & - Consent instead of . ( 4 5 ) J* (^sj c Integration of (43) gives the mass within a coordinate radius cd (46) - 25 - E v a l u a t i n g the i n t e g r a l 2%f i n (46) g i v e s [-^swi^ti t -u>J -/ (47) 3 Now, of J c by use o f e q u a t i o n ( 4 2 ) , we can express N as a f u n c t i o n Z, and we have a r e l a t i o n between the two observed quantities, N and Z. directly - 26 - Chapter VI A New Model Universe Let us go back to the coordinate system ( "t, r, Q , J ), and consider a model i n which the metric i s given by x o<r < r % <£s = dt - & H)^ —~0* where r b > r & (48-1) + T (<Ld + sini dyjj. A T h < T , (48-2) T and the subscripts "1" and. "2" stand f o r the regions. o< r < r & and r ^ < r r e s p e c t i v e l y . 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 t r a n s i t i o n zone with a metric such that the boundary conditions at r = r & and r = r ^ are s a t 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 i n the t r a n s i t i o n zone. to equations Moreover, according (1), (7), (8), (9), and (10), motion of the - 27 - dust i n each Friedmann zone i s independent o f the b e h a v i o r of the o t h e r p a r t s o f the model. We w i l l assume, as the i n i t i a l c o n d i t i o n s of our model, t h a t £,L = (49) & i or e q u i v a l e n t l y (50-1 ) (50-2) X K > where the s u b s c r i p t (50-3) " i " r e f e r s to the v a l u e a t the i n i t i a l epoch t ^ . The c o n d i t i o n (50-3) means t h a t the i n i t i a l value of pro- per mass d e n s i t y o f the o u t e r Friedmann zone i s e q u a l o r l e s s than t h a t o f the i n n e r Friedmann zone. Thus we have the p o s s i b l e model u n i v e r s e s l i s t e d i n Table I . - TABLE I 28 - Possible Model Universes Model. k . Model type of the kp. Model type of the inner Friedmann zone. outer Friedmann zone 1 open -1 open Euclidean -1 0 open Euclidean +1 closed -1 open IV +1 closed V +1 closed I -1 II 0 III 0 +1 Euclidean closed We note that i n the coordinate system ( t,to , 9 , f ), (48-1 ) and (48-2) become /OK , ± 4 <CO A J ( 5 1 .i. _> s for- where O < ed, c* h < (51 - Chapter VII 29 - Number - Red-Shift R e l a t i o n i n the New Model. We w i l l consider two cases: (A) the r e l a t i o n observed from the o r i g i n , (B) the r e l a t i o n observed from a point i n the inner Friedmann zone half-way between the o r i g i n and the boundary of the inner Friedmann zone., Note that the r e l a t i o n applies to quasars under the assumpt i o n that the r a t i o of the number of quasars to the t o t a l number of galaxies i s a constant. (A) The N - Z r e l a t i o n observed from the o r i g i n The expressions derived i n chapter IV and V are true f o r each Friedmann zone, but we w i l l have to make some extra assumptions about the. t r a n s i t i o n zone. ( i ) For a source i n the inner Friedmann zone, the a>— Z r e l a t i o n s are where Z i s the r e d - s h i f t of l i g h t signals emitted at r ci and received at the present epoch at the o r i g i n , the subscript "o" r e f e r s to the present epoch and we have Z < z a i - 30 - ( i i ) For a source i n the outer Friedmann zone, the relations co ~ Z are f" 1 4 V «a (53) I fat fa where t ^ i s the epoch such that l i g h t emitted at t ^ and at r = r ^ w i l l be received at t u>£ are the r e d - s h i f t at the o r i g i n . Z' and and r a d i a l coordinate of the source f o r an observer at r = r ^ and at the epoch t ^ . Let Z^ be the r e d - s h i f t at r = r f e of l i g h t signals emitted and received by an observer at the o r i g i n at the present epoch. The epoch at which these l i g h t signals emitted i s then t ^ . Now are f o r a source i n the outer F r i e d - mann zone, we can imagine the s i t u a t i o n that the signals are emitted with wavelength As and received by an observer at r = r signals b at the epoch t . f e have a wavelength A. The l i g h t signals According to t h i s observer, the and a r e d - s h i f t are then immediately observer at r = r^» & — /% ~ ' • re-emitted by the and reach the observer at the o r i g i n at the present epoch with wavelength X . . By definition, - ^ = / 31 - , and the t o t a l r e d - s h i f t i s (54) The coordinate distance of the source as c a l c u l a t e d by the observer at the o r i g i n i s (55) By means of ( 5 4 ) and ( 5 5 ) , equations ( 5 3 - 1 ) , (53-3) (53-2), and are r e w r i t t e n as CO, - 6 i , zs;»~'/2ht±„ zafate-OO+za if KM+I (56) vo The N — eo r e l a t i o n s are modified to »A7r]*,e f o r r < r , where T - ()£ shim, ^ - f «a, ) , W « £ K j ^ ( t^) . - and 32 - to (58) (59) and takes account of the e f f e c t As a f i r s t transition the of the t r a n s i t i o n a p p r o x i m a t i o n , we assume t h a t zone i s s m a l l and n e g l i g i b l e . T h e n we zone. the have relations: r « r. (60) and N 3 = ( B ) The 0. N — Z relation Friedmann boundary zone h a l f - w a y between of the i n n e r Friedmann As tropic observed from a p o i n t i n the t h e o r i g i n and as inhomogeneous, the zone 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 as w e l l inner i s aniso- and t h e number c o u n t i s a - 33 - function of distance ( or r e d - s h i f t ) and d i r e c t i o n . For s i m p l i c i t y , we consider only the r a d i a l d i r e c t i o n s ( r a d i a l with respect to the o r i g i n ). The W-direction i s the one that passes through the o r i g i n and the V - d i r e c t i o n i s the opposite one, as shown i n the diagram. * * observer o]*ewer A B X. M * ;> * In c a l c u l a t i n g the N — Z r e l a t i o n , we note that we are c a l c u l a t i n g the number of objects per u n i t s o l i d as a f u n c t i o n of Z i n the W- and V - d i r e c t i o n s . angle Because of the d i f f e r e n c e i n density between the inner and outer Friedmann zones, l i g h t rays that cross the boundary i n other than r a d i a l d i r e c t i o n s w i l l be r e f r a c t e d . This means that a bundle of l i g h t rays that subtend a s o l i d angle 'ii- as observed i n the inner. Friedmann zone w i l l subtend a d i f f e r e n t s o l i d angle i n the outer Friedmann zone. However, t h i s e f f e c t i s very small with the density that we may reasonably fio/ f d expect. As w i l l be seen i n Tables I I - VI, i s greater than 6.4x10 have considered. Since /% v i n a l l cases that we 2.0x10" g/cnr and J V ^ -Pi » - 34 - the density i s smaller than 3.1x10 -11 of g/cm at a l l points space-time. The equations (52) — (60) apply f o r both d i r e c t i o n s , but we must remember that the coordinate distance from the observer to the boundary of the inner Friedmann zone i n the W-direction i s three times that i n the V - d i r e c t i o n . This provides a r e l a t i o n between the boundary values of the r e d - s h i f t Z„ i n the two from equation (52). d i r e c t i o n s , which can be c a l c u l a t e d In the f i r s t approximation, where we neglect the t r a n s i t i o n zone, we have the r e l a t i o n s : H 3 - « (61 ) where the s u p e r s c r i p t s (V) and W-directions r e s p e c t i v e l y . (W) r e f e r to the V- and - 35 - Chapter VIII Numerical Calculation As noted i n the introduction, we are t r y i n g to f i n d values of the parameters H and q that f i t the observed d i s t r i b u t i o n of quasars. The f i r s t approximation i n which the t r a n s i t i o n zone i s neglected, w i l l be used i n the lowing numerical work. (60) and fol- Thus we have the r e l a t i o n s given by (61), i n a d d i t i o n to the basic assumptions of the model as given by equation (50). For an observer at the o r i g i n , the value Z, = Z b a w i l l be chosen as 2.0. ( As remarked e a r l i e r , very few quasars are observed with r e d - s h i f t l a r g e r than 2.0. ) For an observer half-way between the o r i g i n 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 r e l a t i o n i n t h i s d i r e c t i o n as does an observer at the o r i g i n i n the other case. V - d i r e c t i o n w i l l then be c a l c u l a t e d from The value i n the (61) and (50). A l l the f i v e model universes given i n Table I w i l l be considered. - 36 A. Procedure to obtain the N — Z r e l a t i o n 1. Determine the value of q » 1o q.j - can be determined the Hubble constant H^ Q "the deceleration parameter. from the observed values of 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 i n the values of 10 both q^ 0 and H^ . + 1 .2 * 0.4. We may Q take the value of q^ as However, as remarked i n the i n t r o d u c t i o n , t h i s value i s very uncertain. In our c a l c u l a t i o n , we will therefore take values of q^ ranging from + 0.025 to + 2.00. The value of H., i s not used i n the c a l c u l a t i o n of the 1o N — Z r e l a t i o n . ( The value of H. i s i n fact between 1o q q 7.5x10 and 19.5x10 years, as r e c e n t l y reported by 10 Sandage . ) Q 2. C a l c u l a t e ///* ~?t I from the equations y 0 - 37 - where Zt i s "the time taken f o r l i g h t to t r a v e l from r to a the observer and also the time taken f o r l i g h t to t r a v e l from r ^ to the observer i n the f i r s t approximation-.have T"i ~ "^o " . lows: We Equation (62) i s derived as f o l - , 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 together with ^/£ )L the models I and I I I , we w i l l take ^/f^ - 0.1, = 1 ( single Friedmann model ) and = 0 ( no matter i n the outer Friedmann zone ). For the model I I , we have q^ = q^ model i s described by parameters H^/.H^ For parameter is 2± q 1± = 0.5/q . ; 1jL The and ^ / j ^ c • the model IV, we have qg^ - <i2± ^/Ju = q / = 0.5. = 0.5. The only - 38 - F o r the model V, s i n c e 2.0 £ ^y>,i = q i 2 / q 1i' w e 4. C a l c u l a t e h\j Then H., Q ( t & h a v e ( t - ^ 1 / * ^ 4 - t L < ) from ± 1 > ^ > 0.5 and * equations (29-1,2,3). ) i s g i v e n by H,o < < - t.) = H,o C*. - V - H l% r _ 4 5. F o r models I, I I I , IV, and V, c a l c u l a t e H . 0 ci from T h i s e q u a t i o n i s o b t a i n e d by e q u a t i n g e q u a t i o n s ( t - t. ) a i 0 (23) and (24). F o r model I I , 6. F o r models I , I I , I I I , and V, c a l c u l a t e q equations 2 b from the (29-1,2). F o r model IV, c a l c u l a t e H 7. C a l c u l a t e a^/a.^ 2 i /H 2 b from (29-3). " f o r models I , I I I , and V from the equa- tion \s V which i s o b t a i n e d from equation (23). - 39 F o r models I I and IV, s i n c e we - are f r e e to choose the v a l u e o f a^ i n a Friedmann zone w i t h k = 0, we can put a^ya^ = T h i s i s e q u i v a l e n t o n l y to a l i n e a r change o f the s c a l e 1. of the r a d i a l c o o r d i n a t e . 8. F o r models I, I I I , IV, and V, calculate T h i s f o r m u l a i s o b t a i n e d by e q u a t i n g e q u a t i o n s (23) and F o r model I I , 9. C a l c u l a t e - % ^ c from 7?* " 10. C a l c u l a t e (61). J?c the e q u a t i o n ~ Ti <£>W and /V(^>CZ)^ c (44): £7> from • equations (52) to (24) - 40 - B. R e s u l t s The of numerical results f o r our.model w i t h neglect t h e t r a n s i t i o n z o n e a r e shown i n T a b l e s I I - V I a n d Figures 2 - 11. are In the tables, t h e m o d e l s w it h simple Friedmann models and those w i t h J M.c = f/J\t = 0 a r e m o d e l s 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 z o n e . the figures, considered S i n c e we two u n i v e r s e s 1 such t h a t an o b s e r v e r In have half-way 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 Friedmann zone i n t h e s e c o n d u n i v e r s e in s e e s t h e same N — Z r e l a t i o 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 first, figures the z! ) a W we p l o t the N — Z r e l a t i o n s ( t h e e v e n numbered boundary value Z^ ^ V ones = 2.00.. The v a l u e s o f z{ ^ a Y direction We n o t e t h a t the ). i n t h e same In the V-direction, i s determined by t h e c h o i c e o f shown i n c o l u m n 3 o f t h e t a b l e s . this f o r them are plotted depend on q Mo r 1 and a r e The N - Z r e l a t i o n s i n i n t h e odd numbered the curves i n the figures figures. and t h e models i n 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 m o d e l numbers. - 41 - The Quantities L i s t e d On The' Tables &\o : The present value of the deceleration parameter of the inner Friedmann zone. Z : The maximum r e d - s h i f t of the objects i n the inner Friedmann zone. For an observer at the o r i g i n , 2 ^ = . For an observer half-way between the o r i g i n and the boundary of the inner Friedmann zone, the value i n (ur) the W-clirection i s also taken as Z & : The maximum value 2.^ = 2.0 . i n the V - d i r e c t i o n . i s determined from c?iL : The. i n i t i a l value of the deceleration parameter of the inner Friedmann zone. ?aj : The i n i t i a l value of the deceleration parameter of the outer Friedmann zone. The r a t i o of the mass density per u n i t coordinate volume of the outer Friedmann zone to that of.the inner Friedmann zone. ^yp^ > The r a t i o of the present proper density to the i n i t i a l proper density i n the inner Friedmann zone. - 42 - This i s the r a t i o of the number of galaxies (quasars) seen by an observer at the o r i g i n , with red s h i f t s between 2.00 and 3.00, to the number with r e d - s h i f t s smaller than 2.00.~ This i s the r a t i o of the number of galaxies (quasars) seen by an observer half-way between the o r i g i n and the boundary of the inner Friedmann zone i n the V - d i r e c t i o n , with r e d - s h i f t s between 1.50 and 2.00, to the same quantity i n the W-direction. 43 - Table II Model I k 1 = 1, k t,i 2 = - 1 -12 1 . .025 2.00 .435 .499 .0499 1.02x10 -5 1 .17x10 2. .025 2.00 .435 .450 .0450 3.63x10 -3 2.00x10 -7 3. .025 2.00 .435 .300 .0300 2.77x10 -2 4.32x10 -5 4. .025 2.00 1 .00 5. .025 2.00 0.00 6. .100 2.00 .415 .499 .0499 1 .02x10 -5 1.26x10 7. .100 2.00 .415 " .450 .0450 3.63x10 -3 2.1 4x10 -5 8. .100 2.00 .41 5 .300 .0300 2.77x10 -2 4.63x10 -3 9. .100 2.00 1 .00 10. .100 2.00 0.00 11 . .400 2.00 .366 .499 .0499 1 .02x10-5 5.1 5x10 -7 12. .400 2.00 .366 .490 .0490 3.31x10 -4 5.44x10 -4 13. .400 2.00 1 .00 14. .400 2.00 0.00 -10 Table I I X (continued) i 1. .0057 .0032 2. .043 .018 3. .086 .062 6. .0064 .0039 7. .049 .028 8. .11 .095 11 . .01 4 .0080 12. .048 .038 4. 5. 9. 10. 13. 45 - Table I I I Model I I k., = 0, 4" trio k 2 = - tax 1. .500 2.00 .360 .500 .499 .998 6.40x10 -17 2. .500 2.00 .360 .500 .400 .800 6 .40x10 -17 3. .500 2.00 .360 .500 .010 .020 6 .40x10 -17 4. .500 2.00 .500 .500 1 .00 5. .500 2.00 6. .500 2.00 .360 .500 .499 .998 6 .40x10 -11 7. .500 2.00 .360 .500 .400 .800 6 .40x10 -11 8. .500 2.00 .360 .500 .010 .020 6 .40x10 -11 9. .500 2.00 .500 .500 1 .00 10. .500 2.00 11 . ' .500 2.00 .360 .500 .499 .998 6 .40x10 -5 12. .500 2.00 .360 .500 .400 .800 6 .40x10 -5 13. .500 2.00 .360 .500 .010 .020 6 .40x10 -5 14. .500 2.00 .500 .500 1 .00 15. .500 2.00 0.00 0.00 0.00 Table I I I (continued) 2 1. .063 .035 2. .0050 .0031 3. .0000 .0000 6. .65 .63 7. .050 .032 8. .0006 .0001 11.- 4.3 1 90 12. .23 1 .8 13. .0026 .0020 4. 5. 9. 10. 14. - 47 - T a b l e IV Model I I I k 2 = - 1 1. .600 2.00 .154 501 .0501 1.02x10~ 2. .600 2.00 .154 510 .0510 3.31x10" 3. .600 2.00 1 .00 4. .600 2.00 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 10. 2.00 2.00 .118 .600 .0600 5 4 1.02x10 -5 -2 1.08x10 11 2.00 2 .00 1.00 12 2.00 2 .00 0.00 1.72x10 -6 1.63x10~ 3 6.36x10 -8 1.10x10' T a b l e IV * (continued) 3 1. .014 .012 2. .049 .055 5. .0067 .0075 6. .056 .084 .0043 .0070 .068 .11 3. 4. 7. 8. 9. 10. 11 . 12. - 49 - Table V Model IV L ^ = + 1, z^-zt & k 2 = 0 'V* 1. .600 2.00 .154 .501 .500 .998 1 .72x10" 2. .600 2.00 .154 .510 .500 .980 1 .63x10" 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 • 8. 1 .00 2.00 9. 2 .00 2.00 .118 .501 10. 2.00 2.00 .118 .600 . .500 11 . 2.00 2.00 1 .00 12. 2.00 0.00 2.00 b 3 1 .00 0.00 1 .500 .998 . 1.88x10" .833 8 1 .10x10" 2 Table V (continued) x 1 1. .60 1 .01 2. .60 1.01 5. .50 1.00 6. .50 1.00 9. .39 .99 10. .35 .98 3. 4. 7. 8. 11 . 12. - 51 - T a b l e VI Model V k 1 = + 1 , k = +1 2 1. .600 2.00 2. .600 2.00 1 .00 3. .600 2.00 0.00 4. 1.00 2.00 5. 1.00 2.00 1 .00 6. 1.00 2.00 0.00 7. 2.00 2.00 8. 2.00 2.00 1 .00 9. 2.00 2.00 0.00 .154 .138 .118 .510 .510 .510 .501 .501 .501 31.1 1.63x10" 31.1 .6.03x10"' 31.1 1.78x10~* T a b l e VI 1. (continued) .53 1 .38 .50 1.30 .40 1.19 2. 3. 4. 5. 6. 7. 8. 9. - 56 - T" or 1 'V. H 1 s I - 61 - - 63 - Column 8 of the tables gives the r a t i o of the number of quasars with r e d - s h i f t s between 2.00 the number with- r e d - s h i f t s smaller than 2.00 observer at the o r i g i n . l a r g e l y on the values o f f o r a f i x e d value of fix J\v and 3.00_to as seen by an I t i s seen that the r a t i o depends s i . 0 / s - | i ' o r e ( l i u v a l e n t l y of This allows us i n p r i n c i p l e to f o r our model by comparison with observation.. However, at present, there are only 136 quasars with known . r e d - s h i f t s , and i t i s hard to determine the proper value Z a of the r e d - s h i f t at the boundary. we could perhaps choose Z I f we had more quasars, i n a non-arbitrary fashion as the value of Z f o r which the slope of the N — Z curve change suddenly. But we cannot draw any s i g n i f i c a n t conclusions from only 136 quasars. I f we assume that the presently observed d i s t r i b u t i o n of quasars i s c o r r e c t , then the above described r a t i o i s about 0.1, and we may exclude those mo- dels i n which the r a t i o d i f f e r s too much from 0.1 . In Chapter I, we discussed how are d i s t r i b u t e d over the sky. the known quasars I t i s c l e a r that i f the s i - t u a t i o n remains unchanged as more observational data are accumulated, then our simple model, c o n s i s t i n g of an inner Friedmann zone and an outer Friedmann zone, must be wrong. In our model, an observer at the o r i g i n sees an - 64 - i s o t r o p i c d i s t r i b u t i o n ; while the observed quasars with Z > 1.5 d i s t r i b u t i o n of i s anisotropic. For an observer half-way between the o r i g i n and the boundary of the inner Friedmann zone, the d i s t r i b u t i o n i s anisotropic but i n the wrong way. l a r g e s t Z are concentrated The quasars with the i n the W-direction instead of i n two almost antipodal d i r e c t i o n s as i s observed. (See F i g . 1) Column 9 of the tables shows the c a l c u l a t e d r a t i o y of the number of quasars with r e d - s h i f t s between 1 .50 2.00 and i n the V - d i r e c t i o n to the number i n the W-direction. It i s possible to divide the sky into sixteen s i m i l a r regions, each subtending a s o l i d angle of 3% » in such a manner that a l l known quasars with r e d - s h i f t s greater than 1 .50 l i e i n just two of the regions. are a n t i p o d a l . One These two regions of them contains 1 4 of the quasars with r e d - s h i f t s greater than 1.50, and the other contains 7. We i d e n t i f y the d i r e c t i o n 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 - d i r e c t i o n . Assume that at any epoch t and i n any region at a distance r from the center of the universe, the p r o b a b i l i t y of a galaxy being a quasar i s p ( r , t ) . p ( r , t ) on our observed We have to determine past light-cones f o r r 1 c < r< r ? n , - 65 - where r^ ^ and Q correspond to distances f o r which Z = 1 .5 and Z = 2.0 r e s p e c t i v e l y . As a rough approximation, we assume that p ( r , t ) i s constant i n t h i s region and equal to the r a t i o i n t h i s region of the t o t a l number of quasars to the t o t a l number of g a l a x i e s . event TE We w i l l c a l c u l a t e the p r o b a b i l i t y of the that the r a t i o of the number of quasars with r e d - s h i f t s 1.5< Z < 2.0 i n the V - d i r e c t i o n to the number i n the W-direction i s greater than 1/2. Consider f i r s t the case where the observer i s h a l f way between the o r i g i n and the boundary of the inner Friedmann zone. i ) From the assumptions of our model, and by comparing our model with observations, we f i n d the N — Z r e l a t i o n i n the inner Friedmann zone. s o l i d angle of r„ n 1.5 < r < r Then N , the number of galaxies i n a w i n the W-direction at distances ~, i s c a l c u l a t e d to be about 5x10 c. .U 0 corresponding number N y 9 , and the i n the V - d i r e c t i o n i s yN , where w y i s given i n column 9 of tables II — VI. • i i ) To determine two cases. the constant value p.of p ( r , t ) , we consider In the f i r s t case, we take account of the r a d i o - quiet quasars and have p •~ 1 .0x10" . 4 In the second case, only the observed quasars with known r e d - s h i f t are taken —9 into account and p/-v 4.0x10 . - 66 - 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"^, N p and N p w are large, v the Le Moivre Laplace approximation can be made. It i s found that f o r a l l the cases we have considered, fit, < ^ . _q In the second case where p = 4.0x10 , both N p w and N p are small, the Poisson d i s t r i b u t i o n can be used. y The p r o b a b i l i t y becomes * s . rt In the numerical c a l c u l a t i o n f o r Pg, we exclude the cases where y i s greater than 0.5, f o r they are accompanied by - 67 - large values of x and contradict the dearth of quasars with r e d - s h i f t greater than 2.00. The p r o b a b i l i t y i s again very small. y4 For the case where y = 0.11 (the l a r g e s t value of 0.5 that we have considered), P <>/ 1 .37x10 2 and P 2 decreases very r a p i d l y as y decreases. For the universe with the observer at the o r i g i n , consider the event J that the r a t i o of the number o f IT/ quasars i n an a r b i t r a r y d i r e c t i o n , i n a s o l i d angle of / i * , with r e d - s h i f t s 1.5 < Z < 2.0, to the number i n any other . d i r e c t i o n i s greater than 1/2. Since the number of galaxies i s the same i n a l l the d i r e c t i o n s , the p r o b a b i l i t y of the event ^ i- s "very small by the same argument as given above f o r the other universe. This means that the p r o b a b i l i t y of f i n d i n g that the quasars with r e d - s h i f t s 1 . 5 < Z < 2.0 are concentrated i n two almost antipodal groups i s very small. - Chapter IX It 68 - Conclusion i s seen from the n u m e r i c a l r e s u l t s that model can s u c c e s s f u l l y account f o r the d e a r t h of with red-shift initial l a r g e r t h a n 2.0 distributed An quasars choose v a l u e of the d e c e l e r a t i o n parameter, d e s c r i b e t h e way a suitable but i t does n o t i n which quasars of l a r g e s t r e d - s h i f t i n the are sky. obvious m o d i f i c a t i o n i s to c o n s i d e r a model i n which the i n n e r Friedmann into p r o v i d e d we our an e l l i p s o i d . zone i s deformed A schematic diagram from a i s plotted sphere as follows: 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 and the dashed curve corresponds to a r e d - s h i f t ellipsoid of Z = 2.0. - 69 - An a l t e r n a t i v e i s to consider an universe i n which there are inner Friedmann zones d i s t r i b u t e d randomly i n an outer Friedmann zone as suggested by Rees and Sciama ^' ^. 4 4 We may or may not be l i v i n g i n one of the inner;Friedmann zones. A schematic diagram i s as f o l l o w s : It i s assumed that by chance, there are two zones i n the W- and V - d i r e c t i o n s with distances corresponding to a r e d - s h i f t of Z = 2.0. The p o s s i b i l i t y of formation of such l a r g e - s c a l e 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. A s t r . S o c , 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, P r o c Lond. Math. S o c , (2), 42, 90. 7. De S i t t e r , W., 1917, Mon. Not. R. A s t r . S o c , 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 N a r l i k a r , J.V., 1966, Reports on Progr. i n Phys., V. 29, p t . 2, 594. 12. Hoyle, F., 1968, Proc. Roy. S o c , Ser. A, 308, 1. 13. Ryle, M., and Clarke, R.W.,.1961, Mon. Not. R. A s t r . S o c , 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 N a r l i k a r , J.V., 1962, Observatory, 82, 13. 18. Partridge, R.B., 1969., American S c i e n t i s t , 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, Q u a s i - S t e l l a r Objects, Freeman and Company, P. 10. 22. Burbidge, G.R., and Burbidge, E.M.', 1969, Nature, 224, Oct. 4, 21. 23. c . f . 21, P. 188. 24. Hoyle, F., Burbidge, G.R., and Sargent, W.L.W., 1966, Nature, 209, 751. 25. S e t t i , G., and Woltjer, I., 1966, Ap. J . , 1_44, 838. 26. B a h c a l l , 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. S o c , 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., Ap. J . , 1_44, 447. and Stockton, A.N., 1966, - 72 - 34. S t o c k t o n , A, N., and l o y d s , 35. B u r b i d g e , E.M., Loyds, C.R., Ap. 36. B a h c a l l , J . , 1_52, J.N., 1968, Ap. 40. B a h c a l l , Ap. J.N., R.T., J . L e t t . , 156, Nature, 1969, 1969, 7. and S p i t z e r , L., 41. S t r i t t m a t t e r , P.A., 1969, 1047. and P e e b l e s , P.J.E., J.N., 1969, L. 63.. F a u l k n e r , J . , and Walmsley, 21_2, M., 1441. 90. 43. Penston, M.V., and Robinson, Nature, 21_3, 44. Tolraan, R.C., 1968, 735. J . 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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. |
DOI | 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 |
Aggregated Source Repository | DSpace |
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