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UBC Theses and Dissertations

New cosmological model Shen, Po-Yu 1970

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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. 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