R E L A T I V I S T I C H I E R A R C H I C A L C O S M O L O G Y

II: Some Classes o f Mode l Universes

P A U L S. WESSON St. John's College, and Institute of Astronomy, Cambridge University, Cambridge, England

(Received 29 May, 1974)

Abstract. Following the re-expression of the metric for a hierarchical cosmology in Section 1 (Intro- duction), the metric curvature is considered in Section 2 in showing that the hierarchy models are not, in general, related to the Robertson/Walker models even though the metrics can be made to appear superficially similar. The classes of models examined are quasi-Robertson/Walker models with A =0 (Section 3), quasi-Robertson/Walker models with A C0 (Section 4), constant curvature (k(t)= constant) models (Section 5) and zero-curvature (k(t)=0) models (Section 6). The first group have Qo(t)ozt -3, S(t)oct a in a suitable limit; the second group have solutions expressible as incomplete elliptic integrals of the first and third kind; the next group (k(t)= const.) includes an oscillating model with 0o(t)~[sin (t/2T)] -z, S(t)oz[sin (t/2T)] 2/3 and a model with k(t)ocA (~OoCZ;t-2), and a model with k(t) ocA --/~2/3 (0o cce at, fl< 0); the last group comprises three models, one having A =0 (0ooz t-2), one having A = --}T 2 where 2~zT is the bounce period of the (oscillating) model, and an analogue of the de Sitter Universe with Qo(t)oce - ' ~ t and S(t)oce -'/(--~7~)̀ . Section 7 contains our conclusions.

1. Introduction

I t has been shown in Par t I that a plausible hierarchical cosmology can be represented

by the metr ic I(18), which is o f R o b e r t s o n / W a l k e r fo rm and can be rewri t ten with the

pa rame te r Ro(t) replaced by S(t) as

S2(t) ds2 = dt2 - ,_,,2,~2C2 [dR'- + R 2 d0 2 + R 2 sin 2 0 d~b2], (1) (1+

where

and

20"o(t) (Oo( t ) ] 2 k( t ) =- A + 3flo(t-----~ - \O~o(t)] (2)

s ( t ) -- [;,co(t)] (3)

The behav iour o f the dus t with t ime thus determines the effective scale fac tor S'( t ) as

the term outs ide the b racke t in (1), which reduces in cer tain cases to S(t ) of (3). The

curvature will, in general , a lso change as the dus t evolves, according to (2). Cer ta in

classes o f mode l Universes with metr ic (1) are pa r t i cu la r ly wor thy o f s tudy: quasi-

R o b e r t s o n / W a l k e r models have k ( t )S2 ( t )=cons tan t and mer i t a t ten t ion because in

this case the scale fac tor and Hubb le pa r ame te r etc. become par t i cu la r ly s imple in

fo rm (S ' ( t ) -> S(t) , H = S/S). The Q R W cases with A = 0 are much easier to t rea t than

Astrophysics and Space Science 32 (1975) 305-314. All Rights Reserved Copyright �9 1975 by D. Reidel Publishing Company, Dordrecht-Holland

3 0 6 PAULS. W~SON

those with A r and different approaches are needed to examine either case. These models, it will be shown below in Section 2, are not generally equivalent to the conventional Robertson/Walker models. This is seen by considering the curvature of the metric, and leads to the study of constant-curvature solutions (k(t)=constant). Models with zero curvature (k(t)=O) are particularly simple cases of all classes. It should be emphasized, however, that while one can find justification for studying QRW, constant-curvature and flat solutions, there is, in hierarchical cosmology, no reason to think a~priori that the Universe might be describable by any one such model: from another point of view, the simplest models (Section 13) are those with dust density evolving according to ~o(t) oc t - q, where q is a constant. Except for the case q = 2 (when k(t)=A), these models do not lead to any particularly simple formulation as regards metric. Bearing this in mind, I will first evaluate the space-curvature of the hierarchical metric and then proceed to examine QRW (A = 0), QRW (A r 0), constant- curvature solutions and, lastly, flat solutions.

2. The Metric Curvature

If K denotes the curvature scalar of a given space-time, it is known (Weinberg, 1972, p. 381) that two maximally symmetric metrics which have the same K and the same number of eigenvalues of each sign are connected by a co-ordinate transformation (i.e. it is possible to go over from one to another using only a co-ordinate trans- formation: this means that two such metrics are essentially the same). The metric (1) has maximally symmetric subspaces, and this can be used to show that in general (1) is not equivalent to the Robertson/Walker metric even though it looks superficially similar. With the metric in the form of 1(16), the three-dimensional curvature scalar (3/s is compounded of the term outside the bracket and the coefficient of dr 2, so that the hierarchy described by (1) and I(16) is

3K = k/R~ oc k, (4)

i.e.,

2 o(t) [0o(t)] 3K oc A + 3~Oo(t ~ - [Q-g~] �9 (5)

The corresponding 3K for Robertson/Walker models is kS-2(t), where k here means k = + 1, 0 and S(t) is given by the Friedmann equations. Since, in general, 3K of (5) is not going to behave like S-2(t) of a Robertson/Walker model, the two metrics are not equivalent.

3. QRW (A = 0) Models

These models, if they exist, are solutions of the equation

+ 2fro(t) ~bo(t) _ G, (6) Vg/a(t) 3~oS/a(t) ~o*/3(t)

RELATIVISTIC HIERARCHICAL COSMOLOGY~ II -q07

where G is a constant. One can check by resubstitution that an implicit solution of (6) in the variable z(t) ~ 4o 1/3(t) is

o 8 ~ s i n - -

where D and F are constants of integration, and I have not been able to find an explicit solution of (6). In practical work, a useful relation obeyed by z(t) is found to be

z2 = (Dz -- 16Gz2) I/2 (8) 2

The constant G of (7), (8) is not arbitrary but is determined by physical considerations. The constant F can be found to be F=3~D/16G 3/z by taking the condition that ao---> oo as z--~ 0: this, in the limit, amounts to saying that there was something similar to a big-bang in the past history of the Universe and is a restatement of assumption I(iii) on which the metric is based. The solutio~ with this assumption taken is

8Gz(t)D - 1 + s i n [ ~ + _.___ff_SGt/2 (_D_~__ Gz2)~/2+ ~ ] . (9)

The constants G and D are seen to be related, and so connected to the physics of the model, by the realization that the solution becomes unphysical when 4G/D=o~/3. This means that the density cannot decrease indefinitely with time unless G is very small or D is very large: i.e., ~o o -+ 0 as the Universe evolves needs an asymptotically flat geometry, since G is proportional to k(t). Put another way: QRW Universes with finite curvature cannot become indefinitely ratified.

The condition that G/D be very small enables (9) to be simplified to

8Gz(t) [ ~ 8G,/2 (~__fz )2/2] D - 1 - c o s + T - GZ2 " ( 1 0 )

This relation is accurate to an arbitrarily high degree depending on how small 00(t) ultimately becomes. Since (6) can be rewritten in terms of t /=00 2/3 and ~ - l f l / x (dr//dt) 2 as

dE r/~-~ + ~ = 4(At/ - G), (11)

it is feasible to iterate and so obtain a soIution a~alogous to (10) with A#0 . This would proceed by solving (10) for z=rl I/2, using this to cMculate ~, and substituting into (11) to derive a new solution for z=z(t, G, A) or r/=r/(t, G, A). Such an iteration might be more profitable in working out practical aspects of QRW solutions with A # 0 than the expressions found in the next Section, with latter would require the evaluation of incomplete elliptic integrals of the first and third kind. A more amenable solution for A = 0 can be obtained from (10) by taking [G/D]~I and choosing the

308 PAUL S. WESSON

constant ~ of I(16) to allow G to take values consistent with IGz/DI ,~ 1. In this case, one finds that

( D ; } Oo(t) ~ ~ t -~, (12)

S2( t ) oc t ~.

These simple results show that S2(t)Oo(t)oct-1, which is consistent with the charac- teristic property of all QRW solutions that Oo2/3(t)S2(t)=constant (see Figure 2). The Hubble parameter for QRW (A=0) models, by I(21), is H=-Oo(t)/3o~o(t)= +2(t)/z(t), so that H ~_ 1/t. Thus, the inverse of the Hubble parameter gives a direct measure of the age of the Universe in this model.

4. QRW (A # 0) Models

The solution of (6) for A r 0 is considerably more complicated than, and requires a different approach to that of the A = 0 case. When A ~0, in place of Equation (8) in the variable z - 6 o 1/3 one has

2z~ = (Dz - 4 (G - A--~--~)z2) 1/z (13)

The magnitude of the constant D can be seen to be large if G r 0, A ~ 0 from (18), since the function in the bracket should be positive if Oo(t) -+ 0 as t --> m is to hold. The minimum of the function concerned occurs when Az 2 = G, with value at the minimum

of [ D - 8G3/2/3~/A]. Since this should be >/0, then D > 8G3/Z/3~/-A, which suggests that A and G ought to be of the same sign. Alternatively, ~o may not tend to zero as t ~ oo.

The Equation (13) can be integrated to give

~/'3- f z dz -- F ' A (3D_4_~z___f_+3Gz 2 z 4)'/2 = t , (14)

where F ' is an arbitrary constant, later seen to be of the nature of the lifetime of the Universe. The integral in (14) can be reduced to a sum of elliptic integrals by writing the equation as

z~

~/~ f zdz - F ' [(Z - - ~ I ) ( Z - - 0~2~Z ~ 0~3)(Z - - 0~4)] 1/2 = t, ( 1 5 )

0

where

~3 >/ z* > ~4 > cq > c~2. (16)

RELATIVISTIC HIERARCHICAL COSMOLOGY, 1i 309

The constants are defined as

~1 - - ~ ( s , + s2) + - T - ( & - s , ) ,

(g2 ~ - - l ( S t -4- 82) -- T (S1 -- 32) '

~3 ~ S1 -~ 82,

c,4 - 0;

(17)

and

$1-- ~ + \64A-

( 9 1 ) 2

G3~ 1/21 1/3

G3~ 112] 113

A31 J

(18)

The integral in (15) runs from z = 0 (60=00) up to some value of z*=(~o*) -1/3, and so corresponds to evolution from a proper big-bang. The step from (14) to (15) holds if S~ is replaced by ]S~[ and $2 by ISz[. Employing (17), (18) in (15) shows that the A r 0 solution is

t = [A0q _ es)(c~4 - ~2)] 1/z (:r - cq)/7 2, [~3 ~1] ' F "-~

+~lF(2, r)} - F' , (19)

where H is the elliptic integral of the third kind and F is the elliptic integral of the first kind (both incomplete). The new parameters in (19) are

2 ~ arc sin [ (~3 - ~1)(z* - ~4)1 ~/2 L(== =+) (z . ~,~1 '

[(== - =+)(=i - ==)i ',= r~L(== ==)(=+ ~J "

(20)

In using the result (19) from (15) it is essential that the inequalities of (16) be satisfied. This means the cases G = D , A = 0 and G=0, D = 0 must be excluded from con- sideration. It is also necessary that the units and the constant 7 from the original metric be so chosen that cq, which is positive, be always larger than z* =(~o*)-~/s This is possible in general, so that no problems arise in this instance. The two singular cases noted are also not serious omissions because the first is covered in Section 3 and the second can be quickly treated as a separate case here.

When G = 0 and D=0 , the Equation (13) reduces to a simple form that can be

310 PAUL S. WESSON

immediately integrated to give

z(t) = (const) e " / ~ t

or (21)

Oo(t) = (const) e v~ t

The Hubble parameter for this model is obtained from S(t), which is S(t)w. too'~2 or

S(t) oc e "/FZ~)t, as H = SIS = ~/A--/3, and so the G = 0, D = 0 case of(13) is recognized as the analogue of the de Sitter Universe.

The result (19), for practical purposes, can be converted into normal trigonometrical form, by putting z=s in 2 when (19) becomes

t = [Acq(~-3 7 ~ 0 ] u 2 ~/1 - 02 sin 2 r - 0

f de } - (1 + n ' s i n 2 ~ ) ~ / 1 - 02sin 2r - T', (22)

0

where

r _ - ]

/ 0 = [ 0(3(0~1 - - ~ 2 ) ] 1 ' 2

/~r ~ ~3

(23)

In (22), F ' has been evaluated as T', the lifetime of the Universe, by letting e3 --> co (since ca>z* by (16), and z* --> co as 00 --> 0). At this stage it is not known whether t --> T' is finite or not. To decide this, it is to be noticed that the elliptic integral of the first kind diverges if 0, the modulus, is greater than unity: for 02 > 1, (23) gives

> 1 ,

which, by (16), means that divergence requires

1~1---~ < 1. J 2t

Since, by (16), le~[> [e2], this does not hold, and so in general t will not tend to infinity as a result of possible choices of D, G or A, provided the constants cq, e2, e3 can be chosen in a suitable way to prevent other kinds of divergencies from appearing in (19): i.e., el ~ ca, e2 ~ 0 are necessary, for instance.

Since the limit r of (22) cannot ever be larger than the value r = 1 obtained by

RELATIVISTIC HIERARCHICAL COSMOLOGY, II 311

letting c~s --> oo (as z* ~ os), it is not possible to convert the integrals in (22) or (19) into complete elliptic integrals (for which ~*= ~r/2 is needed). This means that the evolution of QRW models (A r 0) must be studied numerically by using tables of the incomplete elliptic integrals of the first and third kind (Larsen, 1953; Eagle, 1958). The study of QRW models (A 50) is rather intractable for this reason and also because the H integral in particular cannot be expressed as theta and zeta functions of real argument, as might be suspected, because this would require 0 < - n ' < 02, or

~1. ~3 2 < - - , - - < 0 , ~2 ~3 ~ ~1

which, by (16) will not hold in any practical case. For these reasons, QRW (Ar models seem to be most suitably approached by using King's (1924) numerical solutions of elliptic integrals.

5. Constant Curvature Models

It was shown in Section 2 that the curvature of hierarchical cosmologies is a charac- teristic of these models, this being especially interesting because in general k=k(t) . For k to be a constant, (2) shows that ~Oo(t ) must be a solution of the equation

28o(t) [~o(t)] 2 1 30o(0 [ 0 - ~ J = const -- ~T--~, (24)

where T z is a constant with the dimensions of a [time] 2 that is so chosen by a posteriori adjustment. The density behaviour satisfying (24) is

O O o ( t ) - , ( 2 5 )

sin2 (2T)

where only one arbitrary constant of integration (O) remains because I have chosen to have 0o = oo at t = 0: if this big-bang origin is not wanted, (t/2T) can be replaced by (t/2T+ a phase factor). The behaviour (25) clearly presents a model of an oscillating Universe, although the old problem of how to define the bounce when it involves a singularity still remains as an irritating aspect of the solution. The scale-factor corresponding to (25) is found from (3) to be

(') S(t) = (const) sin 2/s ~-~ �9 (26)

This gives a positive Hubble parameter at the present epoch, unlike the formally equally-acceptable solution with a cosine factor in (25) instead of a sine. Explicitly, (26) gives

H = ~Tcot ~ , (27)

where the model as a whole has a bounce period of 2roT. The evolution of O(t) by (25)

312

25

PAUL S. WESSON

20

15

Po 10

0-75

0.5

0'25

Fig. 1.

,,2

\

"n- IT

The evolution with time (t) of the density (Qo) and scale factor (S) in an oscillating, constant curvature hierarchical cosmology. The constant T is defined by Equation (24) of the text, while Oo

and S are given by (25) and (26) respectively.

Jp 5

, o

I~ k: const)

23 s S(QRW)~

1

1 2 3 t ~ 4

Fig. 2. The density (Oo) and scale factor (S) in constant curvature hierarchies in which 0o oc t-2; and in quasi-Robertson/Walker (A--0) cosmologies as discussed in Section 3 of the text and specified by

Equation (12).

RELATIVISTIC HIERARCHICAL COSMOLOGY, n 313

Fig. 3.

=

I I I I I I I

-2 -1 0 1 2 3 4 5 t

FIG. 3

The density (~Oo) and scale factor (5') in a cons tant curvature hierarchy possessing ~oo(t) oc e at (fl< 0) and as specified in Section 3 and Equat ions (21)-(23) of Paper I.

and S(t) by (26) are shown in Figure 1. Alternative constant k(t) solutions to that depicted in Figure 1 are provided by the ~o(t) oct -2 model of Section I3 (see Figure 2),

which is seen by I(23) to have k(t)ocA=constant, and also (Figure 3) by ~o(t)oce ~ (/~< 0) where k(t) ocA -fi2/3 = constant.

Particularly simple forms of QRW models are those with k(t)= 0. There are three

elementary cases where the cosmology possesses a flat space-time: (a) The model with ~oo(t)oct -2 mentioned in Section I3 and at the end of the last

paragraph is seen to be flat with k ( t ) = 0 for all t when A =0.

(b) The analogue of the de Sitter Universe with ~o(t):~te-'/g-2t and S(t)oce "/(-57~)t mentioned at the end of Section I5.

(c) The oscillating model of this section with A = - 1/3T 2 chosen.

6. Conclusion

It is apparent from the preceding sections that there are available several types of plausible hierarchical model for the Universe. The best approach to pursue in choosing a model for the whole Universe would be to observe some restricted part of it and so try to infer ~o = ~o(t). In practice, this does not seem to be possible, although some restricted progress might be made by observing the dynamics and evolution stages of clusters of galaxies. Alternatively, one can appeal to what observational data there are, perhaps supplementing these with postulates of simplicity and aesthetics, to try

314 PAUL S. WESSON

and arrive at a workable cosmology. From the latter point of view, zero-curvature,

constant-curvature, QRW (A = 0) and QRW (A r 0) models might be considered as a

series of possibilities in order of decreasing simplicity.

References

Eagle, A.: 1958, The Elliptic Functions as They Should Be, Cambridge, 194 pp. King, L. V. : 1924, On the Direct Numerical Calculation of Elliptic Functions and lntegrals, Cambridge

Univ. Press, 241 pp. Larsen, H. D.: 1953, Reinhart Mathematical Tables, Formulas and Curves, Reinhart, Chapman and

Hall Ltd., New York, London, 336 pp. Weinberg, S. : 1972, Gravitation and Cosmology, J. Wiley Ltd., London, 657 pp.