metric conditions for clusters in hierarchical cosmologies

M E T R I C C O N D I T I O N S F O R C L U S T E R S I N

H I E R A R C H I C A L C O S M O L O G I E S

PAUL S. WESSON St. John's College, Cambridge, England

(Received 16 January, 1974)

Abstract. Algebraic conditions on the continuity of the components of the metric tensor are employed to get an approximate metric in four limiting forms relevant to a condensation in an expanding Ein- stein/de Sitter substratum. The metric of the condensation is in general spherically-symmetric, non- static and asymptotically fiat, passing over into the usual Friedmann solution at large distances and late times. The line-element derived supersedes an earlier incorrect formulation of the problem by Einstein and Straus. The metric is applicable in particular to clusters of galaxies, Wich cannot avoid being involved in the expansion of the Universe for the density-distributions relevant to average loose dusters as presently observed. It is likely that all clusters, including compact ones, are in a state of dynamical evolution, a conclusion which may remove the missing mass problem. The results found agree, in this respect, with recent work by Noerdlinger and Petrosian, and give effective Hubble par- meters for systems in an expanding substratum.

1. Introduction

Although the large-scale structure of the Universe appears to be closely Fr iedmann

in nature, its actual character is that o f a mosaic o f spacetime domains each of which

differs slightly f rom its neighbours. In particular, galaxies are embedded in clusters, clusters are probably embedded in superclusters and so on, the whole approximating

a simple Rober tson/Walker model. A metric for a hierarchical cosmology does not exist: by their very nature, hierarchical cosmologies are o f an ensemble type in which

each level of clustering is expected to possess its own metric. The influence of adjacent

regions o f spacetime on each other and their aggregation to form a composite Universe

has not attracted much attention, but these effects must represent the criterion on wich the efficacy of hierarchical cosmology is to be judged.

General relativistic mechanics is based on the assumption that spacetime can be

broked up into overlapping domains, each of which possesses a system of admissible

co-ordinates: between adjacent domains a group of C a t ransformations is presumed

to exist (Synge, 1964, p. 2), while across 3-spaces of discontinuity within the domains

the g,~ are at least C 1. There exist basically two methods o f joining metric domains

smoothly together: firstly, a somewhat intuitive scheme based on the postulated con-

tinuity of physical (especially mechanical) processes; and secondly an algebraic method based on properties o f the metric tensor. The first method has been used by Lindquist and Wheeler to build a closed lattice Universe out o f Schwarzschild cells (Lindquist

and Wheeler, 1957; Wheeler, 1964a, pp. 369-387; Wheeler, 1964b, pp. 228-232; Harr ison et al., 1965, pp. 71, 127, 133-134). The second method has not been used

much in practice, a l though Bonner (1956) has employed it in considering the format ion

Astrophysics and Space Science 30 (1974) 71-94. All Rights Reserved Copyright �9 1974 by D. Reidel Publishing Company, Dordrecht-Holland

72 PAUL S. WESSON

of galaxies in the early Universe (the 'Swiss cheese' model). Some of the pertinent formulae are noted by Schild (1967, pp. 61-81), while the work of Lichenowicz (1955) is more extensive but somewhat obscure due to the use of non-admissible co-ordinates. (A similar contribution to obscuring the usefulness of the algebraic method is the misleading statement by Zel'dovich and Novikov (1971, p. 111) concerning a claimed discontinuity in the metric tensor.) The most comprehensive account is that of O'Brien and Synge (1952), summarised in Synge (1964, pp. 39-41): for the purpose of what follows, i note that

g~p; g~p, ~; g~, ~k (1)

are continuous across any 3-space of discontinuity, where Greek subscripts run 1 ~ 4 and Latin ones run 1 --+ 3.

Using the algebraic method, I aim to decide in subsequent sections whether sub- systems of a hierarchical cosmology are dynamically affected by the expansion of the Universe as a whole. Particularly, it is of great astrophysical importance to know if clusters of galaxies are affected by the expansion of the spacetime continuum outside them; similarly for superclusters and for lesser systems (such as groups of galaxies) in possibly expanding clusters. Since the missing mass problem and other dynamical properties of clusters may be elucidated by considering them as expanding Einstein/ de Sitter (k = 0) sub-Universes, I will embed each system to be investigated in a larger (k = 0) Robertson/Walker background. An approximate metric will then be derived to describe the condensation-plus-background, from which the dynamical state of the condensation can be obtained. It is important to realise that, a priori , there is no theoretical basis in conventional cosmological models based on general relativity for automatically deciding that relatively dense astronomical systems are unaffected by the general expansion of the Universe. The reason for the widespread but mainly un- voiced opinion that clusters, galaxies etc. are unaffected by the expansion seems to lie in the existence of analagous Newtonain cosmological models, which are known to treat of bound systems. This correspondence, however, is not sufficient evidence to warrant ignoring the possibility that the expansion of space may affect dense bodies in relativistic cosmology: an explicit attempt to decide whether stars expand with the Universe was made by Einstein and Straus (1945). This paper, however, seems to me to be unsatisfactory in several aspects: (i) the junction conditions for the metrics used are not complete; (ii) there are numerous misprints present, some, but not all, of which were corrected by Einstein and Straus (1946); (iii) the authors surround their hypothetical star with a vacuum region in which the Schwarzschild metric holds, and join it onto a conformally Euclidean homogeneous universal metric, not realising that by doing so they have abrogated the whole problem, since Birkhoff's theorem of 1923 in one form states that any spherically symmetric field in vacuo is static (see Synge, 1964, p. 276). The last point, namely the failure to take account of the existence of Birkhoff's theorem, also applies to the work of McVittie (1932, 1933), which has been held to indirectly support the viewpoint that dense astronomical systems do not expand with the Universe.

METRIC CONDITIONS FOR CLUSTERS IN HIERARCHICAL COSMOLOGIES 73

Clearly, to analyse the problem properly there is no point in using the Schwarzschild

metric for the test body, since such an assumption answers the question in the negative

before one has begun - by definition, as it were. The space around stars and within galaxies certainly cannot be considered to be a vacuum: the interstellar medium in our own Galaxy has a density of order 10-23 gm cm-3, while the density of the averaged Universe is within an order of 10 -3o gm cm -3. The effect of the expansion of spacetime on galaxies and stars at any given time would be minute as measured by their appropriate standards, though it might well be significant over a time span of 1 0 a o yr or so. (If the Earth, for instance, were expanding radially at the equivalent Hubble velocity, the rate would be ~_6x 10 -2 c m y r -1 compared to a present radius of -~ 6 x 108 cm (Wesson, 1973); such a small effect (10-1 o, ratio) is beyond direct detec-

tion.) To derive an approximate metric for the condensation and the Friedmann background is a job requiring careful and often tedious work, but insofar as there does not exist any satisfactory account of the problem it would seem to be a worthwhile under- taking.

The main application of the approximate metrics found is expected to be directed towards clusters of galaxies, which can be treated accurately as clouds of dust embedded in a Friedmann background. Before embarking on the derivation of the approximate metrics, it is necessary to ask in the first instance whether or not the metric for such a cloud of dust can be consistently joined onto a Robertson/Walker substratum. The answer to this, using the general Tolman metric for the dust (Tolman, 1934a), is in the affirmative. A proof that such a join can be made for any spherically, symmetric, non-static metric with unrestricted energy-momentum tensor T ~ emerges in the following Section 2, which is an account of the method to be used in the general prob-

lem as it concerns the field equations; Section 3 develops the method as it applies to the approximation scheme used; Section 4 gives an approximate form of the metric for small values of the radial co-ordinate drawn from the centre of the condensation; Section 5 gives a form of the metric for large r; Section 6 gives an approximate form of the metric for small values of the scale factor R ( t ) ; Section 7 gives a metric form for large R ( t ) . Section 8 is concerned with the applications of the results of the previous Sections 4-7 to clusters of galaxies, while Section 9 is a conclusion.

2. Method (I): Field Equations

The system is taken to consist of a condensation and a Friedmann background. The condensation is characterised by a spherically symmetric, non-vacuum (in general), non-static metric given in isotropic co-ordinates by

ds 2 = e v dt z _ eU(dr 2 + r 2 d02 + r 2 sin20 dq~2), (2)

where v = v ( r , t ) , # = # ( r , t ) until shown to be otherwise, and the other co-ordinates have their usual meanings. The cosmological background is taken to be described by the usual metric

7 4 PAUL S. WESSON

R z ( t ) [dr 2 + r z dO 2 + r 2 sin 2 0 dq~ 2] d s 2 = d t 2 - (3)

( 1 + k r 2 / 4 ) 2

where specialisation to flat-space (k = 0) is deferred until Section 3. It is the object of the present section to see if (2) and (3) can be matched consistently for any value of the co-ordinates (t, r, 0, qS) without invoking subsidiary assumptions or incurring the necessity of extreme physical conditions in getting a fit. For the sake of definiteness, the join of (2), (3) can be taken to be across some 3-surface 2 , but in general the precise location of 27 need not be specified.

The algebraic conditions (1) applied to (2), (3) give tile equations of continuity

e 2

e # - (1 + k r 2 / 4 ) 2 '

- r k R 2 ~r eu - - (1 + k r 2 / 4 ) 3 '

2R/~ #'e'~ = (1 + k r 2 / 4 ) z ;

(4)

e v = 1, v~e ~ = 0, h a l = 0 ; (5)

( rkR2 (#,e"), = (1 + k r Z / a ) a J , . '

( # t d ' ) ~ = \ ((1 2R/~

+ kr2/4)U/

(6)

In these Equations (4) - (6), subscripts on/z or v denote partial differentiation with respect to r or t, while a dot as in/~ denotes differentiation with respect to t only. For use in the field equations, (4) - (6) give

rk

#r = (i +kr2/4) '

2R R '

])r ~--- 0 ,

V t = O,

2 R r k #tl~, = R (1 + kr2/4) '

1 ( 3r2k 2

u" - (1. + kr2/4) 2 (1 + kr2/4) k) -

(7)


The field equations (Tolman, 1934b, p. 252) for the metric (2) take the form

8~T) -- - e - " + 2 + + e-~ #t' + �88

_ Vrr+ Vr 8~T~ = 8~T~ = - e ~ + 2 4 + 2r /I +

+ e-~ ~ttt -~- N[At 2

#r 8~rT2 = - e -u #,, + ~ + + ke-Vyz -- A ,

#tVt A) 2

(8)

The cosmological constant A is taken finite initially; all other components of the energy momentum tensor in (8) vanish, where

TaP = (0o + Po) u~ua --g~PPo, (9)

and units in which c - 1 and G - 1 are being hereafter employed. This describes a perfect fluid, and can be written with e lowered as

d x ~ d x p T~ = (0o + Po) g,~ ds ds 9~Po. (10)

By writing (2) in an alternative form in which the coefficient of dr 2 is e z with 2 = 2(r, t), it is possible to see that Birkhoff's theorem holds since T~ - 0 => 2 t - 0 (Tolman, 1934b, p. 253). The procedure so far is the same as that given by Einstein and Straus (1946), except for a much wider generality: following this method I will proceed to manipulate (7) and (8), with the expectation that an acceptable form of the (so far undefined) function R ( t ) will emerge if the join between (2) and (3) is valid.

In comoving co-ordinates, by definition,

~t]~r Y,t - , (11)

2

and this holds in the present case. Using (4a), (3a) and (7) in (8a, b, c) we find that

76 PAUL S. WESSON

1 < ) 3R2

8rcT2Z : 8roT33 - - R2 :/'1 kr2~N:

3~ 2 + #tt "~ ~ -- A ,

+ 4 ) 3 r2k z -'~ 8roT44

1 +

:(,

3k 2 3k + ~ - - A.

+

(12)

From (12c) the form of R(t) is obtained directly as

87cT4R 2 = 3k + 3R 2 - AR 2. (13)

But since T 4 = ~o, the Equation (13) is the same as the Friedmann equation, so that the behaviour of R(t) is cosmologically satisfactory and (2) is seen to be compatible with (3) in that they can be joined together consistently. The implication is, however, that the matching is only truly valid of ~o has the cosmological value (which is independent of position) at the radius of the join, since this is a consequence of (1 3). In what follows I will treat Co in such a way that it is not a uniform constant but reduces to a constant in suitable cases: this means that the application of an approximate metric to compact clusters will not be as accurate as that of the application to open clusters, but is not inconsistent because I have shown elsewhere that hierarchical cosmologies in which ~o falls off as an inverse power of the radial Co-ordinate admit of the formulation of a characteristic equation which is of Friedmann form (with variable uo0), and reduces to

it in a suitable limit.

3. Method (II): Approximation Scheme

Having seen that a spherically symmetric condensation can be embedded consistently in a Friedmann background, I will now put k = 0 in (3) and proceed to obtain an approximate metric for the combined system as joined across the 3-surface 22 The present section derives the basic equation up to a specified order in the approximation, from which various limiting forms of the combined-system metric are derived in the succeed- ing four sections. To proceed, I approximate the metric coefficients in (4), (5) by putting

e u = R2(t) + a, (14) e V = l + %


where a and z are small quantities that can be thought of as series

a = ao + a l r -~ /2 q- a2r -t- ..., z = b 0 + blr -1/2 + b2r + ...,

(15)

where the a i and bi are functions only of t. The form (15) is similar to that of Einstein and Straus (1945) but more general; the region in which 2; lies is not a vacuum, and A is taken as initially finite. The generalisation to non-vacuo means that the method of approximation used by Einstein and Straus (1945) will not work, since ignorance prevails as to what order in r the quantities 7"11, T2(= T 3) and T 2 are expected to be.

The metric conditions (1) when applied to (14) give the following results:

O'p

#" - ( R 2 + a ) '

2 O'rr O" r

(R 2 + (R +

2Rf i + a, /'it ( R 2 + o-) '

Idtt =

2JR z + 2RfR + att (2Rf~ + ~,~2

R 2 + a \ ~ + ~ J '

~r - , ( 1 6 ) v, 1 + ~

z , , ( z , ' ~ 2 Yrr - -

1 + ~ \ 1 + U

"C t

V , = l + . c ,

_ Z t t ( ' r ] t ' ~ 2

vtt l + z \ l + z / "

In comoving co-ordinates, the field equations

B "p - �89 + A g ~ = xT ~p (K = 8u)

give the relations (8) when the perfect-fluid energy momentum tensor (10) is employed, the specification of co-ordinates which are comoving leaving only (8a, b, c) as non- singular equations. The use of (16) in the latter gives a trio of very long expressions that can be restricted by neglect of the terms involving products of o- and z, resulting in

ar Zr 2J~ 2fi~ f izz 2J~z

8~zT] - R%" R 2 r + R R 3 R 2 R +

k 2 k~, k~, cr. + R 2 R 3 R + R 2 - A, (17)

78 :PAUL S. WESSON

= 3 . . . . ff rr Trr (Yr Tr f~2 2R art

2R 4 2R 2 2rR 4 2rR 2 + ~ + - ~ + R ~

/~2T 2/~ ' r Rift 2Ra R'r t A,

R 2 R R 3 e 3 R

8roT2- Gr 2G 3R z 3Rat 6R2o " 3RZz R4 rR 4 + ~ - + R ~ R4 R2 A .

(18)

(19)

The specification of comoving co-ordinates by (11) can also be used in the form

Rz r 2RrT r trtr 0 = ~ r Ra R2, (20)

which is obtained by a combination of (11) and (16). The Equations (17)-(20) are the basic relations expressing the behaviour of the

matter forming the condensation in a frame which is at rest relative to the matter. The components of T ~p, and A, are thought of as specified, but atl other parameters including the precise form of R ( t ) are unknown.

4. Small r Behaviour

Near the centre of the condensation r ~ 0 and the Equations (17)-(20) become three relations in T~, G, zr, R, R, R, %, %,, r and A. The relation (20) does not involve r, but can be used as an auxiliary equation in certain cases. If second derivatives of a and z are small compared to first derivatives, (17)-(19) simplify further so that (19) and (17) can be used, respectively, with A =0 to give

G = - 4rcR4rT2,

% = 41rR2r (T44 -- 2T~). (21)

For the sake of illustration, assume T44 and TI do not depend on r, so that (21) can

be integrated to give a metric, in the r ~ 0 limit,

ds z = [-1 + 2gR2 (T~ - 2T~)r 2 + f i (t)~ dt z +

- - [ R 2 - - 2 7 ~ R 4 T 4 r 2 § (/)] [dr 2 + r 2 dO z + r z sin20 dr (22)

The functions f~ (t) and f 2 ( t ) are unknown. In a first approximation the space can be considered as fiat, so that the mass within a radial distance r from the centre of the condensation is

t 3 M ~ ~/rr3Oo ~ k r , (23)

where k' = k ' ( t ) is independent ofr . If we define other functions r (R, T[, T4)= r and r 1 6 2 the metric of the body is approximately given by


x [dr 2 + r z d0 z + r 2 sin 20 dq~2].

~bzM f 2 ] k'r t - ~ x

(24)

This metric is asymptotically flat and non-static to a cursory examination, and in one sense closely resembles the line element of a spherically symmetric distribution of matter whose Newtonian gravitational potential is t?; this in a first approximation being

ds 2 = (1 - 20) dt 2 - (1 + 2Q) [r 2 + r 2 dO 2 + r 2 sin20 dO2].

The absence in this Newtonian metric of the time-dependent terms which enter through R=R(t ) in (24) is, however, crucial, since it by these terms that (24) admits of the possibility of expansion. The line element (24) also has certain similarities to one used by McVittie (1932). The work of McVittie (1932, 1933) refers to an early epoch in the Universe, where he finds that galaxies and other condensations tend to collapse as the substratum expands around them. The metric (24) presumably becomes etablished some time after this initial collapse phase, when the condensations have reached internal equilibrium and the expanding background metric can make itself felt: (24) shows that in general the metric is of a perturbed Friedmann type, the scale factor R(t) of the usual cosmology being reduced by the presence of the condensation (mass M). The exact Friedmann (k=0) Universe is recovered as M-~0 and f~(t), f2(t)-+0. The scenario of McVittie (1932) in which the Universe started expanding from a static Einstein state with consequent collapse of condensations may be valid for early epochs, but (24) shows that in the general case condensations are affected by the expansion of the present-day Friedmann-type Universe.

A metric for a collection of many condensations has been found by McVittie (1931, 1956) and is very similar indeed to (24), excepting the absence of the time-dependent (via R) terms: these are not present in McVittie's multiple-condensation model due to the explicit use of an approximation that removes any possibility of such dependency (McVittie, 1931). McVittie made, in ~11, five explicit assumptions, and so it is not surprising that (24) goes somewhat beyond his result. The same comment applies to the work of Ygrnefelt (1940), Shticking (1954), and Lemaitre (1931), although the latter has mentioned Birkhoff's theorem in connection with the realization that it does not preclude pure radial pulsations in a perfect vacuum being compatible with the Schwarzschild line element outside the body. (This does not, o f course, mean that v= v(r, t), #= l~(r, t) of (2) are compatible with the Schwarzschild interior solution, which joins smoothly onto the exterior one; Lemaitre's work is awkward in claiming that there exist discontinuities in the first spatial derivative of the metric coefficient of dr 2 in his postulated metric.) The treatment by Noerdlinger and Petrosian (1971) does not suffer from these noted shortcomings, and concludes (qualitatively) that systems of cluster size and larger will be affected by the expansion of the Universe, especially at an early epoch. This should not be confused with the possibility of infall

80 PAUL S. WESSON

(Gunn and Gott, 1972) as a result of explicitly assumed cosmogonical processes, of which no mention has so far been made.

While (24) appears to be non-static, is there yet any way that it can be transformed to a static form? If there is, then, by the alternative formulation of Birkhoff's theorem, the metric could be brought into congruence with the Schwarzschild line element. Vice versa, if (24) cannot be brought into equivalence with the Schwarzschild metric it cannot be transformed to a static form. If we consider the Schwarzschild field in conformally Euclidean notation,

b 2 d s 2 = - a46~p dx'Mx '~ + ~ dt'2,

(m) (o) a = 1 + - - b - 1 - - - p r l / 2 ~ r , l / 2 ' (25)

with the co-ordinate transformation which leaves the field centrally symmetric, as given by Einstein and Straus (1954, p. 123), it is seen that in general the metric (23) cannot be transformed into the Schwarzschild field.

The approximation made above depended on the smallness of o-,r z,,, and oSr compared to G/r. By (15), this presupposes that

3al a2 2ar~5/2

,'3bt[4~ 7~ < 'a; 2ar (26)

which are conditions not certain to hold. It is, however, reasonable to assume with Einstein and Straus (1945) that differentiation with respect to t increases the order of smallness by a half, so that in (20)

atr 2ar R~ = O, % - - R2 . (27)

Employed in (17), Equation (27) gives

a r=R4r 8zcT~ R2 ~ + A . (28)

Differentiating (28) and using the result in (19) with A = 0 we obtain a form involving

T 2 as ( 4R2 R% _ 8~rT 2 + 8~T) + + - 4ztr2R" (T(),. (29)

a" = ~ ~ R2, ]

The equivalent form of % can be simply obtained from (27). Plausible dependencies


of pressure and density on the radius vector are

T) = A r "-p ,

= = B r - " ,

7"44= Cr-". (30)

Substitution into (29) and the corresponding equation for vr, followed by integration, gives

R4rZ ( 8ztCr-n 8zcAr-P /~ 2/~2"~ O" = ~ - - - ( 2 -- I'1) "}- (2 -- p ) 8 ~ Z r - P "]- ~ "[- ~ - J "]- F2 ( t ) ,

R 2 r 2 ( 8 ~ C r _ n 8~AF_ p f~ 2/~2 ~ (31) ~: = - - + 8 7 t A r -p R2 + F l ( t) .

The coefficients A, B, C can be functions of t, while F 1 and F 2 are undetermined. If the density falls off as T [ ( = Co)-- Cr- 2, the first terms in (31) are replaced by ~ 8ztC log e r, respectively. This is an important case, since there is evidence (Yahil, 1973) that QoCd r - 2 in rich clusters of galaxies.

From (31) it is seen that the presence of matter tends to make a negative and thus decrease the effective scale factor given by $ 2 = R z + ~. This is as expected; but it only holds if n<2 . If n>~2, the effect of matter would appear from (31a) to be one of increasing a and accelerating the expansion of the Universe. What is the explanation of this anomalous behaviour ? Consider, for the moment, what would happen if T4*( = C0) az r -3; then a c d - r20o(r) and the Newtonian potential of the whole body out to some radius I would contain a part of the form

G4rrr 2 drQo (r) df2~ l -

r and

f dr (2o~ t _ ~ '

0

which clearly is unbounded, as is any distribution falling off faster than r -2 , in which limiting case O~lS~dr/r 2 for an idealised condensation. Again, the potential gradient for distributions in which r would contain a term dO~dr ez_ 1/r 2. The origin of the peculiar behaviour for 0oegr -", n>~2, is seen to lie in the presence of a potential gradient which causes matter to be in unstable equilibrium unless it can move out towards spatial infinity. Of course, clusters of galaxies are not simple spheres of fluid, but the argument just given does seem to suggest that bodies in which 00 falls off faster than r - 2 should not be found in Nature. The full correct form of the potential for an expanding cluster is derived in the Appendix, where the instability for n >~ 2 is further investigated.

The influence of the expansion of the Universe on the condensation is expressed via (31) in comoving co-ordinates. Following the suggestion of McVittie (1932), it is worth

82 PAUL S. WESSON

pointing out that a Galilean form of the metric in the limit r ~ 0 can be obtained by suitable re-parametrisation. For a pressure-free fluid in which R and /~ may be neglected, the previously-found metric with T, 4 = C r - 2 takes the form

ds 2 = [1 + 8z~CR a loger] dt a - R2(1 - 4 n C R z lOge r ) X X [dr 2 + r z d0 z + r z sin20 d~b2]. (33)

I f x u denotes any of the co-ordinates t, r, 0, q~ and e u are Gali lean co-ordinates at a specified point P ( t o , r o, 0o, qSo), then

. . . .

and the metric (33) becomes

ds 2 = [~+ (ro, to)] d~1 - R 2 (to) if- (ro, to) x

x [a t E + ro 2 de22 + r~ sin 2 0o de32], (34)

~+ (ro, to) = 1 -t- 8zcC (to, to) R e (ro, to) log e ro,

~_ (ro, to) = 1 - 47cC(ro, t o )R 2 (ro, to ) log e r o.

Define new co-ordinates by:

t/4 = (1+/2 (to, to ) e4, r/1 = R (to) (1_/2 (to, to ) e l ,

t/2 = R (to) ~_/2 (ro, to ) roe2, q3 = R (to) ~ /2 (to, to) ro sin 00e 3 .

In these co-ordinates the metric (33) becomes Galilean at P ( t o, r o, 00, q~o): i.e.,

2 : + + (35)

For any point of interest, it is now possible to find the local velocity of the fluid in Galilean co-ordinates by using the usual tensor t ransformat ion and comparing the result with the general formula

~Gal u T14= 1 - u 2' c - - -1 , (36)

which is valid in any frame. While this procedure is possible, I think it preferable to continue to formulate the limiting forms of the metric in comoving co-ordinates (or approximately comoving, that is, since g44 is not exactly equal to unity, though 944 -~ 1 to a very good approximation), since in this form a ready comparison with Fr iedmann cosmology is available. The question of choice of co-ordinates is discussed further in Section 8.

I t will be seen that the Hubble parameter is reduced in the ne ighbourhood o f the condensat ion; a comoving observer at infinity therefore sees an apparent collapse near the condensation. This does not mean that the mot ion of the cosmological stratum near the condensation has reversed, because an observer on the condensat ion still finds a positive Hubble parameter. This will be seen in Equations (45), (46) below


which give an approximate metric for a cluster. It should be noted at this stage that only first approximations are given throughout: these approximations can be re- substituted into the field equations (8) to get a better approximation via a better T~. The result is expected to be a condensation surrounded by a region where the density is less than that of the cosmological background, as can be appreciated by considering Newtonian arguments. It can be verified by direct re-substitution that the metrics found are indeed approximate solutions of the field equations (8). For instance, taking (46), one can calculate/#,/z, vr, vr~ and re-substitute back into (8c): it is found that a trivial identity results in the (large r, large R) limit in which (46) is valid.

5. L a r g e r B e h a v i o u r

Far away from the condensation, the metric as derived in the last section becomes asymptotically fiat. In the limit r ~ oo, Equations (17)-(20), it transpires, do not give any non-trivial solution unless it is assumed (on the grounds mentioned in the last section as regards the smallness of derivatives with respect to t) that o-., at, and r t can be put equal to zero. Under these conditions, it is seen that

2R 2/~a /~2z 2/~z fi2 8roT? = 8~T22 = 8~T 3 - R R 3 R 2 ~ - + R 5 - A,

31~2 6/~2 a 3R2. c (37)

8~rT2- R2 R4 R2 A.

Eliminating -r by use of (37b) in (37a), and resubstituting, we obtain

R 4 4 2 RR\ A a = 4zcTt 1 - 1 1 -

2 (R2 + RE) + + 3- ' A R 2 8~T2R 2 R 2

z = 1 x (38) 3R2 3/~2 (/~2 q_ R/~)

x [47rT~ 4~T44 (1 + 2R/~'~ A(I .

The relations (38) show, as in the r ~ 0 limit, that the scale factor far away from the centre of the condensation is reduced by the matter present, whereas the existence of a positive cosmological constant A tends to augment the expansion.

Except for an implicit dependency via T~ and T44, neither a nor z of (38) involve r. This is not surprising, since the metric (24) for r -+ 0 is asymptotically flat, and the effect of the condensation is expected to be negligible as a function of radial distance far away from its centre. The approximations made in deriving (38) do not preclude the dependence of a and -c on the epoch, and in general the scale factor far away from the condensation changes with time roughly as the factor outside the bracket in (38a), with a ~-0 as expected when the condensation is absent (TI , T44 ~ 0) and the cosmological constant is small.

84 PAUL S. WESSON

6. Small R(t) Behaviour

It is to be expected from the form (3) that small R(t) corresponds to small t, when the combined system of condensation and expanding background was in a relatively com- pressed state. In the limit of R ( t ) ~ 0, bearing in mind that the sizes of T~, T~(= T~), T~ and A are unknown, the Equations (17)-(19) give four relations in T~, G, crt, o, or, R,/~, R, r and A. Firstly, consider terms involving 1/R 4, to which order only the first three equations (17), (18), (19) are pertinent. None of these equations contain z, and so to this order z-~0. Using (17) for G in (18) to obtain G,, and putting the or and G~ so found into (19) gives o-, with no assumptions having been made about T~, as

AR 4 4~R4 (27"22 + T~ - 7"44) + 3~ T , with r = 0 (39)

O ' ~ 3 ~ ~ -

In practice, AR4(t) probably tends to zero rapidly as R ( t ) ~ O, but the same does not hold for the components of the energy momentum tensor, since if the system started off from an initially condensed state, T~ in particular would have been very large.

A more accurate form that brings out other features of the behaviour of the composite system may be obtained by taking terms including 1/R 3 and 1/R 4. Since it is not expected that/~(t) = 0 early on, (20) implies that G = 0, which used in the other relations of (17)-(19) enables one to form 3 x (18) + (19) combined with (17)-(18), giving ( A - 0) the assumption-free equations

2R 4 ~ - 3 (R/~ + R~) (4~T# - 10~T? - 2~T~), (40)

~ 0 .

Comparison of the solutions (39) and (40) shows that near the state R(t)~-0 the effect of A is negligible and that the presence of matter tends to inhibit the expansion by making o smaller and so decreasing the effective scale factor given in (14) by $2= = R 2 +a. When R has grown somewhat, a term in/~ appears in (40) which is not present initially in (39). In both forms the scale factor S(t ) tends to become close to the equivalent Friedmann one of (3) when the velocity of the expansion (Tzk) is high, but the pressure terms T~, T2(= T]) change in relative importance as the condensation evolves under the influence of the comsological background. Without knowledge on the behaviour of R4/R z as R ~ 0, it is impossible to say what the original form of the condensation would look like.

If, alternatively,/~(t)=0 in (20) with G # 0 ; and /~ ( t )=0 so that the system is not evolving, the equivalent forms for o-r and z (with A-~ 0) are given by

tT, = -- 87rR4T~r ,

= 8rcR 4 (27"22 - T2) r , (41)

~ 0 .


These two expressions can easily be integrated when the dependence of T~, T22 and T~ on r is known, but there will in general be an unknown function of time present in the final metric.

7. Large R(t) Behaviour

Since large R(t) corresponds to a late epoch, it is to be expected that the line element in the limit R(t) ~ ~ will closely resemble that of a Friedmann (k = 0) Universe. In this limit, the last two equations of the basic set (17)-(20) give respectively 8re T 3 = - A and [~zr/R=O up to terms in 1/R, suggesting that T~--0, A~-0 and zfl?---0. This is indeed as expected. To this order, Equations (17) and (18) are identical, implying that 8n r ~ "-~ 8 rc T ~ ( = 8n T 33) ~- 0, again as expected. To extract a useful result from (17)-(20) one could elect to neglect terms involving derivatives with respect to t, as done in earlier sections: this still leaves (17)---(18) to order 1/R with/~ = 0. Thus, irrespective of whether a depends on t in the limit R(t)~ 0% there remains essentially only one equation, of the form

2/~ 2/~ 8rcT? = snr2 z (= sTzr 3) = A. (42)

R R

This, with the somewhat trivial results of the other Equations (19) and (20), gives

R r = 1 - - ~ . (8~r~ + A),

2R 0 " ~ 0 ,

8 ~ T 2 = - A ,

rrfi ~- O.

(43)

Equivalently, by (43a) and (43c), the cosmological constant can be eliminated to give the metric as

[ 8nR ] R2 r2 r2 ds z= 2 - 2R (T~- 7744) dt z - (t)[dr 2 + dO z+ sin z0dq52].

(44)

This confirms the expectation that the system has relaxed at late times to a state in which the line element is almost exactly that of a Robertson-Walker Universe.

8. Application to Clusters

Clusters of galaxies are often approximately spherically symmetric with an equivalent matter density that falls off continually from the centre and merges into the cosmological background. Compact clusters seem to come close to obeying Oo(r)e~r -2, and appear to have been formed at an epoch z < 3 (Yahil, 1973). Non-compact clusters are probably not centrally condensed, and for the sake of illustration I will restrict attention to a modelcluster in which 0o ( r ) eZr- 1. Do such clusters expand with the Universe ?

86 PAUL S, WESSON

To examine this question, the appropriate metrics are those for r ~ 0 (centre of the cluster) and r--* m (perimeter of the cluster) at a late cosmological epoch. To simplify matters, I assume that there does not exist any hot intergalactic gas or other popula- tion of objects that could give rise to a significant pressure, and in (31) I assume (on the grounds only of ignorance) that the unknown functions Fl ( t ) and F2(t ) can be taken as negligibly small to the order of the approximation. The late epoch ( R ( t ) large) limits of (31) and (38), are metrics relevant to present-day clusters, so that the combined-system line elements become

ds z = [1 + R28~zCr] dt z - R z [1 - R 2.41rCr] x

x Edr 2 + r z dO 2 + r z sin z 0 dq~2], (45) (r'-, 0)

I 4rcC R2 (1 ---- ?/~2) ] [ 4rcCR2"] ds 2 = 1 + r / ~ 2 ( k 2 + R R ) j d t 2 - R2 1 3/~2 r j x

x [dr 2 + r 2 d02 + r 2 sin 2 0 d~b2]. (46) (r-~ ~)

In both (45) and (46) the cosmological constant has been put equal to zero. This is a good approximation for (45), and is reasonably satisfactory also for (46) if C is large compared to A. (C = C (t) is a measure of the density of matter.) These approximations do not necessarily mean that the rest of the actual Universe has A = 0: rather, (45) and (46) describe a system consisting of a cluster embedded in a background which is evolving cosmologically as determined by R ( t ) . The function R ( t ) is essentially a property of the Universe as a whole, and its behaviour could well be dominated by a non-zero A on the scale of the entire cosmos. The behaviour of R ( t ) in cosmologies of the type being considered is determined by the Friedmann relation, which (for units in which c = 1, G - 1) gives the two equations

3k 2 3/~ 2 8~Zeo = ~ T + ~ - -- A,

k /~2 2/~ (47)

8rrP~ - R 2 R 2 R + A.

For a pressure-free (Po - 0 ) , flat universe as used previously, these give

4nooR 2 = 1~ 2 - R R , (48)

which is a convenient relationship between R,/~ and/~ for any A, and can be used as an auxiliary equation to assess acceptable behaviours of R ( t ) in the approximate cluster metrics (45), (46) given above.

To investigate whether or not clusters expand with the rest of the Universe, I will define effective scale factors S (t) in place of the R ( t ) of the usual Friedmann models. In the limit r ~ 0, from (45), it is seen that the modified scale factor is given by

S 2 = R 2 - 4nCRgr, (49)


where the influence of matter would seem at first to be purely one of retarding the expansion (density= C). Differentiating (49) and dividing the result by S 2 gives

2SS 2 R R - 167rCR3Rr - 47zR*rC

S 2 R 2 _ 47zCR4r '

i.e., 2 H ~ t r ~ ( 7 1 6 7 r C R R r - 4 ~ R Z r C ) x ( l + 4 ~ C R 2 r ) ,

where H~fe is an effective Hubble parameter, pertinent to the dynamics of the system as a whole. The conventional Hubble parameter, which would prevail if there were no condensation, is f~/R, and in general H~f~ r R /R , but rather

2k 2H,~ . . . . 8nCR/~r - 4nR2rC - 16~z2R4rZCC, (50)

R

where a term in C 2 has been neglected. The problem now arises of evaluating the rate of variation with time of C.

If the cluster is assumed, a priori, not to be partaking of the expansion of the Uni- verse (or if some version of the steady-state theory is operative), then ~ in (50) is identically zero and

k RR Hef f= - - 4nGC ~ - r, (51)

R c

on replacing the conventional units. (In (51), C is the density of the matter in (say) gm cm -a at unit distance from the origin, while r is a dimensionless radius vector measure: r = f/re, where r is a physical length and r c is the characteristic dimension of the cluster.) At the centre of the condensation Hef~ = R/R , from which the Hubble parameter decreases outwards as more matter is encompassed within the sphere being considered of radius r. This behaviour in itself suggests that the cluster cannot be static. Ultimately, (51) would seem to imply that Herr could be zero, at a point where (intuitively) the effect of the condensation cancels the influence of the expansion of the the Universe. Before this interpretation can be accepted, it is necessary to know what domain of validity (51) and the (r-+ 0) metric (31) encompass. Looking at equation (21), from which the metric (31) is derived, and in particular at (21a), we see that the assumption was made that the terms (z,/RZr) and (/~z/R2) were of the same order of magnitude, so that, replacing conventional units,

Zr .~ R2 HoR2 r ~ ~ ~ ~ c ~ ' (52)

where Ho is the Hubble parameter for a Friedmann Universe with scale factor R( t ) , without the presence of the condensation. H o is known to be roughly 50-100 km s- a Mpc -1, and while this parameter no longer has a precisely-defined significance in

88 PAUL S. WESSON

hierarchical cosmology, for the purposes of the present order-of-magnitude calculation H o ~ 10-17-10-18 s-1 can be taken. By use of R ~ c T, where T is the age of the visible pait of the Universe (~ 101~ y r ~ 3 x 1017s..~ 1/Ho, of course), an estimate can be obtained of "c/r 2. But by the basic Equation (14), it has been assumed that z ~ 1 throughout, so that (51) and this inequality give

1 r H - - ~- 1. (53)

H o T

The metric (31) and the expressions (50) and (51) for H~ff thus only hold near the centre of the cluster (r --- fire), say within its innermost one-tenth-radius sphere. For comparison, the region where (51) predicts that H e e - 0 is of size

c 2 1 r o _ 41rGRZ C = -47rGTZ C _ 10-100 (54)

for clusters of mean density in the range 1 0 - 2 9 - 1 0 -30 gm cm -3. While some small,

high-density clusters might be able to resist the expansion of the Universe by having a small ro, in most cases it is clear that the metric (31) and the expressions (50), (51) have ceased to be valid before the critical distance r o is reached. This means that the expansion of the Universe does have a noticeable effect, and that H,~ is not zero any- where within an average cluster, being instead a positive parameter.

If the cluster does little to inhibit the expansion of the substratum in its neighbour- hood, then I can consider a portion of comoving volume R 3 containing a mass M = = R 3 C of matter. (r ~ 1 is required strictly, so that r will be greater than C, but with Oo = C r - 1 as being used, the calculation to be given will not be out by more than an order of magnitude or so.) Since matter is neither lost nor gained, Nf = 0, so that

3k = - - - ( 5 5 )

R"

Inserting (55) into (50) we find that

2 R 2H~ff = - ~ + 4 ~ C R k r + 48zc2CR3Rr 2 , (56)

which cannot be negative under any circumstances since the Universe as a whole is known to be expanding. Only if R(t) = 0 would the effective Hubble parameter be zero, and it is seen than in general Heir > k / R : indeed Heff increases with distance from the centre of the condensation, though the effect is small.

In the central regions of a cluster, where (31) holds, it has been seen that H~e>0 if the density of matter is constant or decreasing. The only way to obtain H,e < 0, and so stop the expansion or turn it into a collapsing motion, would be to postulate that some cosmogenic process is operative in clusters with the consequence of systemat- ically increasing the density of their central regions over long time spans. There does


not seem to be any basis for suggesting that this is so, and any such process would have to be cataclysmic on an astrophysical time scale to seriously alter Hoe from its natural cosmologically-determined value. Outside the inner core of the cluster, where (45) holds, is a transition region (in which the conclusion that Hee >0 must still hold, since the inner region expands anyway and so, therefore, must the outer regions), before the outer portion of the cluster is reached and (46) takes over. No explicit criterion for the domain of validity of (38) and (46) can be obtained from (37) in the way as done above for the r ~ 0 line element. While (45) holds in the realm r ~ 1, with the transition region at r -~ 1, it is expected that (46) will hold firmly when r > 1 and (in view of the low density of the outer regions) will be a reasonably true description of things for any region where r > 1.

The modified scale factor in the limit r ~ oe is given by (46) as

4rcCR 4 S 2 = R 2 - - (57)

3/~2r �9

Proceeding as in the previous case to differentiate and divide by S z, using (55) as a value for C which certainly holds in this domain, we obtain

2S$ 2 R R - l&rCR3/3f?r + 47cRa/Rr + 87rCR*ff~/3R3r S 2

H ~ ~- 3/~r

R 2 _ 4rcCR4/3f{2r

4rcR 8rcCR2~ ( 4rcCR2'~ +~rr + 3/~2r ,] \ 1 + 3~- r ]'

where Heir now refers to the r --+ oo region. Neglect of terms in C 2 and in CR (as compared to R), gives Heir as

2/~ 4 r c R 8~zCR2( ~ _ ) 2HO =R-+Ty+ . (58)

The effective Hubble constant is found to be generally larger than that for a perfectly isotropic model, although its actual size depends on/~ and/~. While/~ is positive, could well be negative for the Universe as a whole, and this would lead to a negative term in Hell. A cosmological constant that is not large and positive would cause/~ to be negative, as can be seen from (47) and (48), and while this is likely to be true, the presence of the factor C in (57) means that Heir is not much affected by this possibility. The cluster expands essentially at the same rate as the rest of the Universe, ultimately merging into it.

It should be noted for the purposes of the use of the results given in this section that the co-ordinates in which (45), (46) are expressed are no longer perfectly comoving ones, as were those of the original Robertson/Walker metric. Perfectly co-moving co-ordinates, as mentioned in Section 4, should have #44 = 1, and a suitable re-definition of co-ordinates can be made to accomplish this (Weinberg, 1972, pp. 338-341).

90 PAUL S. WESSON

In the two limits in which (45), (46) hold, not much error is incurred in neglecting the extra terms in g44 compared to those in g~p(a = fi-- 1, 2, 3), and this has the further desirable property of giving results in terms of R(t) as pertinent to a uniform Robert- son/Walker background, this being a familiar parameter. This procedure is further justified by the fact that k - -0 Universes (A = 0) approach asymptotically to a static end-state as R(t) ~ oe : the components g44 of (45), (46) then tend to constants in the limit in which they are valid, and for this situation (Weinberg, 1972, p. 339) the co- ordinates are accurately co-moving.

9. Conclusion

By considering conditions on the components of the metric tensor, line elements for a cluster of galaxies embedded in a cosmological Einstein/de Sitter background have been obtained. These are (23, (31) in the r--> 0 limit; (38) in the r ~ oo limit; (39), (40), (41) for the small R(t) limit; (43), (44) in the large R(t) limit. These various forms correspond to the assumption of various possibilities for the limiting behaviour of functions like R(t) in extreme situations. The metric near the centre of the condensation (which may be finite in extent if a suitable density law is substituted into o-r, ~r) is in general a spherically symmetric, non-static, asymptotically flat one. Applied to clusters which do not possess any intergalactic gas to give an appreciable pressure, and assumed to have a density distribution of the form Qo~C(t) (f/rc) -1, the limiting forms of the metric give a typical cluster line element as (45), (46). The density of presently-observed clusters is not sufficient to seriously inhibit the expansion of the cosmological substratum as specified by R(t), and it is to be expected that loose clusters at least are expanding with the Universe. This conclusion is not likely to be altered by the adoption of a density law for compact clusters of ~oo~tC(t)(f/rc) -2, which are in any case unstable (Section 4), or by the introduction of a hot intergalactic gas (which would give rise to a pressure that would merely augment the expansion). The only ways in which clusters as presently observed could be rendered static are by the confinement pressure of a hypothetical hot inter-cluster gas or by the assumption that clusters are much denser than usually supposed. Since the missing mass problem may not arise for expanding clusters, the latter suggestion is unjustified while the former is ad hoc; so it is concluded that clusters are, in general, in a state of expansion. This conclusion is in agreement with, but goes considerably beyond, recent work by Noerd- linger and Petrosian (1971) that deduces the presence of expansion for systems of cluster size and larger.

Appendix: Consequences of an Expanding Cluster Potential

Since the missing mass problem can be circumvented if clusters of galaxies are expanding, and since clusters of the presently-observed density are in any ease expected under some circumstances to be expanding with the Universe, it seems desirable to find a potential for the interior of an expanding system of pressure-free particles so that the


dynamics of clusters may be investigated. I have derived elsewhere, starting from a potential due to Zel'dovich and Novikov (197l, pp. 66-68, 80-82) for an expanding relativistic shell, the proper cluster potential

l-r" rr ~ - , 2r z - . ] r . 0 , (A1) 47~Gmv~176 ) (1 n) (3 - n) /

4' (r) = - 4rcGmvor o - r log e . (n2)

Of these two equations, (A1) holds for any n such that n ~ 1, n ~ 3; and (A2) holds for n = 1. The exponent n determines the density profile of the cluster, according to

((o)- v = Vo , (A3)

where v is the number density of galaxies (each of mass m) at a distance ? from the centre of the cluster (which has a radius ro), and v0 is a constant. The potential for n = 3 is singular in the sense that 4' ( r ) ~ - oo as r -+ 0. Examination of (A1) and (A2) shows that clusters as now observed cannot be in equilibrium, in the sense noted in the text and at the end of this Appendix.

Radiation emerging from a cluster is in general redshifted by different amounts according as it originated from different places. For a difference in potential of A4', the frequency is shifted from the rest frequency f0 to

f = f o 1 - ,

giving a frequency change of

foEZ~f d f = c2 ,

and a change in wavelength of

4rtGv~176 r"oA[ rr~-" 2r2-" 1 [d21- ca [_ ( i - -n ) ( 1 - n ) ( 3 - n ) ' I n # l , 3 1 ; (A4)

o r

Id21 4nGv~176 [ r (~o)1 - c2 roA ~ - r l o g e , In = iI . (A5)

Consider, as an illustration, a dense cluster with n = 0. Light that originated near the centre of the duster is unshifted, while that emitted from the perimeter (r = ro) is redder by an amount

4nGvom2o r2 [d21 - 3c 2 (A6)

92 PAUL S. WESSON

Galaxies in the cluster would be spaced by about ten times their intrinsic sizes, so that the linear separation would be ~-100 kpc, and the density of galaxies about 3 per (100 kpc) 3. Thus v= 10 -7~ and the masses of the galaxies can be taken as 1011 M o with a radius of to=0.5 Mpc. The shift in wavelength (A6) for 2o=5000 A is then found to be Id21 ~-0.05 A. This is for light emitted from the perimeter of the cluster at r = ro, and a slightly higher shift would be found for light emit ted at r ~-0.75 r o and also for light emitted at different radii for clusters with n #0. The maximum shift possible is of the order of 0.1 A as far as the visible light of the spectrum is concerned.

Similarly, it is easy to work out the expected wavelength shift for the local supercluster, which, using data given by de Vaucouleurs (1958), has rc ~- 15 Mpc and a mass of 1015 M o or so. Assuming each galaxy to have a mass of 1011 Mo, and putting n = 2, gives the shift in visible light (that emitted from the centre of the supercluster compared to that emitted near the Galaxy's position in the supercluster' soutlying regions) as Id)~l =0.014 A. The shift for the supercluster is thus less than, but of the same order of magnitude as, that from a dense cluster.

The effects noted are in the sense that an exterior observer would see the outer por- tions of a cluster as redder than its centre. However, there are four other effects operative that make slim the hopes of detecting the ~-0.01-0.1 A changes noted: (i) the component galaxies of the cluster have dispersion velocities running up to adisp ~ 3000 km s-1, causing a shift of frequency for line-of-sight velocities of (fo - f ) ,

where

O~disp/

so that

Id21 _ aaisp _-_ 10_ 2 . (A7) 2 c

This is to be compared to the shifts found above, which are of size 0.1/5000 = 0.2 x 10- 4.

The difference between these two quantities is an unavoidable one, as can be seen by considering the virial theorem. The redshift of those galaxies with dispersion velocities directed transverse to the line of sight is of order [d21/2 = 10-4/2. To these accuracies it is still justified to neglect the factor ( 1 - vz/ez) -1/z in the original potential. (ii) If the clusters are expanding, they are doing so with their own effective Hubble parameters, which might be expected to be of order 50-100 km s -1 Mpc -1. There are thus systematic outward velocity components (superposed on the dispersion velocities) of size 50-100 km s -1, leading to wavelength shifts of size Id~.[/2~ 10 -4. (iii) If light is propagating right through clusters from points located beyond them, its wavelength will be altered in a complicated manner as evaluated by Rees and Sciama (1968); these shifts are generally toward the red or blue at different places of the cluster's cross- section. (iv) Superposed on the three previous effects is an overall Hubble recession redshift of the cluster with respect to the Earth corresponding to typical redshifts z =

= 1d21/2-0.2.


Of the four effects (i)-(iv), the first three affect the colour of the radiation emitted from a cluster as its cross-section is scanned. While the velocities in (i) are random, the large size of thes hifts involved probably swamps the gravitational one. A qualitively similar effect to that of gravity is produced by (ii), viz. : reddening of the perimeter with respect to the centre of a cluster, although the expansion effect is located further out than the gravitational one, being largest at r = r o.

At longer wavelengths, the absolute shift Id2J is larger than in the visible, since 1d2[/2 is a constant for a given cluster. At 2=21 cm, [d2] = 4 x 10 - 4 cm, with an equivalent frequency shift of ]df[ = 3 x 104 Hz. This is, in theory, measurable with a radio telescope, but in practice is swamped by the dispersion shifts (ii) above.

If clusters expand roughly as Einstein/de Sitter systems, they should possess their own, peculiar black-body background radiation. If this microwave radiation prop- agates towards the Earth, it should be detectable, the cluster showing up as a temperature anisotropy in the local black-body background. For photons propagating toward the Earth out of a cluster, the change in wavelength produces an equivalent change in the temperature of the radiation given by

dT [d21

T (A8)

If a cluster has a local black-body background of temperature T, a wavelength change of order 1d2[/2=0.1/5000 gives d T / T ~ - 2 x 10 -s. Further, if by analogy with familiar measurements, T ~- 2.7 K, then d T-~ 5 x 10- s K. The presently observed dipole varia- tions in the microwave background are of a size d T / T = O . 0 3 ( + O . 0 7 ) x 10 -z over a 24-h period (Misner, 1968). If these measurements could be increased in accuracy by one order of magnitude, it should be possible to detect the expected d T / T - ~ 2 x 10-5 anisotropies in the background due to clusters. Unfortunately, the interpretation of any positive result of such a search for anisotropies would be confused by the peculiar velocity of the Solar System relative to the background. The latter effect, being one of accurately-known period, could, however, be removed from the data: what would be left, assuming it to be significant, would be the anisotropies due to clusters and similar congregations of objects. (The small scale anisotropies in the visible region (Shect- man, 1973) have already been detected and identified with dusters.) The dispersion difficulty (ii) is not expected to enter the microwave anisotropy determination, because the observations would be of a propagated ambient field. Likewise, (ii) and (iii) are unimportant, and so a positive result might be looked-for in the near future.

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94 PAUL S. WESSON

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Chicago, U.S.A.

metric conditions for clusters in hierarchical cosmologies

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