The stability borders of cosmological density perturbations in
de Donder gauge; scales of superclusters of galaxies being favoured (*)
B. ROSE
Universität Konstanz, Fakultät für Physik - Postfach 5560, D-78434 Konstanz, Germany
(ricevuto il 30 Luglio 1996; approvato il 29 Aprile 1997)
Summary. — For the propagation of density perturbations on a flat Robertson-Walker background the Stability-Instability Borders (SIBs) are determined in the de Donder gauge. A subtler definition of the SIBs leads to a modification of former results and a new class of SIBs near which perturbation growing is very strong. In general, there are SIBs with the character of lower-mass borders of perturbations for gravitationally growing and upper-mass borders. A physically realistic model for the cosmological substratum at the epoch of hydrogen recombination and times before is considered. For the matter perturbations on it their ability of radiation capturing is explicitly taken into account, which leads to a scale-dependent velocity of sound. The calculation of the SIBs in this concrete model shows that the important cosmological epoch for density growing is given by z 6800 and that by a pure gravitational point of view superclusters and clusters of galaxies are preferred for condensation. This is independent of the special matterOradiation coupling and corresponds to a scenario “from the top to the bottom”.
PACS 98.80 – Cosmology.
1. – Introduction
In a series of previous papers ([1-4]) and references given there, we studied the propagation of self-gravitating density perturbations on a flat Robertson-Walker background. The analysis was made with the linearized Einstein equations for the perturbations on the background. A central point was the usage of the de Donder gauge for the metric perturbations on the unperturbed background. We have outlined the ultimative necessity for fixing a certain gauge and have given physical arguments for favouring the de Donder gauge against other gauges, especially the synchronous one. Although critical contributions to the gauge-invariant formalism and the physical meaning of gauge-solutions are given in our cited papers, we shall conclude here the basic points.
One of the central points of the gauge-invariant formalism (see [5] and [6]) is the proposition that physical meaning comes only from gauge-invariant quantities or at
(*) The author of this paper has agreed to not receive the proofs for correction.
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least quantities which are gauge invariant in a certain region (physical meaning in that region). Since a gauge is understood here as the choice of a certain coordinate system, gauge-invariant variables are free from unphysical special coordinate effects.
However, the quantities constructed, for example, by Bardeen are indeed gauge invariant but in general do not have a physical meaning. Only in some certain gauges these quantities become physical ones, for example density contrasts. So in the not-at-all unique construction of the gauge-invariant variables lies a hidden choice of a gauge.
The basic proposition of the gauge-invariant formalism therefore should be rewritten in the following sense: If one wants to study density perturbations, only those gauges lead to physical results, in which the density perturbations are gauge-invariant variables. These variables are not, in general, density perturbations in other gauges. This is a strong restriction of the choice of gauges.
The density contrast in the de Donder gauge is not a gauge-invariant variable. The reason for this is the derivative character of the de Donder gauge. For studying density contrasts, therefore, the de Donder gauge is forbidden from the point of view of the gauge-invariant formalism.
The crux, however, in the whole impressing deduction of the gauge-invariant formalism lies in the interpretation of a gauge as choosing a special coordinate system. Gauge transformations are seen as coordinate transformations at fixed background. If then there are effects which are present only in a certain gauge or in a certain class of gauges, these are considered as coordinate effects and therefore as unphysical. Only the results which are common to all gauges are therefore defined as “physical”. Exactly these results are picked out by studying the propagation of gauge-invariant variables, which are for example density perturbations in certain gauges.
The situation changes drastically if a gauge is not viewed as taking a special coordinate system but as specifying a certain observer. The choice of a gauge consists in four conditions at the metric perturbations on an unperturbed background manifold. It therefore can be interpreted as the choice of an observer, who sees the metric perturbations just so that the gauge conditions are satisfied. This interpretation is supported and becomes evident by considering, for example, the gauge conditions in the synchronous gauge. Among them is the condition of vanishing of the 00-component of the metric perturbation. This corresponds to an observer seeing no Newtonian gravitational potential of the density perturbation. This observer therefore falls freely into the center of the perturbation.
The task now is not to get rid of unphysical solutions as is done in the gauge-invariant formalism. Every solution is physical here in the sense that it is
observed. But there are suitable observers—say gauges—and non-suitable ones.
Instead of constructing gauge-invariant variables, observers must be selected, who satisfy specific conditions. If the conditions are motivated and the observers—say gauges—fulfilling these conditions are found, there is no need to worry about the physicality of solutions. All solutions are physical and must be taken seriously.
As we have deduced above, the gauge-invariant formalism too distinguishes certain gauges, but the conditions for the distinction are totally different from those in the gauge-observer formalism. There those gauges are preferred in which the density perturbation is identical to a gauge-invariant variable, which has physical meaning just and only in this gauge. The selection of a gauge is therefore made there by reference to
an artificial, unphysical quantity. Density contrasts in other gauges are different
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In our “gauge-observer” formalism gauge-invariant variables (and even worse with no physical meaning) are irrelevant. The gauge is selected by manifest physical
arguments with reference to physical quantities. Having distinguished the gauge, all solutions are physical.
In our cited earlier papers we have given arguments for choosing the de Donder gauge. The observer conditions, which are satisfied by the de Donder gauge, are among others: 1) the correct Newtonian limit for the perturbations and 2) the correct asymptotic behaviour, i.e. perturbations with infinite scales should represent an unperturbed but time-shifted Robertson-Walker background. It is remarkable that just gauge solutions (which are treated as unphysical in the gauge-invariant formalism) guarantee that the above conditions are fulfilled. Clearly—and it must be so—the results in the de Donder gauge are different from that in the gauge-invariant formalism, since the density perturbation in the de Donder gauge is not a gauge-invariant variable. The gauge-invariant formalism excludes physical results by taking reference to unphysical quantities.
Mainly in [4] we gave a definition about what is meant by Stability-Instability Borders (SIBs) of the perturbations with respect to gravitational amplification. We developed a method for calculating the SIBs and gave results for the extreme cases of a dust and a radiation background.
In this paper we give a more subtle definition of SIBs, which firstly leads to a modification of the formerly found results especially in the pure radiation case and secondly gives rise to a new class of SIBs in the vicinity of which perturbations growing or decaying is strongly favoured.
We here concretely calculate the SIBs of perturbations on a realistic model of the cosmological substratum [7]. It consists of radiation and partially ionized hydrogen. The degree of ionization is given by the Saha equation and is dependent on the cosmological red-shift factor z (corresponding to cosmological time). The evolution of matter perturbations on this substratum—especially its SIBs—is determined too by the perturbations’ ability to capture the radiation. We take care of this ability by a phenomenological function f, which depends on the ratio of the spatial perturbation scale and the mean free path of the photons due to Thomson scattering. This corresponds to the phenomenological picture of the microscope base. The coupling function satisfies the limiting cases of no coupling and full coupling in the physical right sense. f takes both into account: the loss of a hydrodynamical treatment for scales smaller than the photon mean free path and the variability of matter-radiation coupling near recombination. Surely one cannot expect from a phenomenological treatment all details gained by a microscopic treatment. But the rough features—say the most important contributions—should be covered. And they are! The steep decreasing of the velocity of sound in the recombination area, its physically right scale dependence, and, last but not least, the correct Jeans instability. The advantage of this phenomenological ansatz against others, especially the kinetic one, consists in its “easy to use” character and its simplicity. The very large characteristic scales of superclusters of galaxies however, which are the important ones in this paper, are unaffected by the special radiation-matter coupling, since they are much larger than the mean free path of photons. Radiation is coupled completely.
As a result, we get a whole spectrum of SIBs in the de Donger gauge—one of them being nearly identical to the SIB in synchronous gauge in a certain z-region. In general, there are SIBs with the character of a lower border in mass of the perturbation and the character of an upper border for amplification from a purely
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gravitational point of view. Especially important are the SIBs with very strong density growing in their vicinity. They lead to the distinction of scales of growing density contrasts which correspond to superclusters of galaxies; clusters of galaxies are also favoured. Therefore for perturbation growing (by gravitation only) a scenario “from top to bottom” is preferred—first condensation of large scales and then the smaller scales. The relevant cosmological epoch for perturbation growing in the above sense lasts from z 6800 down to the post-recombination epoch. Dominant cosmological radiation stages therefore can be neglected with respect to structure formation.
The new rather large characteristic scales of superclusters are calculated mainly
before recombination and just in the recombination and are clearly much larger than
the photon mean free path there. So radiation is coupled completely and the hydrodynamical treatment is justified.
The reason for the existence of these large stability borders is therefore not the matter-radiation coupling mechanism but a consequence of the chosen de Donder
gauge and therefore a purely general relativistic effect. Since the evolution of
cosmological density perturbations was studied in the de Donder gauge only by me and my coworkers, it is not astonishing that the new stability borders were not found earlier. Whether the stability analysis is based on a kinetic theory or not is therefore
irrelevant for the existence of the large scales.
The organization of the paper is as follows: in sect. 2, we give in subsect. 2.1 a short summary of the important equations for perturbation propagation in the de Donder gauge and in subsect. 2.2 we introduce the concrete substratum model and the perturbations on it especially with respect to the velocity of sound.
In sect. 3 we give the general definition of the SIBs and the way to calculate them. Especial attention is paid to the “locally static” metric perturbations at the SIBs generated by the density perturbations. These “gravitation potentials” are compared with the Newtonian case. Of profound interest is further the determination of the character of the SIBs, i.e. whether they are lower or upper borders for density growing. This and the comparison of the corresponding results in synchronous gauge is a further subject of this section. We localize the therm in the SIB equation which is responsible for the differences in the two gauges and investigate its influence on some special cases analytically. This also leads to analytically statements to the character of the SIBs in these special cases.
In sect. 4 we present and discuss the numerical results with respect to the SIBs of the substratum model given in sect. 2. Finally, in sect. 5 we give the conclusions.
2. – Perturbations on a cosmological background in the de Donder gauge
2.1. The perturbation equations. – The set of equations in the de Donder gauge for the cosmological perturbations in their contrast form is
(
see [4], eqs. (3.1) to (3.3))
1 H2 h n n tt1 5 H h n tt1 Ahtt2 6 H h n jj1 Bhjj4 Cd , (2.1) 1 H2 h n n jj1 5 H h n jj1 Dhjj2 2 3 1 H h n tt1 Ehtt4 Fdz, (2.2) 2 1 H2d O 2
u
2 H n n H2Hn 1 8 1 H 1 3 1 Hbv
d n 1Phjj2 1 2u
6 1 H 1 H n n H2Hnv
h n tt1 Qhtt4 Sd (2.3)1319 with A 461y , (2.4a) B 426
u
2 1 H n H2v
, (2.4b) C 42 H n H2( 1 13b) , (2.4c) D 4221y , (2.4d) E 422u
2 1 H n H2v
, (2.4e) F 422 H n H2( 1 2b) , (2.4f ) P 43u
3 1 H n n H H nv
, (2.4g) Q 423u
4 1 H n n H H n 1 H n H2v
, (2.4h) S 4 H n n n H2Hn 1 8 H n n H H n 1 4 H n H2 1 15 1 y 1 bu
6 H n H2 1 3 H n n H H n 1 15 1 6 1 H vn s vsv
. (2.4i)The meaning of the symbols is as follows: htt, hjj and d are the perturbations in their contrast forms of the cosmological background. The cosmological background metric is taken as a flat Robertson-Walker metric. So if gmn is the perturbed metric, it is decomposed in gmn4 gmn( 0 )1 hmn with gmn( 0 ) the background metric ( gtt( 0 )4 21 , gi j( 0 )4
R2di j, R 4cosmic scales factor, indices i, j run from 1 to 3) and hmnits perturbations. The metric perturbation contrasts are given by htt4 httOgtt( 0 )4 2 httand hjj4 hjjOgjj( 0 )4
hjjO3 R2, here summation from j 41 to j43 is understood. The density contrast d is given by d 4r1O(r 1 p) with r1 the density perturbation of the background
substratum and r , p the energy density and pressure of the background substratum. The dots in eqs. (2.1) to (2.4) mean derivation with respect to time.
H is the Hubble constant, vsthe speed of sound in the perturbations, b 4vs2Oc2with
c the velocity of light. y »4 (k2c2) O(R2H2) 4Hor2Or2 with ROk4»r the spatial scale of
perturbations and Hor4 cOH the horizon in the cosmological evolution of the
background. (Clearly r is dependent on the background expansion.)
Equations (2.1) to (2.3) correspond to linearizations in the perturbations in the Einstein equations on a flat Robertson-Walker background if the de Donder gauge is used ([1-4]). Further for the space-dependent part of the perturbations the ansatz of Fourier amplitudes is used (A eikx).
For the concrete calculations it is convenient to express all time dependences and derivations in (2.1) to (2.4i) by dependences and derivations with respect to the
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cosmological red-shift factor z 4R(T0)
OR (R(T0)
4 cosmic scale factor today, R 4 cosmic scale factor in the cosmological past under investigation, here R b R(T0)). The resulting equations are h 9tt4 2 1 z2Ahtt2
y
H, z H 2 4 zz
h 8tt2 1 z2Bhjj2 6 zh 8jj1 1 z2Cd , (2.5) h 9jj4 2 1 z2Dhjj2y
H, z H 2 4 zz
h 8jj2 1 z2Ehtt2 2 3 zh 8tt1 1 z2Fd , (2.6) d 941 1 z2Phjj1 1 z2Qhtt1 1 2 zu
6 1 H n n H H nv
h 8tt2 (2.7) 2y
H, z H 2 3( 1 1 b) 1 z 2 4 1 z 2 2 H n n zH H nz
d 82 1 z2Sd . Here we used zn 4 2Hz (2.8)and by the Einstein equations for the background
H 4H(T0)
o
e r(T0) B c 2 (2.9) with H(T0)4 Hubble constant today, e 4 energy density of the background in dependence on z, rB(T0)4 density of baryonic matter today. So all background quantities can be expressed in dependence on z if the energy density and pressure of the substratum are known in dependence on z.
2.2. The background substratum and the velocity of sound with respect to matter
perturbations. – A more detailed analysis to this point can be found in [7]. Here we
restrict ourselves to the basic equations and ideas. For the cosmological-background substratum we take an ideal gas of nonrelativistic partially ionized hydrogen and radiation neglecting the influence of primordial helium. The pressure and energy density are given by
p 4 (11x) nBkBT 1 1 3sT 4, (2.10) e 4nBmHc21 3 2( 1 1x) nBkBT 1sT 4 (2.11)
with nB4 baryon number density, T 4 temperature , mH4 mass of hydrogen atom,
c 4speed of light, kB4 1 .38 Q 10216ergOK (Boltzmann constant). The degree of
ioniz-ation is given by the Saha equioniz-ation as
x 4 ne nB 4 a3 /2 e 2xeOkBT nB ]
k
1 12a23 /2nBe1xe/kBT2 1 ( , (2.12) a »4g
mekBT 2 pˇ2h
(2.12a)1321
with ne4 electron number density, xe4 2 .18 Q 10211erg (ionization energy of hydrogen),
me4 electron mass.
Because of the expansion of the universe we set nB4 nB(T0)z3 with nB(T0)4 baryon
number density today. Being mainly interested in the vicinity of the recombination epoch and earlier times, we can set in good approximation the temperature T equal to the radiation temperature, so
T 4T(T0)
rad z
(2.13) with Trad
(T0)
4 2 .7 K . The results therefore should not be extrapolated too much to times later than the recombination epoch.
For calculating the speed of sound with respect to perturbations of this substratum we must carefully take into account the influence of radiation in the considered matter perturbations. This is done in [7] by a simple self-consistent phenomenological description of matter perturbations in the cosmological recombination area. The basic idea is to treat the coupling of radiation to matter by a function f, which depends on the ratio of perturbation scale to the mean free path of photons via Thomson scattering.
This phenomenological coupling function can be deduced in two ways. First by the relation between the virial and the thermic energy of matter distribution with constant density in the volume ( 4 pO3) R3. The particles get an additional force transmission by
the collisions with photons via Thomson scattering. The virial theorem for constant mass densities in a volume ( 4 pO3) R3reads
3
2 ( 1 1x) nBkT 4p2Vi (2.14)
with x 4degree of ionization, nB4 density of baryons, T 4 gas temperature, p 4
pressure , R 4radius of the matter distribution. The magnitude Vi is the virial of the transmission of force density of radiation to the matter particles via Thomson scattering and is given by
Vi4
0
R
fRMdr
(2.15)
with fRMthe force-density transmission (radiationOmatter):
fRM4 nesef
1
c
(2.16)
(
ne4 electron number density, se4 Thomson cross-section (se4 6 .66 Q 10225cm2),c 4speed of light, f4radiation flux
)
.The radiation flux at r is given by the absorption law:
f 4f0e2neser.
(2.17)
The virial in (2.15) reads with (2.16) and (2.17)
Vi4
1
c f0( 1 2e
2neseR) . (2.18)
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no radiation pressure contribution, and R KQ full radiation) must be satisfied, so we get ( 1 Oc) f04 ( 1 O3 ) sT4. We have set TRadiation4 TMatter, which is a good approximation
in the vicinity of recombination. We get finally
Prad/mat4 1 3sT 4f (q) (2.19) with f (q) 412e2q (2.20)
and q is the ratio of the spatial scale to the mean free path of the photons in matter with respect to Thomson scattering,
q »4 R lph with lph4 1 nese . (2.21)
So we have deduced the coupling function f. The energy density in the matter distribution related to the radiation-matter interaction is then given in the same way by
e 4sT4f (q) .
(2.22)
In [7] the coupling function f is obtained in another way. For the “captured” radiation energy density ecaptvia Thomson scattering in a matter density perturbation of scale R
we made the ansatz
decapt4 (eQ2 ecapt) dq
(2.23)
with eQ the total cosmological radiation energy density of the background. The solution of this differential equation leads to the same function f as in (2.20).
The energy density and pressure in the perturbations are given now by
e(p) 4 nBmHc21 3 2 ( 1 1x) nBkBT 1sT 4f (q) , (2.24) p(p)4 ( 1 1 x) nBkBT 1 1 3 sT 4 f (q) . (2.25)
With nB as perturbed and nB( 0 ) as background baryon number density we set for the
perturbation itself n1»4 nB2 nB( 0 ). The corresponding adiabatic temperature
perturbation reads T14 (e(p) 1 p(p)) /n( 0 ) B 2 ¯e(p)/¯nBNnB( 0 ) ¯e(p)/¯TNT( 0 ) n1 (2.26)
with nB( 0 ) and T( 0 ) background quantities and e( p), p( p) are given by background
quantities. The velocity of sound vsor the ratio b 4vs2Oc2is now given by
b 4 p1
e1
1323 with p14 ¯p(p) ¯nB
N
nB( 0 ) n11 ¯p(p) ¯TN
T0 T1, (2.28) e14 ¯e(p) ¯nBN
nB( 0 ) n11 ¯e(p) ¯TN
T0 T1. (2.29)In all derivatives ¯e( p)
O¯nB, ¯e( p)O¯T , ¯p( p)O¯nB, ¯p( p)O¯T we do not take derivatives
of f, which means that everything with respect to perturbations coming from radiation is weighted with f.
Because of the f-coupling of radiation to matter in the matter perturbations, the speed of sound is clearly scale dependent.
Now the model for the background substratum in the vicinity of the recombination epoch and times before is completed as for the matter perturbations on it and the corresponding speed of sound. The propagation of the perturbations is governed by the differential equations (2.1) to (2.3), respectively, (2.5) to (2.7). The stability borders of these self-gravitating perturbations are most interesting as is the propagation of the perturbations near these borders. The Jeans stability border is calculated in [7]. We now calculate the stability borders, however, in full generality, i.e. without the “Newtonian restrictions” of vanishing background pressure, small perturbation scales in comparison to the horizon and nonrelativistic speed of sound.
3. – The stability borders of the perturbations
The stability borders of self-gravitating matter perturbations at every instant of cosmological time t are certain spatial scales of the perturbations (corresponding to their matter content given by the background matter density rB). It is convenient to
parametrize the cosmological time t by the red-shift factor z
(
as is done in (2.5)-(2.7))
. The spatial scale r of perturbations appears in (2.5)-(2.7) in y 4H2orOr2 and q 4rOlph
and can therefore be expressed for example by y. The general spatial scale r of the perturbation can be expressed (see [7]) by its (z-independent!) matter content in units of solar masses by Sk»4 lg
g
M MUh
4 lgk
1 MU 4 p 3 r (T0) B z3r3l
. (3.1)The task of this section, therefore, is to give precise definitions of what is meant by stability border (or Stability-Instability Border = SIB) and calculate ySIB(z),
respec-tively, rSIB(z) or Sk( SIB )(z).
3.1. The meaning of stability borders (SIBs). – Let d( 0 ), h
jj( 0 ) and htt( 0 ) be perturbations at a cosmological red-shift z0. Assume that no initial perturbation grow
or decay is there, i.e. d( 0 )
8 , hjj( 0 )8 and htt( 0 )8 vanish too. That means that the perturbations are awoked at z0 with no initial growing of decaying and then left to
themselves. A necessary condition for stability borders at z0 is that the metric
perturbations should be as static as possible there, that means also hjj( 0 )9 and htt( 0 )9 should be zero there. This corresponds to the Jeans picture that at the SIBs we have locally no dynamics of the perturbations—attractive “forces” and respulsive “forces”
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are in equilibrium. For this locally static case we get with (2.5) and (2.6) conditions—say initial conditions—for the metric perturbations
htt( 0 )4 CD 2BF AD 2EBd ( 0 ) , (3.2a) h( 0 ) jj 4 AF 2EC AD 2EBd ( 0 ) (3.2b)
with A 2F given by (2.4a) to (2.4f ). These are just the relations (3.8) and (3.9) in [4]. We are now ready to give the definition for a stability border (SIB):
That spatial perturbation scale r or mass content Skat a given z0at which—under
the conditions d 8( 0 )
4 hjj( 0 )8 4 htt( 0 )8 4 0 and (3.2a), (3.2b)—a change in sign of d( 0 )9 occurs, is a stability border (SIB). If we insert (3.2a) and (3.2b) in (2.7) we obtain
1 z2
k
AF 2EC AD 2DBP 1 CD 2BF AD 2EBQ 2Sl
d ( 0 ) 4 d( 0 )9 . (3.3)For spatial scales such that d( 0 )
9 D 0 at given z0the density contrast will start to grow
in time, in the opposite case it will start to decay. So that scale at which a change in sign of d 9 occurs clearly is a stability border.
The procedure given here for finding the stability borders is very similar to that given in [4]
(
see especially (3.10) and (3.11) there)
but some caution is advisable. Settingd( 0 )
9 zero in (3.3) can fail in finding the SIBs. There are two reasons for this. First, it could be that d( 0 )
9 is zero at a certain scale without change of sign by scale variation,
i.e. d( 0 )
9 attains an extremum there in dependence on the scale. This scale clearly is not a stability border. Second, it could be that, at a certain z0 and scale, htt( 0 ) and hjj( 0 ) get singular accompanied by a sign change when the scale is varied. This would imply a sign change for d( 0 )
9 in (3.3) at a certain scale without being zero. Because of the sign change this is clearly a SIB, but it could not be found by setting d( 0 )
9 4 0 in (3.3). So in the first case one would misinterpret a scale as a SIB, whereas in the second case one would lose the SIBs. In this way the procedure given in this paper for finding the SIBs is refined and subtler.
3.2. The locally static metric perturbations in comparison to Newtonian
potentials. – Equations (3.2a) and (3.2b) for locally static metric perturbations at
arbitrary scales can be investigated in some special cases. i) Newtonian conditions ( p 40, yc1, bb1).
(2.4a) to (2.4f ) give A 4y, B423, C423, D4y, E421 and F43, so we get approximately h( 0 ) tt 4 2 3 yd ( 0 ), (3.4a) hjj( 0 )4 2htt( 0 ) (3.4b)
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and for the ratio qu»4 htt( 0 )O hjj( 0 ) we obtain qu4 21 . The metric perturbations in (3.4a) and (3.4b) are exactly the Newtonian static gravitational potentials being the solutions of the Poisson equation for Fourier amplitudes,
Df 424pGr1
g
D 4¯i¯i4 1 R2¯i¯ih
(3.5) with F 4 c 2 2 h ( 0 ) tt 4 2 c2 2 htA e ikx, (3.5a) h( 0 ) jj 4 3 R2htt( 0 ), (3.5b) r 1p42 1 4 pG H n , H 4 2 3t 21, Hn 4 23 2H 2 , (3.5c) d 4r1/(r 1p) , (3.5d) y 4 k 2c2 R2H2 . (3.5e)So under Newtonian conditions the locally static metric perturbations are Newtonian gravitational potentials.
ii) Pure radiation perturbations on a radiation background but again small scales
(
p 4 (1O3) e, b41O3, yc1)
.(2.4a) to (2.4f ) give A 4y, B40, C428, D4y, E40 and F48O3, so we get
h( 0 ) tt 4 2 8 yd ( 0 ), (3.6) h( 0 ) jj 4 2 1 3h ( 0 ) tt . (3.7)
The ratio qu4 htt( 0 )O hjj( 0 ) is again negative and has the value 23.
The solutions (3.6) and (3.7) are again “Newtonian” gravitational potentials. But here the modification must be made that the pressure itself acts gravitationally in the Poisson equation. So r1in (3.5) must be replaced by r11 3 p1. Further the source of hjj( 0 ) is not A2(r11 3 p1) as for htt( 0 ) but A1(r12 p1) as can be seen from (2.5) and (2.6).
With those modified sources (due to the gravitational action of pressure) the Poisson equation gives exactly (3.6) and (3.7).
iii) Pure radiation perturbations on a radiation background, but now y K0 (scale
much larger than the horizon).
(2.4a) to (2.4f ) give A 46, B40, C428, D422, E40 and F48O3. As was to be expected we now get no Newtonian solutions for htt( 0 ) and hjj( 0 ) in (3.2a) and (3.2b). The
1326 solutions are htt( 0 )4 2 4 3 d ( 0 ) 4 hjj( 0 ). (3.8)
The ratio quis now positive and has the value 11, hjj( 0 )has changed its sign. Whenever some metric perturbation has changed its sign in comparison to the Newtonian case it seems reasonable to expect some antigravitational influence from it.
So, finally it is to say that in determining the SIBs it is interesting to compare the locally static metric perturbations with the Newtonian case. This means studying the signs of that metric perturbations or the sign of qu. The metric perturbations (3.2a) and (3.2b) are the generalizations of static Newtonian potentials in dependence on background and scale.
3.3. Propagation near the stability borders—character of the SIBs. – Whether the density contrast will grow or decay for scales or mass contents greater than the SIB determines the character of the SIB. If the density contrast will grow for scales greater than the SIB, the SIB has the character of a lower limit for density contrast growing. In the opposite case the character is that of an upper limit. To get the character of the SIBs it is only necessary to take scales slightly differing from them, take the locally static metric perturbations at that scales and calculate in (3.3) the sign of d( 0 )
9 . If the SIB comes from a sign change in d( 0 )9 without going through zero, the values for d( 0 )
9 , htt( 0 ) and hjj( 0 ) near that SIB will be very large, which means that the growing, respectively decaying, of the density contrast will be very strong. So scales near that SIB are preferred for growing of perturbations.
3.4. Comparison of the SIB condition in de Donder and synchronous gauge. – The condition for a SIB is a change in sign of d( 0 )9 with locally static metric perturbations (contrasts!). The corresponding equation for d( 0 )
9 in de Donder gauge in (3.3), it reads 1 z2[ (Ph ( 0 ) jj 1 Qhtt( 0 )) 2S02 by] d( 0 )4 d( 0 )9 (3.9) with S0»4 S 2 by . (3.9a)
The corresponding equation in synchronous gauge is (see [4]) 1
z2[2S02 by] d ( 0 )
4 d( 0 )9 . (3.10)
So difference in the SIBs in the two gauges come from
j »41 [Phjj( 0 )1 Qhtt( 0 )]
(3.11)
in the de Donder gauge. We now calculate j for some special cases and investigate its influence on the SIBs.
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a) Pressure 40, bb1.
(2.4a) to (2.4i) give P 40, Q43O2 and (3.2a) reads
h( 0 ) tt B 3( 5 2y) y2 1 4 y 2 15 d( 0 ). (3.12)
Because of P 40 we have no hjj( 0 )-term:
S04 2 3 2 2 9 b B 2 3 2 . (3.13) So (3.9) reads
y
3 2 3( 5 2y) y2 1 4 y 2 15 1 3 2 2 byz
d ( 0 ) 4 d( 0 )9 (3.14) and j 4 3 2 3( 5 2y) y2 1 4 y 2 15 . (3.15)If j can be neglected, (3.14) goes just to the Jeans condition (see [7]) and is identical to (3.10). j can be neglected for very large y—say small scales. So we get the result that in the Newtonian case (vanishing background pressure, b b 1 and very small scales) the SIB in the two gauges is approximately identical and attains approximately the Jeans value. In (3.14) 2S04 3 O2 gives the self-gravitating term governed by the background
and 2by gives the repulsive pressure term which grows for smaller scales. We now study the influence of j , i.e. we expand j in the vicinity of the stability border ySIB.
Setting y 4ySIB1 Dy in j and linearizing in Dy gives
j 4jSIB2 jSIB 2 ySIB1 4 y2 SIB1 4 ySIB2 15 Dy 2 9 2 1 y2 SIB1 4 ySIB2 15 Dy (3.16)
with jSIB»4 j( y 4 ySIB).
For the case ySIBc1 we get jSIBB 2 ( 9 O2 )( 1 Oy) E 0 . That means j acts like a
repulsive pressure term and is antigravitating. So the SIBs are slightly moved to larger scales compared with the synchronous gauge and the Jeans limit.
In the above case ySIBc1 we obtain for j
j BjSIB( 1 1ySIBDy) .
(3.17)
Greater scales than the SIB mean
!
y E0, so NjNENjSIBN , which corresponds to aweakening of the repulsive influence of j . For smaller scales the repulsion is amplified. So we get the result: For vanishing background pressure, b b 1 and ySIBc1 the
j-term is small and has an antigravitating character. In its scale dependence in the vicinity of a SIB it acts like a pressure term, so the SIB has the character of a lower bound.
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Since ySIBc1 is reasonable by the Jeans analysis and the results in synchronous
gauge, one sees from (3.14) that there is a SIB too which corresponds to a sign change in d( 0 )
9 without going through zero. This SIB gets the value ySIB4 2 2 1k19 B2.36.
Since now jSIB gets singular, the above expansion (3.16) makes no sense. We get by
(3.15) for j at y 4ySIB1 Dy in linear approximation
j 4a 1
Dy with a D0 .
(3.18)
So for smaller scales than ySIB(i.e. Dy D0) j acts gravitatively, i.e. attractively and for
greater scales (i.e. Dy E0) it acts repulsively. Therefore the SIB considered here has the character of an upper bound for density contrast growing.
The final result here for the case of negligible background pressure and velocity of sound is that there are at last two SIBs—the one for small scales, which corresponds to the Jeans limit (and that found in the synchronous gauge) and another for scales comparable to the horizon ( ySIBB 2 .36 ). The first has the character of a lower bound
for growing of density contrast, the second of an upper bound. In the vicinity of the upper bound the metric perturbations and d( 0 )
9 get very large values, so the region just below the upper bound (masses just below the upper bound) is preferred for density contrast growing. The existence of an upper boundary has its origin purely in j , that term which is specific in the de Donder gauge and not present in the synchronous gauge. Clearly the j-term can give rise to even more SIBs and a corresponding complex structure of stability borders. This will be investigated numerically in the next section. Especially on this level here it cannot be guaranteed that b stays small and constant for large scales, since radiation coupling can get important. Before discus-sing the results of the numerics we will study another special case analytically.
b) Radiation perturbations in pure radiation (b 41O3).
(2.4a) to (2.4i) give P 423, Q416, S4241 (1O3) y, so that
h( 0 ) tt 4 28 1 6 1yd ( 0 ), (3.19) hjj( 0 )4 1 38 1 y 22d ( 0 ) , (3.20) S04 24 , (3.21) j 42
g
8 1 y 22 1 48 1 6 1yh
, (3.22) [j 142by] d( 0 ) 4 d( 0 )9 . (3.23)Again 2S0 gives the self-gravitating term governed by the background and 2by the
repulsive pressure term. We immediately obtain two SIBs analytically: y(1)
SIB40, ySIB(2)42
(note, since y »4H2
orOr2, only positive y make sense). Since ySIB( 1 )4 0 is a SIB with a zero
for d( 0 )
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former paper [4] ySIB4 0 was found as the only SIB in the radiation case. The reason
for this was the strong SIB condition of vanishing d( 0 )9 , herewith the SIB ySIB( 2 )4 2
clearly was excluded. Further it was argued in [4] that ySIB4 0 means infinite
perturbation scales. Thus is only true if the horizon Horis finite. This is the case in a
model universe of pure radiation. For the substratum we discussed in sect. 2, this
however is not the case. Really pure radiation is there only at the big bang. Since in pure radiation there we have Hor2A t2 and r2A t , y 4 0 implies non-infinite scales. We
will discuss this in the next section when the numerical results are presented. Let us study now the influence of j at the two found SIBs y( 1 )
SIB and ySIB( 2 ). We take
again y 4ySIB( 1 ), ( 2 )1 Dy with small Dy . In the case of ySIB( 2 )4 2 we obtain
j B28 1
Dy . (3.24)
So for scales above the SIB (i.e. Dy E0) j is positive and has therefore a gravitating (attractive) character; for scales below (i.e. DyD0) j is negative and is therefore repulsive. We conclude that the SIB with ySIB4 2 is a lower boundary (in mass) for density
contrast growing.
In the case of ySIB4 ySIB( 1 )4 0 we expand and linearize j in the vicinity of ySIB( 1 ) with
y 4y( 1 )
SIB1 Dy and obtain
j 4jSIB1
10
3 Dy with (3.25)
jSIB4 j(y 4 ySIB( 1 )) 424 antigravitating .
(3.25a)
So for scales above the SIB (i.e. Dy E0) the repulsive character of j is amplified, in the opposite case it is weakened. We therefore conclude that the SIB with ySIB4 0 is an
upper boundary for density contrast growing. Since in a pure radiation substratum y 4 0 corresponds to the infinite perturbation scale, we get the final result that above the lower-boundary SIB (i.e. ySIB( 2 )4 2 ) all density contrasts will grow. This is in contrast to
a result in [4], which stated that no stability borders do exist in a pure radiation substratum. We mentioned that the reason for this misinterpretation lies in the loss of one SIB because of the too strong SIB condition used there.
The SIB in the synchronous gauge
(
see (3.10) and (3.21))
is ySIB4 12 . This is theonly SIB and has the character of a lower boundary. Since in the pure radiation case in the de Donder gauge too no really upper-boundary SIB does exist, the only difference between the two gauges manifests in different lower-boundary SIBs. The SIB in the de Donder gauge is at slightly greater scales. This is the same effect as for the lower-boundary SIB in the pure-dust case. This effect is easily understandable. Since the observer in the synchronous gauge sees no metric perturbation htt, he falls freely towards the center of the perturbations, whereas the de Donder observer is fixed at the background. A decaying density contrast as seen from the de Donder observer can be seen as growing by the synchronous gauge observer if he is only falling in the perturbation swifter than the perturbation decay. So as a consequence the synchronous-gauge SIBs are at smaller scales than the corresponding ones in the de Donder synchronous-gauge.
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As we have mentioned in the dust case analysis the j-term in the SIB condition in the de Donder gauge
(
(3.9) and (3.11))
can give reason to a very complex SIB structure different from the results in the synchronous gauge. We will now turn to the results related to the SIB structure of the realistic cosmological model presented in sect. 2.4. – Numerical results
The model for the background substratum and matter perturbations on it in the vicinity of the recombination epoch and times before was presented in sect. 2. The relevant equations for pressure, energy density and velocity of sound coming from the matter perturbations are (2.18) and (2.24). With these equations we calculate at given z those masses of perturbations—given by Sk4 lg (MOMU), see (3.1)—at which d( 0 )9 in
(3.3) (de Donder gauge), respectively, in (3.10) (synchronous gauge) changes its sign. The found curves Sk(z) are the stability borders (SIBs) of the matter perturbations. The results of the numerical calculations are presented in fig. 1.
Further we tested whether the SIBs are lower, respectively upper, borders for growing of density contrasts. As mentioned in the preceding section, this can be done by calculating d 9 at fixed z for masses Skjust above and just below the SIB. In the case
Fig. 1. – The Stability-Instability Borders (SIBs) Sk(lg (z)) with Sk»4 lg (MOMU) and z the
cosmological red-shift factor. The curves 1 to 4 are the SIBs in the de Donder gauge, 5 is the one in the synchronous gauge. The letters “l” and “u” mark the character of the SIBs; “l” stands for lower border and “u” for upper border. The “anormal” SIB with very strong growing or decaying of density contrasts in its vicinity is additionally marked by “1”. Whether the strong growing occurs below or above the SIB is given by the position of the “1” below or above the curve. The curve 6 gives the horizon in mass scales calculated by the z-dependent background density.
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Fig. 2. – Character of the SIBs in the de Donder gauge given in fig. 1. The numbers correspond to the curve numbers in fig. 1. The magnitude CHARis defined in sect. 5. Positive values of CHARstand
for lower-character SIBs, negative values for upper-character SIBs. Additionally, the character is marked by “l”, respectively, “u”. The absolute value of CHARis the curve number.
of the de Donder gauge the static expressions for htt and hjj
(
see (3.2a) and (3.2b))
must be inserted in (3.3). If the SIB is a lower border, then d 9(z, Sk(1)) D0 for Sk(1)D Sk( SIB ) (mass for stability border). If the SIB is an upper border, then d 9(z, Sk(1)) E0. ForSk(2)E Sk( SIB ) we have the opposite cases. As test quantity for the character of the SIB we constructed CHAR»4 cu sign [d 9(z, S(1) k ) ] 2sign [d9(z, Sk(2)) ] 2 . (4.1)
We choose Sk(1)4 Sk( SIB )1 0 .05 and Sk(2)4 Sk( SIB )2 0 .05 . The quantity cu counts the different curves, its values are 1, 2, 3, 4 (see fig. 1). If the SIB is a lower border, then
CHAR4 cu, if it is an upper border, then CHAR4 2 cu. The results for the de Donder SIBs are given in fig. 2 and are indicated at the curves in fig. 1 and fig. 2 as letters “l” for lower and “u” for upper. The SIB in the synchronous gauge is a lower border.
As shown in the previous section, there are two kinds of sign changes—i.e. SIBs—of d 9; a sign change caused by a zero of d9 and a sign change caused by a pole. The latter is the case for the de Donder SIB labeled by “4” in fig. 1 and 2. The static metric perturbations and d 9 become very large near this SIB, so density growing is
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Fig. 3. – The quantity qu defined in sect. 4 and 5 is a measure for how “Newtonian” the locally static metric perturbations (gravitational potentials) near the SIBs are. Here qu(Sk(2)) in dependence on lg (z) gives the situation just below the SIBs in the de Donder gauge. Negative qu correspond to Newtonian or quasi-Newtonian potentials, positive qu correspond to “anti-Newtonian” potentials. The numbers correspond to the SIB numbers in fig. 1. Additionally, the character of the SIBs is again marked by “l” and “u”.
preferred there: in the case of lower limit character just above this SIB, in the case of upper limit character just below. These distinguished regions are marked in fig. 1 by “1” signs near the corresponding SIB.
In the previous section we introduced the quantity qu4 htt( 0 )Ohjj( 0 ) with htt( 0 ), hjj( 0 ) as the locally static metric perturbations at the SIBs or near them. We calculated these metric perturbations in the special case of a pure dust background with small perturbation scales and a pure radiation background with small and large perturbation scales. We found that qu4 21 corresponds to Newtonian metric perturbations—say gravitational potentials—whereas qu4 11 corresponds to the extreme anti-Newton case. The first case appeared for the pure dust case and very small scales (SIBs), the second for the pure radiation case and scales (SIBs) comparable to the horizon.
In fig. 3 and 4 the exact numerical results for qufor the different de Donder SIBs are presented. In fig. 3 qu(2)4 qu(Sk(2)4 Sk( SIB )2 0 .05 ) is given—i.e. qu for masses just below the SIB. Figure 4 shows qu(1)4 qu(Sk(1)4 Sk( SIB )1 0 .05 )—i.e. qu for masses just above the SIB. The above-mentioned analytic results for the special cases are fully recovered.
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Fig. 4. – Like fig. 3, but now just above the de Donder SIBs. So qu(Sk(1)) is plotted in dependence on lg (z).
5. – Conclusions
We have studied a cosmological substratum built of hydrogen, plasma and radiation in the recombination epoch of hydrogen and times before. In this model the degree of ionization of hydrogen is considered by Saha’s equation. We investigated matter perturbations on this substratum and took care of their ability to capture radiation by the phenomenological function f, which depends on the ratio of the perturbation scale to the mean free path of the photons via Thomson scattering. The equations of motion for the relevant metric and density contrasts are given by the linearized Einstein equations in the de Donder gauge. Since we are mainly interested in the stability borders of the perturbations with respect to self-gravity and pressure “forces”, we did not study the dynamics of the perturbations explicitly. Instead we gave a definition of a Stability-Instability Border (SIB) and a calculus to find them, which is a refinement of one given in a former paper [4]. This refinement leads to one more SIB, which was lost in the former too rough calculus. The vicinity of this new SIB is characterized by very strong growing or decaying of density contrasts.
All SIBs found in the de Donder gauge are given in fig. 1. Here we have also plotted the SIB calculated in the synchronous gauge (there is only one) and the cosmological horizon. Further on we have studied the character of the SIBs (lower or upper border) and marked the regions in the vicinity of that SIB in which density contrasts grow very strongly. Lastly we have discussed the locally static perturbation contrasts in the vicinity of the SIBs with respect to whether they have Newtonian character or not (see fig. 3 and 4). The result is that the only SIBs with Newtonian or quasi-Newtonian character are the lower character SIB 1 and the “normal” upper character piece of SIB 3.
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The most striking result is that in the de Donder gauge there exists a SIB below the horizon, which has the character of an upper border (between z B6800 and
z EzRecombination). This SIB is in the lower part of the z-interval of that kind with very
large density contrast growing or decaying in its vicinity. So density contrast growing is favoured for masses between 1017
MUand 1018MUfor z 3700 . This is the scale of
superclusters of galaxies. Between z B3700 and zB6800 the upper character SIB is a
normal one (very small density growing, respectively, decaying, below, respectively, above it).
Below the whole upper-character SIB for z 6800 there is a “normal” lower-character SIB, which is nearly identical to the SIB found in synchronous gauge (including the steep slope during recombination, which can already be found in the Jeans analysis). The synchronous gauge SIB is at slightly smaller scales—especially for larger z (i.e. growing importance of radiation) and larger masses (scales). This result was already analytically found and interpreted in sect. 3.
So from a pure gravitational point of view (this is the argumentation with SIBs) we have in the de Donder gauge for z 6800 and perturbations below the horizon the following situation.
Before recombination all masses smaller than B1015.5M
Udecay. For 3700 z 6800
masses between 1015.5MU and 1017MU grow with preference to B1016MU (scale of
clusters of galaxies). This growing is cosmologically early but slow. Below z B3700 preference in density contrast growing is given for masses between 1017M
U and
1018
MU (scale of superclusters of galaxies). This growing is cosmologically a bit later
than that for 1016M
Ubut much more rapid. So we have a scenario from the top to the
bottom. First superclusters of galaxies will come to appearance, then clusters (either by the earlier but slow growing or by fragmentation of the superclusters).
During the recombination (as in the Jeans and the synchronous gauge case) the lower character SIB drops drastically to 104M
U (because of the drastic slope of the
speed of sound in the matter perturbations). So all masses between 104M
Uand 1018MU
are allowed to grow. A mass spectrum for galaxies, which ends at 1012MU, cannot be
deduced by pure gravitational considerations.
For very large z (z 6800 ) the SIB situation is as follows: while the synchronous gauge SIB (only one and lower character) is just a continuation of the case z 6800 , in the de Donder gauge we have a different picture. Here we have as lower-character border the continuation of the “anormal” upper-character border in the region z 3700 . This lower-character border is anormal too. Therefore just above this SIB the density contrast growing is very strong. The entire “anormal” SIB (labeled by “4” in fig. 1) is by the way below and parallel (nearly!) to the horizon and lies above the SIB in the synchronous gauge. Above the horizon there exists a “normal” upper-character SIB.
So for very large z (z 6800 ) there exist in the de Donder gauge two SIBs, an “anormal” lower-character one and a “normal” upper-character one. This situation especially appears in the case of pure radiation (large z; see sect. 3). The lower-character one was not found in a former paper [4], because the calculus for finding SIBs there did not include the possibility of “anormal” SIBs. The upper-character one was found in this paper as infinite. This is only true if the horizon is not zero (see sect. 3). In our model, however, really pure radiation exists only at the big bang and there the horizon is zero. Especially in the radiation case the horizon is proportional to t, whereas the scale is proportional to t1 O2(with t 4cosmological time).
So y 4 ( Horizon )2O( Scale )2goes to zero for t K0 with no need of infinite scales. Therefore the conclusion given in [4], that in the pure radiation case there are no
1335
instabilities of radiation perturbations possible, is not true. Instead there always exists an “anormal” lower-character SIB (below the horizon but above the synchronous gauge SIB) and a normal upper-character SIB above the horizon. This SIB is infinite in the case of a nonvanishing horizon. In realistic models, however, really pure radiation occurs only at the big bang and there the horizon vanishes. So the SIB structure for large z in fig. 1 is explained.
For the growing of density contrast, however, these cosmological epochs with large
z are not important. On the one hand, density contrasts with masses just above the “anormal” lower-character SIB are strongly amplified, but these masses, on the other
hand, quickly fall below the SIB during the cosmological evolution. The contrasts therefore do strongly decay.
So we conclude that the important region for structure formation below the horizon in the universe lies between the SIBs labeled by 1, 4 and 3 in fig. 1 and z 6800 . From a pure gravitational point of view the scenario is one from the top to the bottom. First, superclusters of galaxies did appear with masses Msupercl .B ( 1017–1018) MU, then
clusters with MclusterB 1016MU. The mass spectrum of galaxies cannot be deduced. This
can be done by taking the explicit dynamics of density contrasts into account, especially near the region of the steep slope of the lower-character SIB labeled by 1). For an explanation of the galaxy mass spectrum a pure gravitational point of view—which manifests in SIBs—therefore is not adequate. This will be treated in a subsequent paper.
R E F E R E N C E S
[1] ROSEB., RAHMSTORFS. and DEHNENH., Gen. Relativ. Gravit., 20 (1988) 1193. [2] ROSEB. and DEHNENH., Gen. Relativ. Gravit., 21 (1988) 705.
[3] ROSEB. and CORONAM., Gen. Relativ. Gravit., 23 (1991) 1317. [4] ROSEB., Gen. Relativ. Gravit., 25 (1993) 503.
[5] MUNKHANOVV. F., FELDMANH. A. and BRANDENBERGERR. H., Phys. Rep., 215 (1992) 203. [6] BARDEENJ. M., Phys. Rev. D, 22 (1980) 1882.
