fulltext

The amplification of cosmological density perturbations

in the recombination area; upper limits for galaxy masses

B. ROSE

Universität Konstanz, Fakultät für Physik - Postfach 5560, D-78434 Konstanz, Germany (ricevuto il 31 Luglio 1996; approvato il 29 Aprile 1997)

Summary. — Cosmological density perturbations with masses below 1015_{MUare the} subject. Their evolution from cosmological times where they are stable (before hydrogen recombination) to times where they are unstable (after recombination) is studied. It is found that not the amplification of the density contrasts by pure self-gravity in the unstable phase is dominant but the amplification during times of stability just before the perturbations become unstable. The decoupling of the radiation from the matter perturbations during the recombination plays a crucial role, which leads to a steep drop of the speed of sound. The larger the masses are, the smaller the amplification is. Above 1012_MU–1013_{MUthe amplification is so small that} this can be regarded as upper limits for galaxies.

PACS 98.80 – Cosmology.

1. – Introduction

The growing of cosmological density perturbations has been the subject of many contributions in the literature ([1-15], for a review see especially [11]). But the mass spectrum of galaxies coming out of the perturbations is not explained till now.

For the possibility of matter perturbations to grow during the cosmological evolution, their Stability-Instability, Border (SIB) is an essential concept. The SIB for a matter perturbation is its spatial scale or its mass content, which marks the border between its dissolving or oscillating and its growing.

In a series of former papers [16-21] tha SIBs of matter perturbations were calculated in a full general relativistic treatment by linearization in the perturbations. We got a spectrum of scales which are SIBs. We found that at least before recombination perturbations with scales of superclusters are preferred for growing and therefore a scenario “from the top to the bottom” is favoured [21]. Concerning the mass spectrum of galaxies, we could only recover the old result that after recombination all masses above 104MU are possible for growing perturbations. The

truth is, however, that above 1012_M

Uno galaxies do exist.

The SIB before and after the recombination common to the General Relativistic treatment in synchronous—and de Donder—gauge and common to a Newtonian

treatment on an expanding background is the important one for perturbations up to galaxies masses and above. Before recombination it is nearly constant in dependence on the cosmic red-shift factor z and has a value of about 1015_M

U. After the

recombination it falls with a steep slope to values of about 104_M

U. The whole SIB has

the character of a lower border. So all perturbations with masses above the SIB masses grow and beneath do dissolve or oscillate. The steep slope of the SIB is a consequence of the decoupling of the radiation from the matter perturbations, which implies a corresponding fall with a steep slope of the velocity of sound in the matter perturbations.

By arguing alone with the SIBs—i.e. arguing alone with growing by self-gravity— before recombination perturbations with masses above 1015_M

U grow and after

recombination all perturbations with masses above 104MUgrow. So the SIBs—i.e. pure

self-gravity—do not explain the mass spectrum of galaxies.

What is the time evolution of a matter perturbation with a mass below 1015MU

starting before the steep slope of the SIB and passing the SIB? This is a question we study in this paper. As a result, we get an oscillation till the SIB followed by a growing. The oscillation however can be amplified in amplitude and gradient strongly till the perturbation in its time evolution reaches the SIB. The possible degree of amplification at the SIB depends strongly on the mass explicitly, and implicitly by the kind of the steep slope of the velocity of sound. So the decoupling of radiation from matter perturbations must be examined carefully. The resulting amplification is much greater than the corresponding one if one starts at the SIB with the same initial conditions as before, i.e. amplification by pure self-gravity.

We have therefore found a powerful amplification mechanism which acts at cosmological times where the perturbation is stable and has its root in the hydrodynamics of the perturbations. Small masses are amplified much stronger than large ones and the amplification of masses beyond 1012_–1013_M

Uis very small. So this

could be an explanation as to why galaxies beyond 1012_M

U do not exist. The

amplification till the SIB is reached is just too small as is the amplification in the following time by self-gravity alone. There is just not time enough for such high masses to build galaxies, because the amplification at the SIB is so small.

The plan of the paper is as follows. In sect. 2 we give a detailed phenomenological description of the decoupling of radiation from matter perturbations in the recombination area. This is an improved version of that in ref. [ 20 , 21 ] and leads to a scale-dependent velocity of sound.

In sect. 3 we then study the propagation of density contrasts which are stable at the beginning and then pass the SIB. We deduce some analytical implications about the possible amplification of the perturbations till the SIB is reached. We find the mass and the drastic drop-down of the velocity of sound to be crucial. Smaller masses are more amplified than larger ones.

In sect. 4 we give the numerical results and conclusions.

2. – The coupling of radiation to matter perturbations

In two former papers [ 20 , 21 ] the coupling of radiation to cosmological matter perturbations was described by the function f:

f 412e2q

with

q 4 r

lph

. (2.2)

Herein r is the spatial scale of the matter perturbation, and

lph4

1

xnBse

(2.3)

is the mean free path of the photons with respect to Thomson scattering; nB is the

baryon number density, se the Thomson cross-section and x the degree of hydrogen

ionization given by the Saha equation,

x 4 ne nB 4 a3 O2 e 2 xe kBT nB ]

k

g

h

The symbols mean: T 4temperature; mH4 mass of hydrogen atom; c 4 speed of light;

kB4 1 .38 Q 10216ergOK (Boltzmann constant); ne4 electron number density; xe4 2 .18 Q

10211_{erg (ionization energy of hydrogen); m}

e4 electron mass.

The coupling of radiation to matter in the matter perturbations then results in a weightening of radiation contributions by the function f. So let be energy density and pressure for the cosmological background substratum be

e 4nBmHc21 3 2( 1 1x) nBkBT 1sT 4_, (2.5) P 4 (11x) nBkBT 1 1 3sT 4 (2.6) (s 47.56Q10215_{erg cm}23_K24_).

For linear matter perturbations of scale r we then obtain

e14 ¯e ¯nB n11 ¯e ¯TT1, (2.7) p14 ¯p ¯nB n11 ¯p ¯TT1, (2.8)

but all radiation contributions in the partial differential quotients are weighted (multiplied) by the function f. The basic perturbations are n1 and T1. For adiabatic

perturbations we get T14 (e 1p)OnB2 ¯eO¯nB ¯_eO¯T n1. (2.9)

Here again all radiation contributions are weighted by f. So we obtain for the adiabatic speed of sound b 4 v 2 s c2 4 p1 e1 . (2.10)

The coupling function f describes how much radiation can be trapped by a matter perturbation via Thomson scattering. The crucial feature of the perturbation is its spatial scale compared with the mean free path of the photons. Photons not trapped are in no interaction with the matter perturbation.

But for the time evolution of the perturbations their ability alone for trapping radiation just by their spatial scale and Thomson scattering is not enough to describe the matterOradiation coupling. The trapped radiation, moreover, should be kept trapped during a characteristic time for the time evolution of the perturbations. Then the trapped radiation is important dynamically in the matter perturbation. Since for the trapping ability q4rOlph is the decisive quantity, for the ability to stay trapped

q14

TESC

T

(2.11)

is the corresponding quantity. TESCis the time the photons need to leave a perturbation

and T is some typical time for the evolution of the perturbation. The corresponding weighting function is

f14 1 2 e2q1

(2.12)

and the total weighting function for radiation contributions reads

ftot4 ff14 ( 1 2 e2q)( 1 2q2q1) .

(2.13)

An approximation for TESCcan be given as

TESC4 T0( 1 1q) ,

(2.14)

where T04 rOc (r 4 scale of the perturbation) is the time the photons need to leave the

perturbation when no interaction with matter is there. The quantity q 4rOlphthen is a

measure for the number of photon collisions with electrons on their way to leave the perturbation.

For the typical time T for perturbation propagation we set

T 4 TH

1 1THOTper

, (2.15)

where THB 1 OH is the characteristic time for background expansion (H 4 Hubble

constant) and Tper is the “oscillation time” of the perturbation with self-gravity

influence included. The expression (2.15) for T is an interpolation between the case far away from a stability border and the case at the stability border. In the first case we have TperB rOvsbTH(vsis the speed of sound, self-gravity influences can be neglected)

and therefore T BTper. In the second case TpercTH(TperK Q) and therefore T B TH

(background expansion dominates).

Great influence of f1is to be expected if f1b1 . This is the case for stability borders

So just for the propagation of density perturbations under Newtonian conditions

r b horizon , vs2Oc2b1 O3, background pressure negligible) the correction by f1 gets

important. All results got in [21] beyond the Newtonian region stay valid. The Newtonian equation for the propagation of a density contrast on an expanding matter-dominated background reads (see [ 21 , 22 ] )

dO_{1 2 H}. d._1H2

k

yb 2 3

2

l

A dot means time derivative, d 4r1O(r 1 p) with r1a density perturbation, b 4vs2Oc2

and y 4r2

OrHor2 with rHorthe horizon.

This equation can be rewritten in dependence on the cosmic red-shift factor z instead of cosmological time t:

d 91 1 2 1 zd 81v 2 effd 40 (2.17) with v2 eff4 1 z2

k

l

Here z0means the initial cosmological instant from which the d-propagation is studied

and y0means the value of y at this z0.

Fig. 1. – Stability border SK(z) with f as radiation coupling function. SK is given by lg (MOMU) with M as the perturbation mass (MU4 mass of the sun), which is, at the corresponding cosmic red-shift factor z, the stability border of the perturbation.

Fig. 2. – Same as fig. 1, but now ftotis the radiation coupling function.

From (2.17), respectively (2.17a), one sees that the “oscillation time” for d including self-gravity influence reads

Tper4

1

k

.

The Newtonian stability border (SIB) is obtained by setting veff4 0 and therefore

Tper4 Q . In fig. 1 and 2 this stability border is plotted (for the old radiation coupling

function f and the corrected ftot) as critical mass content of a perturbation in

dependence on z. The f-correction leads to a steeper decreasing of the SIB at greater z (earlier cosmological time).

3. – The propagation of density contrasts

We now consider the propagation of density contrasts with scales up to galaxies. This means: we take density contrasts with mass contents smaller than 1015_M

U at

initial cosmological z0 before the recombination epoch and study their time evolution

through the recombination epoch and beyond. At the initial z0we take (ddOdz)(z0) 40,

and normalize d (z0) to 1. So the density contrasts start below the horizontal part of the

Newtonian stability border and are therefore stable—an oscillation is to be expected. In the recombination epoch they pass the steep slope of the stability border and get unstable and therefore grow slowly by self-gravity alone. The differential equation governing the time evolution of d is therefore (2.17) with (2.17a) and the corrected

Fig. 3. – b 4v2

sOc2with vsthe velocity of sound with respect to the perturbations and c 4velocity of light. b is given in dependence on the cosmic red-shift factor z for different perturbation masses SK. SK 45 to 9 here. In all calculations the radiation coupling function ftotis used. Further it is marked roughly that z 4zSIBat which the steep slope of the stability border occurs (see fig. 2).

Fig. 5. – Same as fig. 3, but for SK 411 to 15.

Fig. 6. – Stability border SK(z) with ftot as radiation coupling function as in fig. 2. But now additionally the part of curve SK(z) is plotted, which marks b 9(z) 40 in the velocity of hydrogen recombination.

coupling function ftot(see 2.13): d 91 1 2 1 zd 81v 2 effd 40 , (3.1) v2 eff4 1 z2

k

l

In the region of stability we get v2

effD 0 , at the SIB: v2eff4 0 and in the region of

instability v2 effE 0 .

Before making any numerical calculations one can see that the evolution of d depends crucially on the z-dependence of b (b 4vs2Oc2). If one starts at a z0in the stable

region, then v2

eff(z0) D0. If b(z) increases with z not stronger than z, then in the cosmic

evolution (going to smaller values of z) v2

effwill grow and stay positive for all z. A stable

contrast would stay stable and no steep slope of the SIB would occur.

A decreasing of veff (with decreasing z) can only appear in that z-region in which

b(z) grows stronger than z. For each scale SK (mass content) below the horizontal part

of the Newtonian SIB this is only the case for z out of the region: zSIBG z G zs. Here zSIB

means that z for which SK is the SIB and zsis that z at which, for given SK, 2( d2bOdz2)

becomes zero. Energy conservation in the sound wave then implies growing of amplitudes in that region.

In fig. 3 to 5 b(z) is given with SK as parameter. In fig. 6 the SIB is given by SK(z) and additionally, for each z, that SK for which the z is that z at which b 9 is zero. So for each SK the z-interval for which the amplitude amplification of d is to be expected is given by the intersection with these two curves. One sees that this z-interval shrinks drastically for larger SK, whereas b8(zs)—which is a measure for the degree of

amplification—grows weaker and weaker for large SK. So large SK are not favoured by the amplitude amplification of d. From (3.2) one sees further that higher-scales

SK—i.e. lower y0—lead to lower v2eff. The higher the frequencies, however, the higher

the possible gradients of d at the SIB, which lead to a higher amplification by self-gravity in the subsequent evolution. So again we conclude that larger scales are discriminated against amplification.

4. – Numerical results and conclusions

In fig. 7 to 9 d (z) is plotted for different mass scales SK »4lg(MOMU). The initial

values for integration are chosen as z04 3000

(

(z B1500) and the steep slope of the SIB (zSIBB 1218 )

)

sees that the analytical results of the preceding section are fully recovered. Amplitude amplification occurs for all z at which b(z) grows stronger than linearly. The substantial amplification however occurs in the z-region (z E1600) readable from fig. 6. The amplification gets less for higher masses since the relevant z-region for amplitude amplification shrinks and the d-gradients delivered for amplification by the oscillations in the stable part decrease when going to higher masses.

The actual amplification of the initial d (z0) 41 in dependence on z (zGzSIB)

depends on the value of d and its gradient d 8 at the steep slope of the SIB as a result of the z-evolution of d mentioned before (see fig. 7 to 9 for example).

Fig. 7. – Density contrast d of a perturbation in dependence on z with initial values d04 1 , d 804 0 at z04 3000 . The curves are parametrized by the perturbation mass magnitude SK. Here SK 45 to 9. In all calculations ftotis used as radiation coupling function. Further it is marked roughly that z 4zSIBat which the steep slope of the stability border occurs (see fig. 2).

Fig. 9. – Same as fig. 7, but for SK 411 to 15.

Fig. 10. – Maximal possible density contrast d_maxat the z corresponding to the stability border in dependence on the perturbation mass magnitude SK. This dmaxis the result of the time evolution of d as in fig. 7-9, when the initial value zinis varied in the interval [z02 zosc, z0] with z04 3000 and zosc4 1 Oveff(z0) (see sect. 4).

Fig. 11. – Like fig. 10, but now dmax(SK) is calculated at z 4100 instead of the stability border zSIB.

Fig. 12. – d (z) as in fig. 7 with the same initial values d04 1 and d 804 0 but now at the starting point z04 zSIB. Here zSIB is again the z which corresponds to the stability border at given perturbation mass magnitude SK. Here SK 45 to 9.

Fig. 13. – Same as fig. 12, but with SK 49 to 13.

One may ask now for the maximal possible amplification of d at a given z (the examples considered here are z 4zSIBand z 4100) in dependence on its mass scale SK.

This maximum of amplification can be calculated by varying the initial starting point z0

of the d-evolution in the scale of the “oscillation time” there: zosc4 1 O

(

)

initial point variation then leads to different values of d and d 8 at the SIB and one can pick out the maximum of d. This is done and the results are presented in fig. 10 and 11. There dmaxis plotted against the mass scale SK. Figure 10 gives the situation at z 4zSIB

and fig. 11 at z 4100.

The above-mentioned tendency that large mass scales are hardly amplified is strongly confirmed by these figures. There is no sharp upper boundary, but masses greater than 1012_M

U are amplified less by a factor of 10 than, for example, 106MU.

Additionally, there is a relative maximum for contrast amplification at M B106MU.

The calculations and considerations throughout this paper are made in a linear approximation for the perturbations. So only the tendency of mass-dependent amplification is extracted from this formalism. This tendency is that masses greater than 1012_M

U are hardly amplified. A nonlinear analysis, which can cover the

cosmological area from z 43000 till now, could well show that with respect to the small amplification of M D1012_M

U, there is simply not time enough for formation of galaxies

with such great masses.

All the amplifications treated till now mainly have their origin in the time evolution of d in the stable region below the horizontal part of the SIB (zSIBG z G 3000 ). For the

resulting amplification at the SIB the scale of the perturbation and the kind of descending of the speed of sound is crucial—i.e. the kind of coupling of radiation to matter perturbations. So for the contrast amplification it is not self-gravity that plays the dominant part, but the hydrodynamics in the perturbation of the substratum. Only after the steep slope of the SIB is crossed (zSIBB 1218 ) self-gravity governs purely the

further evolution of the contrast d. But there the overwhelming part of

amplification—both in d and d 8—is already done and large d and d8 are already given at the SIB.

How soft the amplification is by pure self-gravity in comparison to the “hydrodynamical one” is illustrated in fig. 12 to 14. We let the density contrast start at the zSIB (corresponding to the chosen mass scale) with d (zSIB) 41, d8(zSIB) 40. At

z 4800 we get dB1.1 which corresponds to an amplification of 1.1. From fig. 7 to 9 one

sees that the hydrodynamical amplification in the important z-region (fig. 6) is however already about 10.

The seeds for galaxies are sown far before recombination. What seeds can be amplified so much that eventually galaxies can form out of them is decided before recombination when they are stable. The amplification mechanism has its roots in the steep fall of the velocity of sound in the matter perturbations. This finally is caused by the kind of coupling of radiation to the matter perturbations.

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