# Structure Formation in the Early Universe

###### Abstract

The evolution of the perturbations in the energy density and the particle number density in a flat Friedmann-Lemaître-Robertson-Walker universe in the radiation-dominated era and in the epoch after decoupling of matter and radiation is studied. For large-scale perturbations the outcome is in accordance with treatments in the literature. For small-scale perturbations the differences are conspicuous. Firstly, in the radiation-dominated era small-scale perturbations grew proportional to the square root of time. Secondly, perturbations in the Cold Dark Matter particle number density were, due to gravitation, coupled to perturbations in the total energy density. This implies that structure formation could have begun successfully only after decoupling of matter and radiation. Finally, after decoupling density perturbations evolved diabatically, i.e., they exchanged heat with their environment. This heat exchange may have enhanced the growth rate of their mass sufficiently to explain structure formation in the early universe, a phenomenon which cannot be understood from adiabatic density perturbations.

pacs: 98.62.Ai; 98.80.-k; 97.10Bt

keywords: Perturbation theory; cosmology; diabatic density perturbations; structure formation

###### Contents

## 1 Introduction

The global properties of our universe are very well described by a model with a flat Friedmann-Lemaître-Robertson-Walker (flrw) metric within the context of the General Theory of Relativity. To explain structure formation after decoupling of matter and radiation in this model, one has to assume that before decoupling Cold Dark Matter (cdm) has already contracted to form seeds into which the baryons (i.e., ordinary matter) could fall after decoupling. In this article it will be shown that cdm did not contract faster than baryons before decoupling and that structure formation started off successfully only after decoupling.

In the companion article^{2}^{2}2Section and equation numbers with a
refer to sections and equations in the companion
article [1]. it has been shown that there are
two unique gauge-invariant quantities
and which are the real, measurable, perturbations
to the energy density and the particle number density, respectively.
Evolution equations for the corresponding contrast functions
and have been derived for closed, flat
and open flrw universes. In this article these evolution
equations will be applied to a flat flrw universe in its
three main phases, namely the radiation-dominated era, the plasma era,
and the epoch after decoupling of matter and radiation. In the
derivation of the evolution equations, an equation of state for the
pressure of the form has been taken into account,
as is required by thermodynamics. As a consequence, in addition to a
usual second-order evolution equation (3a) for density
perturbations, a first-order evolution equation (3b) for
entropy perturbations follows also from the perturbed Einstein
equations. This entropy evolution equation is absent in former
treatments of the subject. Therefore, the system (3)
leads to further reaching conclusions than is possible from treatments
in the literature.

Analytic expressions for the fluctuations in the energy density and the particle number density in the radiation-dominated era and the epoch after decoupling will be determined. It is shown that the evolution equations (3) corroborate the standard perturbation theory in both eras in the limiting case of infinite-scale perturbations. For finite scales, however, the differences are conspicuous. Therefore, only finite-scale perturbations are considered in detail.

A first result is that in the radiation-dominated era oscillating density perturbations with an increasing amplitude proportional to are found, whereas the standard perturbation equation (61) yields oscillating density perturbations with a constant amplitude. This difference is due to the fact that in the perturbation equations (3) the divergence of the spatial part of the fluid four-velocity is taken into account, whereas is missing in the standard equation. In the appendix it is made clear why is important.

In the radiation-dominated era and the plasma era baryons were tightly coupled to radiation via Thomson scattering until decoupling. A second result is that cdm was also tightly coupled to radiation, not through Thomson scattering, but through gravitation. This implies that before decoupling perturbations in cdm have contracted as fast as perturbations in the baryon density. As a consequence, cdm could not have triggered structure formation after decoupling. This result follows from the entropy evolution equation (3b) since , (5), throughout the history of the universe as will be shown in Section 3.

From observations [2] of the Cosmic Microwave Background it follows that perturbations were adiabatic at the moment of decoupling, and density fluctuations and were of the order of or less. Since the growth rate of adiabatic perturbations in the era after decoupling was too small to explain structure in the universe, there must have been, in addition to gravitation, some other mechanism which has enhanced the growth rate sufficiently to form the first stars from small density perturbations. A final result of the present study is that it has been demonstrated that after decoupling such a mechanism did indeed exist in the early universe.

At the moment of decoupling of matter and radiation, photons could not ionize matter any more and the two constituents fell out of thermal equilibrium. As a consequence, the pressure dropped from a very high radiation pressure just before decoupling to a very low gas pressure after decoupling. This fast and chaotic transition from a high pressure epoch to a very low pressure era may have resulted in large relative diabatic pressure perturbations due to very small fluctuations in the kinetic energy density. It is found that the growth of a density perturbation has not only been governed by gravitation, but also by heat exchange of a perturbation with its environment. The growth rate depended strongly on the scale of a perturbation. For perturbations with a scale of (see the peak value in Figure 1) gravity and heat exchange worked perfectly together, resulting in a fast growth rate. Perturbations larger than this scale reached, despite their stronger gravitational field, their non-linear phase at a later time since heat exchange was low due to their larger scales. On the other hand, for perturbations with scales smaller than gravity was too weak and heat exchange was not sufficient to let perturbations grow. Therefore, density perturbations with scales smaller than did not reach the non-linear regime within , the age of the universe. Since there was a sharp decline in growth rate below a scale of , this scale will be called the relativistic Jeans scale.

The conclusion of the present article is that the model of the universe and its evolution equations for density perturbations (3) explain the so-called (hypothetical) Population iii stars and larger structures in the universe, which came into existence several hundreds of million years after the Big Bang [3, 4].

## 2 Einstein Equations for a Flat FLRW Universe

In this section the equations needed for the study of the evolution of density perturbations in the early universe are written down for an equation of state for the pressure, .

### 2.1 Background Equations

The set of zeroth-order Einstein equations and conservation laws for a flat, i.e., , flrw universe filled with a perfect fluid with energy-momentum tensor

(1) |

is given by

(2a) | ||||||

(2b) | ||||||

(2c) |

The evolution of density perturbations took place in the early universe shortly after decoupling, when . Therefore, the cosmological constant has been neglected.

### 2.2 Evolution Equations for Density Perturbations

The complete set of perturbation equations for the two independent density contrast functions and is given by (44)

(3a) | ||||

(3b) |

where the coefficients , and , (45), are for a flat flrw universe filled with a perfect fluid described by an equation of state given by

(4a) | ||||

(4b) | ||||

(4c) |

where and are the partial derivatives of the equation of state :

(5) |

The symbol denotes the Laplace operator. The quantity is defined by . Using that and the conservation laws (2b) and (2c) one gets

(6) |

From the definitions and and the energy conservation law (2b), one finds for the time-derivative of

(7) |

This expression holds true independent of the equation of state.

The pressure perturbation is given by (49)

(8) |

where the first term, , is the adiabatic part and the second term the diabatic part of the pressure perturbation.

The combined First and Second Law of Thermodynamics reads (57)

(9) |

Density perturbations evolve adiabatically if and only if the source term of the evolution equation (3a) vanishes, so that this equation is homogeneous and describes, therefore, a closed system that does not exchange heat with its environment. This can only be achieved for , or, equivalently, , i.e., if the particle number density does not contribute to the pressure. In this case, the coefficient , (4c), vanishes.

## 3 Analytic Solutions

In this section analytic solutions of equations (3) are derived for a flat flrw universe with a vanishing cosmological constant in its radiation-dominated phase and in the era after decoupling of matter and radiation. It is shown that throughout the history of the universe. In this case, the entropy evolution equation (3b) implies that fluctuations in the particle number density, , are coupled to fluctuations in the total energy density, , through gravitation, irrespective of the nature of the particles. In particular, this holds true for perturbations in cdm. Consequently, cdm fluctuations have evolved in the same way as perturbations in ordinary matter. This may rule out cdm as a means to facilitate the formation of structure in the universe after decoupling. The same conclusion has also been reached by Nieuwenhuizen et al. [5], on different grounds. Therefore, structure formation could start only after decoupling.

### 3.1 Radiation-dominated Era

At very high temperatures, radiation and ordinary matter are in thermal equilibrium, coupled via Thomson scattering with the photons dominating over the nucleons (). Therefore the primordial fluid can be treated as radiation-dominated with equations of state

(10) |

where is the black body constant and the radiation temperature. The equations of state (10) imply the equation of state for the pressure , so that, with (5),

(11) |

Therefore, one has from (6),

(12) |

Using (11) and (12), the perturbation equations (3) reduce to

(13a) | |||

(13b) |

where (58) has been used. Since the right-hand side of (13a) vanishes, implying that density perturbations evolved adiabatically: they did not exchange heat with their environment. Moreover, baryons were tightly coupled to radiation through Thomson scattering, i.e., baryons obey . Thus, for baryons (13b) is identically satisfied. In contrast to baryons, cdm is not coupled to radiation through Thomson scattering. However, equation (13b) follows from the General Theory of Relativity, Section 2.7. As a consequence, equation (13b) should be obeyed by all kinds of particles that interact through gravitation. In other words, equation (13b) holds true for baryons as well as cdm. Since cdm interacts only via gravity with baryons and radiation, the fluctuations in cdm are coupled through gravitation to fluctuations in the energy density, so that fluctuations in cdm also satisfy equation (13b).

In order to solve equation (13a) it will first be rewritten in a form using dimensionless quantities. The solutions of the background equations (2) are given by

(14) |

implying that . The dimensionless time is defined by . Since , one finds that

(15) |

Substituting into equation (13a) and using (15) yields

(16) |

where a prime denotes differentiation with respect to . The parameter is given by

(17) |

with the physical scale of a perturbation at time (), and . To solve equation (16), replace by . After transforming back to , one finds

(18) |

where the ‘constants’ of integration and are given by

(19) |

For large-scale perturbations (), it follows from (18) and (19) that

(20) |

The energy density contrast has two contributions to the growth rate, one proportional to and one proportional to . These two solutions have been found, with the exception of the precise factors of proportionality, by a large number of authors [6, 7, 8, 9, 10, 11]. Consequently, the evolution equations (13) corroborates for large-scale perturbations the results of the literature.

Small-scale perturbations () oscillate with an increasing amplitude according to

(21) |

as follows from (18) and (19). Thus, the evolution equations (13) yield oscillating density perturbations with an increasing amplitude, since in these equations , as follows from their derivation in Section 2.7. In contrast, the standard equation (61), which has , yields oscillating density perturbations with a constant amplitude.

Finally, the plasma era has begun at time , when the energy density of ordinary matter was equal to the energy density of radiation, (58), and ends at , the time of decoupling of matter and radiation. In the plasma era the matter-radiation mixture can be characterized by the equations of state (Kodama and Sasaki [12], Chapter V)

(22) |

where the contributions to the pressure of ordinary matter and cdm have not been taken into account, since these contributions are negligible with respect to the radiation energy density. Eliminating from (22), one finds for the equation of state for the pressure, Section 2.1,

(23) |

so that with (5) one gets

(24) |

Since , equation (3b) implies that fluctuations in the particle number density, , were coupled to fluctuations in the total energy density, , through gravitation, irrespective of the nature of the particles.

### 3.2 Era after Decoupling of Matter and Radiation

Once protons and electrons combined to yield hydrogen, the radiation pressure was negligible, and the equations of state have become those of a non-relativistic monatomic perfect gas with three degrees of freedom

(25) |

where is Boltzmann’s constant, the mean particle mass, and the temperature of the matter. For the calculations in this subsection it is only needed that the cdm particle mass is such that for the mean particle mass one has , so that . Therefore, as follows from the background equations (2a) and (2b), one may neglect the pressure and the kinetic energy density with respect to the rest mass energy density in the unperturbed universe. However, neglecting the pressure in the perturbed universe yields non-evolving density perturbations with a static gravitational field, as is shown in Section 4. Consequently, it is important to take the pressure perturbations into account.

Eliminating from (25) yields, Section 2.1, the equation of state for the pressure

(26) |

so that with (5) one has

(27) |

Substituting , and (25) into (6) on finds, using that ,

(28) |

with the adiabatic speed of sound and the matter temperature. Using that and , expression (7) reduces to , so that with one has . This implies that the matter temperature decays as

(29) |

This, in turn, implies with (28) that . The system (3) can now be rewritten as

(30a) | |||

(30b) |

where and have been neglected with respect to constants of order unity. From equation (30b) it follows with that

(31) |

Since the system (30) is derived from the General Theory of Relativity, it should be obeyed by all kinds of particles which interact through gravity, in particular baryons and cdm.

It will now be shown that the right-hand side of equation (30a) is proportional to the mean kinetic energy density fluctuation of the particles of a density perturbation. To that end, an expression for will be derived from (25). Multiplying by and subtracting the result from , one finds

(32) |

where also the definitions (40a) and (52) have been used. Dividing the result by , (25), and using that , one finds

(33) |

to a very good approximation. In this expression is the relative perturbation in the total energy density. Since , it follows from the derivation of (33) that can be considered as the relative perturbation in the rest energy density. Consequently, the second term is the fluctuation in the kinetic energy density, i.e., . The relative kinetic energy density perturbation occurs in the source term of the evolution equation (30a) and is of the same order of magnitude as the term with in the left-hand side.

Combining (29) and (31) one finds from (33) that is constant

(34) |

to a very good approximation, so that the kinetic energy density fluctuation is given by

(35) |

In Section 4 it will be shown that the kinetic energy density fluctuation has played, in addition to gravitation, a role in the evolution of density perturbations. In fact, if a density perturbation was somewhat cooler than its environment, i.e., , its growth rate was, depending on its scale, enhanced.

Using (27) and (33), one finds from (8)

(36) |

where is the relative pressure perturbation defined by , with given by (25). The term is the adiabatic part and is the diabatic part of the relative pressure perturbation. The factor is the so-called adiabatic index for a monatomic ideal gas with three degrees of freedom. Thus, relative kinetic energy density perturbations give rise to diabatic pressure fluctuations.

Finally, the perturbed entropy per particle follows from (9) and (33)

(37) |

In Section 3.2 it has been shown that the background entropy per particle is independent of time. In a linear perturbation theory the perturbed entropy per particle is approximately constant, i.e., . Therefore, heat exchange of a perturbation with its environment decays proportional to the temperature, i.e., , as follows from (29).

In order to solve equation (30a) it will first be rewritten in a form using dimensionless quantities. The solutions of the background equations (2) are given by

(38) |

where the kinetic energy density and pressure have been neglected with respect to the rest mass energy density. The dimensionless time is defined by . Using that , one gets

(39) |

Substituting , , (28) and (35) into equations (30) and using (29) and (39) one finds that equations (30) can be combined into one equation

(40) |

where a prime denotes differentiation with respect to . The parameter is given by

(41) |

with the physical scale of a perturbation at time (), and . To solve equation (40) replace by . After transforming back to , one finds for the general solution of the evolution equation (40)

(42) |

where are Bessel functions of the first kind and and are the ‘constants’ of integration, calculated with the help of Maxima [13]:

(43) |

The particle number density contrast follows from equation (33), (34) and (42). In (42) the first term (i.e., the solution of the homogeneous equation) is the adiabatic part of a density perturbation, whereas the second term (i.e., the particular solution) is the diabatic part.

In the large-scale limit terms with vanish. Therefore, the general solution of equation (40) becomes

(44) |

Thus, for large-scale perturbations the diabatic pressure fluctuation did not play a role during the evolution: large-scale perturbations were adiabatic and evolved only under the influence of gravity. These perturbations were so large that heat exchange did not play a role during their evolution in the linear phase. For perturbations much larger than the Jeans scale (i.e., the peak value in Figure 1), gravity alone was insufficient to explain structure formation within , since they grow as .

The solution proportional to is a standard result [6, 7, 8, 9, 10, 11]. Since is gauge-invariant, the standard non-physical gauge mode proportional to is absent from the solution set of the evolution equations (30). Instead, a physical mode proportional to is found. This mode follows also from the standard perturbation equations if one does not neglect the divergence , as is shown in the appendix. Consequently, only the growing mode of (44) is in agreement with results given in the literature.

In the small-scale limit , one finds from (42) and (3.2)

(45a) | ||||

(45b) |

where (36) has been used to calculate the fluctuation in the pressure. Thus, density perturbations with scales smaller than the Jeans scale oscillated with a decaying amplitude which was smaller than unity: these perturbations were so small that gravity was insufficient to let perturbations grow. Heat exchange alone was not enough for the growth of density perturbations. Consequently, perturbations with scales smaller than the Jeans scale did never reach the non-linear regime.

In the next section it is shown that for density perturbations with scales of the order of the Jeans scale, the action of both gravity and heat exchange together may result in massive structures several hundred million years after decoupling of matter and radiation.

## 4 Structure Formation after Decoupling of Matter and Radiation

In this section it is demonstrated that the relativistic evolution equations, which include a realistic equation of state for the pressure yields that in the era after decoupling of matter and radiation density perturbations may have grown fast.

Up till now it is only assumed that for baryons and cdm, without specifying the mass of the baryon and cdm particles. From now on it is convenient to assume that the mass of a cdm particle is of the order of magnitude of the proton mass.

### 4.1 Observable Quantities

The parameter (41) will be expressed in observable quantities, namely the present values of the background radiation temperature, , the Hubble parameter, , and the redshift at decoupling, . From now on the initial time is taken to be the time at decoupling of matter and radiation: , so that .

The redshift as a function of the scale factor is given by

(46) |

where is the present value of the scale factor and . For a flat flrw universe one may take . Using the background solutions (38), one finds from (46)

(47a) | ||||

(47b) | ||||

(47c) |

where it is used that after decoupling, as follows from (10) and (14).

### 4.2 Initial Values from the Planck Satellite

The physical quantities measured by Planck [14] and needed in the parameter (50) of the evolution equation (40) are the redshift at decoupling, the present values of the Hubble function and the background radiation temperature, the age of the universe and the fluctuations in the background radiation temperature. The numerical values of these quantities are

(51a) | ||||

(51b) | ||||

(51c) | ||||

(51d) | ||||

(51e) |

Substituting the observed values (51a)–(51c) into (50), one finds

(52) |

where it is used that the proton mass is , , the speed of light and Boltzmann’s constant .

The Planck observations of the fluctuations , (51e), in the background radiation temperature yield for the initial value of the fluctuations in the energy density

(53) |

In addition, it is assumed that

(54) |

i.e., during the transition from the radiation-dominated era to the era after decoupling, perturbations in the energy density were approximately constant with respect to time.

During the linear phase of the evolution, follows from (33) so that the initial values and need not be specified.

### 4.3 Diabatic Pressure Perturbations

At the moment of decoupling of matter and radiation, photons could not ionize matter any more and the two constituents fell out of thermal equilibrium. As a consequence, the high radiation pressure just before decoupling did go over into the low gas pressure after decoupling. In fact, from (47c) and (59) it follows that at decoupling one has

(55) |

where it is used that at the moment of decoupling the matter temperature was equal to the radiation temperature . The redshift at matter-radiation equality was , Planck [14]. The fast and chaotic transition from a high pressure epoch to a very low pressure era may have resulted in large relative diabatic pressure perturbations , (36), due to very small fluctuations , (35), in the kinetic energy density. It will be shown in Section 4.4 that density perturbations which were cooler than their environments may have collapsed fast, depending on their scales. In fact, perturbations for which

(56) |

may have resulted in primordial stars, the so-called (hypothetical) Population iii stars, and larger structures, several hundred million years after the Big Bang.

### 4.4 Structure Formation in the Early Universe

In this section the evolution equation (40) is solved numerically [15, 16] and the results are summarized in Figure 1, which is constructed as follows. For each choice of in the range , , , …, equation (40) is integrated for a large number of values for the initial perturbation scale using the initial values (53) and (54). The integration starts at , i.e., at and will be halted if either , i.e., , see (48), or for has been reached. One integration run yields one point on the curve for a particular choice of the scale if has been reached for . If the integration halts at and still , then the perturbation pertaining to that particular scale has not yet reached its non-linear phase today, i.e., at . On the other hand, if the integration is stopped at and , then the perturbation has become non-linear within . Each curve denotes the time and scale for which for a particular value of .

The growth of a perturbation was governed by gravity as well as heat exchange. From Figure 1 one may infer that the optimal scale for growth was around . At this scale, which is independent of the initial value of the diabatic pressure perturbation , see (8) and (36), heat exchange and gravity worked together perfectly, resulting in a fast growth. Perturbations with scales smaller than reached their non-linear phase at a much later time, because their internal gravity was weaker than for large-scale perturbations and heat exchange was insufficient to enhance the growth. On the other hand, perturbations with scales larger than exchanged heat at a slower rate due to their large scales, resulting also in a smaller growth rate. Perturbations larger than grew proportional to , (44), a well-known result. Since the growth rate decreased rapidly for perturbations with scales below , this scale will be considered as the relativistic counterpart of the classical Jeans scale. The relativistic Jeans scale at decoupling, , was much smaller than the horizon size at decoupling, .

### 4.5 Relativistic Jeans Mass

The Jeans mass at decoupling, , can be estimated by assuming that a density perturbation has a spherical symmetry with diameter the relativistic Jeans scale . The relativistic Jeans mass at decoupling is then given by

(57) |

The particle number density can be calculated from its value at the end of the radiation-dominated era. By definition, at the end of the radiation-domination era the matter energy density was equal to the energy density of the radiation:

(58) |

Since and , one finds, using (46), (47c) and (58), for the particle number density at the time of decoupling