Structure Formation in the Early Universe P. G. Miedema∗ Netherlands Defence Academy (NLDA) Hogeschoollaan 2 NL-4818CR Breda The Netherlands April 20, 2016 6 1 0 2 r Abstract p A The evolution of the perturbations in the energy density and the particle number den- 0 sity in a flat Friedmann-Lemaître-Robertson-Walker universe in the radiation-dominated 2 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- ] c scale perturbations the differences are conspicuous. Firstly, in the radiation-dominated era q small-scale perturbations grew proportional to the square root of time. Secondly, perturba- - r tions in the Cold Dark Matter particle number density were, due to gravitation, coupled to g [ 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 2 v densityperturbationsevolveddiabatically,i.e.,theyexchangedheatwiththeirenvironment. 0 This heat exchange may have enhanced the growth rate of their mass sufficiently to explain 6 structure formation in the early universe, a phenomenon which cannot be understood from 2 1 adiabatic density perturbations. 0 1. pacs: 98.62.Ai; 98.80.-k; 97.10Bt 0 6 keywords: Perturbation theory; cosmology; diabatic density perturbations; struc- 1 ture formation : v i X r a ∗[email protected] 1 Contents 1 Introduction 2 2 Einstein Equations for a Flat FLRW Universe 4 2.1 Background Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Evolution Equations for Density Perturbations . . . . . . . . . . . . . . . . . . . 4 3 Analytic Solutions 5 3.1 Radiation-dominated Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Era after Decoupling of Matter and Radiation . . . . . . . . . . . . . . . . . . . . 8 4 Structure Formation after Decoupling of Matter and Radiation 11 4.1 Observable Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Initial Values from the Planck Satellite . . . . . . . . . . . . . . . . . . . . . . . . 12 4.3 Diabatic Pressure Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.4 Structure Formation in the Early Universe . . . . . . . . . . . . . . . . . . . . . . 13 4.5 Relativistic Jeans Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 A Why the Standard Equation is inadequate to study Density Perturbations 15 A.1 General Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.2 Newtonian Theory of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A.3 Relativistic versus Newtonian Perturbation Theory . . . . . . . . . . . . . . . . . 19 1 Introduction The global properties of our universe are very well described by a Λcdm model with a flat Friedmann-Lemaître-Robertson-Walker(flrw)metricwithinthecontextoftheGeneralTheory of Relativity. To explain structure formation after decoupling of matter and radiation in this model,onehastoassumethatbeforedecouplingColdDarkMatter(cdm)hasalreadycontracted 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 article1 it has been shown that there are two unique gauge-invariant quantities εgi and ngi which are the real, measurable, perturbations to the energy density and (1) (1) the particle number density, respectively. Evolution equations for the corresponding contrast functions δ and δ have been derived for closed, flat and open flrw universes. In this arti- ε n cle 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 p = p(n,ε) has been taken into account, as is required by thermodynamics. As 1Section and equation numbers with a ∗ refer to sections and equations in the companion article [1]. 2 a consequence, in addition to a usual second-order evolution equation (3a) for density pertur- bations, 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 δ and the particle number ε density δ in the radiation-dominated era and the epoch after decoupling will be determined. It n is shown that the evolution equations (3) corroborate the standard perturbation theory in both erasinthelimitingcaseofinfinite-scaleperturbations. Forfinite scales, however, thedifferences are conspicuous. Therefore, only finite-scale perturbations are considered in detail. Afirstresultisthatintheradiation-dominatederaoscillatingdensityperturbationswithan increasing amplitudeproportionaltot1/2 arefound,whereasthestandardperturbationequation (61) yields oscillating density perturbations with a constant amplitude. This difference is due to the fact that in the perturbation equations (3) the divergence ϑ of the spatial part of the (1) fluid four-velocity is taken into account, whereas ϑ is missing in the standard equation. In the (1) appendix it is made clear why ϑ is important. (1) Intheradiation-dominatederaandtheplasmaerabaryonsweretightlycoupledtoradiation 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 p ≤ 0, (5), throughout the history n of the universe as will be shown in Section 3. From observations [2] of the Cosmic Microwave Background it follows that perturbations wereadiabaticatthemomentofdecoupling, anddensityfluctuationsδ andδ wereoftheorder ε n of 10−5 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 gravita- tion, 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 pres- sure 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 perturba- tion 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 6.5pc ≈ 21ly (see the peak value in Figure 1) gravity and heat exchange worked perfectly together, resulting in a fast growth rate. Perturbations larger than thisscalereached, despitetheirstrongergravitationalfield, theirnon-linearphaseatalatertime since heat exchange was low due to their larger scales. On the other hand, for perturbations with scales smaller than 6.5pc gravity was too weak and heat exchange was not sufficient to let perturbations grow. Therefore, density perturbations with scales smaller than 6.5pc did not 3 reach the non-linear regime within 13.81Gyr, the age of the universe. Since there was a sharp decline in growth rate below a scale of 6.5pc, this scale will be called the relativistic Jeans scale. TheconclusionofthepresentarticleisthattheΛcdmmodeloftheuniverseanditsevolution equations for density perturbations (3) explain the so-called (hypothetical) Population iii stars andlargerstructuresintheuniverse, whichcameintoexistenceseveralhundredsofmillionyears 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, p = p(n,ε). 2.1 Background Equations The set of zeroth-order Einstein equations and conservation laws for a flat, i.e., R = 0, flrw (0) universe filled with a perfect fluid with energy-momentum tensor Tµν = (ε+p)uµuν −pgµν, p = p(n,ε), (1) is given by 3H2 = κε , κ = 8πG /c4, (2a) (0) N ε˙ = −3Hε (1+w), w := p /ε , (2b) (0) (0) (0) (0) n˙ = −3Hn . (2c) (0) (0) The evolution of density perturbations took place in the early universe shortly after decoupling, when Λ (cid:28) κε . Therefore, the cosmological constant Λ has been neglected. (0) 2.2 Evolution Equations for Density Perturbations The complete set of perturbation equations for the two independent density contrast functions δ and δ is given by (44∗) n ε (cid:20) (cid:21) δ δ¨ +b δ˙ +b δ = b δ − ε , (3a) ε 1 ε 2 ε 3 n 1+w (cid:20) (cid:21) (cid:20) (cid:21) 1 d δ 3Hn p δ δ − ε = (0) n δ − ε , (3b) n n cdt 1+w ε (1+w) 1+w (0) where the coefficients b , b and b , (45∗), are for a flat flrw universe filled with a perfect fluid 1 2 3 described by an equation of state p = p(n,ε) given by β˙ b = H(1−3w−3β2)−2 , (4a) 1 β (cid:104) (cid:105) β˙ ∇2 b = κε 2β2(2+3w)− 1(1+18w+9w2) +2H (1+3w)−β2 , (4b) 2 (0) 6 β a2 (cid:40) (cid:34) (cid:35) (cid:41) −2 2p β˙ n ∇2 b = ε p (1+w)+ n +p (p −β2)+n p +p (0) , (4c) 3 1+w (0) εn 3H β n ε (0) nn n ε a2 (0) 4 where p (n,ε) and p (n,ε) are the partial derivatives of the equation of state p(n,ε): n ε (cid:18) (cid:19) (cid:18) (cid:19) ∂p ∂p p := , p := . (5) n ε ∂n ∂ε ε n The symbol ∇2 denotes the Laplace operator. The quantity β(t) is defined by β2 := p˙ /ε˙ . (0) (0) Using that p˙ = p n˙ +p ε˙ and the conservation laws (2b) and (2c) one gets (0) n (0) ε (0) n p β2 = p + (0) n . (6) ε ε (1+w) (0) From the definitions w := p /ε and β2 := p˙ /ε˙ and the energy conservation law (2b), one (0) (0) (0) (0) finds for the time-derivative of w w˙ = 3H(1+w)(w−β2). (7) This expression holds true independent of the equation of state. The pressure perturbation is given by (49∗) (cid:20) (cid:21) δ pgi = β2ε δ +n p δ − ε , (8) (1) (0) ε (0) n n 1+w where the first term, β2ε δ , is the adiabatic part and the second term the diabatic part of the (0) ε pressure perturbation. The combined First and Second Law of Thermodynamics reads (57∗) (cid:20) (cid:21) ε (1+w) δ T sgi = − (0) δ − ε . (9) (0) (1) n n 1+w (0) Densityperturbationsevolveadiabaticallyifandonlyifthesourcetermoftheevolutionequation (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 p ≈ 0, or, n equivalently, p ≈ p(ε), i.e., if the particle number density does not contribute to the pressure. In this case, the coefficient b , (4c), vanishes. 3 3 Analytic Solutions In this section analytic solutions of equations (3) are derived for a flat flrw universe with a vanishingcosmologicalconstantinitsradiation-dominatedphaseandintheeraafterdecoupling of matter and radiation. It is shown that p ≤ 0 throughout the history of the universe. In this n case,theentropyevolutionequation(3b)impliesthatfluctuationsintheparticlenumberdensity, δ , arecoupledtofluctuationsinthetotalenergydensity, δ , throughgravitation, irrespectiveof n ε thenatureoftheparticles. Inparticular,thisholdstrueforperturbationsincdm. 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. 5 3.1 Radiation-dominated Era Atveryhightemperatures,radiationandordinarymatterareinthermalequilibrium,coupledvia Thomson scattering with the photons dominating over the nucleons (n /n ≈ 109). Therefore γ p the primordial fluid can be treated as radiation-dominated with equations of state ε = a T4, p = 1a T4, (10) B γ 3 B γ where a is the black body constant and T the radiation temperature. The equations of state B γ (10) imply the equation of state for the pressure p = 1ε, so that, with (5), 3 p = 0, p = 1. (11) n ε 3 Therefore, one has from (6), β2 = w = 1. (12) 3 Using (11) and (12), the perturbation equations (3) reduce to (cid:20)1∇2 (cid:21) δ¨ −Hδ˙ − − 2κε δ = 0, (13a) ε ε 3 a2 3 (0) ε δ − 3δ = 0, (13b) n 4 ε where (58∗) has been used. Since p = 0 the right-hand side of (13a) vanishes, implying that n densityperturbationsevolvedadiabatically: theydidnotexchangeheatwiththeirenvironment. Moreover, baryons were tightly coupled to radiation through Thomson scattering, i.e., baryons obey δ = 3δ . Thus, for baryons (13b) is identically satisfied. In contrast to baryons, n,baryon 4 ε 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 H ∝ t−1, ε ∝ t−2, n ∝ t−3/2, a ∝ t1/2, (14) (0) (0) implying that T ∝ a−1. The dimensionless time τ is defined by τ := t/t . Since H := a˙/a, (0)γ 0 one finds that dk (cid:20) 1 (cid:21)k dk dk = = [2H(t )]k , k = 1,2. (15) ckdtk ct dτk 0 dτk 0 Substituting δ (t,x) = δ (t,q)exp(iq·x) into equation (13a) and using (15) yields ε ε 1 (cid:20)µ2 1 (cid:21) δ(cid:48)(cid:48)− δ(cid:48) + r + δ = 0, τ ≥ 1, (16) ε 2τ ε 4τ 2τ2 ε where a prime denotes differentiation with respect to τ. The parameter µ is given by r 2π 1 1 µ := √ , λ := λa(t ), (17) r 0 0 λ0 H(t0) 3 6 with λ the physical scale of a perturbation at time t (τ = 1), and |q| = 2π/λ. To solve 0 0 √ equation (16), replace τ by x := µ τ. After transforming back to τ, one finds r (cid:104) √ √ (cid:105)√ (cid:0) (cid:1) (cid:0) (cid:1) δ (τ,q) = A (q)sin µ τ +A (q)cos µ τ τ, (18) ε 1 r 2 r where the ‘constants’ of integration A (q) and A (q) are given by 1 2 (cid:34) (cid:35) sinµ 1 cosµ δ˙ (t ,q) r r ε 0 A (q) = δ (t ,q) ∓ δ (t ,q)− . (19) 1 ε 0 ε 0 cosµ µ sinµ H(t ) 2 r r r 0 For large-scale perturbations (λ → ∞), it follows from (18) and (19) that (cid:34) (cid:35) (cid:34) (cid:35) 1 δ˙ (t ) t δ˙ (t ) (cid:18) t (cid:19)2 ε 0 ε 0 δ (t) = − δ (t )− + 2δ (t )− . (20) ε ε 0 ε 0 H(t ) t H(t ) t 0 0 0 0 The energy density contrast has two contributions to the growth rate, one proportional to t and one proportional to t1/2. These two solutions have been found, with the exception of the precise factors of proportionality, by a large number of authors [6–11]. Consequently, the evolution equations (13) corroborates for large-scale perturbations the results of the literature. Small-scale perturbations (λ → 0) oscillate with an increasing amplitude according to 1  1 (cid:18) (cid:19) (cid:18) (cid:19) t 2 t 2 δε(t,q) ≈ δε(t0,q) cosµr−µr , (21) t t 0 0 as follows from (18) and (19). Thus, the evolution equations (13) yield oscillating density perturbations with an increasing amplitude, since in these equations ϑ (cid:54)= 0, as follows from (1) their derivation in Section 2.7∗. In contrast, the standard equation (61), which has ϑ = 0, (1) yields oscillating density perturbations with a constant amplitude. Finally, the plasma era has begun at time t , when the energy density of ordinary matter eq was equal to the energy density of radiation, (58), and ends at t , the time of decoupling of dec 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) ε(n,T) = nmc2+a T4, p(n,T) = 1a T4, (22) B γ 3 B γ 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 T from (22), one finds for the equation of state for the pressure, Section 2.1∗, γ p(n,ε) = 1(ε−nmc2), (23) 3 so that with (5) one gets p = −1mc2, p = 1. (24) n 3 ε 3 Since p < 0, equation (3b) implies that fluctuations in the particle number density, δ , were n n coupled to fluctuations in the total energy density, δ , through gravitation, irrespective of the ε nature of the particles. 7 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 ε(n,T) = nmc2+ 3nk T, p(n,T) = nk T, (25) 2 B B where k is Boltzmann’s constant, m the mean particle mass, and T the temperature of the B matter. For the calculations in this subsection it is only needed that the cdm particle mass is such that for the mean particle mass m one has mc2 (cid:29) k T, so that w := p /ε (cid:28) 1. B (0) (0) Therefore, asfollowsfromthebackgroundequations(2a)and(2b), onemayneglectthepressure nk T and the kinetic energy density 3nk T with respect to the rest mass energy density nmc2 B 2 B 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 T from (25) yields, Section 2.1∗, the equation of state for the pressure p(n,ε) = 2(ε−nmc2), (26) 3 so that with (5) one has p = −2mc2, p = 2. (27) n 3 ε 3 Substituting p , p and ε (25) into (6) on finds, using that mc2 (cid:29) k T, n ε B (cid:114) v 5k T β ≈ s = B (0), (28) c 3 mc2 with v the adiabatic speed of sound and T the matter temperature. Using that β2 ≈ 5w and s (0) 3 w (cid:28) 1, expression (7) reduces to w˙ ≈ −2Hw, so that with H := a˙/a one has w ∝ a−2. This implies that the matter temperature decays as T ∝ a−2. (29) (0) This, in turn, implies with (28) that β˙/β = −H. The system (3) can now be rewritten as (cid:20) ∇2 (cid:21) 2∇2 δ¨ +3Hδ˙ − β2 + 5κε δ = − (δ −δ ), (30a) ε ε a2 6 (0) ε 3 a2 n ε 1 d (δ −δ ) = −2H(δ −δ ), (30b) n ε n ε cdt where w (cid:28) 1 and β2 (cid:28) 1 have been neglected with respect to constants of order unity. From equation (30b) it follows with H := a˙/a that δ −δ ∝ a−2. (31) n ε 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 8 expression for εgi will be derived from (25). Multiplying ε˙ by θ /θ˙ and subtracting the (1) (0) (1) (0) result from ε , one finds (1) εgi = ngi mc2+ 3ngi k T + 3n k Tgi, (32) (1) (1) 2 (1) B (0) 2 (0) B (1) where also the definitions (40a∗) and (52∗) have been used. Dividing the result by ε , (25), (0) and using that k T (cid:28) mc2, one finds B (0) 3k T δ ≈ δ + B (0)δ , (33) ε n 2 mc2 T to a very good approximation. In this expression δ is the relative perturbation in the total ε energy density. Since mc2 (cid:29) 3k T , it follows from the derivation of (33) that δ can be 2 B (0) n considered as the relative perturbation in the rest energy density. Consequently, the second term is the fluctuation in the kinetic energy density, i.e., δ ≈ δ −δ . The relative kinetic kin ε n 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 β2 in the left-hand side. Combining (29) and (31) one finds from (33) that δ is constant T δ (t,x) ≈ δ (t ,x), (34) T T 0 to a very good approximation, so that the kinetic energy density fluctuation is given by 3k T (t) δ (t,x) ≈ δ (t,x)−δ (t,x) ≈ B (0) δ (t ,x). (35) kin ε n 2 mc2 T 0 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., δ < 0, its growth rate was, depending on its T scale, enhanced. Using (27) and (33), one finds from (8) δ ≈ 5δ +δ , (36) p 3 ε T where δ is the relative pressure perturbation defined by δ := pgi /p , with p given by p p (1) (0) (0) (25). The term 5δ is the adiabatic part and δ is the diabatic part of the relative pressure 3 ε T perturbation. The factor 5 is the so-called adiabatic index for a monatomic ideal gas with three 3 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) sgi ≈ 3k δ . (37) (1) 2 B T In Section 3.2∗ it has been shown that the background entropy per particle s is independent (0) of time. In a linear perturbation theory the perturbed entropy per particle is approximately constant, i.e., s˙gi ≈ 0. Therefore, heat exchange of a perturbation with its environment decays (1) proportional to the temperature, i.e., T sgi ∝ a−2, as follows from (29). (0) (1) 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 H ∝ t−1, ε ∝ t−2, n ∝ t−2, a ∝ t2/3, (38) (0) (0) 9 where the kinetic energy density and pressure have been neglected with respect to the rest mass energy density. The dimensionless time τ is defined by τ := t/t . Using that H := a˙/a, one gets 0 dk = (cid:20) 1 (cid:21)k dk = (cid:2)3H(t )(cid:3)k dk , k = 1,2. (39) ckdtk ct dτk 2 0 dτk 0 Substituting δ (t,x) = δ (t,q)exp(iq · x), δ (t,x) = δ (t,q)exp(iq · x), (28) and (35) into ε ε n n equations (30) and using (29) and (39) one finds that equations (30) can be combined into one equation 2 (cid:20)4 µ2 10 (cid:21) 4 µ2 δ(cid:48)(cid:48)+ δ(cid:48) + m − δ = − m δ (t ,q), τ ≥ 1, (40) ε τ ε 9τ8/3 9τ2 ε 15τ8/3 T 0 where a prime denotes differentiation with respect to τ. The parameter µ is given by m 2π 1 v (t ) µ := s 0 , λ := λa(t ), (41) m 0 0 λ H(t ) c 0 0 with λ the physical scale of a perturbation at time t (τ = 1), and |q| = 2π/λ. To solve 0 0 equation (40) replace τ by x := 2µ τ−1/3. After transforming back to τ, one finds for the m general solution of the evolution equation (40) (cid:34) (cid:35) δ (τ,q) =(cid:104)B (q)J (cid:0)2µ τ−1/3(cid:1)+B (q)J (cid:0)2µ τ−1/3(cid:1)(cid:105)τ−1/2− 3 1+ 5τ2/3 δ (t ,q), ε 1 +72 m 2 −72 m 5 2µ2m T 0 (42) where J (x) are Bessel functions of the first kind and B (q) and B (q) are the ‘constants’ of ±7/2 1 2 integration, calculated with the help of Maxima [13]: √ (cid:20) (cid:21) B (q) = 3 π (cid:0)4µ2 −5(cid:1)cos2µm ∓10µ sin2µm δ (t ,q) + 1 3/2 m sin2µ mcos2µ T 0 2 20µm m m √ (cid:20) (cid:21) π (cid:0)8µ4 −30µ2 +15(cid:1)cos2µm ∓(cid:0)20µ3 −30µ (cid:1)sin2µm δ (t ,q) + 7/2 m m sin2µ m m cos2µ ε 0 8µm m m √ π (cid:20)(cid:0)24µ2 −15(cid:1)cos2µm ±(cid:0)8µ3 −30µ (cid:1)sin2µm(cid:21) δ˙ε(t0,q). (43) 7/2 m sin2µ m m cos2µ H(t ) 8µm m m 0 The particle number density contrast δ (t,q) follows from equation (33), (34) and (42). In (42) n 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 λ → ∞ terms with ∇2 vanish. Therefore, the general solution of equation (40) becomes (cid:34) (cid:35) 2 (cid:34) (cid:35) 5 1 2δ˙ (t ) (cid:18) t (cid:19)3 2 δ˙ (t ) (cid:18) t (cid:19)−3 ε 0 ε 0 δ (t) = 5δ (t )+ + δ (t )− . (44) ε ε 0 ε 0 7 H(t ) t 7 H(t ) t 0 0 0 0 Thus, for large-scale perturbations the diabatic pressure fluctuation δ (t ,q) did not play a T 0 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 10