Structure Formation in the Early Universe

Home , Newtonian gauge, Structure formation

P. G. Miedema∗ Netherlands Defence Academy Hogeschoollaan 2 NL-4818CR Breda The Netherlands March 9, 2021

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: Cosmology; structure formation; perturbation theory; diabatic density perturbations arXiv:1601.01260v3 [gr-qc] 7 Mar 2021

∗[email protected]

1 Contents

1 Introduction 2

2 Einstein Equations and Conservation Laws for a Flat FLRW Universe3 2.1 Background Equations ...... 3 2.2 Evolution Equations for Density Perturbations ...... 3

3 Analytical Solutions 4 3.1 Radiation-dominated Era ...... 4 3.2 Plasma Era ...... 6 3.3 Era after Decoupling of Matter and Radiation ...... 6

4 Structure Formation after Decoupling of Matter and Radiation9 4.1 Observable Quantities ...... 9 4.2 Initial Values from the Planck Satellite ...... 10 4.3 Initial Values of Diabatic Pressure Fluctuations ...... 11 4.4 Structure Formation ...... 12 4.4.1 Relativistic Jeans Scale ...... 13 4.4.2 Classical Jeans Scale ...... 13 4.5 Growth Rates ...... 13

5 Conclusion 14

A Why the Standard Equation is inadequate to study Density Perturbations 15 A.1 Radiation-dominated Era ...... 17 A.2 Era after Decoupling of Matter and Radiation ...... 17 A.3 Dust-ﬁlled Universe ...... 19 A.4 Newtonian Theory of Gravity ...... 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) 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 Miedema(2014) 1 expressions for the true, physical energy density and particle phys phys number density perturbations ε(1) and n(1) have been found. Evolution equations for the corresponding contrast functions δε and δn 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 p = p(n, ε) has been taken into account, as is required by thermodynamics. As a consequence, in addition to a second-order evolution equation (3a) for density perturbations, a first-order evolution equation (3b) for entropy perturbations

1Section and equation numbers with a ∗ refer to sections and equations in the companion article.

2 follows also from the perturbed Einstein equations. The coeﬃcients of the second-order evolution equation depend on the equation of state for the pressure and diﬀers from equations found in the literature. The 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.

Analytical expressions for the fluctuations in the energy density δε and the particle number density δn 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. In the appendix it is shown in detail why the standard perturbation equation is inadequate to study the evolution of cosmological density perturbations. Furthermore, it is shown that the Jeans theory applied to an expanding universe, i.e., the Newtonian cosmological perturbation theory, is invalid.

2 Einstein Equations and Conservation Laws 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) = 0, flrw universe filled with a perfect fluid with energy-momentum tensor

T µν = (ε + p)uµuν − pgµν , p = p(n, ε), (1) is given by (8∗)

2 4 3H = κε(0), κ = 8πGN/c , (2a)

ε˙(0) = −3Hε(0)(1 + w), w := p(0)/ε(0), (2b)

n˙ (0) = −3Hn(0). (2c) The evolution of density perturbations took place in the early universe shortly after decoupling, when

Λ κε(0). Therefore, the cosmological constant Λ has been neglected. An over-dot denotes diﬀerentiation with respect to ct.

2.2 Evolution Equations for Density Perturbations

The complete set of perturbation equations for the two independent density contrast functions δn and δε is given by (65∗) δ δ¨ + b δ˙ + b δ = b δ − ε , (3a) ε 1 ε 2 ε 3 n 1 + w 1 d δε 3Hn(0)pn δε δn − = δn − , (3b) c dt 1 + w ε(0)(1 + w) 1 + w ∗ where the coefficients b1, b2 and b3, (66 ), are for a flat flrw universe filled with a perfect fluid described by an equation of state for the pressure p = p(n, ε) given by β˙ b = H(1 − 3w − 3β2) − 2 , (4a) 1 β h i β˙ ∇2 b = κε 2β2(2 + 3w) − 1 (1 + 18w + 9w2) + 2H (1 + 3w) − β2 , (4b) 2 (0) 6 β a2 ( " ˙ # ) 2 −2 2pn β 2 n(0) ∇ b3 = ε(0)pεn(1 + w) + + pn(pε − β ) + n(0)pnn + pn 2 , (4c) 1 + w 3H β ε(0) a

3 where the partial derivatives of the equation of state for the pressure p(n, ε) are given by

∂p ∂p ∂2p ∂2p : : : : pn = , pε = , pεn = , pnn = 2 . (5) ∂n ε ∂ε n ∂ε ∂n ∂n

2 2 The symbol ∇ denotes the Laplace operator. The quantity β(t) is deﬁned by β :=p ˙(0)/ε˙(0). Using that p˙(0) = pnn˙ (0) + pεε˙(0) and the conservation laws (2b) and (2c) one gets

2 n(0)pn β = pε + . (6) ε(0)(1 + w)

2 From the deﬁnitions w := p(0)/ε(0) and β :=p ˙(0)/ε˙(0) and the energy conservation law (2b), one ﬁnds for the time-derivative of w w˙ = 3H(1 + w)(w − β2). (7) This expression holds true independent of the precise form of the equation of state. The pressure perturbation is given by (71∗)

δ pphys = β2ε δ + n p δ − ε , (8) (1) (0) ε (0) n n 1 + w

2 where the ﬁrst term, β ε(0)δε, is the adiabatic part and the second term the diabatic part of the pressure perturbation. The combined First and Second Law of Thermodynamics reads (79∗) phys ε(0)(1 + w) δε T(0)s(1) = − δn − . (9) n(0) 1 + w

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 pn ≈ 0, or, equivalently, p ≈ p(ε), i.e., if the particle number density does not contribute to the pressure. In this case, the coeﬃcient b3,(4c), vanishes. In view of (8) and (9), the right-hand side of equation (3a) will be referred to as the entropy term and equation (3b) is considered as the entropy evolution equation.

3 Analytical Solutions

In this section analytical 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 pn ≤ 0 throughout the history of the universe. In this case, the entropy evolution equation (3b) implies that fluctuations in the particle number density, δn, are coupled to fluctuations in the total energy density, δε, 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.(2009), 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 9 scattering with the photons dominating over the nucleons (nγ /np ≈ 10 ). Therefore the primordial ﬂuid can be treated as radiation-dominated with equations of state

4 1 4 ε = aBTγ , p = 3 aBTγ , (10)

4 where aB is the black body constant and Tγ the radiation temperature. The equations of state (10) imply 1 the equation of state for the pressure p = 3 ε, so that, with (5), 1 pn = 0, pε = 3 . (11) Therefore, one has from (6), 2 1 β = w = 3 . (12) Using (11) and (12), the perturbation equations (3) reduce to 1 ∇2 δ¨ − Hδ˙ − − 2 κε δ = 0, (13a) ε ε 3 a2 3 (0) ε 3 δn − 4 δε = 0, (13b) ∗ where (80 ) has been used. Since pn = 0 the right-hand side of (13a) vanishes, implying that density perturbations evolved adiabatically: they did not exchange heat with their environment. Moreover, baryons 3 were tightly coupled to radiation through Thomson scattering, i.e., baryons obey δn, baryon = 4 δε. 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 8∗. 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

−1 −2 −3/2 1/2 H ∝ t , ε(0) ∝ t , n(0) ∝ t , a ∝ t , (14)

−1 implying that T(0)γ ∝ a . The dimensionless time τ is deﬁned by τ := t/t0. Since H :=a/a ˙ , one ﬁnds that k k k k d 1 d k d k k = k = [2H(t0)] k , k = 1, 2. (15) c dt ct0 dτ dτ

Substituting δε(t, x) = δε(t, q) exp(iq · x) into equation (13a) and using (15) yields 1 µ2 1 δ00 − δ0 + r + δ = 0, τ ≥ 1, (16) ε 2τ ε 4τ 2τ 2 ε where a prime denotes diﬀerentiation with respect to τ. The parameter µr is given by 2π 1 1 µr := √ , λ0 := λa(t0), (17) λ0 H(t0) 3 with λ the physical scale of a perturbation at time t (τ = 1), and |q| = 2π/λ. To solve equation (16), 0 √ 0 replace τ by x := µr τ. After transforming back to τ, one ﬁnds h √ √ i√ δε(τ, q) = A1(q) sin µr τ + A2(q) cos µr τ τ, (18) where the ‘constants’ of integration A1(q) and A2(q) are given by " ˙ # sin µr 1 cos µr δε(t0, q) A 1 (q) = δε(t0, q) ∓ δε(t0, q) − . (19) 2 cos µr µr sin µr H(t0)

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

1 " ˙ # " ˙ # δε(t0) t δε(t0) t 2 δε(t) = − δε(t0) − + 2δε(t0) − . (20) H(t0) t0 H(t0) t0

5 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 (Lifshitz and Khalatnikov, 1963; Adams and Canuto, 1975; Olson, 1976; Peebles, 1980; Kolb and Turner, 1994; Press and Vishniac, 1980). If one makes the ˙ plausible assumption that density perturbations do not evolve when they arise, i.e., δε(t0, q) ≈ 0, then only perturbations with λ → ∞ will die out, as follows from (20). In contrast, ﬁnite-scale density perturbations grow proportional to the square root of time (18). For example, λ → 0 yields

1  1  t 2 t 2 δε(t, q) ≈ δε(t0, q) cos µr − µr  , (21) t0 t0 as follows from (18) and (19). Thus, the evolution equations (13) yield oscillating density perturbations with an increasing amplitude.

3.2 Plasma Era

The plasma era has begun at time teq, when the energy density of ordinary matter was equal to the energy density of radiation, (56), and ends at tdec, the time of decoupling of matter and radiation. In the plasma era the matter-radiation mixture can be characterised by the equations of state (see Kodama and Sasaki (1984), Chapter V) 2 4 1 4 ε(n, T ) = nmc + aBTγ , p(n, T ) = 3 aBTγ , (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 Tγ from (22), one ﬁnds for the equation of state for the pressure

1 2 p(n, ε) = 3 (ε − nmc ), (23) so that with (5) one gets 1 2 1 pn = − 3 mc , pε = 3 . (24)

Since pn < 0, equation (3b) implies that ﬂuctuations in the particle number density, δn, were coupled to ﬂuctuations in the total energy density, δε, through gravitation, irrespective of the nature of the particles.

3.3 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

2 3 ε(n, T ) = nmc + 2 nkBT, p(n, T ) = nkBT, (25) where kB is Boltzmann’s constant, m the mean particle mass, and T 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 2 mean particle mass m one has mc kBT , so that w := p(0)/ε(0) 1. Therefore, as follows from the background equations (2a) and (2b), one may neglect the pressure nkBT and the kinetic energy density 3 2 2 nkBT with respect to the rest mass energy density nmc 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 7∗ on the non-relativistic limit. Consequently, it is important to take the pressure perturbations into account. Eliminating T from (25) yields the equation of state for the pressure

2 2 p(n, ε) = 3 (ε − nmc ), (26) so that with (5) one has 2 2 2 pn = − 3 mc , pε = 3 . (27)

6 2 Substituting pn, pε, and ε given by (25) into (6) on ﬁnds, using that mc kBT , r v 5 k T k T β ≈ s = B (0) , w ≈ B (0) , (28) c 3 mc2 mc2

2 5 with vs the adiabatic speed of sound and T(0) the matter temperature. Using that β ≈ 3 w and w 1, expression (7) reduces to w˙ ≈ −2Hw, so that with H :=a/a ˙ one has w ∝ a−2. This implies with (28) that the matter temperature decays as −2 T(0) ∝ a . (29) This, in turn, implies with (28) that β/β˙ = −H. The system (3) can now be rewritten as

∇2 2 ∇2 δ¨ + 3Hδ˙ − β2 + 5 κε δ = − (δ − δ ) , (30a) ε ε a2 6 (0) ε 3 a2 n ε 1 d (δ − δ ) = −2H (δ − δ ) , (30b) c dt n ε n ε where w 1 and β2 1 have been neglected with respect to constants of order unity. From equation (30b) it follows with H :=a/a ˙ that −2 δn − δε ∝ a . (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 phys energy density ﬂuctuation of the particles of a density perturbation. To that end, an expression for ε(1) ˙ will be derived from (25). Multiplying ε˙(0) by θ(1)/θ(0) and subtracting the result from ε(1), one ﬁnds

phys phys 2 3 phys 3 phys ε(1) = n(1) mc + 2 n(1) kBT(0) + 2 n(0)kBT(1) , (32)

∗ ∗ where also the expressions (60 ) and (74 ) have been used. Dividing the result by ε(0),(25), and using that 2 kBT(0) mc , one ﬁnds 3 k T δ ≈ δ + B (0) δ , (33) ε n 2 mc2 T to a very good approximation. In this expression δε is the relative perturbation in the total energy density. 2 3 Since mc 2 kBT(0), it follows from the derivation of (33) that δn is the relative perturbation in the rest energy density. Consequently, the second term is the ﬂuctuation in the kinetic energy density, i.e.,

δkin ≈ δε − δn. 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 β2 in the left-hand side. This is the very reason that pressure perturbations should not be neglected in the perturbed universe.

Combining (28), (29) and (31) one ﬁnds from (33) that δT is constant

δT (t, x) ≈ δT (t0, x), (34) to a very good approximation, so that the kinetic energy density fluctuation is given by 3 k 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 Section4 it will be shown that the kinetic energy density fluctuation has played, in addition to gravitation, a role in the evolution of density perturbations. Using (27) and (33), one finds from (8)

5 δp ≈ 3 δε + δT , (36) phys where δp is the relative pressure perturbation deﬁned by δp := p(1) /p(0), with p(0) given by (25). The term 5 5 3 δε is the adiabatic part and δT is the diabatic part of the relative pressure perturbation. The factor 3

7 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)

phys 3 s(1) ≈ 2 kBδT . (37) ∗ In Section 9.2 it has been shown that the background entropy per particle s(0) is independent of time. In the linear regime after decoupling of matter and radiation, the perturbed entropy per particle is approximately phys constant, i.e., s˙(1) ≈ 0, as follows from (34) and (37). Therefore, heat exchange of a perturbation with its phys −2 environment decays proportional to the temperature, i.e., T(0)s(1) ∝ a , 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

−1 −2 −2 2/3 H ∝ t , ε(0) ∝ t , n(0) ∝ t , a ∝ t , (38) 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/t0. Using that H :=a/a ˙ , one gets k k k k d 1 d 3 k d k k = k = 2 H(t0) k , k = 1, 2. (39) c dt ct0 dτ dτ

Substituting δε(t, x) = δε(t, q) exp(iq · x), δn(t, x) = δn(t, q) exp(iq · x),(28) and (35) into equations (30) and using (29) and (39) one ﬁnds that equations (30) can be combined into one equation 2 4 µ2 10 4 µ2 δ00 + δ0 + m − δ = − m δ (t , q), τ ≥ 1, (40) ε τ ε 9 τ 8/3 9τ 2 ε 15 τ 8/3 T 0 where a prime denotes diﬀerentiation with respect to τ. The parameter µm is given by

2π 1 vs(t0) µm := , λ0 := λa(t0), (41) λ0 H(t0) c with λ0 the physical scale of a perturbation at time t0 i.e., τ = 1, and |q| = 2π/λ. In equation (40) four terms can be recognised that increase or decrease the growth, i.e., 2 4 µ2 Expansion: + δ0 , Pressure: + m δ , (42a) τ ε 9 τ 8/3 ε 10 4 µ2 Gravity: − δ , Entropy: − m δ (t , q). (42b) 9τ 2 ε 15 τ 8/3 T 0

When δε > 0 the pressure counteracts the growth of density perturbations and gravity increases its growth. 0 0 The expansion counteracts the growth when δε > 0 and increases the growth when δε < 0. The entropy increases the growth if δT (t0, q) < 0. −1/3 To derive an analytical solution of equation (40) replace τ by x := 2µmτ . After transforming back to τ, one ﬁnds for the general solution of the evolution equation (40) 2/3 h −1/3 −1/3i −1/2 3 5 τ δε(τ, q) = B1(q)J 7 2µmτ + B2(q)J 7 2µmτ τ − 1 + δT (t0, q), (43) + 2 − 2 2 5 2 µm where J±7/2(x) are Bessel functions of the ﬁrst kind and B1(q) and B2(q) are the ‘constants’ of integration, calculated with the help of Maxima(2020): √ 3 π 2 cos 2µm sin 2µm B 1 (q) = 4µ − 5 ∓ 10µ δ (t , q) + 3/2 m m T 0 2 20µ sin 2µm cos 2µm √ m π cos 2µ sin 2µ 8µ4 − 30µ2 + 15 m ∓ 20µ3 − 30µ m δ (t , q) + 7/2 m m m m ε 0 8µm sin 2µm cos 2µm √ π cos 2µ sin 2µ δ˙ (t , q) 24µ2 − 15 m ± 8µ3 − 30µ m ε 0 . (44) 7/2 m m m 8µm sin 2µm cos 2µm H(t0)

8 The particle number density contrast δn(t, q) follows from equation (33), (34) and (43). In (43) 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 2 5 " ˙ # " ˙ # − 1 2δε(t0) t 3 2 δε(t0) t 3 δε(t) = 5δε(t0) + + δε(t0) − , (45) 7 H(t0) t0 7 H(t0) t0

Thus, for large-scale perturbations the diabatic pressure fluctuation δT (t0, q) 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 relativistic Jeans scale, i.e., the peak value at 6.5 pc in Figure1, gravity alone was insufficient to explain structure formation within 13.79 Gyr, since they grow 2/3 as δε ∝ t . The solution proportional to t2/3 is a standard result (Lifshitz and Khalatnikov, 1963; Adams and

Canuto, 1975; Olson, 1976; Peebles, 1980; Kolb and Turner, 1994; Press and Vishniac, 1980). Since δε is gauge-invariant, the standard non-physical gauge mode proportional to t−1 is absent from the solution set of the evolution equations (30). Instead, a physical mode proportional to t−5/3 is found. Consequently, only the growing mode of (45) is in agreement with results given in the literature. In the small-scale limit λ → 0, one ﬁnds from (43) and (44)

1  1  − 3 − 3 3 t h 3 i t δε(t, q) ≈ − 5 δT (t0, q) + δε(t0, q) + 5 δT (t0, q) cos 2µm − 2µm  . (46) t0 t0

Thus, density perturbations with scales smaller than the relativistic 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 relativistic 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 relativistic 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 p = p(n, ε) yields that in the era after decoupling of matter and radiation density perturbations may have grown fast. 2 Up till now it is only assumed that mc kBT 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 µm (41) will be expressed in observable quantities, namely the present values of the background radiation temperature, T(0)γ (tp), the Hubble parameter, H(tp), and the redshift at decoupling, z(tdec). From now on the initial time is taken to be the time at decoupling of matter and radiation: t0 = tdec, so that τ := t/tdec. The redshift z(t) as a function of the scale factor a(t) is given by a(t ) z(t) = p − 1, (47) a(t)

9 Table 1: Planck satellite results.

z(tdec) = 1090 z(teq) = 3387 −1 −1 cH(tp) = 67.66 km s Mpc

T(0)γ (tp) = 2.725 K tp = 13.79 Gyr −5 δTγ (tdec) = . 10

where a(tp) is the present value of the scale factor and z(tp) = 0. For a ﬂat flrw universe one may take a(tp) = 1. Using the background solutions (38), one ﬁnds from (47)

3/2 H(t) = H(tp) z(t) + 1 , (48a) −3/2 t = tp z(t) + 1 , (48b) T(0)γ (t) = T(0)γ (tp) z(t) + 1 , (48c)

−1 where it is used that T(0)γ ∝ a after decoupling, as follows from (10) and (14). The dimensionless time τ := t/tdec can be expressed in the redshift

z(t ) + 13/2 τ = dec , (49) z(t) + 1 by using that τ = (t/tp)(tp/tdec) and (48b). Substituting (28) into (41), one gets r 2π 1 5 k T (t ) B (0) dec : µm = 2 , λdec = λa(tdec), (50) λdec H(tdec) 3 mc where tdec is the time when matter decouples from radiation and λdec is the physical scale of a perturbation at time tdec. From (48) one ﬁnds r 2π 1 1 5 kBT(0)γ (tp) µm = , (51) λdec cH(tp) z(tdec) + 1 3 m

where it is used that for the matter temperature T(0) at decoupling one has T(0)(tdec) = T(0)γ (tdec). With (51) the parameter µm is expressed in observable quantities.

4.2 Initial Values from the Planck Satellite The physical quantities measured by Planck Collaboration(2020) and physical constants needed in the parameter µm (51) of the evolution equation (40) are given in Table1 and2. Substituting these values into (51), one ﬁnds 16.48 µm = , λdec in pc, (52) λdec

The Planck observations of the ﬂuctuations δTγ (tdec) in the background radiation temperature yield for the initial value of the ﬂuctuations in the energy density

−5 |δε(tdec, q)| . 10 . (53)

In addition, it is assumed that ˙ δε(tdec, q) ≈ 0, (54)

10 Table 2: Physical constants.

m = 1.6726 × 10−27 kg pc = 3.0857 × 1016 m = 3.2616 ly c = 2.9979 × 108 m s−1 −23 −1 kB = 1.3806 × 10 JK −11 3 −1 −2 GN = 6.6743 × 10 m kg s 30 M = 1.9889 × 10 kg −16 −3 −4 aB = 7.5657 × 10 J m K implying that 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. The initial values (53) and (54) imply the initial values for equation (40)

−5 0 |δε(τ = 1, q)| . 10 , δε(τ = 1, q) = 0. (55)

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

4.3 Initial Values of Diabatic Pressure Fluctuations

The initial value of a diabatic pressure ﬂuctuation δT (tdec, q),(36), just after decoupling of matter and radiation cannot be obtained directly from observation, since its value depends on how a small area under- goes the transition from a high pressure, radiation-dominated, era to a low pressure, matter-dominated, era. In fact, δT (tdec, q) is a quantity that takes random values. This section describes the circumstances of this transition, so that a estimates of the δT (tdec, q) can be provided. To that end, the pressure just before decoupling is compared with the pressure just after decoupling.

The particle number density n(0)(tdec) at the time of decoupling can be calculated from its value n(0)(teq) at the end of the radiation-dominated era. By deﬁnition, at the end of the radiation-domination 2 era the matter energy density n(0)mc was equal to the energy density of the radiation:

2 4 n(0)(teq)mc = aBT(0)γ (teq). (56)

−3 −1 Since n(0) ∝ a and T(0)γ ∝ a , one ﬁnds, using (47), (48c) and (56), for the particle number density

4 aBT (tp) 3 n (t ) = (0)γ z(t ) + 1z(t ) + 1 . (57) (0) dec mc2 eq dec At the moment of decoupling of matter and radiation, photons could not ionise matter any more and the 1 4 two constituents fell out of thermal equilibrium. As a consequence, the high radiation pressure p = 3 aBTγ just before decoupling did go over into the low gas pressure p = nkBT after decoupling. In fact, from (48c) and (57) it follows that at decoupling one has, using also the values in Tables1 and2, n (t )k T (t ) 3k T (t ) (0) dec B (0) dec = B (0)γ p z(t ) + 1 ≈ 8.2 × 10−10, (58) 1 4 mc2 eq 3 aBT(0)γ (tdec) where it is used that at the moment of decoupling the matter temperature T(0)(tdec) was equal to the radiation temperature T(0)γ (tdec). The fast and chaotic transition from a high pressure epoch to a very low pressure era may have resulted in positive or negative relative diabatic pressure perturbations δT ,(36), with widely diﬀering values due to very small ﬂuctuations δkin,(35), in the kinetic energy density. Large voids were created if δT > 0. In Section 4.4 it will be shown that density perturbations which were cooler than their environments may have collapsed fast, depending on their scales. In fact, perturbations for which

δT (tdec, q) . −0.005, (59)

11 Structure Formation starting at z = 1090 24 0.11 −0.10 22 0.13 20 −0.09 0.14 18 −0.08 0.17

16 −0.07 0.20 14 0.24 −0.06 12 0.29 −0.05 10 0.38 8 −0.04 0.51 in Gyr Time −0.03

Cosmological Redshift 6 0.74 4 −0.02 1.23

2 −0.01 2.65 0 −0.005 13.79 0 10 20 30 40 50 Perturbation Scale λdec in parsec

Figure 1: The curves give the redshift and time, as a function of λdec, when a linear perturbation in the −5 ˙ energy density with initial values δε(tdec, q) . 10 and δε(tdec, q) ≈ 0 starting to grow at an initial redshift of z(tdec) = 1090 has become non-linear, i.e., δε(t, q) = 1. The curves are labeled with the initial values of the relative perturbations δT (tdec, q) in the diabatic part of the pressure. For each curve, the Jeans scale, i.e., the peak value, is at 6.5 pc.

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 this section the evolution equation (40) is solved numerically (Soetaert et al., 2010; R Core Team, 2020) and the results are summarised in Figure1, which is constructed as follows. For each choice of δT (tdec, q) in the range −0.005, −0.01, −0.02,..., −0.1 equation (40) is integrated for a large number of values for the initial perturbation scale λdec using the initial values (55). The integration starts at τ = 1, i.e., at 3/2 z(tdec) = 1090 and will be halted if either z = 0, i.e., τ = [z(tdec) + 1] , see (49), or δε(t, q) = 1 for z > 0 has been reached. One integration run yields one point on the curve for a particular choice of the scale

λdec if δε(t, q) = 1 has been reached for z > 0. If the integration halts at z = 0 and still δε(tp, q) < 1, then the perturbation pertaining to that particular scale λdec has not yet reached its non-linear phase today, i.e., at tp = 13.79 Gyr. On the other hand, if the integration is stopped at δε(t, q) = 1 and z > 0, then the perturbation has become non-linear within 13.79 Gyr. Each curve denotes the time and scale for which

δε(t, q) = 1 for a particular value of δT (tdec, q). The growth of a perturbation was governed by gravity as well as heat exchange. From Figure1 one may infer that the optimal scale for growth was around 6.5 pc. At this scale, which is independent of the initial value of the diabatic pressure perturbation δT (tdec, q), see (8) and (36), heat exchange and gravity worked together perfectly, resulting in a fast growth. Perturbations with scales smaller than 6.5 pc 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 6.5 pc exchanged heat at a slower rate due to their large scales, so

12 that they reached the non-linear phase much later. Perturbations larger than 50 pc grew proportional to t2/3,(45), a well-known result. These large-scale perturbations did not reach the non-linear regime within 13.79 Gyr.

4.4.1 Relativistic Jeans Scale Since the growth rate decreased rapidly for perturbations with scales below 6.5 pc, this scale is the relativistic Jeans scale in an expanding universe at decoupling:

λJ(tdec) := λJa(tdec) ≈ 6.5 pc = 21 ly. (60)

The value of λJ(tdec) was much smaller than the horizon size at decoupling, dH(tdec) = 3ctdec ≈ 3.5 × 5 6 10 pc ≈ 1.1 × 10 ly. The Jeans mass at decoupling, MJ(tdec), can be estimated by assuming that a density perturbation has a spherical symmetry with diameter the relativistic Jeans scale λJ(tdec). The relativistic Jeans mass at decoupling is then given by

4π 3 M (t ) = 1 λ (t ) n (t )m ≈ 4.3 × 103M , (61) J dec 3 2 J dec (0) dec where (57) and the values in Table1 and2 have been used.

4.4.2 Classical Jeans Scale The usual Jeans scale in the literature follows from the standard equation for the gauge-dependent contrast function δ := ε(1)/ε(0), i.e.,

∇2 δ¨ + 2Hδ˙ − β2 + 1 κε (1 + w)(1 + 3w) δ = 0. (62) a2 2 (0) by taking H = 0, w = 0 and β = vs/c, see equations (1.1) and (2.11)–(2.13) in the classic article of Bonnor (1957). At decoupling its value is given by

1 1 2 2 2 π dp πvs (tdec) λJ,H=0(tdec) = = ≈ 31 pc = 103 ly, (63) GNρ dρ GNn(0)(tdec)m where (57) and the values in Tables1 and2 are used. The corresponding Jeans mass is

4π 3 M (t ) = 1 λ (t ) n (t )m = 4.9 × 105M . (64) J,H=0 dec 3 2 J,H=0 dec (0) dec As a result of the expansion, the relativistic Jeans mass (61) is a factor 114 smaller than the usual Jeans mass. It will be shown in the appendix that equation (62) is questionable, whereas taking H = 0 is simply wrong because this violates the Friedmann equation (2a). Consequently, one may not apply the Jeans theory to a non-static universe.

4.5 Growth Rates 0 −1 The growth rate δε in equation (40) is expressed in τ , where τ := t/tdec is the dimensionless time. To express the growth rate in Gyr−1, one needs (39), i.e.,

dδ (t, q) cδ˙ (t, q) = ε = 3 cH(t )δ0 (τ, q)[s−1] = 0.1038 δ0 (τ, q)[Gyr−1], (65) ε dt 2 dec ε ε where (48a) and Tables1 and2 are used.

In Figure2 the growth rates are shown for δT (tdec, q) = −0.1 for four diﬀerent values of λdec. Since the initial values (55) were very small, the entropy in equation (40) had to be positive i.e., δT < 0, to start the growth of a density perturbation after decoupling. A negative diabatic pressure ﬂuctuation δT < 0 and

13 Time in Myr 0.38 0.51 0.74 1.2 2.6 14

λ = 0.20 dec 2

δT = − 0.1 0.15

0.10

λdec = 4.1 0.05 Growth Rate Growth

λdec = 6.5 λdec = 31 0.00

−0.05

1100 900 700 500 300 100 Cosmological Redshift

0 −1 0 Figure 2: The curves give the growth rates δε, expressed in τ , with initial value δε(tdec, q) = 0, as function of the redshift z, or time in million of years. The evolution of perturbations started at z = 1090.

The initial dimensions λdec of the perturbations are measured in parsec.

5 −5 a small adiabatic pressure fluctuation 3 δε(tdec, q) < 10 implied that the total pressure fluctuation (36) was negative δp < 0, so that a density perturbation could have started to grow. The largest initial growth rate occurred for perturbations which were smaller than 6.5 pc, the peak value in Figure1. The smallest perturbation which had become non-linear within 13.79 Gyr had a dimension of λdec = 2 pc, as can be seen in Figure1. All perturbations with initial dimensions 2 pc ≤ λdec < 4.1 pc oscillated towards the non-linear phase within 13.79 Gyr. These perturbations grew initially so fast that after a period of time the pressure in equation (40) had become larger than the entropy, so that the growth rate decreased rather fast and had become over time negative and the density perturbation started to decay. In the meantime, the density perturbation δε had grown so that gravity in (40) did come into play. When the expansion term had become 0 negative, i.e., when δε < 0, and since the pressure decayed faster than the expansion term the expansion had taken over and the growth rate increased again until the next, lower, peak in the growth rate. For all 3 perturbations the growth rate started to decrease when δp > 0, i.e., when δε > − 5 δT , as follows from (36). In this case, the pressure in equation (40) had become larger than the entropy. However, gravity decayed slower than the pressure, so that gravity had become gradually stronger than the counteracting pressure and a density perturbation could continue to grow. After about 15 million years, the pressure and entropy no longer played a significant role so that the most turbulent growth phase of a density perturbation was over and a phase of slow and steady growth by its own gravity towards the non-linear regime had begun.

5 Conclusion

The new perturbation theory developed for density perturbations in closed, flat and open flrw universes (Miedema, 2014) has been applied to a flat flrw universe in the radiation-dominated era and in the epoch after decoupling of matter and radiation. A first result is that in the radiation-dominated era oscillating density perturbations with an increasing amplitude proportional to t1/2 are found, whereas the standard

14 perturbation equation (62) with w = 0 yields oscillating density perturbations with a constant amplitude.

This difference is due to the fact that in the perturbation equations (3) the divergence ϑ(1) of the spatial part of the fluid four-velocity is taken into account, whereas ϑ(1) is missing in the standard equation. 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. The fact that cdm and radiation are coupled by gravitation follows from the entropy evolution equation (3b) since pn ≤ 0 throughout the history of the universe as has been shown in Section3. 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. From observations (Komatsu et al., 2011) of the Cosmic Microwave Background it follows that perturbations were adiabatic at the moment of decoupling, and density fluctuations δε and δn were of the order 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 gravitation, some mechanism other than cdm 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 ionise 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 6.5 pc ≈ 21 ly, the peak value in Figure1, 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 much smaller than 6.5 pc gravity was too weak and heat exchange was not sufficient to let perturbations grow. Therefore, density perturbations with scales much smaller than 6.5 pc did not reach the non-linear regime within 13.79 Gyr, the age of the universe. Since there was a sharp decline in growth rate below a scale of 6.5 pc, this scale will be called the relativistic Jeans scale. The conclusion of the present article is that the Λcdm 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 as has been observed by Watson et al.(2015); Sobral et al.(2015).

A Why the Standard Equation is inadequate to study Density Perturbations

The standard evolution equation for relative density perturbations δ(t, x) in a flat, R(0) = 0, flrw universe with vanishing cosmological constant, Λ = 0, is given by (62). In the radiation-dominated universe one has 2 1 1 β = 3 and w = 3 and this equation is identical to the relativistic equation (15.10.57) in the textbook of Weinberg(1972). In order to arrive at equation (62) Weinberg(1972) applies the large-scale limit q → 0, implying that ∇2δ → 0, so that the pressure term with β2 does not occur. However, q → 0 yields, in 2 : k addition to ∇ δ → 0, also q · U 1 → 0, which is equivalent to ϑ(1) = u(1)|k → 0, so that there is no fluid ∗ flow at all. In Section 7 it is shown that ϑ(1) → 0 yields the non-relativistic limit. Therefore, the limit

ϑ(1) → 0 is not allowed. In the epoch after decoupling of matter and radiation β is given by (28), so that 3 2 w ≈ 5 β 1. In this case (62) is identical to equation (15.9.23) of Weinberg(1972) which has been derived using the Newtonian Theory of Gravity.

15 In this section it will be shown that the standard relativistic equation (62) is inadequate to study the evolution of density perturbations in the universe. To that end, equation (62) will be derived from the full set of linearized Einstein equations and conservation laws (32∗), or, equivalently, (62∗) in the companion article. Since the source term of (62) is zero, this equation describes adiabatic perturbations, Section 9.3∗, which evolve only under the influence of their own gravitational field. Therefore, the equation of state is given by p = p(ε). This implies that pn = 0, so that p˙(0) = pεε˙(0) and p(1) = pεε(1). Consequently, the evolution equations for the background particle number density n(0),(2c), and its first-order perturbation ∗ 2 2 n(1), (62b ), need not be considered. From (6) one finds that pε = β so that p(1) = β ε(1). Using the ∗ definition δ := ε(1)/ε(0) and R(0) = 0 equations (62 ) for scalar perturbations can be written in the form

h i R δ˙ + 3Hδ β2 + 1 (1 − w) + (1 + w) ϑ + (1) = 0, (A.1a) 2 (1) 4H β2 ∇2δ ϑ˙ + H(2 − 3β2)ϑ + = 0, (A.1b) (1) (1) 1 + w a2 ˙ R(1) + 2HR(1) − 2κε(0)(1 + w)ϑ(1) = 0, (A.1c) where also the background equations (2) have been used. These equations include the complete set of Einstein equations and conservation laws for scalar perturbations as follows from their derivation in the companion article (Miedema, 2014) in Sections 5∗ and 8∗. In fact, equation (A.1a) is the energy density conservation law combined with the energy density constraint equation. Equation (A.1b) is the momentum conservation law or continuity equation, and (A.1c) is the momentum constraint equation. As is shown ∗ i in Section 5 , the dynamical equations G(1)j are not needed, since they are automatically fulﬁlled by the system (A.1).

Diﬀerentiating (A.1a) with respect to time and eliminating the time-derivatives of H, ε(0), ϑ(1) and R(1) with the help of the system of equations (2) and perturbation equations (A.1b) and (A.1c), respectively, and, subsequently, eliminating R(1) with the help of (A.1a), one ﬁnds, using Maxima(2020), that the set of equations (A.1) can be recast in the form

h i ∇2 δ¨ + 2Hδ˙ 1 + 3β2 − 3w − β2 + 1 κε (1 + w)(1 + 3w) a2 2 (0) 2 2 2 4 ˙ 2 + 4w − 6w + 12β w − 4β − 6β − 6ββH δ = −3Hβ (1 + w)ϑ(1), (A.2a)

β2 ∇2δ ϑ˙ + H(2 − 3β2)ϑ + = 0, (A.2b) (1) (1) 1 + w a2 where w˙ has been eliminated using (7). The system (A.2) consists of two relativistic equations for the ∗ unknown quantities δ and ϑ(1). Thus, the relativistic perturbation equations (62 ) for open, ﬂat or closed flrw universes and a general equation of state for the pressure p = p(n, ε) reduce for a ﬂat flrw universe and a barotropic equation of state p = p(ε) to the system (A.2). Therefore, the system (A.2) describes adiabatic density perturbations, notwithstanding the non-zero source term in (A.2a). The gauge modes (42∗)

ψ(x)ε ˙ (t) ∇2ψ(x) ˆ : (0) ˆ δ(t, x) = = −3H(t)ψ(x) 1 + w(t) , ϑ(1)(t, x) = − 2 , (A.3) ε(0)(t) a (t) are, for all scales, solutions of equations (A.2), with w˙ given by (7), as can be veriﬁed by substitution. phys ˙phys The usual way to solve the system (A.2), is to impose physical initial conditions δ (t0, x), δ (t0, x) phys and ϑ(1) (t0, x), so that the initial values are written as linear combinations of the constants of integration. This standard procedure is not feasible for the cosmological perturbation equations (A.2), since the gauge modes (A.3) are solutions of the system (A.2), so that some integration constants are in fact gauge modes,

16 which cannot be fixed by physical initial conditions. Consequently, the presence of gauge modes in the linearized Einstein equations and conservation laws hinders the formulation of a correct perturbation theory. A striking difference between equations (62) and (A.2a) is that the right-hand side of the standard equation (62) is zero, whereas the right-hand side of equation (A.2a) contains ϑ(1). The divergence ϑ(1) is an intrinsic property of a density perturbation and thus should be incorporated in the left-hand side of : k a perturbation equation as is the case in equation (3a). The fact that ϑ(1) = u(1)|k is indeed part of the left-hand side of equation (3a) follows from its derivation in the appendix of Miedema(2014). The source term in equation (A.2a) displaces the actual source term, namely the entropy. In the next subsections it : k will be shown, using the system (A.2) why the three-divergence, ϑ(1) = u(1)|k, of the spatial part of the fluid velocity should not be neglected and why, after decoupling of matter and radiation, the entropy is of the utmost importance.

A.1 Radiation-dominated Era 1 1 In this era, the pressure is given by a linear barotropic equation of state p = 3 ε, so that pn = 0 and pε = 3 . 2 1 From (6), one ﬁnds that β = 3 , so that (7) is identically satisﬁed. In this case equations (A.2) reduce to 1 ∇2 δ¨ + 2Hδ˙ − + 4 κε δ = − 4 Hϑ , (A.4a) 3 a2 3 (0) 3 (1) 1 ∇2δ ϑ˙ + Hϑ + = 0. (A.4b) (1) (1) 4 a2 1 The gauge modes (A.3) are solutions of the system (A.4) for w = 3 . For large-scale perturbations one has ∇2δphys → 0, so that the solution of the system (A.4) is

−1 9 1/2 δ = (c1t − 2ψt ) + 8 t , (A.5a) ∇2ψ ϑ = − + 9 t−1/2, (A.5b) (1) a2 8 2 where it is used that 3H = κε(0) and H = 1/2t. The expression between brackets in (A.5a) is a solution 9 1/2 of the homogeneous part of equation (A.4a), whereas the particular solution 8 t comes from the physical part of ϑ(1). Since the physical part of ϑ(1) in (A.5b) is positive, the fluid flows out of a density perturbation, so that a density perturbation will be diluted, in accordance the remark below expression (20). The solution (A.5a) will be compared with the solution of the system (13) for large-scale perturbations, which reads 1/2 δε(t) = d1t + d2t , (A.6) which does not contain the gauge mode. The particular solution of equation (A.4a) is a solution of the homogeneous equation (13a) since ϑ(1) is part of equation (13a). Hence the differences in appearance between (13a) and (A.4a). Comparing (A.5a) and (A.6) it must be concluded that the gauge mode has displaced the physical solution from its proper place, since ϑ(1) occurs in the source term of (A.4a).

The fact that ϑ(1) is absent in (62) is detrimental to our view of cosmological density perturbations in the radiation-dominated universe. Since ∇2δphys could have been large for small-scale perturbations, it phys may have a large inﬂuence on ϑ(1) and this may have, in turn, a major impact on the evolution of δ . That is why (13a) yields oscillating density perturbations (18) with an increasing amplitude, instead of a constant amplitude as follows from (62). The solutions of the standard equation (62) and the system (A.4) contain gauge modes. Moreover, equation (62) is incomplete. Therefore, both (62) and (A.4) are inadequate to study small-scale density perturbations in the radiation-dominated universe.

A.2 Era after Decoupling of Matter and Radiation In this era, the equation of state for the pressure is, according to thermodynamics, given by (26), so that in this case one should have p = p(n, ε). In the literature one takes p = 0 in the background as well as in the

17 perturbed universe. Obviously, the pressure can be neglected in the background universe since w 1 in equation (2b). However, the pressure in the perturbed universe should not be neglected, since p = 0 yields the non-relativistic limit, as is shown in Section 7∗. Therefore, the equations of state (25) will be used to 3 2 ˙ derive (62). From (28) it follows that w ≈ 5 β 1, so that with (29) one ﬁnds β/β = −H. Using that 2 ˙ 2 3H = κε(0) one gets 6ββH = −2κε(0)β . Substituting the latter expression into (A.2a) and neglecting w and β2 with respect to constants of order unity, the system (A.2) reduces to

∇2 δ¨ + 2Hδ˙ − β2 + 1 κε δ = −3Hβ2ϑ , (A.7a) a2 2 (0) (1) ∇2δ ϑ˙ + 2Hϑ + β2 = 0. (A.7b) (1) (1) a2 The gauge modes (A.3) are solutions of the system (A.7) for w 1 and ∇2ψ = 0, as can be veriﬁed by substitution. Consequently, for the system (A.7) ψ is an arbitrary inﬁnitesimal constant C. This implies phys ˆ that ϑ(1) = ϑ(1) is a physical quantity, since its gauge mode ϑ(1),(A.3), vanishes identically. Consequently, for large-scale perturbations the solution of the system (A.7) is

2/3 −1 7 −5/3 δ = (c1t − 2Ct ) + 9 t , (A.8a) phys 7 −4/3 ϑ(1) = − 9 t , (A.8b)

2 2 −4/3 where it is used that 3H = κε(0), H = 2/3t, and β ∝ t as follows from (28) and (29). The expression between brackets in (A.8a) is a solution of the homogeneous part of equation (A.7a), whereas the particular 7 −5/3 phys phys solution 9 t comes from ϑ(1) . The fact that ϑ(1) < 0, implies that the ﬂuid ﬂow is towards the density perturbation, so that the density perturbation grows (45). Since ψ = C, the gauge transformation (2∗) and (4∗) reduces to the residual gauge transformation (51∗) in the non-relativistic limit, i.e.,

x0 → x00 = x0 − C, xi → x0i = xi − χi(x). (A.9)

This is to be expected, since a cosmological fluid for which w 1 and β2 1 can be described by non-relativistic equations of state (25). The solution (A.8a) will be compared with the solution of the system (30) for large-scale perturbations, which reads 2/3 −5/3 δε(t) = d1t + d2t . (A.10) This solution does not contain the gauge mode. The particular solution of (A.7a) is a solution of the homogeneous part of equation (30a) since ϑ(1) is part of the left-hand side of equation (30a). Hence the differences in appearance between the left-hand side of (30a) and (A.7a). Comparing (A.8a) and (A.10) it must be concluded that the gauge mode has displaced the physical solution from its proper place, due to the fact that ϑ(1) occurs in the source term of (A.7a). phys Just as in the radiation-dominated era, the standard equation (62) lacks the quantity ϑ(1) in its source term. Although ∇2δphys could have been large for small-scale density perturbations in the early universe phys 2 the absence of ϑ(1) is not as harmful as it is in the radiation-dominated phase: due to the smallness of β phys phys and the non-relativistic particle velocities after decoupling, the impact of ϑ(1) on the evolution of δ is low, but non-zero. This explains why both (62) and the homogeneous part of (30a) yield perturbations which grow to slow to explain structure formation after decoupling as can be seen from the analytical solution (43) with δT = 0. The usual perturbation theory for density perturbations in the era after decoupling does not take the entropy and its evolution equation into account. This is a serious flaw, since the entropy, which act as a source for the growth of density perturbations, plays an important role in structure formation. Therefore, equations (62) and (A.7) are incomplete and, as a consequence, do not explain structure formation in the universe. In contrast, the perturbation theory developed in Miedema(2014) yields for equations of

18 state (25) the system (30). The entropy term in equation (30a) is of the same order of magnitude as the pressure term, as follows from equation (40). The calculations in Section4 show that the entropy term has in the first 15 million years after the Big Bang a major influence on the growth of density perturbations. This growth is sufficient to explain structure formation after decoupling of matter and radiation.

A.3 Dust-filled Universe Using the system (A.2) it will now be shown that the standard equation (62) is not valid in a universe filled with a pressure-less fluid, also known as ‘dust’. In a dust-filled universe the pressure vanishes, so that β = 0 and w = 0. The system (A.2) reduces to

¨ ˙ 1 δ + 2Hδ − 2 κε(0)δ = 0, (A.11a) ˙ ϑ(1) + 2Hϑ(1) = 0. (A.11b)

Using (2a) and (38), the solution of the system (A.11) is

2/3 −1 δ = c1t − 2ψt , (A.12a) ∇2ψ ϑ = − . (A.12b) (1) a2 Since equations (A.11a) and (A.11b) are decoupled, the fluid flow has no influence on the evolution of δ. phys Therefore, ϑ(1) = 0 so that ϑ(1) is equal to its gauge mode. This is to be expected, since in a pressure-less fluid there is no fluid flow. Notwithstanding the fact that there is no local fluid flow, equation (A.11a) yields ∗ phys a growing mode. This, however, is impossible. As has been shown in Section 7 (Miedema, 2014), ϑ(1) = 0 and w = 0 yields the non-relativistic limit in which the gravitational field is constant. Consequently, equation (A.11a) is not correct. It must be concluded that the standard equation (A.11a) cannot be considered as the non-relativistic limit of a cosmological perturbation theory, as is assumed by Hwang and Noh(1997, 2006). This fact will be further discussed in the next subsection.

A.4 Newtonian Theory of Gravity It is generally assumed that if the energy density is dominated by non-relativistic particles, so that w 1 and β2 1, and if the linear scales involved are small compared with the characteristic scale H−1 of the universe, then one may safely use the Newtonian Theory of Gravity to study the evolution of density perturbations. The perturbation equations of the (Newtonian) Jeans theory adapted to an expanding universe after decoupling are given by Weinberg(1972), equations (15.9.12)–(15.9.16). See also Mukhanov(2005),

Section 6.2 on the Jeans theory. Substituting v1 := au(1), ρ := ε(0) and ρ1 := ε(1) = ε(0)δ in equations (15.9.12)–(15.9.16) and taking the divergence of (15.9.13), one arrives at the Newtonian equations in the notation used in the present article:

˙ δ + ϑ(1) = 0, (A.13a) ∇2δ ∇2φ ϑ˙ + 2Hϑ + β2 + = 0, (A.13b) (1) (1) a2 a2 ∇2φ = 1 κε δ, (A.13c) a2 2 (0) where the energy conservation law (2b) with w 1 has been used. Diﬀerentiating (A.13a) with respect to ˙ 2 time and eliminating ϑ(1) with the help of (A.13b) and, subsequently, eliminating ϑ(1) and ∇ φ with the help of (A.13a) and (A.13c), respectively, yields

∇2 δ¨ + 2Hδ˙ − β2 + 1 κε δ = 0, (A.14) a2 2 (0)

19 which is precisely the left-hand side of the standard equation (62) for w 1 and β2 1. Just as the system of equations (A.7), equation (A.14) is invariant under the Newtonian gauge transformation (A.9). This implies that the general solution of equation (A.14) is gauge-dependent and the gauge mode δˆ given by (A.3) is for w 1 and ψ = C one of the two linear independent solutions. Consequently, the right-hand side of equation (A.13c) is gauge-dependent and φ depends on time. As a consequence, equation (A.13c) is not the real Poisson equation. It has been shown in Section 7∗ (Miedema, 2014) on the non-relativistic limit that the source term of the Poisson equation is invariant under the Newtonian gauge transformation (A.9), i.e., is gauge-invariant, and the potential is independent of time. The system of equations (A.13) will now be compared with the relativistic system (A.1) with w 1 2 ∗ and β 1. Using the expression (45 ) for R(1), the system (A.1) becomes

4 ∇2φ 1 δ˙ + 3 Hδ + ϑ − = 0, (A.15a) 2 (1) c2 a2 4H ∇2δ ϑ˙ + 2Hϑ + β2 = 0, (A.15b) (1) (1) a2 ∇2φ˙ + 2κε ϑ = 0. (A.15c) a2 (0) (1) Just as the system (A.2) follows from (A.1), the system (A.7) follows from (A.15). Comparing the sys- tems (A.13) and (A.15) it must be concluded that the Newtonian system (A.13) is not correct.

In the non-relativistic limit, i.e., ϑ(1) → 0 and vanishing pressure, the system (A.15) reduces to

4 ∇2φ 1 δ˙ + 3 Hδ − = 0, (A.16a) 2 c2 a2 4H ∇2φ˙ = 0. (A.16b)

Equation (A.16b) is equal to equation (53b∗) and implies that the Newtonian potential is independent of time, whereas equation (A.16a) is equal to equation (53c∗). This can be shown as follows. First, eliminate 2 ∗ ∗ ∇ φ using the expression (45 ). Next, use the perturbed Friedmann equation (32a ) with ϑ(1) = 0 to eliminate R(1). Finally, substitute δ := ε(1)/ε(0) and use the background equations (2). The non-relativistic limit is discussed in detail in Section 7∗ (Miedema, 2014).

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