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Structure Formation in the Early Universe 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 accor- dance 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 grav- itation, 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 perturba- tions 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 "(1) and n(1) have been found. Evolution equations for the corresponding contrast functions δ" 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; ") 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 δ" 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; "). 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 = (" + p)uµuν − pgµν ; p = p(n; "); (1) is given by (8∗) 2 4 3H = κε(0); κ = 8πGN=c ; (2a) "_(0) = −3H"(0)(1 + w); w := p(0)="(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 δ" is given by (65∗) δ δ¨ + b δ_ + b δ = b δ − " ; (3a) " 1 " 2 " 3 n 1 + w 1 d δ" 3Hn(0)pn δ" δn − = δn − ; (3b) c dt 1 + w "(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; ") given by β_ b = H(1 − 3w − 3β2) − 2 ; (4a) 1 β h i β_ r2 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) r b3 = "(0)p"n(1 + w) + + pn(p" − β ) + n(0)pnn + pn 2 ; (4c) 1 + w 3H β "(0) a 3 where the partial derivatives of the equation of state for the pressure p(n; ") are given by @p @p @2p @2p : : : : pn = ; p" = ; p"n = ; pnn = 2 : (5) @n " @" n @" @n @n 2 2 The symbol r denotes the Laplace operator. The quantity β(t) is defined by β :=p _(0)="_(0). Using that p_(0) = pnn_ (0) + p""_(0) and the conservation laws (2b) and (2c) one gets 2 n(0)pn β = p" + : (6) "(0)(1 + w) 2 From the definitions w := p(0)="(0) and β :=p _(0)="_(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 = β2" δ + n p δ − " ; (8) (1) (0) " (0) n n 1 + w 2 where the first term, β "(0)δ", 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 "(0)(1 + w) δ" 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("), 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, δ", 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 " = aBTγ ; p = 3 aBTγ ; (10) 4 where aB is the black body constant and Tγ the radiation temperature.
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