1 Growth of Primeval H 2 and HD Inhomogeneities in the Early Universe Denis Puy 1,2, Daniel Pfenniger 1 1 Geneva Observatory (Switzerland), 2 University of Montpellier II (France) Abstract The chemistry of the early Universe (Standard Big Bang Chemistry, SBBC) has been the source of several studies in the conventional frame of a homogeneous Universe. One of their most important consequences is the existence of a significant abundance of molecules, which play a crucial role on the dynamical evolution of the first collapsing structures appearing at temperatures below a few hundred degrees K. In a Universe made of baryonic and non-baryonic dark matter, we need to discuss how each component couples with the other, and how this coupling may modify the SBBC during the gravitational growth of structures. Since dark matter is assumed to be affected only by gravity and is collisionless, there is no effective pressure term in its equation of evolution. The linearized continuity equation in Fourier modes describes the dark matter and baryon fluids by two second-order differential equations which couple the baryon chemistry and gas dynamics to dark-matter by gravity. In this article we present calculations in the linear approximation of density fluctuations, but in the full nonlinear regime of chemical abundances about the primordial molecule formation in a uniform medium perturbed by small density inhomogeneities at various spatial wavelengths. We analyze the differential abundances of the primordial molecules H 2, HD and LiH. As the Universe expands, the baryonic fluctuations increase and induce strong contrasts on the primordial molecular abundances. The main result is that the chemical abundances at the transition between the linear and non-linear regimes of density fluctuations (such as in proto-collapsing structures) are already very inhomogeneous and scale dependent. These results indicate that pronounced inhomogeneous chemical abundances are present already before and during the dark age. This must have a direct consequence on the mass spectrum of the first bound objects since gas cooling depends then mainly on the particular abundances of H 2 and HD. 1. INTRODUCTION The standard Big Bang nucleo-synthesis (SBBN) model for the Universe predicts the nuclei abundances of mainly hydrogen, helium and lithium and their isotopes. Similarly the subsequent chemistry of these light elements and their respective isotopic forms may be called the standard Big Bang chemistry (SBBC). Although the matter dilution acts against molecule formation, the temperature drop is sufficiently fast for allowing the formation of simple molecules, see Lepp & Shull , Puy et al. , Galli & Palla . All these authors clearly pointed out that trace amounts of molecules (the primordial molecules) such as H 2 and HD were formed during the post-recombination period of the Universe. The ongoing physical reactions are numerous after the nuclei recombination, partly due to the cosmic microwave background radiation (CMBR), see Stancil & Dalgarno . Early studies of the chemistry in the first density inhomogeneities focused on highly idealized collapse models of primordial clouds including their chemical evolution and cooling (see Palla et al. , Mac Low & Shull , Lahav  and Puy & Signore ), or in spherical symmetry (see Aninos & Norman , Abel et al.  and Bromm et al. ). However, very little is known about the chemistry in the initial linear density fluctuations. Indeed, chemical reactions are highly non-linear, and it is a priori not obvious that chemical abundance fluctuations remains linear for a time as long as the density fluctuations. In fact we show below that chemistry becomes complex well before the non-linear density fluctuations. The formation of large scale structure is today one of the outstanding problems in cosmology. Structure formation initiates from the growth of small positive density fluctuations. The linear theory of perturbations, applied to the uniform isotropic cosmological situation, is now well understood. In this paper we have investigated in the linear regime of density fluctuations the full chemistry in Fourier modes characterized by their wave-numbers. This approach allows us to quantify the differential growth of molecular abundances and provides the initial conditions of perturbation in the non-linear density fluctuation regime. This last point is crucial to reach a better understanding of the conditions prevailing before the period where the first stars formed. The plan of this paper is as follow: in Sect. 2 we analyze the evolution of matter fluctuations, in Sect. 3 we perform an analysis of the chemical evolution in pure Fourier modes, still possible and convenient in the linear regime. Finally, in Sect. 4, we discuss some possible implications of this study. 2. EVOLUTION OF MATTER FLUCTUATIONS Here we analyze the full non-linear evolution of the molecular abundances in the linear regime of the density fluctuations including baryons and dark matter.
2 species value species value species value [H].45 [H + ].44 [H ] [H 2 ] [H + 2 ] 3 18 [H + 3 ] [He].11 [He + ] [He 2+ ] [HeH + ] [D] [D + ] [HD] [HD + ] [H 2 D + ] [Li] [Li + ] 3.34 [Li ] [LiH] [LiH + ] Table 1: Relative chemical abundances at the redshift of hydrogen recombination z rec,h = 3, from Puy & Pfenniger  Equations of evolution The mathematical description of a lumpy Universe revolves around the dimensionless density perturbation, δ( r), which is obtained from the spatially varying baryons ρ b ( r) and dark matter ρ d ( r) mass density via δ b ( r) = ρ b( r) ρ b ρ b and δ d ( r) = ρ d( r) ρ d ρ d (1) where r is the comoving coordinate, ρ d and ρ b are respectively the average mass density of baryons and of dark matter. Below the index b is always relative to baryons, and d to dark matter. It is useful to expand the density contrast in Fourier series, in which the fluctuations δ b and δ d are treated as a superposition of plane waves δ b ( r) = k δ d ( r) = k δ b ( k)exp(i k r) δ d ( k)exp(i k r), (2) where the comoving wavenumber k is in h Mpc 1 units with h = H/H. The expansion is characterized by the Hubble parameter, H, which depends on the energetic component of the Universe, taken here as flat: H(t) = ȧ a = H Ωtot, where Ω tot = Ω r + Ω m + Ω Λ, (3) each component is coupled to the other by gravity. Since cold dark matter is affected only by gravity and is presumably collisionless, in the fluid limit we can ignore the velocity dispersion of the cold dark matter particles. There is no effective pressure term in the equation of evolution for dark matter. Moreover we can neglect in the linear approximation the influence of a peculiar velocity, which introduce second order transport terms. Thus from the linearized continuity equation in the Fourier modes, the dark matter-baryon fluid is governed by two second-order differential equations (see Peebles ): d 2 δ b ( k) dt 2 + 2H(t) dδ b( k) = dt [ 4πG ρ b δ b ( k) + ρ d δ d ( ] k) c2 s a 2 k2 δ b ( k), (5) d 2 δ d ( k) dt 2 + 2H(t) dδ d( k) = [ dt ] 4πG ρ b δ b ( k) + ρ d δ d ( k), (6) where G is the gravitational constant, c s is the soundspeed at which baryonic disturbances propagate, expressed by the equation of state for a mono-atomic ideal gas. The system of second-order differential equations for the perturbations are coupled with the chemical and baryons evolution equations of temperature and of density, see Hui & Gnedin , Puy & Pfenniger . and where a is the scale factor, and H, the Hubble constant, is the present value of the Hubble parameter. Ω r is the radiation and Ω m the matter density parameter, and Ω Λ the cosmological constant; they depend on the redshift z. The density fluctuation Fourier transform contains both an amplitude δ( k) and a phase φ k, with δ b ( k) = δ b ( k) exp(iφ b,k ) δ d ( k) = δ d ( k) exp(iφ d,k ). (4) The density perturbations are small, thus the evolution of density contrast can be obtained separately for each k through the linear perturbation theory. In a Universe made of non-baryonic dark matter and baryons, we need to discuss both components together, since 2.2. Initial conditions We start our simulation well before the redshift of hydrogen recombination at z s = z rec,h = 3. We pull the initial abundances out of the SBBC, given by Puy & Pfenniger  (see Table 1). We consider a cold dark matter power spectrum P( k), see , such as: P(k) k, (7) where the proportionality constant is chosen to be a numerically convenient value of 6. Thus the initial perturbations, at z s, are defined by, for each of the baryons and dark matter: δ( k) t=ts = 6 k 1/2. (8)
3 -1 k=1 h Mpc -4-1 k= h Mpc -6-1 k=5 h Mpc -6-6 k=99 h Mpc (A) e-6-1e (B) (C) (D) Figure 1: Evolution of the pure Fourier modes of the density fluctuations, as a function of redshift z for different comoving wave-numbers and elementary initial conditions. The black lines are the dark matter density fluctuations, the red lines are the baryons density fluctuations. Each column defines a particular scale (k = 1, k =, k = 5 and k = 99 h Mpc 1 ), each row is relative to a particular initial condition such as (A): δ b,s = 6 / k and δ d,s = δ b,s = δ d,s =, (B): δ b,s = 6 H/ k and δ b,s = δ d,s = δ d,s =, (C): δ d,s = 6 H/ k and δ b,s = δ b,s = δ d,s =, (D): δ d,s = 6 H/ k and δ b,s = δ b,s = δ b,s =. The initial time variation of the density perturbation reads δ( k) t=ts = 6 H(t s )k 1/2, (9) where H(z s ) is the Hubble parameter at z s. t = t s is the age of the Universe at redshift z s, defined by (in a flat Universe 1 ): t s = 1 H dx 1+z s x. () Ω r, x 4 + Ω m, x 3 + Ω Λ We suppose that the initial abundance fluctuations δ ξ vanish for all k: ξ, δ ξ t=ts = = [ξ] = [ξ] SBBC at z = z s, (11) where [ξ] is the relative abundance of the species ξ in the baryonic fluctuations Evolution of density perturbations The solutions for the baryons change from oscillatory to growing modes if k is larger or smaller than the critical 1 Ω r, is the present radiation parameter, Ω m, the present matter density parameter. Jeans wavenumber k J, where c 2 sk 2 J = 4πG ρ b. (12) Fig. (1) shows the evolution of density fluctuations (matter and dark matter) for different comoving wavenumbers k and different initial conditions. From k = 1h Mpc 1 to k = 5h Mpc 1, i.e., from large to medium scales, the evolution of baryons fluctuation are strongly coupled with the dark matter fluctuations. For scales k > h Mpc 1 the oscillations of baryonic fluctuations are important and are decoupled from the dark matter evolution. At the smallest scales, k = 99h Mpc 1, as expected the baryons fluctuate fast, since the pressure term in Eq.(6) is proportional to k 2. This pressure effect is, of course, absent for collisionless dark matter. 3. MOLECULAR FLUCTUATIONS We solve and analyze the full chemical equations for different Fourier modes and take the difference of the solutions for the abundances of H 2 and HD with the ones of the unperturbed SBBC case. Doing this way we preserve the full
4 Figure 2: The top panels show the H 2 density and the lower panels show the HD density in slices cut through the simulation box (comoving size 2π h 1 Mpc, i.e., 8.85Mpc in the real space) at redshifts z = 3, z = 1425, z =, z = from left to right. The scale of the pixel values are indicated at right (from 15 to +15). non-linearities of the chemical network in the linear density perturbations. To compare the chemical fluctuations with eventual future observations it is useful to calculate typical differential abundances in physical space. For the sake of simplicity and display convenience, we consider here only a 2D map. A 3D generalization of the following experiment is straightforward, but would be much more computer intensive without bringing much different results. The fluctuation δ( x) in physical space is the Fourier transform of δ( k), δ( x) = 1 2N 1 2N 1 δ( 2N ( k) exp i ) k x. (13) k 1= k 2= For each wavenumber k with modulus k = k k2 2 N, we take an initial vector δ s = (δ b,s, δ b,s,δ d,s, δ d,s ) of random phase but fixed modulus. The purpose of restricting k N is to eliminate high frequency waves introducing grid anisotropies. By Fourier transforming the integration results between z s = 3 and z f = we obtain the density fluctuations δ b ( x), δ d ( x) as functions of z, as well as the abundance fluctuations δ ξ ( x) for each species ξ. The series of such 2D maps of density or molecular abundance fluctuations can then be combined to produce movies of the abundance fluctuations. In Fig. (2), top, we present different frames of the H 2 fluctuations in order to illustrate the evolution of molecular abundances in real space (box of length l = 2π h Mpc) at the successive redshifts z = 3, z = 1425, z = 6 and z =. At the initial redshift z s the H 2 distribution is relatively homogeneous. However as the Universe expands, baryonic fluctuations increase and induce strong contrasts of H 2. A very lumpy distribution of H 2 and HD abundances results at z f =. Fig. (2), bottom, illustrates some frames in the evolution of the HD abundance in real space (box of length l = 2π h Mpc) at the successive redshifts z = 3, z = 1425, z = and z =. At the initial redshift z s, the HD distribution is less homogeneous than the H 2 distribution. Apparently the HD chemistry is more sensitive to the differential growth of the baryonic fluctuations. The contrast becomes more important and leads, at z f =, to a more lumpy distribution than the H 2 distribution. We notice that some regions reveal HD fluctuations close to the value 15, see Fig.2). 4. DISCUSSIONS The results indicate that a lumpy distribution of molecules is already present during the dark age period of the Universe (roughly < z < 3). The clear conclusion is that well before the first stars form the molecular Universe is already strongly inhomogeneous. Moreover the peak of molecular contrasts appears precisely in the range of redshift z = 5. These redshifts are the typical turnaround redshifts between the linear and non-linear regime of small-scale structures leading to stars, see Padmanabhan . Thus it is crucial to have a precise view of the chemistry at this period, which determines the fragmentation conditions and the properties and mass spectrum of the first collapsing objects. The picture of a molecular lumpy Universe suggests that in the non-linear regime of perturbations, strong molecular abundances contrasts could be very important. Pfenniger & Combes  pointed out the possible fractal distribution of cold molecular gas in the interstellar medium,
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