The thermal balance of the first structures in the primordial gas

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1 Mon. Not. R. Astron. Soc. 323, 672± ) The thermal balance of the first structures in the primordial gas D. R. Flower 1w and G. Pineau des ForeÃts 2w 1 Physics Department, The University, Durham DH1 3LE 2 DAEC, Observatoire de Paris, F Meudon Principal Cedex, France Accepted 2000 November 24. Received 2000 November 24;in original form 2000 August 30 ABSTRACT The contraction of matter in the primordial medium, to form the first gravitationally bound structures, was mediated by radiative cooling of the gas by H 2 and HD. We have computed the initial phases of free-fall collapse, incorporating the results of quantum mechanical calculations of rate coefficients for collisional excitation of H 2 and HD by the principal perturbers, H, He, H 2 and H +. The structure of shock waves produced when the collapse speed exceeds the local sound speed is determined. In the post-shock gas, radiative cooling by H 2 exceeds that by HD, but by a factor of only 4. The intensities of the strongest emission lines of H 2 ± rotational transitions within the vibrational ground state ± are calculated. Even with coarse spectral and angular resolution, these transitions might be observable as inhomogeneities in the cosmic background radiation. Key words: gravitation ± molecular processes ± shock waves ± early Universe. 1 INTRODUCTION The formation of gravitationally bound objects from inhomogeneities in the primordial gas presupposes the existence of cooling agents to remove the heat generated during the gravitational collapse. Chemical models of the primordial medium Latter & Black 1991;Puy et al. 1993;Galli & Palla 1998;Lepp, Stancil & Dalgarno 1998;Stancil, Lepp & Dalgarno 1998) have shown that the only coolants present at significant levels of abundance were H 2 and HD. The abundance of HD, relative to H 2, was enhanced by about 2 orders of magnitude, compared with the elemental D:H ratio, by chemical fractionation. Furthermore, the closer rotational energy level spacing and finite dipole moment of HD ensured that the contribution of HD to the thermal balance of the primordial gas was comparable to that of H 2 Flower 2000). It is important to establish whether HD was also significant as a moderator of the initial phases of gravitational collapse;this is one of the questions that will be addressed below. Gravitational collapse and the fragmentation of clouds in the primordial medium have been studied by Rees & Ostriker 1977), Carlberg 1981), Palla, Salpeter & Stahler 1983), Lepp & Shull 1984) and Stahler, Palla & Salpeter 1986), and more recently by Tegmark et al. 1997), Teyssier, ChieÁze & Alimi 1998), Omukai & Nishi 1998), Nakamura & Umemura 1999), Abel & Haiman 2000) and Palla & Galli 2000). In the last two references, the role of H 2 cooling was considered specifically. Struck-Marcell 1982) modelled the formation of shock waves hydrodynamically, through cloud collisions in protogalaxies, and Teyssier et al. 1998) have extended the hydrodynamical treatments in order to w david.flower@dur.ac.uk DRF);forets@obspm.fr GPdF) study media in which the temperatures of the electrons, ions and neutrals may differ. We shall consider the initial phases of gravitational collapse, incorporating the results of recent determinations of the rates of cooling of the gas, not only by H 2 Le Bourlot, Pineau des ForeÃts & Flower 1999) but also by HD Flower et al. 2000). We calculate the spectrum of H 2 that is produced by collisional excitation in shock waves, which are generated when the speed of collapse becomes locally supersonic. It may be possible to detect the emission in pure rotational transitions of H 2, in the early, optically thin phase of gravitational collapse, as anisotropies in the cosmic background radiation. In Section 2, we describe the model that we have used to study the initial phases of collapse. Our results are presented and discussed in Section 3 and concluding remarks are given in Section 4. 2 THE MO DEL Flower & Pineau des ForeÃts 2000) studied the physical and chemical evolution of the homogeneous and uniformly expanding primordial gas. The contribution of H 2 to the thermal balance of the medium was calculated, incorporating fits to recently computed rate coefficients for rovibrational transitions, induced by collisions of H 2 with H and He Le Bourlot et al. 1999). Rotationally inelastic transitions induced by H + Gerlich 1990) were shown to be important, even dominant in collisional population transfer within the vibrational ground state of H 2. Furthermore, such transitions determine the evolution of the ortho:para H 2 ratio, from its initial, statistical value of 3 to a value of approximately 0.25 for z < 20; subsequent to which it is `frozen'. As the populations of the energy levels of ortho and para q 2001 RAS

2 Thermal balance in primordial gas 673 H 2 remain separately) coupled to the blackbody radiation field beyond the point z < 300 at which the temperatures of radiation, T r, and matter, T m, decouple, and T r. T m ; the net effect of collisions with H 2 is to heat, rather than cool, the gas cf. Puy et al. 1993). A similar statement applies to collisions with HD molecules, and the contribution of HD to the heating of the gas is comparable to that of H 2 Flower 2000). The model developed by Flower & Pineau des ForeÃts 2000) and Flower 2000) has been adapted, as will now be described, in order to study the initial phases of gravitational collapse. The equation which determines the evolution of the temperature of the neutral gas, T n, may be written Flower & Pineau des ForeÃts 2000) 1 dt n ˆ T n dt B n 2 d n nu n dt N n n n n nkt n 1 U n 2 kt n R dr dt ; where B n is the rate of gain of energy by the neutrals, per unit volume, n n is the number density of the neutrals, and N n is the rate of creation of neutral particles, per unit volume; n n U n is the internal energy of the neutral gas per unit volume, given by n n U n ˆ X n vj E vj ; vj where n vj are the population densities of the rovibrational levels vj of H 2, of energy E vj relative to the ground state, v ˆ 0 ˆ J: The only significant contribution to the internal energy of the gas arises from molecular hydrogen. Equations analogous to 1) hold for the temperatures of the ions, T i, and of the electrons, T e.in practice, the differences between T n, T i and T e are found to be small Flower & Pineau des ForeÃts 2000), increasingly so with increasing gas density. Henceforth, we use a single temperature for matter, T m ; T n ˆ T i ˆ T e : We consider the free-fall collapse of a spherical condensation of initial radius a and current radius R, given by 1 dr R dt ; 1 dx x dt ˆ 2 p 1 1=2 2t f x x 2 1 ; 2 where x ˆ R=a # 1 and 3p 1=2 t f ˆ 32Gr 0 is the free-fall time; G is the gravitational constant and r 0) is the initial mass density of the gas Spitzer 1978). The variation of the mass density r is determined by dlnr=dt ˆ 23dlnR=dt for a spherical collapse, with d ln R=dt given by equation 2). Noting that R 3 r ˆ a 3 r 0 ; from mass conservation, we have x ˆ R=a ˆ r 0 =rš 1=3 ; and it follows that, when x! 1; dlnr ˆ 3p dt 2 1 r 1=2 : r 0 t f Exchange of energy between matter and the blackbody radiation field occurs through Thomson scattering of the photons on electrons and through emission and absorption of radiation in rotational transitions of H 2 and HD. Flower & Pineau des ForeÃts 2000) solved the time-dependent rate equations for the populations n vj of the rovibrational levels of H 2, allowing for spontaneous emission and stimulated emission and absorption of 1 3 radiation and for population transfer in collisions with H, He, H +, e 2, and other H 2 molecules. We follow the same procedure here. In the case of HD, which has a small but finite dipole moment, both the radiative and collisional rates are larger than for H 2.In practice, neutral particles H, He, H 2 ) dominate collisional population transfer in HD. We have assumed a steady state when calculating the populations of the rotational levels of the ground vibrational state of HD;this assumption is increasingly valid with increasing gas density. The free-fall time, t f, is greater than the initial value of 1=H z ˆ1= H 0 1 z 3=2 Š; where H 0 ˆ 67 km s 21 Mpc 21 is the Hubble constant, and so, when applying equation 2), we have allowed for the variation of the temperature of the blackbody radiation according to 1 dt r ˆ 2 s T T r dt m e c k T r 2 T e n e 2 H z ; where the first term on the right-hand side is energy loss by the radiation field through Thomson scattering, and the second term represents the expansion of the Universe. The chemical rate equations take the form 1 dn X ˆ N X n X dt n X 2 3 dr R dt ; where N X) is the rate of creation of species X, per unit volume, and n X) is the density of this species. As in our previous study, we used the `reduced' chemical reaction network of Galli & Palla 1998), subject to minor changes. We also included the formation of H 2 in the three-body association reactions H H H! H 2 H and H H H 2! H 2 H 2 ; 4 5 6a 6b which become important for densities n H cm 23 : The rate constants adopted for these reactions 1: T m = cm 6 s 21 and 2: T m = cm 6 s 21 ; respectively] derive from Jacobs, Giedt & Cohen 1967) and were used by Palla et al. 1983) and by other authors subsequently. The analogous reactions leading to the formation of HD, D H H! HD H 7a and D H H 2! HD H 2 7b were also included, adopting the same rate constants as for the formation of H 2. 3 RESULTS We consider first the evolution of a cloud experiencing free-fall collapse, up to the point at which the gas becomes molecular owing to three-body association reactions), and then the collisional excitation of H 2 in shock waves generated by the collapse. 3.1 Free-fall collapse In Fig. 1, we plot, as functions of the gas density, the temperature of matter, T m, and the fractional abundances [relative to n H < n H 2n H 2 Š of atomic and molecular hydrogen and of D and

3 674 D. R. Flower and G. Pineau des ForeÃts Figure 1. The variations of the temperatures of radiation, T r, and matter, T m, and of the fractional abundances of H, H 2, D, and HD with the gas density, assuming a free-fall collapse initiated at z < 40 and tending asymptotically to z < 10 and V b ˆ 0:1: Figure 2. a) The principal heating rates for the model in Fig. 1. The bold continuous line is the rate of heating of the gas owing to the contraction; the broken line is the rate of heating owing to chemical reactions, notably three-body association to form H 2. The kinetic temperature of the gas is also plotted. b) The corresponding rates of cooling of the gas, by H 2 bold continuous line) and by HD broken line). HD, for a baryonic density V b ˆ 0:1 of the critical density. The radiation temperature, T r, is also plotted. In this model, the collapse is initiated at z < 40; and z < 10 is attained asymptotically. It transpires that the profiles are almost independent of the initial value of z, for reasons that will be given below. As is characteristic of free-fall, most of the evolution of the gas takes place at values of z approaching the asymptote, z < 10: The kinetic temperature first rises and then falls towards a minimum of approximately 470 K at a gas density of the order of 10 4 cm 23. This minimum in T m is much greater than the corresponding value of T r ;the latter has reached its asymptotic value of 28.5 K by this stage in the contraction, when, as may be seen from Fig. 2, the heating of the gas is dominated by the contraction, and the cooling is shared about equally between H 2 and HD. For gas densities in excess of cm 23, heating arising from the formation of H 2 approaches the rate of heating owing to the contraction. The levelling out of the initial increase in the fractional abundance of H 2 and of HD) occurs because the recombination of H + and e 2 closes the chemical routes to H 2 through the binary reactions H + H, hn H 2 ; H 2 H; H+ )H 2 and H e 2, hn)h 2, H 2 H, e 2 )H 2. Eventually, for densities in excess of about 10 9 cm 23, the three-body association reactions, 6) and 7) above, transform the gas into a molecular form;h 2 and HD) formation is complete when the gas density <10 12 cm 23. The effects of optical depth in the lines of H 2 then lead to a reduction in the rate of cooling by H 2 and a rise in the kinetic temperature cf. Palla et al. 1983). The three-body association of H into H 2 and D into HD is accompanied by a decrease in the ratio n HD =n H 2 to 2n D =n H ˆ ; corresponding to the HD:H 2 ratio being no longer enhanced by chemical fractionation. For gas densities between about 10 cm 23 and cm 23, the ratio HD:H 2 remains at approximately ; having attained a maximum value of at a density of 1 cm 23. The energy liberated in the three-body association to form H ev) may appear as kinetic energy of the products of this reaction, as assumed in these calculations, or as radiation emitted following the formation of H 2 in excited states. However, for densities of the order of cm 23, where the formation of H 2 is a significant source of heating, collisional de-excitation tends to dominate population transfer, even between the vibrational levels of H 2, reducing the fraction of the binding energy that is lost to the radiation field. Thus, it is a good approximation to assume that the entire binding energy of H 2 is recovered as kinetic energy of the gas. The starting conditions for the free-fall collapse are determined by the initial value of z. The Universe expands adiabatically as z decreases, and the radiation and matter temperatures and the matter density decrease accordingly. The first phase of the freefall collapse, on the other hand, is the reverse process of adiabatic contraction. Thus, commencing the free-fall collapse at a smaller value of z results only in a somewhat longer duration of the adiabatic contraction phase, and the same evolutionary track is followed subsequently. In particular, the initial value of z is not material to the formation and the structure of shock waves, which we now consider. 3.2 J-type shock waves The acceleration in the rate of collapse that occurs in the free-fall model results in the collapse speed becoming strongly supersonic shortly after the initial maximum of the kinetic temperature has

4 Thermal balance in primordial gas 675 Figure 3. Speed of collapse, dr=dt; the adiabatic sound speed, c s, and the kinetic temperature, T m, plotted as functions of the gas density; V b ˆ 0:1: The locations of the discontinuities of the J-type shocks illustrated in Figs 4 a) and b) are indicated by the arrows. been attained and beyond the first, adiabatic contraction phase. This phenomenon is illustrated in Fig. 3, where the speed of collapse, dr=dt; the adiabatic sound speed and the kinetic temperature are plotted as functions of the gas density. As supersonic flow leads to the generation of shock waves, we have investigated the consequences of shock wave heating and compression of the gas. In the absence of a magnetic field, or when the field is weak, compression and heating of the gas is quasi-discontinuous and governed by the Rankine±Hugoniot relations: a `jump' or J-type shock obtains. The gas subsequently cools and is further compressed until, in steady state, the post-shock equilibrium is attained. As little is known of the magnetic field strength in the primordial medium, we shall consider J-type shock waves, neglecting the magnetic field. We have followed the evolution of the gas behind the shock discontinuity, taken to be located just beyond the initial temperature maximum in Fig. 3, where the collapse speed diverges from the sound speed. In Fig. 4 a), the pre-shock temperature was T m ˆ 631 K; the adiabatic sound speed was 2.67 km s 21, and the collapse speed was 6.35 km s 21 ;in Fig. 4 b), the pre-shock temperature was T m ˆ 1140 K; the adiabatic sound speed was 3.59 km s 21 and the collapse speed was 4.26 km s 21. Thus, the Mach number of the shock wave is twice as large in a) as in b), which permits the dependence of the results on the strength of the shock wave to be investigated. Fig. 4 shows that, whilst the cooling of the hot post-shock gas is dominated by H 2, cooling by HD takes over as the kinetic temperature begins to fall. In both a) and b), the integrals of the cooling rates with respect to distance l behind the discontinuity, l ˆ v dt; where v is the flow speed, show that the contribution of HD to the cooling of the gas is about a quarter of that of H 2. Cooling occurs principally through collisional excitation of rotational transitions within the vibrational ground state. In Table 1 the most intense lines are listed, together with their rest wavelengths. The total intensity of radiation in these transitions is a) I H 2 ˆ1: erg cm 22 s 21 sr 21 ; I HD ˆ 4: erg cm 22 s 21 sr 21 ; and b) I H 2 ˆ8: erg cm 22 s 21 sr 21, I HD ˆ1: erg cm 22 s 21 sr 21 : Evidently, Figure 4. The kinetic temperature of the gas and the rates of cooling by H 2 and HD molecules, as functions of the flow time of the gas behind the J- type shock discontinuity. In a) the pre-shock temperature was T m ˆ 631 K; the adiabatic sound speed was 2.67 km s 21, and the collapse speed was 6.35 km s 21 ;in b) the pre-shock temperature was T m ˆ 1140 K; the adiabatic sound speed was 3.59 km s 21, and the collapse speed was 4.26 km s 21. Note that, when equilibrium is reached in the post-shock gas, the kinetic temperature T m ˆ T r ; the local temperature of the blackbody radiation field. the stronger shock a) gives rise to higher total intensities than b), by a factor of approximately 20. Although the cooling rates erg cm 23 s 21 ) are much larger in a) than in b), the width of the cooling zone is larger in b) than in a), as may be deduced from Fig. 4. Consequently, the computed intensity of the 0±0 S 1) transition, which is one of the strongest lines, is only 5.5 times greater in a) than in b). It is instructive to compare the intensity of the H 2 0±0 S 1) line, at its rest wavelength of mm, with the intensity of the blackbody radiation field, at its local temperature of 28.6 K. The intensity of blackbody radiation in a band of width 10 cm 21, centred on this line, is 3: erg cm 22 s 21 sr 21 : When redshifted from 1 z ˆ 10 to 1 z ˆ 1; this bandwidth contracts to 1 cm 21, the resolution of the COBE instrument Mather et al. 1990). Thus, the intensity of the background radiation is only about 1 per cent of even the lower determination b) of the 0±0 S 1) line intensity. The COBE differential microwave radiometer detected structure in the background radiation at a level of

5 676 D. R. Flower and G. Pineau des ForeÃts Table 1. Emitted intensities of pure rotational transitions within the vibrational ground state of H 2, corresponding to the J-type shock models in Figs 4 a) and b), respectively. Numbers in parentheses are powers of 10. Transition Rest wavelength Intensity erg cm 22 s 21 sr 21 ) mm) a) b) S 0) ) ) S 1) ) ) S 2) ) ) S 3) ) ) S 4) ) ) S 5) ) ) DT=T < at wavelengths extending to 3: mm Smoot et al. 1992). If this sensitivity were attainable at the redshifted wavelength 170 mm) of the 0±0 S 1) transition, our calculations imply that it would be detectable even with a beam dilution factor of CONCLUDING REMARKS We have studied the early phases of gravitational collapse of a cloud of gas in the primordial medium. The cooling resulting from H 2 and HD molecules was computed using determinations of their cooling rates Le Bourlot et al. 1999;Flower et al. 2000) which derive from recent quantum mechanical calculations. We find that, independent of the value of z at which the collapse begins, the infall velocity becomes strongly supersonic at a gas density <100 cm 23. Shock waves produced at this optically thin) stage of the collapse result in compression and heating of the gas, which we have assumed to occur quasi-discontinuously J-type shock wave). The cooling of the gas behind this discontinuity is regulated by H 2, but HD contributes about 20 per cent of the integral cooling rate. Pure rotational transitions of H 2, which are responsible for most of the cooling, are much more intense than the local blackbody radiation field and might be observable as inhomogeneities in the cosmic background radiation. Even if the magnetic field was sufficiently strong to give rise to continuous C-type) shock waves, rather than the J-type considered above, our conclusions would be substantially unchanged. The mechanical energy of a shock wave, of either type, is dissipated principally as radiation in the pure rotational transitions of H 2, and these lines are optically thin in the early phases of free-fall collapse. ACKNOWLEDGMENTS One of the authors DRF) gratefully acknowledges the award of a research fellowship by the Leverhulme Trust. REFERENCES Abel T., Haiman Z., 2000, in Combes F., Pineau des ForeÃts G., eds, H 2 in Space. Cambridge Univ. Press, Cambridge, p. 237 Carlberg R. G., 1981, MNRAS, 197, 1021 Flower D. R., 2000, MNRAS, 318, 875 Flower D. R., Pineau des ForeÃts G., 2000, MNRAS, 316, 901 Flower D. R., Le Bourlot J., Pineau des ForeÃts G., Roueff E., 2000, MNRAS, 314, 753 Galli D., Palla F., 1998, A&A, 335, 403 Gerlich D., 1990, J. Chem. Phys., 92, 2377 Jacobs T. A., Giedt R. R., Cohen N., 1967, J. Chem. Phys., 47, 54 Latter W. B., Black J. H., 1991, ApJ, 372, 161 Le Bourlot J., Pineau des ForeÃts G., Flower D. R., 1999, MNRAS, 305, 802 Lepp S., Shull J. M., 1984, ApJ, 280, 465 Lepp S., Stancil P. C., Dalgarno A., 1998, Mem. Soc. Astron. It., 69, 331 Mather J. C. et al., 1990, ApJ, 354, L37 Nakamura F., Umemura M., 1999, ApJ, 515, 239 Omukai K., Nishi R., 1998, ApJ, 508, 141 Palla F., Galli D., 2000, in Combes F., Pineau des ForeÃts G., eds, H 2 in Space. Cambridge Univ. Press, Cambridge, p. 247 Palla F., Salpeter E. E., Stahler S. W., 1983, ApJ, 271, 632 Puy D., Alecian G., Le Bourlot J., Lorat J., Pineau des ForeÃts G., 1993, A&A, 267, 337 Rees M. J., Ostriker J. P., 1977, MNRAS, 179, 541 Smoot G. F. et al., 1992, ApJ, 396, L1 Spitzer L., 1978, Physical Processes in the Interstellar Medium. Wiley, New York Stahler S. W., Palla F., Salpeter E. E., 1986, ApJ, 302, 590 Stancil P. C., Lepp S., Dalgarno A., 1998, ApJ, 509, 1 Struck-Marcell C., 1982, ApJ, 259, 116 Tegmark M., Silk J., Rees M. J., Blanchard A., Abel T., Palla F., 1997, ApJ, 474, 1 Teyssier R., ChieÁze J.-P., Alimi J.-M., 1998, ApJ, 509, 62 This paper has been typeset from a Microsoft Word file prepared by the author.