The ortho:para H 2 ratio in the primordial gas

Mon. Not. R. Astron. Soc. 16, 901±905 (2000) The ortho:para H 2 ratio in the primordial gas D. R. Flower 1w and G. Pineau des ForeÃts 2w 1 Physics Department, The University, Durham DH1 LE 2 DAEC, Observatoire de Paris, F-92195 Meudon Principal Cedex, France Accepted 2000 April. Received 2000 March 20; in original form 1999 October 4 ABSTRACT We have solved the time-dependent equations for the populations of the rovibrational levels of H 2, from redshift z 5000 to the present time. Population transfer, including ortho±para interconversion, is determined principally by collisions with H 1. Simultaneously, we solved the chemical rate equations and for the temperatures of the radiation field and matter, distinguishing between electrons, ions and neutrals. The ortho:para H 2 ratio is found to decrease from the statistical value of at z < 100 until it freezes at a value of 0.25 for z, 20. The significance of proton collisions for the rates of heating and cooling of the gas by H 2 is investigated. Collisions with protons are found to be increasingly important with decreasing temperature. At z 20, the net rate of heating of the gas, including proton collisions, exceeds the net heating rate neglecting proton collisions by approximately three orders of magnitude. Key words: molecular processes ± early Universe. 1 INTRODUCTION Several independent studies of the physical and chemical conditions in the early Universe (Latter & Black 1991; Puy et al. 199; Galli & Palla 1998; Lepp, Stancil & Dalgarno 1998) have established that trace amounts of molecular hydrogen were present in the primordial gas. This small fraction of molecular hydrogen may have played a role in the formation of the first condensations of matter. Accordingly, the contribution of H 2 to the cooling of the primordial gas has been considered recently by Galli & Palla (1998), TineÂ, Lepp & Dalgarno (1998), Le Bourlot, Pineau des ForeÃts & Flower (1999) and Flower et al. (2000). As is well known, H 2 exists in ortho and para forms, and radiative transitions between these forms are very slow (Raich & Good 1964). Indeed, the lifetimes against ortho±para interconversion by this mechanism are comparable to or greater than the age of the Universe. It follows that ortho± para transitions may occur only in reactive collisions with H 1 or H. Reactions with H have a substantial activation energy and occur at a significant rate only at high temperatures (T. 1000 K) and therefore at high redshifts (z. 400). It will be shown that, once freeze-out of H 1 has occurred, collisions with protons dominate ortho±para transfer and, more generally, population transfer within the H 2 molecule. Then, the rate of cooling by H 2 is also determined by H 1 collisions and is dependent on the ortho:para ratio. The lowest rotational transition in para-h 2, J 0±2, has an energy of 510 K, whereas the lowest in ortho-h 2, J 1±, requires 845 K. In Section 2, we outline the model that we have used for our calculations. We solved separately for the thermal evolution of the neutrals, the positive ions and the electrons. In Section, the H 2 cooling function is calculated as a function of z. In fact, where the kinetic temperature of the neutral gas T n, T r, the background radiation temperature (a condition which applies over the relevant range of z), the net effect of collisions with H 2 is to heat, rather than cool, the gas. 2 THE MODEL Our approach generalizes that of Puy et al. (199) to four `fluids': neutrals, positive ions, electrons and photons. The equation that determines the variation of the neutral temperature, T n, with time, t, is 1 dt n T n dt B n 2 d(n nu n ) dt N n 2 2 2 n n n nkt n 1 1 U n 2 kt n R dr dt ; where the rate of expansion of the Universe is given by 1 w E-mail: david.flower@durham.ac.uk (DF); forets@obspm.fr (GPdF) q 2000 RAS 1 dr R dt H(t) H 0(1 1 z) =2 2

902 D. R. Flower and G. Pineau des ForeÃts and we take H 0 67 km s 21 Mpc 21. In equation (1), N n (cm 2 s 21 ) is the rate of creation of neutral particles per unit volume, at kinetic temperature T n, in chemical reactions, considered below; n n (cm 2 ) is the number density of the neutrals; n n U n is the internal energy of the neutral gas per unit volume, n n U n X n vj E vj ; vj where the summation extends over the rovibrational levels vj of H 2, of energy E vj relative to the v 0 J ground state, and n vj is the population density of the level; B n is the rate of energy gain by the neutrals per unit volume. This term (B n ) comprises chemical heating, considered by Puy et al. (199), owing to exothermic reactions which yield neutral products; it also includes the exchange of energy, owing to elastic scattering, between the neutrals and both the ions and the electrons (see Flower, Pineau des ForeÃts & Hartquist 1985). The rate of variation of the internal energy of the neutral fluid per unit volume is evaluated from d(n n U n ) dt X dn vj E vj ; dt vj where the equations for the temporal variation of the H 2 population densities, dn vj /dt, were solved in parallel, allowing for radiative (spontaneous and induced) and collisional population transfer (Le Bourlot et al. 1999), as well as the expansion of the Universe (cf. equation 6 below). In particular, transfers of population, resulting from collisions with protons, for rotational levels J # 9 in the v 0 vibrational ground state were included; the rate coefficients were taken from Gerlich (1990). Spontaneous radiative transitions between ortho and para levels in the v 0 state were also included. Although the corresponding transition probabilities are very small, they increase rapidly with J (A / J 5 for large J: Raich & Good 1964). We made the simplifying assumption that, when H 2 molecules form or are destroyed in chemical reactions, the population of any given state (v, J) changes in proportion to the fractional population of that level, n vj /n(h 2 ). Subsequent to freeze-out of the fractional abundance of H 2 (z, 100), the results are independent of this assumption. The analogous equations for the kinetic temperatures, T i and T e, of the positive ions and the electrons are 1 dt i T i dt and Bi 2 2 n ikt i N i 2 2 dr n i R dt arising from Thomson scattering of the photons of the blackbody radiation field on the electrons. Finally, the variation of the temperature, T r, of the blackbody radiation field is given by 1 dt r T r dt 2 s T m e c k T r 2 T e n e 2 1 dr R dt ; where the first term on the right-hand side (energy loss resulting from Thomson scattering) can, in practice, be neglected. To the above equations must be added the chemical rate equations, 1 dn(x) N(X) n(x) dt n(x) 2 dr R dt ; where N(X) is the rate of creation of species X, of density n(x), per unit volume. We used the `reduced' chemical reaction network of Galli & Palla (1998), subject to minor changes. The complete set of coupled differential equations was then solved by means of the Gear algorithm (Hindmarsh 1974). RESULTS AND DISCUSSION We have integrated the above equations, starting at z 5000, for which the corresponding time is t 8.68 10 11 s (2.75 10 4 yr). Assuming the Hubble constant H 0 67 km s 21 Mpc 21, the present age of the Universe is 9.7 10 9 yr. The present blackbody radiation temperature was taken to be T r 2.7 K and the proton density n H n(h 1 ) 1 n(h) 1 2n(H 2 ) 1.5 10 27 cm 2, corresponding to a baryonic density V b 0.067 of the critical density (cf. Galli & Palla 1998). We have also considered a larger value of V b 0.1. The fractional number density of helium nuclei was n He /n H 0.08, and the fractional number density of deuterium n D /n H 4 10 25 (Vidal-Madjar 198). The initial value of the ortho:para-h 2 ratio was. In Fig. 1 are plotted the variations with z of the temperatures of 5 6 1 dt e T e dt Be 2 2 n ekt e N e 2 2 dr n e R dt ; 4 where n denotes a number density, N a rate of creation per unit volume, and B is the rate of energy gain per unit volume, which includes the energy exchange through elastic collisions with particles in other fluids; B e specifically includes a term 4s T at 4 r m e c k(t r 2 T e )n e ; Figure 1. The variations of the blackbody radiation temperature (solid curve) and of the temperatures of the neutrals, ions, and electrons (superposed: broken curve) with redshift, z.

The ortho:para H 2 ratio 90 Figure 2. (a) Variation with redshift of the fractional abundances of H 1,H, and H 2 ; the temperature of the neutrals is also plotted. (b) As (a), but for D 1, D, and HD. the radiation field and the gas. Although small differences exist at later times between the temperature profiles of the neutrals, ions and electrons, in the sense T e < T i. T n, these differences are not discernible on the scale of this figure. The basic features of the temperature profiles plotted in Fig. 1 may also be see in fig. 4 of Puy et al. (199). The temperatures of radiation and matter decouple at z < 00, and T r continues to decrease as T r T r (0) (1 1 z) owing to the adiabatic expansion, whereas the temperature of the gas tends towards T m T m (0)(1 1 z) 2. The coupling between the radiation and matter temperatures is determined principally by Thomson scattering. It transpires that the coupling between the electrons and the ions, through Coulomb scattering, and between the ions and the neutrals, are sufficiently strong at the epoch in which the temperatures of matter and radiation decouple to ensure that no significant differences develop between T e, T i and T n. The fractional abundances, n(x)/n H,ofX H 1,H,H 2,D 1, D, and HD are plotted in Fig. 2, where the temperature profile of the neutral fluid is also shown, for reference. The results in Fig. 2 are very similar to those of Galli & Palla (1998; their fig. 4), as is to be expected, given that we adopted essentially the same chemical network. Initially, H 2 is produced principally through H 1 (H, hn) H 2 1, H 1 2 (H, H1 )H 2 and subsequently by H(e 2, hn)h 2, H 2 (H, e 2 ) H 2 (cf. Black Figure. The ortho:para density ratio (broken curve), plotted against redshift, z. The temperature of the neutrals, T n, is also shown (full curve). (b) The ortho:para density ratio, together with the fractional abundances of H 1, H, and H 2, plotted against T n. 1990). Chemical fractionation of HD occurs through the reaction D 1 (H 2, HD)H 1, which, in the reverse direction, is endoergic by 464 K, when the product (H 2 ) and reactant (HD) molecules are in their ground states. The variation of the ortho:para ratio is plotted in Fig. (a) as a function of redshift, z, and in Fig. (b) as a function of the temperature of the neutrals, T n ; the fractional abundances of H 1, H, and H 2 are also plotted in Fig. (b). Initially (i.e. at high z), the fractional abundance of H 2 is extremely small and the ortho:para ratio is determined by collisions with protons. As the fractional abundance of H 1 falls, that of H rises and the ortho:para ratio tends towards the statistical value of ; this trend is reinforced by the principal process of formation of H 2,H 2 (H, e 2 )H 2 (Launay, Le Dourneuf & Zeippen 1991). As the temperature of the neutrals falls to values T n, 1000 K and the fractional abundance of H 2 begins to level out, proton collisions take over once again and cause the ortho:para ratio to fall until, for z, 20, freeze-out of the ratio occurs at a value of approximately 0.25. The occurrence of freeze-out implies that the time-scale for population transfer through proton collisions becomes greater than the evolutionary time-scale. For example, at z < 6, the time-scale for ortho to para J (1! 0) conversion in reactive collisions with protons becomes larger than the assumed age of the Universe (<10 10 yr). Under these circumstances, it is essential to solve the time-dependent

904 D. R. Flower and G. Pineau des ForeÃts Figure 4. The net rate of heating of the gas, G 2 L (erg cm 2 s 21 ), including (full curve) and excluding (broken curve) proton collisions. The gas temperature is also plotted (bold curve). Figure 5. Contributions to the thermal balance of the gas: rate of cooling resulting from the adiabatic expansion of the Universe (full curve); the net rate of heating, G ± L, owing to collisions with H 2 molecules (broken curve). The gas temperature is also plotted (bold curve). equations for the level populations, as here, rather than assuming steady-state (dn vj /dt 0), as in previous studies. The level populations within ortho or para-h 2 are determined by the blackbody radiation field, but, as radiative ortho±para transitions are very slow, the ortho:para ratio at `freeze-out' is determined instead by n(h 1 ). However, n(h 1 )/n H [< n e /n(h)] at freeze-out is inversely proportional to the gas density, n H (cf. table 6 of Galli & Palla 1998), and so the density of ionized hydrogen, n(h 1 ), is insensitive to the value of V b. Thus, calculations with V b 0.067 and V b 0.1 yield practically the same value of the ortho:para ratio at freeze-out. Collisions with H 2 molecules give rise to the exchange of energy with the background radiation field. The net rate of energy transfer per unit volume is given by (G 2 L) X p n ( p)x i. j h n i q ( p) i! j (T) 2 n ijq ( p) j! i (T) i (E i 2 E j ); 7 where n( p) is the number density of the perturber, n i is the number density of H 2 molecules in level i, and E i is the energy of this level; q (p) i!j (T) is a collisional rate coefficient. When the temperature of matter T, T r, the blackbody radiation temperature, the net effect is heating of the gas owing to the preponderance of radiative excitation, followed by collisional de-excitation (Puy et al. 199; Galli & Palla 1998). The importance of collisions with protons may be seen by comparing the net rate of heating of the gas, including and excluding proton collisions. At low temperatures, the effect of proton collisions is two-fold. First, the abundance of para-h 2 is increased, relative to ortho-h 2, through reactive collisions; this enhances the rate of heating, as the excitation energy (510 K) of the J 0! 2 transition in para-h 2 is less than that (845 K) of the lowest rotational transition J 1! in ortho-h 2. Secondly, the rate of de-excitation J 2! 0 is enhanced by collisions with protons. It follows that the rate of heating of the gas is substantially greater in the presence of protons, as may be seen in Fig. 4. The contribution of collisions with H 2 molecules, including proton collisions, to the thermal balance of the gas is small compared with the rate of cooling resulting from the adiabatic expansion of the Universe, as is shown in Fig. 5. The rate of heating that we compute here is less than was predicted by Puy et al. (199), owing to a lower gas density at a given value of z (in the present model), a lower fractional abundance of H 2 subsequent to freeze-out, and a much smaller rate coefficient for the collisional de-excitation J 2! 0 (and J! 1) of H 2 by H, for which Puy et al. used the formulae of Elitzur & Watson (1978). Thus, H 2 molecules played an insignificant role in the thermal evolution of the uniformly expanding Universe. Collisions with molecular hydrogen must have made a major contribution to the rate of cooling of the gas during the formation of the first condensations. In this context, the relevant comparison is between the H 2 cooling rate and the rate of heating resulting from gravitational contraction. Our calculations show that, at z < 50, for example, when n H 0.02 cm 2, the gas temperature T < 50 K, and the rate of cooling by H 2 molecules L(H 2 ) 8.4 10 28 erg cm 2 s 21 ; that of HD is L(HD) 1.4 10 28 erg cm 2 s 21, i.e. only 6 times smaller. Thus, the cooling of the gas by HD should also be taken into account when computing the thermal balance. ACKNOWLEDGMENTS We are grateful to the Royal Society and the CNRS for financial support under the European Science Exchange Programme. One of the authors (DRF) gratefully acknowledges the award of a research fellowship by the University of Durham. REFERENCES Black J. H., 1990, in Hartquist T. W., ed., Molecular Astrophysics. Cambridge Univ. Press, Cambridge, p. 47 Elitzur M., Watson W. D., 1978, A&A, 70, 44 Flower D. R., Le Bourlot J., Pineau des ForeÃts G., Roueff E., 2000, MNRAS, 14, 75 Flower D. R., Pineau des ForeÃts G., Hartquist T. W., 1985, MNRAS, 216, 775 Galli D., Palla F., 1998, A&A, 5, 40 Gerlich D., 1990, J. Chem. Phys., 92, 277 Hindmarsh A. C., 1974, Lawrence Livermore Lab. Report, UCID- 0001

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