Deuterium chemistry in the primordial gas

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Planetary and Space Science 50 (2002) 1197–1204

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Deuterium chemistry in the primordial gas

Daniele Galli∗, Francesco PallaOsservatorio Astro�sico di Arcetri, Largo Enrico Fermi 5, I-50125 Firenze, Italy

Received 12 November 2001; received in revised form 4 February 2002; accepted 4 February 2002

Abstract

We review and update some aspects of deuterium chemistry in the post-recombination Universe with particular emphasis on the formationand destruction of HD. We examine in detail the available theoretical and experimental data for the leading reactions of deuteriumchemistry and we highlight the areas where improvements in the determination of rate coe0cients are necessary to reduce the remaininguncertainties. We discuss the cooling properties of HD and the modi1cations to the standard cooling function introduced by the presenceof the cosmological radiation 1eld. Finally, we consider the e2ects of deuterium chemistry on the dynamical collapse of primordial cloudsin a simple “top-hat” scenario, and we speculate on the minimum mass a cloud must have in order to be able to cool in a Hubble time.? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Atomic and molecular processes—early Universe

1. Introduction

The formation of H2 and HD molecules in the post-recombination Universe plays a central role in the evolu-tion of gas condensations. Even trace abundances of thesemolecules strongly a2ect the cooling properties of the pri-mordial gas which would be otherwise an extremely poorradiator (cooling by Ly-� photons is ine2ective at tempera-tures less than ∼ 104 K).The chemistry of the early Universe has been investi-

gated in several studies starting with the seminal paper byLepp and Shull (1984). We mention in particular the workof Puy et al. (1993), Galli and Palla (1998; hereafter GP)and the comprehensive analysis of deuterium chemistry byStancil et al. (1998; hereafter SLD). The abundances ofH2 and HD predicted at low redshift in these studies areof the order of 10−6 and 10−9, respectively, depending onthe cosmological model adopted (see Palla et al. (1995) forthe variation of the chemical abundances with the assumedbaryon-to-photon ratio). A comparison of the abundance ofHD obtained at z = 10 by GP (n[HD]=n[H] = 1:1 × 10−9)and SLD (n[HD]=n[H] = 1:6 × 10−9) for the same cosmo-logical model (h = 0:5, �0 = 1, �b = 0:0367) shows sat-isfactory agreement, although this might not be completely

∗ Corresponding author. Tel.: +39-055-22-0039; fax:+039-055-2752-249.

E-mail address: [email protected] (D. Galli).

signi1cant since both calculations were based on a com-pilation of reaction rates obtained basically from the samesources. To repeat the words of SLD, “further studies areneeded to reduce the uncertainty in the HD abundance”.One of the goals of this paper is to review the progress madein our understanding of HD chemistry in the three years sub-sequent to the publication of these studies. We will outlinethe improvements which have occurred in the meantime anddiscuss the remaining uncertainties.From an observational standpoint, the abundance of

atomic deuterium and molecular hydrogen has been mea-sured in several cosmological clouds at redshift z � 2–3 inabsorption along the line of sight to bright quasars. Deu-terium in particular has attracted renewed attention becauseof the controversy about its abundance in high-redshift Ly-�clouds (see Hogan (1998) for an account and a resolution ofthe controversy). Molecular hydrogen at high redshift was1rst detected by Levshakov and Varshalovich (1985) in adamped Ly-� system at z= 2:8. Since then, the presence ofH2 has been con1rmed in at least four additional systems(see e.g. Levshakov et al., 2000 and references therein).The presence of HD in an absorption system at z = 2:3 hasbeen reported by Varshalovich et al. (2001).In Fig. 1, we compare observational data on D and H2

at high redshift with the corresponding abundances calcu-lated with the standard model of GP that follows the ho-mogeneous expansion of the universe. The agreement be-tween the theoretical and observed deuterium abundance is

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1198 D. Galli, F. Palla / Planetary and Space Science 50 (2002) 1197–1204

Fig. 1. Comparison of the abundances of D, D+ and H2 in the primordialgas relative to hydrogen, as function of redshift z calculated with thestandard model of GP. Data points represent abundance measurements ofH2 and D in damped Ly-� systems. The H2 data are taken from the listcompiled by Levshakov et al. (2000) (detections, dots with errorbars;dashes: upper limits), whereas D data (triangles with errorbars) are fromBurles and Tytler (1998a, b), O’Meara et al. (2001) and D’Odoricoet al. (2001).

excellent. In the case of H2, one should keep in mind thesensitivity of this molecule to ambient conditions in Ly-�clouds such as the stronger ultraviolet radiation 1eld and thelower dust content with respect to local interstellar medium.These factors can account for the considerable spread ofobserved abundances. Despite the complex phenomenologyassociated with damped Ly-� systems and the resulting un-certainty in the interpretation of the results, it is encourag-ing, to say the least, to witness the emergence of the obser-vational foundations of the highly theoretical discipline ofprimordial chemistry.

2. Chemical reactions

The formation of HD in the primordial gas follows twomain routes involving a deuteron exchange with H2:

D + H2 → HD+ H (1)

and

D+ + H2 → HD+ H+: (2)

Being an isotopic modi1cation of the most fundamen-tal three-electron interaction namely the H + H2 reaction(1) has received considerable interest. Thermal rate con-stants for this reaction have been measured by Ridley etal. (1966), Westenberg and de Haas (1967), Mitchell andLeRoy (1973), and Michael and Fisher (1990) over a wide

Fig. 2. Rate coe0cient for the reaction D + H2 → HD + H accordingto the calculations of Charutz et al. (1997) (short-dashed line), Zhangand Miller (1989) (long-dashed line), Mielke et al. (1994) (solid line).Experimental data are from Mitchell and LeRoy (1973) (1lled squares),Westenberg and deHaas (1967) (open triangles), Ridley et al. (1966)(asterisks).

range of temperatures. Theoretical calculations employingstatistical, semiclassical and quantal method have been per-formed by several groups, and show very good agreementwith each other and with the experimental data. The mostrecent studies are by Zhang and Miller (1989), Michael etal. (1990), Mielke et al. (1994), and Charutz et al. (1997).These results are compared in the usual Arrhenius plotshown in Fig. 2 (the high-temperature experimental resultsof Michael and Fisher, 1990 are not shown). In this paperwe adopt the reaction rate computed by Mielke et al. (1994)with the DMBE surface (Varandas et al. 1987) which pre-dicts much more accurate low-temperature kinetics thanother surfaces. Their results agree with the values computedby Michael and Fisher (1990) (with the same surface)within ∼ 20%. We also note that Zhang and Miller (1989)computed state-by-state cross section for vJ → v′J ′ tran-sitions and state-to-state rate constants in the temperaturerange 200–1000 K. The agreement with the experimentalcross section values at E � 1 eV (Phillips et al., 1989) iswithin a factor ∼ 2.

Reaction (2) represents the major source of HD in di2useinterstellar clouds (Dalgarno et al., 1973). Its rate coe0-cient is almost constant and close to the Langevin value(2× 10−9 cm3 s−1, see Fig. 3). The reaction rate has beenmeasured in the laboratory by Fehsenfeld et al. (1973; 1974,at T =200 and 278K) using a Rowing afterglow technique,and by Henchman et al. (1981, at T = 205 and 295K) us-ing a variable-temperature selected ion Row tube technique(see also Smith et al., 1982). Gerlich (1982) performedaccurate quantum–mechanical calculations of the ratecoe0cient, which are in excellent agreement with the ex-perimental results of Henchman et al. (1981) at T =295 K,

D. Galli, F. Palla / Planetary and Space Science 50 (2002) 1197–1204 1199

Fig. 3. Rate coe0cient for the reaction D+ +H2 → HD+H+ accordingto the calculations of Gerlich (1982) (1lled dots) and the measurementsby Fehsenfeld et al. (1973; 1974) (1lled triangles) and Henchman et al.(1981) (empty triangles). The dashed line shows the Langevin value ofthe rate coe0cient. The solid line shows our 1t to Gerlich’s results (seeTable 1).

but less at T = 205 K (still within a factor ∼ 2). There isa discrepancy with the results of Fehsenfeld et al. (1973;1974) at both temperatures.The destruction of HD occurs mainly via the reverse re-

actions of (1) and (2)

HD + H → D+ H2 (3)

and

HD + H+ → D+ + H2 (4)

In general, the rate coe0cients for the forward and reversechemical reactions are related by the standard thermody-namic expression (e.g. Berry et al., 1980)

ln(kfkr

)=−SH 0

RT+

SS0

R; (5)

where R is the universal gas constant, and SH 0, SS0 arethe enthalpy and entropy changes. Thus, if the rate kf isknown the rate for the reverse reaction kr can be obtaineddirectly from Eq. (5). From the di2erences in the zero-pointvibrational energies of H2 and HD and in the ionizationpotentials of H and D, one obtains the enthalpy changesfor reactions (1) and (2): SH 0=R = −412 K and −462 K,respectively. The entropy change SS0=R can be calculatedon statistical grounds (see e.g. Flower, 2000) and is the samefor both reactions: SS0=R=ln 2 for para-H2, SS0=R=ln(2=3)for ortho-H2. These values are for reactant and productmolecules in their ground states. Both the entropy and theenthalpy changes are modi1ed when rovibrationally excitedmolecular states are involved (Flower, 2000).

Rather than using Eq. (5) in this paper we prefer to pro-vide independent 1tting formulae for both the direct and

Fig. 4. Rate coe0cient for the reaction HD + H → D + H2 accordingto the calculations of Shavitt (1959) for two values of the asymmetricstretching force constant Au (1lled dots: Au =−0:358× 105 dyne cm−1;empty dots: Au = −0:732 × 105 dyne cm−1). The experimental resultsof Boato et al. (1956) and van Meersche (1951) are shown by 1lledtriangles and empty triangles, respectively. The solid line is our 1t to thedata of Shavitt (1959) for Au =−0:358× 105 dyne=cm−1 (see Table 1).

reverse reactions that dominate the chemistry of deuteriumin the primordial gas. For speci1c applications, the readermay adopt the reaction rate of the forward (or reverse, ifbetter constrained) reaction and compute the rate of the re-verse (forward) reaction from the thermodynamic relationexpressed by Eq. (5).The rate coe0cient of reaction (3) has been calculated

by Shavitt (1959) using a semiempirical H3 energy sur-face. Only sparse laboratory data exist for this reaction. AtT = 103 K, the experimental result of Boato et al. (1956)is within ∼ 20% from the value calculated by Shavitt(1959). At lower temperatures (720–880 K), the theoreti-cal rate is a factor ∼ 2 lower than the experimental data byvan Meersche (1951) but the extrapolation of the adoptedH3 energy surface introduces signi1cant uncertainty in theresults at low temperatures (see Fig. 4). As a challenge tochemical physicists, we recall the words of Shavitt (1959):“it is unfortunate that for a reaction as basically impor-tant as the one considered here, the experimental data areso incomplete and inconclusive”.Since reaction (4) is endothermic by 0:0398 eV (462 K),

the removal of HD at low temperatures is reduced by a fac-tor exp(−462=T ), and this can lead to signi1cant enhance-ment of the HD=H2 ratio (fractionation). The rate coe0-cient of this reaction has been measured in the laboratory byHenchman et al. (1981) at T=205 and 295 K, and the resultsare in good agreement with the values obtained by applyingthe principle of detailed balance to the reverse reaction (2).As in GP, we adopt the rate coe0cient calculated by Gerlich(1982) over the temperature range 30–600 K (see Fig. 5).


Fig. 5. Rate coe0cient for the reaction HD+H+ → D+ +H2 accordingto the calculations of Gerlich (1982) (1lled dots and the experimentalresults of Henchman et al. (1981) at T = 205 and 295 K (triangles witherrorbars). The solid line shows our 1t to Gerlich’s results (see Table 1).

Finally, the relative abundance of D and D+ is determinedby the charge exchange reactions

H+ + D → H + D+ (6)

and its reverse

H + D+ → H+ + D: (7)

The cross section for reaction (6) was computed byMatveenko (1974) and Hunter and Kuriyan (1977) forenergies from 10−3 to 7:5 eV. The two results di2er by afactor ∼ 2 at low energies for reasons unclear. Subsequentcalculations by Hodges and Breig (1993), and, more re-cently, by Igarashi and Lin (1999) and Zhao et al. (2000)con1rm the validity of the results of Hunter and Kuriyan(1977) and improve signi1cantly the accuracy of the cal-culations around ∼ 10−3 eV. Good agreement was alsofound with the experimental measurements of Newmanet al. (1982) and Esry et al. (2000). The cross section ofreaction (7) was also calculated by Igarashi and Lin (1999)and Zhao et al. (2000). Unfortunately, no experimental dataare available for this reaction.Watson (1976) estimated the rate of reactions (6) and (7)

and Watson et al. (1978) calculated the rate coe0cient onthe basis of the cross section obtained by Hunter and Kuriyan(1977). These rates have been widely adopted in studiesof deuterium chemistry. Since reaction (6) has a thresholdof 43 K, the rate coe0cient for reaction (7) is usually ob-tained by multiplying that of reaction (6) by exp(43=T ).The situation has been reanalyzed recently by Savin (2002),who computed accurate rates for reactions (6) and (7) fromthe cross sections of Igarashi and Lin (1999) and Zhaoet al. (2000). These results are compared to those ofWatson et al. (1978) in Figs. 6 and 7.

Fig. 6. Rate coe0cient for the reaction H+ + D → H + D+ accord-ing to the calculations of Savin (2002) (1lled dots) and Watson et al.(1978) (empty dots). The solid line shows the 1t obtained by Savin(2002) to his results (see Table 1).

Fig. 7. Rate coe0cient for the reaction D+ + H → D + H+ accord-ing to the calculations of Savin (2002) (1lled dots) and Watson et al.(1978) (empty dots). The solid line shows the 1t obtained by Savin(2002) to his results (see Table 1).

In Table 1, we collect accurate 1tting formulae (computedby us or by the authors quoted in the references) for thechemical reactions discussed in this section. These expres-sions update and replace the corresponding formulae givenin Table 2 of GP.In addition to these reactions, additional contributions to

the formation of HD in the early Universe come from theassociative detachment reactions

D + H− → HD+ e (8)


Table 1Rate coe0cients

No. Reaction Rate coe0cient (cm3 s−1) Reference

(1) D + H2 → HD + H 1:69× 10−10e(−4680=T+198800=T 2) Mielke et al. (1994)(2) D+ + H2 → HD + H+ 10−9 × [0:417 + 0:846 log T − 0:137(log T )2] Gerlich (1982)(3) HD + H → D + H2 5:25× 10−11e(−4430=T+173900=T 2) Shavitt (1959)(4) HD + H+ → D+ + H2 1:1× 10−9e−488=T Gerlich (1982)(5) H+ + D → H + D+ 2:00× 10−10T 0:402e−37:1=T − 3:31× 10−17T 1:48 Savin (2002)(6) H + D+ → H+ + D 2:06× 10−10T 0:396e−33:0=T − 2:03× 10−9T−0:332 Savin (2002)

and

D− +H → HD+ e (9)

whose rate coe0cients however are not known, and can onlybe estimated from the corresponding H reactions (see SLD).Finally, minor contributions to the formation of HD comefrom the radiative association reaction

H + D → HD+ h� (10)

whose rate coe0cient was computed by Stancil andDalgarno (1997), and from reactions involving less abun-dant deuterated species like HD+ and H2D+ (see SLD fordetails).Generally, the chemistry of HD in the primordial gas is

dominated by the ion–neutral reactions (2) and (4) in a gasof low density (e.g. the primordial gas before the formationof the 1rst structures), whereas the neutral–neutral reactions(1) and (3) become more important in conditions of highdensity and temperature (cloud collapse, shocked gas).

3. Heating and cooling

In order to calculate the cooling properties of HD, oneshould know the population of all rovibrational levels. Insteady-state, these are obtained by solving the balance equa-tions

xJ∑J ′

[RJJ ′(Trad) + CJJ ′(n; Tgas)]

=∑J ′xJ ′ [RJ ′J (Trad) + CJ ′J (n; Tgas)]; (11)

where J and J ′ indicate a generic couple of rovibrational lev-els. The collisional transition probabilities CJJ ′(n; Tgas) andCJ ′J (n; Tgas) are obtained by multiplying the correspondingexcitation coe0cients, �JJ ′(Tgas) and �J ′J (Tgas), and the den-sity of the colliding species. The terms RJJ ′ and RJ ′J are theradiative excitation and de-excitation rates that can be ex-pressed in terms of the Einstein coe0cients AJJ ′ and BJJ ′ as

RJJ ′ =

{AJJ ′ + BJJ ′u(�JJ ′ ; Trad); J ′¡J;

BJJ ′u(�JJ ′ ; Trad); J ′¿J;(12)

where u(�JJ ′ ; Trad) is the energy density of the cosmic back-ground radiation (CBR) per unit frequency at the tempera-ture Trad:

u(�JJ ′ ; Trad) =8�h�3JJ ′c2

[exp(h�JJ ′ =kTrad)− 1]−1 : (13)

In the primordial gas, collisional excitation of HD isdominated by collisions with H and to a less extent withHe. The coe0cients for inelastic scattering of He–HD werecomputed by Green (1974) and Schaefer (1990) at tem-peratures T6 600 K for 06 J6 3 and SJ = +1;+2.Collisional coe0cients for rotational excitation of the sys-tem H–HD were usually derived by scaling the He–HD val-ues with the square root of the ratio of the reduced massesof the two systems, �H–HD = (�He–HD=�H–HD)1=2�He–HD,where (�He–HD=�H–HD)1=2 = 1:51 (see e.g. Timmermann,1996, GP). In the words of Timmermann (1996), “thisassumption is however an educated guess, and dataon H–HD are urgently needed”. These data were re-cently provided by Flower and Roue2 (1999) (for Hand H2) and Roue2 and Zeippen (1999) (for He), us-ing full quantum-mechanical methods and updated en-ergy potential surfaces to evaluate rovibrational ex-citation coe0cients for collisions of HD with H, H2

and He. The results for H–HD collisions computed byFlower and Roue2 (1999) in the temperature range1006Tgas6 2000 K for v6 2 and J6 9 for a total of30 rovibrational levels are available from the CCP7 serverhttp://ccp7.dur.ac.uk/.The energy levels and the Einstein coe0cients AJJ ′ of

HD were calculated by Abgrall et al. (1982), who consid-ered both dipole and quadrupole transitions and included alarge number of rovibrational levels. Since the energy spac-ing of the rotational levels of HD is quite large, E1=k =128 K, E2=k =383 K, E3=k =764 K, etc. GP computed theHD cooling function with a simple four-level system (J =0–3) adopting the collisional coe0cients of Schaefer (1990).Flower et al. (2000) updated the calculations of GP adoptingthe collisional rate coe0cients of Flower and Roue2 (1999)and Roue2 and Zeippen (1999). The HD cooling functioncomputed by Flower et al. (2000) is also available fromthe CCP7 server. A useful approximation in the low-density

http://ccp7.dur.ac.uk/


Fig. 8. The cooling rate per HD molecule computed for n(H) = 1 to108 cm−3 (solid lines: Flower et al., 2000; dashed lines: GP). Onlycollisions of HD with H have been considered.

limit is the expression

�HD[n(H → 0)] = 2�10(Tgas)E10e−E10=kTgas

+(5=3)�21(Tgas)E21e−E21=kTgas ; (14)

where E10=k=128 K, E21=k=255 K and the collisional ratecoe0cients �JJ ′ are given by Flower and Roue2 (1999):

�10(Tgas)

=4:4× 10−12 + 3:6× 10−13(log Tgas)0:77 cm3 s−1 (15)

and

�21(Tgas)

=4:1× 10−12 + 2:0× 10−13(log Tgas)0:92 cm3 s−1: (16)

The comparison between the HD cooling rate calculatedby GP and Flower et al. (2000) shown in Fig. 8 is instruc-tive. A simple four-level molecule is able to predict quiteaccurately the cooling rate in a wide range of temperatures(Tgas . 2000 K) and densities (n[H] . 107 cm−3) butof course fails badly in the high-temperature, high-densityregime. However, for cosmological applications, it is impor-tant to keep in mind that the HD cooling function of Floweret al. (2000) has been computed assuming that the temper-ature of the CBR is much smaller than the gas temperature.This approximation is valid at low redshifts (where the twotemperatures di2er by more than a factor ∼ 10 for z .20), but becomes increasingly inaccurate at higher redshifts.Since most cosmological scenarios of structure formationbegin at z � 100 when Trad � 300 K, the level populationof molecules is strongly a2ected by stimulated emission andabsorption. In such conditions, molecules may become ane2ective heating source for the gas because the rate of colli-sional de-excitation of the rovibrational levels is faster thantheir radiative decay (Khersonskii, 1986; Puy et al., 1993).

Fig. 9. The heat transfer function ( −�)HD for n(H) = 1 cm−3 versusgas temperature at selected redshifts. The solid line is computed ignoringthe e2ects of the CBR (z = 0). When Tgas¿Trad the heat exchange is acooling term (solid lines). In the opposite case, the transfer becomes aheating source for the gas (dashed lines).

As an illustration of this e2ect, we plot in Fig. 9 the netheat transfer function ( − �)HD computed with the GPmodel for n(H) = 1 cm−3 as function of gas temperature atthree selected redshifts. Note that for Tgas¿Trad the coolingfunction is signi1cantly decreased from the value computedwith Trad=0 because of the radiative depopulation of excitedstates. The sudden drop signals the condition Tgas =Trad. Fi-nally, for Tgas¡Trad the function changes sign and becomesa net heating term for the gas. In cosmological simulationsthe heat transfer should be computed self-consistently ateach redshift.

4. Application: cloud collapse at z � 100

In both cold dark matter and baryonic dark matter cos-mological scenarios the 1rst objects predicted to enter in thenonlinear stage are the smallest ones. In cold dark mattermodels it is expected that overdense regions with masses105–107 M� 1rst collapse in the redshift range 10 . z .100 (see e.g. Cen et al., 1993). The crucial question iswhether molecular cooling allows the baryonic componentto dissipate its kinetic energy and collapse on a time scaleshorter than a Hubble time. This important question is fullyaddressed e.g. in the 1-D hydrodynamical calculations byHaiman et al. (1996) and in the 3-D numerical simulationspresented by Abel et al. (1997) and Bromm et al. (1999).Here, we consider a particular issue related to the choice ofthe initial conditions for collapse calculations.Following Tegmark et al. (1997) we consider the growth

of a “top-hat” overdensity region, an isolated spherical


Fig. 10. The time evolution of gas in a primordial cloud for zvir = 110,Tvir = 2000 K, h = 0:5, and �b = 0:06. The mass of the cloud is1:35×106M�. The top panel shows the gas density n (in cm−3), the gastemperature Tgas (in K) and the temperature of the cosmic backgroundradiation Trad (in K) as function of redshift z. The bottom panel showsthe abundances of electrons, D, D+, H2 and HD, relative to H, for thesame model.

perturbation in a uniform density Universe (see e.g.Padmanabhan, 1993). The radius of this region increasesat a slower rate than the scale factor of the Universe (butstill obeys the Friedman’s equations) and after reachinga maximum value (turnaround) recollapses to a point.The formation of a singularity is clearly an artifact of thesimpli1ed assumptions of the “top-hat” model. In a re-alistic situation gas dynamical process (internal pressure,shocks) will eventually halt the collapse of the baryoniccomponent at some 1nite value of the density. The result-ing quasi-equilibrium con1guration is a virialized “halo”,and its subsequent evolution depends on the ability of thegas to cool on a time scale shorter than a dynamical timescale.We show in Fig. 10 the results of a sample run of the

Tegmark et al. (1997) evolutionary model obtained with ourchemical network and molecular cooling prescriptions. Forthe case shown, the cloud reaches virialization at zvir = 110,where the temperature is Tvir = 2000 K. After virializationthe gas density is assumed to remain constant and uniformfor the rest of the run, and we follow the cooling of the gasdue to H2 and HD molecules. This is clearly a poor approx-imation since the cooling of the gas will of course inducean increase in the density, and the post-virialization evolu-tion of the cloud must be followed with a full hydrodynam-ical calculation (Galli et al., in preparation). Nevertheless,it is instructive to consider how rapidly the baryons are able

to dissipate the thermal energy of the cloud via molecularcooling. As we see in Fig. 10, recombination reduces theionization fraction of the cloud to negligible values, weak-ening the Compton cooling of the gas (since Trad¡Tgas).The rapid rise of H2 (via the H− channel) around zvir causesthe sudden cooling of the gas from Tvir down to a few hun-dred degrees, and the subsequent increase in the HD abun-dance further reduces the gas temperature down to few tensof degrees establishing again the thermal coupling of gasand cosmic radiation at redshift z � 10.

Loosely speaking, a Hubble time corresponds to the red-shift dropping by a factor 22=3 � 1:6. For the particularcase considered in this illustrative example, at a redshiftz = zvir=1:6 � 70 the cloud temperature has dropped toTgas � 350, i.e. of a factor ∼ 6 with respect to the virialtemperature. Molecular cooling (mostly H2 in this range ofredshift) therefore enables the cloud to cool signi1cantlywithin a Hubble time and eventually to collapse and formluminous objects. Cooling by HD appears to induce a fur-ther substantial reduction of the temperature at later times,but the subsequent evolution of the cloud can only be fol-lowed with a realistic calculation. However, it must be keptin mind that the mass scale of the fragments formed by thecollapse of the cloud depends sensitively on the gas temper-ature (MJ ∼ "−1=2T 3=2

gas ). The cooling induced by the forma-tion of HD may in fact reduce the gas temperature below∼ 100 K and therefore enable the formation of primordiallow-mass stars or even brown dwarfs, a possibility recentlyexplored by Uehara and Inutsuka (2000) (see also Flowerand Pineau des Forets, 2001). These exciting results repre-sent a radical change of perspective in the 1eld of primordialstar formation and deserve further investigation.

5. Conclusions

We have examined the most e2ective chemical reactionsleading to the formation/destruction of HD molecules in theearly Universe, and presented a list of accurate reaction ratecoe0cients for primordial chemistry calculations that up-date those adopted in previous studies. We have analyzedthe heating/cooling properties of HD molecules, stressingthe relevance of a self-consistent calculation of the energytransfer rate between gas and radiation for cosmological ap-plications. As an illustration we have presented a simpli-1ed model for the evolution of density perturbations in theexpanding Universe, following previous investigations byHaiman et al. (1996) and Tegmark et al. (1997) but in-cluding our comprehensive and updated chemical network.These results together with recent 1ndings by Uehara andInutsuka (2000) and Flower and Pineau des Forets (2001)underline the substantial contribution of HD to gas coolingduring the collapse of primordial clouds. A preliminary con-clusion from these studies is that HD is at least as impor-tant as H2 in determining the thermal balance of zero-metalclouds.


Acknowledgements

This work is supported by the Italian Ministry for theUniversity and for Scienti1c and Technological Research(MURST) through a COFIN-2000 Grant. It is a pleasure tothank Guillaume Pineau des Forets and Malcolm Walmsleyfor informative dicussions on deuterium chemistry. We alsothank A. Dalgarno, S.N. Shore and an anonymous refereefor useful comments on an earlier version of this paper.

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Deuterium chemistry in the primordial gas