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MNRAS 434, 114­122 (2013)

doi:10.1093/mnras/stt1007

Advance Access publication 2013 June 24

Non-thermal photons and H2 formation in the early Universe
C. M. Coppola,
1 2 3 4 5

1,2,3 <

D. Galli,3 F. Palla,3 S. Longo

2,3,4

and J. Chluba

5

Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK ` Dipartimento di Chimica, Universita degli Studi di Bari, Via Orabona 4, I-70126 Bari, Italy INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy IMIP-CNR, Section of Bari, via Amendola 122/D, I-70126 Bari, Italy Johns Hopkins University, Bloomberg Center 435, 3400 N. Charles St, Baltimore, MD 21218, USA

Accepted 2013 June 5. Received 2013 June 5; in original form 2013 April 22

ABSTRACT

The cosmological recombination of H and He at z 103 and the formation of H2 during the dark ages produce a non-thermal photon excess in the Wien tail of the cosmic microwave background blackbody spectrum. Here, we compute the effect of these photons on the H- photodetachment and H+ photodissociation processes. We discuss the implications for the chemical evolution of 2 the Universe in the post-recombination epoch, emphasizing how important a detailed account of the full vibrational manifold of H2 and H+ in the chemical network is. We find that the 2 final abundances of H2 ,H+ ,H+ and HD are significantly smaller than in previous calculations 2 3 that neglected the effect of non-thermal photons. The suppression is mainly caused by extra hydrogen recombination photons and could affect the formation rate of first stars. We provide simple analytical approximations for the relevant rate coefficients and briefly discuss the additional effect of dark matter annihilation on the considered reaction rates. Key words: molecular processes ­ early Universe.

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1 I NTR O DUCTION In the era of precision cosmology, the determination of the chemical composition of the early Universe requires an accurate evaluation of the reaction rates of the main chemical processes involved. A detailed chemical­kinetic model for the evolution of the homogeneously expanding Universe in the post-recombination epoch is also needed to follow the collapse of primordial clouds and hence to study the formation of the first-generation stars (Tegmark et al. 1997; Abel, Bryan & Norman 2002; Bromm, Coppi & Larson 2002). In particular, H2 represents a `key' element because of its abundance and coolant properties. For this reason, the line emissions associated with molecular hydrogen could in principle give information about the matter distribution during the phase of pre-reionization of the Universe (e.g. Ciardi & Ferrara 2001; Gong, Cooray & Santos 2013). Over the past years, continuous improvements have been made to the modelling of the early Universe chemistry under nonequilibrium conditions. For example, Coppola et al. (2011, hereafter C11), Longo et al. (2011) and Coppola et al. (2012) demonstrated the importance of taking into account the complete internal states of chemical partners in a chemical network for the primordial gas, as well as all non-equilibrium processes occurring at high redshift z. In this paper, we compute the abundance of the main chemical species (such as H2 , H+ , H- , HD and H+ ), including the effect of 2 3 non-thermal photons due to cosmological recombination of H and
E-mail: carla.coppola@chimica.uniba.it

He (see Sunyaev & Chluba 2009 for overview), and the radiative cascade following the formation of H2 . The non-thermal photons appear as an excess in the Wien tail of the cosmic microwave background (CMB) blackbody spectrum and thus significantly affect the H- photodetachment and H+ photodissociation processes dur2 ing the dark ages. As we show here, the main effect is caused by the extra H I recombination photons released at z 100, limiting the formation of H2 , H+ , H+ and HD. We also estimate the effect 2 3 of extra ionizations from annihilating dark matter particles on the H- photodetachment rate, finding a sensitivity of the early Universe chemistry to this process (see Appendix B). The paper is organized as follows: in Sections 2 and 3, we describe the computational methods, providing expressions for the spectral distortion of the CMB introduced by emission processes occurring in past epochs of the expanding Universe. The distortion spectra resulting from primordial atomic recombination and non-equilibrium H2 radiative cascade are then used to evaluate non-thermal reaction rates for photoprocesses involving the main `catalytic' species for H2 formation (H+ and H- ). In Section 3, we describe the time2 dependent kinetics and summarize the reaction rates and cosmological parameters introduced. The resulting fractional abundances of several atomic and molecular species adopting the updated rate coefficients of Coppola et al. (2011) are discussed in Section 4.

2 E FFECT OF NON-THERMAL P HO T O NS Every radiative transition from an upper atomic level i to a lower level j is associated with the emission of a photon, causing a

C 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society


Primordial chemistry
spectral distortion Iij ( ). Assuming a very narrow emission-profile, the observing frequency at some redshift, z < zem , is related to the rest-frame frequency, ij , of the transition i j by = ij (1 + z)/(1 + zem ). For this reason, the redshift at which the transition happens is labelled as zem . The spectral distortion produced by the emission process at zem and observed at redshift z < zem can be ~ written as (e.g. see Rubino-Martin, Chluba & Sunyaev 2006)
z Iij ( ) =

115

hc 4

Rij (zem )(1 + z)3 , H (zem )(1 + zem )3

(1)

]1/2 where H (z) = H0 [ r (1 + z)4 + m (1 + z)3 + k (1 + z)2 + is the Hubble function and Rij is related to the level populations, Ni and Nj of the ith and jth levels by Rij = pij Aij Ni ehij
/kB Tr

ehij /kB Tr

-1

1-

gi Nj -hij e gj Ni

/kB Tr

,

(2)

where pij is the Sobolev-escape probability, gi and gj the degeneracy of upper and lower levels, respectively (both factors are equal to 1 in the case of transitions occurring among the vibrational manifold), Aij is the Einstein coefficient of the transition and Tr = 2.726 (1 + zem ) K (Fixsen et al. 1996; Fixsen 2009). To evaluate the contribution of spectral distortions to the reaction rate of a photoreaction at a given redshift z, the integration over the actual photon distribution should be carried out: ( ) z Bz ( ) + Iij ( ) d. (3) kph (z) = 4 h 0 i j Here, ( ) is the cross-section of the photoreaction as a function of frequency, Bz ( ) the Planck distribution at Tr corresponding to z the redshift z at which the reaction rate is calculated, and Iij ( )the spectral distortion. Several physical and chemical processes can modify the pure blackbody shape of the CMB (see Chluba & Sunyaev 2012 for some example related to early energy release); in the present calculations, we consider the primordial recombination of H and He and the non-equilibrium radiative cascade of H2 as sources of distortion photons. For the former, the outputs of COSMOREC1 (Chluba & Thomas 2011) are used to evaluate the non-thermal photon contribution. For the latter, the values of Aij were calculated as in C11 averaging over the initial rotational levels and summing over the final ones the rovibrationally resolved Einstein coefficients computed by Wolniewicz, Simbotin & Dalgarno (1998); the non-equilibrium level populations calculated in C11 at several z have been used, following the treatment of Coppola et al. (2012). To estimate the effect of non-thermal photons on the H2 chemistry, the main formation channels for molecular hydrogen, namely the H+ and H- pathways, should be considered separately. Fig. 1 2 shows the spectrum of the CMB at several redshifts, including the distortion photons produced by the cosmological H recombination and the H2 radiative cascade. For the cosmological recombination radiation, only the emission of the Ly ,Ly and Ly lines and the 2s­1s two-photon continuum are shown, with the computation including detailed radiative transfer effects, such as line feedback and two-photon processes. The position of the Ly line can be noticed in the upper panel of Fig. 1, while photons above 10.2 eV are caused by the Ly and Ly transitions. Relative to the Ly line these resonances only add a small number of photons in the far Wien tail of the CMB (Chluba & Sunyaev 2007) and thus do not affect the final
1

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Figure 1. Photons spectra at several z, corresponding to the values shown in the figure (dashed lines: partial contributions; red and black lines: CMB and distortions, respectively; solid lines: total spectra). Top panel: blackbody and distortion photons produced by the cosmological recombination of H; bottom panel: blackbody plus photons produced by the radiative cascade following the non-equilibrium formation of H2 at the same redshifts. The vertical blue lines represent the thresholds for the processes considered in this work: from the lowest energy, the threshold for H- photodetachment (0.754 eV), H+ (v = 0) and H+ (v = 6) photodissociation (2.65 and 2 2 1.247 eV, respectively). The value for the highest vibrational level, H+ (18), 2 is 0.0029 eV, out of the range of the present figure.

www.chluba.de/CosmoRec

results for the reaction rates at a significant level. We also omitted the emission caused by transitions among exited states (Chluba & ~ ~ Sunyaev 2006; Rubino-Martin et al. 2006; Chluba, Rubino-Mart´ in & Sunyaev 2007), since these only give rise to a tiny derivation relative to the CMB blackbody spectrum. The reprocessing of helium photons emitted at z 2000 was also taken into account for the computation of the distortion (Chluba & Sunyaev 2010), producing a pre-recombination feature in the H I Ly recombination spectrum ~ (see also Rubino-Mart´ Chluba & Sunyaev 2008). in, Helium photons only directly affect the H+ and H- formation 2 rates close to z 2400, so that we did not present their contribution to the CMB spectrum separately. The lower panel of Fig. 1 shows the distortion produced by the radiative cascade following the non-equilibrium formation of H2 . At the highest redshift, the largest contribution comes from the most energetic 4.7 eV transition between vibrational state v = 14 and the vibrational ground level of H2 molecules. At lower redshifts, the high-v transitions become progressively less important because


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of the expansion of the Universe that shifts them to lower energies. The features present in the spectra produced by molecular radiative cascade lines reflect the presence of many transitions with v 1. As a consequence, the spectra are broader than the ones obtained for the atomic recombination (Fig. 1). As for the cosmological recombination distortion, the photons produced by H2 radiative transitions give rise to an excess in the far Wien tail of the CMB. The extra photons are introduced at late times, during the dark ages, when most of the H2 is forming. In comparison to the recombination radiation it is, however, much smaller and thus only leads to a tiny correction to the reaction rates. 2.1 H+ channel 2 Charge transfer between H+ and H, 2 H+ (v ) + H H2 (v ) + H+ , 2 (4)
Figure 2. H+ photodissociation cross-sections averaged at several temper2 atures, from T = 104 to 50 K. The intermediate curves are for T = 8400, 8000, 7000, 6300, 6000, 5000, 4200, 4000, 3500, 3000, 2500, 2000, 1500, 1000, 750, 500, 300, 200 and 100 K. Calculations have been performed by ´ Mihajlov and Ignjatovic using the theoretical method described in Mihajlov et al. (2007) on an extended temperature grid (private communication).

represents the dominant formation channel of H2 at high z. The reaction (4) is exoergic for all vibrational states, v , unlike the charge transfer between H2 and H+ that is endoergic for v 3, although ´ ´ with low threshold energies (e.g. Krstic & Schultz 1999; Krstic, ´ Schultz & Janev 2002; Krstic 2002, 2003, 2005). For the conditions of the primordial Universe a very efficient collisional way to destroy H+ is by dissociative recombination (Takagi 2002; Motapon et al. 2 2008). Among the photoprocesses, the reaction H+ (v ) + h H + H+ 2 (5)

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represents a favourable destruction pathway and it has been the subject of several theoretical quantum chemistry studies (Dunn 1968; Argyros 1974; Stancil 1994; Lebedev, Presnyakov & Sobel'man 2000, 2003). In this work, we adopt the cross-sections calculated by Mihajlov et al. (2007) using a quantum mechanical method in which the photodissociation process is assumed to be the result of radiative transitions between the ground and the first excited adiabatic electronic state of the molecular ion H+ . These transitions 2 are the results of the interaction of the electron component of the ion­atom system with the free electromagnetic field in the dipole approximation. The cross-sections are given as Maxwell­Boltzmann averages (i.e. assuming statistical equilibrium populations for the Jsubstates) of the state-to-state resolved cross-sections v, J () over the rovibrational distribution function at each temperature: ph (, T ) = 1 Z +
J even

been assumed to make calculations more feasible. This is justified considering the faster relaxation times of rotation with respect to the other molecular degrees of freedom. In Fig. 2, we show the adopted set of average cross-sections. Both the CMB blackbody and non-thermal reaction rates have been calculated using equation (3). The former is compared to the fit given by Galli & Palla (1998, hereafter GP98), obtained using data by Stancil (1994) and Argyros (1974). The results shown in Fig. 3 indicate that the comparison between the thermal reaction rates is satisfactory at all redshifts. At low temperatures (corresponding to z< 300), the dominant effect is due to the non-thermal tails deriving

v

J odd

3 - (2J + 1) e 2

EvJ -E00 kB T

vJ ()

1 - (2J + 1) e 2

EvJ -E00 kB T

v,J () ,

(6)

where Z is the partition function, Z=
v J odd

3 - (2J + 1) e 2

EvJ -E0,0 kB T

+
J even

1 - (2J + 1) e 2

EvJ -E00 kB T

.

(7)

The temperature used in these equations is the radiation temperature because of the conditions present in the early Universe. As a check in support of this assumption, it can be shown that for the majority of the pairs (v , v ), the critical density ncr = Av,v /v,v (i.e. the ratio between radiative and collisional de-excitation coefficients) is much above the mean density of the primordial Universe. Moreover, the hypothesis of thermalization of rovibrational levels has

Figure 3. H+ photodissociation rate coefficient as function of redshift z 2 (lower scale) and radiation temperature Trad (upper scale). Black solid curve: thermal reaction rate calculated using the cross-sections shown in Fig. 1; blue dashed curve: fit by GP98. Green dotted curve: non-thermal contribution due to cosmological recombination photons computed with the cross-sections of Fig. 1. Red small dashed curve: same as above, with photons produced by the H2 radiative cascade. It is evident from the figure that distortion photons coming from primordial atomic recombination represent the main non-thermal contribution to the reaction rate.


Primordial chemistry
equations (see e.g. GP98) has to be solved: dNi =- dt +
n

117

~ kl Ni -
l j

kij Ni Nj jm ki Nj Nm ,
j m

~ kn Nn +

(10)

where Ni is the abundance of the ith species relative to the total ~ baryon density, kl are the photodestruction rate coefficients of the ith species via the lth photoprocess; kij are the destruction rate coefficients for the ith species for collisions with the jth chemical ~ partner; kn are the formation rate coefficients of the ith species due jm to the nth photodestruction process for the Nn species and ki are the formation rate coefficients of the ith species due to collisions between the jth and mth species. Each reaction rate is proportional to the variation of the baryonic density as a function of time
Figure 4. H- photodetachment rate coefficient. Separate contribution of the blackbody spectrum, H and He recombination photons, and H2 radiative cascade are shown. As in the case of H+ photodissociation, atomic 2 recombination represents the dominant non-thermal contribution.

nb ( z ) =

b

2 3H0 (1 + z)3 , 8GmH

(11)
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from atomic recombination. The contribution from the H2 cascade in negligible across the temperature interval.

2.2 H- channel The formation of H2 at low redshifts is controlled by the associa i ´ tive detachment process (Pagel 1959; C´zek, Horacek & Domcke 1998; Flower 2000; Dalgarno 2005; Kreckel et al. 2010; Schlemmer 2011): H- + H H2 (v ) + e- . (8)

where b is the baryon fraction, H0 = 100 h km s-1 Mpc-1 is the Hubble constant with h = 0.705, G is the gravitational constant, mH is the atomic hydrogen mass and = 4/(4 - 3Yp ) is the mean atomic weight of the gas, Yp denoting the helium fractional abundance by mass. The equations for the radiation and gas temperatures are solved in order to evaluate the specific velocity of each chemical process in the kinetics. For the present calculations, the cosmological parameters from Wilkinson Microwave Anisotropy Probe 7andstandard BBN data have been used (Komatsu et al. 2011; Iocco 2012). For our network, we adopted the rate coefficients summarized in Table B1. The table also provides polynomial fits to the non-thermal contributions to the photodetachment of H- and photodissociation of H+ rates. When applicable, the complete vibrational manifolds 2 of H2 and its cation were used, both in LTE approximation in the entrance channel and as sum of contributions in the exit channel. 4 R ESUL TS Using the new rates and the improved rate coefficients reported in C11, we determined the fractional abundances of several atomic and molecular species with the kinetic model described in the previous section. Fig. 5 shows the evolution of H2 , H+ , H+ and HD, 2 3 along with that of H- and D- . The main differences with respect to previous studies can be summarized as follows: starting at high redshifts, the abundances are affected by the enhanced H2 destruction channels (H2 /H+ charge transfer, dissociation via H and H+ collisions, photodestruction), H- photodetachment and modified H/H- associative detachment. The first process results in a reduced fractional abundance of H2 at redshifts z 1000, where it reaches values roughly one order of magnitude smaller than in previous calculations (e.g. Schleicher et al. 2008). Consequently, the fractional abundances of HD and H+ are reduced in the same redshift range. 3 This effect is caused by the inclusion of the entire vibrational manifold, as also found by Capitelli et al. (2007) for the dissociative attachment process of H2 (see figs 4­6 of C11). At lower redshifts, the abundances are affected by the combined effect of the enhanced photodetachment of H- and the decrease of the efficiency of associative detachment due to non-thermal photons. This result qualitatively agrees with what was found in the steady-state model by Hirata & Padmanabhan (2006), where however no expression for the non-thermal rate coefficient was given, and here a more detailed treatment for the recombination spectrum is used. The effect of the contribution of non-thermal photons to the

The reaction responsible for the loss of H- is the photodetachment process: H- + h H + e- . (9)

For reaction (9), we have adopted the analytical fit of Tegmark et al. (1997) to the cross-section data computed by Wishart (1979). In Fig. 4, we show the H- photodetachment rate coefficient obtained considering both the integration over the CMB and the distortion photons. The former clearly provides the largest contribution at redshifts greater than z 100, whereas at lower redshift hydrogen recombination photons contribute significantly. The effect of helium photons is restricted to very early times (z 2400) when only very insignificant amounts of chemical elements have formed. The feedback of helium photons on hydrogen also creates extra features in the Ly recombination spectrum that leads to non-monotonic behaviour of the H- photodetachment rate at z 1800. It is also important that at high redshifts (z 1300) half of the non-thermal reaction rate is caused by the 2s­1s continuum emission, while in the post-recombination epoch only the H I Ly distortion is important. As for H+ photodissociation, the process of H2 radiative cascade 2 remains negligible at all redshifts.

3 R EA CTION R ATES AND KINETICS The chemistry of the early Universe at z < 103 can be described as the kinetics of an H­He plasma in an expanding medium. For this reason, the following time-dependent system of ordinary differential


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C. M. Coppola et al.

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Figure 5. Fractional abundances of selected species: H2 , HD and H+ (top 3 panel), H- and D- (bottom panel), with and without non-thermal photons (green dotted and blue solid curves, respectively.)

photodetachment of H- can be appreciated in Fig. 5 (bottom panel) at z < 100. Although the freeze-out value of H- at low z remains unchanged, the abundance of H2 is reduced by about 70 per cent at the epochs when the H- channel is dominant. Importantly, the new evolution reduces the final rise of the H2 abundance at z 100 that characterized all previous calculations. It is also worth noting that, despite the huge effect of non-thermal photons on the photodissociation of H+ , its abundance is not significantly affected. This can be 2 understood considering the relatively high threshold energy for the photodissociation process compared to the photodetachment of H- and to the mean thermal energy available. Indeed, the integration over the high-frequency part of the distortion spectrum is much more favourable for lower thresholds, as it can be derived from Fig. 1. To compare the efficiency of each process included in the kinetic model, Fig. 6 shows the destruction and formation rates for H2 as a function of z. It should be noted that the dissociation of H2 via H+ collisions (process labelled `6d' in the figure) is one of the most efficient destruction mechanisms, although it is usually neglected. For the formation channels, the effect of non-thermal photons is most evident in the channel of associative detachment (process `2f '). 5 C ONCLUSIONS We followed the formation and destruction of the main molecules and molecular ions in the early Universe, focusing on the effect

Figure 6. H2 destruction (top panel) and formation (bottom panel) rates as a function of redshift z. Destruction processes: D+ + H2 HD + H+ (1d); D + H2 HD + H (2d); H+ + H2 H+ + H (3d); H2 + h 2 2H (indirect, 4d); H2 + H 3H (5d); H2 + H+ 2H + H+ (6d); H2 + e H + H- (7d); H2 + e 2H + e(8d);H2 + h H+ + e (9d); 2 H+ + H2 H+ + h (10d). Formation processes: H+ + H H2 + H+ 3 2 (1f); H- + H H2 + e (2f); HD + H+ D+ + H2 (3f); HD + H D + H2 (4f); 2H + H H2 + H (5f); H+ + H- H + H2 (6f). 2

of non-thermal photons produced by the recombination of H and He and by the non-equilibrium formation of H2 . We computed the changes in the fractional abundances of H2 , H- , H+ , H+ and on 2 3 deuterated species such as D- and HD. We find that because of high-energy tails in the photon spectra at several z, the efficiency of photodestruction is greatly enhanced, yielding lower fractional elemental abundances than in the standard thermal treatment of the chemical kinetics. We also showed that the inclusion of vibrational levels in the calculation of reaction rates is critical for their determination at high temperatures when excited levels are more populated. At high z, where these conditions apply, the resulting fractional abundances of H2 and H+ are reduced by a factor of 10. However, if used 2 in other environments where molecular hydrogen is more abundant (e.g. during the collapse of primordial clouds), these new rates are expected to affect more significantly the final molecular abundances.

A C KNO W LEDGEMENTS We acknowledge the referee Steve Lepp for a careful reading of the paper. We are very grateful to Anatolij A. Mihajlov and


Primordial chemistry
Lj. M. Ignjatovic (Insitute of Physics, University of Belgrade) for providing photodissociation cross-section data computed over a wide range of wavelengths and temperatures. CMC and SL ` acknowledge financial support of MIUR-Universita degli Studi di Bari (`fondi di Ateneo 2010') and MIUR-PRIN (grant no. 2010ERFKXL). JC was supported by DoE SC-0008108 and NASA NNX12AE86G. This work has also been partially supported by the FP7 project `Phys4Entry' - grant agreement n. 242311.

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REFERENCES
Abel T., Anninos P., Zhang Y., Norman M. L., 1997, New Astron., 2, 181 (AAZN97) Abel T., Bryan G. L., Norman M. L., 2002, Sci, 295, 93 Argyros J. D., 1974, J. Phys. B, 7, 2025 Bovino S., Tacconi M., Gianturco F. A., Galli D., 2011, A&A, 529, A140 (BTGG11) Bromm V., Coppi P. S., Larson R. B., 2002, ApJ, 564, 23 Capitelli M., Coppola C. M., Diomede P., Longo S., 2007, A&A, 470, 811 (CCDL07) Chen X., Kamionkowski M., 2004, Phys. Rev. D, 70, 043502 Chluba J., 2010, MNRAS, 402, 1195 Chluba J., Sunyaev R. A., 2006, A&A, 458, L29 Chluba J., Sunyaev R. A., 2007, A&A, 475, 109 Chluba J., Sunyaev R. A., 2010, MNRAS, 402, 1221 Chluba J., Sunyaev R. A., 2012, MNRAS, 419, 1294 Chluba J., Thomas R. M., 2011, MNRAS, 412, 748 ~ Chluba J., Rubino-Mart´ J. A., Sunyaev R. A., 2007, MNRAS, 374, 1310 in Ciardi B., Ferrara A., 2001, MNRAS, 324, 648 Coppola C. M., Longo S., Capitelli M., Palla F., Galli D., 2011, ApJS, 193, 7 (C11) Coppola C. M., D'Introno R., Galli D., Tennyson J., Longo S., 2012, ApJS, 199, 16 i ´ C´zek M., Horacek J., Domcke W., 1998, J. Phys. B: At., Mol. Opt. Phys., 31, 2571 Dalgarno A., 2005, J. Phys.: Conf. Ser., 4, 10 Dickinson A. S., 2008, J. Phys. B: At. Mol. Opt. Phys., 41, 049801 (DI08) Dunn G., 1968, Phys. Rev., 172, 1 Fixsen D. J., 2009, ApJ, 707, 916 Fixsen D. J., Cheng E. S., Gales J. M., Shafer R. A., Wright E. L., 1996, ApJ, 473, 576 Flower D., 2000, A&A, 362, 774 Galli D., Palla F., 1998, A&A, 335, 403 (GP98) Galli D., Palla F., 2002, Planet. Space Sci., 50, 1197 (GP02) Gong Y., Cooray A., Santos M. G., 2013, ApJ, 768, 130 Hirata C. M., Padmanabhan N., 2006, MNRAS, 372, 1175 Iocco F., 2012, Mem. Soc. Astron. Ital. Suppl., 22, 19 Ju M., Spirko V., Kraemer W. P., 1995, Chem. Phys., 193, 287 (JSK95) rek Komatsu E. et al., 2011, ApJS, 192, 18 i Kreckel H., Bruhns H., C´zek M., Glover S. C. O., Miller K. A., Urbain X., Savin D. W., 2010, Sci., 329, 69 ´ Krstic P. S., 2002, Phys. Rev. A, 66, 042717 ´ Krstic P. S., 2003, Phys. Rev. A, 66, 022708 ´ Krstic P. S., 2005, Nucl. Instr. Meth. Phys. Res. B, 241, 58 ´ Krstic P. S., Schultz D. R., 1999, J. Phys. B: At., Mol. Opt. Phys., 32, 2415 ´ Krstic P. S., Schultz D. R., Janev R. K., 2002, Phys. Scripta, T96, 61 Lebedev V. S., Presnyakov L. P., Sobel'man I. I., 2000, Astron. Rep., 11, 5, 338 Lebedev V. S., Presnyakov L. P., Sobel'man I. I., 2003, Phys.-Usp., 46, 5, 473 Lepp S., Stancil P. C., Dalgarno A., 2002, J. Phys. B, 35, 57 (LSD02) Longo S., Coppola C. M., Galli D., Palla F., Capitelli M., 2011, Rend. Fis. Accad. Lincei, 22, 119 ´ ´ Mihajlov A. A., Ignjatovic Lj. M., Sakan N. M., Dimitrijevic M. S., 2007, A&A, 469, 2, 749

Motapon O., Tamo F. O. W., Urbain X., Schneider I. F., 2008, Phys. Rev. A, 77, 5 Padmanabhan N., Finkbeiner D. P., 2005, Phys. Rev. D, 72, 023508 Pagel B. E., 1959, MNRAS, 119, 609 Palla F., Salpeter E. E., Stahler S. W., 1983, ApJ, 271, 632 (PSS83) Peebles P. J. E., Seager S., Hu W., 2000, ApJ, 539, L1 ~ Rubino-Martin J. A., Chluba J., Sunyaev R. A., 2006, MNRAS, 371, 1939 ~ Rubino-Mart´ J. A., Chluba J., Sunyaev R. A., 2008, A&A, 485, 377 in Savin D. W., 2002, ApJ, 566, 599 (SA02) Schleicher D. R. G., Galli D., Palla F., Camenzind M., Klessen R. S., Bartelmann M., Glover S. C. O., 2008, A&A, 490, 521 (S08) Schlemmer S., 2011, Angew. Chem., Int. Ed., 50, 2214 Stancil P. C., 1994, ApJ, 430, 360 Stancil P. C., Lepp S., Dalgarno A., 1998, ApJ, 509, (SLD98) Sunyaev R. A., Chluba J., 2009, Astron. Nachr., 330, 657 Takagi, H., 2002, Phys. Scripta, T96, 52 Tegmark M., Silk J., Rees M. J., Blanchard A., Abel T., Palla F., 1997, ApJ, 474, 1 Trevisan C. S., Tennyson J., 2002, Plasma Phys. Control. Fusion, 44, 1263 (TT02) Wishart A. W., 1979, MNRAS, 187, 59p Wolniewicz L., Simbotin I., Dalgarno A., 1998, ApJS, 115, 293 Zygelman B., Dalgarno A., Kimura M., Lane N. F., 1989, Phys. Rev. A, 40, 2340 (ZDKL89) Zygelman B., Stancil P. C., Dalgarno A., 1998, ApJ, 508, 151 (ZSD98)

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APPENDIX A : F ITTING FORMULAE FOR T HE RATE COEFFICIENTS We fitted the reaction rate coefficients for the photodissociation of H+ and the photo detachment of H- with logarithmic polynomials 2 of the form log k (Tr ) =
n

an (log Tr )n .

(A1)

The coefficients of equation (A1) are given in Table B1, together with the complete set of reaction rates employed in the kinetic model. The temperature of gas and radiation are indicated by Tg and Tr , respectively, and are expressed in K. Natural logarithms are indicated as ln, logarithms to base 10 as log . The results of the fitting formulae are compared to the numerical results in Figs A1 and A2.

Figure A1. Fits (dashed lines) for the contribution of non-thermal photons to the photodissociation of H+ . Upper curve: primordial hydrogen recom2 bination contribution; lower curve: H2 radiative cascade contribution.


120

C. M. Coppola et al.

Figure A2. Fits (dashed lines) for the contribution of non-thermal photons to the photodetachment of H- . Upper curve and x-axis: primordial hydrogen recombination contribution; lower curve and x-axis: H2 radiative cascade contribution. Their validity corresponds to the temperature ranges of each axis.

Figure B1. H- photodetachment: effect of DM annihilation on the reaction rate. Calculations are reported for different annihilating efficiency, namely -23 , 10-24 , 5 â 10-24 eV s-1 . 0 = 10

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on the thermally averaged cross-section v for that process: APPENDIX B: D ARK MATTER (DM) ANNIHILATION Dark matter annihilation or decay leads to extra ionizations of hydrogen and helium atoms in the early Universe (Chen & Kamionkowski 2004; Padmanabhan & Finkbeiner 2005), delaying the cosmological recombination process (Peebles, Seager & Hu 2000) and causing emission of extra recombination photons (Chluba 2010). The additional injection of energy and photons should be taken into account when considering the physical phenomena occurring in the primeval plasma as well as the chemistry. Here, we evaluate the effect of extra photons produced by the direct reprocessing of annihilation energy by hydrogen on the rate coefficient of H- photodetachment. Details on the equations employed and their derivation can be found in Chluba (2010). The energy release associated with the annihilation of some DM particle with its antiparticle depends on the mass of the particles ¯ involved, on the fractional abundances of particle/antiparticle and
Table B1. Reaction rates. Process (1) H + e- - H- + h (2) + - H + (3) H- + H - 2H + e- (4) H- + H+ - 2H H- e- H- e- 2e- Reaction rates [MKS] 1.4 â 10-24 Tg0.
928 e-Tg /16 200

dE M c2 v N N (eV cm-3 s-1 ). (B1) ¯ dt Here, results are given as a function of the annihilating efficiency of the particle/antiparticle pair 0 : dEd = dt
0

NH (z)(1 + z)3 .

(B2)

In Fig. B1, we show the cases 0 = 10-23 ,10-24 and 5 â 10-24 eV s-1 . The presence of extra photons increases the H- photodetachment rate considerably: the larger the annihilating efficiency the stronger the destruction process. In particular, in the range of annihilating efficiencies shown in the figure and for z < 70, the enhancement to the rate coefficient scales as: f ( 0 ) 1 + 0.44( 0 /10-24 ). (B3)

This implies that the early Universe chemistry is not only sensitive to direct ionizations induced by the annihilation products, but also to the reprocessed energy causing additional ionizations of abundant neutral hydrogen atoms and reemission of Ly photons.

Ref

a

GP98 AAZN97 AAZN97 LSD02 GP98 This work

Fit from reference Fit from reference 1.40 â 10-13 (Tg /300)-

0.487 eTg /29 300

(5) + h - H + Thermal Non-thermal: atom. recombin.

Non-thermal: H2 radiative cascade

0.11Tr 2.13 e-8823/Tr log k = 6 =0 an (log Tr )n n a0 =-26.6463 a1 = 3.359 98 a2 = 25.729 a3 =-31.6442 a4 = 15.9545 a5 =-3.600 13 a6 = 0.298 272 log k = 5 =0 bn (logTr )n n b0 =-81.12

This work


Primordial chemistry
Table B1 Reaction rates. Process Reaction rates [MKS] b1 b2 b3 b4 b5 As = 139.379 =-137.531 = 73.0553 =-19.4282 = 1.997 68 fit for reaction (5)
32 400/Tr 32 400/Tr

121

Ref

a

(6) D- + h - D + e- (7) HD+ + h - D + H+ (8) HD+ + h - D+ + H (9) HD+ + h - H+ + D+ + e- (10) HD + h - HD + (11) D + H+ - D+ + H (12) D+ + H - D + H+ (13) D + H - HD + h (14) HD + H - HD +
+ +

S08 S08 S08 S08 S08 - 3.31 â 10-23 Tg
1.48 -0.332

(1/2) â 1.63 â 107 e- (1/2) â 1.63 â 107 e- 9.0 â 101 Tr 2.9 â 10 Tr
2

1.48 -335 000/Tr e 1.56 -178 500Tr e

e-

2.0 â 10 10-32

-16

Tg0.

402 e-37.1/Tg

SA02 SA02 DI08 SLD98
2

2.06 â 10

-16

Tg

0.396 -33.0/Tg e

+ 2.03 â 10-15 Tg

H+

[2.259 - 0.6(Tg /10 + 0.101(Tg /10 -0.015 35(Tg /103 )-2 + 5.3 â 10-5 (Tg /103 )] 6.4 â 10-16
3 )0.5

3 )-1.5 -3

(15) D + H+ - HD+ + h (16) D+ + H - HD+ + h (17) HD+ + e- - D + H (18) D + - + h (19) D+ + D- - 2D (20) H+ + D- - D + H (21) H- + D - H + D- (22) D- + H - D + H- (23) D- + H - HD + e- (24) D + H- - HD + e- (25) H- + D+ - D + H (26) He + H+ - He+ + H
+

log k/10-6 =-19.38 - 1.523 log Tg + 1.118(log Tg ) -0.1269(log Tg )3 As fit for reaction (15) 7.2 â 10-14 Tg- 10-22
0.5

Downloaded from http://mnras.oxfordjournals.org/ at inaf on August 16, 2013

GP98 GP98 SLD98 SLD98 LSD02 LSD02 SLD98 SLD98 SLD98 S08 LSD02

e-

D-

3. 0 â (Tg /300)0.95 e-Tg /9320 -13 (T /300)-0.487 eTg /29 1.96 â 10 g 1.61 â 10-13 (Tg /300)- 6.4 â 10 6.4 â 10 1.5 â 10
-15 -15 -15

300

0.487 eTg /29 300

(Tg

/300)0.41
0.1

(Tg /300)0.41 (Tg /300)-

As fit for reaction (22) 1.61 â 10-13 (Tg /300)-0.487 eTg /29 300 4.0 â 10-43 Tg4.74 for Tg > 104 1.26 â 10-15 Tg-0.75 e-127 500/Tg for Tg < 10 10-21 /300)0.25

4

S08 ZDKL89 SLD98 JSK95, ZSD98 SLD98 SLD98 JSK95 GP98 C11
2

(27) He + H - He + 1.25 â (Tg (28) He + H+ - HeH+ + h 8.0 â 10-26 (Tg /300)-0.24 e-Tg /4000 (29) He + H+ + h - HeH+ + h 3.2 â 10-26 Tg1.8 e-Tg /4000 (1 + 2 â 10 (30) He + H - HeH + h (31) He+ + e- - He + H (32) HeH+ + h - He + H+ (33) HeH+ + h - He+ + H (34) + H - H2 + (35) H+ + H - H+ + h 2 (36) H2 + + H - H2 + H+ (37) 2H + H - H2 + H (38) H2 + H+ - H+ + H 2 H- e-
+ +

H+

-4

Tr1.1 )(1 + 0.1Tg2.04 )

-1

4

3 2 7.8 â 103 Tr1.2 e-

.16 â Tg-0.37 e-Tg /87 600 -14 (T /300)-0.47 .0 â 10 g .20 â 102 e-22 740/Tr 10-22
240 000/Tr

log k =-14.4 - 0.15(log Tg )2 - 7.9 â 10-3 (log Tg )4 log (k/10-6 ) = -19.38 - 1.523 log Tg + 1.118(log Tg ) -0.1269(log Tg )3 6.4 â 10-16 5.5 â 10-35 Tg-1 ln k = a0 + a1 Tg + a2 Tg-1 + a3 Tg2 a0 =-33.081 a1 = 6.3173 â 10-5 a2 =-2.3478 â 104 a3 =-1.8691 â 10-9 1.91 â 10 6.9 â 10 9.6 â 10 k a a a a a a
0 -15

GP98 GP98 PSS83 C11

(39) H2 + e- - 2H + e- (40) H- + H+ - H+ + e- 2 (41) H2 + + e- - 2H

Tg0.

136 e-53 407.1/Tg

TT02 GP98 C11

-15 -13

Tg- Tg-

0.35 0.9

for Tg < 8000 for Tg > 8000

= = 4.2278 â 10-14 -17 1 =-2.3088 â 10 -21 2 = 7.3428 â 10 -25 3 =-7.5474 â 10 -29 4 = 3.3468 â 10 -34 5 =-5.528 â 10

5 n n=0 an Tg


122

C. M. Coppola et al.
Table B1 ­ continued Process (42) H2 + + H- - H + H2 (43) H2 + e- - H + H- (44) H2 + + h - H + H+ Thermal Non-thermal: atom. recombin. Reaction rates [MKS] 5 â 10
-12

Ref

a

3.67 â 10

Tg-0.5 for Tg < 100 -5 T -2.28 e-47 172/Tg g

AAZN97 CCDL07 GP98 This work

Non-thermal: H2 radiative cascade

1.63 â 107 e-32 400/Tr log k = 5 =0 an (logTr ) n a0 =-257.413 a1 = 294.406 a2 =-93.7846 a3 =-13.1607 a4 = 11.3683 a5 =-1.467 34 log k = 5 =0 bn (logTr ) n b0 =-1084.08 b1 = 1990.32 b2 =-1447.42 b3 = 503.994 b4 =-83.6462 b5 = 5.288 98 3.065 87 â 109 e- 9 â 10
1

n

n

This work

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(45) H2 + h - H2 + + e- (46) (47) H2 + h - H2 - 2H (48) H2 + H H + H + H (49) D + H2 - HD + H (50) D+ + H2 - HD + H+ (51) HD + H - D + H2 (52) HD + H+ - D+ + H2 H+ 2 + h - 2H+ + e-

18 948.1/Tr

C11 GP98
1

Tr1.48 e-335 000/Tr

ln k = 17.555 + 7.2643 â 10-6 Tr - 1.4194 â 105 Tr- 1.9535 â 10-10 Tg-0.932 67 e-497 43/Tg 1.69 â 10-16 e- 9 â 10 10
-15
2 4680Tg +198 800/Tg

C11 C11 GP02 GP98 GP02 GP02 GP98 GP02 GP98 BTGG11 GP98 GP98 C11

for Tg > 200

-17 e-3876/Tg

for Tg < 200 for Tg > 200

[0.417 + 0.846log Tg - 0.137(log Tg )2 ]
2 -17 e-4430/Tg +173 900/Tg

5.25 â 10 3.2 â 10 1.1 â 10 3 â 10 10 ln a0 a1 a2 a3 ln a0 a1 a2 a3
-18

-17 e-3624/Tg -15 e-488/Tg

for Tg < 200

(53) He + H+ - HeH+ + H 2 (54) HeH+ + H - He + H+ 2 (56) H+ + e- - H + H2 3 (55) H+ + H2 - H+ + h 3

-16 e-6717/Tg 110 373 e-31.5396/Tg

4.3489 â 10-16 Tg0. 4.6 â 10
-12

(57) H+ + H - H + H+ + H 2

Tg-

0.65

(58) H2 + H+ - H + H + H+

k = a0 + a1 Tg + a2 Tg-1 + a3 Tg2 =-32.912 = 6.9498 â 10-5 =-3.3248 â 104 =-4.08 â 10-9 k = a0 + a1 Tg + a2 Tg-1 + a3 Tg2 =-33.404 = 2.0148 â 10-4 =-5.2674 â 104 =-1.0196 â 10-8

C11

a

References: Galli & Palla (1998) ­ GP98; Abel et al. (1997) ­ AAZN97; Lepp, Stancil & Dalgarno (2002) ­ LSD02; Schleicher et al. (2008) ­ S08; Savin (2002) ­ SA02; Dickinson (2008) ­ DI08; Stancil, Lepp & Dalgarno (1998) ­ SLD98; Zygelman (1989) ­ ZDKL89; JurZek, SZpirko & Kraemer (1995) ­ JSK95; Zygelman, Stancil & Dalgarno (1998) ­ ZSD98; Coppola et al. (2011) ­ C11; Palla, Salpeter & Stahler (1983) ­ PSS83; Trevisan & Tennyson (2002) ­ TT02; Capitelli (2007) ­ CCDL07; Galli & Palla (2002) ­ GP02; Bovino et al. (2011) ­ BTGG11.

A This paper has been typeset from a TEX/L TEX file prepared by the author.