Draft version November 4, 2018
Preprint typeset using LATEX style emulateapj v. 08/22/09
FEEDBACK FROM FIRST RADIATION SOURCES: H− PHOTODISSOCIATION
Leonid Chuzhoy
, Michael Kuhlen
and Paul R. Shapiro
Draft version November 4, 2018
ABSTRACT
During the epoch of reionization, the formation of radiation sources is accompanied by the growth
of a H− photodissociating flux. We estimate the impact of this flux on the formation of molecular
hydrogen and cooling in the first galaxies, assuming different types of radiation sources (e.g. Pop
II and Pop III stars, miniquasars). We find that H− photodissociation reduces the formation of H2
molecules by a factor of Fs ∼ 1 + 10
3ksxf
esc δ
−1, where x is the mean ionized fraction in the IGM,
fesc is the fraction of ionizing photons that escape from their progenitor halos, δ is the local gas
overdensity and ks is an order unity constant which depends on the type of radiation source. By the
time a significant fraction of the universe becomes ionized, H− photodissociation may significantly
reduce the H2 abundance and, with it, the primordial star formation rate, delaying the progress of
reionization.
Subject headings: cosmology: theory – early universe – galaxies: formation – galaxies: high redshift
1. INTRODUCTION
The first stars in the ΛCDM universe are believed
to have formed inside dark-matter-dominated minihalos
filled with mostly neutral, metal-free gas of virial tem-
perature Tvir < 10
4 K, when H2 molecules formed in
sufficient abundance to cool the gas radiatively to ∼ 102
K. If, as currently thought, these stars were massive, hot,
and luminous, they may have contributed significantly to
the reionization of the universe, which CMB polarization
observations by WMAP indicate was highly ionized by
z ∼ 10 (Spergel et al. 2006). The release of ionizing UV
radiation by minihalos and other sources (e.g. stars in
more massive halos, with Tvir > 10
4 K, or miniquasars),
required to explain reionization, must have been accom-
panied by radiation release at energies below the H Ly-
man limit, as well, however. This may, in turn, have lim-
ited the H2 abundance inside minihalos and their ability
to form stars, thereby limiting their contribution to cos-
mic reionization.
In the absence of dust and at densities below the three-
body formation regime (n . 1010 cm−3), the most im-
portant reaction for the production of H2 is
H− +H → H2 + e
−, (1)
(e.g., Shapiro & Kang 1987 and refs. therein) with re-
action rate k− = 1.3× 10
−9 cm3 s−1 (Schmetekopf et al.
1967). Once formed, H2 can be destroyed by collisions
with other species
H2 +H
+ → H+2 +H, (2)
H2 +H → H+H+H, (3)
H2 + e
− → H+H+ e−, (4)
or by photodissociation via Lyman-Werner band photon
absorption
H2 + γ → H+H. (5)
1 McDonald Observatory and Department of Astronomy, The
University of Texas at Austin, RLM 16.206, Austin, TX 78712,
USA; chuzhoy@astro.as.utexas.edu
2 Institute for Advanced Study, Princeton, NJ, 08540, USA;
mqk@ias.edu
The latter process becomes dominant once a substan-
tial UV background is built up between 912 and 1110
Å, providing a feedback mechanism against the forma-
tion of new radiation sources (e.g. Haiman et al. 1997;
Haiman et al. 2000; Ciardi et al. 2000; Machacek et al.
2001; Mesinger et al. 2006).
In this paper we explore the impact of another feedback
mechanism, the photodissociation of H−,
H− + γ → H+ e−. (6)
The cross-section for photodissociation of H− is well fit-
ted by (Wishart 1979)
σ−(ǫ) = 2.1× 10
−16 (ǫ− 0.75)
ǫ3.11
cm2, (7)
where ǫ is the photon energy in eV. The cross section is
zero below a threshold of ǫ < 0.755eV, the binding en-
ergy of the second electron. In the absence of the UV
background, the primary mode of H− destruction is the
formation of H2 (Eq. [1]),
3 so introducing the H− pho-
todissociating flux reduces the H2 formation rate by a
factor
Fs = 1 +
, (8)
where ζ− =
nγ(ǫ)σ−(ǫ)cdǫ is the photodissociation rate
per H− ion, nH is the hydrogen atom number density
and nγ(ǫ) is the number density of photons with energy
ǫ.4 Hence the importance of this mechanism depends pri-
marily on the local density ratio of H− photodissociating
photons and hydrogen atoms.
3 When gas fractional ionization is high (x & 0.01) mutual neu-
tralization with H+ can provide another efficient channel for H−
destruction. However, typically the fractional ionization of mini-
halos is much lower.
4 This approximation for Fs breaks down when its value exceeds
∼ 50, since for such UV intensities H+ +H → H
+ γ reaction
becomes a dominant channel of H2 production (assuming reaction
rates given by Shapiro & Kang (1987)). Note also that k
is still
uncertain to within a factor of a few (see Glover et al. 2006), and
this uncertainty carries over to Fs when Fs ≫ 1.
http://arxiv.org/abs/0704.0426v2
The impact of H− photodissociation differs from that
of H2 by two fundamental characteristics. First, the time
required for H− abundance to approach equilibrium is
very short (typically less than 10000 years), while for
H2 the equilibration time can exceed the Hubble time.
Therefore, when gas is exposed to a transient UV flux,
produced by nearby Pop III stars, for example, H− pho-
todissociation can generally be ignored, as it does not
affect the subsequent thermal and chemical evolution.
Secondly, photons that make up the H2 photodissociat-
ing background are destroyed after a few percent of the
Hubble time, as they redshift into one of the Lyman se-
ries resonances, and must be replenished continuously.
By contrast, photons that constitute the H− photodis-
sociating background are very rarely destroyed, which
allows them to accumulate over time. Consequently the
importance of H− photodissociation increases over time,
and as we show in this paper, by the time a significant
(∼ 10 %) fraction of the Universe is ionized, H− pho-
todissociation may result in a drastic reduction of the
molecular hydrogen abundance. This in turn may lead
to a reduced star formation rate and delay the progress
of reionization.
Recently, Glover (2007) considered the suppression of
H2 formation due to the photodissociation of H
− and
H+2 . Whereas Glover (2007) focused on the local feed-
back around and inside HII regions created by Pop III
stars, we treat the problem globally and also consider
long range effects due to the much lower optical depth of
the universe below the Lyman limit.
The paper is organized as following. In §2 and 3, we
estimate the intensity of H− photodissociating flux pro-
duced by UV and X-ray sources, respectively. In §4, we
discuss the implication of our results for gas cooling in
minihalos.
2. H− PHOTODISSOCIATING BACKGROUND - UV
SOURCES
2.1. Recombination products
Since the first radiation sources are expected to form
within overdense gas clouds, only the escaping fraction
of their ionizing photons, fesc, was available for ioniza-
tion of the diffuse IGM. The rest was absorbed within
the host halos and, via the process of radiative recombi-
nation, converted into lower energy UV photons. Since
the universe during that epoch is transparent to most
non-ionizing UV photons, 5 almost all of them add to
the H− photodissociation background.
Neglecting recombinations in the diffuse IGM, the
mean ionization is x = Nibfesc, where Nib is the to-
tal number of ionizing photons per baryon produced up
to this point. Inside halos, the recombination time is
quite short, and so the number of ionizations taking place
there, Nib(1− fesc) = x(1− fesc)/fesc, is almost equal to
5 An exception occurs for photons whose frequency is close to
one of the high (n > 2) Lyman resonances, which, following their
absorption by hydrogen atoms, are further split into two or more
lower energy photons. For Lyα photons, the optical depth is also
very high, but in their case the absorption in almost all cases is
followed by reemission, with the destruction probability being ex-
tremely low (e.g. Furlanetto & Pritchard 2006). Also, at the very
early stage of reionization (x ≪ 1) the presence of H2 molecules
makes the universe opaque in the Lyman-Werner range. However,
since their initial abundance (∼ 10−6) is already very low, the
number of photons they destroy is negligible.
the number of electron recombinations to n ≥ 2 states
(i.e., recombinations which do not result in emission of
additional ionizing photons), Nrec. Therefore the average
H− photodissociating rate is given by
ζ− = Nrecnbarc〈σ−〉 = xnbarc〈σ−〉
1− fesc
, (9)
where nbar is the mean baryon density and 〈σ−〉 is the av-
erage cross-section per recombination photon times the
average number of photons per recombination, 〈σ−〉 =
(jǫ/αrecnenpǫ)σ−(ǫ)dǫ. Note that since emissivity, jǫ,
is proportional to nenp, 〈σ−〉 is in fact independent of
ne and np. Using Osterbrock’s (1989, Sec. 4.3) calcula-
tion of the recombination spectrum, (jǫ/αrecnenp), and
assuming that the temperature of the recombining gas is
close to 104 K, we find 〈σ−〉 = 3.4× 10
−17cm2.
By combining equations (8) and (9), we can estimate
the importance of the H− photodissociation due to re-
combination radiation. Assuming that most of the re-
combinations occurred recently, we find that, the recom-
bination radiation alone will suppress the H2 formation
rate by
Fs = 1 + 800δ
(1− fesc), (10)
where δ = 1.08nH/nbar is the local overdensity. Here we
have neglected recombinations in the diffuse intergalactic
medium (IGM) and the associated H− dissociating pho-
tons from these recombinations, but these would only
further increase Fs.
Cosmological redshift can affect the photodissociation
rate by shifting the spectrum to longer wavelengths.
Initially this leads to an increase in 〈σ−〉 due to the
ǫ−3/2 dependence of the cross-section for ǫ ≫ 0.755
eV. Eventually, as more and more of the spectrum is
shifted below the threshold, the cosmological redshift
begins to decrease the dissociation rate. For recombi-
nation photons this redshift effect is small, and the tran-
sition to 〈σ−〉-depression occurs at a redshift factor of
(1 + zi)/(1 + z) ≈ 2.5, see Figure 1.
2.2. Direct emission
Unlike ionizing photons, whose intensity is heavily at-
tenuated both in stellar atmospheres and in their host
galaxies, most of the photons with frequencies below the
Lyman limit escape freely into the IGM. From then on,
photons with frequency below Lyβ undergo no evolution
apart from cosmological redshift. By contrast, within a
small fraction of the Hubble time, most photons with
frequency between Lyβ and the Lyman limit are split by
cascade into two or more photons after being redshifted
into one of the hydrogen resonances. Most of the cas-
cade products, which include lines such as Lyα, Hα, and
Hβ, as well as a continuum spectrum produced by the
two photon transition 2s → 1s, are above the 0.755 eV
threshold for H− photodissociation.
The relative importance of these directly emitted H−
dissociating photons depends on the nature of the UV
sources. Figure 2 shows the increase of the H− dissocia-
tion rate due to inclusion of direct emission from metal-
poor Pop III stars, which we calculated using the stellar
atmosphere models of Schaerer (2002). Predictably, for
1 1.5 2 2.5 3 3.5 4
)/(1+z)
Fig. 1.— Redshift evolution of the average H− photodissociation
cross-section of the UV photons produced by recombination (dot-
ted line), excitations by non-thermal electrons (dashed line) and
massive Pop III stars (solid line).
4 6 8 10 12 14
Fig. 2.— The ratio between the total H− photodissociation rate
and the photodissociation by recombination products alone for star
with different effective surface temperature.
very massive Pop III stars, with surface temperatures
∼ 105 K, adding the stellar continuum below the Lyman
limit to the recombination spectrum increases the pho-
todissociation rate by only ∼ 10%. If, on the other hand,
most of the early ionizing flux was produced by stars
with masses below 10M⊙, whose continuum emission is
stronger at lower frequencies, then the total H− disso-
ciation rate would be tripled at least. Likewise, direct
emission may be important if most of the UV photons
were produced by miniquasars. For example, assuming
that their spectrum can be approximated by a power
law, Lν ∝ ν
−1.7, with a cutoff below 0.75 eV, adding the
directly emitted photons to the recombination products
increases the total photodissociation rate by a factor of
3. H− PHOTODISSOCIATING BACKGROUND - X-RAY
SOURCES
It has been suggested that X-ray photons could con-
tribute a large fraction of the energy emitted by the
first radiation sources (e.g. Ricotti & Ostriker 2004).
By increasing the number of free electrons, X-rays can
boost the production of H−, and thus of H2, provid-
ing a positive feedback to the formation of new sources
(Haiman et al. 2000; Kuhlen & Madau 2005). This ef-
fect, however, would be at least partially offset by an in-
crease of the H− photodissociating background, caused
by conversion of X-rays into UV photons.
The absorption of an X-ray photon is followed by re-
lease of a non-thermal electron, which then loses some of
its energy by inelastic collisions with atoms before it can
thermalize its energy by elastic scattering with ions and
other electrons. When the gas ionization fraction is low
(x . 0.05), the photoelectron splits most of its energy
evenly between collisional ionizations and excitations of
hydrogen atoms (Shull & van Steenberg 1985). Using
electron-hydrogen excitation cross-sections (Grafe et al.
2001; Stone et al. 2002), we find that around ∼ 5/6 of
the excitations are to the 2p level, which are followed by
emission of a Lyα photon. Most of the remaining excita-
tions are to the 3p level, which decays via emission of one
Hα photon and a subsequent two-photon decay from the
2s level. The Lyα, Hα and two-photon continuum each
produce roughly equal contributions to H− photodisso-
ciation. Per ionization, the average intensity-weighted
cross-section for these photons is 〈σ−〉 = 1.6×10
−17cm2.
Due to the low number of UV photons produced during
this phase, the formation of H2 is not strongly affected
Fs = 1 + 4δ
. (11)
After the ionized fraction climbs above x ∼ 0.05, most
of the energy of the non-thermal electrons is converted
to heat. However, simultaneously with the growth of the
ionized fraction, the temperature of the gas rises, and as
it crosses 104 K, the collisions between thermal electrons
and atoms begin to dissipate the energy added by X-
rays, mainly via emission of Lyα photons. Neglecting
gas clumping, we find that the number of emitted Lyα
photons per hydrogen atom is
Nα = 4.6× 10
−8 cm3 s−1
x(1 − x)e−1.18×10
5/T nH dt.
Assuming for simplicity that x and T are constants, we
can rewrite the equation (12) as
Nα = 11.2
(τe,X
(1− x)e−1.18×10
, (13)
where τe,X =
xnσT dt is the Thompson optical depth
from the epoch of partial ionization by X-rays. If X-
ray preionization contributes at least half of the τe ∼ 0.1
measured by WMAP (i.e. τe,X ≈ 0.05), hydrogen atomic
de-excitations in the diffuse IGM may produce & 30 Lyα
photons per baryon.
The suppression of H2 formation due to H
− photodis-
sociation by Lyα photons is
Fs ≈ 1 + 100Nαδ
−1. (14)
Since the energy of Lyα photons (10.2 eV) is far above
the H− photodissociation threshold (0.75 eV), the pho-
todissociation rate grows roughly as (1+zi)
1.5/(1+z)1.5,
where zi is the redshift at which the photon was emitted.
In the case of an extended period of partial ionization, Fs
may be increased by a factor of a few, possibly exceeding
104δ−1.
Since, when the IGM temperature rises above 104 K,
the formation of new minihalos is suppressed, the impact
of H− photodissociating flux produced by X-ray conver-
sion is relevant only for minihalos which have formed
some time ago or for halos with Tvir > 10
4 K, which also
rely on H2 cooling to form stars.
4. DISCUSSION
As shown by our calculations, H− photodissociation
reduces the formation of H2 molecules by a factor of
Fs ∼ 1 + 10
3ksxf
esc δ
−1, (15)
where ks is a constant of order a few, whose value de-
pends on the type of radiation source and the growth
history of the radiation background. Thus, by the time a
significant fraction (& 0.1) of the universe becomes ion-
ized, H− photodissociation can significantly reduce the
H2 formation rate in regions with overdensities of up to
a few thousands, i.e. in the interior regions of miniha-
los. The equilibrium abundance of molecular hydrogen
during this stage would be determined by the balance be-
tween its formation and destruction rates (Eqs. [1] and
nH2 =
k− nH nH−
, (16)
where kLW is the H2 destruction rate by the Lyman-
Werner photons. Thus a reduction of H− abundance by
a factor Fs translates into the same reduction of the H2
abundance and, in minihalos, a comparable increase of
the cooling time.
Indirectly, H− photodissociation may affect the cool-
ing in the central regions of minihalos even during the
early stages of reionization. The maximum density that
gas can reach in the core region of a minihalo is lim-
ited by the amount of entropy it is able to radiate away
during collapse. The lower density gas prevalent dur-
ing the early collapse phase would be susceptible to H−
dissociation from even a relatively low intensity H− dis-
sociating flux, and the resulting lowered H2 abundance
would limit its ability to radiate away entropy via H2
cooling. Furthermore, the density and H2 abundance at
the center depend on the conditions in the low density
outer regions, through their contributions to both the to-
tal pressure and the self-shielding ability of the halo. We
plan to investigate these effects further with numerical
radiation-hydrodynamic simulations in the future.
LC thanks the McDonald Observatory for the W.J.
McDonald Fellowship. MK gratefully acknowledges sup-
port from the Institute for Advanced Study. This work
was partially supported by NASA Astrophysical Theory
Program grants NAG5-10825 and NNG04G177G to P.
R. S.
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http://arxiv.org/abs/astro-ph/0703716
http://arxiv.org/abs/astro-ph/0603449

  During the epoch of reionization, the formation of radiation sources is
accompanied by the growth of a H- photodissociating flux. We estimate the
impact of this flux on the formation of molecular hydrogen and cooling in the
first galaxies, assuming different types of radiation sources (e.g. Pop II and
Pop III stars, miniquasars). We find that H- photodissociation reduces the
formation of H2 molecules by a factor of ~1+1000k_s*x/(f_esc*delta), where x is
the mean ionized fraction in the IGM, f_esc is the fraction of ionizing photons
that escape from their progenitor halos, delta is the local gas overdensity and
k_s is an order unity constant which depends on the type of radiation source.
By the time a significant fraction of the universe becomes ionized, H-
photodissociation may significantly reduce the H2 abundance and, with it, the
primordial star formation rate, delaying the progress of reionization.

Draft version November 4, 2018
Preprint typeset using LATEX style emulateapj v. 08/22/09
FEEDBACK FROM FIRST RADIATION SOURCES: H− PHOTODISSOCIATION
Leonid Chuzhoy
, Michael Kuhlen
and Paul R. Shapiro
Draft version November 4, 2018
ABSTRACT
During the epoch of reionization, the formation of radiation sources is accompanied by the growth
of a H− photodissociating flux. We estimate the impact of this flux on the formation of molecular
hydrogen and cooling in the first galaxies, assuming different types of radiation sources (e.g. Pop
II and Pop III stars, miniquasars). We find that H− photodissociation reduces the formation of H2
molecules by a factor of Fs ∼ 1 + 10
3ksxf
esc δ
−1, where x is the mean ionized fraction in the IGM,
fesc is the fraction of ionizing photons that escape from their progenitor halos, δ is the local gas
overdensity and ks is an order unity constant which depends on the type of radiation source. By the
time a significant fraction of the universe becomes ionized, H− photodissociation may significantly
reduce the H2 abundance and, with it, the primordial star formation rate, delaying the progress of
reionization.
Subject headings: cosmology: theory – early universe – galaxies: formation – galaxies: high redshift
1. INTRODUCTION
The first stars in the ΛCDM universe are believed
to have formed inside dark-matter-dominated minihalos
filled with mostly neutral, metal-free gas of virial tem-
perature Tvir < 10
4 K, when H2 molecules formed in
sufficient abundance to cool the gas radiatively to ∼ 102
K. If, as currently thought, these stars were massive, hot,
and luminous, they may have contributed significantly to
the reionization of the universe, which CMB polarization
observations by WMAP indicate was highly ionized by
z ∼ 10 (Spergel et al. 2006). The release of ionizing UV
radiation by minihalos and other sources (e.g. stars in
more massive halos, with Tvir > 10
4 K, or miniquasars),
required to explain reionization, must have been accom-
panied by radiation release at energies below the H Ly-
man limit, as well, however. This may, in turn, have lim-
ited the H2 abundance inside minihalos and their ability
to form stars, thereby limiting their contribution to cos-
mic reionization.
In the absence of dust and at densities below the three-
body formation regime (n . 1010 cm−3), the most im-
portant reaction for the production of H2 is
H− +H → H2 + e
−, (1)
(e.g., Shapiro & Kang 1987 and refs. therein) with re-
action rate k− = 1.3× 10
−9 cm3 s−1 (Schmetekopf et al.
1967). Once formed, H2 can be destroyed by collisions
with other species
H2 +H
+ → H+2 +H, (2)
H2 +H → H+H+H, (3)
H2 + e
− → H+H+ e−, (4)
or by photodissociation via Lyman-Werner band photon
absorption
H2 + γ → H+H. (5)
1 McDonald Observatory and Department of Astronomy, The
University of Texas at Austin, RLM 16.206, Austin, TX 78712,
USA; chuzhoy@astro.as.utexas.edu
2 Institute for Advanced Study, Princeton, NJ, 08540, USA;
mqk@ias.edu
The latter process becomes dominant once a substan-
tial UV background is built up between 912 and 1110
Å, providing a feedback mechanism against the forma-
tion of new radiation sources (e.g. Haiman et al. 1997;
Haiman et al. 2000; Ciardi et al. 2000; Machacek et al.
2001; Mesinger et al. 2006).
In this paper we explore the impact of another feedback
mechanism, the photodissociation of H−,
H− + γ → H+ e−. (6)
The cross-section for photodissociation of H− is well fit-
ted by (Wishart 1979)
σ−(ǫ) = 2.1× 10
−16 (ǫ− 0.75)
ǫ3.11
cm2, (7)
where ǫ is the photon energy in eV. The cross section is
zero below a threshold of ǫ < 0.755eV, the binding en-
ergy of the second electron. In the absence of the UV
background, the primary mode of H− destruction is the
formation of H2 (Eq. [1]),
3 so introducing the H− pho-
todissociating flux reduces the H2 formation rate by a
factor
Fs = 1 +
, (8)
where ζ− =
nγ(ǫ)σ−(ǫ)cdǫ is the photodissociation rate
per H− ion, nH is the hydrogen atom number density
and nγ(ǫ) is the number density of photons with energy
ǫ.4 Hence the importance of this mechanism depends pri-
marily on the local density ratio of H− photodissociating
photons and hydrogen atoms.
3 When gas fractional ionization is high (x & 0.01) mutual neu-
tralization with H+ can provide another efficient channel for H−
destruction. However, typically the fractional ionization of mini-
halos is much lower.
4 This approximation for Fs breaks down when its value exceeds
∼ 50, since for such UV intensities H+ +H → H
+ γ reaction
becomes a dominant channel of H2 production (assuming reaction
rates given by Shapiro & Kang (1987)). Note also that k
is still
uncertain to within a factor of a few (see Glover et al. 2006), and
this uncertainty carries over to Fs when Fs ≫ 1.
http://arxiv.org/abs/0704.0426v2
The impact of H− photodissociation differs from that
of H2 by two fundamental characteristics. First, the time
required for H− abundance to approach equilibrium is
very short (typically less than 10000 years), while for
H2 the equilibration time can exceed the Hubble time.
Therefore, when gas is exposed to a transient UV flux,
produced by nearby Pop III stars, for example, H− pho-
todissociation can generally be ignored, as it does not
affect the subsequent thermal and chemical evolution.
Secondly, photons that make up the H2 photodissociat-
ing background are destroyed after a few percent of the
Hubble time, as they redshift into one of the Lyman se-
ries resonances, and must be replenished continuously.
By contrast, photons that constitute the H− photodis-
sociating background are very rarely destroyed, which
allows them to accumulate over time. Consequently the
importance of H− photodissociation increases over time,
and as we show in this paper, by the time a significant
(∼ 10 %) fraction of the Universe is ionized, H− pho-
todissociation may result in a drastic reduction of the
molecular hydrogen abundance. This in turn may lead
to a reduced star formation rate and delay the progress
of reionization.
Recently, Glover (2007) considered the suppression of
H2 formation due to the photodissociation of H
− and
H+2 . Whereas Glover (2007) focused on the local feed-
back around and inside HII regions created by Pop III
stars, we treat the problem globally and also consider
long range effects due to the much lower optical depth of
the universe below the Lyman limit.
The paper is organized as following. In §2 and 3, we
estimate the intensity of H− photodissociating flux pro-
duced by UV and X-ray sources, respectively. In §4, we
discuss the implication of our results for gas cooling in
minihalos.
2. H− PHOTODISSOCIATING BACKGROUND - UV
SOURCES
2.1. Recombination products
Since the first radiation sources are expected to form
within overdense gas clouds, only the escaping fraction
of their ionizing photons, fesc, was available for ioniza-
tion of the diffuse IGM. The rest was absorbed within
the host halos and, via the process of radiative recombi-
nation, converted into lower energy UV photons. Since
the universe during that epoch is transparent to most
non-ionizing UV photons, 5 almost all of them add to
the H− photodissociation background.
Neglecting recombinations in the diffuse IGM, the
mean ionization is x = Nibfesc, where Nib is the to-
tal number of ionizing photons per baryon produced up
to this point. Inside halos, the recombination time is
quite short, and so the number of ionizations taking place
there, Nib(1− fesc) = x(1− fesc)/fesc, is almost equal to
5 An exception occurs for photons whose frequency is close to
one of the high (n > 2) Lyman resonances, which, following their
absorption by hydrogen atoms, are further split into two or more
lower energy photons. For Lyα photons, the optical depth is also
very high, but in their case the absorption in almost all cases is
followed by reemission, with the destruction probability being ex-
tremely low (e.g. Furlanetto & Pritchard 2006). Also, at the very
early stage of reionization (x ≪ 1) the presence of H2 molecules
makes the universe opaque in the Lyman-Werner range. However,
since their initial abundance (∼ 10−6) is already very low, the
number of photons they destroy is negligible.
the number of electron recombinations to n ≥ 2 states
(i.e., recombinations which do not result in emission of
additional ionizing photons), Nrec. Therefore the average
H− photodissociating rate is given by
ζ− = Nrecnbarc〈σ−〉 = xnbarc〈σ−〉
1− fesc
, (9)
where nbar is the mean baryon density and 〈σ−〉 is the av-
erage cross-section per recombination photon times the
average number of photons per recombination, 〈σ−〉 =
(jǫ/αrecnenpǫ)σ−(ǫ)dǫ. Note that since emissivity, jǫ,
is proportional to nenp, 〈σ−〉 is in fact independent of
ne and np. Using Osterbrock’s (1989, Sec. 4.3) calcula-
tion of the recombination spectrum, (jǫ/αrecnenp), and
assuming that the temperature of the recombining gas is
close to 104 K, we find 〈σ−〉 = 3.4× 10
−17cm2.
By combining equations (8) and (9), we can estimate
the importance of the H− photodissociation due to re-
combination radiation. Assuming that most of the re-
combinations occurred recently, we find that, the recom-
bination radiation alone will suppress the H2 formation
rate by
Fs = 1 + 800δ
(1− fesc), (10)
where δ = 1.08nH/nbar is the local overdensity. Here we
have neglected recombinations in the diffuse intergalactic
medium (IGM) and the associated H− dissociating pho-
tons from these recombinations, but these would only
further increase Fs.
Cosmological redshift can affect the photodissociation
rate by shifting the spectrum to longer wavelengths.
Initially this leads to an increase in 〈σ−〉 due to the
ǫ−3/2 dependence of the cross-section for ǫ ≫ 0.755
eV. Eventually, as more and more of the spectrum is
shifted below the threshold, the cosmological redshift
begins to decrease the dissociation rate. For recombi-
nation photons this redshift effect is small, and the tran-
sition to 〈σ−〉-depression occurs at a redshift factor of
(1 + zi)/(1 + z) ≈ 2.5, see Figure 1.
2.2. Direct emission
Unlike ionizing photons, whose intensity is heavily at-
tenuated both in stellar atmospheres and in their host
galaxies, most of the photons with frequencies below the
Lyman limit escape freely into the IGM. From then on,
photons with frequency below Lyβ undergo no evolution
apart from cosmological redshift. By contrast, within a
small fraction of the Hubble time, most photons with
frequency between Lyβ and the Lyman limit are split by
cascade into two or more photons after being redshifted
into one of the hydrogen resonances. Most of the cas-
cade products, which include lines such as Lyα, Hα, and
Hβ, as well as a continuum spectrum produced by the
two photon transition 2s → 1s, are above the 0.755 eV
threshold for H− photodissociation.
The relative importance of these directly emitted H−
dissociating photons depends on the nature of the UV
sources. Figure 2 shows the increase of the H− dissocia-
tion rate due to inclusion of direct emission from metal-
poor Pop III stars, which we calculated using the stellar
atmosphere models of Schaerer (2002). Predictably, for
1 1.5 2 2.5 3 3.5 4
)/(1+z)
Fig. 1.— Redshift evolution of the average H− photodissociation
cross-section of the UV photons produced by recombination (dot-
ted line), excitations by non-thermal electrons (dashed line) and
massive Pop III stars (solid line).
4 6 8 10 12 14
Fig. 2.— The ratio between the total H− photodissociation rate
and the photodissociation by recombination products alone for star
with different effective surface temperature.
very massive Pop III stars, with surface temperatures
∼ 105 K, adding the stellar continuum below the Lyman
limit to the recombination spectrum increases the pho-
todissociation rate by only ∼ 10%. If, on the other hand,
most of the early ionizing flux was produced by stars
with masses below 10M⊙, whose continuum emission is
stronger at lower frequencies, then the total H− disso-
ciation rate would be tripled at least. Likewise, direct
emission may be important if most of the UV photons
were produced by miniquasars. For example, assuming
that their spectrum can be approximated by a power
law, Lν ∝ ν
−1.7, with a cutoff below 0.75 eV, adding the
directly emitted photons to the recombination products
increases the total photodissociation rate by a factor of
3. H− PHOTODISSOCIATING BACKGROUND - X-RAY
SOURCES
It has been suggested that X-ray photons could con-
tribute a large fraction of the energy emitted by the
first radiation sources (e.g. Ricotti & Ostriker 2004).
By increasing the number of free electrons, X-rays can
boost the production of H−, and thus of H2, provid-
ing a positive feedback to the formation of new sources
(Haiman et al. 2000; Kuhlen & Madau 2005). This ef-
fect, however, would be at least partially offset by an in-
crease of the H− photodissociating background, caused
by conversion of X-rays into UV photons.
The absorption of an X-ray photon is followed by re-
lease of a non-thermal electron, which then loses some of
its energy by inelastic collisions with atoms before it can
thermalize its energy by elastic scattering with ions and
other electrons. When the gas ionization fraction is low
(x . 0.05), the photoelectron splits most of its energy
evenly between collisional ionizations and excitations of
hydrogen atoms (Shull & van Steenberg 1985). Using
electron-hydrogen excitation cross-sections (Grafe et al.
2001; Stone et al. 2002), we find that around ∼ 5/6 of
the excitations are to the 2p level, which are followed by
emission of a Lyα photon. Most of the remaining excita-
tions are to the 3p level, which decays via emission of one
Hα photon and a subsequent two-photon decay from the
2s level. The Lyα, Hα and two-photon continuum each
produce roughly equal contributions to H− photodisso-
ciation. Per ionization, the average intensity-weighted
cross-section for these photons is 〈σ−〉 = 1.6×10
−17cm2.
Due to the low number of UV photons produced during
this phase, the formation of H2 is not strongly affected
Fs = 1 + 4δ
. (11)
After the ionized fraction climbs above x ∼ 0.05, most
of the energy of the non-thermal electrons is converted
to heat. However, simultaneously with the growth of the
ionized fraction, the temperature of the gas rises, and as
it crosses 104 K, the collisions between thermal electrons
and atoms begin to dissipate the energy added by X-
rays, mainly via emission of Lyα photons. Neglecting
gas clumping, we find that the number of emitted Lyα
photons per hydrogen atom is
Nα = 4.6× 10
−8 cm3 s−1
x(1 − x)e−1.18×10
5/T nH dt.
Assuming for simplicity that x and T are constants, we
can rewrite the equation (12) as
Nα = 11.2
(τe,X
(1− x)e−1.18×10
, (13)
where τe,X =
xnσT dt is the Thompson optical depth
from the epoch of partial ionization by X-rays. If X-
ray preionization contributes at least half of the τe ∼ 0.1
measured by WMAP (i.e. τe,X ≈ 0.05), hydrogen atomic
de-excitations in the diffuse IGM may produce & 30 Lyα
photons per baryon.
The suppression of H2 formation due to H
− photodis-
sociation by Lyα photons is
Fs ≈ 1 + 100Nαδ
−1. (14)
Since the energy of Lyα photons (10.2 eV) is far above
the H− photodissociation threshold (0.75 eV), the pho-
todissociation rate grows roughly as (1+zi)
1.5/(1+z)1.5,
where zi is the redshift at which the photon was emitted.
In the case of an extended period of partial ionization, Fs
may be increased by a factor of a few, possibly exceeding
104δ−1.
Since, when the IGM temperature rises above 104 K,
the formation of new minihalos is suppressed, the impact
of H− photodissociating flux produced by X-ray conver-
sion is relevant only for minihalos which have formed
some time ago or for halos with Tvir > 10
4 K, which also
rely on H2 cooling to form stars.
4. DISCUSSION
As shown by our calculations, H− photodissociation
reduces the formation of H2 molecules by a factor of
Fs ∼ 1 + 10
3ksxf
esc δ
−1, (15)
where ks is a constant of order a few, whose value de-
pends on the type of radiation source and the growth
history of the radiation background. Thus, by the time a
significant fraction (& 0.1) of the universe becomes ion-
ized, H− photodissociation can significantly reduce the
H2 formation rate in regions with overdensities of up to
a few thousands, i.e. in the interior regions of miniha-
los. The equilibrium abundance of molecular hydrogen
during this stage would be determined by the balance be-
tween its formation and destruction rates (Eqs. [1] and
nH2 =
k− nH nH−
, (16)
where kLW is the H2 destruction rate by the Lyman-
Werner photons. Thus a reduction of H− abundance by
a factor Fs translates into the same reduction of the H2
abundance and, in minihalos, a comparable increase of
the cooling time.
Indirectly, H− photodissociation may affect the cool-
ing in the central regions of minihalos even during the
early stages of reionization. The maximum density that
gas can reach in the core region of a minihalo is lim-
ited by the amount of entropy it is able to radiate away
during collapse. The lower density gas prevalent dur-
ing the early collapse phase would be susceptible to H−
dissociation from even a relatively low intensity H− dis-
sociating flux, and the resulting lowered H2 abundance
would limit its ability to radiate away entropy via H2
cooling. Furthermore, the density and H2 abundance at
the center depend on the conditions in the low density
outer regions, through their contributions to both the to-
tal pressure and the self-shielding ability of the halo. We
plan to investigate these effects further with numerical
radiation-hydrodynamic simulations in the future.
LC thanks the McDonald Observatory for the W.J.
McDonald Fellowship. MK gratefully acknowledges sup-
port from the Institute for Advanced Study. This work
was partially supported by NASA Astrophysical Theory
Program grants NAG5-10825 and NNG04G177G to P.
R. S.
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