Hydrogen 2p–2s transition: signals from the
epochs of recombination and reionization
Shiv. K. Sethi, Ravi Subrahmanyan, D. Anish Roshi
Raman Research Institute , Sadashivanagar, Bangalore 560080, India
email: sethi@rri.res.in, rsubrahm@rri.res.in, anish@rri.res.in
ABSTRACT
We propose a method to study the epoch of reionization based on the possible
observation of 2p–2s fine structure lines from the neutral hydrogen outside the cosmo-
logical H II regions enveloping QSOs and other ionizing sources in the reionization
era. We show that for parameters typical of luminous sources observed at z ≃ 6.3
the strength of this signal, which is proportional to the H I fraction, has a brightness
temperature ≃ 20µK for a fully neutral medium. The fine structure line from this
redshift is observable at ν ≃ 1GHz and we discuss prospects for the detection with
several operational and future radio telescopes. We also compute the characteristics
of this signal from the epoch of recombination: the peak brightness is expected to be
≃ 100µK; this signal appears in the frequency range 5-10 MHz. The signal from the
recombination era is nearly impossible to detect owing to the extreme brightness of
the Galactic emission at these frequencies.
Subject headings: cosmic microwave background—radio lines:general—line:formation—
radiative transfer
1. Introduction
Even though the existence of Hydrogen fine structure lines and their explanation using Dirac’s
atomic theory has been known for close to a century, a Hydrogen fine structure line has never been
detected from an astrophysical object. An interesting Hydrogen fine structure line is the 2p–2s
transition. The main difficulty in detecting this line is that the line strength is proportional to the
population of either the 2p or 2s states which, being excited states, are not so readily populated
in most astrophysical circumstances. Moreover, the line width of the excited 2p state, which is
determined by its decay time, is large (99.8 MHz), making the detection of the fine structure line
a difficult observation. One astrophysical setting where the feasibility of detecting such a line has
been studied is the interstellar H II regions (see, e.g., Dennison, Turner, & Minter 2005 and refer-
ences therein; Ershov 1987); in H II regions, the excited levels are populated by recombination.
http://arxiv.org/abs/0704.1226v1
– 2 –
Here we consider two cosmological settings in which the excited levels are populated by either
recombination or pumping by Lyman-α photons from an external source: (a) The Recombination
epoch: The Universe makes a transition from a fully ionized to an almost fully neutral medium at
z ≃ 1089 (Spergel et al. 2006; for details see, e.g., Peebles 1993 and references therein). During
this era, as the density and temperature of the Universe drops, recombination is stalled owing to a
high Lyman-α radiation density and progresses either by the depopulation of the 2p state owing to
redshifting of the photons out of the line width or the 2-photon decay of the 2s state. This results
in a significant 2s and 2p level population during the recombination era. (b) The Reionization
epoch: Recent observations suggest that the universe made a transition from nearly fully neutral to
fully ionized within the redshift range 6 <∼ z
∼ 15 (Page et al. 2006; White et al. 2003; Fan et al.
2002; Djorgovski et al. 2001; Becker et al. 2001). It is widely believed that this ‘reionization’
was achieved by the percolation of individual H II regions around the sources of reionization. The
nature of these sources is not well understood: they might be Pop III stars, active galactic nuclei
or star-forming galaxies. During this epoch, a signal from the 2p − 2s fine-structure transition
might originate from either within the cosmological H II regions or from the almost fully neutral
medium surrounding the H II region. The level population of the first excited state in the former
case would be largely determined by recombinations and in the latter case by Lyman-α photons
from the central source. We shall show below that for most cases of interest the fine-structure line
from within the cosmological H II region might be negligible as compared to the signal from the
regions immediately surrounding the H II region.
Throughout this work we adopt the currently-favoured ΛCDM model: spatially flat with
Ωm = 0.3 and ΩΛ = 0.7 (Spergel et al. 2006; Riess et al. 2004; Perlmutter et al. 1999) with
2 = 0.022 (Spergel et al. 2006; Tytler et al. 2000) and h = 0.7 (Freedman et al. 2001).
2. Fine-structure lines from the reionization epoch
Subsequent to the recombination of the primeval baryon gas at redshift z ≃ 1089 (Spergel
et al. 2006) and the transformation of the gas to an almost completely neutral state, it is believed
that the gas was reionized during epochs corresponding to the redshift range 6 <∼ z
∼ 15. WMAP
measurements of cosmic microwave background radiation (CMBR) anisotropy in total intensity
and polarization have been used to infer that the baryons were likely neutral at redshifts z >∼ 12 −
15; however, the detection of CMB polarization anisotropy requires substantial ionization by about
z ≃ 11 (Page et al. 2006). Observationally, the Gunn-Peterson (GP) test shows that the universe is
highly ionized at redshifts lower than z ≃ 5.5; the detection of GP absorption at greater redshifts
suggests that the neutral fraction of the intergalactic hydrogen gas rises to at least 10−3 in the
redshift range 5.5 <∼ z
∼ 6, and that reionization was not complete till about z ≃ 6 (White et al.
– 3 –
2003; Fan et al. 2002; Djorgovski et al. 2001; Becker et al. 2001). However, from the GP test
alone it is not possible to infer the neutral fraction of the medium; it only gives a rather weak lower
limit of ≃ 10−3 on the neutral fraction of the universe for z >∼ 6. From other considerations it is
possible to put more stringent bounds on the neutral fraction; for example, Wyithe & Loeb (2004)
obtain a lower limit of 0.1 on the neutral fraction of the universe at z ≃ 6.3 (see also Mesinger &
Haiman 2004).
Our understanding of the nature of the sources that caused the reionization is far from com-
plete. The transition from an almost completely neutral gas to a highly ionized gas during redshifts
6 <∼ z
∼ 15 is a key problem in modern cosmology and considerable theoretical and experimental
efforts are currently directed at this unsolved problem. Here we propose a new method, based
on the 2p − 2s fine-structure transition, to determine the evolution of the neutral fraction of the
intergalactic medium within this epoch.
Owing to fine structure splitting, the two possible transitions between the 2s and the 2p states
are: 2p1/2–2s1/2 at a frequency ≃ 1058MHz that has an Einstein A coefficient 1.6× 10
−9 s−1 and
2p3/2–2s1/2 at a frequency νps ≃ 9911MHz that has an Einstein A coefficient 8.78 × 10
−7 s−1.
The Einstein A coefficient for the latter transition is more than an order of magnitude greater than
the former; therefore, in this work we consider only the 2p3/2–2s1/2 transition and hereinafter we
refer to this specific transition simply as the 2p–2s transition.
The ionizing UV photons from sources in the reionization era create ‘Stromgren spheres’.
Whereas the gas in the cosmological H II regions are highly ionized by the photons, the ioniza-
tion level of the gas beyond the Stromgren spheres is determined by the history of the gas, the
density, and the mean specific intensity of the background ionizing photons, which includes both
the photons diffusing out of the Stromgren spheres as well as the background radiation field. In
this work we assume that the ionizing sources at these high redshifts are AGNs and in illustrative
examples adopt parameters of a few QSOs that have been observed at z ≃ 6. The photons at
the Lyman-α transition frequency (here and throughout, unless otherwise specified, we shall con-
tinue to refer to frequencies between Lyman-α and Lyman-limit as ’Lyman-α) from a high redshift
QSO escape the mostly-ionized Stromgren sphere and are strongly scattered and absorbed in the
medium beyond. The population of the 2p level in this region is determined by (a) the intensity of
Lyman-α photons from the central source, (b) recombination rate of free electrons, (c) absorption
of CMBR photons by electrons in the 2s state (it is assumed here and throughout this work that the
only radio source at high redshifts is the CMBR) and (d) collisional transition from the 2s state.
The 2s state is populated via (a) the recombination rate of free electrons, (b) collisional transfer
of atoms from the 2p state, (c) the spontaneous decay of the 2p state, and (d) transition from the
2p state stimulated by CMBR photons. Additionally, the absorption of photons from the central
sources, with energy equal to or in excess of the Lyman-β transition, would result in electronic
– 4 –
transitions to the second excited state, which could be followed by spontaneous decay to the 2s
state. (It might be pointed out here that both the 2s and 2p states could also be populated by atoms
cascading from excited states with n > 3. In particular, all photons absorbed from 1s states to any
excited state can directly de-excite to the 2s level. However, the rate of transition from 1s to any
excited state is roughly ∝ 1/n3 (e.g. Rybicki & Lightman 1979) and, therefore, we include only
the most dominant transition in each case.)
The population of the ground state is denoted by n1s. We denote the level populations of the
the two states—2s1/2 and 2p3/2—by the number densities n2s and n2p; these may be solved for,
respectively, from the following two equations of detailed balance:
i + cB2p2sn2pnCMBR + Cpsnin2p + A2p2sn2p + cn1sp32
B13,βφ13(ν)nα(ν)dν
= A2s1sn2s + Cspnin2s + cB2s2pn2snCMBR, (1)
(1− 2f)αBn
i + cB2s2pn2snCMBR + Cspnin2s + cn1s
B12,αφ12(ν)nα(ν)dν
= A2p1sn2p + Cpsnin2p + cB2p2sn2pnCMBR. (2)
Here f is the fraction of all the atoms that recombine to the 2s state. In equilibrium f = 1/3
as the n = 2 state splits into three doublets: 2p1/2, 2p3/2 and 2s1/2. αB is the recombination
coefficient and ni is the density of the ionized gas. B2p2s = B2s2p = c
2/(8πν3ps)A2p2s is the Einstein
B coefficient for the 2p3/2–2s1/2 transition in terms of the corresponding Einstein A coefficient
A2p2s (note that the two B coefficients are equal as the two states have the same degeneracy).
Cps = Csp = 5.31 × 10
−4 cm3 s−1 is the rate coefficient of transition owing to collisions with
electrons. nCMBR is the number density of CMBR photons within the transition line width. nα(ν)
is the number density (per unit frequency) of photons with frequency equal to or larger than the
Lyman-α frequency (and smaller than the Lyman-limit frequency) at any location; φ13 and φ12
are, respectively, the line profiles corresponding to the Lyman-β and Lyman-α transitions, and
p32 is the probability for the electron transition to the 2s state following excitation to n = 3 via
absorption of a Lyman-β photon. We have not included the induced Lyman-α transition because
the number density of atoms in the 2p state is negligible as compared to that in the 1s state. p32 is
the probability that an atom in the third excited state (3p) will decay to the 2s state. A2s1s is the
Einstein A coefficient corresponding to the 2-photon decay of the 2s state. Other symbols have
their usual meanings.
Owing to the fact that the mostly neutral medium in the vicinity of the cosmological Strom-
gren spheres is optically thick to Lyman-α scattering, the Lyman-α photons from the decay of
the 2p state are strongly scattered by the gas. Therefore, nα(ν) will contain contributions from
both the Lyman-α photons from the central source as well as the Lyman-α photons that arise
– 5 –
from recombinations outside the Stromgren sphere and are multiply-scattered therein: nα(ν) =
nsourceα (ν) + n
α (ν). We neglect the multiply-scattered Lyman-α photons from the Stromgren
sphere that have been reprocessed via recombination within the Stromgren sphere because these
would be redshifted redward of the Lyman-α line before encountering the boundary of the Strom-
gren sphere. The scattering of recombination photons in an optically thick, expanding medium is a
complex problem (Field 1959; Rybicki & Dell’Antonio 1994). One of its first applications was to
study the recombination of primeval plasma (Peebles 1968; Zeldovich, Kurt & Sunyaev 1969). In
these analyzes it was implicitly assumed that apart from 2-photon decay, in an expanding universe
the dominant effect that results in resonant photons ceasing interaction with the gas, and leaving
the system, is its redshifting out of the line profile. The effect of scattering off the moving atoms
was deemed to be either negligible or at best comparable. This assumption has been borne out by
more recent detailed analysis that have taken into account the effect of scattering on the photon
escape (Krolik 1990). Taking only the redshift as the main agent of photon escape, it can be shown
that the net effect of the scattering of a resonance photon before it drops out of consideration is to
reduce the decay time of the 2p state from A21 = 6.2× 10
8 s−1 to A21/τGP (Zeldovich et al. 1969;
for more recent work see, e.g., Chluba, Rubino-Martin & Sunyaev 2007, Seager, Sasselov & Scott
1999 and references therein; we give a concise derivation in Appendix A). In this expression, the
Gunn-Peterson optical depth τGP = [3/(8πH)]A2p1sλ
αn1s.
Similar complications exist in computing p32, the probability that an atom in the third excited
state will decay spontaneously to the 2s state, in an optically thick medium. In optically thin media,
p32 = A32/(A32 + A31); on the other hand, in an optically thick medium, we would need to take
into account the ‘trapping’ of the Lyman-β photon owing to resonant scattering. The effect of this
scattering in an optically thick medium would be that the fraction of photons that decay directly
to the ground state are reabsorbed ‘locally’ to the third excited state and, therefore, all photons
absorbed to the third excited state result in an Hα photon and an atom in the 2s state. This means
that the appropriate value of p32 is close to unity in an optically thick medium: in this work we
assume p32 = 1.
The astrophysical setting in which we seek solutions to the algebraic equations above is cos-
mological H II regions at high redshift. In particular, we are interested in the signal from the neutral
region surrounding the cosmological H II region. For a fully neutral inter-galactic medium (IGM)
at z ≃ 6.5, τGP ≃ 6 × 10
5. If we adopt spectral luminosities corresponding to QSOs observed at
these high redshifts, it may be shown that the populating of the 2p state via direct recombinations
from the free-free state, collisional transfer from the 2s state, and upward transitions from the 2s
to 2p state arising from absorption of background CMBR photons may all be neglected. (The rele-
vant parameters are: ni ≃ nb ≃ 2.8(1 + z)
3 cm−3 in the H II region surrounding the sources, with
ni expected to be much smaller in the neighboring mostly neutral medium; the number density
of CMBR photons that might cause a 2p–2s transition is ≃ 5(1 + z) cm−3; the number density
– 6 –
of Lyman-α photons from the central source, assuming luminosities typical of SDSS quasars at
z ≃ 6.5 (more details in §4), is nα ≃ 10
−4 cm−3. First, for these parameters, the dominant process
that populates the excited state is the pumping by Lyman-α photons. Second, it may be readily
verified that for these plausible values for the parameters the signal expected from the H II region
surrounding the central source is negligible as compared to the signal from the surrounding neu-
tral region.) Given that the Lyman-α flux from the central QSO is the dominant causative factor
for populating the 2p state, the two equations (1 and 2) that determine the level populations is
simplified. The number density of atoms in the 2p state is given approximately by:
n2p ≃ n1sΓατGP/A2p1s, (3)
where n1s = fneunb, with fneu denoting the neutral fraction and nb ≃ 2.7 × 10
−7(1 + z)3 cm−3
is the number density of baryons in the IGM. Γα =
B2p1sφ12(ν)n
source
α (ν) (in units of s
−1) is
the transition rate to the 2p state owing to the Lyman-α photons from the central source, where
nsourceα (ν) is the number density of Lyman-α photons from the central source alone.
Similarly, the dominant process that determines the population of the 2s state is the absorption
of Lyman-β photons (for details, see the discussion above and §4) and the subsequent decay to the
2s state. The number density of atoms in the 2s state is:
n2s ≃ n1sΓβ/A2s1s, (4)
where Γβ =
B3p1sφ13(ν)n
source
α (ν) (in units of s
−1) is the transition rate to the 2s state owing to
the Lyman-β photons from the central source.
3. Fine structure lines from the epoch of recombination
The universe made a transition from fully ionized to nearly fully neutral at z ≃ 1089 (Spergel
et al. 2006; Peebles 1968; Zeldovich et al. 1969). This transition is mainly accomplished by the
2-photon decay of the 2s state and the slow redshifting of the Lyman-α photons which deplete the
2p state (see Peebles 1993 and references therein for a detailed discussion). The Saha ionization
formula, valid for thermodynamic equilibrium conditions between the hydrogen level populations
and free electrons, is a poor approximation for studying the epoch of recombination. A good ap-
proximation for studying this transition is to assume that all states, excepting 1s, are in equilibrium
with the CMBR (matter temperature to a very good approximation remains equal to the CMBR
temperature throughout this transition) (Seager et al. 1999; Peebles 1968). However, in this ap-
proximation, where the matter temperature and all transitions, excepting the Lyman-α line, are in
thermal equilibrium, the 2p–2s signal is unobservable because the excitation temperature for this
transition equals the background radiation temperature. Even though this second approximation
– 7 –
might be useful for studying the evolution in ionization fraction, it will be strictly true only if the
dominant mechanisms that determine the level populations of the 2p and the 2s states are either
interaction with the CMBR photons or collisions between atoms. As there are a variety of other
processes relevant to the determination of the level populations—for example, the free decay of
either of the two states—a small deviation from equilibrium is expected in the 2s and 2p level
populations and it is our aim here to compute it.
One approach to this problem is to simultaneously solve for the level populations of the 2p
and the 2s states as well as the change in the ionization fraction. However, assuming thermal
equilibrium between these two states is a good approximation for solving the evolution of ioniza-
tion. Therefore, the approach we have adopted is to solve for the evolution of ionization using the
method of Peebles (1968), and use the resulting ionized/neutral fraction to solve for the populations
of the 2s and the 2p states using detailed balance. The resulting equations are:
i + cB2p2sn2pnCMBR + Cpsnin2p + A2p2sn2p + n1sA2s1s exp(−(B1 − B2)/(kBTCMBR))
= A2s1sn2s + Cspnin2s + cB2s2pn2snCMBR + βcn2s (5)
(1− 2f)αBn
i + cB2s2pn2snCMBR + Cspnin2s
= A2p1sn2p/τGP + Cpsnin2p + cB2p2sn2pnCMBR + βcn2p,(6)
where B1 = 13.6 eV and B2 = 2.4 eV are, respectively, the ionization potentials of the ground
and the first excited states, kB is the Boltzmann’s constant, and TCMBR is the temperature of the
CMBR. βc is the rate at which the CMBR photons cause a bound-free transition of electrons from
the n = 2 state (2s or 2p states). The various other terms in these equations have the same meanings
as in the previous case. The main difference is that the CMBR photons and baryonic matter at the
time of recombination are hot and dense enough to directly affect the level populations of the
excited states by ionizing the excited state, and the two-photon capture to the excited state is not
completely negligible (see Peebles 1993 and 1968 for details of the different physical processes
that are relevant at this epoch).
4. Expected brightness of the fine structure lines
The brightness temperature in the 2p–2s transition is:
∆Tb ≡ Tb − TCMBR = g⋆
n2p(0)hpA2p2sλ
e(1 + z)
8πkBH(z)
(1− TCMBR/Tex) ≡ τex(Tex − TCMBR) (7)
Here n2p(0) = n2p/(1 + z)
3 and λe is the rest wavelength of the 2p–2s transition. Tex =
(hpνe/kB)[n2p/(n2s − n2p)] is the excitation temperature corresponding to the transition, where
– 8 –
hp is the Planck constant. τex is the optical depth of the source in the 2p–2s signal. g⋆ takes into
account the selection rules for transitions between the 2p3/2 and 2s1/2 levels. Given the selec-
tion rules there are three allowed transitions between these two states; they occur at frequencies
ν ≃ 9852, 9875, and 10029 MHz (see, e.g., Ershov 1987). The first two transitions are blended by
the natural width of the line, which is approximately 100 MHz, but the third should be observable
as a distinct line. This implies that g⋆ = 2/3 if the observing frequency is ≃ 9900/(1 + z) MHz.
4.1. Expectations for signals from cosmological H II regions
For computing the strength of this signal in a typical case, we adopt observed parameters
of QSO SDSS J1030+0524 (see, e.g., Wyithe & Loeb 2004 and references therein), which is at
redshift z = 6.28 and shows no detectable flux beyond the QSO Stromgren sphere: the radius
of the Stromgren sphere has been estimated to be R ≃ 4.5Mpc (Mesinger & Haiman (2004)
argue that the size of the Stromgren sphere could be roughly 30% higher; this makes no essential
difference to our results). It may be noted here that the QSO SDSS J1148+5251 also has similar
parameters. We arrive at an estimate of nsourceα (ν) by assuming that Lα photons per second, with
wavelengths corresponding to the Lyman-α transition, are emitted by the central source in an
effective frequency range ∆νsource, and that the photons are absorbed at a radial distance R. This
leads to:
nsourceα = Lα/(4πR
∆νsource
. (8)
We assume typical values: Lα = 10
58 s−1, R = 4.5Mpc and ∆νsource = 30, 000 km s
−1. These
lead to the following estimate for the transition rate, Γα, in the gas at the boundary of the Stromgren
sphere arising due to the photons from the central QSO:
Γα ≃ 8× 10
−10 s−1. (9)
It should be noted that the mean specific intensity of Lyman-α photons in the IGM would also
give a non-zero signal. Assuming a mean specific intensity of ≃ 10−21 erg cm−2 sec−1 Hz−1 sr−1
(this might be needed to couple the HI spin temperature to the matter temperature; see e.g. Madau,
Meiksin, & Rees 1997), the expected signal is many orders of magnitude smaller than we have
computed from the outskirt of bright sources. If we assume that the central sources are continuum
emitters in the frequency range between Lyman-α and Lyman-limit frequencies, we may assume
similar parameter values for computing the expectations for Γβ. In that case,
Γβ ≃ Γα
, (10)
– 9 –
where fβ and fα are the oscillator strengths of the Lyman-β and Lyman-α transitions, respectively.
From Eqs. (3), (4) and (10):
A2p1s
A2s1sτGP
. (11)
For a completely neutral medium at z ≃ 6.4, τGP ≃ 6 × 10
5, which may be used to show that
n2s ≫ n2p for the parameters of IGM in the redshift range of interest: 6.4 <∼ z
∼ 10. This implies
that in the outskirts of the Stromgren sphere surrounding the QSO, the transition is expected to
be observable as an absorption feature against the CMBR. Using Eqs. (3) and (4) in Eq. (7), the
observable brightness temperature is estimated to be:
∆TB ≃ −20 µK
, (12)
where fneu is the neutral fraction of hydrogen outside the Stromgren sphere, and might be close to
unity.
We have adopted parameters typical of QSOs observed at the edge of the reionization epoch
in estimating the above temperature decrement. The main uncertainty above is in the estimation of
the ‘Lyman-α’ flux from the central source, and as the observed temperature decrement is directly
proportional to this flux from the central source, this constitutes a major uncertainty in reliably
computing the expectations for the signal. For QSOs that have strong Lyman-α and Lyman-β
lines, the blueward side of the lines will be strongly absorbed in the medium just beyond the
Stromgren sphere (and this has been observed to happen in many cases), provided that the blueward
side photons have not been redshifted to frequencies smaller than the Lyman-α frequency while
transversing the Stromgren sphere. In the case of QSOs that have large line fluxes and small
Stromgren spheres, the Lyman-α luminosity Lα might be underestimated.
4.2. Expectations for the fine-structure line from recombination
Using equations (5) and (6) the level populations of the 2p and the 2s states may be computed.
Solving for the level populations and using equation (7), we have computed the expected signal
from the recombination epoch; the expected signal is shown in Fig. 1. The fine structure line
transition is expected to be an absorption feature, with a maximum temperature decrement of order
100 µK at an observing frequency of about 7 MHz. The width of the decrement in frequency space
corresponds to a redshift span ∆z ≃ 200, which is roughly the width of the visibility function at
recombination.
– 10 –
5. Prospects for the detection of the cosmological fine structure lines
As Eq. (12) shows, a detection of the fine structure line in the outer regions of cosmological
H II regions is potentially a probe of the cosmological neutral hydrogen density in the vicinity of
QSOs in the reionization epoch, and might be a tool for the investigation of the evolution in the
neutral fraction with cosmic epoch through the reionization era. Given the cosmological impor-
tance of such a measurement, and the fact that there does not exist many reliable methods for the
detection of H I at high redshifts (see, e.g., Barkana & Loeb 2001), the detection of the 2p–2s line
transition in the cosmological context assumes additional importance.
In deriving equation (7) the Hubble expansion was assumed to be the only cause for the
velocity width in the observed line. However, an important contribution to the velocity dispersion
in the line in this case is the natural width of the fine structure line: owing to the rapid decay of
the 2p state, the natural Lorentzian width in the rest frame of the gas is VLor ≃ 100MHz (see,
e.g., Dennison et al. 2005). Therefore, the peak brightness in the observed line profile might
be suppressed by a factor Vexp/VLor, where Vexp is the line-of-sight peculiar velocity dispersion
owing to the Hubble flow across the region being observed. However, in the case Vexp >∼ VLor this
suppression has a negligible effect. For QSOs at redshift z ≈ 6.5, the natural Lorentzian width
of the fine structure line is equivalent to the peculiar Hubble flow across a proper line-of-sight
distance of ≃ 4Mpc. This distance is approximately the size of the Stromgren spheres around
QSOs at that redshift; therefore, the natural width of the line does not significantly diminish the
expectations, given by equation (7), for the peak brightness temperature. A second inference is
that in the case of a QSO that has a smaller Stromgren sphere, the increase in mean brightness
temperature is roughly proportion to the inverse of the radius of the Stromgren sphere: 1/R, and
not 1/R2.
Another assumption that was made while deriving Eq. (7) is that the only radio frequency
radiation that needs to be considered for the determination of the level populations and brightness
temperature decrement is the CMBR. It may be that the central ionizing source, which may be a
QSO, is radio loud. There may also exist radio sources behind the observed Stromgren sphere and
within the angular region over which the Lyman-α flux from the QSO is appreciable. Equation
(7) may be modified, to account for this, by replacing TCMBR with TCMBR + TB, where TB is the
brightness temperature of the radio source at wavelengths corresponding to the rest frequency of
the transition, which is ≃ 9GHz. The result would be to enhance the brightness temperature of
the line signal by ≃ TB/TCMBR. If the radio source is unresolved, it is appropriate to estimate the
expected signal in terms of the optical depth. The optical depth corresponding to equation (12) is
≃ 10−6.
We now discuss the feasibility of the detection of the fine structure line towards SDSS J1030+0524.
We shall assume that the neutral fraction, fneu, outside the Stromgren sphere of this QSO is unity,
– 11 –
consistent with the measured GP trough. The redshifted fine structure line would be expected
at ≃ 1.36GHz. The observed line ‘width’, considering natural broadening and the Hubble flow
across the Stromgren sphere, is expected to be approximately ∆ν ≃ 100MHz/(1 + z); using
z = 6.28, we obtain ∆ν = 13.7MHz. The fine structure line would be expected to originate in a
shell that is roughly the size of the Stromgren sphere, and fall of as 1/r2 beyond this shell, where r
is the distance from the QSO. In the case of SDSS J1030+0524, the angular size of the Stromgren
sphere is expected to be 15 arcmin.
The frequency of the redshifted line is in the observing bands, and the line width is within the
spectral line capabilities, of several currently operational telescopes. However, large-collecting-
area arrays like the Giant Metrewave Radio Telescope (GMRT) have large aperture antennas of
45-m diameter and, therefore, poor surface brightness sensitivity for such extended structures.
The Australia Telescope Compact Array (ATCA), with 22-m antennas, has reasonable brightness
sensitivity for this problem and, additionally, operates with 128-MHz bandwidths in spectral line
mode. At ν ≃ 1.3GHz, the ATCA has a system temperature Tsys ≃ 25K and antenna efficient
K = 0.1K Jy−1. The five movable 22-m diameter antennas may be configured into an ultra-
compact 2-D close-packed H75 (75-m maximum baseline) array, and this would yield a number
of baselines sensitive to the 15-arcmin scale fine-structure line signal. The brightness sensitivity
of this array, for a 8-arcmin scale structure, is 400 µK in 6 hr integration time. The brightness
sensitivity in Fourier synthesis images could be enhanced somewhat, by factors of a few, by ap-
propriately weighting the baselines to match the synthetic beam to the expected structure scale;
however, the required integration times for a detection are still in the ball park of 103 hr.
The detection of the signal from cosmological H II regions, however, might be feasible using
facilities under construction or planned for the near future, like the xNTD in Australia or the Square
Kilometer Array (SKA). These arrays would have smaller antenna sizes—making the detection of
these large-angular-scale structures detectable in interferometers—and significantly more numbers
of antennas, giving more numbers of short baselines that would usefully respond to the large-
angular-scale fine-structure line. However, the array configuration designs would have to factor in
the extraordinary brightness temperature sensitivity requirements for this demanding observation.
As an example, the SKA might consist of about 104 12–15 m class reflector antennas, with
aperture efficiency of 60%, and the system temperature at 1.4 GHz might be about 20 K. Assuming
that the visibilities are optimally weighted so that the synthetic beam of the Fourier synthesis array
is matched to the source size of about 15 arcmin FWHM, the line strength would have a peak of
about 20 µJy. A 5-σ detection of the fine structure line towards a typical QSO, in a reasonable
integration time of about 1 hr, will require that about 250 baselines (just 0.001% of the total) be
within about 50 m.
The signal from the recombination epoch is observable in the frequency band 6–8 MHz as
– 12 –
a broad decrement in the brightness temperature of the extragalactic background sky, and would
be extremely small compared to the orders of magnitude more intense Galactic non-thermal emis-
sion as well as the average low-frequency background brightness temperature arising from the
numerous extragalactic radio sources. This decrement may be considered to be a distortion to the
CMBR spectrum at long wavelengths, and would be an all-sky cosmological signal. However,
owing to the ionosphere, the frequency range in which this feature is expected to appear is too
low a radio frequency to be easily accessible using ground-based observatories. Therefore, even
though the observation of this signal might be yet another tool to probe the epoch of recombi-
nation, new custom-made instruments, which should presumably operate from space and above
the ionosphere, will require to be built if a detection of this signal is to be attempted. An addi-
tional cause for concern is that the Galactic and Extragalactic background radiations might have
low-frequency spectral turnovers at these frequencies, as a result of free-free absorption as well as
synchrotron self-absorption, and these would result in significant spectral features in the band that
would require a careful modelling in order to detect any CMBR decrement feature arising from
fine structure transition absorption. Interference from terrestrial man-made transmitters, as well as
from auroral phenomena and solar system objects would also be an issue.
6. Summary and Discussion
We have discussed the possibility of detecting the hitherto undetected fine structure line of
2p–2s transition in two cosmological settings: the epoch of recombination at z ≃ 1089 and the
epoch of reionization at z ≃6–15.
The expectations for the line from the environments of ionization sources in the epoch of
reionization are interesting and worthy of attention as a novel tool for the investigation of the
reionization process and the cosmological evolution of the gas. The signal is expected to be ob-
servable as an extended and weakly absorbing source, which causes a decrement in the brightness
of the background CMBR, and may be detected by interferometers as a negative source akin to
the Sunyaev-Zeldovich decrements observed along the lines of sight through hot gas in clusters
of galaxies. Detection of the 2p–2s line signal from the outskirts of cosmological H II regions at
different redshifts within the reionization era may serve to determine the neutral fraction of the
medium during the epoch of reionization, which is a quantity of significant interest in modern cos-
mology. In particular, we have computed a representative signal strength by adopting parameters
typical of a QSO observed at z ≃ 6. These QSOs show GP troughs in their spectra; however,
owing to the weakness of the GP test, the spectra have only been useful in setting weak limits
on the neutral fraction (constraining the neutral fraction to be >∼ 10
−3) outside the observed H II
region. We have shown that for a fully neutral medium the line peak may reach ≃ 20µK, which is
– 13 –
potentially observable by radio interferometers that are being designed today.
Other interesting probes of the reionization epoch include detecting OI line from this epoch
(Hernandez-Monteagudo et al. 2006).
It is of interest to compare the relative difficulty associated with detecting the neutral hy-
drogen in this indirect way, using the redshifted fine structure line, with direct imaging of the
redshifted 21-cm line from neutral hydrogen during the epoch of reionization (see, e.g., Sethi
2005; Zaldarriaga, Furlanetto, & Hernquist 2004). The ‘all-sky’ H I signal might be detectable
with a peak strength of ≃ 50mK; however, it could be very difficult to detect owing to calibra-
tion and foreground contamination issues (e.g., Zaldarriaga et al. 2004; Shaver et al. 1999, and
references therein). A better approach might be to attempt to detect the fluctuating component
of the sky signal, which could have peak intensities of ≃ a fewmK at observing frequencies of
ν ≃ 100–200MHz (Zaldarriga et al. 2004; Shaver et al. 1999). This translates to roughly the
same signal strength (specific intensity) as we have obtained in Eq. (12) for the fine structure line.
This is not entirely unexpected: even though the level populations of the excited states are much
smaller as compared to the ground level population needed in computing the HI signal, the A coef-
ficient of the fine structure transition we consider here is roughly 8 orders of magnitude larger than
the HI hyperfine transition A coefficient. Currently, there are many ongoing and planned radio
interferometer experiments for the detection of the redshifted H I emission/absorption from the
epoch of reionization (e.g., Pen, Wu, & Peterson 2004; the LOFAR project at www.lofar.org;
the MWA project at www.haystack.mit.edu/ast/arrays/mwa/site/index.html).
The detection of the H I signal is firstly more difficult—requiring greater sensitivity—because the
typical frequency width of the signal ≃ 0.5MHz (Zaldarriga et al. 2004), which is far smaller than
the typical width expected for the fine structure line (≃ 10MHz). Second, the fine structure line
is expected at 1.4 GHz, where the sky background temperatures are significantly lower than in the
100–200MHz band, making the telescope system temperatures lower. An additional advantage
of the the indirect detection is that the redshifted fine structure line appears at higher frequencies
( >∼ 1GHz), which are relativity free of interference as compared to the low frequency band of
100–200MHz. The main advantage of the direct detection is that unlike the method we suggest
here, it is independent of the existence of strong Lyman-α emitters at high redshifts. Another ad-
vantage of direct detection is that it may be detected ‘statistically’ and such a detection might be
achieved with greater ease than direct ‘imaging’ (e.g Zaldarriaga et al. 2004; Bharadwaj & Sethi
2001); however, the foreground subtraction problem becomes a severe constraint for a statistical
detection. To summarize: if strong Lyman-α emitting sources are present at high redshifts, they
would facilitate the indirect detection of the neutral hydrogen via enabling the detectability of the
fine structure line. The imaging issues and problems associated with the detection of this signal
appears to be less of a challenge as comparison to the direct detection of redshifted H I from those
epochs.
– 14 –
Another astrophysical context in which the fine structure line might be detectable is the envi-
ronments of high redshift galaxies, which are strong Lyman-α emitters. As equation (12) shows,
the observed signal depends on the neutral fraction outside the Stromgren sphere. Therefore, de-
tection of this signal would constitute an alternate probe of the neutral fraction of the IGM at large
redshifts.
The aim of this work has been to examine the detectability of the fine structure line in cos-
mological contexts, to point out the cosmological significance of detections, and spawn work that
may refine the modelling presented herein and improve the case for appropriate design of future
telescopes, which might enable the detection of the fine structure line towards multiple sources in
the reionization era.
Appendix A: Photon distribution function
The evolution of the photon distribution function, neglecting the effect of scattering off mov-
ing atoms, is (see e.g. Rybicki & Dell’Antonio 1994):
= A2p1sn2pφν − cBνn1sφνnν . (13)
Here Bν = 3/(8π)c
2/ν2αA2p1s and H = ȧ/a. The equation may be written as:
= −1/τ(nν − n⋆). (14)
Here τ = 1/(cBνφνn1s) and f⋆ = A2p1sn2p/(Bνn1sc). The equation above lends itself to a ready
interpretation. If the second term on the left hand side (which is owing to the expansion of the
Universe) was absent, the distribution function will approach f⋆ on a time scale ≃ τ , where τ ≃
3 × n1s s ≪ ȧ/a ( n1s here has units cm
−3). It may be readily verified that for the recombination
epoch and also the epoch of reionization, τ ≪ 1/H , excepting when the neutral fraction of the
medium is very small. We work here with the assumption that the neutral fraction is always large
enough so that τ ≪ 1/H . In this case the solution to Eq. (14) may be simplified: to leading order
the distribution function approaches f⋆ and the first order term (of the order of τH) represents the
slow time variation of the distribution function owing to the expansion of the Universe. In this
approximation, one may write the solution to Eq. (14) as:
nν ≃ n⋆ − τH(t)n⋆. (15)
Using these equations we may proceed to prove the contention that the net effect of the ‘trapping’
of Lyman-α photons is to reduce the decay time of the 2p state by a factor τGP . The 1s–2p transition
– 15 –
rate, which is given by Eq. (6), may be solved using Eq. (15):
Bνφ(ν)n(ν)dν ≃ A2p1sn2p − A2p1sn2p/τGP . (16)
The first term on the right hand side cancels with the decay term of the 2p state on the right hand
side of Eq. (6), and, therefore, the net effect of the scattering of recombination photons is to reduce
the decay time of the 2p state by a factor τGP . It may be pointed out that the condition needed to
derive the above expression roughly translates to the condition that τGP ≫ 1. For the reionization
case, this requires that the neutral fraction >∼ 10
−5. In the case of primordial recombination, it
leads to an even weaker condition that the neutral fraction is >∼ 10
Acknowledgment
One of us (SKS) would like to thank Jens Chluba for many useful discussions and to Zoltan
Haiman for many useful comments on the manuscript.
REFERENCES
Barkana, R. and Loeb, A. 2001, Phys. Rep., 349, 125
Becker, R. H. et al. 2001, AJ, 122, 2850
Bharadwaj, S. & Sethi, S. K. 2001, J. Astrophys. Astr., 22, 293
Chluba, J., Rubino-Martin, J. A., & Sunyaev, R. A. 2007, MNRAS, 374, 1310
Dennison, B., Turner, B. E., & Minter, A. H. 2005, ApJ, 633, 309
Djorgovski, S. G., Castro, S., Stern, D., & Mahabal, A. A. 2001, ApJL, 560, L5
Ershov, A. A. 1987, Sov. Astron. Lett. 13(2), 115
Fan, X., Narayanan, V. K., Strauss, M. A., White, R. L., Becker, R. H., Pentericci, L., & Rix, H.
2002, AJ, 123, 1247
Field, G. 1959, ApJ, 129, 551
Freedman, W. L. et al. 2001, ApJ, 553, 47
Hernandez-Monteagudo, C., Haiman, Z. Jimenez, R., & Verde, L. 2006, astro-ph/0612363
http://arxiv.org/abs/astro-ph/0612363
– 16 –
Krolik, J. H. 1990, ApJ, 353, 21
Madau, P., Meiksin, A., & Rees, M. J. 1997, 475, 429
Mesinger, A. and Haiman, Z. 2004, ApJL, 611, 69
Page, L. et al. 2006, astro-ph/0603450
Peebles, P. J. E. 1993, Principles of Physical Cosmology, Princeton University Press
Peebles, P. J. E. 1968, ApJ, 153, 1
Pen, U.-L., Wu, X.-P., Peterson, J. 2004, astro-ph/0404083
Perlmutter, S. et al. 1999, ApJ, 517, 565
Riess, A. G. et al. 2004, ApJ, 607, 665
Rybicki, G. B. & Dell’Antonio, I. P. 1994, ApJ, 427, 603
Rybicki, G. B. & Lightman, A. 1979, Radiative Processes in Astrophysics, Wiley-Interscience
Seager, S., Sasselov, D. D., & Scott, D. 1999, ApJL, 523, L1
Sethi, S. K. 2005, MNRAS, 363, 818
Shaver, P. A., Windhorst, R. A., Madau, P. & de Bruyn, A. G. 1999, A&A, 345, 380
Spergel, D. N. et al. 2006, astro-ph/0603449
Tytler, D., O’Meara, J. M., Suzuki, N., & Lubin, D. 2000, Physics Reports, 333, 409
White, R. L., Becker, R. H., Fan, X. & Strauss, M. A. 2003, ApJ, 126, 1R
Wyithe, J. S. B. & Loeb, A. 2004, Nature, 432, 194
Zaldarriaga, M., Furlanetto, S. R. & Hernquist, L. 2004, ApJ, 608, 622
Zeldovich, Ya. B., Kurt, V. G. & Sunyaev, R. A. 1969, JETP Lett, 28, 146
This preprint was prepared with the AAS LATEX macros v5.2.
http://arxiv.org/abs/astro-ph/0603450
http://arxiv.org/abs/astro-ph/0404083
http://arxiv.org/abs/astro-ph/0603449
– 17 –
Fig. 1.— The expected brightness temperature decrement in the background radiation, owing
to the fine structure transition in gas at the recombination epoch, is plotted versus the observing
frequency.
	Introduction
	Fine-structure lines from the reionization epoch
	Fine structure lines from the epoch of recombination
	Expected brightness of the fine structure lines
	Expectations for signals from cosmological H ii regions
	Expectations for the fine-structure line from recombination
	Prospects for the detection of the cosmological fine structure lines
	Summary and Discussion

  We propose a method to study the epoch of reionization based on the possible
observation of 2p--2s fine structure lines from the neutral hydrogen outside
the cosmological H {\sc ii} regions enveloping QSOs and other ionizing sources
in the reionization era. We show that for parameters typical of luminous
sources observed at $z \simeq 6.3$ the strength of this signal, which is
proportional to the H {\sc i} fraction, has a brightness temperature $\simeq 20
  \mu K$ for a fully neutral medium. The fine structure line from this redshift
is observable at $\nu \simeq 1 \rm GHz$ and we discuss prospects for the
detection with several operational and future radio telescopes. We also compute
the characteristics of this signal from the epoch of recombination: the peak
brightness is expected to be $\simeq 100 \mu K$; this signal appears in the
frequency range 5-10 MHz. The signal from the recombination era is nearly
impossible to detect owing to the extreme brightness of the Galactic emission
at these frequencies.

Introduction
Even though the existence of Hydrogen fine structure lines and their explanation using Dirac’s
atomic theory has been known for close to a century, a Hydrogen fine structure line has never been
detected from an astrophysical object. An interesting Hydrogen fine structure line is the 2p–2s
transition. The main difficulty in detecting this line is that the line strength is proportional to the
population of either the 2p or 2s states which, being excited states, are not so readily populated
in most astrophysical circumstances. Moreover, the line width of the excited 2p state, which is
determined by its decay time, is large (99.8 MHz), making the detection of the fine structure line
a difficult observation. One astrophysical setting where the feasibility of detecting such a line has
been studied is the interstellar H II regions (see, e.g., Dennison, Turner, & Minter 2005 and refer-
ences therein; Ershov 1987); in H II regions, the excited levels are populated by recombination.
http://arxiv.org/abs/0704.1226v1
– 2 –
Here we consider two cosmological settings in which the excited levels are populated by either
recombination or pumping by Lyman-α photons from an external source: (a) The Recombination
epoch: The Universe makes a transition from a fully ionized to an almost fully neutral medium at
z ≃ 1089 (Spergel et al. 2006; for details see, e.g., Peebles 1993 and references therein). During
this era, as the density and temperature of the Universe drops, recombination is stalled owing to a
high Lyman-α radiation density and progresses either by the depopulation of the 2p state owing to
redshifting of the photons out of the line width or the 2-photon decay of the 2s state. This results
in a significant 2s and 2p level population during the recombination era. (b) The Reionization
epoch: Recent observations suggest that the universe made a transition from nearly fully neutral to
fully ionized within the redshift range 6 <∼ z
∼ 15 (Page et al. 2006; White et al. 2003; Fan et al.
2002; Djorgovski et al. 2001; Becker et al. 2001). It is widely believed that this ‘reionization’
was achieved by the percolation of individual H II regions around the sources of reionization. The
nature of these sources is not well understood: they might be Pop III stars, active galactic nuclei
or star-forming galaxies. During this epoch, a signal from the 2p − 2s fine-structure transition
might originate from either within the cosmological H II regions or from the almost fully neutral
medium surrounding the H II region. The level population of the first excited state in the former
case would be largely determined by recombinations and in the latter case by Lyman-α photons
from the central source. We shall show below that for most cases of interest the fine-structure line
from within the cosmological H II region might be negligible as compared to the signal from the
regions immediately surrounding the H II region.
Throughout this work we adopt the currently-favoured ΛCDM model: spatially flat with
Ωm = 0.3 and ΩΛ = 0.7 (Spergel et al. 2006; Riess et al. 2004; Perlmutter et al. 1999) with
2 = 0.022 (Spergel et al. 2006; Tytler et al. 2000) and h = 0.7 (Freedman et al. 2001).
2. Fine-structure lines from the reionization epoch
Subsequent to the recombination of the primeval baryon gas at redshift z ≃ 1089 (Spergel
et al. 2006) and the transformation of the gas to an almost completely neutral state, it is believed
that the gas was reionized during epochs corresponding to the redshift range 6 <∼ z
∼ 15. WMAP
measurements of cosmic microwave background radiation (CMBR) anisotropy in total intensity
and polarization have been used to infer that the baryons were likely neutral at redshifts z >∼ 12 −
15; however, the detection of CMB polarization anisotropy requires substantial ionization by about
z ≃ 11 (Page et al. 2006). Observationally, the Gunn-Peterson (GP) test shows that the universe is
highly ionized at redshifts lower than z ≃ 5.5; the detection of GP absorption at greater redshifts
suggests that the neutral fraction of the intergalactic hydrogen gas rises to at least 10−3 in the
redshift range 5.5 <∼ z
∼ 6, and that reionization was not complete till about z ≃ 6 (White et al.
– 3 –
2003; Fan et al. 2002; Djorgovski et al. 2001; Becker et al. 2001). However, from the GP test
alone it is not possible to infer the neutral fraction of the medium; it only gives a rather weak lower
limit of ≃ 10−3 on the neutral fraction of the universe for z >∼ 6. From other considerations it is
possible to put more stringent bounds on the neutral fraction; for example, Wyithe & Loeb (2004)
obtain a lower limit of 0.1 on the neutral fraction of the universe at z ≃ 6.3 (see also Mesinger &
Haiman 2004).
Our understanding of the nature of the sources that caused the reionization is far from com-
plete. The transition from an almost completely neutral gas to a highly ionized gas during redshifts
6 <∼ z
∼ 15 is a key problem in modern cosmology and considerable theoretical and experimental
efforts are currently directed at this unsolved problem. Here we propose a new method, based
on the 2p − 2s fine-structure transition, to determine the evolution of the neutral fraction of the
intergalactic medium within this epoch.
Owing to fine structure splitting, the two possible transitions between the 2s and the 2p states
are: 2p1/2–2s1/2 at a frequency ≃ 1058MHz that has an Einstein A coefficient 1.6× 10
−9 s−1 and
2p3/2–2s1/2 at a frequency νps ≃ 9911MHz that has an Einstein A coefficient 8.78 × 10
−7 s−1.
The Einstein A coefficient for the latter transition is more than an order of magnitude greater than
the former; therefore, in this work we consider only the 2p3/2–2s1/2 transition and hereinafter we
refer to this specific transition simply as the 2p–2s transition.
The ionizing UV photons from sources in the reionization era create ‘Stromgren spheres’.
Whereas the gas in the cosmological H II regions are highly ionized by the photons, the ioniza-
tion level of the gas beyond the Stromgren spheres is determined by the history of the gas, the
density, and the mean specific intensity of the background ionizing photons, which includes both
the photons diffusing out of the Stromgren spheres as well as the background radiation field. In
this work we assume that the ionizing sources at these high redshifts are AGNs and in illustrative
examples adopt parameters of a few QSOs that have been observed at z ≃ 6. The photons at
the Lyman-α transition frequency (here and throughout, unless otherwise specified, we shall con-
tinue to refer to frequencies between Lyman-α and Lyman-limit as ’Lyman-α) from a high redshift
QSO escape the mostly-ionized Stromgren sphere and are strongly scattered and absorbed in the
medium beyond. The population of the 2p level in this region is determined by (a) the intensity of
Lyman-α photons from the central source, (b) recombination rate of free electrons, (c) absorption
of CMBR photons by electrons in the 2s state (it is assumed here and throughout this work that the
only radio source at high redshifts is the CMBR) and (d) collisional transition from the 2s state.
The 2s state is populated via (a) the recombination rate of free electrons, (b) collisional transfer
of atoms from the 2p state, (c) the spontaneous decay of the 2p state, and (d) transition from the
2p state stimulated by CMBR photons. Additionally, the absorption of photons from the central
sources, with energy equal to or in excess of the Lyman-β transition, would result in electronic
– 4 –
transitions to the second excited state, which could be followed by spontaneous decay to the 2s
state. (It might be pointed out here that both the 2s and 2p states could also be populated by atoms
cascading from excited states with n > 3. In particular, all photons absorbed from 1s states to any
excited state can directly de-excite to the 2s level. However, the rate of transition from 1s to any
excited state is roughly ∝ 1/n3 (e.g. Rybicki & Lightman 1979) and, therefore, we include only
the most dominant transition in each case.)
The population of the ground state is denoted by n1s. We denote the level populations of the
the two states—2s1/2 and 2p3/2—by the number densities n2s and n2p; these may be solved for,
respectively, from the following two equations of detailed balance:
i + cB2p2sn2pnCMBR + Cpsnin2p + A2p2sn2p + cn1sp32
B13,βφ13(ν)nα(ν)dν
= A2s1sn2s + Cspnin2s + cB2s2pn2snCMBR, (1)
(1− 2f)αBn
i + cB2s2pn2snCMBR + Cspnin2s + cn1s
B12,αφ12(ν)nα(ν)dν
= A2p1sn2p + Cpsnin2p + cB2p2sn2pnCMBR. (2)
Here f is the fraction of all the atoms that recombine to the 2s state. In equilibrium f = 1/3
as the n = 2 state splits into three doublets: 2p1/2, 2p3/2 and 2s1/2. αB is the recombination
coefficient and ni is the density of the ionized gas. B2p2s = B2s2p = c
2/(8πν3ps)A2p2s is the Einstein
B coefficient for the 2p3/2–2s1/2 transition in terms of the corresponding Einstein A coefficient
A2p2s (note that the two B coefficients are equal as the two states have the same degeneracy).
Cps = Csp = 5.31 × 10
−4 cm3 s−1 is the rate coefficient of transition owing to collisions with
electrons. nCMBR is the number density of CMBR photons within the transition line width. nα(ν)
is the number density (per unit frequency) of photons with frequency equal to or larger than the
Lyman-α frequency (and smaller than the Lyman-limit frequency) at any location; φ13 and φ12
are, respectively, the line profiles corresponding to the Lyman-β and Lyman-α transitions, and
p32 is the probability for the electron transition to the 2s state following excitation to n = 3 via
absorption of a Lyman-β photon. We have not included the induced Lyman-α transition because
the number density of atoms in the 2p state is negligible as compared to that in the 1s state. p32 is
the probability that an atom in the third excited state (3p) will decay to the 2s state. A2s1s is the
Einstein A coefficient corresponding to the 2-photon decay of the 2s state. Other symbols have
their usual meanings.
Owing to the fact that the mostly neutral medium in the vicinity of the cosmological Strom-
gren spheres is optically thick to Lyman-α scattering, the Lyman-α photons from the decay of
the 2p state are strongly scattered by the gas. Therefore, nα(ν) will contain contributions from
both the Lyman-α photons from the central source as well as the Lyman-α photons that arise
– 5 –
from recombinations outside the Stromgren sphere and are multiply-scattered therein: nα(ν) =
nsourceα (ν) + n
α (ν). We neglect the multiply-scattered Lyman-α photons from the Stromgren
sphere that have been reprocessed via recombination within the Stromgren sphere because these
would be redshifted redward of the Lyman-α line before encountering the boundary of the Strom-
gren sphere. The scattering of recombination photons in an optically thick, expanding medium is a
complex problem (Field 1959; Rybicki & Dell’Antonio 1994). One of its first applications was to
study the recombination of primeval plasma (Peebles 1968; Zeldovich, Kurt & Sunyaev 1969). In
these analyzes it was implicitly assumed that apart from 2-photon decay, in an expanding universe
the dominant effect that results in resonant photons ceasing interaction with the gas, and leaving
the system, is its redshifting out of the line profile. The effect of scattering off the moving atoms
was deemed to be either negligible or at best comparable. This assumption has been borne out by
more recent detailed analysis that have taken into account the effect of scattering on the photon
escape (Krolik 1990). Taking only the redshift as the main agent of photon escape, it can be shown
that the net effect of the scattering of a resonance photon before it drops out of consideration is to
reduce the decay time of the 2p state from A21 = 6.2× 10
8 s−1 to A21/τGP (Zeldovich et al. 1969;
for more recent work see, e.g., Chluba, Rubino-Martin & Sunyaev 2007, Seager, Sasselov & Scott
1999 and references therein; we give a concise derivation in Appendix A). In this expression, the
Gunn-Peterson optical depth τGP = [3/(8πH)]A2p1sλ
αn1s.
Similar complications exist in computing p32, the probability that an atom in the third excited
state will decay spontaneously to the 2s state, in an optically thick medium. In optically thin media,
p32 = A32/(A32 + A31); on the other hand, in an optically thick medium, we would need to take
into account the ‘trapping’ of the Lyman-β photon owing to resonant scattering. The effect of this
scattering in an optically thick medium would be that the fraction of photons that decay directly
to the ground state are reabsorbed ‘locally’ to the third excited state and, therefore, all photons
absorbed to the third excited state result in an Hα photon and an atom in the 2s state. This means
that the appropriate value of p32 is close to unity in an optically thick medium: in this work we
assume p32 = 1.
The astrophysical setting in which we seek solutions to the algebraic equations above is cos-
mological H II regions at high redshift. In particular, we are interested in the signal from the neutral
region surrounding the cosmological H II region. For a fully neutral inter-galactic medium (IGM)
at z ≃ 6.5, τGP ≃ 6 × 10
5. If we adopt spectral luminosities corresponding to QSOs observed at
these high redshifts, it may be shown that the populating of the 2p state via direct recombinations
from the free-free state, collisional transfer from the 2s state, and upward transitions from the 2s
to 2p state arising from absorption of background CMBR photons may all be neglected. (The rele-
vant parameters are: ni ≃ nb ≃ 2.8(1 + z)
3 cm−3 in the H II region surrounding the sources, with
ni expected to be much smaller in the neighboring mostly neutral medium; the number density
of CMBR photons that might cause a 2p–2s transition is ≃ 5(1 + z) cm−3; the number density
– 6 –
of Lyman-α photons from the central source, assuming luminosities typical of SDSS quasars at
z ≃ 6.5 (more details in §4), is nα ≃ 10
−4 cm−3. First, for these parameters, the dominant process
that populates the excited state is the pumping by Lyman-α photons. Second, it may be readily
verified that for these plausible values for the parameters the signal expected from the H II region
surrounding the central source is negligible as compared to the signal from the surrounding neu-
tral region.) Given that the Lyman-α flux from the central QSO is the dominant causative factor
for populating the 2p state, the two equations (1 and 2) that determine the level populations is
simplified. The number density of atoms in the 2p state is given approximately by:
n2p ≃ n1sΓατGP/A2p1s, (3)
where n1s = fneunb, with fneu denoting the neutral fraction and nb ≃ 2.7 × 10
−7(1 + z)3 cm−3
is the number density of baryons in the IGM. Γα =
B2p1sφ12(ν)n
source
α (ν) (in units of s
−1) is
the transition rate to the 2p state owing to the Lyman-α photons from the central source, where
nsourceα (ν) is the number density of Lyman-α photons from the central source alone.
Similarly, the dominant process that determines the population of the 2s state is the absorption
of Lyman-β photons (for details, see the discussion above and §4) and the subsequent decay to the
2s state. The number density of atoms in the 2s state is:
n2s ≃ n1sΓβ/A2s1s, (4)
where Γβ =
B3p1sφ13(ν)n
source
α (ν) (in units of s
−1) is the transition rate to the 2s state owing to
the Lyman-β photons from the central source.
3. Fine structure lines from the epoch of recombination
The universe made a transition from fully ionized to nearly fully neutral at z ≃ 1089 (Spergel
et al. 2006; Peebles 1968; Zeldovich et al. 1969). This transition is mainly accomplished by the
2-photon decay of the 2s state and the slow redshifting of the Lyman-α photons which deplete the
2p state (see Peebles 1993 and references therein for a detailed discussion). The Saha ionization
formula, valid for thermodynamic equilibrium conditions between the hydrogen level populations
and free electrons, is a poor approximation for studying the epoch of recombination. A good ap-
proximation for studying this transition is to assume that all states, excepting 1s, are in equilibrium
with the CMBR (matter temperature to a very good approximation remains equal to the CMBR
temperature throughout this transition) (Seager et al. 1999; Peebles 1968). However, in this ap-
proximation, where the matter temperature and all transitions, excepting the Lyman-α line, are in
thermal equilibrium, the 2p–2s signal is unobservable because the excitation temperature for this
transition equals the background radiation temperature. Even though this second approximation
– 7 –
might be useful for studying the evolution in ionization fraction, it will be strictly true only if the
dominant mechanisms that determine the level populations of the 2p and the 2s states are either
interaction with the CMBR photons or collisions between atoms. As there are a variety of other
processes relevant to the determination of the level populations—for example, the free decay of
either of the two states—a small deviation from equilibrium is expected in the 2s and 2p level
populations and it is our aim here to compute it.
One approach to this problem is to simultaneously solve for the level populations of the 2p
and the 2s states as well as the change in the ionization fraction. However, assuming thermal
equilibrium between these two states is a good approximation for solving the evolution of ioniza-
tion. Therefore, the approach we have adopted is to solve for the evolution of ionization using the
method of Peebles (1968), and use the resulting ionized/neutral fraction to solve for the populations
of the 2s and the 2p states using detailed balance. The resulting equations are:
i + cB2p2sn2pnCMBR + Cpsnin2p + A2p2sn2p + n1sA2s1s exp(−(B1 − B2)/(kBTCMBR))
= A2s1sn2s + Cspnin2s + cB2s2pn2snCMBR + βcn2s (5)
(1− 2f)αBn
i + cB2s2pn2snCMBR + Cspnin2s
= A2p1sn2p/τGP + Cpsnin2p + cB2p2sn2pnCMBR + βcn2p,(6)
where B1 = 13.6 eV and B2 = 2.4 eV are, respectively, the ionization potentials of the ground
and the first excited states, kB is the Boltzmann’s constant, and TCMBR is the temperature of the
CMBR. βc is the rate at which the CMBR photons cause a bound-free transition of electrons from
the n = 2 state (2s or 2p states). The various other terms in these equations have the same meanings
as in the previous case. The main difference is that the CMBR photons and baryonic matter at the
time of recombination are hot and dense enough to directly affect the level populations of the
excited states by ionizing the excited state, and the two-photon capture to the excited state is not
completely negligible (see Peebles 1993 and 1968 for details of the different physical processes
that are relevant at this epoch).
4. Expected brightness of the fine structure lines
The brightness temperature in the 2p–2s transition is:
∆Tb ≡ Tb − TCMBR = g⋆
n2p(0)hpA2p2sλ
e(1 + z)
8πkBH(z)
(1− TCMBR/Tex) ≡ τex(Tex − TCMBR) (7)
Here n2p(0) = n2p/(1 + z)
3 and λe is the rest wavelength of the 2p–2s transition. Tex =
(hpνe/kB)[n2p/(n2s − n2p)] is the excitation temperature corresponding to the transition, where
– 8 –
hp is the Planck constant. τex is the optical depth of the source in the 2p–2s signal. g⋆ takes into
account the selection rules for transitions between the 2p3/2 and 2s1/2 levels. Given the selec-
tion rules there are three allowed transitions between these two states; they occur at frequencies
ν ≃ 9852, 9875, and 10029 MHz (see, e.g., Ershov 1987). The first two transitions are blended by
the natural width of the line, which is approximately 100 MHz, but the third should be observable
as a distinct line. This implies that g⋆ = 2/3 if the observing frequency is ≃ 9900/(1 + z) MHz.
4.1. Expectations for signals from cosmological H II regions
For computing the strength of this signal in a typical case, we adopt observed parameters
of QSO SDSS J1030+0524 (see, e.g., Wyithe & Loeb 2004 and references therein), which is at
redshift z = 6.28 and shows no detectable flux beyond the QSO Stromgren sphere: the radius
of the Stromgren sphere has been estimated to be R ≃ 4.5Mpc (Mesinger & Haiman (2004)
argue that the size of the Stromgren sphere could be roughly 30% higher; this makes no essential
difference to our results). It may be noted here that the QSO SDSS J1148+5251 also has similar
parameters. We arrive at an estimate of nsourceα (ν) by assuming that Lα photons per second, with
wavelengths corresponding to the Lyman-α transition, are emitted by the central source in an
effective frequency range ∆νsource, and that the photons are absorbed at a radial distance R. This
leads to:
nsourceα = Lα/(4πR
∆νsource
. (8)
We assume typical values: Lα = 10
58 s−1, R = 4.5Mpc and ∆νsource = 30, 000 km s
−1. These
lead to the following estimate for the transition rate, Γα, in the gas at the boundary of the Stromgren
sphere arising due to the photons from the central QSO:
Γα ≃ 8× 10
−10 s−1. (9)
It should be noted that the mean specific intensity of Lyman-α photons in the IGM would also
give a non-zero signal. Assuming a mean specific intensity of ≃ 10−21 erg cm−2 sec−1 Hz−1 sr−1
(this might be needed to couple the HI spin temperature to the matter temperature; see e.g. Madau,
Meiksin, & Rees 1997), the expected signal is many orders of magnitude smaller than we have
computed from the outskirt of bright sources. If we assume that the central sources are continuum
emitters in the frequency range between Lyman-α and Lyman-limit frequencies, we may assume
similar parameter values for computing the expectations for Γβ. In that case,
Γβ ≃ Γα
, (10)
– 9 –
where fβ and fα are the oscillator strengths of the Lyman-β and Lyman-α transitions, respectively.
From Eqs. (3), (4) and (10):
A2p1s
A2s1sτGP
. (11)
For a completely neutral medium at z ≃ 6.4, τGP ≃ 6 × 10
5, which may be used to show that
n2s ≫ n2p for the parameters of IGM in the redshift range of interest: 6.4 <∼ z
∼ 10. This implies
that in the outskirts of the Stromgren sphere surrounding the QSO, the transition is expected to
be observable as an absorption feature against the CMBR. Using Eqs. (3) and (4) in Eq. (7), the
observable brightness temperature is estimated to be:
∆TB ≃ −20 µK
, (12)
where fneu is the neutral fraction of hydrogen outside the Stromgren sphere, and might be close to
unity.
We have adopted parameters typical of QSOs observed at the edge of the reionization epoch
in estimating the above temperature decrement. The main uncertainty above is in the estimation of
the ‘Lyman-α’ flux from the central source, and as the observed temperature decrement is directly
proportional to this flux from the central source, this constitutes a major uncertainty in reliably
computing the expectations for the signal. For QSOs that have strong Lyman-α and Lyman-β
lines, the blueward side of the lines will be strongly absorbed in the medium just beyond the
Stromgren sphere (and this has been observed to happen in many cases), provided that the blueward
side photons have not been redshifted to frequencies smaller than the Lyman-α frequency while
transversing the Stromgren sphere. In the case of QSOs that have large line fluxes and small
Stromgren spheres, the Lyman-α luminosity Lα might be underestimated.
4.2. Expectations for the fine-structure line from recombination
Using equations (5) and (6) the level populations of the 2p and the 2s states may be computed.
Solving for the level populations and using equation (7), we have computed the expected signal
from the recombination epoch; the expected signal is shown in Fig. 1. The fine structure line
transition is expected to be an absorption feature, with a maximum temperature decrement of order
100 µK at an observing frequency of about 7 MHz. The width of the decrement in frequency space
corresponds to a redshift span ∆z ≃ 200, which is roughly the width of the visibility function at
recombination.
– 10 –
5. Prospects for the detection of the cosmological fine structure lines
As Eq. (12) shows, a detection of the fine structure line in the outer regions of cosmological
H II regions is potentially a probe of the cosmological neutral hydrogen density in the vicinity of
QSOs in the reionization epoch, and might be a tool for the investigation of the evolution in the
neutral fraction with cosmic epoch through the reionization era. Given the cosmological impor-
tance of such a measurement, and the fact that there does not exist many reliable methods for the
detection of H I at high redshifts (see, e.g., Barkana & Loeb 2001), the detection of the 2p–2s line
transition in the cosmological context assumes additional importance.
In deriving equation (7) the Hubble expansion was assumed to be the only cause for the
velocity width in the observed line. However, an important contribution to the velocity dispersion
in the line in this case is the natural width of the fine structure line: owing to the rapid decay of
the 2p state, the natural Lorentzian width in the rest frame of the gas is VLor ≃ 100MHz (see,
e.g., Dennison et al. 2005). Therefore, the peak brightness in the observed line profile might
be suppressed by a factor Vexp/VLor, where Vexp is the line-of-sight peculiar velocity dispersion
owing to the Hubble flow across the region being observed. However, in the case Vexp >∼ VLor this
suppression has a negligible effect. For QSOs at redshift z ≈ 6.5, the natural Lorentzian width
of the fine structure line is equivalent to the peculiar Hubble flow across a proper line-of-sight
distance of ≃ 4Mpc. This distance is approximately the size of the Stromgren spheres around
QSOs at that redshift; therefore, the natural width of the line does not significantly diminish the
expectations, given by equation (7), for the peak brightness temperature. A second inference is
that in the case of a QSO that has a smaller Stromgren sphere, the increase in mean brightness
temperature is roughly proportion to the inverse of the radius of the Stromgren sphere: 1/R, and
not 1/R2.
Another assumption that was made while deriving Eq. (7) is that the only radio frequency
radiation that needs to be considered for the determination of the level populations and brightness
temperature decrement is the CMBR. It may be that the central ionizing source, which may be a
QSO, is radio loud. There may also exist radio sources behind the observed Stromgren sphere and
within the angular region over which the Lyman-α flux from the QSO is appreciable. Equation
(7) may be modified, to account for this, by replacing TCMBR with TCMBR + TB, where TB is the
brightness temperature of the radio source at wavelengths corresponding to the rest frequency of
the transition, which is ≃ 9GHz. The result would be to enhance the brightness temperature of
the line signal by ≃ TB/TCMBR. If the radio source is unresolved, it is appropriate to estimate the
expected signal in terms of the optical depth. The optical depth corresponding to equation (12) is
≃ 10−6.
We now discuss the feasibility of the detection of the fine structure line towards SDSS J1030+0524.
We shall assume that the neutral fraction, fneu, outside the Stromgren sphere of this QSO is unity,
– 11 –
consistent with the measured GP trough. The redshifted fine structure line would be expected
at ≃ 1.36GHz. The observed line ‘width’, considering natural broadening and the Hubble flow
across the Stromgren sphere, is expected to be approximately ∆ν ≃ 100MHz/(1 + z); using
z = 6.28, we obtain ∆ν = 13.7MHz. The fine structure line would be expected to originate in a
shell that is roughly the size of the Stromgren sphere, and fall of as 1/r2 beyond this shell, where r
is the distance from the QSO. In the case of SDSS J1030+0524, the angular size of the Stromgren
sphere is expected to be 15 arcmin.
The frequency of the redshifted line is in the observing bands, and the line width is within the
spectral line capabilities, of several currently operational telescopes. However, large-collecting-
area arrays like the Giant Metrewave Radio Telescope (GMRT) have large aperture antennas of
45-m diameter and, therefore, poor surface brightness sensitivity for such extended structures.
The Australia Telescope Compact Array (ATCA), with 22-m antennas, has reasonable brightness
sensitivity for this problem and, additionally, operates with 128-MHz bandwidths in spectral line
mode. At ν ≃ 1.3GHz, the ATCA has a system temperature Tsys ≃ 25K and antenna efficient
K = 0.1K Jy−1. The five movable 22-m diameter antennas may be configured into an ultra-
compact 2-D close-packed H75 (75-m maximum baseline) array, and this would yield a number
of baselines sensitive to the 15-arcmin scale fine-structure line signal. The brightness sensitivity
of this array, for a 8-arcmin scale structure, is 400 µK in 6 hr integration time. The brightness
sensitivity in Fourier synthesis images could be enhanced somewhat, by factors of a few, by ap-
propriately weighting the baselines to match the synthetic beam to the expected structure scale;
however, the required integration times for a detection are still in the ball park of 103 hr.
The detection of the signal from cosmological H II regions, however, might be feasible using
facilities under construction or planned for the near future, like the xNTD in Australia or the Square
Kilometer Array (SKA). These arrays would have smaller antenna sizes—making the detection of
these large-angular-scale structures detectable in interferometers—and significantly more numbers
of antennas, giving more numbers of short baselines that would usefully respond to the large-
angular-scale fine-structure line. However, the array configuration designs would have to factor in
the extraordinary brightness temperature sensitivity requirements for this demanding observation.
As an example, the SKA might consist of about 104 12–15 m class reflector antennas, with
aperture efficiency of 60%, and the system temperature at 1.4 GHz might be about 20 K. Assuming
that the visibilities are optimally weighted so that the synthetic beam of the Fourier synthesis array
is matched to the source size of about 15 arcmin FWHM, the line strength would have a peak of
about 20 µJy. A 5-σ detection of the fine structure line towards a typical QSO, in a reasonable
integration time of about 1 hr, will require that about 250 baselines (just 0.001% of the total) be
within about 50 m.
The signal from the recombination epoch is observable in the frequency band 6–8 MHz as
– 12 –
a broad decrement in the brightness temperature of the extragalactic background sky, and would
be extremely small compared to the orders of magnitude more intense Galactic non-thermal emis-
sion as well as the average low-frequency background brightness temperature arising from the
numerous extragalactic radio sources. This decrement may be considered to be a distortion to the
CMBR spectrum at long wavelengths, and would be an all-sky cosmological signal. However,
owing to the ionosphere, the frequency range in which this feature is expected to appear is too
low a radio frequency to be easily accessible using ground-based observatories. Therefore, even
though the observation of this signal might be yet another tool to probe the epoch of recombi-
nation, new custom-made instruments, which should presumably operate from space and above
the ionosphere, will require to be built if a detection of this signal is to be attempted. An addi-
tional cause for concern is that the Galactic and Extragalactic background radiations might have
low-frequency spectral turnovers at these frequencies, as a result of free-free absorption as well as
synchrotron self-absorption, and these would result in significant spectral features in the band that
would require a careful modelling in order to detect any CMBR decrement feature arising from
fine structure transition absorption. Interference from terrestrial man-made transmitters, as well as
from auroral phenomena and solar system objects would also be an issue.
6. Summary and Discussion
We have discussed the possibility of detecting the hitherto undetected fine structure line of
2p–2s transition in two cosmological settings: the epoch of recombination at z ≃ 1089 and the
epoch of reionization at z ≃6–15.
The expectations for the line from the environments of ionization sources in the epoch of
reionization are interesting and worthy of attention as a novel tool for the investigation of the
reionization process and the cosmological evolution of the gas. The signal is expected to be ob-
servable as an extended and weakly absorbing source, which causes a decrement in the brightness
of the background CMBR, and may be detected by interferometers as a negative source akin to
the Sunyaev-Zeldovich decrements observed along the lines of sight through hot gas in clusters
of galaxies. Detection of the 2p–2s line signal from the outskirts of cosmological H II regions at
different redshifts within the reionization era may serve to determine the neutral fraction of the
medium during the epoch of reionization, which is a quantity of significant interest in modern cos-
mology. In particular, we have computed a representative signal strength by adopting parameters
typical of a QSO observed at z ≃ 6. These QSOs show GP troughs in their spectra; however,
owing to the weakness of the GP test, the spectra have only been useful in setting weak limits
on the neutral fraction (constraining the neutral fraction to be >∼ 10
−3) outside the observed H II
region. We have shown that for a fully neutral medium the line peak may reach ≃ 20µK, which is
– 13 –
potentially observable by radio interferometers that are being designed today.
Other interesting probes of the reionization epoch include detecting OI line from this epoch
(Hernandez-Monteagudo et al. 2006).
It is of interest to compare the relative difficulty associated with detecting the neutral hy-
drogen in this indirect way, using the redshifted fine structure line, with direct imaging of the
redshifted 21-cm line from neutral hydrogen during the epoch of reionization (see, e.g., Sethi
2005; Zaldarriaga, Furlanetto, & Hernquist 2004). The ‘all-sky’ H I signal might be detectable
with a peak strength of ≃ 50mK; however, it could be very difficult to detect owing to calibra-
tion and foreground contamination issues (e.g., Zaldarriaga et al. 2004; Shaver et al. 1999, and
references therein). A better approach might be to attempt to detect the fluctuating component
of the sky signal, which could have peak intensities of ≃ a fewmK at observing frequencies of
ν ≃ 100–200MHz (Zaldarriga et al. 2004; Shaver et al. 1999). This translates to roughly the
same signal strength (specific intensity) as we have obtained in Eq. (12) for the fine structure line.
This is not entirely unexpected: even though the level populations of the excited states are much
smaller as compared to the ground level population needed in computing the HI signal, the A coef-
ficient of the fine structure transition we consider here is roughly 8 orders of magnitude larger than
the HI hyperfine transition A coefficient. Currently, there are many ongoing and planned radio
interferometer experiments for the detection of the redshifted H I emission/absorption from the
epoch of reionization (e.g., Pen, Wu, & Peterson 2004; the LOFAR project at www.lofar.org;
the MWA project at www.haystack.mit.edu/ast/arrays/mwa/site/index.html).
The detection of the H I signal is firstly more difficult—requiring greater sensitivity—because the
typical frequency width of the signal ≃ 0.5MHz (Zaldarriga et al. 2004), which is far smaller than
the typical width expected for the fine structure line (≃ 10MHz). Second, the fine structure line
is expected at 1.4 GHz, where the sky background temperatures are significantly lower than in the
100–200MHz band, making the telescope system temperatures lower. An additional advantage
of the the indirect detection is that the redshifted fine structure line appears at higher frequencies
( >∼ 1GHz), which are relativity free of interference as compared to the low frequency band of
100–200MHz. The main advantage of the direct detection is that unlike the method we suggest
here, it is independent of the existence of strong Lyman-α emitters at high redshifts. Another ad-
vantage of direct detection is that it may be detected ‘statistically’ and such a detection might be
achieved with greater ease than direct ‘imaging’ (e.g Zaldarriaga et al. 2004; Bharadwaj & Sethi
2001); however, the foreground subtraction problem becomes a severe constraint for a statistical
detection. To summarize: if strong Lyman-α emitting sources are present at high redshifts, they
would facilitate the indirect detection of the neutral hydrogen via enabling the detectability of the
fine structure line. The imaging issues and problems associated with the detection of this signal
appears to be less of a challenge as comparison to the direct detection of redshifted H I from those
epochs.
– 14 –
Another astrophysical context in which the fine structure line might be detectable is the envi-
ronments of high redshift galaxies, which are strong Lyman-α emitters. As equation (12) shows,
the observed signal depends on the neutral fraction outside the Stromgren sphere. Therefore, de-
tection of this signal would constitute an alternate probe of the neutral fraction of the IGM at large
redshifts.
The aim of this work has been to examine the detectability of the fine structure line in cos-
mological contexts, to point out the cosmological significance of detections, and spawn work that
may refine the modelling presented herein and improve the case for appropriate design of future
telescopes, which might enable the detection of the fine structure line towards multiple sources in
the reionization era.
Appendix A: Photon distribution function
The evolution of the photon distribution function, neglecting the effect of scattering off mov-
ing atoms, is (see e.g. Rybicki & Dell’Antonio 1994):
= A2p1sn2pφν − cBνn1sφνnν . (13)
Here Bν = 3/(8π)c
2/ν2αA2p1s and H = ȧ/a. The equation may be written as:
= −1/τ(nν − n⋆). (14)
Here τ = 1/(cBνφνn1s) and f⋆ = A2p1sn2p/(Bνn1sc). The equation above lends itself to a ready
interpretation. If the second term on the left hand side (which is owing to the expansion of the
Universe) was absent, the distribution function will approach f⋆ on a time scale ≃ τ , where τ ≃
3 × n1s s ≪ ȧ/a ( n1s here has units cm
−3). It may be readily verified that for the recombination
epoch and also the epoch of reionization, τ ≪ 1/H , excepting when the neutral fraction of the
medium is very small. We work here with the assumption that the neutral fraction is always large
enough so that τ ≪ 1/H . In this case the solution to Eq. (14) may be simplified: to leading order
the distribution function approaches f⋆ and the first order term (of the order of τH) represents the
slow time variation of the distribution function owing to the expansion of the Universe. In this
approximation, one may write the solution to Eq. (14) as:
nν ≃ n⋆ − τH(t)n⋆. (15)
Using these equations we may proceed to prove the contention that the net effect of the ‘trapping’
of Lyman-α photons is to reduce the decay time of the 2p state by a factor τGP . The 1s–2p transition
– 15 –
rate, which is given by Eq. (6), may be solved using Eq. (15):
Bνφ(ν)n(ν)dν ≃ A2p1sn2p − A2p1sn2p/τGP . (16)
The first term on the right hand side cancels with the decay term of the 2p state on the right hand
side of Eq. (6), and, therefore, the net effect of the scattering of recombination photons is to reduce
the decay time of the 2p state by a factor τGP . It may be pointed out that the condition needed to
derive the above expression roughly translates to the condition that τGP ≫ 1. For the reionization
case, this requires that the neutral fraction >∼ 10
−5. In the case of primordial recombination, it
leads to an even weaker condition that the neutral fraction is >∼ 10
Acknowledgment
One of us (SKS) would like to thank Jens Chluba for many useful discussions and to Zoltan
Haiman for many useful comments on the manuscript.
REFERENCES
Barkana, R. and Loeb, A. 2001, Phys. Rep., 349, 125
Becker, R. H. et al. 2001, AJ, 122, 2850
Bharadwaj, S. & Sethi, S. K. 2001, J. Astrophys. Astr., 22, 293
Chluba, J., Rubino-Martin, J. A., & Sunyaev, R. A. 2007, MNRAS, 374, 1310
Dennison, B., Turner, B. E., & Minter, A. H. 2005, ApJ, 633, 309
Djorgovski, S. G., Castro, S., Stern, D., & Mahabal, A. A. 2001, ApJL, 560, L5
Ershov, A. A. 1987, Sov. Astron. Lett. 13(2), 115
Fan, X., Narayanan, V. K., Strauss, M. A., White, R. L., Becker, R. H., Pentericci, L., & Rix, H.
2002, AJ, 123, 1247
Field, G. 1959, ApJ, 129, 551
Freedman, W. L. et al. 2001, ApJ, 553, 47
Hernandez-Monteagudo, C., Haiman, Z. Jimenez, R., & Verde, L. 2006, astro-ph/0612363
http://arxiv.org/abs/astro-ph/0612363
– 16 –
Krolik, J. H. 1990, ApJ, 353, 21
Madau, P., Meiksin, A., & Rees, M. J. 1997, 475, 429
Mesinger, A. and Haiman, Z. 2004, ApJL, 611, 69
Page, L. et al. 2006, astro-ph/0603450
Peebles, P. J. E. 1993, Principles of Physical Cosmology, Princeton University Press
Peebles, P. J. E. 1968, ApJ, 153, 1
Pen, U.-L., Wu, X.-P., Peterson, J. 2004, astro-ph/0404083
Perlmutter, S. et al. 1999, ApJ, 517, 565
Riess, A. G. et al. 2004, ApJ, 607, 665
Rybicki, G. B. & Dell’Antonio, I. P. 1994, ApJ, 427, 603
Rybicki, G. B. & Lightman, A. 1979, Radiative Processes in Astrophysics, Wiley-Interscience
Seager, S., Sasselov, D. D., & Scott, D. 1999, ApJL, 523, L1
Sethi, S. K. 2005, MNRAS, 363, 818
Shaver, P. A., Windhorst, R. A., Madau, P. & de Bruyn, A. G. 1999, A&A, 345, 380
Spergel, D. N. et al. 2006, astro-ph/0603449
Tytler, D., O’Meara, J. M., Suzuki, N., & Lubin, D. 2000, Physics Reports, 333, 409
White, R. L., Becker, R. H., Fan, X. & Strauss, M. A. 2003, ApJ, 126, 1R
Wyithe, J. S. B. & Loeb, A. 2004, Nature, 432, 194
Zaldarriaga, M., Furlanetto, S. R. & Hernquist, L. 2004, ApJ, 608, 622
Zeldovich, Ya. B., Kurt, V. G. & Sunyaev, R. A. 1969, JETP Lett, 28, 146
This preprint was prepared with the AAS LATEX macros v5.2.
http://arxiv.org/abs/astro-ph/0603450
http://arxiv.org/abs/astro-ph/0404083
http://arxiv.org/abs/astro-ph/0603449
– 17 –
Fig. 1.— The expected brightness temperature decrement in the background radiation, owing
to the fine structure transition in gas at the recombination epoch, is plotted versus the observing
frequency.
	Introduction
	Fine-structure lines from the reionization epoch
	Fine structure lines from the epoch of recombination
	Expected brightness of the fine structure lines
	Expectations for signals from cosmological H ii regions
	Expectations for the fine-structure line from recombination
	Prospects for the detection of the cosmological fine structure lines
	Summary and Discussion
