# Metals at the surface of last scatter

###### Abstract

Standard big-bang nucleosynthesis (BBN) predicts only a trace
abundance of lithium and no heavier elements, but some
alternatives predict a nonzero primordial
metallicity. Here we explore whether CMB measurements may set
useful constraints to the primordial metallicity and/or whether
the standard CMB calculations are robust, within the tolerance
of forthcoming CMB maps, to the possibility of primordial
metals. Metals would affect the recombination history (and thus
CMB power spectra) in three ways: (1) Ly photons can be
removed (and recombination thus accelerated) by photoionizing
metals. (2) The Bowen resonance-fluorescence mechanism may
degrade Ly photons and thus enhance the Ly escape probability and speed up recombination. (3) Metals could affect the low-redshift tail
of the CMB visibility function by providing additional free
electrons. The last two of these provide the strongest CMB signal.
However, the effects are detectable in the *Planck* satellite
only if the primordial metal abundance is at least a few hundredths
of solar for (2) and a few tenths of solar for (3). We thus conclude that *Planck* will not be able
to improve upon current constraints to primordial metallicity, at the
level of a thousandth of solar, from the Lyman- forest and ultra-metal-poor halo stars,
and that the CMB power-spectrum predictions for *Planck*
suffer no uncertainty arising from the possibility that there
may be primordial metals.

###### pacs:

98.80.Ft, 26.35.+c, 95.30.Jx## I Introduction

Big-bang nucleosynthesis (BBN) is one of the pillars of the hot standard cosmological model. Comparison of BBN theoretical predictions to observed abundances of the lightest nuclei (D, He, He and Li) uniquely determines the only free parameter of standard BBN, the baryon-to-photon ratio , or equivalently, given the cosmic microwave background (CMB) temperature today K, the baryon abundance (see, e.g., Ref. Iocco:2008va ). The latest results from CMB anisotropy measurements by the WMAP satellite are in excellent agreement, with Komatsu:2010fb .

In the standard BBN scenario, elements heavier than lithium are only produced with trace abundances Iocco:2007km . It has been shown, however, that significant amounts of heavy elements may be produced in inhomogeneous BBN (IBBN) models Jedamzik:1994de ; Matsuura:2005rb ; Matsuura:2007sb . IBBN may take place if some non-standard mechanism leads to large baryon-abundance inhomogeneities on small scales, which are allowed by current observations. It is possible to adjust the IBBN parameters to reproduce the observed abundances of light elements, while producing heavier elements with abundances as large as those in the Sun Matsuura:2007sb ; Nakamura:2010is . More generally, it cannot be excluded that some unknown processes may lead to a significant production of metals heavier than lithium. It may therefore be profitable to inquire what empirical constraints to primordial metals might be possible.

Standard methods to constrain metal abundances at high redshifts
rely on line emission or absorption measurements, and therefore
require some sources to have already formed (typically,
high-redshift quasars). The intergalactic medium (IGM) is
therefore already partially enriched by metals produced in the
first stars, and extracting limits on the primordial abundances
relies on understanding the complex physics of galactic outflows
and gas mixing and correctly modeling the ambient radiation
field. It would be of great interest to be able to probe the
abundance of metals *before* the formation of Population
III stars that enriched the IGM. A few ideas were put forward to
probe the metallicity during the dark ages: Ref. Loeb:2001ax suggested using
resonant scattering of CMB photons off neutral lithium atoms (later shown to be unobservable because
lithium is kept ionized by redshifted Lyman- photons
emitted during primordial hydrogen recombination
Switzer:2005nd ); Ref. Basu:2004 studied the effect of fine-structure transitions of heavy elements in atomic or ionized states on CMB anisotropies; Ref. Harwit:2003in considered the spectral signatures of carbon and
oxygen. In this paper we assess whether
heavy elements present during primordial recombination could be
detectable from upcoming CMB experiments.

Primordial recombination has recently been the subject of a
renewed interest, due to the impact of uncertainties in the
standard theory on the predicted CMB temperature and
polarization anisotropy power spectrum. Errors in the
free-electron fraction as small as a few tenths of a
percent near the peak of the visibility function at redshifts would
induce biases of several standard deviations for cosmological
parameters estimated from *Planck* data
Planck ; RubinoMartin:2009ry . This accuracy requirement has
motivated abundant
work on radiative transfer in the Lyman lines, in particular
Lyman- (see for example
Refs. Hirata:2008ny ; Chluba:2009us ; Hirata:2009qy ; AliHaimoud:2010ym and
references therein). The
tails of the visibility function are less important, but an
accuracy of about a percent is still needed, which required
implementing an accurate multi-level–atom formulation of the
recombination problem
Chluba:2006bc ; Grin:2009ik ; Chluba:2010fy ; AliHaimoud:2010ab . Such
a high sensitivity to
the recombination history can be turned into an asset and serve
to probe unusual physics taking place during the recombination
history as, for example, the presence of primordial heavy
elements. In this paper, we explore this idea, and quantify the
impact of neutral metals on the Ly and Ly net decay rates, and
of ionized metals on the low-redshift tail of the visibility
function.

Below we consider three effects of metals on the recombination
history and thus on the CMB visibility function: (1) The removal of
Lyman- photons (and thus acceleration of recombination)
by photoionization of metals (Section II); (2)
the degradation of
Lyman- photons (and thus acceleration of recombination) by the Bowen resonance-fluorescence mechanism
(Section III); and (3)
the contribution to the free-electron abundance at late times by
low-ionization metals (Section IV). We
find that effects (2) and (3) provide the biggest impact on CMB power spectra.
However, the effects are visible in *Planck* only if the
primordial metal abundance is at least a few hundredths of solar for (2) and a few tenths of solar for (3).
Given that the Lyman-alpha forest Wolfe:2005 and ultra-metal-poor halo stars Beers:2005 constrain the primordial
metal abundance to be at least a few orders of magnitude
smaller than solar, we conclude that *Planck* will be
unable to improve upon current constraints to the primordial
metal abundance or, alternatively, that the standard CMB
predictions for *Planck* are robust to primordial metals at
the levels allowed by current empirical constraints.

## Ii Effect of neutral metals on the Lyman- decay rate

All metals (in the proper chemical sense of the term, i.e. not including noble gases, halogens and other nonmetals) have a first ionization energy below 10.2 eV, which corresponds to the Ly transition in hydrogen. This means that neutral metals can provide continuum opacity in the vicinity of the Ly line by absorbing Ly photons in photoionization events. Since the photoejected electrons rapidly thermalize their energy, this results in a net loss of resonant Ly photons, which would have otherwise been reabsorbed by ground state hydrogen atoms. The presence of metals can therefore speed up hydrogen recombination by increasing the net rate of Lyman- decays. A similar process was investigated for primordial helium recombination Kholupenko:2007 ; Switzer:2007sn : in that case the presence of neutral hydrogen leads to continuum opacity in the He I line. To estimate the impact of continuum opacity on the Lyman- line, we use the analytic treatment presented for He I recombination in Ref. AliHaimoud:2010dx .

### ii.1 Continuum opacity in Ly due to photoionization of neutral metals

The radiative-transfer equation in the vicinity of Ly for the photon occupation number , including only true absorptions and emissions (i.e. neglecting resonant scatterings) and continuum opacity is

(1) | |||||

where we approximated in the prefactor on the left-hand-side, is the Sobolev optical depth for true absorption in the Ly line, is the line profile, and is the continuum differential optical depth, given by

(2) |

In Eq. (2), is the abundance of neutral metal M, and is the photoionization cross section of M at frequency . We have assumed that varies slowly over the Ly resonance (specifically, over the region which is optically thick for true absorption, which corresponds to a few tens of Doppler widths Hirata:2008ny ), so we can approximate . Note that Eq. (1) assumes that the ionization state of M is given by the Saha equilibrium equation (this translates in a ratio of continuum emission to absorption rates equal to ), even though this is not strictly correct (see Sec. II.2).

The net rate of decays is then obtained as follows:

(3) |

where the prefactor converts photon occupation numbers to photons per unit frequency per hydrogen atom, and we have approximated in the multiplicative factor. Ref. AliHaimoud:2010dx showed that the net decay rate in Lyman- can be written in the following form:

(4) |

where

(5) |

is the standard net decay rate in Ly, in the Sobolev approximation, for a large optical depth and assuming an incoming blackbody radiation field, and is a correction factor accounting for continuum absorption in the line. The correction factor depends on the single parameter

(6) |

where is the total inverse lifetime of the state. The dimensionless parameter can be interpreted as the continuum optical depth within the part of the Ly line which is optically thick for true absorption. For , , and for , , which is what one would expect as continuum opacity increases the net rate of decays in the line, as explained above. For , we have the following approximate expansion (see Eq. (117) of Ref. AliHaimoud:2010dx ):

(7) |

In order for primordial metals to change the net decay rate in
Ly by (which is roughly the level detectable
by *Planck*), we therefore need near
the peak of the visibility function. Extracting the relevant
parameters from the multilevel atom code of Hirata
Hirata:2008ny , we obtain, for ,

(8) |

where is the abundance of neutral metals relative to hydrogen. We see that for a characteristic photoionization cross section cm, a fractional abundance of neutral metals per hydrogen atom as low as would be potentially detectable.

### ii.2 Ionization state of metals and results

We now turn to the evaluation of the fraction of neutral metals . As a first approximation we use the Saha equilibrium value:

(9) |

where is the ionization energy of M, is the free electron abundance, and the ’s are the degeneracy factors for each species. For a standard recombination history, at , Eq. (9) gives (taking the ratio of degeneracy factors to be unity) and for 10, 9, 8 and 7 eV, respectively, and we can anticipate that only metals with eV may have some impact on Ly.

Saha equilibrium assumes that the ionizing radiation field is thermal. During hydrogen recombination, the radiation field develops large distortions in the vicinity of the Ly line, due to the slow escape of Ly photons (in fact, thermalization of these distorsions is so inefficient that they survive until today Peebles ). These non-thermal photons increase the ionization rate with respect to its thermal value, and the neutral fraction of metals is therefore smaller than predicted by the Saha equation (see for example Ref. Switzer:2005nd for the case of lithium). The ionization state of the metal M is therefore rather determined by the balance of recombinations and photoionizations (this assumes the steady-state limit, valid so long as the photoionization rate is much larger than the Hubble expansion rate, which is a very good approximation around the peak of the visibility function)

(10) |

where is the MM recombination coefficient and is the MM photoionization rate. From Eq. (10), we obtain the neutral fraction,

(11) |

The photoionization rate comprises a thermal part, due to
photoionizations by CMB photons (from the ground state
*and* excited states), which is related to the
recombination coefficient through the detailed balance relation

(12) |

and a non-thermal part, due to photoionizations from the ground state by distortion photons,

(13) |

where and is the non-thermal distortion to the photon occupation number. The Lyman- distortion peaks around . We can therefore expect that distortions may start significantly affecting the ionization state of the metal M around redshift . As a consequence, we expect the Saha equilibrium approximation to be quite accurate around for metals with ionization threshold lower than eV. For the more interesting metals with eV, however, spectral distortions will lower the neutral fraction with respect to the Saha value at , making their detection more difficult through the effect considered here (we will consider the effect of additional free electrons due to the presence of ionized metals in Sec. IV).

We have computed the neutral fraction of several metals with
atomic number , using the fits of
Ref. Verner:1996th for the photoionization cross sections,
and the chianti database for the recombination
coefficients Chianti1 ; Chianti6 . We have extracted the
Ly distortion^{1}^{1}1In principle, to be
self-consistent, one should account for the continuum optical
depth due to metal photoionization above Ly and between
and . Given that we find that
this effect should not be detectable anyway, we have not
implemented a more subtle treatment. from the two-photon code
of Hirata Hirata:2008ny . We show the ionization state of
beryllium ( eV), boron ( eV) and silicon ( eV) as a function
of redshift, for a standard recombination history, in
Fig. 1.

We show in Fig. 2 the minimal abundance of metals detectable through its effect on Ly (i.e. such that at redshift 1100). We see that the smallest detectable abundance would be . Due to lack of data, we have not treated the case of other metals with eV, such as zinc (), arsenic () and gold (), but do not expect significantly lower detectability thresholds unless they have unusually high photoionization cross sections.

## Iii The Bowen resonance-fluorescence mechanism for oxygen

There is an accidental coincidence between the hydrogen Ly line at 1025.72 Å and the O I line at 1025.76 Å. Ly photons may therefore excite neutral oxygen instead of being reabsorbed in the hydrogen line. Neutral oxygen in the state can then either directly decay back to the ground state or first decay to the state by emitting an infrared photon at 1.13 m, and subsequently cascade down to the ground state (in principle, atoms in the state can also be excited or photoionized, but there are very few thermal photons energetic enough to do so). The probability of the latter series of events (neglecting stimulated decays) is . Direct decays of excited oxygen to the ground state do not affect radiative transfer in the Ly line, as they do not change the number of Ly photons. On the other hand, absorptions in the 1025.76 Å line followed by emission of infrared photons degrade Ly photons that would otherwise have been reabsorbed by neutral hydrogen. This effect is similar to the continuum opacity in Ly discussed in Section II, except that this is now a resonant process.

The escape^{2}^{2}2The term “escape” is somewhat misleading in this situation: photons near the Ly frequency do not actually escape more from the resonant region (in fact, their overall escape rate is even lower due to a slightly higher optical depth). They rather only “escape” reabsorption by neutral hydrogen. probability in the Ly line is enhanced by the probability that a Ly photon is absorbed by O I (and then degraded) rather than by H I:

(14) |

where the multiplicative factors are the ratios of the degeneracy factors of the excited levels to those of the ground states. Note that the photon occupation number redward of Ly is slightly decreased by this process: . However, as long as this has no detectable impact on radiative transfer redward of Ly.

As oxygen and hydrogen have very similar ionization energies, we can assume that they have the same recombination history. More specifically, the ratio of ionized to neutral oxygen rapidly equilibrates to the corresponding ratio for hydrogen because the continuum above the ionization threshold is optically thick. We therefore have (the 9/8 comes from properly accouting for degeneracy factors, see Eq. (9)) and as a result we obtain . As long as the damping wings of the O I 1025.76 Å line are optically thin, and the Bowen mechanism can only affect the net decay rate in the Doppler core of the Ly line. The latter is very small anyway as the radiation field is very close to equilibrium with the 3p-1s ratio over many Doppler widths near line center. We have modified the escape probability from the Doppler core of Ly in the recombination code HyRec AliHaimoud:2010dx according to Eq. (14), and found that a minimal abundance of oxygen is required to affect the recombination history at a potentially detectable level at . Note that this would correspond to an enhancement by a factor from the standard escape probability in the Ly Doppler core. This stems from the fact that only a tiny fraction of Ly decays ocurr in the Doppler core of the line, whereas most of them take place in the damping wings. The recombination history is therefore highly insensitive to the exact decay rate in the core.

## Iv Additional free electrons due to ionized metals

If metals remain ionized, they can contribute an additional residual free-electron fraction at late times, . In fact we have , as we show below. At late times the evolution of the free-electron fraction is given by

(15) |

Eq. (15) is valid because the free electron fraction is many orders of magnitude above the Saha equilibrium value at late time (for a discussion, see Ref. AliHaimoud:2010dx ). If is the unperturbed free electron fraction (i.e. obtained with ) and , we obtain

(16) |

which asymptotes to .
The *Planck* satellite will be sensitive to fractional
changes at late times. Since for , we conclude that a potential detection by *Planck*
requires a fractional abundance of metals (in the case that metals remain fully ionized). Note
that for a given , the presence of metals also
modifies the total abundance of hydrogen, ,
throughout the recombination history. However these
modifications are degenerate with a mere change of of
at the level of a few times and are
therefore undetectable.

## V Conclusions

We have investigated whether a
primordial metal content could sufficiently affect the
recombination history to be detectable in upcoming CMB data from
*Planck*. We first considered the effect of photoionization
of neutral metals by Ly photons. We showed that although
a very small abundance of neutral metals would be enough to
significantly affect the net decay rate in Ly, metals
with ionization threshold below Ly are mostly ionized at
, and therefore undetectable. We also considered
the Bowen resonance-fluorescence mechanism if primordial oxygen is present. This effect leads to an enhanced escape rate of Ly photons and a speed up of recombination. We showed that it could lead to detectable changes for a primordial oxygen abundance of a couple hundredths of solar .
Finally, we pointed out
that metals that stay ionized until late times provide
additional free electrons and therefore change the late-time
Thomson scattering optical depth. A fractional abundance
of primordial metals could be detectable through this effect. As a reference, the
most abundant metal in the solar photosphere is oxygen (), followed by carbon (), neon (),
nitrogen (), magnesium (), silicon (), iron () and sulfur
(). Other metals have fractional
abundances in the Sun Asplund:2009fu . As
carbon, nitrogen, oxygen and neon are neutral at late times (due
to their high ionization potential), we conclude that
*Planck* could potentially detect primordial metals with an
abundance at least a few tenths of solar. This is moreover an
optimistic estimate, as the effect of metals is likely to be
degenerate with the redshift of reionization or other
cosmological parameters.

Given that Lyman-alpha-forest
measurements and ultra-metal-poor halo stars suggest a primordial metallicity much
smaller than one hundredth solar, we conclude that the CMB can
unfortunately not usefully constrain the abundance of primordial
metals. At the same time, we also conclude that the CMB
predictions for the *Planck* satellite are robust to a
primordial metallicity allowed by current empirical constraints.

###### Acknowledgements.

MK thanks the support of the Miller Institute for Basic Research in Science at the University of California, Berkeley, where part of this work was completed. This work was supported by DoE DE-FG03-92-ER40701, NASA NNX10AD04G (MK), NSF AST-0807337 (YAH and CH), the Gordon and Betty Moore Foundation, the Alfred P. Sloan Foundation (CH) and the David & Lucile Packard Foundation (CH).## References

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