Towards a complete treatment of the cosmological recombination problem

A new approach to the cosmological recombination problem is presented, which completes our previous analysis on the effects of two-photon processes during the epoch of cosmological hydrogen recombination, accounting for ns-1s and nd-1s Raman events and two-photon transitions from levels with n>=2. The recombination problem for hydrogen is described using an effective 400-shell multi-level approach, to which we subsequently add all important recombination corrections discussed in the literature thus far. We explicitly solve the radiative transfer equation of the Lyman-series photon field to obtain the required modifications to the rate equations of the resolved levels. In agreement with earlier computations we find that 2s-1s Raman scattering leads to a delay in recombination by DN_e/N_e~0.9% at z~920. Two-photon decay and Raman scattering from higher levels (n>3) result in a small additional modifications, and precise results can be obtained when including their effect for the first 3-5 shells. This work is a major step towards a new cosmological recombination code (CosmoRec) that supersedes the physical model included in Recfast, and which, owing to its short runtime, can be used in the analysis of future CMB data from the Planck Surveyor.Comment: 17 pages, 8 figures, minor typos corrected, accepted by MNRAS, CosmoRec available at http://www.Chluba.de/CosmoRe

Time-Dependent Corrections to the Ly-alpha Escape Probability During Cosmological Hydrogen Recombination

We consider the effects connected with the detailed radiative transfer during the epoch of cosmological recombination on the ionization history of our Universe. We focus on the escape of photons from the hydrogen Lyman-alpha resonance at redshifts 600<~ z <~ 2000, one of two key mechanisms defining the rate of cosmological recombination. We approach this problem within the standard formulation, and corrections due to two-photon interactions are deferred to another paper. As a main result we show here that within a non-stationary approach to the escape problem, the resulting correction in the free electron fraction, N_e, is about ~1.6-1.8% in the redshift range 800<~z<~1200. Therefore the discussed process results in one of the largest modifications to the ionization history close to the maximum of Thomson-visibility function at z~1100 considered so far. We prove our results both numerically and analytically, deriving the escape probability, and considering both Lyman-alpha line emission and line absorption in a way different from the Sobolev approximation. In particular, we give a detailed derivation of the Sobolev escape probability during hydrogen recombination, and explain the underlying assumptions. We then discuss the escape of photons for the case of coherent scattering in the lab frame, solving this problem analytically in the quasi-stationary approximation and also in the time-dependent case. We show here that during hydrogen recombination the Sobolev approximation for the escape probability is not valid at the level of DP/P~5-10%. This is because during recombination the ionization degree changes significantly over a characteristic time Dz/z~10%, so that at percent level accuracy the photon distribution is not evolving (abridged)Comment: 18 pages, 12 figures, accepted versio

The evolution of CMB spectral distortions in the early Universe

The energy spectrum of the cosmic microwave background (CMB) allows constraining episodes of energy release in the early Universe. In this paper we revisit and refine the computations of the cosmological thermalization problem. For this purpose a new code, called CosmoTherm, was developed that allows solving the coupled photon-electron Boltzmann equation in the expanding, isotropic Universe for small spectral distortion in the CMB. We explicitly compute the shape of the spectral distortions caused by energy release due to (i) annihilating dark matter; (ii) decaying relict particles; (iii) dissipation of acoustic waves; and (iv) quasi-instantaneous heating. We also demonstrate that (v) the continuous interaction of CMB photons with adiabatically cooling non-relativistic electrons and baryons causes a negative mu-type CMB spectral distortion of DI_nu/I_nu ~ 10^{-8} in the GHz spectral band. We solve the thermalization problem including improved approximations for the double Compton and Bremsstrahlung emissivities, as well as the latest treatment of the cosmological recombination process. At redshifts z <~ 10^3 the matter starts to cool significantly below the temperature of the CMB so that at very low frequencies free-free absorption alters the shape of primordial distortions significantly. In addition, the cooling electrons down-scatter CMB photons introducing a small late negative y-type distortion at high frequencies. We also discuss our results in the light of the recently proposed CMB experiment Pixie, for which CosmoTherm should allow detailed forecasting. Our current computations show that for energy injection because of (ii) and (iv) Pixie should allow to improve existing limits, while the CMB distortions caused by the other processes seem to remain unobservable with the currently proposed sensitivities and spectral bands of Pixie.Comment: 22 pages, 19 figures, 1 table, accepted by MNRA

Two-photon transitions in hydrogen and cosmological recombination

We study the two-photon process for the transitions ns --> 1s and nd --> 1s in hydrogen up to large n. For n<=20 we provide simple analytic fitting formulae to describe the non-resonant part of the two-photon emission profiles. Combining these with the analytic form of the cascade-term yields a simple and accurate description of the full two-photon decay spectrum, which only involves a sum over a few intermediate states. We demonstrate that the cascade term naturally leads to a nearly Lorentzian shape of the two-photon profiles in the vicinity of the resonances. However, due to quantum-electrodynamical corrections, the two-photon emission spectra deviate significantly from the Lorentzian shape in the very distant wings of the resonances. We investigate up to which distance the two-photon profiles are close to a Lorentzian and discuss the role of the interference term. We then analyze how the deviation of the two-photon profiles from the Lorentzian shape affects the dynamics of cosmological hydrogen recombination. Since in this context the escape of photons from the Lyman-alpha resonance plays a crucial role, we concentrate on the two-photon corrections in the vicinity of the Lyman-alpha line. Our computations show that the changes in the ionization history due to the additional two-photon process from high shell (n>2) likely do not reach the percent-level. For conservative assumptions we find a correction DN_e/N_e~-0.4% at redshift z~1160. This is numerically similar to the result of another recent study; however, the physics leading to this conclusion is rather different. In particular, our calculations of the effective two-photon decay rates yield significantly different values, where the destructive interference of the resonant and non-resonant terms plays a crucial role in this context (abridged)Comment: 20 pages, 13 figures, 4 tables, accepted versio

Could the Cosmological Recombination Spectrum Help Us Understand Annihilating Dark Matter?

In this paper we explore the potential effects of DM annihilations on the cosmological recombination spectrum. With this example we want to demonstrate that the cosmological recombination spectrum in principle is sensitive to details related to possible extra energy release during recombination. We restrict ourselves to DM models which produce a negligible primordial distortion of the CMB energy spectrum. However, since during the epoch of cosmological recombination a large fraction of the deposited energy can directly go into ionizations and excitations of neutral atoms, both the cosmological recombination spectrum and ionization history can still be affected significantly. We compute the modifications to the cosmological recombination spectrum using our multi-level HI and HeI recombination code, showing that additional photons are created due to uncompensated loops of transitions which are induced by DM annihilations. As we illustrate here, the results depend on the detailed branching of the deposited energy into heating, ionizations and excitations. This dependence in principle should allow us to shed light on the nature of the underlying annihilating DM model (or more generally speaking, the mechanism leading to energy injection) when measuring the cosmological recombination spectrum. However, for current upper limits on the potential DM annihilation rate during recombination the cosmological recombination spectrum is only affected at the level of a few percent. Nevertheless, we argue here that the cosmological recombination spectrum would provide another independent and very direct way of checking for the presence of sources of extra ionizing or exciting photons at high redshifts. This would open an new window to possible (non-standard) processes occurring (abridged)Comment: 14 pages, 11 figure, submitted to MNRA

Ly alpha escape during cosmological hydrogen recombination: the 3d-1s and 3s-1s two-photon processes

We give a formulation of the radiative transfer equation for Lyman alpha photons which allows us to include the two-photon corrections for the 3s-1s and 3d-1s decay channels during cosmological hydrogen recombination. We use this equation to compute the corrections to the Sobolev escape probability for Lyman alpha photons during hydrogen recombination, which then allow us to calculate the changes in the free electron fraction and CMB temperature and polarization power spectra. We show that the effective escape probability changes by DP/P ~+ 11% at z~1400 in comparison with the one obtained using the Sobolev approximation. This speeds up of hydrogen recombination by DN_e/N_e ~- 1.6% at z~1190, implying |DC_l/C_l| ~1%-3% at l >~ 1500 with shifts in the positions of the maxima and minima in the CMB power spectra. These corrections will be important for the analysis of future CMB data. The total correction is the result of the superposition of three independent processes, related to (i) time-dependent aspects of the problem, (ii) corrections due to quantum mechanical deviations in the shape of the emission and absorption profiles in the vicinity of the Lyman alpha line from the normal Lorentzian, and (iii) a thermodynamic correction factor, which occurs to be very important. All these corrections are neglected in the Sobolev-approximation, but they are important in the context of future CMB observations. All three can be naturally obtained in the two-photon formulation of the Lyman alpha absorption process. However, the corrections (i) and (iii) can also be deduced in the normal '1+1' photon language, without necessarily going to the two-photon picture. Therefore only (ii) is really related to the quantum mechanical aspects of the two-photon process (abridged)Comment: 30 pages, 21 figures, submitted to A&

Cosmological hydrogen recombination: Lyn line feedback and continuum escape

We compute the corrections to the cosmological hydrogen recombination history due to delayed feedback of Lyman-series photons and the escape in the Lyman-continuum. The former process is expected to slightly delay recombination, while the latter should allow the medium to recombine a bit faster. It is shown that the subsequent feedback of released Lyman-n photons on the lower lying Lyman-(n-1) transitions yields a maximal correction of DN_e/N_e 0.22% at z~ 1050. Including only Lyman-\beta feedback onto the Lyman-\alpha transition, accounts for most of the effect. We find corrections to the cosmic microwave background TT and EE power spectra \change{with typical peak to peak amplitude |DC^{TT}_l/C^{TT}_l|~0.15% and |\Delta C^{EE}_l/C^{EE}_l|~0.36% at l<~3000. The escape in the Lyman-continuum and feedback of Lyman-\alpha photons on the photoionization rate of the second shell lead to modifications of the ionization history which are very small (less than |DN_e/N_e|~few x 10^{-6}).Comment: 5+epsilon pages, 7 figures, accepted versio

Precise cosmological parameter estimation using CosmoRec

We use the new cosmological recombination code, CosmoRec, for parameter estimation in the context of (future) precise measurements of the CMB temperature and polarization anisotropies. We address the question of how previously neglected physical processes in the recombination model of Recfast affect the determination of key cosmological parameters, for the first time performing a model-by-model computation of the recombination problem. In particular we ask how the biases depend on different combinations of parameters, e.g. when varying the helium abundance or the effective number of neutrino species in addition to the standard six parameters. We also forecast how important the recombination corrections are for a combined Planck, ACTPol and SPTpol data analysis. Furthermore, we ask which recombination corrections are really crucial for CMB parameter estimation, and whether an approach based on a redshift-dependent correction function to Recfast is sufficient in this context.Comment: 12 pages, 7 figures, submitted to MNRA

Semi-blind Eigen-analyses of Recombination Histories Using CMB Data

Cosmological parameter measurements from CMB experiments such as Planck, ACTpol, SPTpol and other high resolution follow-ons fundamentally rely on the accuracy of the assumed recombination model, or one with well prescribed uncertainties. Deviations from the standard recombination history might suggest new particle physics or modified atomic physics. Here we treat possible perturbative fluctuations in the free electron fraction, \Xe(z), by a semi-blind expansion in densely-packed modes in redshift. From these we construct parameter eigenmodes, which we rank order so that the lowest modes provide the most power to probe the \Xe(z) with CMB measurements. Since the eigenmodes are effectively weighed by the fiducial \Xe history, they are localized around the differential visibility peak, allowing for an excellent probe of hydrogen recombination, but a weaker probe of the higher redshift helium recombination and the lower redshift highly neutral freeze-out tail. We use an information-based criterion to truncate the mode hierarchy, and show that with even a few modes the method goes a long way towards morphing a fiducial older {\sc Recfast} $X_{\rm e,i} (z)$ into the new and improved {\sc CosmoRec} and {\sc HyRec} $X_{\rm e,f} (z)$ in the hydrogen recombination regime, though not well in the helium regime. Without such a correction, the derived cosmic parameters are biased. We discuss an iterative approach for updating the eigenmodes to further hone in on $X_{\rm e,f} (z)$ if large deviations are indeed found. We also introduce control parameters that downweight the attention on the visibility peak structure, e.g., focusing the eigenmode probes more strongly on the \Xe (z) freeze-out tail, as would be appropriate when looking for the \Xe signature of annihilating or decaying elementary particles.Comment: 28 pages, 26 Fig