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

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

Evolution of low-frequency features in the CMB spectrum due to stimulated Compton scattering and Doppler-broadening

We discuss a new solution of the Kompaneets-equation for physical situations in which low frequency photons, forming relatively narrow spectral details, are Compton scattered in an isotropic, infinite medium with an intense ambient blackbody field that is very close to full thermodynamic equilibrium with the free electrons. In this situation the background-induced stimulated Compton scattering slows down the motion of photons toward higher frequencies by a factor of 3 in comparison with the solution that only takes into account Doppler-broadening and boosting. This new solution is important for detailed computations of cosmic microwave background spectral distortions arising due to uncompensated atomic transitions of hydrogen and helium in the early Universe. In addition we derive another analytic solution that only includes the background-induced stimulated Compton scattering and is valid for power-law ambient radiation fields. This solution might have interesting applications for radio lines arising inside of bright extra-galactic radio source, where according to our estimates line shifts because of background-induced stimulated scattering could be amplified and even exceed the line broadening due to the Doppler-effect.Comment: 5 pages, 2 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

Pre-recombinational energy release and narrow features in the CMB spectrum

Energy release in the early Universe (z<~ 2x10^6) should lead to some broad spectral distortion of the cosmic microwave background (CMB) radiation field, which can be characterized as y-type distortion when the injection process started at redshifts z<~ 5x10^4. Here we demonstrate that if energy was released before the beginning of cosmological hydrogen recombination (z~1400), closed loops of bound-bound and free-bound transitions in HI and HeII lead to the appearance of (i) characteristic multiple narrow spectral features at dm and cm wavelengths, and (ii) a prominent sub-millimeter feature consisting of absorption and emission parts in the far Wien tail of CMB spectrum. The additional spectral features are generated in the pre-recombinational epoch of HI (z>~1800) and HeII (z>~7000), and therefore differ from those arising due to normal cosmological recombination in the undisturbed CMB blackbody radiation field. We present the results of numerical computations including 25 atomic shells for both HI and HeII, and discuss the contributions of several individual transitions in detail. As examples, we consider the case of instantaneous energy release (e.g. due to phase transitions) and exponential energy release because of long-lived decaying particles. Our computations show that due to possible pre-recombinational atomic transitions the variability of the CMB spectral distortion increases when comparing with the distortions arising in the normal recombination epoch. The existence of these narrow spectral features would open an unique way to separate y-distortions due to pre-recombinational ($1400<~ z <~5x10^4) energy release from those arising in the post-recombinational era at redshifts z<~800. (abridged)Comment: 17 pages, 12 Figures, 1 Table, submitted to A&

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&

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

Cosmological hydrogen recombination: populations of the high level sub-states

We present results for the spectral distortions of the Cosmic Microwave Background (CMB) arising due to bound-bound transitions during the epoch of cosmological hydrogen recombination at frequencies down to nu~100MHz. We extend our previous treatment of the recombination problem now including the main collisional processes and following the evolution of all the hydrogen angular momentum sub-states for up to 100 shells. We show that, due to the low baryon density of the Universe, even within the highest considered shell full statistical equilibrium (SE) is not reached and that at low frequencies the recombination spectrum is significantly different when assuming full SE for n>2. We also directly compare our results for the ionization history to the output of the Recfast code, showing that especially at low redshifts rather big differences arise. In the vicinity of the Thomson visibility function the electron fraction differs by roughly -0.6% which affects the temperature and polarization power spectra by <~1%. Furthermore we shortly discuss the influence of free-free absorption and line broadening due to electron scattering on the bound-bound recombination spectrum and the generation of CMB angular fluctuations due to scattering of photons within the high shells.Comment: 12 pages, 11 figures, submitted to MNRAS, revised version, included two new figures, Sect. 3.4 adde

Cosmological recombination: feedback of helium photons and its effect on the recombination spectrum

In this paper we consider the re-processing of high frequency photons emitted by HeII and HeI during the epoch of cosmological recombination by HeI and HI. We demonstrate that, in comparison to computations which neglect all feedback processes, the number of cosmological recombination photons that are related to the presence of helium in the early Universe could be increased by ~40%-70%. Our computations imply that per helium nucleus ~3-6 additional photons could be produced. Therefore, a total of ~12-14 helium-related photons are emitted during cosmological recombination. This is an important addition to cosmological recombination spectrum which in the future may render it slightly easier to determine the primordial abundance of helium using differential measurements of the CMB energy spectrum. Also, since these photons are the only witnesses of the feedback process at high redshift, observing them in principle offers a way to check our understanding of the recombination physics. Here most interestingly, the feedback of HeII photons on HeI leads to the appearance of several additional, rather narrow spectral features in the HeI recombination spectrum at low frequencies. Consequently, the signatures of helium-related features in the CMB spectral distortion due to cosmological recombination at some given frequency can exceed the average level of ~17% several times. We find that in particular the bands around nu ~10GHz, ~35GHz, ~80GHz, and ~200GHz seem to be affected strongly. In addition, we computed the changes in the cosmological ionization history, finding that only the feedback of primary HeI photons on the dynamics of HeII-->HeI recombination has an effect, producing a change of DN_e/N_e ~+ 0.17% at z~2300. This result seems to be ~2-3 times smaller than the one obtained in earlier computations for this process (abridged).Comment: 28 pages, 24 figures, submitted to MNRA