The subject of cosmological hydrogen recombination has received much attention recently because of its importance to predictions for and cosmological constraints from CMB observations. While the central role of the two-photon decay 2s->1s has been recognized for many decades, high-precision calculations require us to consider two-photon decays from the higher states ns,nd->1s (n>=3). Simple attempts to include these processes in recombination calculations have suffered from physical problems associated with sequences of one-photon decays, e.g. 3d->2p->1s, that technically also produce two photons. These correspond to resonances in the two-photon spectrum that are optically thick, necessitating a radiative transfer calculation. We derive the appropriate equations, develop a numerical code to solve them, and verify the results by finding agreement with analytic approximations to the radiative transfer equation. The related processes of Raman scattering and two-photon recombination are included using similar machinery. Our results show that early in recombination the two-photon decays act to speed up recombination, reducing the free electron abundance by 1.3% relative to the standard calculation at z=1300. However we find that some photons between Ly-alpha and Ly-beta are produced, mainly by 3d->1s two-photon decay and 2s->1s Raman scattering. At later times these photons redshift down to Ly-alpha, excite hydrogen atoms, and act to slow recombination. Thus the free electron abundance is increased by 1.3% relative to the standard calculation at z=900. The implied correction to the CMB power spectrum is neligible for the recently released WMAP and ACBAR data, but at Fisher matrix level will be 7 sigma for Planck. [ABRIDGED]
CMBR distortion concerned with recombination of the primordial plasma is calculated in frequency band from 1 GHz to 100 GHz in the frame of the standard cosmological model for different values of cosmological density parameters: nonrelativistic matter density $Ω_m$ and baryonic matter density $Ω_b$. Comparison of these results with observational data which will be obtained from planned experiments may be used for independent determination of the cosmological parameters $Ω_m$ and $Ω_b$.
Based on the standard cosmological model, we calculate the correction to the rate of two-photon 2s<->1s transitions in the hydrogen atom under primordial hydrogen plasma recombination conditions that arises when the induced transitions under equilibrium background radiation with a blackbody spectrum and plasma recombination radiation are taken into account.
We solve the recombination equation by taking into account the induced recombinations and a physical cut off in the hydrogen spectrum. The effective recombination coefficient is parametrized as a function of temperature and free electron density and is about a factor of four larger than without the induced recombination. This accelerates the last stage of the recombination processes and diminishes the residual ionization by a factor of about 2.6. The number and energy distribution of photons issuing from the hydrogen recombination are calculated. The distortion of the cosmic microwave backgrou…
The dependences on z of fractional number densities of H+ and He+ ions are calculated with a proper allowance for two-photon decays of upper levels of hydrogen and parahelium and radiative transfer in intercombination line 2 3P_1 --> 1 1S_0 of helium. It is shown that for hydrogen this gives corrections for a degree of ionization in no more than a few percents but for helium this leads to a significant acceleration of recombination compared to the recent papers by Seager et al. (1999; 2000) where these effects were ignored.
Precision measurements of the cosmic microwave background temperature anisotropy on scales $\ell > 500$ will be available in the near future. Successful interpretation of these data is dependent on a detailed understanding of the damping tail and cosmological recombination of both hydrogen and helium. This paper and two companion papers are devoted to a precise calculation of helium recombination. We discuss several aspects of the standard recombination picture, and then include feedback, radiative transfer in HeI lines with partial redistribution, and continuum opacity from HI photoionization…
In this paper we compute the emission coming from the direct recombination of free electrons to a given shell (n>=2) during the epoch of cosmological hydrogen recombination. This contribution leads to a total of one photon per recombined hydrogen atom and therefore a ~30-88% increase of the recombination spectrum within the frequency range 1 GHz<=nu<=100 GHz. In particular the Balmer-continuum emission increases the distortion at nu~ 690 GHz by ~92%. With our 100 shell calculations for the hydrogen atom we find that a total of ~5 photons per hydrogen atom are emitted when including all the bou…
It is shown that taking into account a recoil for radiation scattering in the Ly-alpha line can lead to a noticable acceleration of primordial hydrogen recombination. Thus for LambdaCDM model a decrease of ionization degree exceeds 1% for redshifts z in a range 800 - 1050 achieving approximately 1.3% at z=900. Corresponding corrections to the cosmic microwave background power spectra can achieve 1.1% for TT spectra and 1.7% for EE ones. Radiative transfer in these calculations was treated in a quasistationary approximation. Numerical solutions are also obtained in diffusion approximation for a…
A number of recent works in astronomy and cosmology have relied upon theoretical He I emissivities, but we know of no effort to quantify the uncertainties in the atomic data. We analyze and assign uncertainties to all relevant atomic data, perform Monte Carlo analyses, and report standard deviations in the line emissivities. We consider two sets of errors, which we call "optimistic" and "pessimistic." We also consider three different conditions, corresponding to prototypical Galactic and extragalactic H II regions and the epoch of cosmological recombination. In the extragalactic H II case, the…
The HeII->HeI recombination of primordial helium plasma (z = 1500 - 3000) is considered in terms of the standard cosmological model. This process affects the formation of cosmic microwave background anisotropy and spectral distortions. We investigate the effect of neutral hydrogen on the HeII->HeI recombination kinetics with partial and complete redistributions of radiation in frequency in the HeI resonance lines. It is shown that to properly compute the HeII->HeI recombination kinetics, one should take into account not only the wings in the absorption and emission profiles of the HeI resonanc…
Induced emission due to the presence of soft CMB photons slightly increases the two-photon decay rate of the 2s level of hydrogen defining the rate of cosmological recombination. This correspondingly changes the degree of ionization, the visibility function and the resulting primordial temperature anisotropies and polarization of the CMB on the percent level. These changes exceed the precision of the widely used CMBFAST and CAMB codes by more than one order of magnitude and can be easily taken into account.
Recent experimental results on the formation of molecular hydrogen on astrophysically relevant surfaces under conditions similar to those encountered in the interstellar medium provided useful quantitative information about these processes. Rate equation analysis of experiments on olivine and amorphous carbon surfaces provided the activation energy barriers for the diffusion and desorption processes relevant to hydrogen recombination on these surfaces. However, the suitability of rate equations for the simulation of hydrogen recombination on interstellar grains, where there might be very few a…
Interpretation of precision measurements of the cosmic microwave background (CMB) will require a detailed understanding of the recombination era, which determines such quantities as the acoustic oscillation scale and the Silk damping scale. This paper is the second in a series devoted to the subject of helium recombination, with a focus on two-photon processes in He I. The standard treatment of these processes includes only the spontaneous two-photon decay from the 2^1S level. We extend this treatment by including five additional effects, some of which have been suggested in recent papers but …
A knowledge of the recombination time on the grain surfaces has been a major obstacle in deciding the production rate of molecular hydrogen and other molecules in the interstellar medium. We present a numerical study to compute this time for molecular hydrogen for various cloud and grain parameters. We also find the time dependence, particularly when a grain is freshly injected into the system. Apart from the fact that the recombination times seem to be functions of the grain parameters such as the activation barrier energy, temperature etc, our result also shows the dependence on the number o…
It has been suggested that secondary CMB anisotropies generated by neutral lithium could open a new observational window into the universe around the redshift z ~ 400, and permit a determination of the primordial lithium abundance. The effect is due to resonant scattering in the allowed LiI doublet (2s ^2S_{1/2} - 2p ^2P_{1/2,3/2}), so its observability depends on the formation history of neutral lithium. Here we show that the ultraviolet photons produced during hydrogen recombination are sufficient to keep lithium in the LiII ionization stage in the relevant redshift range and suppress the ne…
Recombination of the primordial helium plasma (HeII->HeI, $z \simeq 1500 - 3000$) is considered. This process has an effect on the CMBR anisotropy and CMBR spectrum distortion. In this work an influence of neutral hydrogen on kinetics of HeII->HeI recombination is investigated in the frame of the standard cosmological model. It is shown that small amount of neutral hydrogen ($10^{-5} - 10^{-2}$ of total number of hydrogen ions and atoms) leads to acceleration of HeII->HeI recombination at $z\lesssim 2000$ and at $z\lesssim 1600$ quasi-equilibrium HeII->HeI recombination (according to the Saha …
We have analysed the infrared hydrogen recombination lines of a sample of well studied hot massive stars observed with the Infrared Space Observatory. Our sample contains stars from several classes of objects, whose circumstellar environment is believed to be dominated by an ionized stellar wind (the Luminous Blue Variables) or by a dense disk-like geometry (Be stars and B[e] stars). We show that hydrogen infrared recombination lines can be used as a diagnostic tool to constrain the geometry of the ionized circumstellar material. The line strengths are sensitive to the density of the emitting …
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 frequencie…
The recombination of hydrogen and helium at z~1000-7000 gives unavoidable distortions to the Cosmic Microwave Background (CMB) spectrum. We present a detailed calculation of the line intensities arising from the Ly-alpha (2p-1s) and two-photon (2s-1s) transitions for the recombination of hydrogen, as well as the corresponding lines from helium. We give an approximate formula for the strength of the main recombination line distortion on the CMB in different cosmologies, this peak occurring at about 170 microns. We also find a previously undescribed long wavelength peak (which we call the pre-re…
The perturbations in the electron number density during recombination contributes to the Cosmic Microwave Background bispectrum through second order terms. Perturbations in the electron density can be a factor of ~5 larger than the baryon density fluctuations on large scales as shown in the calculations by Novosyadlyj. This raises the possibility that the contribution to bispectrum arising from perturbations in the optical depth may be non-negligible. We calculate this bispectrum and find it to peak for squeezed triangles and of peak amplitude of the order of primordial non-Gaussianity of loca…
Upcoming precision measurements of the temperature anisotropy of the cosmic microwave background (CMB) at high multipoles will need to be complemented by a more complete understanding of recombination, which determines the damping of anisotropies on these scales. This is the third in a series of papers describing an accurate theory of HeI and HeII recombination. Here we describe the effect of Thomson scattering, the $^3$He isotope shift, the contribution of rare decays, collisional processes, and peculiar motion. These effects are found to be negligible: Thomson and $^3$He scattering modify th…
It is now well known that a primordial chemistry, involving light elements produced during the nucleosynthesis period, might develop during the hydrogen post-recombination era. In particular, molecular ions and primordial molecules such as H2, HD and LiH will be produced. We summarize this primordial chemistry after the recombination epoch, and then present a simple gravitational collapse model of a cloud. The potentiality of fragmentation of this collapsing protoclouds through the thermal instability is also discussed. We suggest that this study could also be extended to the CO molecule, beca…
We report on the first results of experiments to measure the recombination rate of hydrogen on surfaces of astrophysical interest. Our measurements give lower values for the recombination efficiency (sticking probability S x probability of recombination upon H-H encounter $γ$) than model-based estimates. We propose that our results can be reconciled with average estimates of the recombination rate (1/2 n(H) n(g) v(H)A S $γ$) from astronomical observations, if the actual surface of an average grain is rougher, and its area bigger, than the one considered in models.
Molecular beam experiments that use low-temperature bolometers as (energy-) detectors are well suited to the study of physisorption and recombination of hydrogen on low-temperature surfaces. Experiments where this technique is combined with mass spectrometry to examine atoms and molecules released from the surface are summarised and reviewed with reference to astrophysical implications. Hydrogen atoms physisorbed on polycrystalline water ice are shown to be sufficiently mobile to recombine efficiently even at surface temperatures as low as 3 K. Molecules are formed with substantial internal en…
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 f…
We explore the imprint of the cosmological hydrogen recombination lines on the power spectrum of the cosmic microwave background (CMB). In particular, we focus our analysis in the three strongest lines for the Balmer, Paschen and Brackett series of hydrogen. We expect changes in the angular power spectrum due to these lines on the level of $0.3 μK$ for the H$α$ line, being maximum at small angular scales ($\ell \approx 870$). The morphology of the signal is very rich. It leads to relatively narrow spectral features ($Δν/ ν\sim 10^{-1}$), with several regions in the power spectrum showing a cha…
We explore the ways in which primordial magnetic fields influence the thermal and ionization history of the post-recombination universe. After recombination the universe becomes mostly neutral resulting also in a sharp drop in the radiative viscosity. Primordial magnetic fields can then dissipate their energy into the intergalactic medium (IGM) via ambipolar diffusion and, for small enough scales, by generating decaying MHD turbulence. These processes can significantly modify the thermal and ionization history of the post-recombination universe. We show that the dissipation effects of magnetic…
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-βfeedback onto the Lyman-αtransition, accounts for most of the effect. We find corrections to the cosmic microwa…
The major theoretical limitation for extracting cosmological parameters from the CMB sky lies in the precision with which we can calculate the cosmological recombination process. Uncertainty in the details of hydrogen and helium recombination could effectively increase the errors or bias the values of the cosmological parameters derived from the Planck satellite, for example. Here we modify the cosmological recombination code RECFAST by introducing one more parameter to reproduce the recent numerical results for the speed-up of the helium recombination. Together with the existing hydrogen fudg…
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 $\simeq 20 μK$ for a fully neutral medium. The fine structure line from this redshift is observable at $ν\simeq 1 \rm GHz$ and we discuss prospects for …
Unusual patterns of recombination emission from gas ionized by metal-free stars may distinguish early star-forming galaxies from their present-day counterparts. This pattern arises from the harder ionizing spectrum expected from metal-free stars, which strongly enhances the strength of He II recombination lines. Our calculations indicate that line fluxes of He II 1640 and 4686 are sufficiently large to be detected by narrowband and spectroscopic searches for high-redshift emission-line sources at redshift z ~ 5 using current instruments. An unknown fraction of Lya emitters may harbor low-metal…
We study the implications of primordial magnetic fields for the thermal and ionization history of the post-recombination era. In particular we compute the effects of dissipation of primordial magnetic fields owing to ambipolar diffusion and decaying turbulence in the intergalactic medium (IGM) and the collapsing halos and compute the effects of the altered thermal and ionization history on the formation of molecular hydrogen. We show that, for magnetic field strengths in the range $2 \times 10^{-10} {\rm G} \la B_0 \la 2 \times 10^{-9} {\rm G}$, the molecular hydrogen fraction in IGM and colla…
Semi-empirical molecular dynamics is used to simulate several gaseous atomic hydrogen interactions with hydrocarbon grains in space: recoil, adsorption, diffusion, chemisorption and recombination into molecular hydrogen. Their probabilities are determined by multiple numerical experiments, as a function of initial velocity of gas atoms. The equilibrium hydrogen coverage of free grains is then derived. These data can be used in calculations of material and energy balance as well as rates of hydrogen recombination on grains. The role of grain temperature is also considered.
A method is presented to derive electron temperatures and densities of planetary nebulae (PNe) simultaneously, using the observed hydrogen recombination spectrum, which includes continuum and line emission. By matching theoretical spectra to observed spectra around the Balmer jump at about 3646A, we determine electron temperatures and densities for 48 Galactic PNe. The electron temperatures based on this method are found to be systematically lower than those derived from [O III] 4959/4363 and [O III] (88μ+52μ)/4959 ratios. And the electron densities based on this method are found to be systema…
We compute the spectral distortions of the Cosmic Microwave Background (CMB) arising during the epoch of cosmological hydrogen recombination within the standard cosmological (concordance) model for frequencies in the range 1 GHz-3500 GHz. We follow the evolution of the populations of the hydrogen levels including states up to principle quantum number $n=30$ in the redshift range $500\leq z\leq 3500$. All angular momentum sub-states are treated individually, resulting in a total number of 465 hydrogen levels. The evolution of the matter temperature and the fraction of electrons coming from HeII…
The main goal of this work is to calculate the contributions to the cosmological recombination spectrum due to bound-bound transitions of helium. We show that due to the presence of helium in the early Universe unique features appear in the total cosmological recombination spectrum. These may provide a unique observational possibility to determine the relative abundance of primordial helium, well before the formation of first stars. We include the effect of the tiny fraction of neutral hydrogen atoms on the dynamics of HeII -> HeI recombination at redshifts $z\sim 2500$. As discussed recently,…
Thermal and chemical evolution of primordial gas clouds irradiated with far-ultraviolet (FUV; < 13.6 eV) radiation is investigated. In clouds irradiated by intense FUV radiation, sufficient hydrogen molecules to be important for cooling are never formed. However, even without molecular hydrogen, if the clouds are massive enough, they start collapsing via atomic hydrogen line cooling. Such clouds continue to collapse almost isothermally owing to successive cooling by H^{-} free-bound emission up to the number density of 10^{16} cm^{-3}. Inside the clouds, the Jeans mass eventually falls well be…
The role played by a primordial magnetic field during the pre-recombination epoch is analysed through the cyclotron radiation (due to the free electrons) it might produce in the primordial plasma. We discuss the constraint implied by the measurement or lack thereof COBE on this primordial field.
The power spectrum of number density perturbations of free electrons is obtained for the epoch of cosmological recombination of hydrogen. It is shown that amplitude of the electron perturbations power spectrum of scales larger than acoustic horizon exceeds by factor of 17 the amplitude of baryon matter density ones (atoms and ions of hydrogen and helium). In the range of the first and second acoustic peaks such relation is 18, in the range of the third one 16. The dependence of such relations on cosmological parameters is analysed too.
New X-ray observatories (Chandra and XMM-Newton) are providing a wealth of high-resolution X-ray spectra in which hydrogen- and helium-like ions are usually strong features. We present results from a new collisional-radiative plasma code, the Astrophysical Plasma Emission Code (APEC), which uses atomic data in the companion Astrophysical Plasma Emission Database (APED) to calculate spectral models for hot plasmas. APED contains the requisite atomic data such as collisional and radiative rates, recombination cross sections, dielectronic recombination rates, and satellite line wavelengths. We co…
The cosmological recombination history of lithium, produced during Big--Bang nucleosynthesis, is presented using updated chemistry and cosmological parameters consistent with recent cosmic microwave background (CMB) measurements. For the popular set of cosmological parameters, about a fifth of the lithium ions recombine into neutral atoms by a redshift $z\sim 400$. The neutral lithium atoms scatter resonantly the CMB at 6708 Åand distort its intensity and polarization anisotropies at observed wavelengths around $\sim 300 μ$m, as originally suggested by Loeb (2001). The modified anistropies res…
Measurements of fluctuations in the Cosmic Microwave Background Radiation (CMBR) is one of the most promising methods for measuring the fundamental cosmological parameters. However, in order to infer parameters from precision measurements it is necessary to calculate the theoretical fluctuation spectrum to at least the measurement accuracy. Standard treatments assume that electrons, ions and neutral hydrogen are very tightly coupled during the entire recombination history, and that the baryon-photon plasma can be treated as a two-fluid system consisting of baryons and photons interacting via T…
Primordial chemistry began in the recombination epoch when the adiabatic expansion caused the temperature of the radiation to fall below 4000 K. The chemistry of the early Universe involves the elements hydrogen, its isotope deuterium, helium with its isotopic forms and lithium. In this communication I will present results on the evolution of the primordial $H_2$ abundance for different cosmological models and the influence on the thermal decoupling.
We use one-dimensional radiative transfer simulations to study the evolution of H_2 gas-phase (H^- catalyzed) formation and photo-dissociation regions in the primordial universe. We find a new positive feedback mechanism capable of producing shells of H_2 in the intergalactic medium, which are optically thick in some Lyman-Werner bands. While these shells exist, this feedback effect is important in reducing the H_2 dissociating background flux and the size of photo-dissociation spheres around each luminous object. The maximum background opacity of the IGM in the H_2 Lyman-Werner bands is τ_{H_…
This paper investigates the effect of collisional enhancement of the hydrogen lines on the derivation of the helium abundances in low metallicity H II regions. For this, we have constructed a grid of photoionization models relevant for the analysis of giant \hii regions in blue compact galaxies. We show that the effect of collisional excitation on the Halpha/Hbeta ratio can be quite important (up to 8% or more). The impact of this effect on the determination of the helium mass fraction has been tracked on four low-metallicity blue compact galaxies for which Keck spectra are available and which…
Using published measurements of dielectronic recombination (DR) resonance strengths and energies for C V to C IV and O VIII to O VII, we have calculated the DR rate coefficient for these ions. Our derived rates are in good agreement with multiconfiguration, intermediate-coupling and multiconfiguration, fully-relativistic calculations as well as with most LS coupling calculations. Our results are not in agreement with the recommended DR rates commonly used for modeling cosmic plasmas. We have used theoretical radiative recombination (RR) rates in conjunction with our derived DR rates to produce…
More than half of the atoms in the Universe recombined via forbidden transitions, so that accurate treatment of the forbidden channels is important in order to follow the cosmological recombination process with the level of precision required by future microwave anisotropy experiments. We perform a multi-level calculation of the recombination of hydrogen (H) and helium (He) with the addition of the triplet 2P to singlet 1S spin-forbidden transition for neutral helium (He I), plus the nS--1S and nD--1S two-photon transitions for H (up to n=40) and among singlet states of He I (n<=10 and l<=7). …
(abridged) A detailed examination of the evolution of stochastic magnetic fields between high cosmic temperatures and the present epoch is presented. A simple analytical model matching the results of the 3D MHD simulations allows for the prediction of present day magnetic field correlation lengths and energy. Our conclusions are multi fold. (a) Initial primordial fields with only a small amount of helicity are evolving into maximally helical fields. (b) There exists a correlation between the strength of the magnetic field, B, at the peak of it's spectrum and the location of the peak, given at …
We investigate the ionization of hydrogen in a dynamic Solar atmosphere. We show that the time scale for ionization/recombination can be estimated from the eigenvalues of a modified rate matrix where the optically thick Lyman transitions that are in detailed balance have been excluded. We find that the time scale for ionization/recombination is dominated by the slow collisional leakage from the ground state to the first excited state. Throughout the chromosphere the time scale is long ($10^3$-$10^5$ s), except in shocks where the increased temperature and density shorten the time scale for ion…
The non-linear effects operating at the recombination epoch generate a non-Gaussian signal in the CMB anisotropies. Such a contribution is relevant because it represents a major part of the second-order radiation transfer function which must be determined in order to have a complete control of both the primordial and non-primordial part of non-Gaussianity in the CMB anisotropies. We provide an estimate of the level of non-Gaussianity in the CMB arising from the recombination epoch which shows up mainly in the equilateral configuration. We find that it causes a contamination to the possible mea…
In this paper I will discuss the possibility of a reionization of the Universe due to the photons emitted by evaporating primordial black holes (PBHs); this process should happen during the last stages of the PBHs life, when the particle emission is very intense. I will study the time evolution of the ionization degree x, of the plasma temperature $T_{e}$ and of the photon number density $n_γ$ characterizing the Universe after the recombination epoch: a system of coupled differential equations for these variables is solved in an analytical way by assuming, as a photon source, PBHs having an in…
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-electro…
It has been argued that electroweak baryogenesis generates a left-handed primordial magnetic field. The helicity density of the primordial magnetic field today is estimated to be $\sim 10^2 n_b$ where $n_b\sim 10^{-6}$ /cm$^3$ is the present cosmological baryon number density. With certain assumptions about the inverse cascade the field strength at recombination is $\sim 10^{-13}$ G on a coherence scale $\sim 10^{-4}$ pc. Here I discuss the various assumptions made in obtaining these estimates and the concomitant open issues.
