Generalized CMB initial conditions with pre-equality magnetic fields

The most general initial conditions of CMB anisotropies, compatible with the presence of pre-equality magnetic fields, are derived. When the plasma is composed by photons, baryons, electrons, CDM particles and neutrinos, the initial data of the truncated Einstein-Boltzmann hierarchy contemplate one magnetized adiabatic mode and four (magnetized) non-adiabatic modes. After obtaining the analytical form of the various solutions, the Einstein-Boltzmann hierarchy is numerically integrated for the corresponding sets of initial data. The TT, TE and EE angular power spectra are illustrated and discussed for the magnetized generalization of the CDMradiation mode, of the baryon-radiation mode and of the non-adiabatic mode of the neutrino sector. Mixtures of initial conditions are examined by requiring that the magnetized adiabatic mode dominates over the remaining non-adiabatic contributions. In the latter case, possible degeneracies between complementary sets of initial data might be avoided through the combined analysis of the TT, TE and EE angular power spectra at high multipoles (i.e. l > 1000). 1 Pre-equality CMB initial conditions What are the initial conditions of CMB anisotropies? In the pivotal ΛCDM paradigm, the initial conditions are taken to be adiabatic [1, 2, 3, 4, 5]. In this case the fluctuations of the total pressure are proportional, prior to matter-radiation equality, to the fluctuations of the (total) energy density. Already in the absence of large-scale magnetic fields, the initial data of CMB anisotropies are not exhausted by adiabatic solution. In the non-adiabatic case the fluctuations of the total pressure arise because of the compositeness of the pre-equality plasma. Standard Boltzmann solvers include the possibility of having one (or more) non-adiabatic initial conditions. The generalized initial conditions of the Einstein-Boltzmann hierarchy are hereby derived when large-scale magnetic fields are consistently included in the pre-equality plasma. The obtained results complement and extend former analyses only centered around the magnetized adiabatic mode. A stochastic background of large-scale magnetic field can be naturally incorporated in the physics of the adiabatic initial conditions. The TT angular power spectra have been analytically estimated, within the tight-coupling approximation, in [6, 7, 8]. The results of [7, 8] can be used for simplified estimates of the TT angular power spectra, and, partially [6], of the TE and EE correlations. A full numerical approach is however required to confront the present [4, 5] and forthcoming [9] experimental data. This problem was successfully addressed and solved in [10, 11]. Semi-analytical results (obtained via a different treatment of recombination and diffusive effects) agree with the full numerical calculation: the shape of the correlated distortions of the first three acoustic peaks is correctly captured by the analytical discussion even if the numerical approach is intrinsically more accurate especially at high multipoles. This occurrence strengthen the consistency of the numerical approach and allows interesting analytical cross-checks. In [10, 11] the only initial conditions examined were the ones associated with the magnetized adiabatic mode. This choice is prompted by the best fit of the WMAP data alone as well as, for instance, by the best fits of the WMAP data combined with the large-scale structure data [12, 13], with the supernova data [14, 15] and with all cosmological data sets. The strict adiabaticity of the initial conditions will be now relaxed and non-adiabatic modes will be scrutinized in the presence of stochastic magnetic fields. This program is technically mandatory and physically relevant. If the initial data of the Einstein-Boltzmann hierarchy are solely non-adiabatic, the measured TE correlations cannot be reproduced [3]. This result is already apparent from the first 200 multipoles of the TE power spectra where, generically, non-adiabatic modes lead to a positive correlation while a predominant adiabatic mode would imply, instead, a negative cross-correlation. The position of the anticorrelation peak can be related, in the adiabatic case, to the position of the first Doppler peak of the TT power spectra (i.e. lDop ≃ 220). The position of the first anticorrelation peak of the TE angular power spectrum can be estimated as lanti ≃ 3lDop/4 ≃ 150 to first-order in the well known tight-coupling expansion [16] (see also [17, 18, 19]). The experimental evidence does not exclude that a predominant adiabatic mode could be present in combination with sub-dominant non-adiabatic contributions so that the overall fit to the data may even improve [20, 21]. By Einstein-Boltzmann hierarchy we mean the set of kinetic equations written in curved space and supplemented by the contribution of the gravitational inhomogeneities obeying the perturbed Einstein equations. Following a consolidated terminology, the angular power spectra of the temperature autocorrelations will be denoted, with stenographic notation, by TT. In analog terms EE and TE denote, respectively, the angular power spectra of the polarization autocorrelations and of the temperature-polarization cross-correlations.