Parameter dependence of magnetized CMB observables

Pre-decoupling magnetic fields affect the scalar modes of the geometry and produce observable effects which can be constrained also through the use of current (as opposed to forthcoming) data stemming from the Cosmic Microwave Background observations. The dependence of the temperature and polarization angular power spectra upon the parameters of an ambient magnetic field is encoded in the scaling properties of a set of basic integrals whose derivation is simplified in the limit of small angular scales. The magnetically-induced distortions patterns of the relevant observables can be computed analytically by employing scaling considerations which are corroborated by numerical results. The parameter space of the magnetized Cosmic Microwave background anisotropies is also discussed in the light of the obtained analytical results. Electronic address: [email protected] 1 Formulation of the problem There are two complementary approaches to the analysis of the Cosmic Microwave Background (CMB in what follows) observables. The first one is direct and it consists in computing the angular power spectra by faithfully including all the relevant physical effects. The second approach is indirect, i.e. it amounts to deriving the dependence of the (measured) temperature and polarization anisotropies upon the parameters of the the underlying model which needs to be falsified. The recent WMAP 5yr data [1, 2, 3] (see also [4, 5]) have been confronted with a number of theoretical scenarios that are logically organized around the ΛCDM paradigm where Λ stands for the dark-energy component and CDM stands for Cold Dark Matter. Similar statements can be made for other recent CMB data such as the ACBAR observations [6, 7] and the QUAD measurements [8, 9, 10, 11]. A useful bridge between the direct and the indirect approach is represented by a number of scaling relations which serve as a diagnostic for the dependence of the (observed) angular power spectra upon the parameters of a pivotal model. The temperature and polarization autocorrelations (i.e., respectively, TT and EE angular power spectra) and their mutual cross-correlations (i.e. the TE angular power spectra) can be written, with shorthand notation, as G (TT) l = l(l+ 1) 2π C (TT) l , G (EE) l = l(l+ 1) 2π C (EE) l , G (TE) l = l(l+ 1) 2π C (TE) l . (1.1) In the ΛCDM scenario the angular power spectra of Eq. (1.1) are functions of, at least, six physical quantities G (XY) l = G (XY) l (ns, Ωb0, Ωc0,ΩΛ, H0, ǫre), (1.2) where X and Y stand, respectively, for T and E and where the parameters denote, with standard notations, the spectral index of (adiabatic) curvature perturbations (i.e. ns), the critical fractions of baryons, CDM and dark energy (i.e., respectively, Ωb0, Ωc0 and ΩΛ), the Hubble constant H0 and the optical depth at reionization (i.e. ǫre). In the ΛCDM paradigm as well as in it extensions, the known scaling relations are often not the result of a numerical inference but are derived by means of analytical methods. Suppose, for sake of concreteness, that all the parameters of Eq. (1.2) are fixed to the best fit of the WMAP 5yr data alone and just one (e.g. the spectral index) is allowed to scale. From semi-analytic considerations it follows that G (TT) l ∝ ( l lp )ns+1 , G (EE) l ∝ ( l lp )ns+1 , G (TE) l ∝ ( l lp )ns (1.3) where the notation ∝ signifies that the corresponding quantity scales with the multipole in a given manner. When the scalar spectral index changes from the best-fit value (i.e. In Eq. (1.3) lp denotes the pivot multipole at which the initial conditions are customarily set. This scale is largely conventional and it will be hereby chosen to coincide with l = 29 which does correspond to the pivot wavenumber kp = 0.002Mpc .