Any variation of the fundamental physical constants, and more particularly of the fine structure constant, $alpha$, or of the mass of the electron, $m_e$, would affect the recombination history of the Universe and cause an imprint on the cosmic micro
wave background angular power spectra. We show that the Planck data allow one to improve the constraint on the time variation of the fine structure constant at redshift $zsim 10^3$ by about a factor of 5 compared to WMAP data, as well as to break the degeneracy with the Hubble constant, $H_0$. In addition to $alpha$, we can set a constraint on the variation of the mass of the electron, $m_{rm e}$, and on the simultaneous variation of the two constants. We examine in detail the degeneracies between fundamental constants and the cosmological parameters, in order to compare the limits obtained from Planck and WMAP and to determine the constraining power gained by including other cosmological probes. We conclude that independent time variations of the fine structure constant and of the mass of the electron are constrained by Planck to ${Deltaalpha}/{alpha}= (3.6pm 3.7)times10^{-3}$ and ${Delta m_{rm e}}/{m_{rm e}}= (4 pm 11)times10^{-3}$ at the 68% confidence level. We also investigate the possibility of a spatial variation of the fine structure constant. The relative amplitude of a dipolar spatial variation of $alpha$ (corresponding to a gradient across our Hubble volume) is constrained to be $deltaalpha/alpha=(-2.4pm 3.7)times 10^{-2}$.
We introduce a new method to propagate uncertainties in the beam shapes used to measure the cosmic microwave background to cosmological parameters determined from those measurements. The method, which we call Markov Chain Beam Randomization, MCBR, ra
ndomly samples from a set of templates or functions that describe the beam uncertainties. The method is much faster than direct numerical integration over systematic `nuisance parameters, and is not restricted to simple, idealized cases as is analytic marginalization. It does not assume the data are normally distributed, and does not require Gaussian priors on the specific systematic uncertainties. We show that MCBR properly accounts for and provides the marginalized errors of the parameters. The method can be generalized and used to propagate any systematic uncertainties for which a set of templates is available. We apply the method to the Planck satellite, and consider future experiments. Beam measurement errors should have a small effect on cosmological parameters as long as the beam fitting is performed after removal of 1/f noise.
