Probing the Friedmann equation during recombination with future cosmic microwave background experiments

We show that by combining measurements of the temperature and polarization anisotropies of the Cosmic Microwave Background (CMB), future experiments will tightly constrain the expansion rate of the universe during recombination. A change in the expansion rate modifies the way in which the recombination of hydrogen proceeds, altering the shape of the acoustic peaks and the level of CMB polarization. The proposed test is similar in spirit to the examination of abundances of light elements produced during Big Bang Nucleosynthesis and it constitutes a way to study possible departures from standard recombination. For simplicity we parametrize the change in the Friedmann equation by changing the gravitational constant $G$. The main effect on the temperature power spectrum is a change in the degree of damping of the acoustic peaks on small angular scales. The effect can be compensated by a change in the shape of the primordial power spectrum. We show that this degeneracy between the expansion rate and the primordial spectrum can be broken by measuring CMB polarization. In particular we show that the MAP satellite could obtain a constraint for the expansion rate $H$ during recombination of $\delta H/H \simeq 0.09$ or $\delta G/G \simeq 0.18$ after observing for four years, whereas Planck could obtain $\delta H/H \leq 0.014$ or $\delta G/G \leq 0.028$ within two years, even after allowing for further freedom in the shape of the power spectrum of primordial fluctuations.Comment: Replaced to match published PRD version. ACBAR and new Boomerang data included in analysis. References added. 13 pages, 11 figure