On Breaking Cosmic Degeneracy

It has been argued that the power spectrum of the anisotropies in the Cosmic Microwave Background (CMB) may be effectively degenerate, namely that the observable spectrum does not determine a unique set of cosmological parameters. We describe the physical origin of this degeneracy and show that at small angular scales it is broken by gravitational lensing: effectively degenerate spectra become distinguishable at l ∼ 3000 because lensing causes their damping tails to fall at different rates with increasing l. This effect also helps in distinguishing nearly degenerate power spectra such as those of mixed dark matter models. Forthcoming interferometer experiments should provide the means of measuring otherwise degenerate parameters at the 5− 25% level. It has recently been pointed out [21] that in a parameter space including open models the contours of the estimated likelihood function for planned experiments are highly elongated in certain directions. This indicates that there is a near degeneracy between cosmological parameters which will limit the accuracy with which the individual parameters can be measured using only CMB observations. Note that with sufficiently good data as expected from planned satellite experiments, no such degeneracy arises in the more limited subset of flat models, contrary to an earlier claim [4]. Our intent with this letter is to first elucidate the physical origin of this effective degeneracy and then show how it is broken at small angular scales. We will not attempt to make detailed estimates of the precision with which particular future experiments may determine cosmological parameters. The cosmic degeneracy has a simple physical explanation which can be used to easily identify models that have near degenerate CMB power spectra. We restrict our discussion to homogeneous, isotropic cold dark matter models with arbitrary spatial curvature, adiabatic scalar perturbations and no tensor perturbations. A curve that produces degenerate spectra can be found by first specifying hΩm, h Ωb and the primordial spectrum. This fixes the comoving scale of the acoustic peaks. One then varies Ωm (or h) and ΩΛ in such a way that the angular size distance to the surface of last scattering remains constant. In doing so, the change in the redshift of the surface of last scattering must be taken into account, z∗ ∼= 103Ω lnΩb) b (1) [9]. The Hubble parameter is Ho = 100h km/s/Mpc and Ωm, Ωb and ΩΛ are, respectively, the density in matter, baryons and vacuum energy measured in units of the critical density. In contrast to the apparently similar but non-degenerate behavior found for a grid of models with ΩΛ+Ωm = 1 [4], this prescription gives models whose spectra are completely degenerate at small angles, or large multipole number l. This is what we mean by the description “effectively degenerate” as opposed to a case of “near” degeneracy which may arise due to experimental limitations. However, as can be seen in Figure (1) these spectra are not degenerate at large angular scales. This difference is due to the decay in the potential fluctuations in Ωm < 1 models,