The effect of lensing on the large-scale cosmic microwave background anisotropy

We first compare the cosmic microwave background (CMB) lensing model of Seljak with the empirical model of Lieu & Mittaz to determine if the latter approach implies a larger effect on the CMB power spectrum. We find that the empirical model gives significantly higher results for the magnification dispersion, σ , at small scales (θ < 30 arcmin) than that of Seljak, assuming standard cosmological parameters. However, when the empirical foreground model of Lieu & Mittaz is modelled via correlation functions and used in the Seljak formalism, the agreement is considerably improved at small scales. Thus we conclude that the main difference between these results may be in the different assumed foreground mass distributions. We then show that a foreground mass clustering with ξ (r) ∝ r−3 gives an rms lensing magnification which is approximately constant with angle, θ . In Seljak’s formalism, we show this can lead to a smoothing of the CMB power spectrum which is proportional to e−σ 2l2/2 and which may be able to move the first CMB acoustic peak to smaller l, if the mass clustering amplitude is high enough. Evidence for a high amplitude of mass clustering comes from the quasi-stellar object magnification results of Myers et al. who suggest that foreground galaxy groups may also be more massive than expected, implying that m ≈ 1 and that there is strong galaxy antibias, b ≈ 0.2. We combine the above results to demonstrate the potential effect of lensing on the CMB power spectrum by showing that an inflationary model with neither cold dark matter (CDM) nor a cosmological constant and that predicts a primordial first peak at l = 330 might then fit at least the first acoustic peak of the Wilkinson Microwave Anisotropy Probe (WMAP) data. This model may be regarded as somewhat contrived since the fit also requires high-redshift reionization at the upper limit of what is allowed by the WMAP polarization results. However, given the finely tuned nature of the standard CDM model, the contrivance may be small in comparison and certainly the effect of lensing and other foregrounds may still have a considerable influence on the cosmological interpretation of the CMB.