Testing cosmology with cosmic sound waves

Wilkinson Microwave Anisotropy Probe (WMAP) observations have accurately determined the position of the first two peaks and dips in the cosmic microwave background (CMB) temperature power spectrum. These encode information on the ratio of the distance to the last scattering surface to the sound horizon at decoupling. However prerecombination processes can contaminate this distance information. In order to assess the amplitude of these effects, we use the WMAP data and evaluate the relative differences of the CMB peak and dip multipoles. We find that the position of the first peak is largely displaced with respect to the expected position of the sound horizon scale at decoupling. In contrast, the relative spacings of the higher extrema are statistically consistent with those expected from perfect harmonic oscillations. This provides evidence for a scale dependent phase shift of the CMB oscillations which is caused by gravitational driving forces affecting the propagation of sound waves before recombination. By accounting for these effects we have performed a Markov Chain Monte Carlo likelihood analysis of the location of WMAP extrema to constrain, in combination with recent BAO data, a constant dark energy equation of state parameter $w$. For a flat universe we find a strong $2\sigma$ upper limit $w<-1.10$, and including the Hubble Space Telescope prior, we obtain $w<-1.14$, which is only marginally consistent with limits derived from the Supernova Legacy Survey sample. On the other hand, we infer larger limits for nonflat cosmologies. From the full CMB likelihood analysis, we also estimate the values of the shift parameter $R$ and the multipole $l_a$ of the acoustic horizon at decoupling for several cosmologies, to test their dependence on model assumptions. Although the analysis of the full CMB spectra should always be preferred, using the position of the CMB peaks and dips provides a simple and consistent method for combining CMB constraints with other data sets.

Figure 1

WMAP spacings of $l_{3/2}$, $l_2$, and $l_{5/2}$ relative to $l_1$ (black solid circles) and propagated errors. The values expected from the harmonic series are ${\Delta }_{1,3/2}=1/2$, ${\Delta }_{1,2}=1$, and ${\Delta }_{1,5/2}=3/2$ (open circles). Vertical dashed lines delimit the expected interval of variation of the relative spacings obtained by including the shift corrections as parametrized in 18 and evaluated over a conservative range of cosmological parameter values (see text). The dotted vertical lines include the effect of three massless neutrinos.

Figure 2

As in Fig. 1 for $l_{3/2}$, $l_2$, and $l_{5/2}$ relative spacings. The harmonic series values are ${\Delta }_{3/2,2}=1/3$, ${\Delta }_{2,5/2}=1/4$, and ${\Delta }_{3/2,5/2}=2/3$.

Figure 3

Marginalized $1\sigma$ and $2\sigma$ likelihood contours from WMAP extrema (upper panels) and in combination with BAO (lower panels). Dashed lines correspond to contours inferred under HST priors. The dotted lines in the upper left panel correspond to limits inferred assuming ${\Omega }_b=0.023$ and $n_s=0.96$.

Figure 4

Marginalized $1\sigma$ and $2\sigma$ likelihood contours inferred from the full WMAP 3-year spectra.

Figure 5

Marginalized $1\sigma$ and $2\sigma$ contours in the ${\Omega }_m\mathrm{\text{−}}w$ plane from SNLS data (solid lines) and in combination with the location of the CMB extrema (dark and light grey shaded regions).

Figure 6

As in Fig. 5 in the $w_0\mathrm{\text{−}}{w}_1$ plane.