Large scale structure as a probe of gravitational slip

A new time-dependent, scale-independent parameter, $\varpi$, is employed in a phenomenological model of the deviation from general relativity in which the Newtonian and longitudinal gravitational potentials slip apart on cosmological scales as dark energy, assumed to be arising from a new theory of gravitation, appears to dominate the Universe. A comparison is presented between $\varpi$ and other parametrized post-Friedmannian models in the literature. The effect of $\varpi$ on the cosmic microwave background anisotropy spectrum, the growth of large-scale structure, the galaxy weak-lensing correlation function, and cross correlations of cosmic microwave background anisotropy with galaxy clustering are illustrated. Cosmological models with conventional maximum likelihood parameters are shown to find agreement with a narrow range of gravitational slip.

Figure 1

CMB anisotropy power spectra for different ${\varpi }_0$ cosmologies are shown with the binned WMAP 3-year data. All differences are localized to the low multipole moments: the spectra are normalized so that the higher multipole moments for all models are identical to the case of ${\varpi }_0=0$, corresponding to the WMAP3 ML model.

Figure 2

The predicted CMB temperature anisotropy quadrupole power is shown as a function of ${\varpi }_0$ (solid curve). All other parameters are set to the WMAP3 ML cosmology. The central value and $1\sigma$ upper bound of the WMAP 3-year data, $6C_2/2\pi =211\pm 860$ [31], are shown by the dashed and dot-dashed curves.

Figure 3

The likelihood of ${\varpi }_0$ due to the CMB, calculated using the November 2006 version of the WMAP likelihood code [31], is shown. All other cosmological parameters are set to the WMAP3 ML (solid curve) or the $\mathrm{WMAP}3+\mathrm{SNGold}$ ML (dashed curve) model. The results include the TT and TE spectra. The results are consistent with ${\varpi }_0=0$ ($\Lambda \mathrm{CDM}$), but favor positive values of ${\varpi }_0$. The locations of the primary (left) and secondary (right) likelihood peaks correspond to the range of ${\varpi }_0$ for which the predicted quadrupole lies within the $2\sigma$ range of WMAP.

Figure 4

The equivalence of alternative techniques for evolving cosmological perturbations under PPF gravitation is illustrated by the effect on CMB anisotropy spectra. Lines show the spectra produced following our procedure [Eqs. (14, 15, 16)] with the parametrization (4) replaced by that proposed in BZ, $\varpi (a)=-\beta a^s/(1+\beta a^s)$ with $s=3$. Symbols indicate the spectra produced following BZ’s procedure [Eqs. (18, 19, 20)]. The agreement between the techniques is exact.

Figure 5

The effect of the parametrization of gravitational slip on CMB anisotropy spectra is shown. Lines show the spectra produced using our parametrization, Eq. (4); symbols indicate spectra produced using BZ’s parametrization, $\varphi /\psi =1+\beta a^s$ with $s=3$. To compare, the parameter $\beta$ is set to $\beta =-{\varpi }_0{\Omega }_{\Lambda }/{\Omega }_m$. The agreement is excellent in the limit of small $\beta$.

Figure 6

The matter density contrast as a function of scale factor is shown for different values of ${\varpi }_0$. The density contrast is normalized to a ${\varpi }_0=0$ model. All models use WMAP3 ML parameters.

Figure 7

The goodness of fit of the mapping of the growth rate between $\varpi \Lambda \mathrm{CDM}$ and $\Lambda \mathrm{CDM}$ models is illustrated. The percent difference in the density contrast for a model with ${\varpi }_0=0.4$, ${\Omega }_m=0.3$, and a $\Lambda \mathrm{CDM}$ model with ${\Omega }_m=0.35$ is plotted with respect to the cosmic scale factor. These cosmologies are predicted to be equivalent by Eq. (24).

Figure 8

The dimensionless linear growth rate $f$ plotted as a function of $z$ for different ${\varpi }_0$ cosmologies with the WMAP3 ML background. The data point at $z=0.77$ is due to Ref. 60. The remaining, hybrid set of data points are taken from Table 1 of Ref. 9, and are based on estimates of large-scale structure power spectrum growth at different redshifts.

Figure 9

The values of ${\sigma }_8$ resulting from the normalization to the WMAP 3-year anisotropy power spectrum are shown as a function of ${\varpi }_0$. All other cosmological parameters are set to match the WMAP3 ML model.

Figure 10

Model predictions of ${\xi }_E$ for different values of ${\varpi }_0$. Data are taken from Table B.1 of 47. The background cosmology is the WMAP3 ML. The power spectrum amplitude is determined by the normalization to WMAP.

Figure 11

The likelihood of different ${\varpi }_0$ models according to the ${\xi }_E$ data collected by the Canada-France-Hawaii Telescope Legacy Survey and presented in Table B.1 of Ref. 47 is shown with parameters set by the WMAP3 ML model (solid line) as well as the $\mathrm{WMAP}3+\mathrm{SNGold}$ ML model (dashed line).

Figure 12

The cross-correlation angular power spectrum between CMB temperature (ISW) and galaxy distribution is shown as a function of ${\varpi }_0$. A value of $0.5<{\varpi }_0<1.5$ changes the sign of the ISW, resulting in a negative cross correlation, in conflict with observations.

Figure 13

The cross-correlation amplitude at $\theta =6\mathrm{degrees}$ is shown as a function of redshift for different samples of LSS and WMAP data. The curves show the expected correlation with ${\varpi }_0$ varied. The data plotted here come from the compilation in Ref. 59, making use also of data from Gaztanaga et al. [61] (circles), Cabre et al. [62] (squares), and Giannantonio et al. [59] (triangle).

Figure 14

The likelihood of different ${\varpi }_0$ models using current ISW-galaxy cross-correlation data is shown. All other cosmological parameters are fixed to the WMAP3 ML model (solid line) or the $\mathrm{WMAP}3+\mathrm{SNGold}$ ML model (dashed line).

Figure 15

The likelihood distributions for ${\varpi }_0$, based on the CMB (solid curve), weak lensing (dashed curve), and ISW-galaxy cross correlation (dot-dashed curve) using the WMAP3 ML model parameters are shown.

Figure 16

The likelihood distributions for ${\varpi }_0$, based on the CMB (solid curve), weak lensing (dashed curve), and ISW-galaxy cross correlation (dot-dashed curve) using the $\mathrm{WMAP}3+\mathrm{SNGold}$ ML model parameters are shown.