Weak lensing in scalar-tensor theories of gravity

This article investigates the signatures of various models of dark energy on weak gravitational lensing, including the complementarity of the linear and nonlinear regimes. It investigates quintessence models and their extension to scalar-tensor gravity. The various effects induced by this simplest extension of general relativity are discussed. It is shown that, given the constraints in the Solar System, models such as a quadratic nonminimal coupling do not leave any signatures that can be detected while other models, such as a runaway dilaton, which include attraction toward general relativity, can let an imprint of about 10%.

Figure 1

General description of the computation of the lensing observables. The input are generated by a CMB code that deals with scalar-tensor theories and extended quintessence [9]. Once the linear to nonlinear mapping has been applied, we can compute the shear power spectrum and then all the lensing observables with the same normalization as used for the CMB observables.

Figure 2

Matter power spectrum $P_m(k)$ for a flat fiducial $\Lambda \mathrm{CDM}$ model defined by ${\Omega }_{\Lambda }=0.7$, ${\Omega }_b{h}^2=0.019$, $h=0.72$. Linear (solid line) and nonlinear (dashed line) regimes are presented.

Figure 3

(left) Convergence power spectrum $P_{\kappa }(\ell )$ as a function of the multipole $\ell$ and the 2-point statistics of the shear field: (middle) the aperture mass variance and (right) the shear variance in the case of the fiducial $\Lambda \mathrm{CDM}$ model (123).

Figure 4

Comparison between the fiducial $\Lambda \mathrm{CDM}$ model (123) and various quintessence models. (left) the growth factor normalized to its value in an Einstein-de Sitter universe, $D(a)/a$, as a function of the scale factor $a$, normalized at high redshift; (middle) Relative deviation from $\Lambda \mathrm{CDM}$ on the comoving radial distance, $\chi (z)$, and (right) on the geometrical factor, $q(z)$. In all plots, we consider three QCDM models with potential (124) with $m=6,8,11$ from top to bottom.

Figure 5

Ratio of the three dimensional matter power spectrum $P_m(k)$ for two quintessence models for $m=6$ and $m=11$ to the matter power spectrum for the reference $\Lambda \mathrm{CDM}$ model (123).

Figure 6

(left) Convergence power spectrum $P_{\kappa }(\ell )$ as a function of the multipole $\ell$ and the 2-point statistics of the shear field, as a function of the angle $\theta$: (middle) the aperture mass variance and (right) the shear variance for the reference $\Lambda \mathrm{CDM}$ model (123) (solid/long dashed lines) and two quintessence models ($m=6$, short dashed/dotted line; $m=11$, long-dashed-dotted/short-dashed-dotted line).

Figure 7

Deviation from general relativity parametrized by ${\gamma }_{\mathrm{PPN}}(z)-1$ for several nonminimally coupled models with Ratra-Peebles potential, Eq. (124), as a function of redshift $z$. Solid line (long dashed line): nonminimal coupling, $\xi =+0.001$ and $m=6(m=11)$ [potential defined in Jordan frame]. Short dashed line (dotted line): “runaway dilaton,” with $\beta =4,B=0.5$, and $m=6(m=11)$ [potential defined in Einstein frame]. Horizontal lines: level of the upper bounds measured by gravitational experiment on Solar System scales ($z\sim 0$).

Figure 8

Comparison between an extended quintessence model with $\xi ={10}^{-2}$ and $m=6$ and a quintessence model with same potential. (left) convergence power spectrum $([P_{\kappa }{]}_{\mathrm{EQ}-\xi }/[P_{\kappa }{]}_{\mathrm{QCDM}}-1)$, (middle) aperture mass variance and (right) and shear variance. Solid (long dashed): linear (nonlinear) regime for Ratra-Peebles potential with $m=6$. Short dashed (dotted): linear (nonlinear) regime for Ratra-Peebles potential with $m=11$.

Figure 9

Evolution of the scalar field in Einstein frame in a model of massless dilaton with a quadratic coupling (${\alpha }_0={10}^{-4}$, $\beta ={10}^2$). We compare the case of a vanishing potential (solid) and a constant potential (dashed). The solutions differ only recently. The period of the oscillation is compatible with the analytic result, as can be seen on the left plot where ${\phi }_{*}$ has been divided by the damping factor $(1+z_{*}{)}^{3/4}$.

Figure 10

Relative deviation $(P_{\kappa }/[P_{\kappa }{]}_{\mathrm{SCDM}}-1)$ from a SCDM model ($\Lambda =0$) on the convergence power spectrum of the massless scalar fields models, with quadratic coupling with $({\alpha }_0^2,\beta )=({10}^{-4},0.1)$, as a function of the multipole $\ell$. Linear (solid) and nonlinear (dashed) regimes are presented. The field is attracted rapidly toward the minimum of the coupling function so that the model does not differ from a SCDM by more than 0.1%.

Figure 11

Ratio $D(a)/a$ as a function of the scale factor $a$, normalized at high redshift, for extended quintessence models with quadratic ($\xi =+0.01$, left panel) and exponential coupling ($\beta =4,B=0.1$, right panel) and for the corresponding minimally coupled models. Notice that the potential are defined in different frames.

Figure 12

Comparison between a runaway dilaton model with $\beta =4,B=0.1$, and $m=6,11$ and a quintessence model with same potential. (left) convergence power spectrum $([P_{\kappa }{]}_{\mathrm{EQ}-\xi }/[P_{\kappa }{]}_{\mathrm{QCDM}}-1)$, (middle) aperture mass variance and (right) and shear variance. Solid (long dashed): linear (nonlinear) regime for Ratra-Peebles potential with $m=6$. Short dashed (dotted): linear (nonlinear) regime for Ratra-Peebles potential with $m=11$.

Figure 13

Other possible signatures of the runaway dilaton models. (left) The spectrum of the density perturbation $P_m$ differs from the power spectrum of the metric perturbation because of the modification of the Poisson equation. The lensing is sensitive to the deflecting potential $\Phi$ and thus enable to measure ${\delta }_{\mathrm{eff}}$, see Eq. (81). (middle) The anisotropic stress has two contributions [see Eq. (B6)] that are important only on large scales, the ansiotropic stress of the radiation and the contribution of the scalar field. The latter is dominant by a large factor in scalar-tensor theories. (right) The Wilkinson Microwave Anisotropy Probe (WMAP) satellite data (cross) compared to the runaway dilaton model prediction assuming an optical depth of $\tau =0.16$ for $m=11$ (dashed line) and $m=6$ (dotted line) compared to a $\Lambda \mathrm{CDM}$.

Figure 14

CMB angular power spectra: the WMAP data (cross line) compared to two models differing only by their optical depth $\tau =0$ (solid line) and $\tau =0.16$ (dashed line). This shows the influence of the reionization when one chooses to normalize to the CMB temperature anisotropies.