Cosmological constraints on post-Newtonian parameters in effectively massless scalar-tensor theories of gravity

We study the cosmological constraints on the variation of Newton’s constant and on post-Newtonian parameters for simple models of the scalar-tensor theory of gravity beyond the extended Jordan-Brans-Dicke theory. We restrict ourselves to an effectively massless scalar field with a potential $V\propto F^2$, where $F(\sigma )=N_{\mathrm{pl}}^2+\xi {\sigma }^2$ is the coupling to the Ricci scalar considered. We derive the theoretical predictions for cosmic microwave background anisotropies and matter power spectra by requiring that the effective gravitational strength at present is compatible with the one measured in a Cavendish-like experiment and by assuming an adiabatic initial condition for scalar fluctuations. When comparing these models with Planck 2015 and a compilation of baryonic acoustic oscillations data, all these models accommodate a marginalized value for $H_0$ higher than in $\mathrm{\Lambda }\mathrm{CDM}$. We find no evidence for a statistically significant deviation from Einstein’s general relativity. We find $\xi <0.064$ ($|\xi |<0.011$) at 95% CL for $\xi >0$ (for $\xi <0$, $\xi \ne -1/6$). In terms of post-Newtonian parameters, we find $0.995<{\gamma }_{\mathrm{PN}}<1$ and $0.99987<{\beta }_{\mathrm{PN}}<1$ ($0.997<{\gamma }_{\mathrm{PN}}<1$ and $1<{\beta }_{\mathrm{PN}}<1.000011$) for $\xi >0$ (for $\xi <0$). For the particular case of the conformal coupling, i.e., $\xi =-1/6$, we find constraints on the post-Newtonian parameters of similar precision to those within the Solar System.

Figure 1

Top panel: Relative evolution of $\sigma$ for different values of $\xi$. Bottom panel: Evolution of $\sigma$ for different values of $N_{\mathrm{pl}}$ for the CC case, i.e., $\xi =-1/6$.

Figure 2

Evolution of $w_{\mathrm{DE}}$ for different values of $N_{\mathrm{pl}}$ and $\xi$. We plot the effective parameter of state for DE for $\xi >0$ in the upper panel, $\xi <0$ in the central panel, and the CC case $\xi =-1/6$ in the bottom panel.

Figure 3

Evolution of the density parameters ${\mathrm{\Omega }}_i$: radiation in yellow, matter in blue, and effective DE in red. We plot ${\tilde{N}}_{\mathrm{pl}}=1$ (${\tilde{N}}_{\mathrm{pl}}=0.9$) for $\xi ={10}^{-2},{10}^{-3}$ in the top (bottom) panel.

Figure 4

Evolution of the density parameters ${\mathrm{\Omega }}_i$: radiation in yellow, matter in blue, and effective DE in red. We plot ${\tilde{N}}_{\mathrm{pl}}=1.01$ (${\tilde{N}}_{\mathrm{pl}}=1.1$) for $\xi =-{10}^{-2},-{10}^{-3}$ in the top (bottom) panel.

Figure 5

Evolution of the density parameters ${\mathrm{\Omega }}_i$: radiation in yellow, matter in blue, and effective DE in red. We plot the CC case $\xi =-1/6$ for $\mathrm{\Delta }{\tilde{N}}_{\mathrm{pl}}={10}^{-3},{10}^{-4},{10}^{-5}$.

Figure 6

Evolution of the effective gravitational constant $G_{\mathrm{eff}}$ relative to its value today for different values of $N_{\mathrm{pl}}$ and $\xi$. From top to bottom, the cases with $\xi >0$, $\xi <0$, and $\xi =-1/6$ are displayed, respectively.

Figure 7

Evolution of the post-Newtonian parameters ${\gamma }_{\mathrm{PN}}$ and ${\beta }_{\mathrm{PN}}$ for different values of $N_{\mathrm{pl}}$ and $\xi$. We show the case with $\xi >0$.

Figure 8

Evolution of the post-Newtonian parameters ${\gamma }_{\mathrm{PN}}$ and ${\beta }_{\mathrm{PN}}$ for different values of $N_{\mathrm{pl}}$ and $\xi$. We show the NMC case with $\xi >0$.

Figure 9

Evolution of the post-Newtonian parameters ${\gamma }_{\mathrm{PN}}$ and ${\beta }_{\mathrm{PN}}$ for different values of $N_{\mathrm{pl}}$. We show the NMC case $\xi =-1/6$, i.e., the CC case.

Figure 10

Evolution of scalar field perturbations in the synchronous gauge for $k=0.05{\mathrm{Mpc}}^{-1}$.

Figure 11

Evolution of tensor fluctuations $h^T$ for $k=0.01{\mathrm{Mpc}}^{-1}$.

Figure 12

From top to bottom: Relative differences of the TT-EE-TE-$\varphi \varphi$ power spectra with respect to the $\mathrm{\Lambda }\mathrm{CDM}$ model for ${\tilde{N}}_{\mathrm{pl}}=1$, 0.9 and different values of $\xi ={10}^{-2},5\times {10}^{-3}$.

Figure 13

From top to bottom: Relative differences of the TT-EE-TE-$\varphi \varphi$ power spectra with respect to the $\mathrm{\Lambda }\mathrm{CDM}$ model for ${\tilde{N}}_{\mathrm{pl}}=1.01$, 1.1 and different values of $\xi =-{10}^{-2},-5\times {10}^{-3}$.

Figure 14

From top to bottom: Relative differences of the TT-EE-TE-$\varphi \varphi$ power spectra with respect to the $\mathrm{\Lambda }\mathrm{CDM}$ model for the CC case, i.e., $\xi =-1/6$, with different values of $\mathrm{\Delta }{\tilde{N}}_{\mathrm{pl}}={10}^{-4},{10}^{-5},{10}^{-6}$.

Figure 15

From top to bottom: Relative differences of the matter power spectra at $z=0$ with respect to the $\mathrm{\Lambda }\mathrm{CDM}$ model for $\xi >0$, $\xi <0$, and $\xi =-1/6$.

Figure 16

From top to bottom: CMB B-mode polarization band power, relative differences of the tensor contribution, and relative differences of the lensing contribution with respect to the $\mathrm{\Lambda }\mathrm{CDM}$ model for $\xi >0$, $\xi <0$, $\xi =-1/6$, and IG. Dashed lines refer to the lensing contribution to the B-mode polarization angular power spectrum.

Figure 17

2D marginalized confidence levels at 68% and 95% for ($H_0$, $\xi$) for NMC $\xi >0$ (red) and IG (blue) with Planck $\mathrm{TT}+\text{lowP}+\mathrm{lensing}+\text{BAO}$.

Figure 18

2D marginalized confidence levels at 68% and 95% for ($N_{\mathrm{pl}}$, $\xi$) for NMC $\xi >0$ with Planck $\mathrm{TT}+\text{lowP}+\mathrm{lensing}+\text{BAO}$.

Figure 19

2D marginalized confidence levels at 68% and 95% for ($H_0$, $\xi$) for NMC $\xi <0$ with Planck $\mathrm{TT}+\text{lowP}+\mathrm{lensing}+\text{BAO}$.

Figure 20

2D marginalized confidence levels at 68% and 95% for ($N_{\mathrm{pl}}$, $\xi$) for NMC $\xi <0$ with Planck $\mathrm{TT}+\text{lowP}+\mathrm{lensing}+\text{BAO}$.

Figure 21

2D marginalized confidence levels at 68% and 95% for ($H_0$, $N_{\mathrm{pl}}$) for conformal coupling with Planck $\mathrm{TT}+\text{lowP}+\mathrm{lensing}+\text{BAO}$. We include in blue the local estimates of $H_0=73.52\pm 1.62$ [km/s/Mpc] [52].

Figure 22

Redshift evolution for the relative difference between the Hubble parameter $H(z)$ and its $\mathrm{\Lambda }\mathrm{CDM}$ counterpart (upper panel) and the ratio $D_V(z)/r_s$ (lower panel). For $\mathrm{\Lambda }\mathrm{CDM}$ quantities we use Planck $\mathrm{TT}+\text{lowP}+\mathrm{lensing}+\text{BAO}$ best fit. The models plotted are IG, CC, $\xi <0$, $\xi >0$, the $\mathrm{\Lambda }\mathrm{CDM}$ model with $N_{\mathrm{eff}}>3.046$, and the $w\mathrm{CDM}$ with $w_0=\text{const}\ne 0$ for green, red, brown, black, blue, and orange lines, respectively.

Figure 23

2D marginalized confidence levels at 68% and 95% for (${\gamma }_{\mathrm{PN}}$, ${\beta }_{\mathrm{PN}}$) for NMC $\xi >0$ (left panel) and $\xi <0$ (right panel) with Planck $\mathrm{TT}+\text{lowP}+\mathrm{lensing}+\text{BAO}$.

Figure 24

2D marginalized confidence levels at 68% and 95% for ($H_0$, ${\gamma }_{\mathrm{PN}}$) for NMC $\xi >0$ (red) and IG (blue) with Planck $\mathrm{TT}+\text{lowP}+\mathrm{lensing}+\text{BAO}$.