Galaxy clustering constraints on deviations from Newtonian gravity at cosmological scales

In spite of the growing observational evidence for dark matter and dark energy in the universe, their physical nature is largely unknown. In fact, several authors have proposed modifications of Newton’s law of gravity at cosmological scales to account for the apparent acceleration of the cosmic expansion. Inspired by such suggestions, we attempt to constrain possible deviations from Newtonian gravity by means of the clustering of SDSS (Sloan Digital Sky Survey) galaxies. To be specific, we assume a simple modification of Newton’s law with an additional Yukawa-type term characterized by the amplitude $\alpha$ and the length scale $\lambda$. Adopting spatially-flat universes dominated by cold dark matter and/or dark energy, we solve a linear perturbation equation for the growth of density fluctuations. In particular, we find an exact analytic solution for the Einstein-de Sitter case. Following the Peacock-Dodds prescription, we compute the nonlinear power spectra of mass fluctuations, perform a statistical comparison with the SDSS galaxy data, and derive constraints in the $\alpha$-$\lambda$ plane; for instance, we obtain the constraints of $-0.5<\alpha <0.6$ and $-0.8<\alpha <0.9$ ($99.7\%$ confidence level) for $\lambda =5h^{-1}$ Mpc and $10h^{-1}$ Mpc, respectively. We also discuss several future possibilities for improving our analysis.

Figure 1

Linear growth rate in Model I where $G_{\mathrm{e}\mathrm{f}\mathrm{f}}=(1+{\alpha }_0)G_{\mathrm{N}}$ is independent of scale; (a) ${\Omega }_{\mathrm{m}}=1$ and ${\Omega }_{\Lambda }=0$, (b) ${\Omega }_{\mathrm{m}}=0.3$ and ${\Omega }_{\Lambda }=0.7$.

Figure 2

Linear growth rate in Model II; (a) $\alpha$-dependence for $\lambda =10h^{-1}$ Mpc and $k=0.01h{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${\Omega }_{\mathrm{m}}=1$ and ${\Omega }_{\Lambda }=0$), (b) $\lambda$-dependence for $\alpha =1$ and $k=0.01h{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${\Omega }_{\mathrm{m}}=1$ and ${\Omega }_{\Lambda }=0$), (c) $k$-dependence for $\alpha =1$ and $\lambda =10h^{-1}$ Mpc (${\Omega }_{\mathrm{m}}=1$ and ${\Omega }_{\Lambda }=0$), (d) $\alpha$-dependence for $\lambda =10h^{-1}$ Mpc and $k=0.01h{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${\Omega }_{\mathrm{m}}=0.3$ and ${\Omega }_{\Lambda }=0.7$), (e) $\lambda$-dependence for $\alpha =1$ and $k=0.01h{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (${\Omega }_{\mathrm{m}}=0.3$ and ${\Omega }_{\Lambda }=0.7$), (f) $k$-dependence for $\alpha =1$ and $\lambda =10h^{-1}$ Mpc (${\Omega }_{\mathrm{m}}=0.3$ and ${\Omega }_{\Lambda }=0.7$). In all the panels, the results for $\alpha =0$ (Newton’s law of gravity) are plotted in solid curves for reference.

Figure 3

Nonlinear power spectra in Model II; (a) $\alpha =0.5$, (b) $\alpha =-0.5$. We adopt $\Lambda$-CDM and $\lambda =5h^{-1}$ Mpc for definiteness. Dashed and solid curves indicate the linear and nonlinear predictions as discussed in the text. For reference, nonlinear Newtonian predictions ($\alpha =0$) are also shown in dotted curves.

Figure 4

Linearly extrapolated values of ${\sigma }_8(z)$ for different ${\alpha }_0$ in Model I; (a) ${\Omega }_{\mathrm{m}}=1.0$ and ${\Omega }_{\Lambda }=0.0$, (b) ${\Omega }_{\mathrm{m}}=0.3$ and ${\Omega }_{\Lambda }=0.7$.

Figure 5

Comparison of power spectra between Model II predictions and the SDSS galaxies (in the $\Lambda$-CDM model); (a) $\alpha =1$ (dotted) and $-1$ (long-dashed) for $\lambda =10h^{-1}$ Mpc, and (b) $\lambda =5h^{-1}$ Mpc (long-dashed), $10h^{-1}$ Mpc (dotted), and $30h^{-1}$ Mpc (short-dashed) for $\alpha =1$. For reference, the Newtonian case ($\alpha =0$) is plotted in solid curves. The data points with errors bars represent the results of the SDSS galaxies.

Figure 6

Constraints in the $\alpha$-$\lambda$ plane for Model II in $\Lambda$-CDM cosmology; (a) contours of $\Delta {\chi }^2$, (b) contours of ${\sigma }_8$, (c) contours of $b_{*,\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{m}\mathrm{i}\mathrm{n}}$.