Cosmological tests of modified gravity: Constraints on $F(R)$ theories from the galaxy clustering ratio

The clustering ratio $\eta$, a large-scale structure observable originally designed to constrain the shape of the power spectrum of matter density fluctuations, is shown to provide a sensitive probe of the nature of gravity in the cosmological regime. We apply this analysis to $F(R)$ theories of gravity using the luminous red galaxy sample extracted from the spectroscopic Sloan Digital Sky Survey (SDSS) data release 7 and 10 catalogs. We find that general relativity (GR), complemented with a Friedmann-Robertson-Walker (FRW) cosmological model with parameters fixed by the Planck satellite, describes extremely well the clustering of galaxies up to $z\sim 0.6$. On large cosmic scales, the absolute amplitude of deviations from GR, $|f_{R_0}|$, is constrained to be smaller than $4.6\times 10^{-5}$ at the 95% confidence level. This bound makes cosmological probes of gravity almost competitive with the sensitivity of Solar System tests, although still one 1 order of magnitude less effective than astrophysical tests. We also extrapolate our results to future large surveys like Euclid and show that the astrophysical bound will certainly remain out of reach for such a class of modified-gravity models that only differ from $\mathrm{\Lambda }\mathrm{CDM}$ at low redshifts.

Figure 1

Upper panel: clustering ratio ${\eta }_g^{\mathrm{s}}(n,x)$ as a function of the filtering scale $x$ for $n=2$ (squares), $n=3$ (triangles), and $n=4$ (diamonds), at $z=0.29$, the mean redshift of the $s1$ sample. The upper (dotted) line represents the scaling predicted in the reference $\mathrm{\Lambda }\mathrm{CDM}$ model (flat model with ${\mathrm{\Omega }}_{m0}=0.3175$) while the middle (dashed) and lower (solid) lines correspond to the $F(R)$ models with exponent $N=1$ and normalization parameters $f_{R_0}=-10^{-5}$ and $f_{R_0}=-10^{-4}$, respectively. Lower panel: relative deviation of the $F(R)$ models from the $\mathrm{\Lambda }\mathrm{CDM}$ prediction.

Figure 2

Upper panel: clustering ratio ${\eta }_g^{\mathrm{s}}(n,x,z)$ measured in three different redshift intervals centered at $z=0.29$ ($s1$ sample), 0.42 and 0.60 ($s2$ sample). The redshift intervals in which the $s2$ sample is split are defined so that error bars are roughly equivalent to that estimated from the $s1$ sample. We show measurements obtained for the smoothing scale $x=16h^{-1}\mathrm{Mpc}$ and for the correlation indices $n=2$ (squares) and $n=3$ (triangles). We also show the amplitude of the clustering ratio predicted by the reference $\mathrm{\Lambda }\mathrm{CDM}$ scenario (upper dotted lines) and by the $F(R)$ models with exponent $\mathrm{N}=1$ and with normalization parameters $f_{R_0}=-10^{-5}$ (middle dashed lines) and $f_{R_0}=-10^{-4}$ (lower solid lines). We give an example typical of future surveys: the small black error bars on the standard $\mathrm{\Lambda }\mathrm{CDM}$ curve (dotted) show forecasts for measurements in bins of size $\mathrm{\Delta }z\sim 0.2$ from an 15,000 sq. deg. survey of $7\times 10^6$ galaxies, which closely matches what is expected from the Euclid mission [40]. Lower panel: relative deviation of the $F(R)$ models from the $\mathrm{\Lambda }\mathrm{CDM}$ prediction.

Figure 3

Impact of linear redshift-space distortions on the amplitude of the clustering ratio, as a function of scale (upper panel) and redshift (lower panel). We show the relative deviation between ${\eta }_g$ and ${\eta }_g^s$ [computed using the model in Eq. (10)].

Figure 4

Impact of nonlinear random motions on the amplitude of the clustering ratio, as a function of scale (left panel) and redshift (right panel). We show the relative deviation between the $\eta$ amplitude predicted before and after correcting the clustering ratio estimates with a Gaussian model of the galaxy velocity dispersion (cf. Eq. (9) of [31]).

Figure 5

Impact of scale-dependent contributions of the biasing models, here the P and Q models, on the ratio between galactic and matter quantities $Y$ where $Y$ can be either the power spectrum $P(k)$, the 2-point correlation $\xi (x)$ or the clustering ratio $\eta (2,x)$. The characteristic parameters of the biasing models ($a$ and $Q$) are varied within their 95% confidence interval determined by [48]. In all cases, the $\eta$ dependence on bias is less than 2%.

Figure 6

Upper panel: galaxy power spectrum (diamonds) simulated using the reference $\mathrm{\Lambda }\mathrm{CDM}$ model (${\mathrm{\Omega }}_{m0}=0.3175$) and the Q model for describing scale-dependent galaxy biasing ($b_1=1.5$, $Q=9.6h^{-2}{\mathrm{Mpc}}^2$ and $A=1.4h^{-1}\mathrm{Mpc}$). Error bars roughly correspond to what is expected in a survey like BOSS [33]. The thick red dashed line shows the best fit that is obtained if we analyze this data by the P model (i.e., a “wrong” bias model), marginalizing over $0.9<b_1<3$ and $10<a<60$ with flat priors. The minimum value of the ${\chi }^2$ statistic (with $\nu =100-3$ degrees of freedom) is indicated in the inset and suggests that this best fitting model cannot be rejected on statistical grounds. Lower panel: likelihood constraints on ${\mathrm{\Omega }}_{m0}$. The vertical grey dotted line indicates the input value of the mass density parameter ${\mathrm{\Omega }}_{m0}$. As in the upper panel, the thick red dashed line shows the result obtained from the power spectrum analysis by using the “wrong” P model, which leads to a significant overestimation bias. The thin red dot-dot-dot-dashed line shows the one-dimensional likelihood profile obtained from the power spectrum analysis after marginalizing over the biasing parameters of the “true” Q model itself, which gives an unbiased result. These results are compared with those obtained by adopting the clustering ratio (with $x=16h^{-1}\mathrm{Mpc}$ and $n=2$) as observable in the likelihood analysis, without implementing any marginalization scheme. The corresponding likelihood profile is shown by the thick blue long dashed line. In addition, we also display with the blue thin dot-dashed line the likelihood profile obtained from the $\eta$-test when the data are generated with the P model.

Figure 7

Impact of nonlinear contributions the power spectrum of matter on the amplitude of the clustering ratio, as a function of scale (upper panel) and redshift (lower panel). We show the relative deviation between $\eta$ predictions obtained using the linear theory and the nonlinear model described in [16].

Figure 8

${\chi }^2$ profile from the least square analysis of the clustering ratio data $\eta (n,x)$ measured from the SDSS samples $s1$ and $s2$ for $x=16h^{-1}\mathrm{Mpc}$ and in three different redshift bins at $z=0.29,0.42$, and 0.60. We show results for the correlation indices $n=2$ ( upper panel) and $n=3$ ( lower panel). We consider $F(R)$ models with $N=1$ (upper solid lines) and $N=2$ (lower dashed lines). The horizontal dotted lines are the 68% and 95% confidence contours.

Figure 9

Two-dimensional likelihood contours on ${\mathrm{\Omega }}_{m0}$ and $f_{R_0}$ from the least square analysis of the clustering ratio $\eta (n,x)$ of the SDSS samples $s1$ and $s2$. The clustering ratio is estimated for $x=16h^{-1}\mathrm{Mpc}$ and $n=2$ in three different redshift bins ($z=0.29,0.42$, and 0.60). Contours corresponds to 68% and 95% C.L. for a multivariate Gaussian distribution with 2 degrees of freedom. Black contours show the results obtained by fixing the baryon density ${\mathrm{\Omega }}_b{h}^2=0.0221$, the Hubble constant $H_0=67.4\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ and the scalar spectral index $n_s=0.96$ but letting ${\mathrm{\Omega }}_{m0}$ as a free parameter. Green shaded areas show the results after implementing the Planck Gaussian prior ${\mathrm{\Omega }}_{m0}=0.315\pm 0.017$. We consider $F(R)$ models with $N=1$.

Figure 10

Marginalized one-dimensional likelihood constraints on ${\mathrm{\Omega }}_{m0}$ (upper panel) and $f_{R_0}$ (lower panel). The green/blue curves show the results obtained with/without the Gaussian Planck prior on ${\mathrm{\Omega }}_{m0}$.

Figure 11

Two-dimensional likelihood contours on ${\mathrm{\Omega }}_{m0}$ and $f_{R_0}$ from the least square analysis of the clustering ratio $\eta (n,x)$ of the SDSS samples $s1$ and $s2$. The clustering ratio is estimated for $x=16h^{-1}\mathrm{Mpc}$ and $n=2$ in three different redshift bins ($z=0.29,0.42$, and 0.60). Contours corresponds to 68% and 95% C.L. for a multivariate Gaussian distribution with 2 degrees of freedom. Black contours show the results obtained by fixing the baryon density ${\mathrm{\Omega }}_b{h}^2=0.0221$, the Hubble constant $H_0=67.4\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ and the scalar spectral index $n_s=0.96$ but letting ${\mathrm{\Omega }}_{m0}$ as a free parameter. Blue shaded areas show the results after combining with the expected constraints from Euclid. We consider $F(R)$ models with $N=1$.