Validating estimates of the growth rate of structure with modified gravity simulations

We perform a validation of estimates of the growth rate of structure, described by the parameter combination $f{\sigma }_8$, in modified gravity cosmologies. We consider an analysis pipeline based on the redshift-space distortion modeling of the clustering wedges statistic of the galaxy correlation function and apply it to mock catalogues of Lambda cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$) and the normal branch of Dvali-Gabadadze-Porrati (DGP) cosmologies. We employ a halo occupation distribution approach to construct our mocks, which we ensure resemble the CMASS sample from Baryon Oscillation Spectroscopic Survey (BOSS) in terms of the total galaxy number density and large-scale amplitude of the power spectrum monopole. We show that the clustering wedges model successfully recovers the true growth rate difference between DGP and $\mathrm{\Lambda }\mathrm{CDM}$, even for cases with over 40% enhancement in $f{\sigma }_8$ compared to $\mathrm{\Lambda }\mathrm{CDM}$. The unbiased performance of the clustering wedges model allows us to use the growth rate values estimated from the BOSS DR12 data to constrain the crossover scale $r_c$ of DGP gravity to ${[r_c{H}_0]}^{-1}<0.97$ ($2\sigma$) or $r_c>3090\mathrm{Mpc}/h$, cutting into the interesting region of parameter space with $r_c\sim H_0^{-1}$ using constraints from the growth of structure alone.

Figure 1

Cumulative halo mass function of $\mathrm{\Lambda }\mathrm{CDM}$ and the three DGP cases at $z=0.57$, as labeled (the upper panel shows the absolute value, while the lower panel shows the relative difference to $\mathrm{\Lambda }\mathrm{CDM}$). The dots show the simulation results and the solid lines display the best-fitting ST formulas. These halo catalogues were constructed with the Rockstar code [61].

Figure 2

Best-fitting ST linear halo bias of $\mathrm{\Lambda }\mathrm{CDM}$ and the three DGP cases at $z=0.57$, as labeled.

Figure 3

Best-fitting galaxy HOD for $\mathrm{\Lambda }\mathrm{CDM}$ and the three DGP cases studied in this paper, as labeled. The left panel shows $N(M)$ (solid), $N_c(M)$ (dashed) and $N_s(M)$ (dotted) distributions, while the right panel shows these distributions multiplied by the halo mass function to show the relative contribution to the total galaxy number density from halos of a given mass. The panel inset on the right indicates where each best-fitting HOD model lies in $[{\sigma }_{{\log }_{10}M},{\log }_{10}{M}_{\min }]$ space.

Figure 4

Clustering wedges ${\xi }_{\perp }$ (dots) and ${\xi }_{\parallel }$ (squares) measured from the mocks of $\mathrm{\Lambda }\mathrm{CDM}$ and the three DGP cases, as labeled. The error bars show the diagonal entries of the covariance matrix model of Ref. [66]. The solid lines show the best-fitting model outlined in Sec. 4a when the AP and $f{\sigma }_8$ parameters are fixed to their true values.

Figure 5

Growth rate estimates from the analysis of the galaxy mock catalogues of $\mathrm{\Lambda }\mathrm{CDM}$ and the three DGP cases, as labeled. The left panel shows the expected time evolution of $f{\sigma }_8$ (lines) together with the corresponding estimates obtained from the mocks (dots with error bars; all points correspond to $z=0.57$, but have been slightly shifted for better visualization). The purple squares correspond to the estimates obtained in Ref. [21] from a similar analysis applied to the DR12 galaxy sample from BOSS in three redshift bins, $z=0.38$, 0.51, 0.61 (the open symbol used for the mid-redshift point is used to remind that this point is very covariant with the other two). The cyan squares show the estimates obtained in Ref. [19] using a different RSD clustering wedges model applied to the LOWZ ($z=0.32$) and CMASS ($z=0.57$) BOSS galaxy samples from DR11. The right panel shows the expected difference to $\mathrm{\Lambda }\mathrm{CDM}$ in $f{\sigma }_8$ as a function of ${[r_c{H}_0]}^{-1}$ for $z=0.57$ (line), together with the difference measured from the mocks. The two error bar sizes for each DGP model correspond to two different values of $T_{\mathrm{CV}}$ in Eq. (37).

Figure 6

Clustering wedges ${\xi }_{3w,i}$ $i=1$, 2, 3 measured from the mocks of $\mathrm{\Lambda }\mathrm{CDM}$ and the three DGP cases, as labeled. The solid lines show the prediction of the best-fitting model outlined in Sec. 4a constrained using the two wedges ${\xi }_{\perp }$ and ${\xi }_{\parallel }$.

Figure 7

Observational constraints on the DGP parameter $r_c{H}_0$ using the $f{\sigma }_8$ values obtained from the analysis of the BOSS DR12 (purple squares in Fig. 5) and DR11 (cyan squares in Fig. 5) data, as labeled. The results assume the priors ${\mathrm{\Omega }}_{m0}=0.315\pm 0.013$ ($1\sigma$) and ${\sigma }_8^{\mathrm{\Lambda }\mathrm{CDM}}(z=0)=0.829\pm 0.014$ ($1\sigma$), from Planck [60]. The lines shows the resulting probability distributions after marginalizing over ${\mathrm{\Omega }}_{m0}$ and ${\sigma }_8^{\mathrm{\Lambda }\mathrm{CDM}}$. The dotted vertical lines indicate the 68% and 95% percentiles of the distributions.

Figure 8

Time evolution of the critical density for spherical collapse ${\delta }_c^{\mathrm{\Lambda }\mathrm{CDM}}$ (linearly extrapolated to today assuming $\mathrm{\Lambda }\mathrm{CDM}$) for $\mathrm{\Lambda }\mathrm{CDM}$ and the three DGP cases considered in this paper, as labeled.