Convolution Lagrangian perturbation theory for biased tracers beyond general relativity

We compare analytic predictions for real and Fourier space two-point statistics for biased tracers from a variety of Lagrangian perturbation theory approaches against those from state of the art N-body simulations in $f(R)$ Hu-Sawicki and the nDGP braneworld modified gravity theories. We show that the novel physics of gravitational collapse in scalar tensor theories with the chameleon or the Vainshtein screening mechanism can be effectively factored in with bias parameters analytically predicted using the peak-background split formalism when updated to include the environmental sensitivity of modified gravity theories as well as changes to the halo mass function. We demonstrate that convolution Lagrangian perturbation theory (CLPT) and standard perturbation theory (SPT) approaches provide accurate analytic methods to predict the correlation function and power spectra, respectively, for biased tracers in modified gravity models and are able to characterize both the baryon acoustic oscillation, power-law, and small scale regimes needed for upcoming galaxy surveys such as DESI, Euclid, LSST and WFIRST.

Figure 1

Contributions to the CLPT correlation function prediction, $\xi$, given in (37), for the F4 model at $z=0.5$, by the various terms in the expansion: dark matter (no bias prefactors) [blue solid], $b_1$ term [black dash-dot], $b_2$ term [cyan dotted], $b_1^2$ term [red dash], $b_2^2$ term [orange dash-dot] and $b_1{b}_2$ term [brown solid].

Figure 2

Critical overdensity for collapse, ${\delta }_{\mathrm{cr}}$, as a function of halo mass for the F4 [top], the F5 [middle], and the F6 [bottom] MG models at $z=0.5$ and for various environments. The different color curves correspond to the solutions for the following values of the environment overdensity, ${\delta }_{\text{env}}=[-1(\text{purple}),-0.72,-0.43,-0.15,0.13,0.42,0.7,0.98,1.27,1.55(\mathrm{red})]$. The horizontal black line shows the ${\delta }_{\mathrm{cr}}=1.686$ value for GR. The background cosmology is that of the Group I simulations.

Figure 3

First and second order Lagrangian bias factors $b_1$ [Top] and $b_2$ [Bottom], as a function of halo mass M, calculated for GR [solid black], F6 [dash-dot blue], F5 [green dotted], and F4 [red dash], through relationships (54) and (78), using the Sheth-Tormen values $(q,p)=(0.707,0.3)$ at $z=0.5$. For the 3 $f(R)$ models, the biases are the environmentally averaged.

Figure 4

Halo mass function, in the form of number of halos $d{N}_h$, per mass bin $dM$, as a function of halo mass $M$, calculated from the Group I simulations at $z=0.5$, for GR [black dot] in the upper left panel, F4 [red triangle] in the upper right panel, F5 [green square] in the middle left panel, F6 in the middle right panel and $N1$ and $N5$ $n\mathrm{DGP}$ models in the lower left and right panels, respectively. Alongside with the simulations, we show the analytical Sheth-Tormen halo mass functions (47), plotted using the best-fit $(q,p)$ values for each model using the criterion (82). For the $f(R)$ models, the best-fit values correspond to the environmentally averaged halo mass function, through (81), shown with black curves, while we also plot the halo mass functions for individual values for ${\delta }_{\text{env}}$ for each $f(R)$ model. The color scheme and distribution of values for ${\delta }_{\text{env}}$ is the same as in Fig. 2.

Figure 5

[Top] Two-point correlation function prediction form CLPT, through (37), for the F6 model at $z=0.5$ for the various bias values given when ${\delta }_{\text{env}}=[-1(\text{purple}),-0.72,-0.43,-0.15,0.13,0.42,0.7,0.98,1.27,1.55(\mathrm{red})]$ through (79) and the result when averaged over environments [black line] using (81). [Bottom] Fractional deviation, $\frac{{\xi }_{\text{env}}}{{\xi }_{\text{mean}}}-1$, for the CLPT result for each environment in the top panel, with respect to the CLPT curve given by the mean $b_1$ and $b_2$ values (black curve in the top panel).

Figure 6

Two-point correlation functions for the Group II simulation snapshots at $z=1$ for GR [top panel] and F5 [lower panel] models. The predictions from CLPT (37) [solid blue], the Zel’dovich approximation [dashed cyan], LRT (66) [dotted magenta] and the linear theory [dashed green], using the bias values shown in Table 1 are compared to the correlation function extracted from simulations, shown with Poisson error bars, for the three mass bins defined in Sec. 3: the low mass [red triangle], intermediate mass [black dot], and high mass [brown square] bins.

Figure 7

Fractional deviations in the power-law regime of the correlation function predictions with respect to the Zel’dovich approximation for the results shown in Fig. 6. For the GR analysis [top panel], we also plot the ratio of the CLPT prediction using the standard ST values $(q,p)=(0.75,0.3)$ [dashed-dot blue], rather than the best fit ones in Table 1, divided by the Zel’dovich result.

Figure 8

Two-point correlation functions from the Group I simulations, calculated at $z=0.5$, for GR [black dots] in the upper left panel, for F4 [red triangles] in the upper right panel, for F5 [green squares] in the lower left panel, and for F6 [blue right triangles] in the lower right panel. The results are the average over the 5 realizations and the error bars shown are standard deviations. Furthermore, for each model we plot the predictions from CLPT (37) [solid blue], from the Zel’dovich approximation [dashed cyan], from LRT (66) [dotted magenta] and from linear theory [dashed green], using the bias values shown in Table 1. The linear theory result for the F5 model is plotted using a blue dashed line instead, for ease of comparison.

Figure 9

Two-point correlation functions from the Group I simulations, calculated at $z=0.5$, for N1 [orange diamonds] in the upper panel and for N5 [purple hexagons] in the lower panel. The results are the average over the 5 realizations and the error bars shown are standard deviations. Furthermore, for each model we plot the predictions from CLPT (37) [solid black], from the Zel’dovich approximation [dashed cyan], from LRT (66) [dotted magenta], and from linear theory [dashed green], using the bias values shown in Table 1.

Figure 10

Power spectra from the Group II simulations, calculated for F5 at $z=1$, in the low mass [red triangle], intermediate mass [black dot] and high mass [brown square] bins, that were defined in Sec. 3. The error bars shown are Poisson error bars. Furthermore, for each mass bin we plot the predictions from CLPT (37) [dotted magenta], from SPT (64) [solid blue] and from linear theory [dashed green], using the bias values shown in Table 1.

Figure 11

Power spectra from the Group I simulations, calculated at $z=0.5$, for GR [black dots] in the upper left panel, for F4 [red triangles] in the upper middle panel, for F5 [green squares] in the lower left panel, for F6 [blue right triangles] in the lower middle panel, for N1 [orange diamonds] in the upper right panel, and for N5 [purple hexagons] in the lower right panel. The results are the average over the 5 realizations and the error bars shown are standard deviations. Furthermore, for each model we plot the predictions from CLPT (37) [dotted brown], from SPT (64) [solid blue] and from linear theory [dashed green], using the bias values shown in Table 1. The linear theory result for the F5 model is plotted using a pink dashed line instead, for ease of comparison.