Chameleon $f(R)$ gravity in the virialized cluster

Chameleon $f (R)$ gravity in the virialized cluster

Lucas Lombriser, Kazuya Koyama, Gong-Bo Zhao, and Baojiu Li

Phys. Rev. D 85, 124054 – Published 25 June 2012

Abstract

Current constraints on $f (R)$ gravity from the large-scale structure are at the verge of penetrating into a region where the modified forces become nonlinearly suppressed. For a consistent treatment of observables at these scales, we study cluster quantities produced in chameleon and linearized Hu-Sawicki $f (R)$ gravity dark matter $N$ -body simulations. We find that the standard Navarro-Frenk-White halo density profile and the radial power law for the pseudo-phase-space density provide equally good fits for $f (R)$ clusters as they do in the Newtonian scenario. We give qualitative arguments for why this should be the case. For practical applications, we derive analytic relations, e.g., for the $f (R)$ scalar field, the gravitational potential, and the velocity dispersion as seen within the virialized clusters. These functions are based on three degrees of freedom fitted to simulations, i.e., the characteristic density, scale, and velocity dispersion. We further analyze predictions for these fitting parameters from the gravitational collapse and the Jeans equation, which are found to agree well with the simulations. Our analytic results can be used to consistently constrain chameleon $f (R)$ gravity with future observations on virialized cluster scales without the necessity of running a large number of simulations.

Received 30 March 2012

DOI:https://doi.org/10.1103/PhysRevD.85.124054

Authors & Affiliations

Lucas Lombriser¹, Kazuya Koyama¹, Gong-Bo Zhao^1,2, and Baojiu Li³

¹Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom
²National Astronomy Observatories, Chinese Academy of Science, Beijing, 100012, People’s Republic of China
³Institute for Computational Cosmology, Physics Department, University of Durham, South Road, Durham, DH1 3LE, United Kingdom

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 85, Iss. 12 — 15 June 2012

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Properties of the scalar field $δ f_{R}$ for a cluster of $M_{vir} = 1.36 \times 10^{14} M_{⊙} / h$ and a background field amplitude of $| f_{R 0} | = 10^{- 5}$ (long-dashed line) with $n = 1$ . Left: The analytically derived scalar field $δ f_{R}$ (solid line) and its behavior at the limit of large and small scales, respectively. The transition of the linearized field into a chameleon field occurs instantaneously where $δ f_{R} = | f_{R 0} |$ . Middle: Comparison between the instantaneous chameleon transition (solid line) and the $C^{1}$ transition for $δ f_{R}$ (dashed line) with analog semianalytical derivation to 28. Matching $δ f_{R} = δ f_{R, sim}$ at $r_{vir}$ brings the approximations into good agreement with the simulated $δ f_{R} (r)$ (dots). The thick long-dashed lines show numerical solutions to Eq. (16) assuming a NFW profile with integration constraints set by additionally requiring that $δ f_{R}^{'} (r_{vir})$ correspond to the analytic and semianalytic result, respectively. Right: Enhanced force by the linearized (dotted line), the instantaneous chameleon (solid line), and the $C^{1}$ (dashed line) chameleon scalar field $δ f_{R}$ (matched at $r_{vir}$ ), respectively. The lines of the linearized and instantaneous chameleon field overlap beyond the chameleon region.Reuse & Permissions
Figure 2
Same as middle panel of Fig. 1 but with cosmological background density as boundary condition instead of matching to $δ f_{R, sim} (r_{vir})$ for $δ f_{R}$ . For the radial derivatives, we further require that $δ f_{R}^{'} (r_{vir})$ correspond to the analytic and semianalytic result, respectively.Reuse & Permissions
Figure 3
Comparison of radial dependencies of $f (R)$ halo quantities predicted by simulations and the fits constructed in Sec. 3 for $| f_{R 0} | = 10^{- 5}$ ( $n = 1$ ) and $L_{Box} = 128 Mpc / h$ (F-128-5). The simulated halos and their individual fits are stacked for $M = (1.65 - 1.70) M_{⊙} / h$ . The error bars show the standard deviation in the simulation output. Top left: Cluster density profile $δ ρ_{m} / {\bar{ρ}}_{m}$ . Top middle: Halo mass $M$ . Top right: Velocity dispersion $σ$ . Bottom left: Pseudo-phase-space distribution $ρ / σ^{3}$ normalized at $r_{s}$ . Bottom middle: $f_{R}$ scalar field for the stacked instantaneous (solid) and $C^{1}$ (dot-dashed) transition to the chameleon solution. Bottom right: Gravitational potential $Ψ$ for the $δ f_{R}$ solutions in the bottom middle panel along with the limiting assumptions of Newtonian/GR (dotted) and linearized $f (R)$ (dashed) forces.Reuse & Permissions
Figure 4
Stacked halo fitting parameters along with the concentration $c_{vir}$ for the three different box sizes and predictions for them from scaling relations (see Appendix ). In $f (R)$ gravity, halo abundances and therefore mean masses are enhanced. Clearly identifiable, in the linearized $f (R)$ regime, velocity dispersions squared are enhanced by a factor of 4/3 over the Newtonian results, i.e., in addition to smaller effects of different $c_{vir}$ . The chameleon effect returns the cluster abundance at high halo masses and the velocity dispersion to the Newtonian/GR predictions. The predictions from scaling relations based on the spherical collapse and the Jeans function are shown for the Newtonian (GR—solid line) and linearized $f (R)$ (N—dashed line) case, respectively. Rows from top: The matter overdensity at $r_{s}$ (given by $β$ or $ρ_{s}$ ), the characteristic scale $r_{s}$ , and the velocity dispersion squared at $r_{s}$ , $σ_{s}^{2}$ , as well as the concentration $c_{vir}$ . Columns from left: $| f_{R 0} | = 10^{- 4}$ , $10^{- 5}$ , $10^{- 6}$ ( $n = 1$ ).Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology