Cluster abundance in $f(R)$ gravity models

Cluster abundance in $f (R)$ gravity models

Simone Ferraro, Fabian Schmidt, and Wayne Hu

Phys. Rev. D 83, 063503 – Published 3 March 2011

Abstract

As one of the most powerful probes of cosmological structure formation, the abundance of massive galaxy clusters is a sensitive probe of modifications to gravity on cosmological scales. In this paper, we present results from $N$ -body simulations of a general class of $f (R)$ models, which self-consistently solve the nonlinear field equation for the enhanced forces. Within this class we vary the amplitude of the field, which controls the range of the enhanced gravitational forces, both at the present epoch and as a function of redshift. Most models in the literature can be mapped onto the parameter space of this class. Focusing on the abundance of massive dark matter halos, we compare the simulation results to a simple spherical collapse model. Current constraints lie in the large-field regime, where the chameleon mechanism is not important. In this regime, the spherical collapse model works equally well for a wide range of models and can serve as a model-independent tool for placing constraints on $f (R)$ gravity from cluster abundance. Using these results, we show how constraints from the observed local abundance of X-ray clusters on a specific $f (R)$ model can be mapped onto other members of this general class of models.

Received 12 November 2010

DOI:https://doi.org/10.1103/PhysRevD.83.063503

Authors & Affiliations

Simone Ferraro¹, Fabian Schmidt², and Wayne Hu¹

¹Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA
²Theoretical Astrophysics, California Institute of Technology, Mail Code 350-17, Pasadena, California 91125, USA

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Vol. 83, Iss. 6 — 15 March 2011

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Images

Figure 1
Redshift evolution of the chameleon field $f_{R}$ (top panel) and Compton wavelength $λ_{C}$ (bottom panel) in the background for the broken power law class of models. As the scaling index $n$ increases the field amplitude becomes increasingly suppressed leading to stronger chameleon effects for the same gravitational potentials of clusters. The Compton wavelength for a fixed field amplitude today remains relatively constant at $z ≲ 1$ and then also becomes increasingly suppressed with $n$ . An alternative class of models specified by the expansion history and Compton wavelength $B_{0}$ parameter is also shown for comparison (dashed lines).Reuse & Permissions
Figure 2
Mass function enhancement at $z = 0$ in large-field $| f_{R 0} | = 10^{- 4}$ models for scaling index of $n = 1, 2$ relative to $Λ CDM$ . Here and in the following figures, the no-chameleon results have been displaced horizontally for clarity. Enhancement depends mainly on mass due to the increasing rarity of high mass halos. As $n$ increases, the enhancement drops only moderately given the small change in the background Compton wavelength at $z ≲ 1$ , consistent with only a small contribution from the nonlinear chameleon effect. The spherical collapse predictions (shaded range) capture these qualitative trends and provide conservative lower limits to the enhancement.Reuse & Permissions
Figure 3
Mass function enhancement at $z = 0$ as a function of scaling index $n$ for the mass bin centered at $M_{200} = 5.3 \times 10^{14} M_{⊙} / h$ for large-field models $| f_{R 0} | = 10^{- 4}$ . Spherical collapse predictions (shaded) capture the trend in the no-chameleon simulations. Full results for $n = 1, 2$ and consideration of the field evolution suggests that spherical collapse predictions should hold for $n ≲ 4$ . Note that errors are fully correlated in that all simulations use the same initial conditions and are compared against the same set of $Λ CDM$ simulations.Reuse & Permissions
Figure 4
Mass function enhancement at $z = 0.316$ in the large-field $| f_{R 0} | = 10^{- 4}$ for $n = 1, 2$ . Fractional enhancements at a fixed mass remain significant at higher redshift due to the increased rarity of such halos in $Λ CDM$ and the trend remains well captured by spherical collapse predictions (shaded region). Field evolution in the $n = 2$ case makes the chameleon suppression in the full simulations moderately more important.Reuse & Permissions
Figure 5
Mass function enhancement in the small-field regime. The chameleon effect suppresses the enhancement when the background field amplitude $| f_{R 0} |$ drops below the depth of the gravitational potential for an object of mass $M_{200}$ . Comparison of the full and no-chameleon simulations shows that the limiting mass at which the chameleon appears scales roughly as expected: $M_{cham} \propto | f_{R 0} |^{3 / 2}$ nearly independently of the scaling index $n$ . Spherical collapse predictions roughly capture this suppression in the cluster regime $M_{200} ≳ 3 \times 10^{14} M_{⊙} / h$ but fail to model the enhancement below $M_{cham}$ .Reuse & Permissions
Figure 6
Constraints on $| f_{R 0} |$ (upper panel) and the Compton wavelength $λ_{C}$ (lower panel) as function of the index $n$ . We have converted the 95% confidence level upper limits on $| f_{R 0} |$ reported in 1 for $n = 1$ to other values of $n$ using the spherical collapse model as described in the text. The medium shaded band corresponds to the default limit reported in 1, while dark and light shaded areas use more or less conservative assumptions (see text).Reuse & Permissions

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