Nonlinear evolution of $f(R)$ cosmologies. II. Power spectrum

Nonlinear evolution of $f (R)$ cosmologies. II. Power spectrum

Hiroaki Oyaizu, Marcos Lima, and Wayne Hu

Phys. Rev. D 78, 123524 – Published 18 December 2008

Abstract

We carry out a suite of cosmological simulations of modified action $f (R)$ models where cosmic acceleration arises from an alteration of gravity instead of dark energy. These models introduce an extra scalar degree of freedom which enhances the force of gravity below the inverse mass or Compton scale of the scalar. The simulations exhibit the so-called chameleon mechanism, necessary for satisfying local constraints on gravity, where this scale depends on environment, in particular, the depth of the local gravitational potential. We find that the chameleon mechanism can substantially suppress the enhancement of power spectrum in the nonlinear regime if the background field value is comparable to or smaller than the depth of the gravitational potentials of typical structures. Nonetheless power spectrum enhancements at intermediate scales remain at a measurable level for models even when the expansion history is indistinguishable from a cosmological constant, cold dark matter model. Simple scaling relations that take the linear power spectrum into a nonlinear spectrum fail to capture the modifications of $f (R)$ due to the change in collapsed structures, the chameleon mechanism, and the time evolution of the modifications.

Received 17 July 2008

DOI:https://doi.org/10.1103/PhysRevD.78.123524

Authors & Affiliations

Hiroaki Oyaizu^1,2, Marcos Lima^1,3, and Wayne Hu^1,2

¹Kavli Institute for Cosmological Physics, and Enrico Fermi Institute, University of Chicago, Chicago Illinois 60637, USA
²Department of Astronomy and Astrophysics, University of Chicago, Chicago Illinois 60637, USA
³Department of Physics, University of Chicago, Chicago Illinois 60637, USA

Nonlinear evolution of $f (R)$ cosmologies. I. Methodology

Hiroaki Oyaizu

Phys. Rev. D 78, 123523 (2008)

Nonlinear evolution of $f (R)$ cosmologies. III. Halo statistics

Fabian Schmidt, Marcos Lima, Hiroaki Oyaizu, and Wayne Hu

Phys. Rev. D 79, 083518 (2009)

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Vol. 78, Iss. 12 — 15 December 2008

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Images

Figure 1
The mean power spectra $Δ^{2} = k^{3} P (k) / 2 π^{2}$ of the cosmological simulations without $f_{R}$ modifications ( $Λ CDM$ or $| f_{R 0} | = 0$ ). Upper panel: Average power spectrum of all simulations in comparison to linear theory and the Smith et al. nonlinear fit (solid line). Lower panel: Relative power spectrum offset from Smith et al. fit. The individual simulation box sizes averages are plotted in solid, long-dashed, and dashed lines. Results converge at approximately the half-Nyquist wave numbers (vertical lines). Points with errors show the mean and the $1 − σ$ error bars computed using the bootstrap method, weighted by simulation box volume, out to the half-Nyquist wave numbers (see text).Reuse & Permissions
Figure 2
Relative power spectrum enhancement over $Λ CDM$ at $a = 1$ for the full $f_{R}$ simulation compared with the no-chameleon $f_{R}$ simulations and linear theory. At high $k$ , linear theory underestimates the absolute power for both $Λ CDM$ and $f_{R}$ while overestimating the relative enhancement. Without the chameleon, power is sharply enhanced on scales smaller than the Compton scale in the background which increases with $| f_{R 0} |$ [see Eq. (12)]. For $| f_{R 0} | = 10^{- 6}$ the chameleon strongly suppresses these enhancements at high $k$ . For $| f_{R 0} | = 10^{- 4}$ , this suppression is nearly absent except for a residual effect from the chameleon at high redshift. The three highest points in $k$ have increased sampling errors since they utilize only the $64 h^{- 1} Mpc$ boxes. Note that the linear prediction is the fractional enhancement in the linear power spectrum, that is, $(P_{f_{R}, linear} - P_{Λ CDM, linear}) / P_{Λ CDM, linear}$ .Reuse & Permissions
Figure 3
2D slices through simulations at $a = 1$ for various fields: matter overdensity $δ \equiv δ ρ_{m} / {\bar{ρ}}_{m}$ (left); minimum gravitational potential $Ψ$ along the line of sight (middle), and minimum field $f_{R} / f_{R 0}$ along the line of sight (right). Each slice has the two dimensions of the full $64 h^{- 1} Mpc$ box and is projected across $16 h^{- 1} Mpc$ along the line of sight. The shading scale ranges from white to black between the extreme field values as follows: ( $- 5.0 \to 9.8$ ) for $\ln (1 + δ)$ ; $(0.3 \to - 4.3) \times 10^{- 5}$ for $Ψ$ ; ( $1.05 \to 0$ ) for $f_{R} / f_{R 0}$ . In the $f_{R 0} = | 10^{- 6} |$ (top) simulation, the chameleon mechanism suppresses the field and hence the force deviations in the deep potential wells surrounding large overdensities. In the $f_{R 0} = | 10^{- 4} |$ (bottom) simulation, the $f_{R}$ remains stiff and a chameleon does not appear at the present. While the density and potential are slightly enhanced in this run due to the force modification, the appearance of the chameleon depends mainly on the background field value $f_{R 0}$ .Reuse & Permissions
Figure 4
Evolution of power spectrum deviation for $f_{R 0} = | 10^{- 4} |$ for the full $f_{R}$ simulation and the no-chameleon simulation. The appearance of a chameleon at $a ≲ 0.5$ causes a large fractional change in the deviations at earlier epochs. As the deviations grow, this offset remains as a smaller fraction of the total.Reuse & Permissions
Figure 5
Same as in Fig. 3 but at $a = 0.3$ for $| f_{R 0} | = 10^{- 4}$ with the field values shown as a fraction of the background field at that time $f_{\bar{R}} (a = 0.3)$ . The field amplitude is suppressed in deep potential wells exhibiting a chameleon that suppresses the early growth of structure. The shading scale range for the various fields in this case is also identical to that of Fig. 3.Reuse & Permissions
Figure 6
Power spectrum deviations from $Λ CDM$ of the full $f_{R}$ simulations vs the scaling predictions of Smith et al. employing linear $f_{R}$ calculations. Note that in all cases the scaling predictions fail to capture the deviations at high $k$ due both to the change in the abundance and profiles of collapsed objects in the $f_{R}$ simulations and the chameleon mechanism.Reuse & Permissions

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