Models of $f(R)$ cosmic acceleration that evade solar system tests

We study a class of metric-variation $f(R)$ models that accelerates the expansion without a cosmological constant and satisfies both cosmological and solar-system tests in the small-field limit of the parameter space. Solar-system tests alone place only weak bounds on these models, since the additional scalar degree of freedom is locked to the high-curvature general-relativistic prediction across more than 25 orders of magnitude in density, out through the solar corona. This agreement requires that the galactic halo be of sufficient extent to maintain the galaxy at high curvature in the presence of the low-curvature cosmological background. If the galactic halo and local environment in $f(R)$ models do not have substantially deeper potentials than expected in $\Lambda \mathrm{CDM}$, then cosmological field amplitudes $|f_R|\gtrsim {10}^{-6}$ will cause the galactic interior to evolve to low curvature during the acceleration epoch. Viability of large-deviation models therefore rests on the structure and evolution of the galactic halo, requiring cosmological simulations of $f(R)$ models, and not directly on solar-system tests. Even small deviations that conservatively satisfy both galactic and solar-system constraints can still be tested by future, percent-level measurements of the linear power spectrum, while they remain undetectable to cosmological-distance measures. Although we illustrate these effects in a specific class of models, the requirements on $f(R)$ are phrased in a nearly model-independent manner.

Figure 1

Functional form of $f(R)$ for $n=1$, 4, with normalization parameters $c_1$, $c_2$ given by $|f_{R0}|=0.01$ and a matching to $\Lambda \mathrm{CDM}$ densities (see Sec. 2d). These functions transition from zero to a constant as $R$ exceeds $m^2$. The sharpness of the transition increases with $n$ and its position increases with $|f_{R0}|$. During cosmological expansion, the background only reaches $R/m^2\sim 40$ for $|f_{R0}|\ll 1$ and so the functional dependence for smaller $R/m^2$ has no impact on the phenomenology.

Figure 2

Cosmological evolution of the scalar field $f_R$ and the Compton wavelength parameter $B$ for models with $n=1$, 4. Both parameters control observable deviations from general relativity and deviations decline rapidly with redshift as $n$ increases.

Figure 3

Evolution of the effective equation of state for $n=1$, 4 for several values of the cosmological field amplitude today, $f_{R0}$. The effective equation of state crosses the phantom divide $w_{\mathrm{eff}}=-1$ at a redshift that decreases with increasing $n$ leading potentially to a relatively unique observational signature of these models.

Figure 4

Fractional change in the matter power spectrum $P(k)$ relative to $\Lambda \mathrm{CDM}$ for a series of the cosmological field amplitude today, $f_{R0}$, for $n=1$, 4 models. For scales that are below the cosmological Compton wavelength during the acceleration epoch $k\gtrsim (aH)B^{-1/2}$ perturbation dynamics transition to the low-curvature regime where $\gamma =1/2$ and density growth is enhanced. This transition occurs in the linear regime out to field amplitudes of $|f_{R0}|\sim {10}^{-6}–{10}^{-7}$.

Figure 5

Density profile in the solar interior and vicinity (solid curve). Under GR, the curvature $R$ would track the density profile. For the $f(R)$ model with $n=4$, cosmological field amplitude $|f_{R0}|=0.1$, and a galactic field amplitude that minimizes the scalar potential, the curvature tracks the GR or high-curvature limit out to the edge of the solar corona at about 1 AU (dashed line).

Figure 6

Solar solution for $f_R(r)$ with $n=4$ for a series of cosmological field amplitudes $f_{R0}$ with a galactic field that minimizes the potential. The field smoothly transitions from the $f_R\sim 0$ interior value to the galactic values once the thin-shell criterion is violated.

Figure 7

Solar solution for the curvature $R$ for the $n=4$ model and a series of cosmological field amplitudes $f_{R0}$ with a galactic field that minimizes the potential. The solution abruptly transitions from the high-curvature $R\approx {\kappa }^2\rho$ to the low-curvature regime once the thin-shell criterion is violated.

Figure 8

Metric deviation parameter $|\gamma -1|$ for $n=4$ models and a series of cosmological field amplitudes $f_{R0}$ with a galactic field that minimizes the potential. These deviations are unobservably small for the whole range of amplitudes.

Figure 9

Maximum cosmological field $|f_{R0}|$ allowed by the solar-system constraint alone (top curve) under the assumption that the galactic field remains at the potential minimum at the present epoch. Models with $|f_{R0}|\gtrsim {10}^{-6}$ (bottom curve) have galactic fields $|f_{Rg}|$ that evolve to higher values during the acceleration epoch, potentially enabling substantially stronger constraints with structure formation simulations.

Figure 10

A galactic solution for the scalar curvature $R$ for $n=1/2$ with an NFW density profile (69) and conservative parameters $r_s=75\mathrm{kpc}$ and $v_{\max }=300\mathrm{km}{\mathrm{s}}^{-1}$. For $|f_{R0}|\gtrsim 2\times {10}^{-6}$, no static solution can be found which has high curvature, $R\approx {\kappa }^2\rho$, inside the halo. As in the solar case, once the thin-shell criterion is violated the solution abruptly transitions to low curvature and matches onto the cosmological value of $f_{R0}$ outside the halo.