Can background cosmology hold the key for modified gravity tests?

Modified gravity theories are a popular alternative to dark energy as a possible explanation for the observed accelerating cosmic expansion, and their cosmological tests are currently an active research field. Studies in recent years have been increasingly focused on testing these theories in the nonlinear regime, which is computationally demanding. Here we show that, under certain circumstances, a whole class of theories can be ruled out by using background cosmology alone. This is possible because certain classes of models (i) are fundamentally incapable of producing specific background expansion histories, and (ii) said histories are incompatible with local gravity tests. As an example, we demonstrate that a popular class of models, $f(R)$ gravity, would not be viable if observations suggest even a slight deviation of the background expansion history from that of the $\mathrm{\Lambda }\mathrm{CDM}$ paradigm.

Figure 1

Main panel: Dimensionless quantities $f(R)/R$, $f_R$, $R{f}_{RR}$ as functions of $R/3H_0^2$, derived by requiring the expansion history to match that of dark energy with $w(a)=-1.2$ for $a<a^{*}\approx 0.817$. The solid, dotted and dashed curves correspond to $A=0,+1$ and $-1$, respectively (cf. main text for definition of $A$). The result depends on $A$, but the essential feature that $f_{RR}$ diverges at $R_{\min }=R(a^{*})$ is generic. Inset: ${\mathrm{\Omega }}_{\mathrm{DE}}(a)$ for dark energy with $w(a)=-1.2$ (top solid line) and from the reconstructed $f(R)$ (bottom right; solid, dotted and dashed lines for $A=0,+1$ and $-1$, respectively). Note that $f(R)$ is fully determined by the expansion history for $a<a^{*}$, when both approaches give the same result. However, the prefixed $f(R)$ gives rise to a reconstructed ${\mathrm{\Omega }}_{\mathrm{DE}}(a)$ after $a^{*}$ that is incompatible with the curve deduced from $w(a)$, even though one can make it continuous at $a^{*}$ by tuning $A$ (broken line at top right, which corresponds to $A\approx -7.858$).

Figure 2

Examples of the reconstructed $f_R$ as a function of $R$ for a background expansion history matching that of dark energy with $w_0=-1$ and $|w_1|=2\times {10}^{-5}$. The evolution with a positive value of $w_1$ is shown in the left panel, while that with negative $w_1$ appears on the right. The different curves correspond to different values of $A$ (or equivalently $f_{R0}$) and ${\mathrm{\Omega }}_m=0.308$ in all cases. Depending on $A$, $f_R$ may display an extremum, vary by more than $2\times {10}^{-6}$ in magnitude over the whole curvature range, or both.

Figure 3

Main panel: Region of the $w_0-w_1$ plane where $R(a)$ has no extrema at $a\le 1$, allowing the specified expansion history to be consistently reproduced by $f(R)$ gravity. Inset: Region where the reconstructed $f(R)$ model can pass Solar System tests (see the main text for more details). Red crosses correspond to $\mathrm{\Lambda }\mathrm{CDM}$.