Nonlinear resonant oscillation of gravitational potential induced by ultralight axion in $f(R)$ gravity

We study the ultralight axion dark matter with mass around ${10}^{-22}\mathrm{eV}$ in $f(R)$ gravity which might resolve the dark energy problem. In particular, we focus on the fact that the pressure of the axion field oscillating in time produces oscillations of gravitational potentials. We show that the oscillation of the gravitational potential is sensitive to the model of gravity. Remarkably, we find that the detectability of the oscillation through the gravitational wave detectors can be significantly enhanced due to the nonlinear resonance between the ultralight axion and the scalaron.

Figure 1

The functional form of the effective potential $V_{\mathrm{eff}}(\phi )$ in the exponential model (37). $V_{\mathrm{eff}}(\phi )$ has a minimum at $\phi ={\phi }_0$. The point $\phi =0$ is a singularity, at which the gradient of the potential diverges. Since $\phi =0$ corresponds to $R\to +\infty$, this is called the curvature singularity.

Figure 2

Plot of ${\overline{\phi }}^{\prime }(T/2)$ in the $(\mu ,{\overline{\phi }}_i)$ plane for $\lambda ={10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$ from left to right. The red (blue) regions correspond to positive (negative) value of ${\overline{\phi }}^{\prime }(T/2)$. Since ${\overline{\phi }}^{\prime }(T/2)$ is a smooth function of $(\mu ,{\overline{\phi }}_i)$, there exist the boundaries with ${\overline{\phi }}^{\prime }(T/2)=0$, which correspond to closed-orbit solutions.

Figure 3

The resonance curves near the resonance point with different values of $\lambda$. The black dashed line is the amplitude in the $R^2$ model, Eq. (35). The resonance curves in the exponential model cannot be distinguished from those in the $R^2$ model except for $\mu \sim 1$. The maximum amplitude is inversely proportional to $\lambda$ as seen in Eq. (42). The discontinuity of the curve seen in the case $\lambda ={10}^{-2}$ comes from the absence of a friction term, as in the case of the $R^2$ model.

Figure 4

Resonance points with $\mu =q/5(q=1,2,3\dots )$ for $\lambda ={10}^{-3}$. The plot shows the existence of multiple resonance points in the case of the nonlinear model.