Nonlinear dynamical stability of infrared modifications of gravity

Scalar forces “screened” by the Vainshtein mechanism may hold the key to understanding the cosmological expansion of our Universe, while predicting new and exciting features in the interaction between massive bodies. Here we explore the dynamics of the Vainshtein screening mechanism, focusing on the decoupling limit of the DGP braneworld scenario and dRGT massive gravity. We show that there is a vast set of initial conditions whose evolution is well defined and which are driven to the static screening solutions of these theories. Screening solutions are stable and behave coherently under small fluctuations: they oscillate and eventually settle to an equilibrium configuration, the time scale for the oscillations and damping being dictated by the Vainshtein radius of the screening solutions. At very late times, a power-law decay ensues, in agreement with known analytical results. However, we also conjecture that physically interesting processes such as the gravitational collapse of compact stars may not possess a well-posed initial value problem. Finally, we construct solutions with nontrivial multipolar structure describing the screening field of deformed, asymmetric bodies and show that higher multipoles are screened more efficiently than the monopole component.

Figure 1

The two static screening solutions, ${\mathrm{\Pi }}_{+}$ (solid) and ${\mathrm{\Pi }}_{-}$ (dashed), of equation (16). These solutions correspond to a source radius of $R_0=1$ and a Vainshtein radius $R_V\simeq 1000$. Both solutions grow as $\sim r$ inside of the source and decay as $\sim 1/\sqrt{r}$ outside the source for $0\ll r\ll R_V$. For very large distances $r\gg R_V$, ${\mathrm{\Pi }}_{-}$ diverges as $\sim r$, whereas ${\mathrm{\Pi }}_{+}\sim 1/r^2$. Recall that we are rescalling $\pi$ and $r$ in terms of $\mathrm{\Lambda }$ (12), so this plot is actually of ${\mathrm{\Pi }}_{\pm }{\mathrm{\Lambda }}^{-2}$ versus $r\mathrm{\Lambda }$. The same is true for all subsequent plots.

Figure 2

The sound speed profiles corresponding to fluctuations on top of the screening solutions, shown here for $R_0=1$ and $\rho =10^5$, giving $R_V\sim 80$. Superluminal propagation $c_s>1$ can occur in both branches inside and outside of the source. For large $r$, $c_s\to 1$ for the positive branch, while $c_s\to 1/\sqrt{2}$ for the negative branch.

Figure 3

The evolution of $\pi (r,t)$ (solid) for an initial condition of the type (22) with $A=0.25$, $\sigma =0.5$, $r_w=12$. The screening solution ${\mathrm{\Pi }}_{+}$ is characterized by $\rho =100$, $R_0=0.5$, $R_V\simeq 4$. From left to right, top to bottom are snapshots taken at $t=0,10,15,23$. The initial Gaussian fluctuation is seen to propagate away from the domain at close to the speed of light, leaving behind the static screening solution at very late times.

Figure 4

Left panel: The evolution of $\pi (r,t)$ with $\pi (r,0)=0$ and $\rho =100$, $R_0=0.5$, resulting in a Vainshtein radius $R_V\simeq 4$. At late times, the solution approaches the static, asymptotically flat screening solution ${\pi }_{+}(r)$ corresponding to the same parameters. Right: The evolution of $\pi (r,0)=-\frac{3}{4}(r^2-r_0^2)$ for $\rho =100$, $R_0=0.5$, $r_0=50$, $R_V\simeq 4$. The evolution drives the system towards the screening solution ${\pi }_{-}$ at late times.

Figure 5

Top panel: Linearized time evolution of a Gaussian wave packet in the background of ${\mathrm{\Pi }}_{+}$ with amplitude $A=3$, width $\sigma =1$ and localized at $r_w=10$. The radius of the source is at $R_0=1$. The waves are extracted at $r=2$. The intermediate-time behavior consists on an exponentially damped sinusoid—called quasinormal mode—and the late-time behavior is described by a power-law falloff of the field $\delta \pi \sim t^{-8}$. The observed behavior is in agreement with a frequency-domain numerical and analytical calculation; see text for further details. Bottom panel: The same, in the background of ${\mathrm{\Pi }}_{-}$. The time-domain profile suggests that in this case, the quality factor of the fundamental quasinormal mode ${\omega }_R/{\omega }_I$ is smaller than in the background of ${\mathrm{\Pi }}_{+}$; i.e., ${\omega }_R/{\omega }_I<1$.

Figure 6

Effective potential in the background of ${\mathrm{\Pi }}_{+}$ for $R_0=1$, $R_V=10$ and different multipoles $l$. In the background of ${\mathrm{\Pi }}_{-}$, the potential is qualitatively similar.

Figure 7

Fundamental quasinormal modes of the scalar field in the background ${\mathrm{\Pi }}_{+}$. The full lines correspond to the numerical results, whereas the dashed lines show the analytic approximation at low frequencies. The top and bottom panels show the real part, ${\omega }_R{R}_0$, and the imaginary part, ${\omega }_I{R}_0$, of the mode as a function of the Vainsthein radius $R_V/R_0$.

Figure 8

The evolution of $Z^{tt}$ given in (21) to Cauchy breakdown, with the initial condition (22) with $A=1$, $\sigma =0.5$, $r_w=12$. The source is characterized by $\rho =100$, $R_0=0.5$, $R_V\approx 4$.

Figure 9

Log-Log plots of the region in parameter space of initial conditions for which Cauchy breakdown occurs. Breakdown occurs above each of the lines shown, which correspond to the critical amplitude $A_{\text{crit}}$. Recall that the Vainshtein radius is defined by $R_V=\sqrt{\pi }{\rho }^{1/3}{R}_0$, and everything is measured in terms of $\mathrm{\Lambda }$ [see equations (12) and (13)].

Figure 10

For the initial condition $\pi (r,0)=0$, Cauchy breakdown occurs in the region above the curve. The trend shows that above $R_V/R_0\sim 15.7$, there is Cauchy breakdown.

Figure 11

An example of evolving towards Cauchy breakdown for a collapsing source of the form (49) with $\rho =2000$, $R_0=1$ and $\tau =1$. Left panel: the evolution of $\pi (r,t)$ starting from ${\pi }_{+}(r)$. It is driven towards the vacuum $\pi (r,t)\to 0$ as the source collapses (until Cauchy breakdown is reached). Right: The corresponding factor $Z^{tt}$ (39) that crosses zero at $t=5$, resulting in Cauchy breakdown.

Figure 12

A sample evolution for an exploding source that evades Cauchy breakdown. The source has the form (50) with $\rho =1500$ and $R_0=1$. Left panel: the evolution of $\pi (r,t)$ starting from ${\pi }_{+}(r)$. It is driven towards the vacuum $\pi (r,t)\to 0$ as the source explodes, while a localized packet follows the traveling source off to infinity. Right: The corresponding factor $Z^{tt}$ (21). A perturbation is created that safely travels off to infinity without crossing zero, thus avoiding Cauchy breakdown.

Figure 13

Contour plot of the $y=0$ slice of density profile (57) describing two lumps of matter, here for $\rho =1$, $z_0=0.9$, $R_0=1$.

Figure 14

Hairy solutions for different multipoles $l$, here shown for $R_V=100$, $z_0=0.9$, and compared to ${\mathrm{\Pi }}_{+}$. For $R_0\ll r\ll R_V$, the field decays as $\sqrt{\rho }{z}_0^l{r}^{-l/2}$, while for $r\gg R_V$, they decay as $\rho z_0^l{r}^{-(l+1)}$.

Figure 15

The factor $Z_{\mathrm{MG}}^{tt}$ for various values of $\kappa \equiv \alpha \rho$. The solution is unstable for $\kappa <-6$.

Figure 16

Fundamental quasinormal modes of the scalar field in the decoupling limit of massive gravity. The full lines correspond to the numerical results, whereas the dashed lines show the analytic approximation at low frequencies. The top and bottom panels show the real part, ${\omega }_R{R}_0$, and the imaginary part, ${\omega }_I{R}_0$, of the mode as a function of ${\alpha }^{1/3}{R}_V/R_0$.

Figure 17

Hairy solutions for different multipoles $l$, here shown for $R_V=100$, $z_0=0.9$, $\alpha =1/3$, and compared to $\mathrm{\Pi }\equiv {\pi }^{\prime }$. For $R_0\ll r\ll R_V$ the field decays as ${\rho }^{1/3}{z}_0^l{r}^{-l/4}$, while for $r\gg R_V$ they decay as $\rho z_0^l{r}^{-(l+1)}$.

Figure 18

The convergence factor $Q(t)$ defined by Eq. (a1) as a function of time. The left panel refers to initial data of the form $\pi (r,0)=0$, $\rho =50$, $R_0=1$, while the right panel refers to Eq. (22) for $\rho =200$, $R_0=1$, $A=0.002$, $\sigma =1$, $r_w=10$, $\epsilon =0.001$.