Self-gravitating oscillons and new critical behavior

The dynamical evolution of self-interacting scalars is of paramount importance in cosmological settings and can teach us about the content of Einstein’s equations. In flat space, nonlinear scalar field theories can give rise to localized, nonsingular, time-dependent, long-lived solutions called “oscillons.” Here, we discuss the effects of gravity on the properties and formation of these structures, described by a scalar field with a double well potential. We show that oscillons continue to exist even when gravity is turned on, and we conjecture that there exists a sequence of critical solutions with an infinite lifetime. Our results suggest that a new type of critical behavior appears in this theory, characterized by modulations of the lifetime of the oscillon around the scaling law and the modulations of the amplitude of the critical solutions.

Figure 1

$r$ and $\partial r/\partial \tilde{r}$ are schematically depicted as functions of $\tilde{r}$.

Figure 2

Left Panel: The L2 norm of Hamiltonian constraint violation for different grid resolutions. We have rescaled the line in a way appropriate for second-order accurate codes. The near perfect overlap shows that indeed our results display second-order convergence. Right Panel: Study of the conservation of the Kodama mass.

Figure 3

Envelope of the time evolution of $\mathrm{\Phi }(t,r=0)$ for ${\sigma }^{2}G=1.0\times {10}^{-3}$. The shaded area in the left panel is zoomed in and shown in more detail on the right panel.

Figure 4

The panel shows the time evolution of the period $T$ of the scalar field at the origin for ${\sigma }^{2}G=1.0\times {10}^{-3}$.

Figure 5

Time evolution of the energy inside a sphere with radius $r_0=10L$.

Figure 6

Evolution of oscillons for ${\sigma }^{2}G={10}^{-3}$. Left panel: Dependence of oscillon lifetime on initial bubble radius. Right Panel: Envelope of the time evolution of the scalar field at the origin near the third peak. The dotted blue lines show the power-law dissipation of $\propto t^{-3/2}$, the expected power-law decay for massive scalars in Minkowski backgrounds.

Figure 7

Critical behavior in the collapse of scalars for ${\sigma }^{2}G={10}^{-3}$. Left Panel: Lifetime of configuration as a function of the (log) bubble radius, close to the critical point. This study refers to the first peak in the left panel of Fig. 6. Notice how the broad dependence on the bubble radius is independent on whether the critical point is approached from the left or right. Right Panel: Relation between the exponent $\gamma$ and ${\sigma }^{2}G$.

Figure 8

The left panels show the relation between the lifetime $\tau$ and the initial bubble radius near the first three peaks for ${\sigma }^{2}G=2.0\times {10}^{-3}$. A scaling law similar to (2) is obeyed to good precision, but there is clearly a fine structure not captured by the law. The right panel shows the deviation between the lifetime as predicted by (2) and our actual numerical results. To gauge the importance of grid resolution, we show results for two different grid intervals $\mathrm{\Delta }r$.

Figure 9

The left upper panel shows the time evolution of the envelop of the scalar field at the origin for the first peak for ${\sigma }^{2}G=2.0\times {10}^{-3}$. The left lower panel shows the oscillation of the envelop of the scalar field at the origin in the shaded region of the upper panel. The right panel represents the convergence of the modulation in the plateau for different grid intervals $\mathrm{\Delta }r$.

Figure 10

The left upper panel shows the time evolution of the envelope of the scalar field at the origin for the second peak for ${\sigma }^{2}G=2.0\times {10}^{-3}$. The left lower panel shows the oscillation of the envelope of the scalar field at the origin in the shaded region of the upper panel. The right panel represents the convergence of the modulation in the plateau for different grid intervals $\mathrm{\Delta }r$.

Figure 11

The orbit of the scalar field and its conjugate momentum at the center in phase space. The vertical blue lines and green lines correspond to $dV/d\mathrm{\Phi }=0$ and $d^{2}V/d{\mathrm{\Phi }}^2=0$, respectively.

Figure 12

The time evolution of the envelope of the scalar field for ${\sigma }^{2}G=3.0\times {10}^{-3}$. The upper panel shows the envelope of the scalar field at the origin. The lower panel shows the time evolution of the Kodama mass in the sphere of the radius $r_0$.