Self-gravitating scalar breathers with a negative cosmological constant

Breather-type (time-periodic and spatially localized) solutions with spherical symmetry are investigated in a massless scalar field theory coupled to Einstein’s gravity with cosmological constant in $d$ spatial dimensions imposing anti–de Sitter (AdS) asymptotics on space-time. Using a code constructed with the Kadath library that enables the use of spectral methods, the phase space of breather solutions is explored in detail for $d=3$ and $d=4$. It is found that there are discrete families of solutions indexed by an integer and by their frequency. Using a time evolution code these AdS breathers are found to be stable for up to a critical central density, in analogy to boson stars. Using an analytical perturbative expansion small amplitude breathers are worked out for arbitrary dimensions $d$.

Figure 1

The first panel shows the error on the radial derivative as a function of $\omega$, for $d=3$. The location of the zeros corresponds to the true solutions of the small amplitude expansion. The second panel shows the different solutions with an increasing number of nodes, i.e., for increasing values of $\omega$.

Figure 2

Difference between the result for $(N_r,N_t)=(33,17)$ and various lower resolution results. The $y$ axis denotes the difference in the value of the scalar field at the origin and the $x$ axis the number of radial coefficients $N_r$.

Figure 3

Values of the central frequency $\mathrm{\Omega }$ as a function of the expansion parameter $ϵ$ used in [24]. The solid line denotes the results from this paper and the circles the data taken from Table II of [24].

Figure 4

Oscillation frequency $\mathrm{\Omega }$ measured by a central observer, as a function of the frequency $\omega$ observed by a faraway observer, for $d=4$ spatial dimensions. For small amplitude states both frequencies approach the value 4.

Figure 5

Values of the various modes ${\varphi }_n$ at the origin, for $d=3$ (first panel) and $d=4$ (second panel), as a function of $\omega$. The modes are scaled for plotting convenience.

Figure 6

Solution for $d=3$ and $\omega =2.3$. The first panel shows the first modes of the scalar field and the second one the metric fields at $t=0$. The dashed lines are the linear solution (first panel) and the solution for the exact AdS spacetime (second panel).

Figure 7

Mode ${\varphi }_n$ at the origin, around two resonances found for $n=9$ and $d=3$ (first panel) and $n=5$ and $d=4$ (second panel). The solid lines denote the high resolution results $(N_r,N_t)=(33,17)$, the squares the middle resolution ones $(N_r,N_t)=(25,15)$, and the circles the lower resolution ones $(N_r,N_t)=(21,13)$.

Figure 8

Difference from the expected frequency of the oscillations as a function of time, evolving initial data corresponding to $\omega =2.3$ and $d=3$. An initial and a much later period is shown for numerical resolutions from $2^7$ to $2^{11}$ points.

Figure 9

Frequency dependence of the mass of the configurations for $d=4$ spatial dimensions.

Figure 10

Time evolution of the energy density $E$ for an initial data which is in the large amplitude unstable domain, corresponding to $\omega =2.104$. The upper panel shows $E$ at the center, while the lower one is at a radius that is larger than the one where the horizon forms.

Figure 11

The oscillation frequency $\omega$ is shown as a function of the expansion parameter $ϵ$. For comparison, the small-amplitude expansion results, valid to ${ϵ}^2$ and ${ϵ}^4$ order, are also given. According to the time-evolution code, the configurations with frequency below ${\omega }_s=2.253$ are unstable, so they are represented by a dotted line in that region.

Figure 12

Central frequency $\mathrm{\Omega }$ as a function of the parameter $ϵ$. The solid line shows our numerical results, while the dashed lines give the small-amplitude expansion results.

Figure 13

Mass of the configurations as a function of their frequency for $d=3$ spatial dimensions. The ${ϵ}^2$ and ${ϵ}^4$ order results of the small-amplitude expansion are also shown for comparison.