Schwarzschild scalar wigs: Spectral analysis and late time behavior

Using the Green’s function representation technique, the late time behavior of localized scalar field distributions on Schwarzschild spacetimes is studied. Assuming arbitrary initial data we perform a spectral analysis, computing the amplitude of each excited quasibound mode without the necessity of performing dynamical evolutions. The resulting superposition of modes is compared with a traditional numerical evolution with excellent agreement; therefore, we have an efficient way to determine final black hole wigs. The astrophysical relevance of the quasibound modes is discussed in the context of scalar field dark matter models and the axiverse.

Figure 1

The branch cut (broken lines) for the analytic function $\kappa (s)$ defined in Eq. (20) and the deformation of the curve $\gamma$ which allows us to close the integration contour.

Figure 2

Visualization of the complex angle $\phi =\alpha +i\beta$ in the $s$ plane.

Figure 3

The location of the poles of the Green’s function for small $\lambda >0$ are shown by the crosses and the lines emanating from them. There are no poles outside the shaded region.

Figure 4

For a fixed observer at $x_1=45$, we plot the wave function obtained as the contribution of the first five quasibound modes [twice the real part of the sum over $n=1,2,\dots ,N_q=5$ of the modes defined in Eq. (16)] and the numerical solution. The parameters for the potential used in this example are $A=6$, $\mu =0.3$, $a=50$, and the parameters for the initial data are ${\tilde{\varphi }}_0=0.1$, $x_0=15$, $\sigma =1$. The relative numerical error is estimated to be less than 1%.

Figure 5

After some transient period in which the field bounces between the potential barriers, the energy drops from its initial value and settles down around a fixed value. This late contribution is given by the sum of the modes, Eq. (16).

Figure 6

Quasibound modes at $t=104857.6M$ for initial data of the form given in Eq. (45), with $R_1=4M$ and $R_2=8M$. In this case we have fixed $M\mu =0.3$ (left) and $M\mu =0.2$ (right). Top panel: Individual contribution of each excited mode, ${\psi }_n(t=104857.6M,r)={\varphi }_n(t=104857.6M,r)/r$, given the initial data. Bottom panel: Sum over the first nine modes.

Figure 7

The Fourier spectrum for evolutions with initial scalar field configurations with $[R_1,R_2]=[0M,100M]$, $[100M,200M]$ and $[200M,300M]$. The gray shaded area indicates the resonance region, and the vertical lines indicate the modes’ frequencies.

Figure 8

The Fourier spectrum for evolutions with initial scalar field configurations with $[R_1,R_2]=[3M,9M]$, $[4M,8M]$ and $[5M,7M]$. The gray shaded area indicates the resonance region, and the vertical lines indicate the modes’ frequencies.

Figure 9

The scalar field, $\psi (t,r)=\varphi (t,r)/r$, as a function of time at $r=81.05M$ for initial data of the form in Eq. (45), with $R_1=4M$ and $R_2=8M$. Comparison between dynamical evolutions (dots) and semianalytical results containing only the first nine modes (solid line). Left panel: $\ell =1$, $M\mu =0.3$. Right panel: $\ell =1$, $M\mu =0.2$.

Figure 10

The scalar field, $\psi (t,r)=\varphi (t,r)/r$, as a function of radius at $t=104857.6M$ for initial data of the form in Eq. (45), with $R_1=4M$ and $R_2=8M$. Comparison between numerical evolutions (dots) and semianalytical results containing only the first nine modes (solid line). Left panel: $\ell =1$, $M\mu =0.3$. Right panel: $\ell =1$, $M\mu =0.2$.

Figure 11

The difference between semianalytical and numerical solutions, $d_{\text{AN}}$, and their respective errors, $e_{\mathrm{A}}$ and $e_{\mathrm{N}}$, are shown for the case $\ell =1$, $M\mu =0.2$.

Figure 12

Location, $r_{\text{min}}$, of the local minimum of $V(r)$ for $M={10}^8{M}_{\odot }$, $\ell =0$ (continuous line) and $\ell =1$ (dashed line), as a function of $\mu$. The value of $r_{\text{min}}$ gives a lower bound for the size of the scalar wigs.

Figure 13

Evolution of the ratio $E_{\text{modes}}(t)/E_0$ between the total energy contained in the mode part of the solution (first nine modes only) and the energy of the initial data as a function of time. The initial data are of the form given in Eq. (45), with $R_1=4M$ and $R_2=8M$, and different values of $M\mu$. Note that $E_{\text{modes}}(0)/E_0\ne 1$ because Eq. (49) is not an appropriate representation of $E(t)$ at short times.