Bulk and brane decay of a ($4+n$)-dimensional Schwarzschild–de Sitter black hole: Scalar radiation

In this paper, we extend the idea that the spectrum of Hawking radiation can reveal valuable information on a number of parameters that characterize a particular black hole background—such as the dimensionality of spacetime and the value of coupling constants—to gain information on another important aspect: the curvature of spacetime. We investigate the emission of Hawking radiation from a $D$-dimensional Schwarzschild–de Sitter black hole emitted in the form of scalar fields, and employ both analytical and numerical techniques to calculate greybody factors and differential energy emission rates on the brane and in the bulk. The energy emission rate of the black hole is significantly enhanced in the high-energy regime with the number of spacelike dimensions. On the other hand, in the low-energy part of the spectrum, it is the cosmological constant that leaves a clear footprint, through a characteristic, constant emission rate of ultrasoft quanta determined by the values of black hole and cosmological horizons. Our results are applicable to “small” black holes arising in theories with an arbitrary number and size of extra dimensions, as well as to pure 4-dimensional primordial black holes, embedded in a de Sitter spacetime.

Figure 1

The gravitational potential barrier $V(r)$ in the projected 4-dimensional SdS spacetime for $n=1$, $\mu =1$, $\Lambda r_H^2={10}^{-2}$ and $\ell =0,1$ and 2. The barrier vanishes at $r_H=1.0008$ and $r_C=24.474$.

Figure 2

The absorption cross section ${\sigma }_n(\omega )$, for scalar emission on the brane, versus the dimensionless parameter $\omega r_H$: (a) in the first graph, the dimensionality of spacetime is fixed at $n=0$, while $\Lambda r_H^2$ takes the values $\{0,{10}^{-2},{10}^{-1.5},0.05\}$; (b) in the second graph, the cosmological constant is fixed at $\Lambda r_H^2={10}^{-2}$, while $n$ takes the indicative values $\{0,1,5\}$.

Figure 3

The gravitational potential barrier $V(r)$ in the $(4+n)D$ SdS spacetime, for $n=1$, $\mu =1$, $\Lambda r_H^2={10}^{-2}$ and $\ell =0$, 1 and 2. The barrier vanishes again at $r_H=1.0008$ and $r_C=24.474$ but its height is significantly higher than on the brane.

Figure 4

The absorption cross section ${\sigma }_n(\omega )$, for scalar emission in the bulk, versus the dimensionless parameter $\omega r_H$: (a) the dimensionality of spacetime is fixed at $n=1$, while $\Lambda r_H^2$ takes the values $\{0,{10}^{-2},{10}^{-1.5},0.05\}$ in Planck units; (b) the cosmological constant is fixed at $\Lambda r_H^2={10}^{-2}$, and $n$ takes the indicative values $\{1,3,4\}$.

Figure 5

The differential energy emission rate, for scalar emission on the brane, versus the dimensionless parameter $\omega r_H$: (a) the dimensionality of spacetime is fixed at $n=0$, while $\Lambda r_H^2$ takes the values $\{0,{10}^{-2},{10}^{-15},0.05\}$ in Planck units; (b) the cosmological constant is fixed at $\Lambda r_H^2={10}^{-2}$, and $n$ takes the indicative values {0, 1, 5}.

Figure 6

The differential energy emission rate, for scalar emission in the bulk, versus the dimensionless parameter $\omega r_H$: (a) the dimensionality of spacetime is fixed at $n=1$, while $\Lambda r_H^2$ takes the values $\{0,{10}^{-2},{10}^{-1.5},0.05\}$ in Planck units; (b) the cosmological constant is fixed at $\Lambda r_H^2={10}^{-2}$, and $n$ takes the indicative values $\{0,1,2,3\}$.

Figure 7

The bulk-to-brane relative emission rate, for scalar emission, versus the dimensionless parameter $\omega r_H$: (a) the dimensionality of spacetime is fixed at $n=1$, while $\Lambda r_H^2$ takes the values $\{0,{10}^{-2},{10}^{-1.5},0.05\}$ in Planck units; (b) the cosmological constant is fixed at $\Lambda r_H^2={10}^{-2}$, and $n$ takes the indicative values $\{1,3,4\}$.