Cosmological production of noncommutative black holes

We investigate the pair creation of noncommutative black holes in a background with a positive cosmological constant. As a first step we derive the noncommutative geometry inspired Schwarzschild–de Sitter solution. By varying the mass and the cosmological constant parameters, we find several spacetimes compatible with the new solution: positive-mass spacetimes admit one cosmological horizon and two, one, or no black hole horizons, while negative-mass spacetimes have just a cosmological horizon. These new black holes share the properties of the corresponding asymptotically flat solutions, including the nonsingular core and thermodynamic stability in the final phase of the evaporation. As a second step we determine the action which generates the matter sector of gravitational field equations and we construct instantons describing the pair production of black holes and the other admissible topologies. As a result we find that for current values of the cosmological constant the de Sitter background is quantum mechanically stable according to experience. However, positive-mass noncommutative black holes and solitons would have plentifully been produced during inflationary times for Planckian values of the cosmological constant. As a special result we find that, in these early epochs of the Universe, Planck size black holes production would have been largely disfavored. We also find a potential instability for production of negative-mass solitons.

Figure 1

The function $V(r)$ vs $r/\ell$ for $M=10\ell /G$ and $\Lambda /3={10}^{-4}{\ell }^{-2}$.

Figure 2

The function $V(r)$ vs $r/\ell$ for $M=5\ell /G$ and $\Lambda /3=15\times {10}^{-4}{\ell }^{-2}$.

Figure 3

The function $V(r)$ vs $r/\ell$ for $M=1.9\ell$ and $\Lambda /3=6\times {10}^{-4}{\ell }^{-2}$.

Figure 4

The function $V(r)$ vs $r/\ell$ for $M=10\ell /G$ and $\Lambda /3={10}^{-2}{\ell }^{-2}$.

Figure 5

The function $V(r)$ vs $r/\ell$ for $M=1\ell /G$ and $\Lambda /3={10}^{-2}{\ell }^{-2}$.

Figure 6

The function $V(r)$ vs $r/\ell$ for $M=1\ell$ and $\Lambda /3={10}^{-4}{\ell }^{-2}$.

Figure 7

The function $V_{\mathrm{NCSchw}-}(r)$ vs $r/\ell$ for $M=-3\ell$, corresponding to the negative-mass noncommutative geometry inspired Schwarzschild solution.

Figure 8

The function $V_{-}(r)$ vs $r/\ell$ for $M=-3\ell$ and $\Lambda /3=6\times {10}^{-3}{\ell }^{-2}$.

Figure 9

The thin line represents the black hole temperature in a de Sitter background, $T_{+}\times \ell$, as a function of $r_{+}/\ell$ for $\Lambda /3=3\times {10}^{-3}{\ell }^{-2}$. The thick line represents the temperature of the NCSchw black hole in asymptotically flat space.

Figure 10

The function $2\gamma (3/2;x^2/4)-\frac{1}{4}{x}^3{e}^{-x^2/4}$ vs $x$.