In this paper we focus on the gravitational thermodynamics of the far future. Cosmological observations suggest that most matter will be diluted away by the cosmological expansion, with the rest collapsing into supermassive black holes. The likely fu
ture state of our local universe is a supermassive black hole slowly evaporating in an empty universe dominated by a positive cosmological constant. We describe some overlooked features of how the cosmological horizon responds to the black hole evaporation. The presence of a black hole depresses the entropy of the cosmological horizon by an amount proportional to the geometric mean of the entropies of the black hole and cosmological horizons. As the black hole evaporates and loses its mass in the process, the total entropy increases obeying the second law of thermodynamics. The entropy is produced by the heat from the black hole flowing across the extremely cold cosmological horizon. Once the evaporation is complete, the universe becomes empty de Sitter space that (in the presence of a true cosmological constant) is the maximum entropy thermodynamic equilibrium state. We propose that flat Minkowski space is an improper limit of this process which obscures the thermodynamics. The cosmological constant should be regarded not only as an energy scale, but also as a scale for the maximum entropy of a universe. In this context, flat Minkowski space is indistinguishable from de Sitter with extremely small cosmological constant, yielding a divergent entropy. This introduces an unregulated infinity in black hole thermodynamics calculations, giving possibly misleading results.
Thermal noise is expected to be the dominant source of noise in the most sensitive frequency band of second generation ground based gravitational wave detectors. Reshaping the beam to a flatter wider profile which probes more of the mirror surface re
duces this noise. The Mesa beam shape has been proposed for this purpose and was subsequently generalized to a family of hyperboloidal beams with two parameters: twist angle alpha and beam width D. Varying alpha allows a continuous transition from the nearly-flat to the nearly-concentric Mesa beam configurations. We analytically prove that in the limit of infinite D hyperboloidal beams become Gaussians. The Advanced LIGO diffraction loss design constraint is 1 ppm per bounce. In the past the diffraction loss has often been calculated using the clipping approximation that, in general, underestimates the diffraction loss. We develop a code using pseudo-spectral methods to compute the diffraction loss directly from the propagator. We find that the diffraction loss is not a strictly monotonic function of beam width, but has local minima that occur due to finite mirror effects and leads to natural choices of D. For the Mesa beam a local minimum occurs at D = 10.67 cm and leads to a diffraction loss of 1.4 ppm. We find that if one requires a diffraction loss of strictly 1 ppm, the alpha = 0.91 pi hyperboloidal beam is optimal, leading to the coating thermal noise being lower by about 10% than for a Mesa beam while other types of thermal noise decrease as well. We then develop an iterative process that reconstructs the mirror to specifically account for finite mirror effects. This allows us to increase the D parameter and lower the coating noise by about 30% compared to the original Mesa configuration.
