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Register a new userSymmetry breaking and the Goldstone theorem in de Sitter space
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Authors
Tomislav Prokopec
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We consider an O(N) symmetric scalar field model in the mean field (Hartree) approximation and show that the symmetry can be broken in de Sitter space. We find that the phase transition can be of first order, and that its strength depends non-analytically on the parameters of the model. We also show that the would-be Goldstone bosons acquire a mass, effectively becoming pseudo-Goldstone bosons, thus breaking the O(N) symmetry. Our results imply that topological defects can form during inflation.
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We study the response of a classical massless minimally coupled scalar to a static point scalar charge on de Sitter. By considering explicit solutions of the problem we conclude that -- even though the dynamics formally admits dilatation (scaling) symmetry -- the physical scalar field profile necessarily breaks the symmetry. This is an instance of symmetry breaking in classical physics due to large infrared effects. The gravitational backreaction, on the other hand, does respect dilatation symmetry, making this an example of symmetry non-inheritance phenomenon.
We derive a noncovariant but simple representation for the self-energy of a conformally transformed graviton field on the cosmological patch of de Sitter. Our representation involves four structure functions, as opposed to the two that would be necessary for a manifestly de Sitter invariant representation. We work out what the four structure functions are for the one loop correction due to a massless, minimally coupled scalar. And we employ the result to work out what happens to dynamical gravitons.
We employ a recent, general gauge computation of the one loop graviton contribution to the vacuum polarization on de Sitter to solve for one loop corrections to the photon mode function. The vacuum polarization takes the form of a gauge independent, spin 2 contribution and a gauge dependent, spin 0 contribution. We show that the leading secular corrections derive entirely from the spin 2 contribution.
The equation of motion of an extended object in spacetime reduces to an ordinary differential equation in the presence of symmetry. By properly defining of the symmetry with notion of cohomogeneity, we discuss the method for classifying all these extended objects. We carry out the classification for the strings in the five-dimensional anti-de Sitter space by the effective use of the local isomorphism between $SO(4,2)$ and $SU(2,2)$. We present a general method for solving the trajectory of the Nambu-Goto string and apply to a case obtained by the classification, thereby find a new solution which has properties unique to odd-dimensional anti-de Sitter spaces. The geometry of the solution is analized and found to be a timelike helicoid-like surface.
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 future 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.
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