Abstract
We show that the combination of cubic invariants defining five-dimensional quasitopological gravity, when written in four dimensions, reduce to the version of four-dimensional Einsteinian gravity recently proposed by Arciniega, Edelstein & Jaime, that produces second order equations of motion in a FLRW ansatz, with a purely geometrical inflationary period. We introduce a quartic version of the four-dimensional Einsteinian theory with similar properties, and study its consequences. In particular we found that there exists a region on the space of parameters which allows for thermodynamically stable black holes, as well as a well-defined cosmology with geometrically driven inflation. We briefly discuss the cosmological inhomogeneities in this setup. We also provide a combination of quintic invariants with those properties.
Highlights
Any ultra-violet completion of General Relativity will generically induce higher curvature modifications to the space-time dynamics
As recently discovered in [17], improving the theory with a second, ghost-free, cubic combination in four dimensions, which vanishes on the static black hole ansatz [19], leads to interesting cosmological scenarios. It gives second order differential equations in the Friedmann-LemaıtreRobertson-Walker (FLRW) ansatz, with a purely geometrical inflationary period, smoothly matching a matter dominated era followed by late time acceleration [17]
A point to be stressed is that the sign of the cubic coupling that gives rise to black holes, is the opposite to that originating healthy cosmologies [18]
Summary
Any ultra-violet completion of General Relativity will generically induce higher curvature modifications to the space-time dynamics. The authors of [12] proposed a cubic action in which the scalar and massive graviton acquire an infinite mass, and the relative coefficients of the cubic curvature terms are fixed in a dimension-independent manner This theory leads to simple spherically symmetric black holes, characterized by a single metric function which satisfies a third-order equation that admits a first integral [13, 14]. As recently discovered in [17], improving the theory with a second, ghost-free, cubic combination in four dimensions, which vanishes on the static black hole ansatz [19], leads to interesting cosmological scenarios. A static, spherically symmetric ansatz, fulfilling −gttgrr = 1, must lead to third order equations of motion with a trivial first integral These restrictions lead to constraints on the coefficients (see Appendix A), that defines our four-dimensional theory. In Appendix B, we provide a quintic combination with the aforementioned properties
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