D→4 Einstein-Gauss-Bonnet gravity and beyond

Abstract

A 'novel' pure theory of Einstein-Gauss-Bonnet gravity in four-spacetime dimensions can be constructed by rescaling the Gauss-Bonnet coupling constant, seemingly eluding Lovelock's theorem. Recently, however, the well-posedness of this model has been called into question. Here we apply a 'dimensional regularization' technique, first used by Mann and Ross to write down a D → 2 limit of general relativity, to the case of pure Einstein-Gauss-Bonnet gravity. The resulting four-dimensional action is a particular Horndeski theory of gravity matching the result found via a Kaluza-Klein reduction over a flat internal space. Some cosmological solutions of this four-dimensional theory are examined. We further adapt the technique to higher curvature Lovelock theories of gravity, as well as a low-energy effective string action with an α correction. With respect to the D → 4 limit of the α -corrected string action, we find we must also rescale the dilaton to have a non-singular action in four dimensions. Interestingly, when the conformal rescaling Φ is interpreted as another dilaton, the regularized string action appears to be a special case of a covariant multi-Galileon theory of gravity.

The Stringy Origins of Galileons and their Novel Limit

“…We emphasize this novel Horndeski theory is markedly different from other Horndeski theories; namely, the trace of the gravitational equations of motion of the novel Horndeski theory is equivalent to the trace of the equations of motion of the pure novel EGB theory [26]. The second alternative novel theory [39][40][41] is inspired by the D → 2 limit of GR by Mann and Ross [43], in which one performs a type of dimensional regularization involving the subtraction of a conformally transformed pure EGB action, in addition to the rescaling of the Gauss-Bonnet coupling. Remarkably, up to simple field rescalings, the alternative novel EGB models -KK reduction and dimensional regularization -match identically.…”

“…Nevertheless, this approach has come under some scrutiny (c.f. [33][34][35][36][37][38]) leading to two alternative ways to take the D = 4 limit of EGB [35,[39][40][41], which remarkably coincide.…”

“…In [41], moreover, a D → 4 limit of the low-energy effective heterotic string action was written down without performing a dimensional reduction. In order to have a well-defined D → 4 limit of the α -corrected theory, it was demonstrated that one must also perform a dimensional rescaling of the 10-dimensional dilaton, φ → (D − 4)φ.…”

Dilaton invading from infinitesimal extra dimension

“…1 A related observation in four-dimensions was made in the recent paper [12]. The dimensional reduction approach to obtain modified theories of gravity (a.k.a "four-dimensional Gauss-Bonnet gravity") was introduced in [13] [14][15] [16]. See also [17][18] for the similar approach.…”

Cosmic acceleration from string induced Galileons

“…Combining the observation of[28] together with the dimensionful rescaling utilized in[5], an alternative 'novel' theory of D = 4 Einstein-Gauss-Bonnet gravity was discovered in[29] (shown to be equivalent to dimensionally regularized EGB in[30][31][32]), circumventing some of the ill-posedness of the pure theory. The authors of[32] adapted the method of dimensional regularization to generic Lovelock theories of gravity, and α -corrected heterotic string theory.…”