Softly Massive Gravity

Large-distance modification of gravity may be the mechanism for solving the cosmological constant problem. A simple model of the large-distance modification — four-dimensional (4D) gravity with the hard mass term— is problematic from the theoretical standpoint. Here we discuss a different model, the brane-induced gravity, that effectively introduces a soft graviton mass. We study the issues of unitarity, analyticity and causality in this model in more than five dimensions. We show that a consistent prescription for the poles of the Green’s function can be specified so that 4D unitarity is preserved. However, in certain instances 4D analyticity cannot be maintained when theory becomes higher dimensional. As a result, one has to sacrifice 4D causality at distances of the order of the present-day Hubble scale. This is a welcome feature for solving the cosmological constant problem, as was recently argued in the literature. We also show that, unlike the 4D massive gravity, the model has no strong-coupling problem at intermediate scales. ∗ Present address ⋄ Permanent address 1 Large distance modification of gravity: formulating the problem The reason underlying the observed acceleration of the universe is puzzling. It could be a tiny amount of vacuum energy. However, this possibility is hard to reconcile with known particle-physics models. Instead, it might well be that a new physical scale exists in the gravitational sector and the laws of gravity and cosmology are modified at this scale. To be consistent with data and be able to predict the accelerated expansion, the new scale should be roughly equal to H 0 ∼ 10 cm — the present-day value of the Hubble length. In this regard, developing models in which gravity gets modified at cosmological distances, becomes a timely endeavor. A generally covariant theory of the large-distance modification of gravity is the DGP model [1]. The gravity action of the model can be written as follows: S = M Pl 2 ∫ dx √ g R(g) + M ∗ 2 ∫ dx dy √ ḡR4+N (ḡ) , (1) where R and R4+N are the four-dimensional and (4+N)-dimensional Ricci scalars, respectively, and M∗ stands for the gravitational scale of the bulk theory. Extra dimensions are not compactified, they asymptote at infinity to Minkowski space. The higher-dimensional and four-dimensional metric tensors are related as ḡ(x, y = 0) ≡ g(x) . (2) The first term on the right hand side of (1) acts as a kinetic term for a 4D graviton while the second term acts as a gauge invariant mass term. The observable matter is assumed to be localized on a 4D surface y = 0. The present work is devoted to the study of the DGP scenario in the case N ≥ 2 (see Ref. [2], [3]). Such models have a string theory realization [4]. More importantly, these models are potential candidates for solving [5, 6] the cosmological constant problem (see also Refs. [7]–[24] for interesting cosmological and astrophysical studies). The equation of motion for the theory described by the action (1) takes the form δ(y)M Pl G (4) μν δ μ A δ ν B + M 2+N ∗ G (D) AB = −Tμν(x) δ A δ B δ(y) . (3) Our conventions are as follows: ηAB = diag [+−−...−] , A,B = 0, 1, ..., 3 +N , μ, ν = 0, 1, 2, 3 , a, b = 4, 5, ..., 3 +N . (4) G μν and G (D) AB denote the four-dimensional and D-dimensional Einstein tensors, respectively. We choose (for simplicity) a source localized on the brane, Tμν(x)δ (y).