Acausality in nonlocal gravity theory

Published for SISSA by Springer Received: January 22, 2016 Revised: February 2, 2016 Accepted: February 28, 2016 Published: March 7, 2016 Ying-li Zhang,a,b Kazuya Koyama,b Misao Sasakic and Gong-Bo Zhaoa,b a National Astronomy Observatories, Chinese Academy of Science, Beijing 100012, People’s Republic of China b Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, U.K. c Yukawa Institute for Theoretical Physics, yoto University, Kyoto 606-8502, Japan E-mail: , , , Abstract: We investigate the nonlocal gravity theory by deriving nonlocal equations of motion using the traditional variation principle in a homogeneous background. We focus on a class of models with a linear nonlocal modification term in the action. It is found that the resulting equations of motion contain the advanced Green’s function, implying that there is an acausality problem. As a consequence, a divergence arises in the solutions due to contributions from the future infinity unless the Universe will go back to the radiation dominated era or become the Minkowski spacetime in the future. We also discuss the relation between the original nonlocal equations and its biscalar-tensor representation and identify the auxiliary fields with the corresponding original nonlocal terms. Finally, we show that the acusality problem cannot be avoided by any function of nonlocal terms in the action. Keywords: Classical Theories of Gravity, Models of Quantum Gravity ArXiv ePrint: 1601.03808 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP03(2016)039 JHEP03(2016)039 Acausality in nonlocal gravity theory Contents 1 2 A simple example: scalar field with nonlocal operator 3 3 Linear nonlocal gravity in a homogeneous geometry 3.1 Original equations of motion 3.2 A simple case: the trace equation 3.3 The biscalar-tensor representation 4 4 7 8 4 General case 10 5 Conclusion 11 A Appearance of the advanced Green’s function in the FLRW metric 12 1 Introduction The nonlocal gravity theory was initially proposed as a “filter” to eliminate the contribution of the cosmological constant to the spacetime curvature so that it might provide a possible way to relieve the cosmological constant problem [1–4]. As far as cosmological studies are concerned, a model with nonlocal modifications was proposed by Deser and Woodard in 2007 [5]. At the level of action, the nonlocal correction term takes the form of Rf (2−1 R), in which the dimensionless combination 2−1 R is tiny during the radiation-dominated era but gradually increases in the matter-dominated epoch. Hence, this theory could help relieve the “fine-tuning problem” of the dark energy without introducing any small mass scale. Based on this model, the cosmological correspondences were studied extensively (e.g. see refs. [6]–[26]). However, it was found that although at the background level, the evolution of the nonlocal gravity could be designed to be indistinguishable from that of ΛCDM model [27], studies of the structure formation would disfavor this model [28, 29]. Nevertheless, this negative result does not totally rule out the possibility of including nonlocal corrections in the action. Recently, there appear a series of studies of nonlocal modifications: in refs. [30, 32, 33], a term proportional to gµν 2−1 R was introduced into the field equations. It was found that in this model, a mass term could be introduced without any reference metric [30, 31]. Moreover, its equation of state (EoS) is less than −1, hence this model could mimic the phantom dark energy [32, 33], while studies of its linear perturbations showed that this model was statistically comparable with ΛCDM model [34]. Another model was proposed by introducing a term proportional to R2−2 R into the action [35], with its cosmological perturbations studied in [36] which also gave a –1– JHEP03(2016)039 1 Introduction –2– JHEP03(2016)039 positive result. Besides these two interesting models, there are also discussions on other related topics, e.g. interpretations of dark matter as nonlocal effects from the General Relativity (GR) [37–39]. On the other hand, several theoretical aspects of a theory with nonlocal terms remain to be clarified. For instance, in order to transform the original integro-differential equation into differential equations, the nonlocal gravity theory is often written into a biscalar-tensor theory by introducing a scalar field ψ ≡ 2−1 R with a Langrangian multiplier. In this case, the number of degrees of freedom in this theory becomes ambiguous. It was found that in the corresponding biscalar-tensor theory, there would appear a “ghost-like” mode so that the theory could become unhealthy [2–4, 40]. However, it was argued in [32, 41] that in the biscalar-tensor theory, the Green’s function for ψ should be defined in the way where the initial conditions remove the homogeneous solution which satisfies 2ψhom = 0. In this sense, when ψ is quantized, the creation and annihilation coefficients vanish so that ψ is not a “free field”, hence the “ghost-like” mode is physically irrelevant. Another problem is the appearance of acausality in this theory. As discussed in [5, 23, 37, 41], under the replacement x ↔ x0 , the retarded Green’s function GR (x0 , x) becomes an advanced one GA (x, x0 ). Hence, in the Minkowskian background, for a class of theories which contain nonlocal operators acting on scalar fields, it is expected that the advanced Green’s function cannot be eliminated in the equations of motion (EOM) obtained by the traditional variation principle. One of the consequences is that the future information is needed in order to find the solutions, which may imply acusality problems of the theories. In this paper, we consider the acausality problem arising from the nonlocal gravity theory. A similar problem may appear in a class of modified gravity theories that contain nonlocal operators. We start from a linear nonlocal gravity action and derive the EOM in its original formulation by the variation principle. We find that the variation principle will symmetrize the property of Green’s function in the EOM, i.e., no matter whether the nonlocal operator is defined by the retarded Green’s function or the advanced one in the action, both of them symmetrically appear in the EOM. This means that the advanced Green’s function cannot be eliminated by any construction of functions of the nonlocal operator in the action. Hence, future information is needed to find the solutions, i.e. the acusality problem appears in the nonlocal gravity theory. This could imply that the nonlocal gravity theory is not well-defined, or it is not a fundamental theory to derive the causal nonlocal equations. In most literature, especially for the numerical analysis, the analysis is done in the biscalar-tensor representation. We make a comparison of the original EOM to its biscalartensor representation and identify the extra scalars with the non-loca (...truncated)