The Shadows of Gravity
Jose A. R. Cembranos
[This is an invited article based on the author's recently published work -- 2Physics.com]
Author: Jose A. R. Cembranos
Affiliation: William I. Fine Theoretical Physics Institute, University of Minnesota in Minneapolis, USA
Many authors have tried to explain the dark sectors of the cosmological model as modifications of Einstein’s gravity (EG). Dark Matter (DM) and Dark Energy (DE) are the main sources of the cosmological evolution at late times. They dominate the dynamics of the Universe at low densities or low curvatures. Therefore it is reasonable to expect that an infrared (IR) modification of EG can lead to a possible solution of these puzzles. However, it is in the opposite limit, at high energies (HE), where EG needs corrections from a quantum approach. These natural ultraviolet (UV) modifications of gravity are usually thought to be related to inflation or to the Big Bang singularity. In a recent work, I have shown that DM can be explained with HE modifications of EG. I have used an explicit model: R2 gravity and study its possible experimental signatures .
Einstein’s General Relativity describes the classical gravitational interaction in a very successful way by the metric tensor of the space-time through the Einstein-Hilbert action (EHA). This theory is particularly beautiful and the action particularly simple, since it contains only one term proportional to the scalar curvature. The proportionality parameter which multiplies this term, defines the Newton’s constant of gravitation and the typical scale of gravity. This magnitude is known as the Planck scale and its approximated energy value is 1019 Giga-electronvolts, which is equivalent to a distance of 10-35 meters.
However, the inconsistency of quantum computations within the gravitational theory described by the EHA demands its modification at HE. Quantum radiative corrections produced by standard matter provide divergent terms that are constant, linear, and quadratic in the Riemann curvature tensor of the space-time. The constant divergence can be regularized by the renormalization of the cosmological constant, which may explain the Dark Energy. The linear term is absorbed in the renormalization of the Planck scale itself. On the contrary, the quadratic terms are not included in the standard gravitational action. If these quantum corrections are not cancelled by invoking new symmetries, these terms need to be taken into account for the study of gravity at HE . Indeed, these terms are also produced by radiative corrections coming from the own EG. Unfortunately, the gravitational corrections do not stop at this order as the associated with the matter content. There are cubic terms, quartic terms, etc. All these local quantum corrections are divergent and the fact that there is a non finite number of them implies that the theory is non-renormalizable. We know how to deal with gravity as an effective field theory, working order by order, but we cannot access higher energies than the Planck scale by using this effective approach . In any case, the Planck scale is very high, and unreachable experimentally so far.
Inspired by this effective field theory point of view, which identifies higher energy corrections with higher curvature terms, I have studied the viability of a solution to the missing matter problem from the UV completion of gravity. As I have explained above, the first HE modification to EG is provided by the inclusion of quadratic terms in the curvature of the space-time geometry. The most general quadratic action supports, in addition to the usual massless spin-two graviton, a massive spin-two and a massive scalar mode, with a total of eight degrees of freedom (in the physical gauge ). In fact, this gravitational theory is renormalizable . However, the massive spin-two gravitons are ghost-like particles that generate new unitarity violations, breaking of causality, and important instabilities.
In any case, there is a non-trivial quadratic extension of EG that is free of ghosts and phenomenologically viable. It is the so called R2 gravity since it is defined by the only addition of a term proportional to the square of the scalar curvature to the EHA. This term by itself does not improve the UV behaviour of EG but illustrates the idea in a minimal way. This particular HE modification of EG introduces a new scalar graviton that can provide the solution to the DM problem.
In this model, the new scalar graviton has a well defined coupling to the standard matter content and it is possible to study its phenomenology and experimental signatures  . Indeed, this DM candidate could be considered as a superweakly interacting massive particle (superWIMP ) since its interactions are gravitational, i.e. it couples universally to the energy-momentum tensor with Planck suppressed couplings. It means that the new scalar graviton mediates an attractive Yukawa force between two non-relativistic particles with strength similar to Newton’s gravity. Among other differences, this new component of the gravitational force has a finite range, shorter than 0.1 millimeters, since the new scalar graviton is massive.
This is the most constraining lower bound on the mass of the scalar mode and it is independent of any supposition about its abundance. On the contrary, depending on its contribution to the total amount of DM, its mass is constrained from above. I have shown that it cannot be much heavier than twice the mass of the electron. If that is the case, this graviton decays in an electron-positron pair. These positrons annihilate producing a flux of gamma rays that we should have observed. In fact, the SPI spectrometer on the INTEGRAL (International Gamma-ray Astrophysics Laboratory) satellite, has observed a flux of gamma rays coming from the galactic centre (GC), whose characteristics are fully consistent with electron-positron annihilation .
If the mass of the new graviton is tuned close to the electron-positron production threshold, this line could be the first observation of R2 gravity. The same gravitational DM can explain this observation with a less tuned mass and a lower abundance. For heavier masses, the gamma ray spectrum originated by inflight annihilation of the positrons with interstellar electrons is even more constraining than the 511 keV photons .
On the contrary, for lighter masses, the only decay channel that may be observable is in two photons. It is difficult to detect these gravitational decays in the isotropic diffuse photon background (iDPB) . A most promising analysis is associated with the search of gamma-ray lines from localized sources, as the GC. The iDPB is continuum since it suffers the cosmological redshift, but the mono-energetic photons originated by local sources may give a clear signal of R2 gravity .
In conclusion, I have analyzed the possibility that the DM origin resides in UV modifications of gravity . Although, strictly speaking, my results are particular of R2 gravity, I think they are qualitatively general with a minimum set of assumptions about the gravitational sector. In any case, different approaches to try to link our ignorance about gravitation with the dark sectors of standard cosmology can be taken , and it is a very interesting subject which surely deserves further investigations.
This work is supported in part by DOE Grant No. DOE/DE-FG02-94ER40823, FPA 2005-02327 project (DGICYT, Spain), and CAM/UCM 910309 project.
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