I present briefly some critical remarks on recent applications of gravity theories in cosmology. Einstein's general relativity (GR) is exceptional among physical theories: it is formally merely a point in the huge space of all existing and conceivable theories of gravitation. Most interesting of them are metric nonlinear gravity (NLG) theories which differ from GR only in equations of motion: gravity is described by one unifying tensor field which in the initial formulation of the theories is interpreted as a spacetime metric with a Lagrangian L = f (g αβ , R αβμν ), where f is any smooth scalar function. The simplest of them, restricted NLG theories based on L = f (R), have attracted most attention. In cosmology they are used to provide a generalized Friedmann equation for the cosmic scale function in spatially flat Robertson-Walker (R-W) spacetime to account for the accelerated expansion of the universe without resorting to dark energy. Why to employ to this aim NLG theories instead of assuming that the cosmic acceleration is driven in GR by a kind of self-interacting scalar, vector or tensor field? The standard response is that one must then introduce ad hoc some self-interaction of the field, yet applying an NLG theory one may find approximate solutions which well fit the cosmic evolution by making appropriate approximations in the underlying Lagrangian. Let L = 1/R + R + R 2 , then in the early universe with strong curvature the R 2 term is dominant giving rise to some kind of inflation [1]. In the intermediate curvature period (radiation and galactic eras) the R term dominates and the standard cosmological model is valid. At present and in future the decreasing curvature makes the 1/R term dominant what results in an accelerated expansion. This kind of argument is explicitly or implicitly present in most works on NLG theories.
This line of argument, though suggestive, is actually misleading. One should investigate any NLG theory from the viewpoint of field theory and search for its predictions in the whole realm of gravitational physics. A success of a theory only in flat R-W spacetime is by far not enough.
The Lagrangian of a generic NLG theory, L = f (R, R αβ R αβ , R αβμν R αβμν , . . .), depends on all 14 invariants of Riemann tensor. Even in the simplest case of L = f (R) one may invoke the Cantor's theorem in set theory to the effect that the cardinality number of the set of analytic functions f (R) is larger than continuum. If one believes that the accelerated cosmic expansion requires dynamics different from that in GR, then how to select the correct Lagrangian?
The main criticism of typical cosmological approach to the search of the correct theory is as follows:
- The field equations of NLG theories are of 4th order ('higher derivative gravity'), hence the space of solutions is larger than that of GR, also in cosmology. It may contain astonishing solutions.