f(R) is a type of
modified gravity theory which generalizes
Einstein'sgeneral relativity. f(R) gravity is actually a family of theories, each one defined by a different function, f, of the
Ricci scalar, R. The simplest case is just the function being equal to the scalar; this is general relativity. As a consequence of introducing an arbitrary function, there may be freedom to explain the
accelerated expansion and
structure formation of the Universe without adding unknown forms of
dark energy or
dark matter. Some functional forms may be inspired by corrections arising from a
quantum theory of gravity. f(R) gravity was first proposed in 1970 by
Hans Adolph Buchdahl[1] (although ϕ was used rather than f for the name of the arbitrary function). It has become an active field of research following work by
Starobinsky on
cosmic inflation.[2] A wide range of phenomena can be produced from this theory by adopting different functions; however, many functional forms can now be ruled out on observational grounds, or because of pathological theoretical problems.
There are two ways to track the effect of changing to , i.e., to obtain the theory
field equations. The first is to use
metric formalism and the second is to use the
Palatini formalism.[3] While the two formalisms lead to the same field equations for General Relativity, i.e., when , the field equations may differ when .
Metric f(R) gravity
Derivation of field equations
In metric f(R) gravity, one arrives at the field equations by varying the action with respect to the
metric and not treating the
connection independently. For completeness we will now briefly mention the basic steps of the variation of the action. The main steps are the same as in the case of the variation of the
Einstein–Hilbert action (see the article for more details) but there are also some important differences.
Therefore, its variation with respect to the inverse metric is given by
For the second step see the article about the
Einstein–Hilbert action. Since is the difference of two connections, it should transform as a tensor. Therefore, it can be written as
is the
Hubble parameter,
the dot is the derivative with respect to the cosmic time t, and the terms ρm and ρrad represent the matter and radiation densities respectively; these satisfy the
continuity equations:
Modified Newton's constant
An interesting feature of these theories is the fact that the
gravitational constant is time and scale dependent.[4] To see this, add a small scalar perturbation to the metric (in the
Newtonian gauge):
where Φ and Ψ are the Newtonian potentials and use the field equations to first order. After some lengthy calculations, one can define a
Poisson equation in the Fourier space and attribute the extra terms that appear on the right-hand side to an effective gravitational constant Geff. Doing so, we get the gravitational potential (valid on sub-
horizon scales k2 ≫ a2H2):
where δρm is a perturbation in the matter density, k is the Fourier scale and Geff is:
with
Massive gravitational waves
This class of theories when linearized exhibits three polarization modes for the
gravitational waves, of which two correspond to the massless
graviton (helicities ±2) and the third (scalar) is coming from the fact that if we take into account a conformal transformation, the fourth order theory f(R) becomes
general relativity plus a
scalar field. To see this, identify
and use the field equations above to get
Working to first order of perturbation theory:
and after some tedious algebra, one can solve for the metric perturbation, which corresponds to the gravitational waves. A particular frequency component, for a wave propagating in the z-direction, may be written as
where
and vg(ω) = dω/dk is the
group velocity of a
wave packethf centred on wave-vector k. The first two terms correspond to the usual
transverse polarizations from general relativity, while the third corresponds to the new massive polarization mode of f(R) theories. This mode is a mixture of massless transverse breathing mode (but not traceless) and massive longitudinal scalar mode. [5][6] The transverse and traceless modes (also known as tensor modes) propagate at the
speed of light, but the massive scalar mode moves at a speed vG < 1 (in units where c = 1), this mode is dispersive. However, in f(R) gravity metric formalism, for the model (also known as pure model), the third polarization mode is a pure breathing mode and propagate with the speed of light through the spacetime. [7]
Equivalent formalism
Under certain additional conditions[8] we can simplify the analysis of f(R) theories by introducing an
auxiliary fieldΦ. Assuming for all R, let V(Φ) be the
Legendre transformation of f(R) so that and . Then, one obtains the O'Hanlon (1972) action:
Eliminating Φ, we obtain exactly the same equations as before. However, the equations are only second order in the derivatives, instead of fourth order.
We are currently working with the
Jordan frame. By performing a conformal rescaling:
This is general relativity coupled to a real scalar field: using f(R) theories to describe the accelerating universe is practically equivalent to using
quintessence. (At least, equivalent up to the caveat that we have not yet specified matter couplings, so (for example) f(R) gravity in which matter is minimally coupled to the metric (i.e., in Jordan frame) is equivalent to a quintessence theory in which the scalar field mediates a fifth force with gravitational strength.)
Palatini f(R) gravity
In
Palatinif(R) gravity, one treats the metric and
connection independently and varies the action with respect to each of them separately. The matter Lagrangian is assumed to be independent of the connection. These theories have been shown to be equivalent to
Brans–Dicke theory with ω = −3⁄2.[9][10] Due to the structure of the theory, however, Palatini f(R) theories appear to be in conflict with the Standard Model,[9][11] may violate Solar system experiments,[10] and seem to create unwanted singularities.[12]
Metric-affine f(R) gravity
In
metric-affinef(R) gravity, one generalizes things even further, treating both the metric and connection independently, and assuming the matter Lagrangian depends on the connection as well.
Observational tests
As there are many potential forms of f(R) gravity, it is difficult to find generic tests. Additionally, since deviations away from General Relativity can be made arbitrarily small in some cases, it is impossible to conclusively exclude some modifications. Some progress can be made, without assuming a concrete form for the function f(R) by
Taylor expanding
The first term is like the
cosmological constant and must be small. The next coefficient a1 can be set to one as in general relativity. For metric f(R) gravity (as opposed to Palatini or metric-affine f(R) gravity), the quadratic term is best constrained by
fifth force measurements, since it leads to a
Yukawa correction to the gravitational potential. The best current bounds are |a2| < 4×10−9 m2 or equivalently |a2| < 2.3×1022 GeV−2.[13][14]
The
parameterized post-Newtonian formalism is designed to be able to constrain generic modified theories of gravity. However, f(R) gravity shares many of the same values as General Relativity, and is therefore indistinguishable using these tests.[15] In particular light deflection is unchanged, so f(R) gravity, like General Relativity, is entirely consistent with the bounds from
Cassini tracking.[13]
Starobinsky gravity provides a mechanism for the cosmic
inflation, just after the
Big Bang when was still large. However, it is not suited to describe the present
universe acceleration since at present is very small.[17][18][19] This implies that the quadratic term in is negligible, i.e., one tends to which is General Relativity with a null
cosmological constant.
Gogoi-Goswami gravity
Gogoi-Goswami gravity has the following form
where and are two dimensionless positive constants and is a characteristic curvature constant. [20]
Tensorial generalization
f(R) gravity as presented in the previous sections is a scalar modification of general relativity. More generally, we can have a
coupling involving invariants of the
Ricci tensor and the
Weyl tensor. Special cases are f(R) gravity,
conformal gravity,
Gauss–Bonnet gravity and
Lovelock gravity. Notice that with any nontrivial tensorial dependence, we typically have additional massive spin-2 degrees of freedom, in addition to the massless graviton and a massive scalar. An exception is Gauss–Bonnet gravity where the fourth order terms for the spin-2 components cancel out.