() is a type of
modified gravity theory which generalizes
Einstein's general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. () gravity is actually a family of theories, each one defined by a different function, , of the
Ricci scalar, . 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
In physical cosmology, structure formation is the formation of galaxies, galaxy clusters and larger structures from small early density fluctuations. The universe, as is now known from observations of the cosmic microwave background radiation, beg ...
of the Universe without adding unknown forms of
dark energy
In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
or
dark matter
Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
. Some functional forms may be inspired by corrections arising from a
quantum theory of gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
. () gravity was first proposed in 1970 by
Hans Adolph Buchdahl (although was used rather than for the name of the arbitrary function). It has become an active field of research following work by
Starobinsky on
cosmic inflation. 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.
Introduction
In () gravity, one seeks to generalize the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
of the
Einstein–Hilbert action:
to
where
is the determinant of the
metric tensor, and
is some function of the
Ricci scalar.
[L. Amendola and S. Tsujikawa (2013) “Dark Energy, Theory and Observations”]
Cambridge University Press
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.
While the two formalisms lead to the same field equations for General Relativity, i.e., when
, the field equations may differ when
.
Metric () gravity
Derivation of field equations
In metric () 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.
The variation of the determinant is as always:
The
Ricci scalar is defined as
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
Substituting into the equation above:
where
is the
covariant derivative and
is the
d'Alembert operator
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...
.
Denoting
, the variation in the action reads:
Doing
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
on the second and third terms (and neglected the boundary contributions), we get:
By demanding that the action remains invariant under variations of the metric,
, one obtains the field equations:
where
is the
energy–momentum tensor defined as
where
is the matter Lagrangian.
The generalized Friedmann equations
Assuming a
Robertson–Walker metric with scale factor
we can find the generalized
Friedmann equations to be (in units where
):
where
is the
Hubble parameter
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
,
the dot is the derivative with respect to the cosmic time , and the terms
m and
rad 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. To see this, add a small scalar perturbation to the metric (in the
Newtonian gauge):
where and 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
eff. Doing so, we get the gravitational potential (valid on sub-
horizon scales ):
where
m is a perturbation in the matter density, is the Fourier scale and
eff 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
In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathem ...
(helicities ±2) and the third (scalar) is coming from the fact that if we take into account a conformal transformation, the fourth order theory () becomes
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
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 -direction, may be written as
where
and
g() = d/d is the
group velocity of a
wave packet
In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diff ...
centred on wave-vector . The first two terms correspond to the usual
transverse polarizations from general relativity, while the third corresponds to the new massive polarization mode of () theories. This mode is a mixture of massless transverse breathing mode (but not traceless) and massive longitudinal scalar mode. The transverse and traceless modes (also known as tensor modes) propagate at the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, but the massive scalar mode moves at a speed
G < 1 (in units where = 1), this mode is dispersive. However, in () 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.
Equivalent formalism
Under certain additional conditions we can simplify the analysis of () theories by introducing an
auxiliary field In physics, and especially quantum field theory, an auxiliary field is one whose equations of motion admit a single solution. Therefore, the Lagrangian describing such a field A contains an algebraic quadratic term and an arbitrary linear term, wh ...
. Assuming
for all , let () be the
Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
of () so that
and
. Then, one obtains the O'Hanlon (1972) action:
We have the Euler–Lagrange equations
Eliminating , 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
we transform to the
Einstein frame: