풇(푹)Models 247enough dimension in energy units, the counterterms required to cancel all divergences
proliferate to infinite number, and, at first glance, the theory would seem to gain an
infinite number of free parameters and therefore lose all predictive power, becoming
scientifically worthless. Such theories are called nonrenormalizable.
The Standard Model of particle physics contains only renormalizable operators,
but the interactions of general relativity become nonrenormalizable operators if one
attempts to construct a field theory of quantum gravity in the most straightforward
manner, suggesting that perturbation theory is useless in applications to quantum
gravity.
The endeavor to remedy the renormalizability of the Einstein–Hilbert action
implies adding functions of the Ricci scalar푅which become important at late times
and for small values of the curvature. This is required in order to avoid conflicts with
constraints from the solar system or the Galaxy.
Consider an action of the form
푆∝[
∫
d^4 푥√
−푔[푅+푓(푅)] +
∫
d^4 푥√
−푔푚(푔휇휈,훹)
]
(11.28)
where푓(푅)is an unspecified function of푅. The matter Lagrangian푚is minimally
coupled and, therefore the matter fields훹fall along the metric푔휇휈. By varying this
action with respect to푔휇휈one can obtain the field equations of the form
퐺̃휇휈≡푅휇휈−^1
2푅푔휇휈−푄휇휈 (11.29)
where퐺̃휇휈 contains the modifications to the geometry in terms of the functions
푓(푅),푓푅≡휕푓∕휕푅, 푔휇휈.The푓(푅)term leads to extra terms in the Einstein equation
which is of second order in the metric푔휇휈. The constraint equation of GR becomes
a third order equation, and the evolution equations now become fourth order dif-
ferential equations which are difficult to analyze. A more tractable form of extended
gravity can be obtained by using thePalatini variation, briefly discussed at the end of
Chapter 5.
When taking the metric to be of the Robertson–Walker form [Equation (2.32)] the
Friedmann equation becomes
퐻^2 +푓
6
−푎̈
푎
푓푅+퐻푓̇푅=
휅^2 휌
3
(11.30)
where푓≡푓(푅). The Raychauduri equation becomes
푎̈
푎−푓푅퐻^2 +
푓
6
+
푓̈푅
2
=−
휅^2
6
(휌+ 3 푝) (11.31)
and the stress-energy tensor is replaced by
푇̃휇휈=푇휇휈
푓푅
. (11.32)
Any푓(푅)theory designed to achieve cosmic acceleration must satisfy|푓≪푅|and
|푓푅|≪1 at high curvature to be consistent with our knowledge of the high redshift
universe. In order for푓푅not to be tachyonic it must have a positive squared mass,
that is, there must exist a stable high-curvature regime, such as a matter dominated