Marginally deformed Starobinsky gravity
c The Authors.
-Origins & the Danish IAS, University of Southern Denmark
DK-5230 Odense M
We show that quantum-induced marginal deformations of the Starobinsky gravitational action of the form R2(1), with R the Ricci scalar and a positive parameter smaller than one half, can generate sizable primordial tensor modes. natural microscopic sources of these corrections and demonstrate that they generally lead to a nonzero and positive . Furthermore we argue, that within this framework, the scalar spectral index and tensor modes probe theories of grand unification including theories not testable at the electroweak scale.
The fundamental origin of the inflationary paradigm is a central problem in
cosmology . The simplest models of inflation typically introduce new scalar degrees
An intriguing possibility is that gravity itself is directly responsible for the inflationary
period of the universe. This requires one to go beyond the time-honored Einstein action, for
example by adding a R2-term as in the Starobinsky model [1, 2]. This approach is highly
natural since it enables gravity itself to drive inflation without resorting to the introduction
of new ad hoc scalar fields. The Starobinsky model in isolation predicts a nearly vanishing
ratio of tensor to scalar modes (r), a result challenged by BICEP2 results . However,
independently on the validity of the BICEP2 result , it is of fundamental importance
to know the modifications on the Starobinsky model stemming from the embedding of a
matter theory of particle physics in the gravitational theory.
According to  cosmology can be used qualitatively to establish the quantization
of gravity. In fact, by combining cosmological observations with an effective field theory
(EFT) treatment of gravity [9, 10] one can start estimating the parameters entering
gravitys effective action. An actual discovery of primordial tensor modes can therefore be used
to determine these parameters at the inflationary scale, which may turn out to be close to
the grand unification energy scale.
To lowest order, the effective action for gravity can be parametrized as
where Mp is the Planck mass. Beyond an expansion in the Ricci scalar R, we formally
included the Weyl conformal tensor C2 and the Euler four dimensional topological term
E. However we can drop E since it is a total derivative. Furthermore when gravity is
quantized around the Friedmann Lemaitre Robertson Walker metric the Weyl terms are
sub-leading since the geometry is conformally flat . We are left with an f (R) form of the
EFT. Higher powers of R, C2 and E are naturally suppressed by the Planck mass scale. If
inflation occurs at energy scales much below the Planck scale the EFT is accurate. We must,
however, take into account also marginal deformations including, for example, logarithmic
corrections to the action above. Because of the similarity between the EFT description of
gravity and the chiral Lagrangian for Quantum Chromo Dynamics we expect the
quantuminduced logarithmic corrections to play a fundamental role for a coherent understanding of
low energy gravitational dynamics at the inflationary scale. This is exactly what happens
in hadronic processes involving pions at low energies.
We encode these ideas in a simple f (R) form of the gravitational action formulated in
the Jordan frame:
We linearize the action via SJ = R d4x
g [f (y) + f 0(y)(R y)] with
f 00(y) does not vanish.
Introducing the conformal mode = f 0(y) with V () =
vides the explicit relation between (1) and the effective quantum-corrected non-minimally
coupled scalar field theory used in . Here we can simply set the non-minimal coupling
the results obtained in .
An important difference with respect to the non-minimally coupled theory of  is
following conformal transformation of the metric:
This transformation allows us to rewrite both theories in terms of a propagating scalar
field minimally coupled to ordinary Einstein gravity. This is the Einstein frame. Here the
The map from the Jordan frame of f (R) gravity to the Einstein frame with a canonically
and Planck data provided in .
grows exponentially and, as shown in , it leads to a successful inflationary model with
to produce enough e-folds. In figure 1 generic modifications of the Starobinsky model are
confronted with BICEP2 and PLANCK data. We observe that cosmology may constrain
the generic particle content embedded in this gravity model of inflation.
Different extensions of the original Starobinsky model including, for example, new
degrees of freedom, have been investigated . We believe that our approach is a minimal
one which, as we shall see, also leads to relevant testable phenomenological implications.
We now argue that these marginal deformations, needed from a purely
phenomenological standpoint, arise naturally within a field-theoretical approach to quantum gravity. To
The logarithmic term is reminiscent of what one would obtain via trace-log evaluations
of quantum corrections. There are several possible sources for these corrections. They
may arise, for example, by integrating out matter fields, or they can arise directly from
gravity loops. To sum-up the entire series of logarithmic corrections, and hence recover the
disposal. In the absence of a full theory of quantum gravity we start here by comparing
different predictions for the coefficient of the logarithmic term in (8) stemming out from:
i) Integrating out minimally coupled non-interacting NS real scalar fields  (only
nonconformal invariant matter contributes); ii) gravity corrections via the effective field theory
(EFT) approach ; iii) gravity corrections within higher derivative gravity (HDG) .
The action (8) naturally arises after a direct computation of the quantum corrections
This is deduced via heat kernel methods ,
the Einstein-Hilbert action in the effective field theory case or the HDG action.
methods shows  that leading order quantum fluctuations indeed induce a logarithmic
the coefficient of the beta function of the coupling of the R2 term, as a scale derivative with
respect to the mass scale in (8) shows. But we can give a better argument noticing that,
applications of the non-local terms have recently been studied in ; other kinds of
nonlocal actions, not generated by quantum loops, have also been considered in  and .
Here we are interested in the cosmological role of non-analytical terms which are present
in the effective quantum gravity action. Our analysis favors the idea that non-analytic
terms are more significant for inflation than non-local ones. To the lowest order the beta
function is h = (4)2 with C a constant depending on the source of quantum corrections
considered. After an RG improvement, the equation for h reads
Here R0 = 20 is a given renormalization scale. We therefore have = 4(4)2 and the
first logarithmic correction of (8). Using (9) we construct the log-resummed solution
h(R) = h(R0)
evaluation of C gives [16, 17]:
minimally coupled scalars,
Remarkably we deduce a positive exponent regardless of the underlying theory used to
determine the associated quantum corrections to the gravitation action. Massive particles
the mass term is negligible. Smaller renormalisation scales generally tend to reduce the
From (11) we deduce that quantum gravitational contributions can account, at most,
for a 3% increase in r as compared to the original Starobinsky model. Therefore any larger
value of r can only be generated by adding matter corrections. This, in turn, can be used
to constrain particle physics models minimally coupled to f (R) gravity. Furthermore, as
it is evident form figure 1, for small r the spectral index (ns) depends sensitively on the
sigma confidence level on the number of scalar fields minimally coupled to f (R) gravity
NS 85 .
The corresponding r values cannot exceed 0.007. To exemplify the power of our results we
now compare (12) with popular models of grand unification (GUT) such as minimal SU(5)
that features 34 scalars and (non)minimal SO(10) featuring (297) 109 scalars. It is clear
that only models with a low content of scalars are preferred by current experiments.
Values of r around and above 0.2 can be achieved at two sigma confidence level only
by allowing for the presence of thousands of scalars. This corresponds to the upper part of
figure 1. Here one might hope to use non-minimal models of supersymmetric GUTs which
would otherwise be physically excluded within the paradigm investigated here.
We have pointed out that non-analytic terms are presents in the effective quantum
action and can have a role in cosmology. We have also checked that higher order terms of
the type R3 cannot lead to sizable nonzero tensor modes. Therefore we arrive at the general
conclusion, that if inflation is driven by an f (R) theory of gravity, a natural form for this
whose size is related to the microscopic theory dictating the trace-log quantum corrections.
This form can be tested by current and future experimental results and constitutes a natural
generalization of the original Starobinsky action.
Work supported by the Danish National Research Foundation DNRF:90 grant.
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any medium, provided the original author(s) and source are credited.
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