††thanks: E-mail<EMAIL_ADDRESS>
# Is Asymptotically Weyl-Invariant Gravity Viable?

Daniel Coumbe  _The Niels Bohr Institute, Copenhagen University_
_Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark_

###### Abstract

We explore the cosmological viability of a theory of gravity defined by the
Lagrangian $f(\mathcal{R})=\mathcal{R}^{n\left(\mathcal{R}\right)}$ in the
Palatini formalism, where $n\left(\mathcal{R}\right)$ is a dimensionless
function of the Palatini scalar curvature $\mathcal{R}$ that interpolates
between general relativity when $n\left(\mathcal{R}\right)=1$ and a locally
scale-invariant and superficially renormalizable theory when
$n\left(\mathcal{R}\right)=2$. We refer to this model as asymptotically Weyl-
invariant gravity (AWIG).

We analyse perhaps the simplest possible implementation of AWIG. A phase space
analysis yields three fixed points with effective equation of states
corresponding to de Sitter, radiation and matter-dominated phases. An analysis
of the deceleration parameter suggests our model is consistent with an early
and late period of accelerated cosmic expansion, with an intermediate period
of decelerated expansion. We show that the model contains no obvious curvature
singularities. Therefore, AWIG appears to be cosmologically viable, at least
for the simple implementation explored.

PACS numbers: 04.60.-m, 04.60.Bc

##  1 Introduction

The general theory of relativity is currently our best description of gravity.
One reason for this is its explanatory power: assuming little more than a
single symmetry principle general relativity can explain a truly astonishing
range of experimental phenomena [1].

However, it is at best incomplete. It is often said that general relativity
breaks down at high energies _or_ small distances. Yet, it is more accurate to
say high energies _and_ small distances. This is an important distinction
since it highlights the regime in which we must modify general relativity,
namely for large energy densities, or equivalently for large spacetime
curvatures. For example, general relativity predicts its own breakdown at
curvature singularities, where scalar measures of curvature grow without
bound. Furthermore, general relativity is known to become fundamentally
incompatible with quantum field theory at high curvature scales, a failure
known as its non-renormalizability [2, 3]. Theoretical arguments alone are
enough to tell us that general relativity must be modified at high curvature
scales.

Experimental data also indicates that general relativity must either be
augmented or replaced altogether if it is to agree with observation [4]. For
example, general relativity by itself is unable to explain the early phase of
accelerated cosmic expansion, as evidenced by myriad high-precision
measurements [4], and must be supplemented with unobserved exotic energy
sources and scalar fields [5, 6]. Although this top-down approach, as
exemplified by the $\Lambda$CDM model, is currently our best description of
observed cosmological dynamics [4], its _ad hoc_ construction has driven
attempts to replace general relativity from the bottom-up.

Finding a viable replacement of general relativity is challenging. Such a
theory must at the very least be (i) equivalent to general relativity in the
low-curvature limit, (ii) renormalizable in the high-curvature limit, (iii)
unitary, (iv) stable, (v) contain no curvature singularities, (vi) consistent
with observation.

One attempt is that of higher-order gravity, in which the Lagrangian includes
terms quadratic in the curvature tensor. Although this approach is
perturbatively renormalizable, and hence satisfies criterion (ii), such
higher-order theories are not typically unitary or stable, thus failing to
satisfy criteria (iii) and (iv). The only higher-order theories that are
unitary and stable are so-called $f(R)$ theories, in which the Lagrangian is
an arbitrary function of the Ricci scalar only [7].

There are three types of $f(R)$ theory: metric, Palatini and metric-affine
variations [7]. Metric $f(R)$ gravity assumes that the affine connection
uniquely depends on the metric via the Levi-Civita connection, as in standard
general relativity. The Palatini formalism generalises the metric formalism by
relaxing the assumption that the connection must depend on the metric. The
metric-affine formalism is the most general approach since it even drops the
implicit assumption that the matter action is independent of the connection.

Particular metric $f(R)$ models have been shown to conflict with solar system
tests [8], give an incorrect Newtonian limit [9], contradict observed
cosmological dynamics [10, 11], be unable to satisfy big bang nucleosynthesis
constraints [12] and contain fatal Ricci scalar instabilities [13]. Thus,
metric $f(R)$ theories do not typically satisfy criteria (i), (iv) or (vi). As
for the metric-affine formalism, it is not even a metric theory in the usual
sense, meaning diffeomorphism invariance is likely broken [7]. Thus, metric-
affine theories do not even seem to satisfy criterion (i). However, it has
been shown that the Palatini variation is immune to any such Ricci scalar
instability [14]. Palatini formulations also appear to pass solar system tests
and reproduce the correct Newtonian limit [15]. Remarkably, a Palatini action
that is linear in the scalar curvature is identical to regular general
relativity [7]. However, this equivalence does not hold for higher-order
theories [16, 7]. In particular, a Palatini action that is purely quadratic in
the scalar curvature is identical to normal general relativity plus a non-zero
cosmological constant [17].

In Ref. [18] we proposed the theory of asymptotically Weyl-invariant gravity
(AWIG) within the Palatini formalism (see Refs. [19, 20, 21] for the
background to this proposal). By construction AWIG satisfies criteria
(i)-(iv).111In the low-curvature limit AWIG yields
$f\left(\mathcal{R}\right)=\mathcal{R}$, which is identical to general
relativity [7]. AWIG is at least superficially renormalizable because the
coupling constant of the theory becomes dimensionless in the high curvature
limit, as shown in section 2. AWIG is likely to be unitary because states of
negative norm (ghosts) that cause unitarity violations do not appear in $f(R)$
theories [7, 22]. AWIG appears stable since Ostragadsky’s instability is
evaded by any $f(R)$ theory [23], and the Dolgov-Kawasaki instability can not
occur in Palatini $f\left(\mathcal{R}\right)$ gravity [7] (see Ref. [18] for
more details on the construction of AWIG). The present work aims to test
whether this theory also satisfies criteria (v) and (vi), and hence to
determine if it may be a viable replacement of general relativity.

In addition to satisfying criteria (i)-(iv), a major motivation for developing
AWIG was finding a theory with the symmetry of local scale invariance. The
need for local scale invariance can be seen by recognising that all length
measurements are local comparisons. For example, to measure the length of a
rod requires bringing it together with some standard unit of length, say a
metre stick, at the same point in space and time. In this way the local
comparison yields a dimensionless ratio, for example, the rod might be longer
than the metre stick by a factor of two. Repeating this comparison at a
different spacetime point must yield the same result, even if the metric at
this new point were rescaled by an arbitrary factor $\Omega^{2}(x)$. This is
because both the rod and metre stick would be equally rescaled, yielding the
same dimensionless ratio. Such a direct comparison cannot be made for two rods
with a non-zero space-like or time-like separation [24, 25]. Therefore, it has
been argued that the laws of nature must be formulated in such a way as to be
invariant under local rescalings of the metric tensor
$g_{\mu\nu}\rightarrow\Omega^{2}(x)g_{\mu\nu}$, or equivalently under a local
change of units. Moreover, since scale-invariant theories of gravity are gauge
theories [26, 27], unification with the other three fundamental interactions,
which have all been successfully formulated as local gauge theories, becomes
tractable. The theory analysed in this work is invariant with respect to local
changes of scale in the high-curvature limit.

It is important to establish the standard against which we will judge whether
the presented theory is viable. Criterion (v) will be deemed to be satisfied
if at least two different curvature invariants can be shown to be divergence-
free. To satisfy criterion (vi) we make the maximal demand that the theory
reproduces all four observed phases of cosmological evolution in the correct
order [28], namely an early period of accelerated expansion, followed by
radiation and matter-dominated phases, and finally a late period of
accelerated expansion [29].

This paper is organised as follows. In section 2 we define the model of AWIG,
including a detailed exploration of the dimensionless exponent
$n\left(\mathcal{R}\right)$. In section 3 we detail the methodology that will
be used to test the viability of our model. Results are presented in section 4
followed by a concluding discussion in section 5.

##  2 Model

The class of theories to which our model belongs is defined by the action

$\mathcal{S}=\frac{1}{2\kappa}\int f\left(\mathcal{R}\right)\sqrt{-g}d^{4}x,$
(1)

where $\kappa\equiv 8\pi G$ and $G$ is the gravitational coupling.
$f\left(\mathcal{R}\right)$ is an arbitrary function of the Palatini scalar
curvature $\mathcal{R}$ and $g$ is the determinant of the metric tensor.
Varying Eq. (1) with respect to the metric and taking the trace gives the
field equations [7]

$f^{\prime}(\mathcal{R})\mathcal{R}-2f(\mathcal{R})=\kappa T.$ (2)

AWIG is defined by the specific case [18]

$f\left(\mathcal{R}\right)=\mathcal{R}^{n\left(\mathcal{R}\right)},$ (3)

where $n\left(\mathcal{R}\right)$ is a dimensionless function of $\mathcal{R}$
that interpolates between general relativity when
$n\left(\mathcal{R}\right)=1$ and a locally scale-invariant and superficially
renormalizable theory of gravity when $n\left(\mathcal{R}\right)=2$. By
defining $n\left(\mathcal{R}\right)$ in this way the Lagrangian density
$f\left(\mathcal{R}\right)$ is purely a function of scalar curvature, and
hence is guaranteed to be invariant under arbitrary differential coordinate
transformations. In $4$-dimensional spacetime
$\mathcal{R}^{n\left(\mathcal{R}\right)}$ has canonical mass dimension
$2n\left(\mathcal{R}\right)$. Since $\sqrt{-g}$ has mass dimension $-4$,
$\kappa$ must have a mass dimension of $2n\left(\mathcal{R}\right)-4$ if Eq.
(1) is to be dimensionless, which it must be since we are working in units of
$\hbar=c=1$. Thus, in the limit $n\left(\mathcal{R}\right)\to 2$ the
gravitational coupling becomes dimensionless, as demanded by scale-invariance.
Superficially renormalizable field theories are those with dimensionless
coupling constants [30].

To complete the definition of this model we must specify the function
$n\left(\mathcal{R}\right)$. We begin by taking the first derivative of
$f\left(\mathcal{R}\right)$ with respect to $\mathcal{R}$, denoted by
$f^{\prime}\left(\mathcal{R}\right)$, finding

$f^{\prime}\left(\mathcal{R}\right)=\mathcal{R}^{n\left(\mathcal{R}\right)-1}\left(n\left(\mathcal{R}\right)+\mathcal{R}\rm{log}\left(\mathcal{R}\right)n^{\prime}\left(\mathcal{R}\right)\right).$
(4)

Substituting Eqs.(3) and (4) into Eq. (2) and rearranging yields

$n^{\prime}\left(\mathcal{R}\right)=\frac{\kappa
T+\mathcal{R}^{n\left(\mathcal{R}\right)}\left(2-n\left(\mathcal{R}\right)\right)}{\mathcal{R}^{n\left(\mathcal{R}\right)+1}\log{\left(\mathcal{R}\right)}}.$
(5)

We now use the fact that the symmetry of local scale invariance is signalled
by the vanishing of the traced energy tensor [31]. Thus, as
$n\left(\mathcal{R}\right)\to 2$ we must have $T\to 0$. Applying the limits
$n\left(\mathcal{R}\right)\to 2$ and $T\to 0$ to Eq.(5) yields
$n^{\prime}\left(\mathcal{R}\right)=0$. Similarly, as
$n\left(\mathcal{R}\right)\to 1$ we must have $\kappa T\to-\mathcal{R}$, and
so Eq.(5) again yields $n^{\prime}\left(\mathcal{R}\right)=0$.222If
$\mathcal{R}=1$ when $n\left(\mathcal{R}\right)=2$ and $\kappa T=0$ then
$n^{\prime}\left(\mathcal{R}\right)$ is undefined, since the numerator and
denominator of Eq.(5) both equal zero. Likewise, if $\mathcal{R}=0$ when
$n\left(\mathcal{R}\right)=1$ and $\kappa T=-\mathcal{R}$ then
$n^{\prime}\left(\mathcal{R}\right)$ is undefined. However, in the limiting
cases $\mathcal{R}\to 0$ and $\mathcal{R}\to 1$ we have
$n^{\prime}\left(\mathcal{R}\right)=0$. Therefore, the function we seek must
satisfy the condition $n^{\prime}\left(\mathcal{R}\right)=0$ as
$n\left(\mathcal{R}\right)\to 1$ and $n\left(\mathcal{R}\right)\to 2$.

Experiment also supports a near-constant exponent $n\left(\mathcal{R}\right)$
at lower curvature scales. This is because general relativity agrees with
experiment over a wide range of energy or curvature scales [32, 1], indicating
that $n\left(\mathcal{R}\right)$ has at most a very weak dependence on
$\mathcal{R}$ within the range of current experimental sensitivity. Similarly,
the fact that in the high-curvature limit the theory becomes locally scale-
invariant implies a constant $n\left(\mathcal{R}\right)$, since in this limit
there can be no scale with respect to which $n\left(\mathcal{R}\right)$ can
vary.

We now proceed by assuming $n\left(\mathcal{R}\right)$ admits a series
expansion in $\mathcal{R}$ of the form

$n\left(\mathcal{R}_{*}\right)=\sum_{m=0}^{\infty}c_{m}\mathcal{R}_{*}^{m},$
(6)

where $c_{m}$ are dimensionless constants and $\mathcal{R_{*}}$ is defined by
the dimensionless ratio $\mathcal{R_{*}}\equiv\mathcal{R}/\mathcal{R}_{0}$,
with $\mathcal{R}_{0}$ a finite constant of mass dimension two that represents
the maximum value $R$ can take. In this way, $n\left(\mathcal{R}_{*}\right)$
is a purely dimensionless function of the Palatini scalar curvature
$\mathcal{R}$. Truncating to a third-order function we have333It can be shown
that first and second-order functions cannot produce the desired features
[33].

$n\left(\mathcal{R}_{*}\right)=c_{0}+c_{1}\mathcal{R}_{*}+c_{2}\mathcal{R}_{*}^{2}+c_{3}\mathcal{R}_{*}^{3}.$
(7)

Since the low-curvature limit corresponds to $\mathcal{R}_{*}\to 0$, the
constraint $n\left(\mathcal{R}_{*}\to 0\right)=1$ immediately yields
$c_{0}=1$. Similarly, since the high-curvature limit corresponds to
$\mathcal{R}_{*}\to 1$, the constraint $n\left(\mathcal{R}_{*}\to 1\right)=2$
gives $1+c_{1}+c_{2}+c_{3}=2$, or equivalently $c_{1}+c_{2}+c_{3}=1$.

The first derivative of $n\left(\mathcal{R}_{*}\right)$ with respect to
$\mathcal{R}$ is

$n^{\prime}\left(\mathcal{R}_{*}\right)=\frac{c_{1}}{\mathcal{R}_{0}}+2\frac{c_{2}}{\mathcal{R}_{0}^{2}}\mathcal{R}+3\frac{c_{3}}{\mathcal{R}_{0}^{3}}\mathcal{R}^{2}=\frac{c_{1}}{\mathcal{R}}\mathcal{R}_{*}+2\frac{c_{2}}{\mathcal{R}}\mathcal{R}_{*}^{2}+3\frac{c_{3}}{\mathcal{R}}\mathcal{R}_{*}^{3}.$
(8)

Since $\mathcal{R}_{*}\equiv\mathcal{R}/\mathcal{R}_{0}\to 0$ in the low-
curvature limit, Eq. (8) gives
$n^{\prime}\left(\mathcal{R}_{*}\right)=c_{1}/\mathcal{R}_{0}=0$, which
implies $c_{1}=0$ since $\mathcal{R}_{0}$ is assumed to be finite. The high-
curvature limit corresponds to
$\mathcal{R}_{*}\equiv\mathcal{R}/\mathcal{R}_{0}\to 1$, and so Eq. (8) gives
$n^{\prime}\left(\mathcal{R}_{*}\right)=c_{1}/\mathcal{R}_{0}+2c_{2}/\mathcal{R}_{0}+3c_{3}/\mathcal{R}_{0}=0$,
which implies $2c_{2}+3c_{3}=0$ since $c_{1}=0$. The polynomial coefficients
$c_{2}$ and $c_{3}$ can now be determined by solving the system of equations
$2c_{2}+3c_{3}=0$ and $c_{2}+c_{3}=1$, with the result $c_{2}=3,c_{3}=-2$.
Therefore,

$n\left(\mathcal{R}_{*}\right)=1+3\mathcal{R}_{*}^{2}-2\mathcal{R}_{*}^{3}.$
(9)

Eq. (9) is the lowest-order polynomial to satisfy our criteria, but there are
potentially an infinite number of higher-order polynomial functions. Let
$n_{i}\left(\mathcal{R}_{*}\right)$ label this set of polynomial functions,
where the order of the polynomial is given by $2i+1$. One can then generalise
Eq. (9) to any higher-order using [33]

$n_{i}\left(\mathcal{R}_{*}\right)=1+\mathcal{R}_{*}^{i+1}\sum_{j=0}^{i}{{i+j}\choose{j}}{{2i+1}\choose{i-j}}\left(-\mathcal{R}_{*}\right)^{j},\qquad
i\in\mathbb{N}.$ (10)

The first thirteen functions generated by Eq. (10) are shown in Fig. 1 (left).
The Lagrangian density in this case is then

$f_{i}\left(\mathcal{R}\right)=\mathcal{R}^{n_{i}\left(\mathcal{R}_{*}\right)}=\left(\mathcal{R}_{0}\mathcal{R}_{*}\right)^{n_{i}\left(\mathcal{R}_{*}\right)}.$
(11)

For simplicity, we choose $\mathcal{R}_{0}$ to have the value of one when
expressed in some particular unit of mass dimension two. For example, one
possibility is $\mathcal{R}_{0}=1m_{P}$, where $m_{P}$ is the Planck mass. The
term $\mathcal{R}_{0}$ then only acts to set the dimensionality of
$f_{i}\left(\mathcal{R}\right)$. The first thirteen functions
$f_{i}\left(\mathcal{R}\right)$ generated by applying Eq. (10) to Eq. (11) are
shown in Fig. 1 (middle), where we set $\mathcal{R}_{0}=1$ in some appropriate
unit. Differentiating Eq. (11) with respect to $\mathcal{R}$ gives the set of
first derivative functions $f^{\prime}_{i}\left(\mathcal{R}\right)$, with the
first 13 shown in Fig. 1 (right).

Figure 1: The first 13 exponents $n_{i}\left(\mathcal{R}_{*}\right)$ (left),
Lagrangian densities $f_{i}\left(\mathcal{R}\right)$ (middle), and first
derivative functions $f^{\prime}_{i}\left(\mathcal{R}\right)$ (right)
generated by Eq. (10) as a function of $\mathcal{R}_{*}$.

An important feature of Fig. 1 (right) is that the thirteenth function
$f^{\prime}_{13}\left(\mathcal{R}\right)$ becomes negative for certain values
of $\mathcal{R}_{*}$. A well-defined conformal transformation of the metric
tensor $\tilde{g}_{\mu\nu}=f^{\prime}\left(\mathcal{R}\right)g_{\mu\nu}$
requires that $f^{\prime}\left(\mathcal{R}\right)>0$ for all $\mathcal{R}$.
This condition is only satisfied if $i\leq 12$. Thus, we can exclude
Lagrangian densities $f_{i}\left(\mathcal{R}\right)$ with $i\geq 13$. In this
work, we shall focus on the simplest permitted Lagrangian density

$f_{1}\left(\mathcal{R}\right)=\mathcal{R}^{1+3\mathcal{R}_{*}^{2}-2\mathcal{R}_{*}^{3}}.$
(12)

##  3 Method

In this section we detail the method used to test the cosmological viablility
of the model defined by Eq. (12). The methodology presented in this section
follows the work of Refs. [28, 18].

Since cosmological observations by the Planck satellite show that our universe
is consistent with being spatially flat at late times [4], we begin by
assuming a flat Friedmann-Lemaıtre-Robertson-Walker (FLRW) metric

$ds^{2}=-dt^{2}+a^{2}(t)\left(dx^{2}+dy^{2}+dz^{2}\right),$ (13)

where $a(t)$ is the scale factor of the universe, a function of cosmological
time $t$. The evolution of a spatially homogenous and isotropic universe
filled with a cosmological fluid composed of pressureless dust and radiation
can be described by the modified Friedmann equation [7, 34]

$\left(H+\frac{\dot{f}^{\prime}\left(\mathcal{R}\right)}{2f^{\prime}\left(\mathcal{R}\right)}\right)^{2}=\frac{\kappa\left(\rho_{m}+2\rho_{r}\right)+f\left(\mathcal{R}\right)}{6f^{\prime}\left(\mathcal{R}\right)},$
(14)

where the dot notation signifies a time derivative and $H\equiv\dot{a}/a$ is
the Hubble parameter. $\rho_{m}$ and $\rho_{r}$ are the energy density of
matter and radiation, respectively, which satisfy the conservation conditions

$\dot{\rho}_{m}+3H\rho_{m}=0,\qquad\dot{\rho}_{r}+4H\rho_{r}=0.$ (15)

Since the trace of the energy-momentum tensor for radiation is zero, we simply
have $T=-\rho_{m}$ [28]. By using Eq. (14), combined with the conservation
conditions of Eq. (15), we can express the time derivative of the Palatini
scalar curvature as [34, 28]

$\mathcal{\dot{R}}=-\frac{3H\left(f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-2f\left(\mathcal{R}\right)\right)}{f^{\prime\prime}\left(\mathcal{R}\right)\mathcal{R}-f^{\prime}\left(\mathcal{R}\right)}.$
(16)

Using Eq. (16) we can replace $\mathcal{\dot{R}}$ in Eq. (14) to obtain [34,
7]

$H=\sqrt{\frac{2\kappa\left(\rho_{m}+\rho_{r}\right)+f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-f\left(\mathcal{R}\right)}{6f^{\prime}\left(\mathcal{R}\right)\xi}},$
(17)

where $\xi$ is defined by

$\xi=\left(1-\frac{3}{2}\frac{f^{\prime\prime}\left(\mathcal{R}\right)\left(f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-2f\left(\mathcal{R}\right)\right)}{f^{\prime}\left(\mathcal{R}\right)\left(f^{\prime\prime}\left(\mathcal{R}\right)\mathcal{R}-f^{\prime}\left(\mathcal{R}\right)\right)}\right)^{2}.$
(18)

If $\rho_{r}=0$, it is possible to use
$T=-\rho_{m}=\left(f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-2f\left(\mathcal{R}\right)\right)/\kappa$
to obtain the simpler expression

$H=\sqrt{\frac{3f\left(\mathcal{R}\right)-f^{\prime}\left(\mathcal{R}\right)\mathcal{R}}{6f^{\prime}\left(\mathcal{R}\right)\xi}}.$
(19)

In this work, we shall perform a detailed analysis of the phase space of the
model defined by Eq. (12). To facilitate this analysis we establish an
autonomous system of equations defined by the pair of dimensionless variables
[28]

$y_{1}=\frac{f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-f\left(\mathcal{R}\right)}{6f^{\prime}\left(\mathcal{R}\right)\xi
H^{2}},\qquad
y_{2}=\frac{\kappa\rho_{r}}{3f^{\prime}\left(\mathcal{R}\right)\xi H^{2}}.$
(20)

Using Eqs. (20) and (17) it can be shown that $\rho_{r}$ can be expressed in
terms of the variable $y_{2}$ via

$\rho_{r}=\frac{y_{2}}{1-y_{2}}\left(\rho_{m}+\frac{f^{\prime}\left(\mathcal{R}\right)\mathcal{R}}{2\kappa}-\frac{f\left(\mathcal{R}\right)}{2\kappa}\right).$
(21)

The evolution of $y_{1}$ and $y_{2}$ as a function of the cosmic scale factor
$a$ are established by the differential equations

$\frac{dy_{1}}{dN}=y_{1}\left(3-3y_{1}+y_{2}-3\frac{\left(f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-2f\left(\mathcal{R}\right)\right)f^{\prime\prime}\left(\mathcal{R}\right)\mathcal{R}}{\left(f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-f\left(\mathcal{R}\right)\right)\left(f^{\prime\prime}\left(\mathcal{R}\right)\mathcal{R}-f^{\prime}\left(\mathcal{R}\right)\right)}\left(1-y_{1}\right)\right)$
(22)

and

$\frac{dy_{2}}{dN}=y_{2}\left(-1-3y_{1}+y_{2}+3\frac{\left(f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-2f\left(\mathcal{R}\right)\right)f^{\prime\prime}\left(\mathcal{R}\right)\mathcal{R}}{\left(f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-f\left(\mathcal{R}\right)\right)\left(f^{\prime\prime}\left(\mathcal{R}\right)\mathcal{R}-f^{\prime}\left(\mathcal{R}\right)\right)}y_{1}\right),$
(23)

where $N\equiv\rm{ln}(a)$. The fixed points of this system correspond to the
values $\left(y_{1},y_{2}\right)$ that satisfy

$\frac{dy_{1}}{dN}=\frac{dy_{2}}{dN}=0.$ (24)

Note that there is a direct relationship between $\mathcal{R}$ and the
variables $\left(y_{1},y_{2}\right)$ given by [28]

$\frac{f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-2f\left(\mathcal{R}\right)}{f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-f\left(\mathcal{R}\right)}=-\frac{1-y_{1}-y_{2}}{2y_{1}}.$
(25)

By calculating the eigenvalues $\left(\lambda_{1},\lambda_{2}\right)$ of the
Jacobian matrix at each point $\left(y_{1},y_{2}\right)$ the stability of the
fixed points can be determined [28, 35]. The fixed point is stable when both
eigenvalues are real and negative, and unstable when both are real and
positive. The fixed point is a saddle point when both eigenvalues are real and
of opposite sign. The nature of the fixed point for different eigenvalues
$\left(\lambda_{1},\lambda_{2}\right)$ is summarized in Tab. 1.

Eigenvalues | Fixed point
---|---
$\lambda_{1}\neq\lambda_{2}<0$ | Stable
$\lambda_{1}\neq\lambda_{2}>0$ | Unstable
$\lambda_{1}<0<\lambda_{2}$ | Saddle
Table 1: Fixed point type based on eigenvalue pairs
$\left(\lambda_{1},\lambda_{2}\right)$.

The values $\left(y_{1},y_{2}\right)$ for each corresponding fixed point are
then substituted into the effective equation of state $w_{eff}$ given by
[28]444As a cross-check of our methodology and computer code we verified that
we are able to successfully reproduce the cosmological dynamics found in Ref.
[28] for two different models.

$w_{eff}=-y_{1}+\frac{1}{3}y_{2}+\frac{\dot{f}^{\prime}\left(\mathcal{R}\right)}{3Hf^{\prime}\left(\mathcal{R}\right)}+\frac{\dot{\xi}}{3H\xi}-\frac{\dot{f}^{\prime}\left(\mathcal{R}\right)\mathcal{R}}{18f^{\prime}\left(\mathcal{R}\right)\xi
H^{3}},$ (26)

where $\dot{\xi}$ is determined by taking the derivative of Eq. (18) with
respect to time and using Eq. (16). $\dot{f}^{\prime}\left(\mathcal{R}\right)$
is given by [28]

$\dot{f}^{\prime}\left(\mathcal{R}\right)=-\frac{3H\left(f^{\prime}\left(\mathcal{R}\right)\mathcal{R}-2f\left(\mathcal{R}\right)\right)f^{\prime\prime}\left(\mathcal{R}\right)}{f^{\prime\prime}\left(\mathcal{R}\right)\mathcal{R}-f^{\prime}\left(\mathcal{R}\right)}=\mathcal{\dot{R}}f^{\prime\prime}\left(\mathcal{R}\right).$
(27)

It will also prove useful to define the deceleration parameter $q$ in terms of
the effective equation of state $w_{eff}$. Since the deceleration parameter is
defined in terms of the Hubble parameter via

$q\equiv-\left(\frac{\dot{H}}{H^{2}}+1\right),$ (28)

and since [28]

$\frac{\dot{H}}{H^{2}}=-\frac{3}{2}\left(1+w_{eff}\right),$ (29)

we then find

$q=\frac{1}{2}\left(1+3w_{eff}\right).$ (30)

To further evaluate the viability criteria set out in the introduction, we
must also test whether our theory contains scalar curvature singularities
[18]. A local rescaling of the metric tensor by a conformal factor
$\Omega^{2}(x)$ is equivalent to the transformations [36, 37, 38]

$g_{\mu\nu}\rightarrow\tilde{g}_{\mu\nu}=f^{\prime}(\mathcal{R})g_{\mu\nu},\qquad
g^{\mu\nu}\rightarrow\tilde{g}^{\mu\nu}=\left(f^{\prime}(\mathcal{R})\right)^{-1}g^{\mu\nu}.$
(31)

The Ricci scalar $\mathcal{R}$ defines the simplest possible curvature
invariant. Thus, in the Palatini formalism, $\mathcal{R}$ raised to the power
of any positive integer $m$ transforms under (31) via

$\mathcal{R}^{m}\to\frac{\mathcal{R}^{m}}{\left(f^{\prime}\left(\mathcal{R}\right)\right)^{m}}.$
(32)

The next simplest curvature invariant involves the Ricci tensor. Since our
model is defined in the Palatini variation, the connection
$\Gamma^{\nu}_{\mu\sigma}$ is not assumed to depend on the metric
$g_{\mu\nu}$, and so the Ricci tensor

$R_{\mu\nu}=\partial_{\rho}\Gamma^{\rho}_{\nu\mu}-\partial_{\nu}\Gamma^{\rho}_{\rho\mu}+\Gamma^{\rho}_{\rho\lambda}\Gamma^{\lambda}_{\nu\mu}-\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\rho\mu}$
(33)

may remain invariant under the local rescaling transformation of Eq. (31). The
Ricci tensor with upper indices, however, is given by
$R^{\mu\nu}=g^{\mu\rho}g^{\nu\sigma}R_{\rho\sigma}$ and so it does transform
under Eq. (31) according to $R^{\mu\nu}\to R^{\mu\nu}\
\left(f^{\prime}(\mathcal{R})\right)^{-2}$. Therefore, second order curvature
invariants involving the Ricci tensor, namely $R_{\mu\nu}R^{\mu\nu}$, to any
integer power $m$, will transform under Eq. (31) according to

$\left(R_{\mu\nu}R^{\mu\nu}\right)^{m}\to\frac{\left(R_{\mu\nu}R^{\mu\nu}\right)^{m}}{\left(f^{\prime}(\mathcal{R})\right)^{2m}}.$
(34)

It is unclear whether the Kretchmann scalar is a scalar in the Palatini
formalism [39], and so we omit this from our analysis.

##  4 Results

We find that the model defined by the exponent of Eq. (12) contains three
fixed points $P_{1}$, $P_{2}$ and $P_{3}$. The eigenvalues and stability of
these fixed points, defined by the roots $\left(y_{1},y_{2}\right)$ of Eqs.
(22) and (23), are displayed in Tab. 2 in the low and high-curvature limits.
Figure 2 displays how the eigenvalues $\left(\lambda_{1},\lambda_{2}\right)$
vary as a function of $\mathcal{R}_{*}$ for $P_{1}$ (left), $P_{2}$ (middle),
and $P_{3}$ (right).

Fixed point | $\left(y_{1},y_{2}\right)$ |  $\left(\lambda_{1},\lambda_{2}\right)$ $\left(\mathcal{R}_{*}\to 0\right)$ |  $\left(\lambda_{1},\lambda_{2}\right)$ $\left(\mathcal{R}_{*}\to 1\right)$
---|---|---|---
$P_{1}$ | $\left(1,0\right)$ |  $\left(6,5\right)$ Unstable |  $\left(-4,-3\right)$ Stable
$P_{2}$ | $\left(0,1\right)$ |  $\left(1,-5\right)$ Saddle |  $\left(1,4\right)$ Unstable
$P_{3}$ | $\left(0,0\right)$ |  $\left(-1,-6\right)$ Stable |  $\left(-1,3\right)$ Saddle
Table 2: The dimensionless variables $\left(y_{1},y_{2}\right)$, eigenvalues
$\left(\lambda_{1},\lambda_{2}\right)$ in the low $\left(\mathcal{R}_{*}\to
0\right)$ and high-curvature $\left(\mathcal{R}_{*}\to 1\right)$ limits, and
stability of the three fixed points $P_{1}$, $P_{2}$ and $P_{3}$.

Figure 2: The eigenvalues $\left(\lambda_{1},\lambda_{2}\right)$ as a function
of $\mathcal{R}_{*}$ for fixed points $P_{1}$ (left), $P_{2}$ (middle), and
$P_{3}$ (right).

Figure 2 illustrates a potential advantage of AWIG. Unlike most other $f(R)$
theories of gravity, the variable power in the Lagrangian density of AWIG
makes it possible for the eigenvalues and hence stability of each fixed point
to vary with curvature scale, and hence to potentially vary with cosmological
time. So, for example, the stability of the fixed point $P_{1}$ can change
from being stable in the high-curvature limit to being unstable at lower
curvatures, as can be seen in Fig. 2 (left). This feature allows a richer set
of possible cosmological dynamics.

Inserting the obtained coordinate pairs $\left(y_{1},y_{2}\right)$ into Eq.
(26) yields the effective equation of state $w_{eff}$ as a function of the
Palatini scalar curvature. The results are displayed in Fig. 3 for the fixed
points $P_{1}$ and $P_{3}$. Since $y_{2}=1$ for the fixed point $P_{2}$ we can
see from Eq. (21) that $\rho_{r}$ is undefined, and therefore via Eq. (17) $H$
must also be undefined. Consequently, $w_{eff}$ for $P_{2}$ is undefined, as
is evident from Eq. (26). However, we know that as
$n\left(\mathcal{R}_{*}\right)\to 2$ AWIG is equivalent to general relativity
plus a cosmological constant [17]. Thus, $w_{eff}$ for $P_{2}$ can be
determined in the high-curvature limit by an equivalent analysis of the model
$f\left(\mathcal{R}\right)=\mathcal{R}-\Lambda$, where $\Lambda$ is the
cosmological constant. We have repeated the methodology outlined in section 3
for the model $f\left(\mathcal{R}\right)=\mathcal{R}-\Lambda$ finding
eigenvalues $\left(\lambda_{1},\lambda_{2}\right)=\left(1,4\right)$, which
agrees with our result presented in Fig. 2 (middle) in the high-curvature
limit, and an effective equation of state $w_{eff}=1/3$. Identical results are
also found in Ref. [28]. Therefore, $P_{2}$ corresponds to a radiation-like
phase in the high-curvature limit.

Figure 3: The effective equation of state parameter $w_{eff}$ as a function of
$\mathcal{R}_{*}$ for the fixed point $P_{1}$ (left) and $P_{3}$ (right).

The effective equation of state parameter $w_{eff}$ for the fixed points
$P_{1}$, $P_{2}$ and $P_{3}$ in the low and high-curvature limits are
summarised in Tab.3. Thus, we identify $P_{1}$ as a de Sitter-like phase,
$P_{2}$ as a radiation-like phase, and $P_{3}$ as a matter-like phase. Note
that the unknown value of $w_{eff}$ for $P_{2}$ in the limit
$\mathcal{R}_{*}\to 0$ is denoted by $-$. Figures 2 and 3 suggest that if the
matter-dominated phase $P_{3}$ is to transition back to the de Sitter-like
phase $P_{1}$, to account for the currently observed late period of cosmic
acceleration, then this transition must occur at a curvature scale
$\mathcal{R}_{*}\gtrsim 0.28$. This is because if $\mathcal{R}_{*}\lesssim
0.28$ then it is not possible to exit the stable matter-like phase.

Fixed point | $w_{eff}\left(\mathcal{R_{*}}\to 0\right)$ | $w_{eff}\left(\mathcal{R_{*}}\to 1\right)$ | Phase
---|---|---|---
$P_{1}$ | -1 | -1 | De Sitter
$P_{2}$ | - | 1/3 | Radiation
$P_{3}$ | 0 | 0 | Matter
Table 3: The effective equation of state in the low-curvature limit
$w_{eff}\left(\mathcal{R}_{*}\to 0\right)$, high-curvature limit
$w_{eff}\left(\mathcal{R}_{*}\to 1\right)$ and the phase type for the fixed
points $P_{1}$, $P_{2}$ and $P_{3}$.

To further analyse the cosmological evolution of our model we use Eq. (30) to
investigate how the deceleration parameter $q$ varies as a function of
$\mathcal{R}_{*}$ for the de Sitter-like phase. The results are shown in Fig.
4. If $q>0$ then the universe is expanding but decelerating. If $q<0$ then the
universe is expanding but accelerating [40]. Figure 4, therefore, indicates
that the de Sitter-like phase undergoes two periods of accelerated expansion,
one in the high-curvature regime $0.4\lesssim\mathcal{R}<1$ and one in the
low-curvature regime $0\leq\mathcal{R}\lesssim 0.23$, mediated by a period of
decelerated expansion for $0.23\lesssim\mathcal{R}\lesssim 0.4$ (see Fig. 4).
Assuming curvature on cosmological scales decreases with cosmological time,
this implies an early and late period of accelerated cosmic expansion, with an
intermediate period of decelerated expansion. In this sense, the dynamics
appear consistent with cosmological observations, depending on the exact scale
set by $\mathcal{R}_{0}$.

Figure 4: The deceleration parameter $q$ as a function of scalar curvature
$\mathcal{R}_{*}$ for the fixed point $P_{1}$.

We now analyse the phase space of this model, with the results shown in Fig.
5. The phase space of AWIG is 3-dimensional, with each point in the phase
space uniquely specified by the set of coordinates
$\left(y_{1},y_{2},\mathcal{R}_{*}\right)$. Figure 5 shows the
$\left(y_{1},y_{2}\right)$ plane for three different values of constant
curvature. One possible route the system may take through the 3-dimensional
phase space is depicted in the three plots of Fig. 5, where the system evolves
through the closed sequence of fixed points $P_{1}\to P_{2}\to P_{3}\to P_{1}$
with decreasing curvature scale $\mathcal{R}_{*}$. Thus, the model presented
is consistent with the sequence of an early period of accelerated expansion,
intermediate radiation and matter-dominated eras of decelerated expansion,
followed by the return to a period of accelerated expansion at late times.

Figure 5: Slices of constant curvature through the 3-dimensional phase space
of AWIG at $\mathcal{R}_{*}=0.5036$ (left), $\mathcal{R}_{*}=0.35$ (middle)
and $\mathcal{R}_{*}=0.3$ (right). The red trajectory shows one possible way
the system may evolve through the sequence of fixed points $P_{1}\to P_{2}\to
P_{3}\to P_{1}$.

We now present results for various powers of the Ricci scalar curvature under
the local rescaling of Eq. (31). Using Eqs. (4) and (32) we find

$\mathcal{R}^{m}\to\frac{\mathcal{R}^{m}}{\left(f^{\prime}\left(\mathcal{R}\right)\right)^{m}}\underset{\mathcal{R}_{*}\to
1}{=}\frac{1}{2^{m}}.$ (35)

The first three powers of the Ricci scalar curvature ($m=1,2,3$) are shown in
Fig. 6. As can be seen from Fig. 6 each curvature invariant is divergence-free
and approaches a constant in the limit $\mathcal{R}_{*}\to 1$. Similar results
have been shown in Refs. [41, 42]. Likewise, Eqs. (4) and (34) can be used to
show that the curvature invariant $\left(R_{\mu\nu}R^{\mu\nu}\right)^{m}$
formed from the Ricci tensor asymptotically approaches $1/2^{2m}$ as
$\mathcal{R}_{*}\to 1$. Therefore, the model presented contains no curvature
singularities in $\mathcal{R}$ or $R_{\mu\nu}R^{\mu\nu}$, at any order $m$.

Figure 6: The first three powers ($m=1,2,3$) of the transformed Palatini
scalar curvature as a function of $\mathcal{R}_{*}$.

##  5 Discussion

In this work we have shown that one of the simplest possible implementations
of asymptotically Weyl-invariant gravity (AWIG) may be viable, as measured
against criteria (i)$-$(vi) set out in the introduction.

However, the model’s viability cannot yet be definitively established for
several reasons. Firstly, AWIG is by construction superficially
renormalizable, but establishing its renormalizability via explicit
calculation remains an open problem. Secondly, the analysis performed in this
work has raised some unanswered questions. For example, the transition from
the matter-dominated phase to the late phase of cosmic expansion must occur at
a curvature scale $\mathcal{R}\gtrsim 0.28\mathcal{R}_{0}$. It is unknown
whether this is consistent with cosmological observations since the
dimensionful scale $\mathcal{R}_{0}$ is presently unknown. Furthermore, the
effective equation of state parameter $w_{eff}$ for the fixed point $P_{3}$ is
negative for $0<\mathcal{R}_{*}\lesssim 0.2$, the meaning of which is unclear.
Finally, one of the three fixed points $P_{2}$ has an undefined effective
equation of state for $0\leq\mathcal{R}_{*}<1$, however, we can determine
$w_{eff}$ for $\mathcal{R}_{*}\to 1$.

Nevertheless, the model presented contains several encouraging features, such
as the apparent absence of curvature singularities and three fixed points with
effective equation of states corresponding to de Sitter, radiation and matter-
like phases. The model also contains the correct sequence of early and late
periods of accelerated cosmic expansion, with an intermediate period of
decelerated expansion, something that has proven difficult to achieve in other
attempted modifications of general relativity [28]. Moreover, the early
accelerating phase emerges from AWIG without adding a scalar field. This is
because AWIG asymptotically approaches the Palatini formulation of pure
$\mathcal{R}^{2}$ gravity in the high curvature limit, which is equivalent to
general relativity plus a non-zero cosmological constant and no massless
scalar field [17]. Another positive feature of AWIG is that the variable power
in the Lagrangian density seems to permit a richer set of possible
cosmological dynamics, as can be seen from the variable eigenvalues in Fig.
(2).

The dimensionless exponent $n\left(\mathcal{R}_{*}\right)$ explored in this
work is among the simplest possible choices, but it is far from the only
consistent choice. Going forward, we aim to narrow down, or perhaps even
uniquely determine, the functional form of $n\left(\mathcal{R}_{*}\right)$ and
the value of $\mathcal{R}_{0}$ to more robustly test the viability of
asymptotically Weyl-invariant gravity.

##  6 Acknowledgements

I wish to thank Roberto Percacci and Subir Sarkar for their comments on the
manuscript, and the anonymous referee for invaluable corrections.

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