# An emergent cosmological model from running Newton constant

Aknur Zholdasbek<EMAIL_ADDRESS>Department of Physics, Nazarbayev
University, Kabanbay Batyr 53, 010000 Astana, Kazakhstan  Hrishikesh
Chakrabarty<EMAIL_ADDRESS>Department of Physics,
Nazarbayev University, Kabanbay Batyr 53, 010000 Astana, Kazakhstan  Daniele
Malafarina<EMAIL_ADDRESS>Department of Physics, Nazarbayev
University, Kabanbay Batyr 53, 010000 Astana, Kazakhstan  Alfio Bonanno
<EMAIL_ADDRESS>INAF, Osservatorio Astrofisico di Catania, via S.Sofia
78, I-95123 Catania, Italy

###### Abstract

We propose an emergent cosmological model rooted in the Asymptotically Safe
antiscreening behavior of the Newton constant at Planckian energies.
Distinguishing itself from prior approaches, our model encapsulates the
variable nature of $G$ through a multiplicative coupling within the matter
Lagrangian, characterized by a conserved energy-momentum tensor. The universe
emerges from a quasi-de Sitter phase, transitioning to standard cosmological
evolution post-Planck Era. Our analysis demonstrates the feasibility of
constraining the transition scale to nearly classical cosmology using Cosmic
Microwave Background (CMB) data and the potential to empirically probe the
antiscreening trait of Newton’s constant, as predicted by Asymptotic Safety.

##  I Introduction

For decades, the standard model of cosmology has been very successful in
describing the observable universe in accordance with experimental
observations. We are currently living in an era of precision cosmology in
which observations of the cosmic microwave background (CMB) Hinshaw et al.
(2013); Akrami et al. (2020); Aghanim et al. (2020a, b, c), large scale
structure (LSS) Tegmark et al. (2004); Seljak et al. (2005), type-IA
supernovae (SNeIa) Perlmutter et al. (1999); Riess et al. (1998), baryon
acoustic oscillation (BAO) Eisenstein et al. (2005), weak lensing Jain and
Taylor (2003), gravitational waves (GW) Abbott et al. (2017) and more have
allowed us to refine our picture of how the universe evolved by formulating,
analyzing and potentially discarding theoretical models.

However, it is becoming increasingly clear that the picture is not complete
and there are still important open problems and issues that need to be
addressed. Among these the idea of inflation, although extremely successful,
has recently been put under scrutiny. According to inflation, the universe is
believed to have gone through a period of accelerated expansion at very early
times Guth (1981); Linde (1982); Starobinsky (1980). This idea was proposed to
address two problems, known as the horizon and flatness problems, that plagued
the early theory. Besides solving two major problems, the success of the
inflationary paradigm relies on modern observations that show how the initial
conditions for the hot big bang can be produced during this period of
inflation. The main breakthrough came when early CMB observations Hinshaw et
al. (2013) detected a nearly scale-invariant power spectrum which can be
perfectly explained by inflation. Later observations from Planck made
constraints on the model very strong, thus allowing to discard inflationary
models that do not fit the observations. The latest Planck data favors single-
field scalar models of inflation with concave potential coupled to General
Relativity (GR). The data also prefers no significant departure from slow-roll
behavior Akrami et al. (2020). Still, the nature of the inflationary scalar
field is not known at present. Therefore other models for inflation and the
behavior of the universe in the early stages have been proposed over the
years. Some of these models are able to address the same issue and produce a
scale-invariant CMB power spectrum. For example, bouncing scenarios and
ekpyrotic models have been suggested as possible alternatives to inflation Cai
et al. (2010); Battefeld and Peter (2015); Lyth and Wands (2002);
Brandenberger and Peter (2017). This shows that there is room for other viable
models of the universe to be proposed and there are stringent quantitative
tests that can be made to constrain such proposals.

In the present article, we propose a minimal modification to GR at UV scales
inspired by the Asymptotic Safety (AS) paradigm Bonanno (2023) and implemented
through a multiplicative coupling of matter to gravity as proposed in Markov
and Mukhanov (1985). This proposal provides a relation between a variable
Newton coupling $G$ and variable $\Lambda$ and does not involve any additional
unknown fields. Hence we show that the quasi-de Sitter phase in the early
universe can be obtained as a result of the Asymptotically Safe nature of
gravity at high energies. We show that this new mechanism achieves the same
results as scalar field inflation and we provide arguments in support of the
fact that it can also produce a scale-invariant CMB power spectrum.

The current proposal diverges from prior investigations into Renormalization
Group (RG) improved cosmologies Bonanno and Reuter (2002a, b); Cai and Easson
(2011) by adopting a Lagrangian approach, thus preserving diffeomorphism
invariance. While it shares conceptual similarities with the framework
introduced in Bonanno (2012), and subsequently elaborated upon in various
studies (cf. Bonanno and Saueressig (2017) for an overview), it avoids
imposing a cutoff identification based on curvature invariants Bonanno et al.
(2018). Recent findings, notably those derived from the fluctuation approach
Pawlowski and Reichert (2021); Bonanno et al. (2022), illustrate that the
behavior of the Newton constant during high-energy scattering processes is
predominantly governed by the external graviton momenta which should scale in
accordance with the energy density of the system. A promising initial result
stemming from this novel perspective is evidenced in the regular black hole
collapse model discussed in Bonanno et al. (2023).

The article is organized in the following way: In section II we briefly
discuss AS gravity with the non trivial matter-gravity coupling and apply the
formalism to a Friedmann universe in section III. Implications of the model
for the early universe and the correspondence between the proposed model and
scalar field inflation are discussed in section IV. Finally results and future
perspectives are summarized in section V. Throughout the article, we make use
of units with $c=1$ (unless explicitly stated) and metric signature
$(-,+,+,+)$.

##  II Field equations and energy momentum

Figure 1: Left: Behavior of the effective energy density $\tilde{\varepsilon}$
for dust ($w=0$), radiation ($w=1/3$) and stiff fluid ($w=1$) in AS with MM
coupling as a function of the scale factor $a$ of the universe, with the
current epoch given by $a(t_{0})=1$ and energy density
$\varepsilon(t_{0})=\varepsilon_{\rm 0}$. Here we take
$\varepsilon_{c}=10^{-5}$, $\varepsilon_{\rm 0,dust}=0.3$, $\varepsilon_{\rm
0,rad}=10^{-3}$ and $\varepsilon_{\rm 0,stiff}=10^{-10}$ for illustrative
purposes. Right: Behavior of the scale factor $a$ with respect to $t/t_{0}$,
where $t_{0}=13.8\ {\rm Gyr}$ is the age of the universe today, in the
$\Lambda$-cdm model (solid line) as compared to the AS with MM coupling model
(dashed line). For this plot we use $\varepsilon_{k0}/\varepsilon_{c}=10^{-2}$
for illustrative purposes and set the fiducial cosmological parameters from
Aghanim et al. (2020b).

We aim to build a cosmological model based on three main assumptions regarding
the fundamental behavior of gravity at high energies:

  * (i)

There exists a non-minimal matter-gravity coupling that depends on the energy
scale as suggested by Markov and Mukhanov (MM) in Markov and Mukhanov (1985).

  * (ii)

The precise form of $G$ as a function of the energy density $\varepsilon$ can
be obtained from the running of $G$ in Asymptotic Safety under a physically
motivated cut-off identification.

  * (iii)

The running of $G$ alone captures the qualitative physical features of the
model, i.e. we do not explicitly include possible effects from the running of
higher dimensional operators, cosmological constant, or matter fields.

We do not wish to add any additional matter fields besides a standard one such
as dust, radiation or a stiff fluid and the accelerated expansion phase in the
early universe, namely inflation, shall be recovered from the matter-gravity
coupling. Of course, at low energies we must retrieve the standard field
equations of GR. With the above assumptions the field equations can be derived
in the MM prescription from the following action

$S=\int d^{4}x\sqrt{-g}\left(\frac{R}{8\pi
G_{N}}+2\chi(\varepsilon)\mathcal{L}\right),$ (1)

where $g$ is the determinant of the spacetime metric $g_{\mu\nu}$,
$\mathcal{L}$ is the matter Lagrangian and the matter-gravity coupling depends
only on a scalar function $\chi(\varepsilon)$ of the matter-energy density
$\varepsilon$ Markov and Mukhanov (1985). Variation of the matter part of the
action, with respect to the metric leads to

$\frac{1}{\sqrt{-g}}\delta(2\sqrt{-g}\chi\varepsilon)=2(\chi\varepsilon),_{\varepsilon}\delta\varepsilon-\chi\varepsilon
g_{\mu\nu}\delta g^{\mu\nu},$ (2)

and the variation of the full action leads to the modified Einstein equations

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G_{N}\tilde{T}_{\mu\nu},$ (3)

where

$\tilde{T}_{\mu\nu}=\left(\varepsilon\chi\right)_{,\varepsilon}T_{\mu\nu}+(\varepsilon^{2}\chi_{,\varepsilon})g_{\mu\nu},$
(4)

is the effective energy-momentum tensor while

$T_{\mu\nu}=\left(\varepsilon+P(\varepsilon)\right)u_{\mu}u_{\nu}+P(\varepsilon)g_{\mu\nu},$
(5)

is the energy-momentum tensor for a perfect fluid, where $P$ is the fluid’s
pressure which we assume may be obtained from an equation of state
$P=P(\varepsilon)$. Looking at Eq. (3) and(4) we can identify the terms

$G(\varepsilon)=G_{N}(\chi\varepsilon)_{,\varepsilon}\;\;\text{and}\;\;\Lambda(\varepsilon)=-8\pi
G_{N}\varepsilon^{2}\chi_{,\varepsilon}$ (6)

as a running gravitational constant $G(\varepsilon)$ and a running
cosmological constant $\Lambda(\varepsilon)$. In order to retrieve GR at low
densities, for $\chi(\varepsilon)$ we must ensure that as
$\varepsilon\rightarrow 0$ we get $G(\varepsilon)\rightarrow G_{N}$, which
implies $\chi\rightarrow 1$.

Consider a matter fluid of proper density $\varepsilon$, 4-velocity $u^{\mu}$
with $u^{\mu}u_{\mu}=-1$ and rest mass density $\rho$. Mass continuity implies

$(\rho u^{\mu})_{;\mu}=0,$ (7)

and for a non-dissipative fluid, we have

$\frac{\delta\rho}{\rho}=\frac{\delta\varepsilon}{P(\varepsilon)+\varepsilon}.$
(8)

The effective stress-energy tensor needs to be conserved, i.e.

$\nabla_{\mu}\tilde{T}^{\mu}_{\nu}=0.$ (9)

To see how the effective energy density evolves, we can project it along the
four-velocity field $u^{\mu}$ which gives

$\nabla_{\mu}\left(\tilde{\varepsilon}u^{\mu}\right)+\tilde{P}\nabla_{\mu}u^{\mu}=0,$
(10)

where the quantities with tilde are the effective fluid’s energy-density and
pressure that can be obtained from Eq. (4) as

$\displaystyle\tilde{\varepsilon}$ $\displaystyle=$
$\displaystyle(\chi\varepsilon)_{,\varepsilon}\varepsilon-\varepsilon^{2}\chi_{,\varepsilon}=\chi(\varepsilon)\varepsilon,$
(11) $\displaystyle\tilde{P}$ $\displaystyle=$
$\displaystyle(\chi\varepsilon)_{,\varepsilon}P+\varepsilon^{2}\chi_{,\varepsilon}.$
(12)

Now we can examine the weak and strong energy conditions for the effective
fluid. The weak energy condition (w.e.c.) for the perfect fluid in (5) holds
if $\varepsilon+P\geq 0$ and $\varepsilon\geq 0$. Then for the effective
quantities the w.e.c. is

$\tilde{\varepsilon}+\tilde{P}>0,\ \ \ \tilde{\epsilon}\geq 0,$ (13)

which can be written as

$(\varepsilon\chi)_{,\varepsilon}\left(\varepsilon+P\right)>0,\ \ \
\varepsilon\chi\geq 0.$ (14)

Notice that the behavior of $\chi$ determines the validity of the w.e.c. for
the effective fluid. In particular if the w.e.c. holds for the fluid and
$\chi>0$ we always have that $\tilde{\varepsilon}\geq 0$, while
$\tilde{\varepsilon}+\tilde{P}$ may become negative if
$(\varepsilon\chi)_{,\varepsilon}<0$. A similar reasoning holds for the strong
energy condition (s.e.c.) which for the perfect fluid in (5) corresponds to
$\varepsilon+\sum_{i}P_{i}\geq 0$ and $\varepsilon+P_{i}\geq 0$ (with
$i=1,2,3$ being the spatial directions). The s.e.c. for the effective fluid is
then given as

$\tilde{\varepsilon}+3\tilde{P}\geq 0,\ \ \ \tilde{\varepsilon}+\tilde{P}\geq
0,$ (15)

which can be written in terms of the classical fluid components as

$(\varepsilon\chi)_{,\varepsilon}\left(\varepsilon+3P\right)+2{\varepsilon}^{2}\chi_{,\varepsilon}\geq
0,\ \ \ (\varepsilon\chi)_{,\varepsilon}\left(\varepsilon+P\right)\geq 0.$
(16)

Notice that the condition $\tilde{\varepsilon}+3\tilde{P}\geq 0$ must be
violated in order to have a phase of accelerated expansion. Then the first of
Eq. (16) provides a condition on $\chi$ for which the s.e.c. holds for the
classical fluid but is violated for the effective fluid.

The above framework provides a scenario for a variable cosmological constant
that naturally emerges from the matter-gravity coupling. The only available
freedom is in the choice of one of the two functions $\chi(\varepsilon)$ and
$G(\varepsilon)$. In the following, we aim to build a model for the early
universe for which the gravitational coupling $G(\varepsilon)$ is prescribed
from the Asymptotic Safety (AS) program and therefore we will fix the function
$G(\varepsilon)$ from the AS paradigm and obtain $\chi(\varepsilon)$ and
$\Lambda(\varepsilon)$ accordingly.

If gravity is renormalizable around a non-Gaussian fixed point, one would
expect that $G\sim 1/\kappa^{2}$ at very high energies, with $\kappa$ being
the characteristic scale of energy at which the physics is being probed Reuter
(1998); Weinberg (1980); Eichhorn (2019); Pawlowski and Reichert (2021);
Reichert (2020); Reuter and Saueressig (2019); Percacci (2017); Bonanno et al.
(2020a); Bonanno et al. (2022); Bonanno et al. (2020b). Then the variable
Newton coupling $G(\kappa)$ can be taken as a function of this characteristic
energy scale as Bonanno et al. (2022)

$G(\kappa)=\frac{G_{N}}{1+\kappa^{2}/g_{*}},$ (17)

where the scale $\kappa$ is model dependent while $g_{*}=540\pi/833$ is the UV
fixed point. To connect the cutoff scale $\kappa$ and the energy density
$\varepsilon$, we follow the prescription discussed in Bonanno et al. (2020a);
Platania (2019); Bonanno et al. (2023). Thus we consider the simplest relation
between $\kappa$ and $\varepsilon$ involving both Newton’s and Planck
constants which can be obtained from dimensional analysis. This is

$\kappa^{2}=\frac{\hbar G_{N}}{c^{4}}\varepsilon,$ (18)

from which we obtain the following form for $G(\varepsilon)$:

$G(\varepsilon)=\frac{G_{N}}{1+\frac{\varepsilon}{\varepsilon_{c}}},$ (19)

where $\varepsilon_{c}$ is a dimensional parameter in which we absorb $g_{*}$
and the other constants. The information on the cutoff scale $\kappa$ is now
encoded in $\varepsilon_{c}$ which we can interpret as a crossover scale
separating the Reuter fixed point from the Gaussian fixed point: when
$\epsilon/\epsilon_{c}\gg 1$ the dynamics of the system is essentially
determined by the vanishing of the effective coupling constant. In the
opposite limit, standard gravity is recovered. From Eq. (19) we obtain the
expressions for $\chi(\varepsilon)$ and $\Lambda(\varepsilon)$ Bonanno et al.
(2023):

$\displaystyle\chi(\varepsilon)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{c}}{\varepsilon}\log\left(1+\frac{\varepsilon}{\varepsilon_{c}}\right),$
(20) $\displaystyle\frac{\Lambda(\varepsilon)}{8\pi G_{N}}$ $\displaystyle=$
$\displaystyle\varepsilon_{c}\log\left(1+\frac{\varepsilon}{\varepsilon_{c}}\right)-\frac{\varepsilon}{1+\frac{\varepsilon}{\varepsilon_{c}}}.$
(21)

Notice that for $\varepsilon\rightarrow 0$ we get $\chi\rightarrow 1$ and
$\Lambda\rightarrow 0$ thus retrieving the classical GR limit. However for
large $\varepsilon$ we see that $\chi\rightarrow 0$ while $\Lambda$ diverges.
Looking at Eq. (20) we can also see that as $\varepsilon$ diverges
$\tilde{\varepsilon}=\chi\varepsilon$ diverges as well but more slowly.

##  III Cosmology

The evolution of the universe is governed by the Friedmann equations. We start
by considering a Friedmann-Robertson-Walker (FRW) universe with metric

$ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(22)

where $a(t)$ is the scale factor, $k$ is the curvature and $d\Omega^{2}$ is
the line element on the unit sphere. The Friedmann equations for the AS model
with MM prescription are then obtained from the metric (22) with the effective
energy-momentum tensor (4) as

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G_{N}}{3}\varepsilon\chi-\frac{k}{a^{2}}=\frac{8\pi
G_{N}}{3}\tilde{\varepsilon}-\frac{k}{a^{2}},$ (23)
$\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=$ $\displaystyle-\frac{4\pi
G_{N}}{3}\left[(\varepsilon+3P)\chi+3\varepsilon\frac{\partial\chi}{\partial\varepsilon}\left(\varepsilon+P\right)\right]=$
(24) $\displaystyle=$ $\displaystyle-\frac{4\pi
G_{N}}{3}(\tilde{\varepsilon}+3\tilde{P}),$

where $H=\dot{a}/a$ is the Hubble parameter. Then, the evolution of the
effective energy density can be easily obtained from Eq. (10) as

$\dot{\tilde{\varepsilon}}+3\frac{\dot{a}}{a}\left(\tilde{\varepsilon}+\tilde{P}\right)=\left(\varepsilon\chi\right)_{,\varepsilon}\left[\dot{\varepsilon}+3\frac{\dot{a}}{a}\left(\varepsilon+P\right)\right]=0.$
(25)

Since $\left(\varepsilon\chi\right)_{,\varepsilon}\neq 0$ at all times, from
the above equation we see that the effective energy momentum is conserved if
the original energy momentum is. Namely Eq. (10) holds if

$\dot{\varepsilon}+3\frac{\dot{a}}{a}\left(\varepsilon+P\right)=0.$ (26)

In the limit, $\varepsilon\rightarrow 0$ (and for $\chi=1$), both Friedmann
equations of our model reduce to the standard Friedmann equations of GR

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G_{N}}{3}\varepsilon-\frac{k}{a^{2}},$ (27) $\displaystyle\frac{\ddot{a}}{a}$
$\displaystyle=$ $\displaystyle-\frac{4\pi G_{N}}{3}(\varepsilon+3P).$ (28)

On the other hand, at high energies, i.e. for $\varepsilon\rightarrow\infty$,
the function $\chi(\varepsilon)$ goes to zero in such a way that we obtain a
de Sitter-like initial phase for the early universe.

In the following, we will consider a matter content given by a perfect fluid
described by a linear barotropic equation of state (e.o.s.) $P=w\varepsilon$,
and we will pay particular attention to the cases of non relativistic matter
(including both baryonic and dark matter) for which $w=0$, radiation for which
$w=1/3$ and a stiff fluid for which $w=1$. The case for a stiff equation of
state in the early universe was originally proposed by Zel’dovich Zel’dovich
(1961); Zeldovich (1972) in order to describe a big bang scenario for a
universe filled with hot baryons. More recently the same idea has been
reevaluated in the context of a universe with a polytropic ‘dark fluid’ where
the internal energy plays the role of a stiff fluid and dominates at early
times Chavanis (2015). Interestingly, in our model, due to the vanishing of
the gravitation coupling $G(\varepsilon)$, the qualitative behavior of the
cosmological model at high densities does not depend on the specific fluid
model, with the e.o.s. being important in the determination of the energy
scales at which the AS corrections become relevant.

Figure 2: Left: Dependence of condition for accelerated expansion
($\Upsilon<0$) on the energy density for the single fluid models of dust
($w=0$), radiation ($w=1/3$) and stiff fluid ($w=1$) in AS with MM coupling.
Right: Behavior of the effective equation of state $\tilde{w}$ for the three
single fluid models of dust, radiation and stiff fluid in AS with MM coupling
as a function of $\varepsilon/\varepsilon_{c}$.

We can now take the universe age today as $t_{0}$ and set $a(t_{0})=1$. In the
left panel of Fig. 1, we show the effective energy density
$\tilde{\varepsilon}_{s}=\varepsilon_{s}\chi$ for a single fluid (denoted by
the subscript $s$) in the three cases mentioned above (i.e. $w_{s}=0,1/3,1$)
as a function of the scale factor $a(t)$. The classical energy density
$\varepsilon_{s}$ is obtained from integration of the conservation equation as
$\varepsilon_{s}\sim 1/a^{3(1+w_{s})}$ and $\chi$ is given by Eq. (20). Then
for each fluid component of the universe, we define
$\Omega_{s}=\varepsilon_{s}/\varepsilon_{k}$ with $\varepsilon_{k}$ being the
energy density for which the universe is flat, namely
$\varepsilon_{k}=3H^{2}/8\pi G_{N}$. Then in terms of the e.o.s. parameter
$w_{s}$ we can write the first Friedmann equation in the following way

$\displaystyle\frac{H^{2}}{H_{0}^{2}}$ $\displaystyle=$
$\displaystyle\chi\sum_{s}\Omega_{s}-\frac{k}{a^{2}H_{0}^{2}},$ (29)

with

$\displaystyle\Omega_{s}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{s0}}{a^{3(1+w_{s})}},$ (30) $\displaystyle\chi$
$\displaystyle=$
$\displaystyle\frac{\varepsilon_{c}}{\varepsilon_{k0}}\frac{\ln\left(1+\frac{\varepsilon_{k0}}{\varepsilon_{c}}\sum_{s}\Omega_{s}\right)}{\sum_{s}\Omega_{s}},$
(31)

where $\Omega_{s0}$ are the current values of the density parameters of the
respective fluids and $\varepsilon_{k0}=3H_{0}^{2}/8\pi G_{N}$ is the energy
density for the universe to be flat today. To compare the behavior of the
scale factor in the $\Lambda$CDM and AS models, we solve the Friedmann Eq.
(29) numerically assuming the fiducial cosmological parameters from Aghanim et
al. (2020b) to find $a(t)$ with the initial condition $a(t_{0})=1$. The
resulting scale factor is shown in the right panel of Figure 1 with a ratio of
$\varepsilon_{k0}/\varepsilon_{c}$ fixed arbitrarily at $10^{-2}$ for
illustrative purposes. We see that at late times the model follows the
standard $\Lambda$CDM behavior, while at early times the scale factor never
reaches zero and asymptotically decreases to $a\rightarrow 0$ with
$\dot{a}\rightarrow 0$, thus showing that the big bang singularity is avoided
and there is an infinite amount of time in the past for the universe to reach
thermal equilibrium.

We are interested in the behavior of $a$ at high energies, i.e.
$\varepsilon\rightarrow\infty$, with the aim of replacing inflation driven by
a scalar field with an exponential expansion driven by the variable $G$ in AS.
This can be accomplished since the model automatically leads to a quasi-de
Sitter phase for $t\rightarrow-\infty$ and no singularity is present at any
finite co-moving time. The condition for accelerated expansion $\ddot{a}>0$
from Eq. (24) implies $(\tilde{\varepsilon}+3\tilde{P})<0$ which, according to
Eq. (16), for a universe dominated by a single fluid with the equation of
state, $P=w\varepsilon$ becomes

$\Upsilon=(1+3w)\chi+3(1+w)\varepsilon\frac{d\chi}{d\varepsilon}<0.$ (32)

We plot this condition as a function of the energy density in the left panel
of Fig. 2 for dust, radiation and stiff fluid. We can clearly see from the
figure that $\Upsilon<0$ is satisfied as $\varepsilon\rightarrow\infty$ for
all fluid models. We also see that the qualitative behavior for large
densities is the same regardless of the fluid model and the e.o.s. plays a
role only in determining the energy scale at which $\Upsilon$ becomes
negative. To understand this better we can look at how the effective equation
of state relating $\tilde{\varepsilon}$ to $\tilde{P}$ behaves in the early
universe. The effective equation of state parameter $\tilde{w}$ is defined as

$\tilde{w}(\varepsilon)=\frac{\tilde{P}}{\tilde{\varepsilon}}=\frac{\left(\varepsilon\chi\right)_{,\varepsilon}P+\varepsilon^{2}\chi_{,\varepsilon}}{\varepsilon\chi}=w+(1+w)\frac{\varepsilon\chi_{,\varepsilon}}{\chi}.$
(33)

If $\varepsilon\chi_{,\varepsilon}/\chi\rightarrow-1$, as is the case for Eq.
(20), then for large $\varepsilon$ we see that the effective equation of state
behaves as

$\lim_{\varepsilon\to\infty}\tilde{w}\rightarrow-1,$ (34)

for any value of $w$. This proves that at high energies, i.e. in the very
early universe, the effective equation of state parameter $\tilde{w}$ shows a
limiting de Sitter behavior regardless of the value of $w$. We show the
parameter $\tilde{w}$ in the right panel of Fig. 2 for the cases of dust
($w=0$), radiation ($w=1/3$) and stiff fluid ($w=1$). From the figure, we can
also see that at low energies the fluids behave as ordinary perfect fluids.

Figure 3: The comoving Hubble radius as a function of the scale factor $a$. In
the plot, the solid line corresponds to the comoving Hubble radius of the
$\Lambda$CDM model. The dotted and dashed lines represent the comoving Hubble
radius in the AS model for different values of
$\varepsilon_{k0}/\varepsilon_{c}$ (see Eqs. (29)-(31)). All other parameters
are set from the fiducial values obtained in Aghanim et al. (2020b). Notice
that depending on the value of $\varepsilon_{c}$ there is an epoch in the
early universe, with corresponding scale factor $a_{c}$ at which $(aH)^{-1}$
transitions from increasing to decreasing.

To see how the cosmological model in AS solves the flatness problem, we can
use the Friedmann equations (23) and (24) to write

$\frac{d\tilde{\Omega}}{d\ln
a}=(1+3\tilde{w})(\tilde{\Omega}-1)\tilde{\Omega},$ (35)

where $\tilde{\Omega}=\chi\sum_{s}\Omega_{s}$ is the effective density
parameter. We can clearly see that $\tilde{\Omega}=1$ is a stable solution
since $1+3\tilde{w}<0$ as $\varepsilon\rightarrow\infty$ in AS. This can also
be seen easily by introducing small perturbations to the density parameter of
the form $\tilde{\Omega}=1\pm\delta(a)$ with $\delta\ll 1$. At the linear
order, Eq. (35) becomes

$\frac{d\delta(a)}{d\ln a}=(1+3\tilde{w})\delta(a).$ (36)

Solving this differential equation in a regime where $\tilde{w}\simeq{\rm
const}.$ yields the following result,

$\delta(a)=\delta_{i}\left(\frac{a}{a_{i}}\right)^{(1+3\tilde{w})}.$ (37)

Again, since $1+3\tilde{w}<0$ during the AS phase in the early universe, the
perturbations decay and the flat solution is stable.

To show that this model also solves the horizon problem we compare the
evolution of the comoving Hubble radius $(aH)^{-1}$ in the AS cosmological
model with that of the standard $\Lambda$CDM model. In the standard
$\Lambda$CDM model without an inflationary phase, the amount of conformal time
between the initial singularity and the formation of the CMB is much smaller
than the conformal age of the universe today. This leads to patches of angular
size $\theta\sim 2^{\circ}$ on the all-sky projection of the CMB to be
causally disconnected. However, the CMB appears to have a uniform temperature
map, thus suggesting that those same patches were in causal contact before the
emission of the CMB. Adding a phase of exponential expansion to the
$\Lambda$CDM model before the hot Big Bang solves the problem. To see this we
just need to notice that a decreasing comoving Hubble radius during such a
phase, corresponding to increasing the amount of conformal time in the early
universe, allows for the past light cones of two points on the CMB to be in
causal contact in their past (see Fig. 3).

In the AS model, this quasi-de Sitter phase is achieved automatically via the
coupling of $G$ to $\Lambda$. In Fig. 3 we show the evolution of the comoving
Hubble radius $(aH)^{-1}$ with the Hubble parameter given by the first
Friedmann equation (29). It can be clearly seen that in the early universe,
during the quasi-de Sitter phase, the comoving Hubble radius was decreasing.
The size of the universe at the time of the transition from the exponential
expansion to a universe dominated by matter (such as radiation as considered
in the figure) depends on the ratio between the current energy density
($\varepsilon_{k0}$) of the universe and the critical, model dependent,
density of AS, namely $\varepsilon_{c}$. The transition from radiation-
dominated to the matter-dominated epoch instead occurs around the
recombination era and it follows the standard $\Lambda$CDM behavior,
regardless of the value of $\varepsilon_{c}$ (as can be seen from the
inflexion of the solid line at the top right of Fig. 3). For comparison, in
the same figure, we also show the comoving Hubble radius for the standard
$\Lambda$CDM model.

##  IV Modelling the early universe

In order to compare the predictions of the AS model with those of a standard
inflationary scenario we need to look at the Hubble slow-roll parameters which
indicate the conditions for a sustained inflationary period in the early
universe. The Hubble slow-roll parameters are given by Baumann (2022); Lyth
and Riotto (1999)

$\epsilon\equiv-\frac{\dot{H}}{H^{2}}=-\frac{d\ln H}{dN},\ \ \ \ \
\eta\equiv\frac{|\dot{\epsilon}|}{\epsilon H}=\frac{d\ln\epsilon}{dN},$ (38)

where $dN=Hdt$ refers to the number of e-folds during inflation. The first
condition is $\epsilon<1$ and it states that the fractional change of the
Hubble parameter per e-fold has to be small. The second condition is $\eta<1$,
which on the other hand requires $\epsilon$ to remain small over a large
enough number of Hubble times Baumann (2022). For AS cosmology we get

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\frac{3(1+w)}{2}\frac{\left(\varepsilon\chi\right)_{,\varepsilon}}{\chi}=\frac{3}{2}(1+\tilde{w}),$
(39) $\displaystyle\eta$ $\displaystyle=$ $\displaystyle
3(1+w)\left|\frac{\varepsilon\chi}{\left(\varepsilon\chi\right)_{,\varepsilon}}\frac{d}{d\varepsilon}\left(\frac{\varepsilon}{\chi}\frac{d\chi}{d\epsilon}\right)\right|=$
$\displaystyle=$ $\displaystyle 3(1+\tilde{w})(1-\chi).$

The behaviors of $\epsilon$ and $\eta$ for an AS cosmological model with only
radiation are shown in Fig. 4. We can see from the plot that at high energy
densities, i.e. as we approach the very early universe, these Hubble slow-roll
parameters become smaller than unity which signifies a sustained period of
inflation. Notice that from Eqs. (39) and (IV) we see that for $a\rightarrow
0$, corresponding to $\tilde{w}\rightarrow-1$, we have that $\epsilon$ and
$\eta$ both go to zero.

Figure 4:  Behavior of the Hubble slow-roll parameters with respect to the
energy density. In both plots, increasing energy density signifies moving back
in time. We have used $P=\varepsilon/3$ in Eq. (39) and (IV) to plot this
figure.

Now we would like to obtain constraints on the critical density parameter
$\varepsilon_{c}$ by using observational bounds from the primordial power
spectrum generated by curvature perturbations during the inflationary period.
This analysis will not only enable us to estimate some realistic values of the
characteristic energy scale, but will also allow us to test the observational
viability of the AS program of quantum gravity.

The first step in this direction involves the calculation of the two most
important inflationary variables, e.g., the spectral index of scalar curvature
perturbations $n_{s}$ and the ratio of tensor-to-scalar perturbations $r$, in
terms of the model parameters Mukhanov (2005). These two quantities are
important because their values depend on the energy density scale at the onset
of inflation and they can be constrained from observations of the CMB power
spectrum. Usually, most inflationary models are constructed with one or more
scalar fields with some potentials and canonical or non-canonical kinetic
terms (see for example Lyth and Riotto (1999)). The inflationary observables
are then expressed approximately in terms of the slow-roll parameters which in
turn can be represented by the potential $V$ and its derivatives with respect
to the scalar field $\phi$.

However, our model of inflation in the AS framework is expressed in terms of
an effective fluid. It is known that there is a correspondence between scalar
field and fluid models Arroja and Sasaki (2010); Faraoni (2012); Mukhanov
(2013) and therefore in order to write the inflationary observables in terms
of our model parameters we shall consider the fluid description of inflation
following the procedure suggested in Bamba et al. (2014a, b); Bamba and
Odintsov (2016). To this aim in the following we will express the slow-roll
parameters $\epsilon$ and $\eta$ in terms of fluid parameters by relating them
to the Hubble parameter $H$ and its derivatives with respect to the number of
e-folds $N$ during inflation. Then $\epsilon$ and $\eta$ can be directly
related to the inflationary observables $n_{s}$ and $r$ from their definition
in the scalar field case. A very important caveat is in order at this point.
As we evaluate $n_{s}$ and $r$ from the analogy between fluids and scalar
fields we need to keep in mind that while the correspondence is valid at the
background level (namely for the FRW metric) the perturbations may turn out to
be different in the two cases. In a model of slow-roll inflation however we
may expect differences to remain small.

###  IV.1 Spectral index and tensor to scalar ratio for scalar field inflation

First, we briefly review how $n_{s}$ and $r$ are defined in terms of
$\epsilon$ and $\eta$ in a model of inflation driven by a scalar field $\phi$.
The action of the scalar field $\phi$ with the Einstein-Hilbert term is given
by

$S=\int d^{4}x\sqrt{-g}\left(\frac{R}{16\pi
G_{N}}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right).$ (41)

For slow-roll inflation the spectral index of curvature perturbations $n_{s}$
and the tensor-to-scalar ratio of density perturbations $r$ are given by Lyth
and Riotto (1999)

$\displaystyle n_{s}-1=-6\epsilon+2\eta,\ \ \ \ $ $\displaystyle
r=16\epsilon.$ (42)

The slow-roll parameters $\epsilon$ and $\eta$ can be defined in terms of the
scalar field potential $V(\phi)$ and its derivatives as

$\epsilon=\frac{1}{16\pi
G_{N}}\left(\frac{V^{\prime}(\phi)}{V(\phi)}\right)^{2},\ \ \ \ \ \
\eta=\frac{1}{8\pi G_{N}}\frac{V^{\prime\prime}(\phi)}{V(\phi)},$ (43)

Now we redefine the scalar field $\phi$ in terms of a new scalar field
$\varphi$, $\phi=\phi(\varphi)$ that later we identify as the number of
e-folds $N$ during inflation so that the slow-roll parameters are written in
terms of the redefined variable as

$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
G_{N}}\frac{1}{\omega(\varphi)}\left(\frac{V^{\prime}(\varphi)}{V(\varphi)}\right)^{2},$
(44) $\displaystyle\eta$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
G_{N}V(\varphi)}\left[-\frac{\omega^{\prime}(\varphi)}{2\omega(\varphi)^{2}}V^{\prime}(\varphi)+\frac{V^{\prime\prime}(\varphi)}{\omega(\varphi)}\right],$
(45)

where the primes denote derivatives with respect to the argument of the
function on which the prime operates and
$\omega(\varphi)=(d\phi/d\varphi)^{2}$.

If we consider a flat FRW universe with the metric (22) the Friedmann
equations (46) and (47) during inflation become

$\displaystyle\frac{3H^{2}}{8\pi G_{N}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (46)
$\displaystyle-\frac{3H^{2}+2\dot{H}}{8\pi G_{N}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}-V(\phi),$ (47)

and they can be rewritten terms of the redefined scalar field $\varphi$ as
Bamba and Odintsov (2016)

$\displaystyle\frac{1}{2}\omega(\varphi)H(N)^{2}+V(\varphi)$ $\displaystyle=$
$\displaystyle\frac{3}{8\pi G_{N}}H(N)^{2},$ (48)
$\displaystyle\frac{1}{2}\omega(\varphi)H(N)^{2}-V(\varphi)$ $\displaystyle=$
$\displaystyle-\frac{1}{8\pi G_{N}}\left[3H(N)^{2}+\right.$ (49)
$\displaystyle+\left.2H^{\prime}(N)H(N)\right].$

where the number of e-folds $N$ during inflation is

$N=\ln\left(\frac{a_{f}}{a_{i}}\right)=\int_{t_{i}}^{t_{f}}Hdt,$ (50)

and the subscripts $i$ and $f$ denote the beginning and end of inflation. Thus
inverting Eqs. (48) and (49) we can now express $\omega(\varphi)$ and
$V(\varphi)$ in terms of $H(N)$ as

$\displaystyle\omega(\varphi)$ $\displaystyle=$ $\displaystyle-\frac{2}{8\pi
G_{N}}\frac{H^{\prime}(N)}{H(N)}\Bigg{|}_{N=\varphi},$ (51) $\displaystyle
V(\varphi)$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
G_{N}}\left[3H(N)^{2}+H(N)H^{\prime}(N)\right]\Bigg{|}_{N=\varphi}.$ (52)

Now using Eqs.(51)-(52), the slow-roll parameters in Eq.(44) and (45) can be
written in terms of $H(N)$ and its derivatives with respect to $N$. Then
$n_{s}$ and $r$ can be easily derived from Eqs. (42). The advantage of writing
$n_{s}$ and $r$ in terms of $H(N)$ is that we can now obtain the same
quantities for the AS model for which the early universe is described by an
effective fluid instead of a scalar field. The full expressions for the slow-
roll parameters are given in Appendix A.

###  IV.2 Spectral index and tensor to scalar ratio in AS

Figure 5: Constrains on $\varepsilon_{i}/\varepsilon_{c}$ for the AS model of
inflation in a radiation dominated universe from the marginalised contours of
Planck and Planck+BK15 (left panel), ACTPol+WMAP (mid panel) and SPT3G+WMAP
(right panel) data. The contours are obtained for $n_{s}$ and $r$ using
marginalised joint 68% and 95% confidence-level regions from the observational
data.

We have seen that the slow roll approximation holds for the AS cosmological
model. Also, as mentioned, it is well known that there is a correspondence
between fluid models and scalar field inflation. Therefore at the background
level, it makes sense to use the tools of scalar field inflation discussed in
the previous subsection to obtain some indication of our model’s implications
for the early universe. However, we must remember that even though the
correspondence between the effective fluid model and the scalar field is valid
at the background level this does not necessarily imply that there will be a
correspondence between the quantities obtained at the perturbative level.
Nevertheless, in the slow roll limit, with all higher order quantities being
small, we may expect that the departure between the two approaches at a
perturbative level will also remain small and therefore we may estimate
$n_{s}$ and $r$ for the AS model using the tools discussed above. Our aim here
is to express $n_{s}$ and $r$ in terms of $H(N)$ for the AS cosmological
model. Using the effective density and pressure from Eqs. (11) and (12) we can
write the effective e.o.s. as

$\tilde{P}=-\tilde{\varepsilon}+f(\varepsilon),$ (53)

where
$f(\varepsilon)=\left(\varepsilon\chi\right)_{,\varepsilon}\left(\varepsilon+P\right)$.
Without any loss of generality, we can express this effective e.o.s. as

$\tilde{P}(N)=-\tilde{\varepsilon}(N)+f(\varepsilon),$ (54)

where the effective pressure and density are functions of the number of
e-folds $N$. For a flat FRW universe, with $k=0$, the Friedmann equations (23)
and (24) can be rewritten as

$\displaystyle\frac{3H(N)^{2}}{8\pi G_{N}}$ $\displaystyle=$
$\displaystyle\tilde{\varepsilon},$ (55)
$\displaystyle-\frac{2H(N)H^{\prime}(N)}{8\pi G_{N}}$ $\displaystyle=$
$\displaystyle\tilde{\varepsilon}+\tilde{P}=f(\varepsilon).$ (56)

Using the continuity equation we obtain

$\tilde{\varepsilon}^{\prime}(N)=-3f(\varepsilon),\ \ \ \
\varepsilon^{\prime}(N)=-\frac{3f(\varepsilon)}{\left(\varepsilon\chi\right)_{,\varepsilon}},$
(57)

where the primes denote derivatives with respect to the number of e-folds. Now
we take the derivative of Eq. (56) and use the above relations to get

$\frac{2H(N)^{2}}{8\pi
G_{N}}\left[\left(\frac{H^{\prime}(N)}{H(N)}\right)^{2}+\frac{H^{\prime\prime}(N)}{H(N)}\right]=\frac{3ff_{,\varepsilon}}{\left(\varepsilon\chi\right)_{,\varepsilon}},$
(58)

Now using Eq.(55), (56) and (58) we can express the slow-roll and the
observational parameters in terms of the quantities describing the fluid,
namely $\tilde{\varepsilon}(N)$ and $f(\varepsilon(N))$. We report the
explicit forms of the slow-roll parameters in Appendix A.

For some simple estimates, we can write the expressions for inflationary
observables in the limit when $f(\varepsilon)$ and $\tilde{\varepsilon}$ both
vary slowly during the inflationary period, such that
$f(\varepsilon)/\tilde{\varepsilon}(N)\ll 1$. It is straightforward to check
that this condition corresponds to $\tilde{w}\simeq-1$ and consequently a
slowly varying Hubble parameter $H\simeq{\rm const}$. In this approximation,
the observables take the simple form

$n_{s}\approx 1-6\frac{f(\varepsilon)}{\tilde{\varepsilon}(N)},\ \ \ \
r\approx 24\frac{f(\varepsilon)}{\tilde{\varepsilon}(N)}.$ (59)

These expressions can now be used to put constraints from observational data
on the value of $\varepsilon/\varepsilon_{c}$ at the time of inflation.

###  IV.3 Constraints from observations

$w$ | ($n_{s},r$) | $\varepsilon_{i}/\varepsilon_{c}$ |  $\varepsilon_{i}$ GeV4 |  $\tilde{\varepsilon}_{i}$ GeV4 |  $(\varepsilon_{i})^{1/4}$ GeV |  $(\tilde{\varepsilon}_{i})^{1/4}$ GeV | $N_{\rm tot}$
---|---|---|---|---|---|---|---
1/3 | (0.965, 0.13) | $5.08\times 10^{100}$ | $1.05\times 10^{177}$ | $4.8\times 10^{78}$ | $1.8\times 10^{44}$ | $4.86\times 10^{19}$ | 57.62
| (0.975, 0.10) | $1.77\times 10^{136}$ | $3.67\times 10^{212}$ | $6.5\times 10^{78}$ | $1.38\times 10^{53}$ | $5.05\times 10^{19}$ | 78.08
1 | (0.965, 0.13) | $1.14\times 10^{151}$ | $2.36\times 10^{227}$ | $7.21\times 10^{78}$ | $6.97\times 10^{56}$ | $5.18\times 10^{19}$ | 57.51
| (0.975, 0.10) | $1.14\times 10^{205}$ | $2.36\times 10^{281}$ | $9.79\times 10^{78}$ | $2.2\times 10^{70}$ | $5.59\times 10^{19}$ | 78.23
Table 1: Values of the energy density $\varepsilon_{i}$ and its effective
counterpart $\tilde{\varepsilon}_{i}$ at the epoch of horizon crossing from
the fit with the data assuming $\varepsilon_{c}\sim m_{\rm pl}^{4}$ for both
radiation (i,e, $w=1/3$ as in Fig. 5) and stiff fluid ($w=1$). The last column
shows the number of the universe’s e-folds during the quasi-de Sitter period.

To compare our model of inflation in AS with current observational data we
consider data sets obtained from Planck and other observations. We
specifically consider the observational constraints on the scalar spectral
index $n_{s}$ and the tensor-to-scalar ratio $r$ based on the Planck 2018 and
BICEP2/Keck-Array 2015 (Planck 18 + BK 15) combined data sets Aghanim et al.
(2020a, b, c); Akrami et al. (2020); Ade et al. (2018), the Atacama Cosmology
Telescope DR4 likelihood combined with the WMAP satellite data (ACTPol+WMAP)
Aiola et al. (2020); Hinshaw et al. (2013); Forconi et al. (2021) and the
South-Pole Telescope polarization measurements combined with the WMAP
satellite data (SPT3G+WMAP) Dutcher et al. (2021); Hinshaw et al. (2013);
Forconi et al. (2021). Planck temperature, polarization, and lensing data have
determined the spectral index of scalar perturbations to be $n_{s}=0.9649\pm
0.0042$ at $68\%$ confidence level (CL) with no evidence of a scale
dependence. On the other hand, the Planck data (Planck TT,TE,EE+lensing+lowEB)
puts a $95\%$ CL upper limit on the tensor-to-scalar ratio, $r\leq 0.16$ which
is further improved by combining with the BICEP2/Keck Array data to obtain
$r<0.056$ Aghanim et al. (2020b); Akrami et al. (2020). These are the
strongest constraints on $n_{s}$ and $r$ to date.

Now, let us have a look at the explicit expression for
$f(\varepsilon)/\tilde{\varepsilon}$ to find allowed values of
$\varepsilon/\varepsilon_{c}$

$\frac{f(\varepsilon)}{\tilde{\varepsilon}}=\frac{(1+w)\left(\varepsilon\chi\right)_{,\varepsilon}}{\chi}=\frac{\varepsilon}{\varepsilon_{c}}\frac{(1+w)}{\left(1+\frac{\varepsilon}{\varepsilon_{c}}\right)\log\left(1+\frac{\varepsilon}{\varepsilon_{c}}\right)}.$
(60)

If we use Eq. (59) to calculate $n_{s}$ and $r$ for our model, then having
$f(\varepsilon_{i})/\tilde{\varepsilon_{i}}=5.75\times 10^{-3}$ gives
$n_{s}=0.965$ and $r=0.13$, where the quantities with subscript i are
evaluated at the horizon exit. Notice that, in view of the fact that for
$\varepsilon=\varepsilon_{c}$ from Eq. (19) we get
$G(\varepsilon_{c})=G_{N}/2$, we need $\varepsilon_{i}>>\varepsilon_{c}$ in
order to achieve the de Sitter phase for which $G(\varepsilon_{i})\simeq 0$.
For example, for a critical density of the order of Planck density,
$\varepsilon_{c}\sim m_{\rm pl}^{4}$, the energy scale of the universe at the
horizon exit $\varepsilon_{i}$ would have to be above the Planck density, i.e.
$\varepsilon_{i}^{1/4}\sim 10^{44}{\rm GeV}$ corresponding to
$\tilde{\varepsilon}_{i}^{1/4}\sim 10^{19.5}{\rm GeV}$.

We plot the marginalized contours of $n_{s}$ and $r$ for the datasets
considered along with the predictions of the AS model with radiation in Fig.
5. We can see that our model agrees well with all three datasets. In all three
plots of Fig. 5, $\varepsilon_{i}/\varepsilon_{c}$ varies along the straight
line with the points at the ends taking values
$\varepsilon_{i}/\varepsilon_{c}=5.08\times 10^{100}$ and
$\varepsilon_{i}/\varepsilon_{c}=1.77\times 10^{136}$ (assuming $w=1/3$).
Similarly, we can obtain the constraints on the energy scales of inflation for
the stiff fluid model as $\varepsilon_{i}/\varepsilon_{c}=1.14\times 10^{151}$
and $\varepsilon_{i}/\varepsilon_{c}=1.714\times 10^{205}$. These values,
together with the corresponding number of e-folds during the inflation period,
are shown in Table 1 for both radiation and stiff fluid.

The number of e-folds from the time of horizon exit to the end of the quasi-de
Sitter period is given by

$N_{tot}=\ln\left(\frac{a_{e}}{a_{i}}\right).$ (61)

Here $a_{i}$ and $a_{e}$ are the values of the scale factor at the moment of
horizon exit and the end of the quasi-de Sitter period. The end of accelerated
expansion is defined by $\ddot{a}_{e}=0$. We then use Eq. (25) to write

$\frac{\varepsilon_{e}}{\varepsilon_{i}}=\left(\frac{a_{e}}{a_{i}}\right)^{-3(1+w)},$
(62)

where $\varepsilon_{i}$ and $\varepsilon_{e}$ are the energy densities at the
moment of horizon exit and the end of the quasi-de Sitter period. Using the
above equation, the number of e-folds becomes

$N_{tot}=\ln\left(\frac{\varepsilon_{i}}{\varepsilon_{e}}\right)^{1/3(1+w)}.$
(63)

To derive the ratio of energy densities at the two epochs, we use the
constraints we obtained from CMB observations to find
$\varepsilon_{i}/\varepsilon_{c}$ and the condition for the end of accelerated
expansion to find $\varepsilon_{e}/\varepsilon_{c}$. First, from the CMB
constraints discussed above, we found that the ratio
$\varepsilon_{i}/\varepsilon_{c}$ that gives $n_{s}=0.965$, $r=0.13$ for
radiation ($w=1/3$) is

$\frac{\varepsilon_{i}}{\varepsilon_{c}}\simeq 5.08\times 10^{100}.$ (64)

Then from the condition $\tilde{\varepsilon}+3\tilde{P}=0$ which holds at the
end of accelerated expansion, we get

$\frac{\varepsilon_{e}}{\varepsilon_{c}}\simeq 3.921.$ (65)

These two equations give the ratio of the energy density at the horizon exit
and the end of the quasi-de Sitter period

$\frac{\varepsilon_{i}}{\varepsilon_{e}}\simeq 1.3\times 10^{100},$ (66)

from which we can calculate the number of e-folds using Eq. (63), which gives

$N_{tot}\simeq 57.6\ \ \ {\rm for}\ \ w=1/3.$ (67)

Similarly, we can perform the same calculations for dust and stiff fluid, for
which the numbers of e-folds are

$\displaystyle N_{tot}\simeq 57.8\ \ \ {\rm for}\ \ w=0,$ (68) $\displaystyle
N_{tot}\simeq 57.5\ \ \ {\rm for}\ \ w=1.$

Similar calculations for the ratio $\varepsilon_{i}/\varepsilon_{c}$ that
gives $n_{s}\simeq 0.975$, $r\lesssim 0.10$ for the three matter models give
$N_{tot}\simeq 78$. The values are consistent with the estimates of $N$ from
scalar field inflation thus confirming that the model can be considered a
viable alternative to standard inflationary models.

##  V Summary and Discussions

We considered a modification of the standard cosmological model at high
densities that allows for a phase of accelerated expansion without the
introduction of additional fields. This was achieved with the introduction of
a matter-gravity coupling $\chi$ in the action that leads to an effective
theory with varying gravitational coupling $G$ and $\Lambda$ in the field
equations 111It is worth noting that due to the monotonic behavior of the
coupling $\chi$, we can not use the Markov-Mukhanov mechanism to explain the
observed late-time acceleration of the universe within a single framework.
There exist other approaches to variable $G$ and $\Lambda$ that attempts at
explaining the late-time acceleration, see e.g. de Cesare et al. (2016).. We
showed that the choice of $G(\varepsilon)$ consistent with the Asymptotic
Safety paradigm allows us to reproduce a phase of accelerated expansion in the
early universe without the aid of an ad hoc scalar field. We showed that there
are ranges of values for $\varepsilon_{i}/\varepsilon_{c}$, with
$\varepsilon_{i}$ and $\varepsilon_{c}$ being the energy density at the
horizon exit and the critical energy density (the model’s only free parameter)
respectively, for which the observational constraints from the CMB data can be
satisfied.

In the present work the constraints are derived assuming the equivalence
between our effective fluid description and a scalar field model of inflation.
However, an in-depth analysis of the quantum perturbations of the effective
fluid during the inflationary period is necessary in order to properly put
constraints on the model’s parameter. Such an analysis should also allow us to
distinguish between our model and other existing models of inflation derived
from particle physics or alternative theories. We plan to explore this
direction in the future. Finally, we expect that the data obtained from future
experiments such as CMB-S4 Abazajian et al. (2022); Zegeye et al. (2023),
LiteBird Allys et al. (2023), CORE Delabrouille et al. (2018), CMB-Bhārat Adak
et al. (2022), etc. will be able to test the validity of the model presented
here.

## Acknowledgement

AZ would like to thank Catania Astrophysical Observatory - INAF- for its warm
hospitality during the preparation of the manuscript. DM, HC and AZ
acknowledge support from Nazarbayev University Faculty Development Competitive
Research Grant No. 11022021FD2926. The authors thank William Giarè and his
group for giving us access to some of the data products used in this article.
The authors also thank Antonio De Felice for useful comments and discussions.

##  Appendix A Slow-roll parameters and Inflationary Observables

Here we express $\epsilon$ and $\eta$ in terms of $H(N)$ and its derivatives
with respect to $N$.

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle-\frac{H^{3}}{4H^{\prime}(3H+H^{\prime})^{2}}\left[6\frac{H^{\prime}}{H}+\frac{H^{\prime\prime}}{H^{\prime}}+\left(\frac{H^{\prime}}{H}\right)^{2}\right]^{2},$
(69) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{H}{3+H^{\prime}}\right)\left[9\frac{H^{\prime}}{H}+3\frac{H^{\prime\prime}}{H}+\frac{1}{2}\left(\frac{H^{\prime}}{H}\right)^{2}+3\frac{H^{\prime\prime}}{H^{\prime}}+\right.$
(70)
$\displaystyle\left.-\frac{1}{2}\left(\frac{H^{\prime\prime}}{H^{\prime}}\right)^{2}+\frac{H^{\prime\prime\prime}}{H^{\prime}}\right].$

The slow-roll parameters in terms of $\tilde{\varepsilon}(N)$ and
$f(\varepsilon)$ in the AS model are

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\frac{3}{2}\tilde{\varepsilon}f\left(\frac{f_{,\varepsilon}-2\left(\varepsilon\chi\right)_{,\varepsilon}}{\left(\varepsilon\chi\right)_{,\varepsilon}(2\tilde{\varepsilon}-f)}\right)^{2},$
(71) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{3\tilde{\varepsilon}}{2\tilde{\varepsilon}-f}\left[\frac{f}{\tilde{\varepsilon}}+\frac{ff_{,\varepsilon}}{\left(\left(\varepsilon\chi\right)_{,\varepsilon}\right)^{3}}\left(\varepsilon\chi\right)_{,\varepsilon\varepsilon}+\frac{f_{,\varepsilon}}{\left(\varepsilon\chi\right)_{,\epsilon}}+\right.$
(72)
$\displaystyle\left.-\frac{ff_{,\varepsilon}}{2\tilde{\varepsilon}\left(\varepsilon\chi\right)_{,\epsilon}}-\frac{f_{,\varepsilon}^{2}}{2\left(\left(\varepsilon\chi\right)_{,\epsilon}\right)^{2}}-\frac{ff_{,\varepsilon\varepsilon}}{\left(\left(\varepsilon\chi\right)_{,\epsilon}\right)^{2}}\right].$

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