# Emergence of inflaton potential from asymptotically safe gravity

Agustín Silva<EMAIL_ADDRESS>High Energy Physics Department, Institute
for Mathematics, Astrophysics, and Particle Physics, Radboud University,
Nijmegen, The Netherlands.

###### Abstract

The Asymptotic Safety Hypothesis for gravity relies on the existence of an
interacting fixed point of the Wilsonian renormalization group flow, which
controls the microscopic dynamics, and provides a UV completion of the theory.
Connecting such UV completion to observable physics has become an active area
of research in the last decades. In this work we show such connection within
the framework of scalar-tensor models. More specifically, we found that
cosmological inflation naturally emerges from the integration of the RG flow
equations, and that the predicted parameters of the emergent effective
potentials provide a slow-roll model of inflation compatible with current
observations. Furthermore, the RG evolution of the effective action starting
at the UV fixed point, provides a prediction for the initial value of the
inflaton field.

##  I Introduction

Cosmological inflation has become a cornerstone in current theoretical
cosmology since its initial proposal in [1]. It provides a mechanism for
generating the entire observable universe from a single causal region of
spacetime, solving the horizon problem [1, 2]. In addition, it provides an
origin for the primordial density fluctuations that serve as seeds for
structure formation on larger scales [3]. See [4, 5, 6, 7, 8, 9, 10, 11] for
reviews on the topic, and for other consequences of the inflationary scenario.

There are several proposals for its realization, the most studied of them
being single-field inflationary cosmologies [12]. With sufficient fine tuning,
one can construct an effective field theory (EFT) describing the dynamics of
this scalar degree of freedom (inflaton) such that its predictions match
observations [13], but such models often require ad hoc shapes of the
potential for the inflaton field, as well as a fine tuning of the initial
conditions of the field itself [14].

A tantalizing idea would be that such EFTs can be derived from first
principles, and that the initial conditions of the inflaton come as a
prediction of such derivation. Despite how ambitious this might sound, in this
work we will provide a fundamental derivation of the inflaton potential and
its initial conditions, using only non-perturbative Wilsonian renormalization
group techniques [15].

More in particular, we will work under the Asymptotic Safety (AS) hypothesis
for gravity, first proposed by Weinberg, and later implemented in $4$
dimensions in [16]. In a nutshell, the AS hypothesis for gravity is the
assumption the there exist an interacting renormalization group fixed point in
the gravitational sector, typically called a Non-Gaussian fixed point (NGFP).
For this UV completion to be predictive, this fixed point must come with the
property that one only has to fine tune a finite amount of parameters such
that the theory is attracted to it, or in a more technical language, its UV-
critical surface must be finite dimensional. For reviews on the topic see [17,
18, 19]. This provides a UV completion of the theory, where only a finite
amount of measurements are needed to fully determine the theory. Predictions
from such UV completions of gravity are given by the endpoints of the RG
trajectories that start at the NGFP [20, 21, 22, 23, 24, 25].

A primary tool for computing Wilsonian RG flows is the Wetterich equation
[26],

$k\partial_{k}\Gamma_{k}=\frac{1}{2}\mathbf{Tr}\left[\left(\Gamma_{k}^{(2)}+\mathcal{R}_{k}\right)^{-1}k\partial_{k}\mathcal{R}_{k}\right]\,,$
(1)

which encodes the dependence of so called “effective average action”
$\Gamma_{k}$, on the coarse-graining scale $k$. In this equation,
$\mathbf{Tr}$ represents a functional trace over all the degrees of freedom of
the theory, $\mathcal{R}_{k}$ is an IR regulator, and $\Gamma_{k}^{(2)}$ is
the second order functional derivative of $\Gamma_{k}$ with respect to the
field degrees of freedom. $\Gamma_{k}$ contains information of the path
integral modes of energy higher than the momentum cut-off scale $k$. Since it
is out of the scope of this work to dive into the details of the Wetterich
equation, we refer the reader to the extensive bibliography about it, and
references therein [27, 28, 29]. For the purpose of this work, we are
interested in complete (approximate) solutions of (1), $k\mapsto\Gamma_{k}$,
which interpolate between a NGFP in the limit $k\rightarrow\infty$ (asymptotic
safety) and the effective action $\Gamma\equiv\lim_{k\rightarrow 0}\Gamma_{k}$
associated with observables. In this limit where the regulator scale vanishes,
the effective average action becomes the full effective action, containing
information about all the energy modes of the theory. The resulting $\Gamma$
that comes out if this integration is the prediction we are looking for, and
is how we will connect the Asymptotic Safety hypothesis with cosmological
inflation.

In particular, in this work we solve the functional RG flow equations for a
class of scalar-tensor models, integrating them from the NGFP realizing the
Asymptotic Safety hypothesis, up to the limit of vanishing regulator ($k=0$),
where we make predictions for the scalar field interactions.

This paper is divided as follows. In sec. II we describe the used scalar-
tensor model. In sec. III we describe the necessary preliminaries, such as
coordinates choices and the parameterization of the interaction potentials,
that we will use for finding numerical solutions of the RG flow equations. In
sec. IV we show the results of a numerical integration of the flow equations.
In sec. V we show how the previous results naturally give rise to cosmological
inflation, compatible with observations. We also discuss some of the
implications of such results. In sec. VI we present our conclusions and
forthcomings.

##  II The model

More specifically, here we will study the (euclidean) RG flow of scalar-tensor
EFTs of the form

$\Gamma_{k}=\int
d^{4}\tilde{x}\sqrt{g}(-\tilde{F}(k,\tilde{\varphi})\tilde{R}+\tilde{V}(k,\tilde{\varphi})+\frac{1}{2}\partial_{\mu}\tilde{\varphi}\partial^{\mu}\tilde{\varphi})\,,$
(2)

where $\tilde{F}$ and $\tilde{V}$ are undetermined, even functions of the
scalar field, $g$ is the metric determinant and $R$ the Ricci scalar. These
type of models fall into the category of Horndeski theories [30], and have
been largely studied, using particular potentials $\tilde{F}$ and $\tilde{V}$,
in both early and late time cosmology. See for example [31, 32, 33, 34], and
references therein.

Since the flow equations for the potentials $\tilde{F}$ and $\tilde{V}$ have
been determined using (1) in a number of works, we refer the reader to the
bibliography on such computations for details of their derivation [35, 36, 37,
38]. Such computations in general involve a choice for an IR regulator
$cR_{k}$ and a gauge fixing procedure, typically implemented by means of
Faddeev–Popov ghosts. There are multiple methods for the resolution of the
Wetterich equation, an example of those is an expansion in curvature
invariants using heat-kernel methods. Typically, the obtained flow equations
differ in some of their properties depending on the regulator choice, or the
gauge fixing [39]. Predictions coming from these type of calculations should
be taken with a pinch of salt, and every result should eventually be
contrasted with other calculations using different methods, to verify its
robustness regarding the previously mentioned choices.

Having said that, and given that to the best of our knowledge this is the
first time this particular calculations are performed, we will stick to a
particular set of flow equations, and leave the previously mentioned cross-
checks for a future work.

In this paper, we will take equations (4) and (5), derived in [35], and use
them as a starting point. Notice that in (4) and (5) the dimensionless
variables

$\varphi\doteq\frac{\tilde{\varphi}}{k}\,\,\,\,\,\,\,\,\,F\doteq\frac{\tilde{F}}{k^{2}}\,\,\,\,\,\,\,\,\,V\doteq\frac{\tilde{V}}{k^{4}}\,,$
(3)

are being used for convenience. In these equations, the flow is computed at
constant $\varphi$, for all values of $k$.

$\displaystyle k\,V^{(1,0)}(k,\varphi)=\varphi
V^{(0,1)}(k,\varphi)-4V(k,\varphi)+\frac{1}{16\pi^{2}}+\frac{3F^{(0,1)}(k,\varphi)^{2}+F(k,\varphi)}{32\pi^{2}\left(3F^{(0,1)}(k,\varphi)^{2}+F(k,\varphi)\left(V^{(0,2)}(k,\varphi)+1\right)\right)}$
(4) $\displaystyle k$ $\displaystyle\,F^{(1,0)}(k,\varphi)=\varphi
F^{(0,1)}(k,\varphi)-2F(k,\varphi)+\frac{37}{384\pi^{2}}$
$\displaystyle\quad+\frac{F(k,\varphi)\left(\left(3F^{(0,1)}(k,\varphi)^{2}+F(k,\varphi)\right)\left(-3F^{(0,2)}(k,\varphi)+3V^{(0,2)}(k,\varphi)+1\right)+2F(k,\varphi)V^{(0,2)}(k,\varphi)^{2}\right)}{96\pi^{2}\left(3F^{(0,1)}(k,\varphi)^{2}+F(k,\varphi)\left(V^{(0,2)}(k,\varphi)+1\right)\right)^{2}}$
(5)

In [35], it was found that these flow equations present the following shift-
symmetric ($\varphi\to\varphi+c$) NGFP

$F_{*}=\frac{41}{768\pi^{2}}\,\,\,\,\,\,\,\,\,V_{*}=\frac{3}{128\pi^{2}}\,,$
(6)

compatible with the Asymptotic Safety hypothesis for gravity.

In order to asses the predictivity of the UV completion (6) of (2), one must
determine the number of parameters of the theory that are a priori
undetermined by the RG flow. These parameters are often called “relevant
directions”. Relevant directions are the perturbations of the fixed point that
are attracted to it when one lets them evolve following the RF flow.
Perturbations that are repelled from the fixed point are called irrelevant
directions. If at first order in perturbations there are directions that are
neither repelled nor attracted to the fixed point, they are called marginal
directions. These directions often require an analysis at higher order in
perturbations around the fixed point. A determination of such directions is
carried out by making a stability analysis of the fixed point.

In the case of the model (2), to determine its relevant directions, the
authors of [35] perturbed the flow equations (4) and (5) at linear order in
deviations from the NGFP using the ansatz $F\simeq F_{*}+\delta
F\,(\frac{k0}{k})^{\theta}$ and $V\simeq V_{*}+\delta
V\,(\frac{k0}{k})^{\theta}$, where $k0$ is a finite value for the coarse-
graining scale, and $\theta$ is the so called critical exponent. Directions
with $\theta>0$, $\theta<0$ or $\theta=0$ are considered relevant, irrelevant
and marginal, respectively, due to their behavior when increasing $k$ to
infinity. The authors of [35] determined $2$ directions that were relevant,

$(\delta F,\delta V)_{1}=(0,1),\,\theta_{1}=4\,\,\,\,\,\,\,\,(\delta F,\delta
V)_{2}=(1,0),\,\theta_{2}=2\,,$ (7)

and $1$ direction that at linear order seemed to be marginal

$(\delta F,\delta V)_{3}=(-1+32\pi^{2}\varphi^{2},0),\,\theta_{3}=0\,.$ (8)

All other directions were found to be irrelevant.

With this information, one might naively conclude that the only relevant
directions are $(\delta F,\delta V)_{1}$ and $(\delta F,\delta V)_{2}$, and
thus NGFP (6) is not interesting for scalar field phenomenology. This is
because if that were the case, there would be no relevant direction that
creates shift symmetry breaking in the scalar field direction, meaning that
the Ward identities would forbid the emergence of non-trivial interactions for
the scalar field [40].

Because of this, we decided to extend the analysis of $(\delta F,\delta
V)_{3}$ to second order, to determine its relevance, where we found that
$(\delta F,\delta V)_{3}$ is also a relevant direction. The calculations can
be found in the appendix VIII.1. This means that at smaller energies, RG
trajectories emerging from the NGFP (6), can potentially have symmetry
breaking, and create non-trivial interactions in the scalar field sector. Due
to this, the fixed point (6) can have interesting phenomenology in the IR, and
we will solve (4) and (5) using as initial conditions the UV completion given
by (6).

##  III Preliminaries for numerical methods

Since equations (4) and (5) are non-linear, second order, partial differential
equations, we decided that numerical methods are the best approach to find a
solution that uses the NGFP (6) as initial condition at $k\to\infty$.

Despite the simplicity of (4) and (5) using the variables (3), these are not
the best variables to find a numerical solution. First of all, for
computational reasons, one must work with compact domain intervals, and
therefore one must compactify the RG interval $k\in[0,\infty)$ to some finite
interval. Second, it is not convenient to work with variables that require a
scaling with $k$ once approaching the $k\to 0$ limit. One example of this are
the dimensionless field variables $\varphi$, where if one wants to remove the
regulator $k$, and is interested in what happens at finite scalar field
$\tilde{\varphi}$, one has to deal with $\infty\cdot 0$ limits ($\lim_{k\to
0}\varphi\,k=\tilde{\varphi}$).

Because of this, we will search for solutions of $F$ and $V$ that are
parameterized as

$F(k,\phi)=\frac{1+f(g_{k},\phi^{2})}{16\pi
g_{k}^{2}}\,\,\,\,\,V(k,\phi)=\frac{2\lambda(g_{k})+v(g_{k},\phi^{2})}{(16\pi
g_{k}^{2})^{2}}\,,$ (9)

where $g_{k}\equiv\frac{\tilde{g_{k}}}{k}$ is a dimensionless parameter, with
its corresponding dimensionful version $\tilde{g}$, $\lambda_{k}$ is a
dimensionless coupling, and the field variable $\phi$ is defined as

$\phi\equiv g\varphi=\tilde{g}\tilde{\varphi}\,.$ (10)

Notice that $\phi$, $f$ and $v$ are dimensionless without the need of
multiplication by $k$. In this parameterization we separated the scalar field
dependent part into $f$ and $v$, and the purely gravitational part is
parameterized by $g$ and $\lambda$. For this splitting, we assume that
$f(g_{k},0)=v(g_{k},0)=0$, thus at vanishing field we recover the
gravitational action. Furthermore, notice that we exchanged the $k$ dependence
by a dependence on the coupling $g_{k}$. This is because of the following: at
the fixed point (6), in this parameterization, we have the following result

$\displaystyle\lim_{k\to\infty}$ $\displaystyle g_{k}\equiv
g_{*}=\sqrt{\frac{48\pi}{41}}\simeq 1.92\,,$
$\displaystyle\quad\lim_{k\to\infty}$
$\displaystyle\lambda(g_{k})\equiv\lambda_{*}=\sqrt{\frac{995328\pi^{3}}{68921}}\simeq
21.2\,,$ $\displaystyle\quad\lim_{k\to\infty}$
$\displaystyle\\{f(g_{k},\phi^{2}),\,v(g_{k},\phi^{2})\\}\equiv\\{f_{*},\,v_{*}\\}=\\{0,\,0\\}\,,$
(11)

that indicates that the field dependent parts vanish in the UV.

With the splitting given by (9), the two relevant directions (7) are
parameterized by $g$ and $\lambda$. This means that $g$ and $\lambda$ are
undetermined by the RG flow, and should be determined by measurements. In
particular, we have that $\lim_{k\to 0}\frac{g_{k}}{k}=\lim_{k\to
0}\tilde{g}_{k}=\sqrt{\tilde{G_{0}}}$, where $\tilde{G_{0}}$ is the measured
value of Newton’s constant, and implying that $\lim_{k\to 0}g_{k}=0$. This
means that using as an RG parameter the coupling $g$, one can compactify the
RG domain from $k\in[0,\infty)$ to $g\in[0,\sqrt{\frac{48\pi}{41}}]$, thus
solving the issue of compact domain for numerical solutions. On top of that,
we have that $\lim_{k\to 0}\lambda(g_{k})=16\pi\Lambda_{0}\,G_{0}$, where
$\Lambda_{0}$ is the measured Cosmological constant. Since $\lim_{k\to
0}\lambda(g_{k})\simeq 10^{-120}$, we will approximate it by $0$ when making
inflationary predictions.

Furthermore, the previous observations mean that the variable $\phi$ has a
well defined interpretation in the $k\to 0$ limit, since it satisfies
$\lim_{k\to 0}\phi=\sqrt{\tilde{G_{0}}}\tilde{\varphi}$, meaning that at $k=0$
the variable $\phi$ is just the scalar field in Planck mass units.

The main goal of this separation is that now we can encapsulate all irrelevant
directions inside $f$ and $v$, and we would like to make predictions for all
those. In particular, we want to determine the functional form of
$\lim_{k\to\infty}\\{f(g_{k},\phi^{2}),\,v(g_{k},\phi^{2})\\}=\\{f(0,\phi^{2}),\,v(0,\phi^{2})\\}$.
In order to turn the RG flow equations (5) and (4) in terms of $f$ and $v$,
one has to make a series of transformations. Due to their length, those
transformations, and the resulting flow equations can be found in the appendix
VIII.3.

Figure 1: Numerical solutions for the functions $f(g,\phi^{2})$ and
$v(g,\phi^{2})$ in the domain
$(g,\phi)\in[0,\sqrt{\frac{48\pi}{41}}]\times[0,10]$ using boundary conditions
(13) and (14), fixing $m_{0}^{f}=0.42$ and $m_{0}^{v}=2.75\times 10^{-10}$.
The predicted potentials can be extracted from the intersection of the
surfaces with the plane $g=0$. The color coding is a linear function of $g$
that interpolates between red and purple, meant to represent small and high
values of the original coarse-graining scale $k$.

##  IV Numerical solution of RG flow equations

The resulting flow equations for $f(g,\phi^{2})$ and $v(g,\phi^{2})$ are given
by (26) and (27), and they are coupled, non-linear, second-order differential
equations. So far, the only constrains to be compatible with the NGFP (6) are
that $f(g_{*},\phi^{2})=v(g_{*},\phi^{2})=0$, and due to the splitting used,
we have $f(g,0)=v(g,0)=0$. As mentioned in section II, there is one extra
relevant direction given by (8). This direction will come with an associated
coupling in the action that is undetermined by the RG flow. Besides, since
this direction is field dependent, it is contained within $f$. This means we
will have at least $1$ degree of freedom in the resulting $f$, after we
integrate the RG flow equations.

In order to find the number of free parameters that will be undetermined by
the RG flow, and to see if there are any extra degrees of freedom inside $v$,
it is not sufficient to look only at the stability properties around the fixed
point (see section II), but one must also look at differential order the
equation determining the RG flow. In our case, since the flow equations are
second order, but we only have the conditions
$f(g_{*},\phi^{2})=v(g_{*},\phi^{2})=0$ and $f(g,0)=v(g,0)=0$, there is still
one extra boundary condition to be fixed. This can add extra degrees of
freedom to the solutions, apart from the relevant directions around the fixed
point.

In order to close the system of equations, we will impose boundary conditions
by using the derivatives $f^{(0,1)}(g,0)\equiv f_{1}(g)$ and
$v^{(0,1)}(g,0)\equiv v_{1}(g)$.

A priori one does not know which functions $f_{1}(g)$ and $v_{1}(g)$ are
consistent with the equations and would give rise to a solution. In order to
find candidates for these boundary conditions, we expand the flow equations
(26) and (27) near $\phi=0$ up to order $\phi^{2}$, assuming
$f(g,\phi^{2})=f_{1}(g)\phi^{2}$ and $v(g,\phi^{2})=v_{1}(g)\phi^{2}$. To
determine $v_{1}(g)$ and $f_{1}(g)$ analytically, one can assume
$|f_{1}(g)|,\,|v_{1}(g)|<1$, and expand (26) and (27) to the lowest non-
trivial order in $f_{1}$ and $v_{1}$. The resulting equations are

$\displaystyle 82g\,v_{1}(g)+\left(48\pi-41g^{2}\right)v_{1}^{\prime}(g)$
$\displaystyle=0\,,\,$ $\displaystyle
12g\,f_{1}(g){}^{2}+\left(96\pi^{2}-82\pi g^{2}\right)f_{1}^{\prime}(g)$
$\displaystyle=0\,,$ (12)

and the solutions are

$v^{(0,1)}(g,0)=v_{1}(g)=\frac{41\left(\frac{48\pi}{41}-g^{2}\right)m_{0}^{v}}{48\pi}\,,$
(13)

and

$f^{(0,1)}(g,0)=f_{1}(g)=\frac{m_{0}^{f}}{1-\frac{3}{41\pi}\,m_{0}^{f}\,\log\left(\frac{48\pi-41g^{2}}{48\pi}\right)}\,,$
(14)

where $m_{0}^{v},\,m_{0}^{f}$ are free, but smaller than $1$, parameters.
These boundary conditions are compatible with the NGFP, since they satisfy
$\lim_{g\to g_{*}}v_{1}(g)=\lim_{g\to g_{*}}f_{1}(g)=0$. Furthermore, one of
the degrees of freedom given by this equation is $m_{0}^{f}$, which
parameterizes the relevant direction (8), and the existence of the extra
degree of freedom $m_{0}^{v}$ is a consequence of the differential order of
the RG flow equations. These two free parameters $m_{0}^{f}$ and $m_{0}^{v}$
determine the “infrared mass couplings” of both potentials $f$ and $v$, since
they satisfy $\lim_{g\to 0}f_{1}(g)=m_{0}^{f}$ and $\lim_{g\to
0}v_{1}(g)=m_{0}^{v}$.

Notice that this method to determine the boundary conditions is not unique,
and one can come up with other boundary conditions that are well motivated and
are still compatible with the NGFP and the flow equations themselves. In this
paper, we will restrict to these boundary conditions, but modifications of
those can be found in the supplementary Mathematica notebook.

Using these results, one can close the system of equations with the boundary
conditions (13) and (14), plus
$f(g_{*},\phi^{2})=v(g_{*},\phi^{2})=f(g,0)=v(g,0)=0$. The numerical solution
implementation can be found in the supplementary material, where we use
Mathematica to find a solution for equations (27) and (26) on the range
$\phi\in[0,10]$.

The numerical solutions for the particular choice
$m_{0}^{f}=f^{(0,1)}(0,0)=0.42$ and $m_{0}^{v}=v^{(0,1)}(0,0)=2.75\times
10^{-10}$, can be seen in Fig. 1. These values are fine tuned in order to get
the desired results in the next section, and different values would change the
outcome. In a way, these two degrees of freedom are spanning the set of
possible theories emerging from the NGFP.

Figure 2: Numerical result for the predicted functional dependence of $f$ an
$v$ at vanishing regulator ($k=0$), in the range $\phi\in[0,10]$, for the
choices $m_{0}^{f}=0.42$ and $m_{0}^{v}=2.75\times 10^{-10}$.

In Fig. 1 we can see that the shift-symmetry of the NGFP is broken when
$g<g_{*}$, i.e. when the coarse graining scale $k$ is less than infinity. This
type of solutions are only allowed thanks to the symmetry breaking relevant
direction (8), without which the potentials $f$ and $v$ will remain $0$
throughout the entire flow, preserving the shift-symmetry of the NGFP.

The interaction potentials obtained at $k=0$, $f(0,\phi^{2})$ and
$v(0,\phi^{2})$ can be extracted from Fig. 1. Their functional dependence is
depicted in Fig. 2. One can see that the predicted potentials $f(0,\phi^{2})$
and $v(0,\phi^{2})$ have non-trivial dependence with the scalar field in the
studied range. These results are subject to a particular choice of the free
parameters, and different choices give different results. At least for the
analysed values, one can see that the effect of changing the parameters
$m_{0}^{f}$ and $m_{0}^{v}$ is an overall multiplication factor, and does not
change the shape of the potentials (see appendix VIII.2).

Now, we are interested in the phenomenology of the predicted potentials. We
found that this is easier to understand after a change of variables in the
effective action, as we will show in the next section.

##  V Emergence of Inflaton Potential

The phenomenological consequences of the potentials obtained in Fig. 2, can be
easily interpreted after a conformal change of variables. Specifically, the
scalar-tensor model given in (2) is expressed in the so called “Jordan frame”,
but the same action can be transformed to the “Einstein frame” (E) [41, 42].
To change to such frame, one performs a conformal transformation

$g^{E}_{\mu\,\nu}\equiv(1+f(g,\phi^{2}))g_{\mu\,\nu}$ (15)

such that the action (2) is turned into

$\displaystyle\Gamma_{k}=\int d^{4}x\sqrt{g^{E}}(-\frac{R^{E}}{16\pi
g^{2}}+\frac{V_{eff}(g,\phi(\sigma))}{(16\pi
g^{2})^{2}}+\frac{\partial_{\mu}\sigma\partial^{\mu}\sigma}{2\,g^{2}}),$ (16)

where

$V_{eff}(g,\phi(\sigma))\doteq\frac{2\lambda(g)+v(g,\phi^{2}(\sigma))}{(1+f(g,\phi^{2}(\sigma))^{2}},$
(17)

and where the new scalar field satisfies

$(\frac{\partial\,\sigma}{\partial\,\phi})^{2}=\frac{1}{(1+f(g,\phi^{2}))}+\frac{3\,(f^{{}^{\prime}}(g,\phi^{2}))^{2}}{(1+f(g,\phi^{2}))^{2}}.$
(18)

As we said before, we are interested in the phenomenology of the effective
potentials once all quantum fluctuations are integrated out. This is obtained
by looking at $V_{eff}(0,\phi(\sigma))$. In Fig. 3 one can see the resulting
effective potential as a function of the variable $\phi$. In this result we
approximated $\lambda(0)\simeq 10^{-120}\simeq 0$, since we found it made no
difference in the numerical predictions. Presenting the results as a function
of $\phi$ is merely for simplicity, and the phenomenology is equivalent for
the potential as a function of $\sigma$. If one wants to see the shape of the
potential as a function of $\sigma$, one must solve (18) numerically, at
$g=0$. This can be found in the supplementary material.

Figure 3: Obtained effective potential for the inflaton field as a function of
$\phi$, in the range $\phi\in[0,10]$. The zoomed in region represents the
region where a period of cosmological inflation, compatible with observations,
can occur. The initial and final values of the field for generating such
period are $\phi_{i}=1.55$ and $\phi_{f}=0.25$.

As one can immediately see in Fig. 3, the potential presents an approximate
plateau at values of $\phi$ between $1.5$ and $3$, which, as mentioned before,
can be interpreted equivalently to values of the original scalar field
$\tilde{\varphi}$ between $1.5$ and $3$ times the Planck mass.

This lead us to compute inflationary observables associated with a field
starting at some value of such plateau. In the way the potential $V_{eff}$ is
defined, the slow-roll observables [43] take the form

$\displaystyle\epsilon$
$\displaystyle=\frac{1}{8\pi}\frac{1}{2}(\frac{V_{eff}^{{}^{\prime}}}{V_{eff}})^{2}\,\,\,\,\,\,\,\,\eta=\frac{1}{8\pi}(\frac{V_{eff}^{{}^{\prime\prime}}}{V_{eff}})$
$\displaystyle\quad n_{s}$
$\displaystyle=1-6\,\epsilon+2\,\eta\,\,\,\,\,\,\,\,\,\,r=16\,\epsilon$
$\displaystyle\quad A_{s}$
$\displaystyle=\frac{1}{4}\frac{V_{eff}}{24\pi^{2}\epsilon}\,\,\,\,\,\,\,\,\,\,\,\,\,N_{ef}=8\pi\int_{\sigma_{i}}^{\sigma_{f}}\frac{d\,\sigma}{\sqrt{2\,\epsilon}},$
(19)

where the prime denotes a derivative with respect to $\sigma$, $n_{s}$ is the
spectral index, $r$ is the scalar to tensor ratio, $A_{s}$ is the amplitude of
scalar fluctuations and $N_{ef}$ is the number of e-folds between the initial
value $\sigma_{i}$ and the final $\sigma_{f}$. The parameters $\epsilon$ and
$\eta$ are the so called slow-roll parameters. The condition for slow-roll
inflation to happen, is that they are sufficiently small
($|\epsilon|,\,|\eta|<<1$) for a certain range of values of the scalar field.
Here we will compute everything in terms of the field $\phi$, using the chain
rule and the relation between $\sigma$ and $\phi$ given by (18).

If we assume the start (of the end) of inflation is at $\phi_{i}=1.55$, and
determine the final of inflation when one of the slow roll parameters becomes
$1$ (in our case it was $\epsilon$), it gives us the value for the field
$\phi_{f}=025$. For that interval, one can say that the field slowly rolls on
the potential, and one has the following results for the inflationary
observables

$\displaystyle\quad n_{s}$ $\displaystyle\simeq
0.965\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r\simeq 0.005$
$\displaystyle\quad A_{s}$ $\displaystyle\simeq 2.06\times
10^{-9}\,\,\,\,\,\,\,N_{ef}\simeq 66\,,$ (20)

where $n_{s}$, $r$ and $A_{s}$ where computed at $\phi_{i}$, and $N_{ef}$
between $\phi_{i}$ and $\phi_{f}$. These values are in agreement with current
observations made by [13].

At this stage, it is important to emphasise that these results where obtained
on the basis of small amount of fine tuning. Only the numerical value of two
mass couplings ($m_{0}^{f},\,m_{0}^{v}$) had to be fixed in order to obtain
this result. The non-trivial field dependence of the obtained effective
potential, is the result of the interplay between the gravitational degrees of
freedom and the scalar degrees of freedom throughout the renormalization group
flow between the NGFP and the infrared. To put it in other words, its shape is
a consequence of the quantization of gravity plus the scalar degree of
freedom, and its UV completion.

We have achieved what we claimed at the beginning of this work, and obtained,
from a UV completion of gravity, an effective potential for a scalar degree of
freedom that is consistent with a period of cosmological inflation. This
potential is derived, not constructed, from fundamental physics, using only
Wilsonian renormalization group techniques.

Up to now, every result was independent of any identification of the coarse
graining scale with a physical scale, such as the curvature of spacetime, or
with the age of the universe, as is often done in the so called “RG
improvement” procedures. All our results were obtained exactly in the limit of
vanishing regulator. Nevertheless, one can take one more step. In the
following, we will accept that one can relate the renormalization group scale
$k$ with some physical scale, such that there is an identification between $k$
and physical time. This identification should be such that large values of $k$
are associated with early times of the universe, and small values of $k$ with
later times. We do not need any particular identification, since our
discussion does not require it.

If one accepts that there is such identification, an argument that has been
largely used [44, 45, 46, 47, 48, 49, 50], but also very criticised [51, 52],
one can provide an explanation for the initial value of the inflaton field
near the end of inflation. In Fig. 4, one can see snapshots of the RG flow of
the effective potential $V_{eff}$ as a function of $\phi$, in the studied
range. Interestingly, the potential flows from the scale-invariant, and shift
symmetric NGFP, where it takes the value
$\lim_{k\to\infty}V_{eff}=2\lambda_{*}$, given by (11), to the inflaton
potential shown in Fig. 3.

Figure 4: Snapshots of the RG evolution of $V_{eff}$, from the UV to the IR in
logarithmic scale. The adiabatic evolution of the universe is shown by the
dashed line, where it follows a stable minimum from the UV, that becomes
unstable near $k\simeq 0$, around $\phi_{m}\simeq 2.3$.

A tantalizing idea, extracted from the RG evolution of $V_{eff}$, is that if
the very early universe was effectively described by an action close to (2)
near the NGFP, the initial conditions for the inflaton field don’t have to be
fine tuned. This is because the NGFP is scale invariant, so the action at that
fixed point will be independent of the value of the scalar field, and
therefore the universe can have arbitrary initial conditions ($\phi_{b}$) for
the field $\phi$ at its beginning. Furthermore, for values of $k<\infty$,
$V_{eff}$ develops a stable minimum as a function of the scalar field $\phi$.
If the universe begins near the NGFP, and is let to evolve adiabatically, the
scalar field values will follow that minimum. This minimum becomes unstable
near $k\simeq 0$, around $\phi_{m}\simeq 2.3$. This value is slightly above
$\phi_{i}$, so that the field can roll down to $\phi_{i}$, where we start
seeing the primordial fluctuations. Between $\phi_{m}$ and $\phi_{i}$ the
universe experiences more than 70 e-folds, thus this region is not observable.
An adiabatic evolution of the universe, with respect to the RG flow, will
follow this trajectory and might explain the initial conditions of the
inflaton near the end of inflation, without the need to fine tune its initial
value, since it can start at an arbitrary value in the UV ($\phi_{b}$).

In Fig. 4, snapshots of such evolution are presented, where the dashed line
represent the values of the scalar field if the universe evolves
adiabatically, with an arbitrary starting value of the scalar field $\phi_{b}$
at the beginning of the universe. Once all quantum fluctuations are integrated
out, the stable minimum becomes unstable, and the end of inflation begins,
giving rise to the observed fluctuations.

These and similar arguments have been used in several works before to describe
the phenomenology of the RG flow, in particular when using RG improvements
techniques, but also in other approaches, like using scale invariant solutions
of the Wetterich equation [53, 54, 55, 56]. The underlying justification for
the assumption that the universe at its beginning is described by the highest
energy degrees of freedom of the path integral ($\Gamma_{k}$ for large $k$)
relies on the interpretation of $\Gamma_{k}$. Since at large values of $k$,
$\Gamma_{k}$ contains information only of the highest energy modes of the
theory, it is reasonable to believe that a denser universe in its beginnings
will be mostly described by those degrees of freedom. As the universe evolves
in time, it expands and becomes less dense, thus requiring the inclusion of
lower energy modes for an accurate effective description ($\Gamma_{k}$ for
lower values of $k$). We would like to emphasise that we did not use any
particular identification of $k$ with a physical quantity, and if one desires
to study details of the time evolution of the universe using Fig. 4, one
should stick to a particular identification, such as $k\propto\frac{1}{t}$ or
something more sophisticated.

##  VI Conclusions and outlook

In this work, we studied the RG flow of a scalar tensor model given by (2),
with two arbitrary potentials for the scalar field. We showed the existence of
an UV fixed point, with a finite number of relevant directions, and we used
that fixed point to integrate the RG flow towards the limit of vanishing
regulator, where one can make physical predictions.

By means of this integration, we were able to predict the interaction
potentials for the scalar field, fixing only two degrees of freedom, called
$m_{0}^{f}$ and $m_{0}^{v}$ (see section IV). These two degrees of freedom
span a continuum $2$-dimensional subspace of theory space where the Asymptotic
Safety hypothesis is realized.

By fixing $m_{0}^{f}=0.42$ and $m_{0}^{v}=2.75\times 10^{-10}$, we determined
numerically the effective potentials of the scalar field up to field values of
$10$ times the Plank mass (see Fig. 2). Furthermore, by means of a conformal
change of variables, going form the Jordan to the Einstein frame, we showed
that our calculations give rise to an effective potential that is compatible
with a slow roll inflationary period, driven by a single scalar field. The
slow-roll observables obtained from such effective potential are compatible
with current observations, giving the results $n_{s}\simeq 0.965,\,r\simeq
0.005,N_{ef}\simeq 66,\,A_{s}\simeq 2.06\times 10^{-9}$ (see Fig. 3).

Furthermore, since the effective potential is shift-symmetric in the UV, we
showed that a universe that evolves adiabatically towards the IR, and starts
near the NGFP, does not require a fine tuning of the initial condition of the
inflaton field. In fact, following such evolution, we determined that the
inflaton field reaches an inflection point of the potential slightly above the
value necessary to get more than 60 e-folds at the end of inflation, from
where the scalar field can slowly roll down to the true minimum, giving rise
to the observed fluctuations (see Fig. 4).

Our work leaves room for further improvement. All of our calculations of the
RG flow were done in euclidean signature of the metric, and we could only
compute physical quantities because we assumed a Bianchi type-I space-time,
where a Wick rotation can be performed. Furthermore, the used RG flow
equations, rely on a particular gauge choice, and a particular infrared
regulator. On top of that, those equations are an approximation of the full
flow equations, where one neglects the RG scale derivatives of the potentials
in some sectors of the full flow equations. As mentioned before, a detailed
discussion of such gauge elections, regulator and approximations can be found
in [35]. It would be interesting to see if our results continue to hold
varying such choices, for example using Lorentzian flows and the ADM
formalism. If this is the case, these results would be compelling evidence in
favor of Asymptotic Safety scenario for gravity.

##  VII Acknowledgements

The author is especially thankful to Frank Saueressig, for enlightening
discussions, and for pinpointing key details of this work. The author also
appreciates useful comments from Renate Loll, Cristóbal Laporte, and Jian Wang
during the development of this work.

##  VIII Appendix

###  VIII.1 Marginally relevant direction

In order to determine if the direction (8) is relevant at higher order in
perturbations around the fixed point, we investigated it within a truncation
of the action. In particular, if we assume that

$F(k,\varphi)\simeq F0(k)+F1(k)\varphi^{2}\,\,\,\,\,V(k,\varphi)\simeq
V0(k)+V1(k)\varphi^{2}\,,$ (21)

we can expand the flow equations (5) and (4) up to order $\varphi^{2}$, and
obtain the flow equations for the couplings $F0(k)$, $V0(k)$, $F1(k)$ and
$V1(k)$. In particular, the perturbation direction (8) can be translated in
this truncation into the perturbation $F0(k)\simeq F_{*}-\delta F0(k)$ and
$F1(k)\simeq 32\pi\delta F1(k)$, and fixing the other couplings to their fixed
point value $V0_{*}=V_{*},\,V1_{*}=0$, see (6). Such perturbation satisfies
the flow equations

$\displaystyle k\,\frac{\partial\,\delta F0(k)}{\partial k}$
$\displaystyle=-2(\delta F0(k)-\delta F1(k))\,,$ $\displaystyle
k\,\frac{\partial\,\delta F1(k)}{\partial k}$
$\displaystyle=-\frac{3072\pi^{2}\text{$\delta F1(k)$}^{2}}{41}\,,$ (22)

with solutions

$\displaystyle\delta F1(k)$
$\displaystyle=\frac{41}{3072\pi^{2}\log\left(\frac{k}{\text{k1}}\right)}\,,$
$\displaystyle\delta F0(k)$
$\displaystyle=\frac{\text{k0}^{2}}{k^{2}}-\frac{41\text{k1}^{2}\text{Ei}\left(2\log\left(\frac{k}{\text{k1}}\right)\right)}{1536\pi^{2}k^{2}}\,,$
(23)

where $k0$ and $k1$ are finite positive real numbers, Ei is the exponential
integral, and because we are expanding near the NGFP, we assumed $k>>k0,\,k1$.
One can notice that $\lim_{k\to\infty}\\{\delta F0(k),\,\delta
F1(k)\\}=\\{0,\,0\\}$, thus making the perturbation (8) relevant.

###  VIII.2 Different boundary conditions

As was mentioned in the bulk of the paper, the results obtained in Figs. 2,
and therefore in 3 and 4 are subject to a particular choice for the free
parameters $m_{0}^{f}$ and $m_{0}^{v}$. Different choices for these parameters
lead to different outcomes, and its effect in the obtained potentials is non-
trivial. In Fig. 5 and 6 you can observe the result of changing these
parameters to different values on the predicted potentials $f$ and $v$.

One can see that the effect of changing the values of the parameters
$m_{0}^{f}$ and $m_{0}^{v}$ seems to be mostly an overall multiplication
factor, since the shape of these potentials are very similar so each others.
We do not have a proof of this conjecture, but it would be interesting to
determine of this holds on a more formal setting than just looking at the
results for some different values of $m_{0}^{f}$ and $m_{0}^{v}$.

Figure 5: Effective potential $f$ in the limit of vanishing regulator ($k=0$)
for different choices of the boundary condition parameter $m_{0}^{f}$, and
fixing $m_{0}^{v}=2.75\times 10^{-10}$. With dashed lines is the reference
potential used in this paper, corresponding to $m_{0}^{f}=0.42$. Figure 6:
Effective potential $v$ in the limit of vanishing regulator ($k=0$) for
different choices of the boundary condition parameter $m_{0}^{v}$, and fixing
$m_{0}^{f}=0.42$. With dashed lines is the reference potential used in this
paper, corresponding to $m_{0}^{v}=2.75\times 10^{-10}$.

Not all the analysed values give rise to an effective potential in the
Einstein frame that is compatible with observations, although, one can find
other combinations apart from the ones showed in this paper that still work.
The resulting effective potentials have different forms though, so their
impact on other predictions, like primordial black holes, preheating and
reheating can be different.

###  VIII.3 Equations for the unknown scalar-field potentials

Equations (5) and (4) are computed by keeping $\varphi$ independent of $k$,
but we are interested in finding solutions in the variable $\phi$, keeping
$\phi$ independent of $k$. This can be achieved by the following chain of
replacements into equations (5) and (4), applied in this specific order

$\displaystyle kF^{(1,0)}(k,\varphi)$
$\displaystyle\to\frac{\beta_{g}}{g_{k}}\,\phi\,\mathcal{F}^{(0,1)}(k,\phi)+k\mathcal{F}^{(1,0)}(k,\phi)$
$\displaystyle\quad F^{(0,n)}(k,\varphi)$ $\displaystyle\to
g_{k}^{n}\mathcal{F}^{(0,n)}(k,\phi)$ $\displaystyle\quad
kV^{(1,0)}(k,\varphi)$
$\displaystyle\to\frac{\beta_{g}}{g_{k}}\,\phi\,\mathcal{V}^{(0,1)}(k,\phi)+k\mathcal{V}^{(1,0)}(k,\phi)$
$\displaystyle\quad V^{(0,n)}(k,\varphi)$ $\displaystyle\to
g_{k}^{n}\mathcal{V}^{(0,n)}(k,\phi)$ $\displaystyle\quad\varphi$
$\displaystyle\to\frac{\phi}{g_{k}},$ (24)

with $n=0,1,2$, $\beta_{g}=k\frac{\partial g_{k}}{\partial k}$, and
$\mathcal{F}$ and $\mathcal{V}$ being intermediate auxiliary scalar functions.
Once this is done, we will use the splitting given by (9), that can be
implemented by performing the following replacements

$\mathcal{F}(k,\phi)\to\frac{1+f(g_{k},\phi^{2})}{16\pi
g_{k}^{2}}\,\,\,\,\,\mathcal{V}(k,\phi)\to\frac{2\lambda(g_{k})+v(g_{k},\phi^{2})}{(16\pi
g_{k}^{2})^{2}}\,,$ (25)

and the corresponding partial derivatives.

On the resulting equations, one can determine $\beta_{g}$ and
$\frac{\partial\lambda(g)}{\partial g}$ by evaluating each equation at
$\phi=0$ and using that $f(g_{k},0)=v(g_{k},0)=0$. Once this is done, an
interesting feature is that $\lambda$ decouples from the equations, leaving
only partial differential equations for $f$ and $v$ as functions of $g$ and
$\phi$.

As a result of these transformations, one ends up with two coupled, second
order, non-linear, partial differential equations for $f(g,\phi^{2})$ and
$v(g,\phi^{2})$, (26) and (27).

$\displaystyle\frac{256\pi
g^{7}\left(4\pi(f(g,\phi^{2})+1)\left(v^{(0,1)}(g,\phi^{2})+2\phi^{2}v^{(0,2)}(g,\phi^{2})\right)-v^{(0,1)}(g,0)\left(3\phi^{2}f^{(0,1)}(g,\phi^{2})^{2}+4\pi(f(g,\phi^{2})+1)\right)\right)}{\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)\left(32\pi
g^{2}\left(3\phi^{2}f^{(0,1)}(g,\phi^{2})^{2}+4\pi(f(g,\phi^{2})+1)\right)+(f(g,\phi^{2})+1)\left(v^{(0,1)}(g,\phi^{2})+2\phi^{2}v^{(0,2)}(g,\phi^{2})\right)\right)}$
$\displaystyle-\frac{\left(24576\pi^{3}g^{7}f^{(0,1)}(g,0)-g\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)\left(\left(45g^{2}-48\pi\right)v^{(0,1)}(g,0)+128\pi^{2}\left(41g^{2}-48\pi\right)g^{2}\right)\right)\left(2v(g,\phi^{2})\right)}{24\pi\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)^{2}}$
$\displaystyle+\frac{\left(24576\pi^{3}g^{7}f^{(0,1)}(g,0)-g\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)\left(\left(45g^{2}-48\pi\right)v^{(0,1)}(g,0)+128\pi^{2}\left(41g^{2}-48\pi\right)g^{2}\right)\right)\phi^{2}v^{(0,1)}(g,\phi^{2})}{24\pi\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)^{2}}$
$\displaystyle-\frac{g^{2}\left(\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)\left(\left(45g^{2}-48\pi\right)v^{(0,1)}(g,0)+128\pi^{2}\left(41g^{2}-48\pi\right)g^{2}\right)-24576\pi^{3}g^{6}f^{(0,1)}(g,0)\right)v^{(1,0)}(g,\phi^{2})}{48\pi\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)^{2}}$
$\displaystyle+g\left(-2\phi^{2}v^{(0,1)}(g,\phi^{2})+4v(g,\phi^{2})\right)=0$
(26)
$\displaystyle\frac{48\pi\phi^{2}f^{(0,1)}(g,\phi^{2})}{g^{2}}+37-\frac{48\pi(f(g,\phi^{2})+1)}{g^{2}}$
$\displaystyle-\frac{\phi^{2}\left(24576\pi^{3}g^{6}f^{(0,1)}(g,0)-\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)\left(\left(45g^{2}-48\pi\right)v^{(0,1)}(g,0)+128\pi^{2}\left(41g^{2}-48\pi\right)g^{2}\right)\right)f^{(0,1)}(g,\phi^{2})}{g^{2}\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)^{2}}$
$\displaystyle-\frac{\left(\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)\left(\left(45g^{2}-48\pi\right)v^{(0,1)}(g,0)+128\pi^{2}\left(41g^{2}-48\pi\right)g^{2}\right)-24576\pi^{3}g^{6}f^{(0,1)}(g,0)\right)f(g,\phi^{2})}{g^{2}\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)^{2}}$
$\displaystyle-\frac{\left(24576\pi^{3}g^{6}f^{(0,1)}(g,0)-\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)\left(\left(45g^{2}-48\pi\right)v^{(0,1)}(g,0)+128\pi^{2}\left(41g^{2}-48\pi\right)g^{2}\right)\right)f^{(1,0)}(g,\phi^{2})}{2g\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)^{2}}$
$\displaystyle-\frac{2\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)\left(\left(45g^{2}-48\pi\right)v^{(0,1)}(g,0)+128\pi^{2}\left(41g^{2}-48\pi\right)g^{2}\right)-49152\pi^{3}g^{6}f^{(0,1)}(g,0)}{2g^{2}\left(v^{(0,1)}(g,0)+128\pi^{2}g^{2}\right)^{2}}$
$\displaystyle+\frac{8(f(g,\phi^{2})+1)\left(256\pi^{2}g^{4}\left(3\phi^{2}f^{(0,1)}(g,\phi^{2})^{2}+4\pi(f(g,\phi^{2})+1)\right)\left(-3f^{(0,1)}(g,\phi^{2})-6\phi^{2}f^{(0,2)}(g,\phi^{2})+8\pi\right)\right)}{\left(32\pi
g^{2}\left(3\phi^{2}f^{(0,1)}(g,\phi^{2})^{2}+4\pi(f(g,\phi^{2})+1)\right)+(f(g,\phi^{2})+1)\left(v^{(0,1)}(g,\phi^{2})+2\phi^{2}v^{(0,2)}(g,\phi^{2})\right)\right)^{2}}$
$\displaystyle+\frac{8(f(g,\phi^{2})+1)\left(48\pi
g^{2}\left(3\phi^{2}f^{(0,1)}(g,\phi^{2})^{2}+4\pi(f(g,\phi^{2})+1)\right)\left(v^{(0,1)}(g,\phi^{2})+2\phi^{2}v^{(0,2)}(g,\phi^{2})\right)\right)}{\left(32\pi
g^{2}\left(3\phi^{2}f^{(0,1)}(g,\phi^{2})^{2}+4\pi(f(g,\phi^{2})+1)\right)+(f(g,\phi^{2})+1)\left(v^{(0,1)}(g,\phi^{2})+2\phi^{2}v^{(0,2)}(g,\phi^{2})\right)\right)^{2}}$
$\displaystyle+\frac{8(f(g,\phi^{2})+1)\left((f(g,\phi^{2})+1)\left(v^{(0,1)}(g,\phi^{2})+2\phi^{2}v^{(0,2)}(g,\phi^{2})\right)^{2}\right)}{\left(32\pi
g^{2}\left(3\phi^{2}f^{(0,1)}(g,\phi^{2})^{2}+4\pi(f(g,\phi^{2})+1)\right)+(f(g,\phi^{2})+1)\left(v^{(0,1)}(g,\phi^{2})+2\phi^{2}v^{(0,2)}(g,\phi^{2})\right)\right)^{2}}=0$
(27)

An implementation of these replacements can be found in the supplementary
Mathematica notebook. Despite the complex nature of (26) and (27), these
equations are defined on a compact interval $g\in[0,\sqrt{\frac{48\pi}{41}}]$,
and can be solved numerically for any range of values of the scalar field
$\phi$. They do not require any scaling with $k$, and from these equations one
can directly extract the predictions in the limit of vanishing regulator
($k=0$) by evaluating the solutions at $g=0$.

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University of Tripoli
Faculty of Engineering
Department of Electrical and Electronic Engineering
EE491 COMPUTER APPLICATIONS AND DESIGN LAB

Implementation of DNN Based Data Detector for QPSK Systems

Written by: Ahmed Mustafa Badi

ID Number: 22160644

Email<EMAIL_ADDRESS>
Instructor: Dr. Nadia Adam

## Abstract

In this project, the Quadrature Phase Shift Keying (QPSK) digital modulation
scheme was implemented using Software Defined Radios (SDRs). For this system,
a deep learning based detector was proposed and implemented alongside the
conventional method. The implementation was successfully achieved for both the
conventional and deep learning based data detection techniques, despite the
challenges faced. The results show that the proposed deep learning method is
able to outperform the conventional detector. The code of this project is made
publicly accessible at https://github.com/ABadi13/QPSK_SDR_DNN_Detector.

##  1 Introduction

###  1.1 Objectives

To show experimentally and discuss, the implementation of the QPSK digital
modulation scheme. Furthermore, to utilize deep learning techniques to design
and develop a data detection technique for the QPSK modulation scheme.

###  1.2 Background

As the world’s technology becomes ever more sophisticated and powerful,
techniques such as Artificial Intelligence (AI) were once thought of as a pipe
dream, now AI has been applied to almost every field imaginable. The unique
abilities and characteristics of Artificial Neural Networks (ANNs) have made
this possible, most notably, their reduced complexities when compared to
conventional methods with the same performance. This reduced complexity, not
only enables a faster computation time, but also reduced energy consumption
and computational requirements. Furthermore, ANNs can provide better awareness
of their operational environments, with the ability respond to any
imperfections that cannot be accounted for, using conventional methods.

ANNs and Deep Neural Networks (DNNs) have been undergoing substantial
advancements in the field of wireless communication, replacing conventional
methods in many applications and opening up opportunities in many others.
Their reduced complexities allow for cheaper equipment, reduced latency,
reduced power consumption, etc. Their greater awareness enables end-to-end
learning, allowing them to jointly perform multiple functions, even further
simplifying the system and reducing components [1].

Channel estimation, in particular, has benefited greatly from DNNs. For
instance, the authors in [2] proposed a DNN method for joint channel
estimation and symbol detection for OFDM systems. The results of which were
promising, with the DNN performing comparably to the conventional channel
estimation technique Minimum Mean Square Error (MMSE), and outperforming it in
low overhead situations. Additionally, the authors in [3] proposed a DNN pair
for joint channel estimation and signal detection for the developing spatial
modulation scheme. It was concluded that the proposed DNN pair outperforms
conventional methods in highly dynamic channel conditions. Instances such as
these and many others, show the potential of deep learning in channel
estimation.

Cognitive radio is built upon making intelligent decisions to improve the
overall utilization of the frequency spectrum. It is a method that allows an
unlicensed user to locate and utilize unused licensed spectrum, while also
ensuring no interference is caused to the licensed user. One such challenge
faced especially in cognitive radio is Automatic Modulation Classification
(AMC), in [4] a Convolution Neural Network (CNN) based AMC approach is
proposed. The simulation results showed that the proposed CNN based approach
was able to outperform the feature based approach and closely match the much
more complex maximum likelihood approach, while maintaining significantly
lower computational complexity.

There are many downsides to using DNNs in wireless communication, one
important drawback is the DNN’s inherent need for large datasets for training.
The increased number of layers and neurons need more data to learn the
considered system without overfitting. More data is not much of an issue in
other fields, although, in wireless communication this is much more difficult.
As opposed to regular ANNs, where typically features are manually extracted
from the dataset, DNNs typically extract features internally. Meaning that for
DNNs, there is a much larger emphasis on the data used.

To combat the large amounts of data, mathematical model-based channels have
been utilized to simulate real world environments. Given full Channel State
Information (CSI), the model can very effectively estimate the channel.
However, the CSI is not always fully available, and some channels can be very
dynamic in nature. If the data used to train the DNN is not representative of
the real-world channel, then a significant loss in performance is expected.

Software Defined Radios (SDRs), provide the ability to transmit a signal
generated and processed in software, or receive a signal to then manipulate it
in software. This device practically permits the user freedom in creating any
conceivable system, within the realm of the SDRs capability. Real world data
can easily be produced using this device, enabling the implementation of ANNs
and DNNs directly.

The developments and advancements recently made in SDR technology have been
rather rapid in recent years. Several low-cost consumer-focused SDRs have been
released such as HackRF One, RTL-SDR, BladeRF, etc. With such success for
SDRs, more development and competition is bound to happen in the SDR market,
paving the way for technological advancements and accessibility. Academic
research utilizing SDR technology has been seen significant success, for
instance, the authors in [5] have implemented cooperative communication scheme
using SDR technology. The results show that the cooperative approach achieves
significant performance enhancements. Furthermore, the autors in [6] presented
a digitally enhanced SDR receiver robust to out of bound interference.

More recently, ANNs have been implemented using SDR technology. Instances such
as [7] and [8] show the potential of utilizing SDR technology for real world
implementation of wireless communication techniques. The authors in [7]
designed an ANN based AMC that is able to perform over a wide range of Signal
to Noise Ratio (SNR) and then implemented on a SDR test-bed, proving the
feasibility of the proposed technique. While in [8] the authors implement two
ANNs to jointly determine the presence of a jammer as well as its
characteristics. The results in [8] showed that the proposed methods can
detect jamming with an $85\%$ accuracy. Despite instances such as this and
many others, along with deep learning techniques becoming more relevant than
ever in the field of wireless communication, research on the implementation of
ANNs leaves much to be desired.

##  2 Proposed System Architecture

This section will outline and discuss the system architectures and algorithms
necessary to modulate and demodulate a randomly generated bitstream, using a
QPSK digital modulation scheme with a DNN based data detector and implemented
using SDRs.

###  2.1 QPSK System Architecture

The QPSK system is illustrated in the form of a block diagram in Fig. 1. It is
made up of two main sections, the transmitter and the receiver, separated by a
wireless channel. Both the transmit and receive SDRs were connected to the
same computer, the signal processing, however, will remain entirely
independent of each other, with no feedback loop.

Figure 1: QPSK system utilizing SDR with a DNN based detector

At the transmitter side the GNU Radio Companion software is used to generate a
random bit stream, it is then QPSK baseband modulated. The modulated signal is
then sent to the _HackRF SDR_ to be transmitted over the wireless Radio
Frequency (RF) channel.

The implemented channel was a short clear Line Of Site (LOS) channel, about a
meter in length. The reason for this choice was due to the SDR’s incapability
of high power transmission. The sample rate for both SDRs was 2 MHz (samples
per second), the lowest common sample rate both SDRs can operate on.

For synchronization and channel estimation, there will be a pilot every frame,
this pilot will be omitted when carrying out any performance evaluation. The
pilot structure must also remain consistent throughout the entire process.

On the receiver side, the _RTL-SDR_ receives the signal sent by the HackRF and
then directs it to the GNU Radio Companion software. The received signal is
then filtered using an LPF to remove any unnecessary noise. It is then
synchronized using a Polyphase Clock Synchronizer and a Costas loop. Once the
signal is synchronized, it is stored so that it can be decoded using the
conventional method and the proposed DNN. Using the conventional method helps
form a baseline for the performance results of the DNN.

###  2.2 Algorithm for Creating the Datasets

Figure 2: Flowchart outlining the algorithm used for producing the datasets
for training and testing.

The flowchart shown in Fig. 2 illustrates the algorithm this system uses to
create datasets and evaluate them using the conventional and the DNN methods.
In order to adequately train DNNs with supervised learning techniques,
datasets with accurate labels need to be made. These datasets contain the
original bitstream, the received bitstream decoded using the conventional
method, and the synchronized received signal.

The flowchart starts of as was explained in the previous section, a randomly
generated bitstream is modulated, transmitted with the HackRF, received with
the RTL-SDR, and then synchronized.

After this, the synchronized signal is then decoded using conventional
methods. Using the decoded bitstream, a pilot search algorithm is run, if no
pilots are found then an error has occurred, and the process of transmission
is repeated. If the pilots were found, however, another algorithm is run in
order to find the delay of the decoded bitstream. If the delay cannot be found
then there is an error and the transmission is repeated. If both the phase and
the delay can be found, then the training and testing datasets can be created.
The training dataset is fed to the DNN to be trained. Then the testing dataset
is evaluated using both the conventional and the proposed DNN methods.

It is important to note here that the deployment of a trained DNN is not
dependent on the conventional system. It is however, dependent on the
conventional system during training to determine only the system delay, in
order to accurately label the input samples for supervised learning. It does
not use the bitstream decoded using the conventional methods.

###  2.3 GNU Radio Companion Flowchart

The signal processing will be carried out through the open-source software GNU
Radio Companion. This software uses blocks that perform basic operations, when
connected together these blocks make up a flowgraph, like that shown in Fig.
3. With the flexibility this software allows, a QPSK transmitter and receiver
was assembled, each of their corresponding sections are shown in Fig. 3.

Figure 3: GNU Radio Companion flowchart of the considered QPSK system.

Firstly, the random message signal is generated, and for synchronization a
pilot is then inserted once every frame. The random bitstream is then QPSK
modulated at the baseband using the flexible Constellation Modulator, which
modulates according to its constellation object parameter. This block also
uses a Root Raised Cosine (RRC) pulse shaping filter to reduce Inter-Symbol
Interference (ISI). The modulated signal is then sent to the HackRF SDR which
is supported by the OsmoSDR standard. The chosen sample rate was $2$ MHz, with
$8$ samples/symbol in order to reduce the signal processing on the computer.
The operating frequency was set to $1.033$ GHz for no reason other then being
unoccupied.

After the signal passes through the wireless channel, it is received by the
RTL-SDR, and then sent back to the computer for processing. An LPF is used in
order to remove any unnecessary noise from the received signal. After this,
the signal is timing synchronized with the Polyphase Clock Sync. It is
equalized using the Constant Modulus Adaptive (CMA) equalizer, which is used
for constant amplitude (as the name would suggest) signals, such as the QPSK
signal used for this system. Carrier synchronization is achieved using the
Costas Loop, the carrier frequency is extracted and the signal is reverted
back to the baseband, eliminating any frequency offset experienced by the
signal. Finally, after the signal has been synchronized it can be demapped.
This signal was then stored and demapped using both, the conventional method
utilizing the Constellation Decoder block, and the proposed DNN discussed in
the following section. The Unpack blocks are used in order to convert the
symbols into bitstreams, so that they can be more easily manipulated when
creating the datasets.

###  2.4 DNN Architecture

The proposed DNN system attempts to take the place of the Constellation
Decoder. By utilizing the synchronization pilots, the DNN is able to perform
channel estimation and potentially improve on the Bit Error Rate (BER) of the
conventional method. The proposed DNN will perform the decoding on a frame-by-
frame basis. Meaning that the DNN will take the entire frame into
consideration when performing the decoding. Its inputs will be all the samples
of the synchronized signal in a frame separated into their real and imaginary
counterparts. The hidden layers will utilize the _ReLU_ activation function.
The output layer uses the _Sigmoid_ activation function, and attempts to
decode the entire frame as bits, excluding any pilots.

Due to the inherent nature of the DNN requiring large amounts of data for
training, the network will be deployed in two separate stages, namely,
_training_ and _testing_. The training stage consists of producing the
necessary datasets, this can be achieved by transmitting a priorly known
modulated signal, and trained accordingly using supervised learning. After the
training stage has been fully fulfilled, the testing stage will be implemented
using the produced dataset. In this stage the received modulated signal will
be demodulated on a frame-by-frame basis.

##  3 Methodology and Challenges

In this section, the methods used and the challenges encountered during the
implementation of the proposed DNN based data detector for the QPSK digital
modulation scheme will be outlined and discussed.

###  3.1 Deep Learning in Wireless Communications

Deep learning has seen a substantial amount of research regarding its
simulation and how it can be introduced in many applications. Although, real
world implementation of machine learning and deep learning of these suggested
techniques is not as common as one would assume. Implementing deep learning
presents many challenges and potentially allows even further advances in
research, and the techniques that can be used to implement it.

The Python TensorFlow environment was used to train and test the DNN, due to
its robustness, speed, and flexibility.

###  3.2 The SDR Platform

Until very recently, implementation using SDRs could only be accomplished with
the high-cost Universal Software Radio Peripheral (USRP) devices. Fortunately,
low cost SDRs have seen a huge rise in popularity recently, such as the HackRF
One, BladeRF, RTL-SDR, etc. that found their way into not only researchers and
academics, but also enthusiasts and consumers.

As stated earlier, SDRs allow for the signal processing to be performed
through software, opening up many avenues for flexibility in design and
configuration. One such option, is the ability to implement machine learning
systems, which otherwise would need very specific equipment to implement.
However, when using the SDR platform, all that is required is a computer
alongside the SDRs to implement the required system.

The SDRs chosen for this system were the _HackRF One_ as the transmitter and
the _RTL-SDR_ as the receiver. The HackRF One device is a half-duplex
transceiver, it a is a low-cost open-source hardware project, which would
describe its popularity around the SDR platform community. The RTL-SDR on the
other hand, is only a receiver, due to its low price, it also has a lot of
popularity around the SDR community. Together, they can allow for one way
transmission with reasonable pricing.

When implementing custom systems for the SDR platform, there are few software
options to choose from, the most popular is the open-source GNU Radio
software, due to its wide compatibility for operating systems and SDRs. It is
primarily written in Python and C++, and is flexible enough to allow the user
to write their own signal processing blocks.

###  3.3 Synchronization

Synchronization is important for any wireless communication system, it helps
ensure that the received data actually has meaning and can be utilized,
otherwise it may be difficult to make sense of it. Throughout this project,
many types of synchronization were encountered and studied such as, timing and
carrier synchronization for analog signals, and data synchronization for the
decoded digital signal. Despite it being such an important concept, it is
difficult to encounter texts that truly address these challenges when
performing hardware implementation and how to overcome them. This created a
few challenges that were eventually overcome with research and rigorous trial
and error.

The timing and carrier synchronization used in this system is the same as that
in [9], it was also thoroughly explained. The system in [9] in turn became a
starting point for the system in this project. Due to the low-cost RTL-SDR,
there was a significant frequency offset. To overcome this issue the system
was calibrated beforehand, to ensure that the frequency is aligned as best as
possible. The timing synchronization, was mainly addressed by [9]. However, to
fully perform the synchronization and lock onto a phase and the carrier
frequency, it takes a short period. During this period, the signal is not
synchronized and produces incorrect results when decoded, thus truncation of
this period is necessary to ensure that the decoded bitstream was
synchronized.

Phase ambiguity, although easy to solve, presented many challenges.
Classically, the phase ambiguity is addressed by sending preamble that is
known priorly, allowing the bits to be remapped to match the preamble. This
works quite well given that the phase lock is consistent, otherwise, if the
phase lock may change during operation, pilots (overhead) are required to
address this issue, this is called framing synchronization. When the system
was first implemented, only a preamble was used, and since the phase lock was
inconsistent, especially at low SNR values, this method was not able to
properly address this issue. For this reason, pilots were inserted into the
bitstream at specific intervals, in order to assist with this synchronization
issue.

One of the biggest challenges faced when implementing the SDR system was the
delay encountered when transmitting and receiving symbols. Typically, for
simple QPSK systems such as that in [9], the delay is expected to be very
small and consistent given certain channel conditions, thus, the search for
the delay can fairly easily be completed manually. However, for the
implemented SDR system, the delay was between 0.1 to 0.3 seconds, and
considering that the sampling rate was $2$ MHz, manual search was simply not
plausible.

To overcome this issue, the transmitted and received (decoded using the
conventional decoder) bitstreams needed to be studied closely, in order to
understand what was sent and ensure that the data was truly received without
error. A program was then written to iteratively delay the _received_ data
until it would match the _original_ data, this was repeated for several
situations and it was concluded that the delay was inconsistent and needs to
be found for every dataset created.

In summary, whenever a new dataset is to be created, some of the data at the
beginning must be _truncated_ , the _phase ambiguity_ must be solved, and
finally the _delay_ must be found. Using all these techniques, datasets for
training and testing the DNN were produced.

##  4 Implementation Results

In this section the results of the implementation are shown. The frequency
spectrum of the received signal is shown and compared to the transmitted
signal. Then the Bit Error Rate (BER) performance of the QPSK system using the
conventional and the proposed DNN detection methods are evaluated and
compared. Throughout the experiment, the channel was kept stationary at a
length of approximately $1$ m, with a clear LOS between the transmitter and
the receiver. The sample rate was set to $2$ MHz and the samples per symbol
was set to $8$, in order to reduce computation requirements of the system. A
frame size of $4$ symbols was also considered. The proposed DNN uses the
framework discussed in section 2.4, had five layers of sizes $8$, $100$, $50$,
$20$, and $6$, respectively.

###  4.1 The Transmitted and The Received Signal

(a) All frequency components received by the SDR.

(b) Zoomed in for a better view of the modulated signal

Figure 4: Frequency spectrum of the transmitted signal and the received signal
after the LPF.

Figures 4(a) and 4(b) show the frequency spectrum of the transmitted signal
along with the received signal after being low pass filtered. It is recognized
from the transmitted signal that there is a steep dropoff in the magnitude of
the frequency components. This is owed to the RRC pulse shaping filter, which
in turn is used for timing synchronization for optimal sampling, reducing ISI.
The bandwidth of this signal can be increased by increasing the Excess BW
parameter of the Constellation Modulator block. Ideally, there would be no
frequency components after the bandwidth of the signal, however, due to it
being processed digitally, the floating point precision comes into play and
introduces high frequency components that are mostly negligible.

These figures show how well the signal was received with very little
distortion, owing to the short LOS channel. Also only a slight frequency
offset is noticed, due to the SDRs being calibrated beforehand. From the
figures, five impulses are also observed at both the transmitted and received
signals. These impulses are a result of the pilot symbol being sent so
frequently, increasing the power received from the pilots relative to the rest
of the symbols.

(a) All frequency components received by the SDR.

(b) Zoomed in for a better view of the modulated signal.

Figure 5: Frequency spectrum of the received signal before and after being low
pass filtered.

Shown in Fig. 5(a) is the LPF output compared to the received signal. Before
around 200 kHz the two signals very closely resemble each other, above this
frequency however, heavy attenuation from the LPF eliminates high frequency
components, reducing noise and allowing for more accurate demodulation.

###  4.2 Constellation Diagram

Figure 6: Constellation diagram of the received signal.

The constellation diagram at an SNR of approximately 19 dB is shown in Fig. 6,
it shows the in-phase and quadrature components of the synchronized signal.
From this constellation diagram, the clusters can be viewed clearly and
separated very easily. With careful inspection, it also shows a nonuniformity
in the distribution of received symbol in each quarter. In other words, one of
the received symbols is received far more frequently than the others, this is
quite clearly due to the pilot symbol, which takes up to a quarter of the
frame time for the considered system.

###  4.3 BER Performance

Figure 7: BER performance of the proposed DNN data detector.

The BER performance of the conventional and the proposed DNN approaches are
shown in Fig. 7. The BER performance is measured against variation in the SNR
value of the signal. The SNR is measured by finding the difference between the
noise floor and the average power of the received signal, which could then be
adjusted by altering the RF gain value of the RTL-SDR.

$SNR=P_{R}+G_{R}-N$ (1)

where $P_{R}$ is the received average power with no receiver gain, $G_{R}$ is
the gain of the receiver, and $N$ is the noise floor. From Fig. 5(b) the noise
floor was measured to be approximately $N=-68$ dB, and the average power of
the received signal is approximately $P_{R}+G_{R}=-49$ dB with the gain of the
RTL-SDR set to $G_{R}=15$ dB. Therefore, assuming that both the noise floor
and the received power without receiver gain remain constant, the SNR can be
approximately measured directly from the gain of the receiver as such

$SNR=4\text{ dB}+G_{R}$ (2)

It is observed from this figure that the proposed DNN clearly outperforms the
conventional approach. Proving the potential of DNNs to perform well under
real world channels. The figure also shows that the smooth graphs seen in
simulation may not be as easy to maintain in real world channels, further
proving the need for implementation studies.

##  5 Conclusions and Future Work

In this project, a DNN based data detection method for the QPSK digital
modulation scheme was designed and implemented using the HackRF One and the
RTL-SDR. The implementation results show that the signal was received well,
with distinctive clusters for each symbol. The BER performance was also
evaluated, it showed that the DNN based detector outperforms the conventional
method for the considered system. It was also recognized that implementation
presents numerous challenges not encountered when performing simulations,
fortunately these challenges were overcome and discussed in this report.

Future work will expand this methodology to different digital modulation
schemes, such as Amplitude Shift Keying (ASK) and Frequency Shift Keying
(FSK). Spatial modulation can also be implemented, however, doing so would
require multiple transmit SDRs. Furthermore, cognitive radio applications
present numerous avenues to explore, such as modulation classification,
reconfigurable smart transceivers, spectrum sensing, etc. The flexibility of
SDRs and the GNU Radio software allow for endless applications.

##  6 Acknowledgement

This work was partially funded by the Libyan Foundation for Projects
Development.

## References

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