# Is asymptotically safe inflation eternal?

J. Chojnacki,**footnotetext: Corresponding author. J. Krajecka  J. H. Kwapisz
O. Slowik  A. Strag

(December 2020)

###### Abstract

Recently, based on swampland considerations in string theory, the (no) eternal
inflation principle has been put forward. The natural question arises whether
similar conditions hold in other approaches to quantum gravity. In this
article, the asymptotic safety hypothesis is considered in the context of
eternal inflation.
As exemplary inflationary models the SU(N) Yang-Mills in the Veneziano limit
and various RG-improvements of the gravitational action are studied. The
existence of UV fixed point generically flattens the potential and our
findings suggest no tension between eternal inflation and asymptotic safety,
both in the matter and gravitational sector in contradistinction to string
theory. Moreover, the eternal inflation cannot take place in the range of
applicability of effective field quantum gravity theory.
We employ the analytical relations for eternal inflation to some of the models
with single minima, such as Starobinsky inflation, alpha-attractors, or the
RG-improved models and verify them with the massive numerical simulations. The
validity of these constraints is also discussed for a multi-minima model.

##  1 Introduction

Our Universe consists of 4 fundamental forces. Three of these forces have been
consistently described on the quantum level and combined into the Standard
Model of particle physics. Only quantum gravity seems to be elusive and has
not been fully described in terms of quantum theory. This is not only because
gravity is power counting non-renormalizable but also due to the fact that
direct quantum gravity regime cannot be accessed experimentally (for example
an accelerator measuring the quantum gravity effects would have to be big as
our Solar System).
In recent years an alternative strategy has been put forward, namely one
formulates a fundamental quantum gravity theory and then tests, which of the
low energy effective theories can be UV completed by this quantum gravity
model. In string theory this goes under the name of swampland conjectures [2,
3]. Recently widely discussed is so-called de-Sitter conjecture [4, 5] which
states that string theory cannot have de-Sitter vacua and is in tension with
single field inflation [6, 7]. There seems to be also a tension between
standard S-matrix formulation of quantum gravity and existence of stable de-
Sitter space [8, 9, 10, 11]. However it is not established whether asymptotic
safety admits a standard S-matrix formulation [12] due to fractal spacetime
structure in the deep quantum regime [13, 14] In line of these swampland
criteria the no eternal inflation principle has been put forward [1], see also
the further discussions on the subject of eternal inflation [15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26].
On the other hand, the theory of inflation is a well-established model
providing an answer to problems in classical cosmology, such as the flatness
problem, large-scale structures formation, homogeneity, and isotropy of the
universe. A handful of models is in an agreement with the CMB observations. In
the inflationary models, quantum fluctuations play a crucial role in
primordial cosmology, providing a seed for the large-scale structure formation
after inflation and giving a possibility for the eternally inflating
multiverse. Initial fluctuations in the early universe may cause an
exponential expansion in points scattered throughout the space. Such regions,
rapidly grow and dominate the volume of the universe, creating ever-inflating,
disconnected pockets. Since so far there is no way to verify the existence of
the other pockets, we treat them as potential autonomous universes, being part
of the multiverse.
In the light of this tension between string theory and inflationary paradigm
[1], one can be interested in how robust are the swampland criteria for the
various quantum gravity models. In accordance with the inflation theory, we
anticipate that the dynamics of the universe are being determined by the
quantum corrections to the general relativity stemming from the concrete UV
model. The effective treatment led Starobinsky to create a simple inflationary
model taking into account the anomaly contributions to the energy-momentum
tensor.
As pointed out by Donoghue [27] below the Planck scale, for quantum gravity
one can safely take the effective field theory perspective. Yet these quantum
gravity effects can be important below the Planck scale by the inclusion of
higher dimensional operators. The gravitational constant $G_{N}$ has a
vanishing anomalous dimension below the Planck scale and various logarithmic
corrections to the $R^{2}$ have been considered, capturing the main quantum
effects [28, 29, 30, 31]. Yet in order to get the correct 60 e-fold duration
of inflationary period one has to push the scalar field value in the Einstein
frame beyond the Planck mass [1]. Furthermore most of these models do not
possess a flat potential limit (either diverge or have a runaway solutions),
suggesting that eternal inflation can be investigated only if one takes into
account the full quantum corrections to the Starobinsky inflation.
In the effective field theory scheme, the predictive power of the theory is
limited, as the description of gravity at transplanckian scales requires
fixing infinitely many coupling constants from experiments. The idea of
asymptotic safety [32] was introduced by Stephen Weinberg in 1978 as a UV
completion of the quantum theory of gravity. The behavior of an asymptotically
safe theory is characterized by scale invariance in the high-momentum regime.
Scale invariance requires the existence of a non-trivial Renormalization Group
fixed point for dimensionless couplings. There are many possible realizations
of such non-trivial fixed point scenario, such as the canonical vs anomalous
scaling (gravitational fixed point [33, 34, 35, 36]) and one-loop vs two-loop
contributions or gauge vs Yukawa contributions, see [37] for further details
and [38] for current status of asymptotically safe gravity.
The existence of an interacting fixed point and hence the flatness of the
potential in the Einstein frame led Weinberg to discuss [39] cosmological
inflation as a consequence of Asymptotically Safe Gravity, see also [40, 41]
for discussion of AS cosmology. Following this suggestion, we study two types
of models.
The first type relies on the RG-improvement of the gravitational actions and
is based on the asymptotic safety hypothesis that gravity admits a non-trivial
UV fixed point. Since asymptotically safe gravity flattens the scalar field
potentials [42], one can expect that it will result in the eternal inflation
for large enough initial field values. On the other hand, RG-improved actions
can serve as a UV completion of the Starobinsky model. One should also note
that asymptotically safe swampland has been vastly studied [43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67, 68, 69].
The other model relies on the non-trivial fixed point in the pure matter
sector governed by the Yang-Mills dynamics in the Veneziano limit [70, 71],
see also [72, 73, 74, 75, 75]. In this model, we have uncovered a new type of
eternal inflation scenario relying on tunneling to a false vacuum - in the
opposite direction as it was considered in the old-inflation proposal [76].
In contradistinction to string theory, the couplings in the asymptotic safety
paradigm are predicted from the RG-flow of the theory and their fixed point
values rather than as vacuum expectation values (vev’s) of certain scalar
fields. Hence, the asymptotically safe eternally inflating multiverses
landscape is much less vast than the one stemming from the string theory,
making these models much less schismatic [77]. Finally, let us note that
asymptotic safety can argue for the homogeneous and isotropic initial
conditions on its own using the finite action principle [78].
Our work is organized as follows. In Chapter 2 we introduce the idea of
eternal inflation and multiverse. We discuss necessary conditions for eternal
inflation to occur based on the Fokker-Planck equation. In Chapter 3 we show,
how the developed tools work in practice with the two popular inflationary
models. Chapter 4 is devoted to the presence of eternal inflation in
Asymptotically Safe models. In Chapter 5 the results are discussed and
concluded.

##  2 How inflation becomes eternal?

In this section we discuss, under what circumstances the inflation becomes
eternal. Our discussion follows closely [1].

###  2.1 Fokker-Planck equation

Consider a scalar field in the FLRW metric

$\displaystyle S=\int
d^{4}\sqrt{-g}\left(\frac{1}{2}M_{Pl}^{2}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right),$
(2.1)

with $\phi(t,\vec{x})=\phi(t)$ one obtains the following equations of motion:

$\displaystyle\ddot{\phi}+3H\phi+\frac{\partial V}{\partial\phi}=0,\quad
H^{2}M_{P}^{2}=\frac{1}{3}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right),$
(2.2)

which in the slow-roll approximation become [79]:

$\displaystyle 3H\dot{\phi}+\frac{\partial V}{\partial\phi}\approx 0,\quad$
$\displaystyle H^{2}M_{Pl}^{2}\approx\frac{1}{3}V\left(\phi\right).$ (2.3)

Inflation ends once one of the so-called slow-roll parameters becomes of order
one

$\displaystyle\epsilon\simeq\frac{M_{P}^{2}}{2}\left(\frac{V_{,\phi}}{V}\right)^{2},\quad\eta\simeq
M_{P}^{2}\frac{V_{,\phi\phi}}{V},$ (2.4)

and enters the oscillatory, reheating phase. The standard treatment of eternal
inflation relies on the stochastic inflation approach [80]. One splits the
field into classical background and short-wavelength quantum field

$\displaystyle\phi\left(t,\vec{x}\right)=\phi_{cl}\left(t,\vec{x}\right)+\delta\phi\left(t,\vec{x}\right).$
(2.5)

Due to the fact that action is quadratic in the fluctuations, their spatial
average over the Hubble volume is normally distributed. Hence from now on we
shall assume that both background and fluctuations are homogeneous, which is
the standard treatment of eternal inflation (if not otherwise specified). In
the large e-fold limit, equation of motion for the full field takes form of
slow-roll equation with additional classical noise term [1, 81, 82], known as
the Langevin Equation:

$\displaystyle 3H\dot{\phi}+\frac{\partial V}{\partial\phi}=N\left(t\right),$
(2.6)

where $N\left(t\right)$ is a Gaussian distribution with mean equal 0 and
variance $\sigma=\frac{H^{3}t}{4\pi^{2}}$ [83]. Then the probability density
of the inflaton field is then given by the Fokker-Planck equation [1]:

$\displaystyle\dot{P}[\phi,t]=\frac{1}{2}\left(\frac{H^{3}}{4\pi^{2}}\right)\frac{\partial^{2}P[\phi,t]}{\partial\phi\partial\phi}+\frac{1}{3H}\partial_{i}\left(\partial^{i}V\left(\phi\right)P[\phi,t]\right),$
(2.7)

where $\dot{P}[\phi,t]:=\frac{\partial}{\partial t}P[\phi,t]$.

###  2.2 Analytic solutions

To understand better the Fokker-Planck equation, let us now briefly discuss
the analytical solutions.

#### Case 1. Constant potential

$\displaystyle V\left(\phi\right)=V_{0},$ (2.8)

the Fokker-Planck equation reduces to

$\displaystyle\dot{P}[\phi,t]=\frac{1}{2}\left(\frac{H^{3}}{4\pi^{2}}\right)\frac{\partial^{2}P[\phi,t]}{\partial\phi\partial\phi},$
(2.9)

furthermore $H^{2}=\textrm{const}$ by the Friedman equations. Then the Fokker
Planck equation reduces to the standard heat equation, which has a solution
given by a Gaussian distribution:

$\displaystyle
P[\phi,t]=\frac{1}{\sigma\left(t\right)\sqrt{2\pi}}\exp\left[-\frac{\left(\phi-\mu\left(t\right)\right)^{2}}{2\sigma\left(t\right)^{2}}\right],$
(2.10)

with

$\displaystyle\mu\left(t\right)=0,\quad$
$\displaystyle\sigma^{2}\left(t\right)=\frac{H^{3}}{4\pi^{2}}t.$ (2.11)

A delta-function distribution initially centered at $\phi$ = 0 will remain
centered at $\phi=0$ for all time. It will however, spread out by the amount
$\sigma\left(t=H^{-1}\right)=H/2\pi$ after a Hubble time. This represents the
standard “Hubble-sized” quantum fluctuations that are well-known in the
context of inflation, famously imprinted in the CMB and ultimately seeding the
observed large-scale structure.

#### Case 2. Linear potential

For the linear hilltop model the potential is given by

$\displaystyle V\left(\phi\right)=V_{0}-\alpha\phi.$ (2.12)

Fokker-Planck equation is analogously solved by the Gaussian distribution
(2.10) with:

$\displaystyle\mu\left(t\right)=\frac{\alpha}{3H}t,\quad$
$\displaystyle\sigma^{2}\left(t\right)=\frac{H^{3}}{4\pi^{2}}t.$ (2.13)

The time-dependence of $\mu\left(t\right)$ is due to the classical rolling of
the field in the linear potential. The time-dependence of
$\sigma^{2}\left(t\right)$ is purely due to Hubble-sized quantum fluctuations,
and it precisely matches the result in the constant case. In general, for a
linear and quadratic potential the equation simplifies to the heat equation,
hence the solutions are Gaussian. Furthermore, if the potential is
asymptotically flat, a finite limit at infinity exists. One may employ a
series expansion around this point at infinity, approximating the potential up
to the linear term. It is then expected that the probability density is
approximately Gaussian (2.13). In the next section, we describe in detail how
the Gaussian distribution causes the inflaton to decay exponentially.

###  2.3 Eternal inflation conditions

Given an arbitrary field value $\phi_{c}$, one can ask what is the probability
that quantum field $\phi=\phi(t)$ is above this value:

$\displaystyle\mathrm{Pr}[\phi>\phi_{c},t]=\int^{\infty}_{\phi_{c}}d{\phi}P[\phi,t].$
(2.14)

Since the distribution is Gaussian, then for $\phi_{c}$ large enough the
$Pr[\phi>\phi_{c},t]$ can be approximated by an exponential decay:

$\displaystyle\mathrm{Pr}[\phi>\phi_{c},t]\approx C(t)\exp(-\Gamma t),$ (2.15)

where $C(t)$ is polynomial in $t$ and all of the dependence on $\phi_{c}$ is
contained in $C(t)$. Then it seems that inflation cannot last forever since

$\displaystyle\lim_{t\to\infty}\mathrm{Pr}[\phi>\phi_{c},t]=0.$ (2.16)

However, there is an additional effect to be included: expansion of the
universe during inflation. The size of the universe depends on time according
to:

$\displaystyle U\left(t\right)=U_{0}e^{3Ht},$ (2.17)

where $U_{0}$ is the initial volume of the pre-inflationary universe. One can
interpret the probability $Pr[\phi>\phi_{c},t]$ as fraction of the volume
$U_{inf}\left(t\right)$ still inflating, that is:

$\displaystyle U_{inf}\left(t\right)=U_{0}e^{3Ht}Pr[\phi>\phi_{c},t],$ (2.18)

then in order for the Universe to inflate eternally, the positive exponential
factor $3H$ in Eq. (2.18) and the negative exponential factor $-\Gamma$ in
(2.15) must satisfy:

$\displaystyle 3H>\Gamma.$ (2.19)

We shall illustrate this general property on an example of linear potential.
Evaluating the integral for probability density, in the linear case gives:

$\displaystyle
Pr[\phi>\phi_{c},t]=\frac{1}{2}\textrm{erfc}\left({\frac{\frac{\alpha}{3H}t-\phi_{c}}{\frac{H}{2\pi}\sqrt{2Ht}}}\right).$
(2.20)

The error function may be approximated by an exponential:

$\displaystyle
Pr[\phi>\phi_{c},t]=C\left(t\right)\textrm{exp}\left(-\frac{4\pi^{2}\alpha^{2}}{18H^{5}}t\right),$
(2.21)

where $C\left(t\right)$ is power-law in $t$ and $\phi_{c}$ vanished from the
final approximation of the probability, which is a generic feature. By
comparing the exponents we can check, whether $U_{inf}$ will grow or tend to
zero. The condition for eternal inflation to occur becomes:

$\displaystyle 3H>\frac{4\pi^{2}\alpha^{2}}{18H^{5}}.$ (2.22)

For linear potential $\alpha=V^{\prime}\left(\phi\right)$ using the slow-roll
equations equation (2.3), above condition can be rewritten:

$\displaystyle\frac{|V^{\prime}|}{V^{\frac{3}{2}}}<\frac{\sqrt{2}}{2\pi}\frac{1}{M^{2}_{Pl}}.$
(2.23)

This can be interpreted as quantum fluctuations dominating over classical
field rolling. For linear potential, this is satisfied for a large $\phi$.
Similarly, the second condition for the eternal inflation may be derived from
the quadratic hilltop potential:

$-\frac{V^{\prime\prime}}{V}<\frac{3}{M^{2}_{Pl}}.$ (2.24)

Further necessary conditions on p-th derivative with $p>2$ have been derived
in [1] and give:

$\displaystyle[-\textrm{sgn}\left(\partial^{p}V\right)]^{p+1}\frac{|\partial^{p}V|}{V^{(4-p}/2}<\mathcal{N}_{p}M_{Pl}^{p-4},$
(2.25)

where $\mathcal{N}_{p}\gg 1$ is numerically determined coefficient. Eternal
inflation can be understood as a random walk of a field and a diffusion
process on top of the classical motion [Vilenkin, Guth:2000ka, 15]. In order
to cross check the formulas (2.23, 2.24) the numerical simulation has been
developed. To reconstruct the probability distribution one simulates the
discretized version of equation (2.6):

$\displaystyle\phi_{n}=\phi_{n-1}-\frac{1}{3H}V^{\prime}\left(\phi_{n-1}\right)\delta
t+\delta\phi_{q}\left(\delta t\right),$ (2.26)

with $\delta\phi_{q}\left(\delta t\right)$ being random number taken from the
gaussian distribution with mean equal zero, and variance
$\frac{H^{3}}{4\pi^{2}}\delta t$. We further assume the Hubble parameter to be
constant and respecting the slow-roll regime
$H=\frac{1}{M_{Pl}}\sqrt{\frac{V(\phi_{0})}{3}}$, where $V(\phi_{0})$ is a
value of the potential at the start of the simulation. We verified, that the
change of $H$ caused by the field fluctuation does not affect the conclusions
for eternal inflation. The simulation starts at the user-given value
$\phi_{0}$ and follows the Langevin discretized equation (2.26).
If the inflation occurs, the corresponding timestep $t_{n}$ is added to a
list. This happens while the slow-roll conditions are satisfied, meaning
$\epsilon(t_{n})$ and $\eta(t_{n})$ are smaller than one. Violation of one of
these conditions resets the simulation. However, the list containing
information about time $t_{n}$ of the ongoing inflation is stored in the
memory. This large time list is appended in the similar way each evolution.
Its size may be estimated by $N\frac{T_{c}}{\delta t}$, where $N$ is the total
number of simulations and $T_{c}$ is the time of the classical slow-roll
inflation starting at $\phi_{0}$.
It is important to stress, that the duration of a particular evolution may be
too long to compute in any practical time. We employ a large timeout ending
the evolution. This is a good approximation for our purposes. Finally, the
list containing information about every timestep at which inflation was
ongoing in $N$ independent simulations is sorted in an ascending order. A
normalized histogram with 1000 equal-width bins is created from the list. The
number of counts is related to the probability of ongoing inflation, while the
bins correspond to inflationary time. The field’s evolution supports the
Fokker-Planck result (2.15). In the slow-roll regime, the inflaton decays
exponentially with decay parameter $\Gamma$. This is true for every
numerically investigated potential in this work. In order to recognize the
eternally inflating models we search for such initial value of the field
$\phi_{0}$ that $\Gamma<3H$. In the numerical analysis we eliminate $\phi_{c}$
in equation (2.14) and instead check for the slow-roll conditions violation at
each step.

###  2.4 Tunneling and eternal inflation

Most of the inflationary potentials are of the single-minimum type such as
Starobinsky inflation and alpha-attractors. There are however, potentials
which are of type depicted on figure 1 and possess a various minima. In such
models, the inflation can become eternal due to tunnelling to the false vacua.
When the vacua are degenerate enough, the tunneling dominates over quantum
uphill rolling. The tunnelling goes in the opposing direction to the old
inflation scenario [76] as showed on figure 1. As it will turn out this is the
dominant effect for the model discussed in section 4.3. The eternal inflation
mechanism discussed in the previous sections relies on the local shape of the
potential and cannot provide an accurate description in that case. In order to
quantitatively derive predictions for this new effect, we shall rely on the
first passage formalism [84, 85] instead, and apply it to the eternal
inflation considerations.

Figure 1: Left: field initially placed at the maximum of the potential may
decay towards one of the two vacua: at $\phi_{-}$ with probability $p_{-}$ and
at $\phi_{+}$ with probability $p_{+}$.
Right: field initially placed at $\phi_{0}<\phi_{max}$ may tunnel trough the
barrier towards $\phi_{+}$ with probability $p_{+}(\phi_{0})$. Analogous
tunneling from "plus" to "minus" side is also possible.

Given the initial value of the field $\phi_{0}$ being between $\phi_{-}$ and
$\phi_{+}$, the probability that it reaches $\phi_{+}$ before $\phi_{-}$ and
$\phi_{-}$ before $\phi_{+}$, denoted respectively $p_{+}(\phi_{0})$ and
$p_{-}(\phi_{0})$, obeys the following equation:

$\displaystyle
vp^{\prime\prime}_{\pm}(\phi)-\frac{v^{\prime}}{v}p^{\prime}_{\pm}(\phi)=0,$
(2.27)

with initial conditions: $p_{\pm}(\phi_{\pm})=1$, $p_{\pm}(\phi_{\mp})=0$,
where $v=v(\phi)$ is the dimensionless potential:

$\displaystyle v(\phi):=\frac{V(\phi)}{24\pi^{2}M_{Pl}^{4}}.$ (2.28)

The analytical solution is:

$\displaystyle
p_{\pm}(\phi_{0})=\pm\frac{\int^{\phi_{0}}_{\phi_{\pm}}e^{-\frac{1}{v(\phi)}}d\phi}{\int^{\phi_{+}}_{\phi_{-}}e^{-\frac{1}{v(\phi)}}d\phi}.$
(2.29)

One may also define the probability ratio $R$:

$\displaystyle
R(\phi_{0}):=\frac{p_{+}(\phi_{0})}{p_{-}(\phi_{0})}=\frac{\int^{0}_{\phi_{-}}e^{-\frac{1}{v(\phi)}}d\phi}{\int^{\phi_{+}}_{\phi_{0}}e^{-\frac{1}{v(\phi)}}d\phi}.$
(2.30)

The above integrals may be evaluated numerically. However, if the amplitude of
$v(\phi)$ is much smaller than 1, the term $e^{-1/v(\phi)}$ will be extremely
small, possibly exceeding machine precision in the computation.
Yet, one can use the steepest descent approximation. Consider a potential with
the field located initially on the local maximum $\phi_{max}$ with a minimum
on each side, as depicted on figure 1. Then the probability ratio $R$ may be
evaluated approximately, where the leading contributions to (2.30) come from
the values of the field in the neighborhood of $\phi_{max}$. $p_{+}$ and
$p_{-}$ gives a probability of evolution realised respectively by the red and
the green ball. We get:111For the details of the calculation consult [85].

$\displaystyle R(\phi_{max})\approx
1-\frac{2}{3}\frac{\sqrt{2}}{\pi}\frac{v(\phi_{max})v^{\prime\prime\prime}(\phi_{max})}{|v^{\prime\prime}(\phi_{max})|^{3/2}}.$
(2.31)

In this regime, probability of descending into each of the minima $\phi_{-}$
and $\phi_{+}$ is similar, giving $|1-R|\ll 1$. It is possible to start the
inflation in the subset of $[\phi_{-},\phi_{+}]$ that would lead to the
violation of the slow-roll conditions and tunnel through the potential barrier
to the sector dominated by eternal inflation, schematically shown on figure 1.
We further analyze this possibility in Sec. 4.3 for a particular effective
potential with two vacua, stemming from an asymptotically safe theory. We use
equation (2.31) to find the dependence of $R$ on the parameters of the theory
and verify the result with direct numerical simulation of the Langevin
Equation (2.6) given set of parameters.

##  3 Exemplary models

In this section we show the basic application of the conditions (2.23, 2.24)
to simple effective potentials stemming from the $\alpha$-attractor models and
the Starobinsky inflation.

###  3.1 Alpha-attractor models

We start our investigation with the $\alpha$-attractor models [86], a general
class of the inflationary models, originally introduced in the context of
supergravity. They are consistent with the CMB data, and their preheating
phase has been studied on a lattice in [87]. The phenomenological features of
these models are described by the Lagrangian:

$\displaystyle\frac{1}{\sqrt{-g}}\mathcal{L}_{T}=\frac{1}{2}R-\frac{1}{2}\frac{\partial\phi^{2}}{\left(1-\frac{\phi^{2}}{6\alpha}\right)^{2}}-V(\phi).$
(3.1)

Here, $\phi$ is an inflaton and $\alpha$ can take any real, positive value. At
the limit $\alpha\xrightarrow{}\infty$ the scalar field becomes canonically
normalized, and the theory coincides with the chaotic inflation. Canonical and
non-canonical fields are related by the transformation:

$\displaystyle\phi=\sqrt{6\alpha}\tanh{\frac{\varphi}{\sqrt{6\alpha}}}.$ (3.2)

We further consider T-models, in which the potential of canonically normalised
field is given by:

$\displaystyle
V(\phi)=\alpha\mu^{2}\tanh^{2n}{\frac{\varphi}{\sqrt{6\alpha}}},$ (3.3)

where parameter $\mu$ is of order $10^{-5}$. The shape of the potential for
$n=1$ was plotted on figure 2. At the large $\phi$ the potential (3.3) is
asymptotically flat, this creates the possibility for eternal inflation to
occur. Using the first condition (2.23) we have verified, that generally the
space is eternally inflating for all initial values of the $\phi$, above
certain $\phi_{EI}$. The second eternal condition (2.24) as well as the higher
order conditions are satisfied for almost all values of $\phi_{0}$ above $0$,
providing no new information. This is a generic feature for all of the models
we investigate.
For every $\alpha$, $\phi_{0}$ necessary to produce 60 e-folds is safely below
$\phi_{EI}$. We have found $\phi_{0}$ by solving slow-roll equation
numerically. It is shown on figure 2. The time of inflation larger than 60
Planck-times is unlikely according to the Planck Collaboration data [88]. The
values of $\phi_{EI}$ change only slightly with $n$. We may therefore
conclude, that $\alpha$-attractor models are consistent with the beginning of
"our" pocket universe. However, it is not inconceivable that the field
fluctuations in other parts of the early universe had values
$\phi_{0}>\phi_{EI}$, driving the eternal inflation.

Figure 2: Left: the T-model potential with $n=1$ and various $\alpha$ were
depicted.
Right: plot of the initial value $\phi_{0}$ necessary for 60 e-folds, as a
function of $\alpha$ (blue), as well as the lowest initial value $\phi_{EI}$
of the field, at which the eternal inflation "kicks in" (yellow).

###  3.2 Starobinsky model

Solutions stemming from Einstein-Hilbert action predict initial singularity.
In 1980 Starobinsky proposed a model [89], where pure modified gravitational
action can cause non-singular evolution of the universe, namely:

$\displaystyle
S=\frac{1}{2}\int\sqrt{|g|}d^{4}x\left(M^{2}_{p}R+\frac{1}{6M^{2}}R^{2}\right),$
(3.4)

this can be rewritten to the effective potential form, with:

$\displaystyle
V\left(\phi\right)=V_{0}\left(1-\exp\left(-\sqrt{\frac{2}{3}}\frac{\phi}{M_{Pl}}\right)\right)^{2}.$
(3.5)

The inflation begins on a plateau at large $\phi$. The field rolls towards a
minimum at $\phi=0$, where the oscillatory reheating phase occurs. It has been
estimated from the CMB data, that during the inflation the volume of the
universe has grown by approximately 60 e-folds. This corresponds to the
initial condition $\phi_{0}=5.5$ $M_{Pl}$, without taking into account quantum
gravity effects. It is possible to perturbatively recover information about
the shape of the potential from the CMB, for details see [90]. Amplitude of
the scalar power spectrum $A_{s}=2\times 10^{-9}$ fixes the value
$V_{0}=8.12221\times 10^{-11}$ $M^{4}_{Pl}$, via the relation:

$\displaystyle V_{0}=24\pi^{2}\epsilon(\phi_{0})A_{s}.$ (3.6)

Applying the analytical eternal inflation conditions (2.23, 2.24) to the
Starobinsky potential, the initial value of the field, above which the eternal
inflation occurs, has been estimated to be $\phi_{0}=16.7$ $M_{Pl}$. It has
been found, that decay rate $\Gamma$ decreases approximately exponentially
with $\phi_{0}$. Our numerical simulation confirms the analytical prediction
discussed in [1] within a sample of 10000 simulations. They are performed as
described in the previous section. An exemplary numerical evolution of the
Langevin equation (2.6) is shown in the figure 3. The linear fit in the early
times shows the exponential decay of the inflation in the slow-roll regime.
Nevertheless, for eternal inflation scenario and realistic phenomenology, this
model requires the transplanckian values of the fields in order to reproduce
the correct tensor to scalar ratio $r$, amplitudes and spectral tilt $n_{s}$.
Hence the Starobinsky inflation will be affected, possibly invalidated by the
quantum gravity fluctuations. The leading log corrections have been studied in
[31], which we as well study in the context of eternal inflation. Yet, due to
the large field values required for eternal inflation to occur, one should
seek a theory predictive up to an arbitrary large energy scale, which we
discuss in the next section.

Figure 3: Left: exemplary field evolution for the Starobinsky model has been
plotted. Green plot shows the solution to the classical slow-roll equation,
and the black plot is the Langevin solution. The inflation ends, when the
slow-roll parameter reaches 1. Values of the field are given in $M_{Pl}$.
Right: the time dependence of the probability, that inflation still occurs. In
the slow-roll regime, the probability decays exponentially. Linear fit slope,
the decay rate is around $\Gamma=0.15$. Red, dashed line denotes eternal
inflation threshold with slope $3H$ of order $10^{-5}$. Since $3H<\Gamma$,
initial condition $\phi_{0}=3$ $M_{Pl}$ is an example of non-eternally
inflating universe.

##  4 Eternal inflation in asymptotically safe models

In this chapter, as a warm up we study effective corrections to the
Starobinsky inflation, providing a different behavior at the large field
values. Later we show, that RG-improvement of Starobinsky model proposed in
[91], see also [92, 93] for review, closely related to the $R+R^{2}$
renormalizable Fradkin-Tseytlin gravity [94] produces a branch of inflationary
potential entirely dominated by eternal inflation. We find the initial values
of the inflaton such that inflation becomes eternal, for the remaining branch,
as a function of theory parameters. Finally we show, that the possibility of
tunneling through the potential barrier present in [95] becomes a new
mechanism for eternal inflation. In all of the asymptotically safe
inflationary theories, eternal inflation is present as a consequence of
asymptotic flatness of the effective potential.

###  4.1 Quantum corrections to the Starobinsky model

Below Planck scale the gravitational constant $G_{N}$ has a vanishing
anomalous dimension and the $R^{2}$ has a coefficient that runs
logarithmically [96] (this comes from the fact that $R^{2}$ is dimensionless
in $4$ dimensions). Hence, one can motivate various quantum corrected
inflationary models, such as [28, 29, 30, 31]. In particular, the leading-log
corrections to the Starobinsky model are given by [31]:

$\displaystyle\mathcal{L}_{eff}=\frac{M_{Pl}^{2}R}{2}+\frac{\frac{a}{2}R^{2}}{1+b\ln(\frac{R}{\mu^{2}})}+\mathcal{O}(R^{3}).$
(4.1)

In order to find the Einstein frame potential for this model a few steps need
to be taken. First, we use the conformal transformation [97]. Then, following
the next transformation for the Ricci scalar and the metric determinant we get
the Einstein frame action:

$\displaystyle S=\int
d^{4}x\sqrt{-g_{E}}\left[\frac{M_{Pl}^{2}}{2}R_{E}-\frac{1}{2}g_{E}^{\mu\nu}\left(\partial_{\mu}\phi_{E}\right)\left(\partial_{\nu}\phi_{E}\right)-V_{E}(\phi_{E})\right],$
(4.2)

which depends on the sought potential $V_{E}$, that can be further
obtained:222Here, the effective action differs from [31] due to the
introduction of auxiliary numerical parameter
$e\approx 2.81$. Nevertheless, the dynamics stemming from each of the
potentials are equivalent.

$\displaystyle
V_{E}(\Phi)=\frac{M_{Pl}^{4}}{2}\frac{a\Phi^{2}\left(1+b\ln\left(\frac{\Phi}{\mu^{2}}\right)\right)^{2}\left(1+b\ln\left(\frac{\Phi}{e\mu}^{2}\right)\right)}{\left\\{M_{p}^{2}\left(1+b\ln\left(\frac{\Phi}{\mu^{2}}\right)\right)^{2}+2a\Phi\left(1+b\ln\left(\frac{\Phi}{\sqrt{e}\mu^{2}}\right)\right)\right\\}^{2}},$
(4.3)

with $\phi_{E}$ given by

$F(\phi_{E})=M_{Pl}^{2}\exp\left(\sqrt{\frac{2}{3}}\frac{\phi_{E}}{M_{Pl}}\right)=M_{P}^{2}+\frac{a\Phi[2-b+2b\ln(\Phi/\mu^{2})]}{[1+2b\ln(\Phi/\mu^{2})]^{2}},$
(4.4)

yet the transformation between $\Phi$ and $\phi_{E}$ is non-invertible. By
taking into account the COBE normalization, we can treat $b$ as a free
parameter and fix $a(b)$. For $b=0$ one obtains the usual Starobinsky model,
and for $b\ll 1$ one gets the model discussed in [29] $R^{2}(1+\beta\ln R)$
with the potential given by Lambert W function (so the same as for model
discussed in section 4.3) and approximated in the limit $\beta\ll 1$ as

$V\approx\frac{V_{s}}{1+b/(2\alpha)+\beta/\alpha\ln[(e^{\tilde{\chi}}-1)/2\alpha]},$
(4.5)

where $V_{s}$ is the Starobinsky potential, $\tilde{\chi}$ is the Einstein
frame field and $\alpha(\beta)$, where we have kept the original notation.
From the plots 4, 4, one can see that both of the models should give similar
inflationary observables as in the Starobinsky inflation. On the other hand
the eternal inflation in this models will be quite different. These models for
$\beta<0$ and $b>0$ have the potentials that are non-flat for large field
values, while for $\beta>0$ potential depicted on Figure 4 have the runaway
behaviour, so different asymptotic behaviour, which is discussed in Section
4.3. This makes those potentials quantitatively different from the Starobinsky
model in the context of eternal inflation and suggest, that eternal inflation
cannot take place in those models. To be concrete, we have checked that
eternal inflation for model described by 4.3 takes place at $\Phi\approx
1000\,M_{Pl}$, which is far beyond the applicability of the model. So now we
turn to the inflationary models stemming from the asymptotic safety.

Figure 4: Left: Plots of the potential (4.3) for various $b$ parameters.
Right: Potential (4.5) in dependence of $\beta$ parameter values. Approach
towards infinity in case of $\beta>0$ is visible.

###  4.2 RG-improved Starobinsky inflation

Renormalization Group improvement is a procedure of identifying and replacing
the RG scale $k^{2}$ with a physical scale. It incorporates leading-order
quantum effects in the dynamics of classical system. In the case of gravity,
running of coupling constants in Einstein-Hilbert action results in additional
contribution to the field equations from the gravitational energy-momentum
tensor [93]. In the de Sitter-type setting $k^{2}\sim R$ is the unique
identification of the physical scale dictated by Bianchi identities [93]. Such
replacement in the scale-dependent Einstein-Hilbert action generates an
effective $f(R)$ action, whose analytical expression is determined by running
of the gravitational couplings. RG-improvement could solve classical black
hole singularity problem [98, 99], gives finite entanglement entropy [100] and
generates inflationary regime in quantum gravity [37].
In this section, we study the asymptotically safe inflation based on RG-
improved quadratic gravity Lagrangian, considered in [91, 93]:

$\mathcal{L}_{k}=\frac{1}{16\pi
g_{k}}\left(R-2\lambda_{k}k^{2}\right)-\beta_{k}R^{2},$ (4.6)

with the running dimensionless couplings $g_{k}$, $\lambda_{k}$, $\beta_{k}$
being the three relevant directions of the theory, with running given by
[101]:

$g_{k}=\frac{6\pi
c_{1}k^{2}}{6\pi\mu^{2}+23c_{1}(k^{2}-\mu^{2})},\quad\quad\beta_{k}=\beta_{\ast}+b_{0}\left(\frac{k^{2}}{\mu^{2}}\right)^{-\theta_{3}/2},$
(4.7)

where $\mu$ is the infrared renormalization point such that
$c_{1}=g_{k}(k=\mu)$ and $c_{1}$ and $b_{0}$ are integration constants. We
introduce a parameter $\alpha$ as

$\alpha=-2\mu^{\theta_{3}}b_{0}/M_{P}^{2},$ (4.8)

that measures the departure from the non-gaussian fixed point (NGFP). One may
find the behavior of the couplings near the NGFP and substitute the
appropriate expressions to the Lagrangian using the RG-improvement and the
following identification of scale $k^{2}=\xi R$, where $\xi$ is an arbitrary
parameter of order one.

Figure 5: Left: $V_{+}(\phi)$ plot for various $\alpha$ and fixed $\Lambda=1$
is shown.
Right, logarithmic dependence of initial field value above which eternal
inflation occurs on parameter $\alpha$ has been found. Blue points were
evaluated via (2.23).

Following [91] we shall assume $\theta_{3}=1$, then the transformation from
the Jordan to the Einstein frame yields an effective potential [91, 93]:

$\displaystyle\begin{split}V_{\pm}=&\frac{m^{2}e^{-2\sqrt{\frac{2}{3}}\kappa\phi}}{256\kappa}\Bigg{\\{}\vphantom{6\alpha^{3}\sqrt{\alpha^{2}+16e^{\sqrt{\frac{2}{3}}\kappa\phi}-16}}192(e^{\sqrt{\frac{2}{3}}\kappa\phi}-1)^{2}-3\alpha^{4}+128\Lambda\\\
&-\sqrt{32}\alpha\left[(\alpha^{2}+8e^{\sqrt{\frac{2}{3}}\kappa\phi}-8)\pm\alpha\sqrt{\alpha^{2}+16e^{\sqrt{\frac{2}{3}}\kappa\phi}-16}\right]^{\frac{3}{2}}\\\
&-3\alpha^{2}(\alpha^{2}+16e^{\sqrt{\frac{2}{3}}\kappa\phi}-16)\mp
6\alpha^{3}\sqrt{\alpha^{2}+16e^{\sqrt{\frac{2}{3}}\kappa\phi}-16}\Bigg{\\}}\end{split},$
(4.9)

where the only free parameters are cosmological constant $\Lambda$ and
$\alpha$ after the CMB normalization we perform below. The $V_{+}$ branch
predicts the reheating phase, figure 5 shows its plot for various $\alpha$.
Similarly as in the case of Starobinsky inflation, we have denoted $V_{0}$ the
constant part of the potential at infinity
$V(\phi\xrightarrow{}\infty)=V_{0}=\frac{3m^{2}}{4\kappa^{2}}$ and fixed it
with CMB data by the relation (3.6). For example, given $\alpha=2.8$,
$\Lambda=1$ the plateau value is equal to $V_{0}=1.99\times 10^{-10}$
$M^{4}_{Pl}$, hence one may fix the mass parameter $m=2\times 10^{14}$ GeV.
Now we shall investigate the eternal inflation conditions given by (2.23,
2.24). These conditions restricts the initial value of the field. We search
for $\phi_{0}$ above which the eternal inflation occurs, as a function of the
theory parameters. We have also found, that initial value above which eternal
inflation occurs does not depend on the cosmological constant. It is due to
the fact that $\Lambda$ only shifts the minimum of the potential and does not
affect the large-field behavior of the system. Analytical conditions for EI
have been checked for a set of $\alpha$ and depicted on figure 5. The initial
value of the field, depends logarithmically on $\alpha$. The reason for that
behaviour is the following. In the large field expansion:

$\displaystyle V_{\pm}(\phi)=V_{plateau}-128V_{0}\alpha
e^{-\frac{1}{2}\sqrt{\frac{3}{2}}\phi},$ (4.10)

and by the substitution $\tilde{\phi}=e^{-\frac{1}{2}\sqrt{\frac{3}{2}}\phi}$
the potential reduces to the linear hilltop model, which justifies the usage
of formulae (2.23, 2.24) and the functional form of $\phi_{0}(\alpha)$. The
results were also confirmed by the numerical simulations. For example, given
$\Lambda=1,\,\alpha=1.6$, the analytical considerations predict
$\phi_{EI}=22.6\,M_{Pl}$. The direct numerical simulation for this set of
parameters yields $\Gamma=0.0001$ $M_{Pl}$, and $3H=0.0003$ $M_{Pl}$, meaning
that the eternal inflation begins slightly below the expected value
$\phi_{EI}$. The plateau of (4.9) at large field values is a characteristic
feature of effective inflationary potentials stemming from the asymptotically
safe theories. It is dominated by eternal inflation and may suggest a deeper
relation between the asymptotic safety of quantum gravity and multiverse.

###  4.3 Large N-dynamics and (eternal) inflation

In this section we investigate model in which inflation is driven by an
ultraviolet safe and interacting scalar sector stemming from a new class of
non-supersymmetric gauge field theories. We consider a $\mathrm{SU}(N_{C})$
gauge theory, with $N_{F}$ Dirac fermions and interacting with an $N_{F}$
$\times$ $N_{F}$ complex scalar matrix $H_{ij}$ that self interacts, described
in [95]. The Veneziano limit ($N_{F}\to+\infty$, $N_{C}\to+\infty$,
$N_{F}/N_{C}=\mathrm{const}$) is taken such that the ratio $N_{F}/N_{C}$
becomes a continuous parameter [70]. The action in Jordan frame has the
following form:

$S_{J}=\int
d^{4}x\sqrt{-g}\left\\{-\frac{M^{2}+\xi\phi^{2}}{2}R+\frac{g^{\mu\nu}}{2}\partial_{\mu}\phi\partial_{\nu}\phi-
V_{\mathrm{iUVFP}}\right\\},$ (4.11)

where the leading logarithmically resummed potential $V_{iUVFP}$ is given by:

$V_{\mathrm{iUVFP}}(\phi)=\frac{\lambda_{*}\phi^{4}}{4N_{f}^{2}\left(1+W(\phi)\right)}\left(\frac{W(\phi)}{W(\mu_{0})}\right)^{\frac{18}{13\delta}},$
(4.12)

where
$\lambda_{*}=\delta\frac{16\pi^{2}}{19}(\sqrt{20+6\sqrt{23}}-\sqrt{23}-1)$ is
positive quartic coupling at the fixed point, $\phi$ is the real scalar field
along the diagonal of $H_{ij}=\phi\delta_{ij}/\sqrt{2N_{f}}$ and
$\delta=N_{F}/N_{C}-11/2$ is the positive control parameter, $W(\phi)$ is the
Lambert function solving the transcendent equation

$z=W\exp W,$ (4.13)

with

$z(\mu)=\left(\frac{\mu_{0}}{\mu}\right)^{\frac{4}{3}\delta\alpha*}\left(\frac{\alpha*}{\alpha_{0}}-1\right)\exp\left[\frac{\alpha*}{\alpha_{0}}-1\right].$
(4.14)

The parameter $\alpha*=\frac{26}{57}\delta+O(\delta^{2})$ is the gauge
coupling at its UV fixed point value and $\alpha_{0}=\alpha(\mu_{0})$ is the
same coupling at a reference scale $\mu_{0}$.
A conformal transformations allows to rewrite the action from Jordan to
Einstein frame. Assuming a single field slow-roll inflation, we examine
inflationary predictions of the potential and compute the slow-roll
parameters:

$\epsilon=\frac{M_{Pl}^{2}}{2}\left(\frac{dU/d\chi}{U}\right)^{2},\quad\quad\eta=M_{Pl}^{2}\frac{d^{2}U/d\chi^{2}}{U},$
(4.15)

where $U=V_{\mathrm{iUVFP}}/\Omega^{4}$, with
$\Omega^{2}=(M^{2}+\xi\phi^{2})/M_{Pl}^{2}$ being the conformal transformation
of the metric, and $\chi$ is the canonically normalized field in the Einstein
frame. We assume, that $M=M_{Pl}$. Inflation ends when the slow-roll
conditions are violated, that is when $\epsilon(\phi_{end})$ or
|$\eta(\phi_{end})$| = 1. We analyze the non-minimal case, where the coupling
$\xi$ is non-vanishing. The potential $U$ is given by:

$U=\frac{V_{\mathrm{iUVFP}}}{\Omega^{4}}\approx\frac{\lambda_{*}\phi^{4}}{4N_{F}^{2}\left(1+\frac{\xi\phi^{2}}{M_{Pl}^{2}}\right)^{2}}\left(\frac{\phi}{\mu_{0}}\right)^{-\frac{16}{19}\delta}\mathrm{.}$
(4.16)

Figure 6: Left: the non-minimally coupled potential as a function of $\phi$
for $\delta=0.1$, $\xi=1/6$ and $\mu_{0}=10^{-3}M_{Pl}$. There is a maximum at
$\phi_{max}=16.7\,M_{Pl}$.
Right: For the same set of parameters, we plot the first eternal inflation
condition as a function of $\phi$ (blue curve) and the eternal inflation bound
(yellow curve). Inflation becomes eternal if the blue curve is below the
yellow one. At $\phi_{max}$ the first derivative of the potential vanishes and
(2.23) predicts a narrow window for the eternal inflation.

In the large field limit $\phi$ $\gg$ $M_{Pl}/\sqrt{\xi}$ the $\phi^{4}$ term
in the numerator cancels against the term in the denominator. In this limit,
the quantum corrections dictate the behaviour of the potential, which is found
to decrease as:

$\frac{\lambda_{*}M_{Pl}^{4}}{4N_{F}^{2}\xi^{2}}\left(\frac{\phi}{\mu_{0}}\right)^{-\frac{16}{19}\delta}\mathrm{.}$
(4.17)

The non-minimal coupled potential has one local maximum and two minima. The
region to the left of the maximum is the region, where the inflation can be
brought to an end and the reheating takes place [102]. To the right of the
maximum, the inflation becomes classically eternal. For large values of $\phi$
the potential flattens out and the slow-roll conditions are not violated.
Numerical solutions to the FP equation shows that the is no possibility of
eternal inflation in that region, since it is an unstable maximum and hence
any quantum fluctuation will drop it from that position. Furthermore due to
steepness of the potential around this maxima there is no possibility for the
field to remain in that region. Let us now investigate the analytical eternal
inflation conditions. Similarly as in the Starobinsky model, the second
condition (2.24) is always satisfied. The first condition (2.23) is
illustrated on the figure 6. There is a peak for $\phi$ = $\phi_{max}$ = 16.7
$M_{Pl}$, due to the vanishing derivative and if we "zoom in", the analytical
condition allows for eternal inflation in the close neighbourhood of
$\phi_{max}$. We have verified numerically this is not a sustainable attractor
of eternal inflation. A field that starts evolution at $\phi_{max}$ will leave
this region, as it cannot climb further up-hill. Nevertheless, eternal
inflation may still occur due to the quantum tunnelling through the potential
barrier.

#### Tunneling through the potential barrier

As described in section 2.4, if the potential has multiple vacua, quantum
tunneling through the potential barrier is expected. The non-minimal coupling
potential (4.16) belongs in this class. The question is, whether tunnelling
from the non-eternal inflation region of $\phi<\phi_{max}$ to the region of
classical eternal inflation $\phi>\phi_{max}$ is possible.
We start with investigating the fate of the field initially placed at the peak
$\phi_{0}=\phi_{max}$ of the potential depicted on figure 6. By the virtue of
the steepest descent approximation at the maximum equation (2.31) may be
employed. The resulting ratio of probabilities of the right-side descent to
the left-side descent $R(\alpha,\xi)=\frac{p_{+}}{p_{-}}$, as a function of
parameter $\alpha$ and the non-minimal coupling constant $\xi$ was calculated
directly from the formula (2.31) without the need of numerical simulation of
the Langevin equation. It is presented on figure 7. Due to the complexity of
the potential (4.16) its maximum was found numerically and then employed in
(2.31). Figure 7 shows how the maximum changes with parameters. As expected,
the ratio $R(\alpha,\xi)$ is close to 1 and favors the right-side (left-side)
of the potential for large (small) values of the parameters. The biggest ratio
emerges at large values of the parameters $\xi$ and $\delta$, since the
potential is "step-like" and highly asymmetric. It is monotonically decreasing
with the values of $\phi_{max}$, for which the potential has a maximum, see
figure 8.

Figure 7: Left: probability ratio $R=\frac{p_{+}}{p_{-}}$ of descending from
the maximum towards the right minimum ($p_{+}$) and the left minimum
($p_{-}$), as a function of theory parameters $\xi$ and $\delta$. For the
small values of the parameters it is more probable to fall from the maximum
towards $\phi_{-}=0$ with non-eternal inflation, while for the large values of
parameters, minimum at $\phi_{+}=\infty$ is favored, resulting in eternally
inflating universe.
Right: the value of the field at which, the potential is maximal. The above
figures are qualitatively similar because for the small values of
$\phi_{max}$, the effective potential is highly asymmetric ("step-like"). This
brakes the symmetry between the right and left descend probability.

In order to verify the accuracy of the relation (2.31), we have performed
numerical simulation of the discretized Langevin equation (2.26) with initial
condition $\phi_{0}=\phi_{max}$. For example it was found, that for the set of
the parameters $N_{F}=10$, $\mu=10^{-3}M_{Pl}$, $\xi=\frac{1}{6}$,
$\delta=0.1$ the steepest descent approximation yields $R=0.92$, and the
numerical analysis results in $R=0.97$, which proves good accuracy of the
analytical formula.
One may wonder, how does the probability $p_{\pm}(\phi_{0})$ depends on the
departure from the maximum $\phi_{0}\neq\phi_{max}$. The analytical answer is
given by (2.29). As we have checked numerically inflation becomes eternal,
when tunneling probability is non-zero, as depicted on Figure 8.
To bypass the numerical calculation of the integral (2.29) we employ the
direct numerical simulation of the Langevin equation. This time however, we do
not seek for the time evolution of the inflaton. Rather than creating
histograms of the count of inflationary events at a given timestep, we simply
track the probabilities $p_{+}$ and $p_{-}$. We say that the particle
tunnelled through the potential barrier contributing to $p_{+}$ if the
evolution starts at $\phi_{0}<\phi_{max}$ and proceeds to arbitrarily large
field values after a long time. For each point at figure 8 the probability has
been calculated on the sample of 10000 simulations. As expected, choosing
values of $\phi_{0}$ smaller than $\phi_{max}$ lowers the probability of the
tunneling to the right side of the barrier. Moreover, the probability of
tunneling decreases linearly with the distance to the maximum. The result of
the simulation for the set of parameters

$N_{F}=10,\quad\mu=10^{-3}M_{Pl},\quad\xi=\frac{1}{6},\quad\delta=0.1$ (4.18)

is shown on figure 8. The green line corresponds to the green ball on figure 1
and shows the probability of tunneling through the barrier (as in figure 1) as
a function of proximity to the maximum $\phi_{0}\neq\phi_{max}$. The red line
corresponds to the red ball on figure 1 and shows the probability of rolling
towards the minimum at infinity.
The rolling is also a stochastic process, as the tunneling in the opposite
direction is possible. The probability distribution of tunneling in either
direction is not a symmetric process. Notice, that the initial condition, for
which $p_{+}=\frac{1}{2}$ is shifted to the right of $\phi_{max}$. This means,
that starting from the maximum, it is slightly more probable to land in
$\phi_{-}$. There is a point, below which the green ball cannot tunnel
$p_{+}=0$ (for the set of parameters given by (4.18), at $9.2M_{Pl}$), and the
other limiting case (at $24.8$ $M_{P}$), when the red ball cannot tunnel
$p_{-}=0$. Hence, for every initial value of the field above $9.2$ Planck
Masses, there is a non-zero probability of eternal inflation. On the other
hand, for $\phi_{0}=9.2\,M_{Pl}$ and parameters given by (4.18), the inflation
classically produces roughly 54 e-folds, depending on the reheating time
[102], and is in the agreement with CMB data. This shows that the model is on
a verge of being eternally inflating, which may point out to the interesting
phenomenology.
To sum up the critical point of our analysis is that the analytical conditions
(2.23, 2.24) did not allow for eternal inflation, even though the tunneling
process may evolve any initial point above 9.2 Planck Masses to
$\phi_{+}=\infty$, and not violate the slow-roll conditions. This shows that
conditions (2.23,2.24) are not well suited for the multiple minima models and
cannot contain the full information of the global influence of quantum
fluctuations in the early universe.

Figure 8: Left: Linear probability distribution of tunneling (green side) and
rolling (red side) towards the minimum at infinity as. The data points have
been directly simulated.
Right: Probability ratio $R$, evaluated with (2.31), is a monotonically
decreasing with the value of the maximum of the potential in the steepest
descent approximation

##  5 Conclusions

Eternal inflation remains a conceptual issue of the inflationary paradigm. The
creation of scattered, causally disconnected regions of spacetime - the
multiverse is not confirmed observationally and raises question about the
inflationary predictions [77]. Hence, one may impose the no eternal inflation
principle [1] to restrict free parameters and the initial conditions.
We have investigated popular inflationary models, and have found that in
principle, eternal inflation is present at every asymptotically flat effective
potential for large field values, assuming the ergodicity of the system. The
finite inflationary time of our pocket universe serves as a consistency
condition of the multiverse predictions. In section 3.1, we verified that
$\alpha$-attractor T-models are consistent from this point of view.
If the initial value of the scalar field driving inflation is above the Planck
scale, a UV-completeness of a given model is necessary. Starobinsky inflation
stemming from $R^{2}$ gravity gives around 60 e-folds for $\phi_{0}$=5.5
$M_{pl}$. We have considered the effective quantum corrections to the
Starobinsky inflation based on the qualitative behavior of the running
coupling constants. Next, the RG-improvement of $R+R^{2}$ Lagrangian was
studied in this case and we have found that field values required for eternal
inflation are typically higher than the ones for the Starobinsky case. The
flatness of the potential and possibility of eternal inflation seems to be a
signature mark of asymptotically safe UV completions, in contradistinction to
the effective theory corrections. We have checked that [31] $\Phi\sim 1000$
$M_{Pl}$ in order to get eternal inflation, which is far beyond the
applicability of the model.
Furthermore we have found, that for potentials with multiple vacua, tunneling
through potential barriers provides a new mechanism for eternal inflation. So
in order to understand the inflationary dynamics one cannot simply cut the
potential at the maximum. The $\mathrm{SU}(N)$ Gauge theory with Dirac
fermions provides an example for such behavior. The probability of tunneling
to the side dominated by eternal inflation becomes negligible few Planck
Masses away from the peak of the potential. Yet the fixed point values of the
couplings and possibly the shape of the potential can be obscured by the
quantum gravity effects and this shall be investigated elsewhere.
Our analysis reveals that there is no obstruction for the multiverse scenario
in the asymptotically safe models. Yet its occurrence depends on the initial
conditions for the inflationary phase and the matching to the observational
data, tying these three profound issues together. On the other hand, in AS
models these questions can have intriguing answers by the finite action
principle [78].

## Acknowledgments

We thank J. Reszke and J. Łukasik for participating in the early stages of
this project. We thank G. Dvali, A. Eichhorn, M. Pauli, A. Platania, T.
Rudelius, S. Vagnozzi and Z.W. Wang for fruitful discussions and extensive
comments on the manuscript. Work of J.H.K. was supported by the Polish
National Science Center (NCN) grant 2018/29/N/ST2/01743. J.H.K. would like to
acknowledge the CP3-Origins hospitality during this work. The computational
part of this research has been partially supported by the PL-Grid
Infrastructure.

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