# A dynamical inflaton coupled to strongly interacting matter

Christian Ecker  Institut für Theoretische Physik, Goethe Universität, Max-
von-Laue-Str. 1, 60438 Frankfurt am Main, Germany  Elias Kiritsis  Université
Paris Cité, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France
Institute for Theoretical and Computational Physics, Department of Physics,
P.O. Box 2208,University of Crete, 70013, Heraklion, Greece  Wilke van der
Schee  Theoretical Physics Department, CERN, CH-1211 Genève 23, Switzerland
Institute for Theoretical Physics, Utrecht University, 3584 CC Utrecht, The
Netherlands

###### Abstract

According to the inflationary theory of cosmology, most elementary particles
in the current universe were created during a period of reheating after
inflation. In this work we self-consistently couple the Einstein-inflaton
equations to a strongly coupled quantum field theory (QFT) as described by
holography. We show that this leads to an inflating universe, a reheating
phase and finally a universe dominated by the QFT in thermal equilibrium.

gauge/gravity duality, numerical relativity,inflation, reheating

##  I Introduction

Cosmological inflation is a paradigm of extended exponential expansion of our
universe at its earliest moments of existence. Due to the rapid expansion this
leads to a quickly cooling universe, which at the end reheats to the hot
plasma that then forms the Big Bang. One of the main features of inflation is
that the exponential expansion naturally explains why our universe is to a
good approximation homogeneous, even though different parts could not have
been in causal contact since the Big Bang.

One of the main uncertainties in inflation is the so-called ‘exit’ to the hot
Big Bang scenario. Due to the many e-foldings of expansion a natural end state
of inflation would be an empty universe, so the question is how ordinary and
possibly dark matter arise in inflation. _Standard inflation_ posits a
distinct reheating stage where the inflaton undergoes a damped oscillation in
the inflaton potential while interacting with and heating up ordinary matter
Kofman  _et al._ (1994, 1997) (see Amin  _et al._ (2014); Mazumdar and White
(2019); Bassett  _et al._ (2006); Baumann (2011); Allahverdi  _et al._ (2010)
for reviews). A different scenario is called _warm inflation_ Berera and Fang
(1995); Bastero-Gil and Berera (2009); Berghaus  _et al._ (2020). In this case
there is always a subdominant but significant part of the universe made up by
ordinary or dark matter. It is only when the inflaton rolls down the potential
that subsequently ordinary matter becomes dominant, thereby making a smooth
transition to the Big Bang.

Many microscopic models have been proposed for either scenario, all of which
have advantages and disadvantages (see e.g. Mazumdar and Rocher (2011)). In
standard inflation there is often a ‘preheating’ phase, where bosonic fields
undergo an exponential increase in density due to resonant amplification.
This, however, leads to a non-thermal state of which it is not _a priori_
clear if it thermalizes in time for the Big Bang scenario. Recently there has
been renewed interest in warm inflation, since it may avoid some of the
conjectured constraints on consistent quantum gravity theories that arise from
the swampland program Motaharfar  _et al._ (2019); Das  _et al._ (2020).

In this work we present a toy universe in which the inflaton is coupled to a
strongly coupled QFT (see also Kawai and Nakayama (2016) for an earlier
attempt). A unique and important aspect of strongly coupled QFTs is that they
approach hydrodynamics and thermalize as fast as possible Heller  _et al._
(2012); Liu and Sonner (2018). At the relevant energy scales even the strong
coupling constant of Quantum Chromodynamics is small due to asymptotic
freedom, so this QFT can be thought of as a hidden sector that exists at some
high energy scale. The strongly coupled QFT is described using holography,
which is a remarkable duality arising from string theory that can describe
strongly coupled QFTs in terms of a classical anti-de-Sitter (AdS) universe of
one higher dimension. The extra dimension can be thought of as energy scale,
whereby for a thermal state there exists a black hole horizon in the infrared.

While we present a general framework for reheating with a strongly coupled
QFT, in this work we will present a simple model example to illustrate its
dynamics. Quite strikingly we find that the model qualitatively reproduces
many of the features of warm inflation (see Fig. 1 for a cartoon). This
includes an extended period of cooling and exponential expansion, an inflaton
rolling down the potential, heating up the QFT and finally the transition to a
universe dominated by QFT matter in a thermal state.

While in this Letter we use standard inflationary terminology in describing
the evolution of the constructed universe we stress that in this work we do
not attempt to construct a realistic model for our universe. Rather, we focus
on a qualitative general description of an inflaton interacting with a
strongly coupled QFT with a specific evolution as an explicit example.

Figure 1: Illustration of the bulk and boundary during holographic reheating.

##  II Model

In order to model the energy transfer of the inflaton field to matter on a
dynamical spacetime, we evolve self-consistently the Einstein-inflaton
equations together with the energy momentum tensor for strongly coupled matter
described by holography. The total action of this model consists of two
different sectors and an interaction part

$S=S_{\rm EH+inf}+S_{\rm hol}+S_{\rm int}\,.$ (1)

The first sector $S_{\rm EH+inf}$ consists of four-dimenensional Einstein
gravity with a dynamical inflaton field, $S_{\rm hol}$ models the dynamics of
a strongly coupled QFT via the gauge/gravity duality in terms of a five-
dimensional gravity dual and $S_{\rm int}$ accounts for the direct coupling
between these two sectors.

The first term in (1) is the standard Einstein-Hilbert plus Klein-Gordon
action with a non-trivial scalar field potential $V_{\rm inf}(\phi)$ which
together describe the dynamics of the spacetime and the inflaton $\phi$ in the
four boundary dimensions

$S_{\rm
EH+inf}=\int\mathrm{d}^{4}x\sqrt{-\gamma}\left(\frac{R}{2\kappa_{4}}-\frac{1}{2}\gamma^{ij}\partial_{i}\phi\partial_{j}\phi-
V_{\rm inf}(\phi)\right)\,.$ (2)

Here $\kappa_{4}$ parametrizes the strength of the gravitational interaction
and $R$ is the Ricci scalar of the spacetime metric $\gamma_{ij}$. We define
this metric to be of Friedmann–Lemaître–Robertson–Walker (FLRW) type

$\mathrm{d}s^{2}=\gamma_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}=-\mathrm{d}t^{2}+a(t)^{2}\mathrm{d}\vec{x}^{2}\,,$
(3)

where $a(t)$ is the scale factor that determines the expansion of the
spacetime via the Einstein field equations. We consider a generic family of
inflaton potentials given by

$V_{\rm
inf}(\phi)=v_{0}+v_{1}\,e^{v_{2}(\phi-\phi_{m})}-v_{3}\,e^{-v_{4}(\phi-\phi_{m})^{2}}\,,$
(4)

where $\phi_{m}$ and $v_{0}$ are fixed by demanding that the potential and the
inflaton vanish at the global minimum $V_{\rm inf}(0)=V_{\rm
inf}^{\prime}(0)=0$. In Fig. 2 we show the potential we use in our simulation,
where we set $v_{1}=9/8$, $v_{2}=2/3$, $v_{3}=45$ and $v_{4}=1/50$. This
potential allows an inflaton that starts at $\phi=-30$ to go through a
sufficiently long inflating phase followed by an oscillating phase in which it
then reheats the QFT.

Figure 2: We show the inflaton potential. As an initial condition the inflaton
starts at $\phi=-30$ and then slowly rolls down the potential. At the bottom
of the potential the inflaton oscillates and reheats the universe due to its
coupling to the QFT.

The strongly coupled matter sector is defined via the gauge/gravity duality by
the five-dimensional bulk action

$S_{\rm
bulk}=\frac{2}{\kappa_{5}}\int\mathrm{d}^{5}x\sqrt{-g}\left(\frac{1}{4}\mathcal{R}-\frac{1}{2}(\partial\Phi)^{2}-V_{\rm
bulk}(\Phi)\right)\,,$ (5)

where $\kappa_{5}$ denotes the bulk gravitational coupling, $\mathcal{R}$ is
the Ricci scalar associated to the bulk metric $g_{\mu\nu}$ and $\Phi$ is a
bulk scalar field with potential

$V_{\rm
bulk}(\Phi)=\frac{1}{L^{2}}\left(-3-\frac{3\Phi^{2}}{2}-\frac{\Phi^{4}}{3}+\frac{11\Phi^{6}}{96}-\frac{\Phi^{8}}{192}\right)\,,$
(6)

where $L$ denotes the length scale of the asymptotic AdS metric $g_{\mu\nu}$,
which we set to unity. The bare bulk action $S_{\rm bulk}$ needs to be
renomalized by adding appropriate counter terms $S_{\rm bdry}$ which render
the holographic action $S_{\rm hol}=S_{\rm bulk}+S_{\rm bdry}$ in Eq. (1)
finite on-shell. The renormalized action $S_{\rm hol}$ then defines a
holographic bottom-up model Attems  _et al._ (2016) for a strongly coupled QFT
with regular renormalization group flow between its conformal ultraviolet and
infrared fixed points and broken conformal symmetry in between. The mass of
the bulk scalar field
$m^{2}=\frac{\partial^{2}V}{\partial\Phi^{2}}|_{\Phi=0}=-\frac{3}{2}$
determines the conformal scaling dimension $\Delta=3$ of the dual operator
$\mathcal{O}$ via the relation $m=\sqrt{\Delta(\Delta-4)}$.

Figure 3: (left) After an initial stage where the QFT cools down (till about
$t=3$), the inflaton $\phi$ slow-rolls down till it starts oscillating in the
potential well. (middle) Initially the dynamics is dominated by the QFT energy
till about $t=3$. After this the universe inflates till the inflaton reaches
the bottom of the potential at $t=14.3$. The inflaton oscillations then reheat
the QFT universe. (right) Initially the Hubble rate decreases due to the
dilution of the QFT energy till $t\approx 5$. After this the universe inflates
at a constant exponential rate till about $t=14.3$ when the inflaton is at the
bottom of the potential.

Finally, there is an interaction term in the effective action that couples the
inflaton via the vacuum expectation value (VEV) of the scalar operator
$\langle\mathcal{O}\rangle=\mathcal{O}_{\rm QFT}/\kappa_{5}$ to the
holographic sector

$S_{\rm int}=\int\mathrm{d}^{4}x\sqrt{-\gamma}~{}U(\phi)~{}\mathcal{O}_{\rm
QFT}\,,$ (7)

where the free function $U(\phi)$ defines the coupling of the model. In the
QFT the inflaton hence acts as a source for the scalar operator $\mathcal{O}$,
where the source is given by the asymptotic boundary value of the bulk scalar
$\Phi_{(0)}=U(\phi)$.

The total energy momentum tensor in the boundary theory consists of three
parts

$T_{ij}={\rm
diag}\left(\mathcal{E},\mathcal{P},\mathcal{P},\mathcal{P}\right)=T^{\rm
inf}_{ij}+\mathcal{T}_{ij}^{\rm QFT}+T^{\rm int}_{ij}\,,$ (8)

where $\mathcal{E}$ and $\mathcal{P}$ denote energy density and pressure,
respectively. The first part is the usual expression for the energy momentum
tensor of a scalar field

$T^{\rm
inf}_{ij}=\partial_{i}\phi\partial_{j}\phi-\gamma_{ij}\left(\frac{1}{2}\partial_{k}\phi\partial^{k}\phi+V_{\rm
inf}\right)\,.$ (9)

The second term $\mathcal{T}_{ij}^{\rm QFT}={\rm diag}\left(\mathcal{E}_{\rm
QFT},\mathcal{P}_{\rm QFT},\mathcal{P}_{\rm QFT},\mathcal{P}_{\rm QFT}\right)$
is the VEV of the holographic energy momentum tensor, where $\mathcal{E}_{\rm
QFT}$ and $\mathcal{P}_{\rm QFT}$ denote the corresponding energy density and
pressure. The third term results in an energy-momentum contribution due to the
direct coupling between the inflaton and the holographic sector

$T^{\rm int}_{ij}=-\frac{2}{\sqrt{-\gamma}}\frac{\delta S_{\rm
int}}{\delta\gamma^{ij}}=U(\phi)\,\mathcal{O}_{\rm QFT}\,\gamma_{ij}\,.$ (10)

The total energy momentum tensor is covariantly conserved

$\nabla^{i}T_{ij}=\nabla^{i}T^{\rm inf}_{ij}+\nabla^{i}\mathcal{T}_{ij}^{\rm
QFT}+\nabla^{i}T^{\rm int}_{ij}=0\,,$ (11)

when using the on-shell condition for the scalar field $\nabla^{i}T^{\rm
inf}_{ij}=-U(\phi)\,\partial_{j}\mathcal{O}_{\rm QFT}$ together with the Ward
identity for the holographic energy momentum tensor
$\nabla^{i}\mathcal{T}_{ij}^{\rm QFT}=-\partial_{j}\,U(\phi)\,\mathcal{O}_{\rm
QFT}$, where $\nabla^{i}$ is the Levi-Civita connection associated to
$\gamma_{ij}$.

In the standard holographic dictionary the QFT lives on a fixed (curved)
background spacetime $\gamma_{ij}$ and also the scalar source $\Phi_{(0)}$
acts as a free parameter that can be specified arbitrarily. Here, however, we
require them to satisfy the equations of motion that follow from the boundary
action (1). For the FLRW line element (3) one obtains the standard Friedmann
equations together with a scalar field equation for the inflaton that is
coupled to $\mathcal{O}_{\rm QFT}$:

$\displaystyle H(t)^{2}$
$\displaystyle=-\frac{\kappa_{4}}{3}\mathcal{E}(t)\,,$ (12)
$\displaystyle\frac{a^{\prime\prime}(t)}{a(t)}$
$\displaystyle=-\frac{1}{2}\left(\kappa_{4}\mathcal{P}(t)+H(t)^{2}\right)\,,$
(13) $\displaystyle\phi^{\prime\prime}(t)$
$\displaystyle=\partial_{\phi}\,U(\phi(t))\,\mathcal{O}_{\rm
QFT}(t)-3H(t)\phi^{\prime}(t)-\partial_{\phi}V_{\rm inf}(\phi(t))\,,$ (14)

where $H(t)=a^{\prime}(t)/a(t)$ is the Hubble rate and the total energy
density and pressure are given by

$\displaystyle\mathcal{E}$ $\displaystyle=$ $\displaystyle\mathcal{E}_{\rm
QFT}+V_{\rm inf}(\phi)+U(\phi)\,\mathcal{O}_{\rm QFT}+\frac{1}{2}\phi^{\prime
2}\,,$ (15) $\displaystyle\mathcal{P}$ $\displaystyle=$
$\displaystyle\mathcal{P}_{\rm QFT}-V_{\rm
inf}(\phi)-U(\phi)\,\mathcal{O}_{\rm QFT}+\frac{1}{2}\phi^{\prime 2}\,.$ (16)

The $-3\,H(t)\,\phi^{\prime}(t)$ in Eq. (14) is the standard friction term
that brings the inflaton to rest, but we note that with the holographic
coupling the scalar VEV $\mathcal{O}_{\rm QFT}$ also contributes. Importantly,
both $\mathcal{E}_{\rm QFT}$ and $\mathcal{O}_{\rm QFT}$ depend on the full
bulk geometry, including explicit dependencies on $\phi(t)$,
$\phi^{\prime}(t)$, $a(t)$ and $a^{\prime}(t)$. In addition, $\mathcal{O}_{\rm
QFT}$, $\mathcal{E}_{\rm QFT}$ and $\mathcal{P}_{\rm QFT}$ are not
independent, but related via the trace Ward identity

$\gamma^{ij}\mathcal{T}_{ij}^{\rm QFT}=\mathcal{E}_{\rm
QFT}-3\,\mathcal{P}_{\rm QFT}=-U(\phi)\,\mathcal{O}_{\rm QFT}+\mathcal{A}\,,$
(17)

where $\mathcal{A}$ is the conformal anomaly Henningson and Skenderis (1998);
Papadimitriou (2011). The variational principle in holography with dynamical
boundary conditions, the holographic renormalization of $S_{\rm hol}$,
together with the resulting expressions for $\mathcal{E}_{\rm QFT}$,
$\mathcal{P}_{\rm QFT}$, $\mathcal{O}_{\rm QFT}$ and the corresponding anomaly
corrected Ward identities as well as the thermodynamic properties of the
holographic QFT are given in Appendices A \- C.

Figure 4: (left) The pressure over energy density of the QFT together with the
predictions from ideal and viscous hydrodynamics. After a brief initial
hydrodynamisation period, the QFT is well described by viscous hydrodynamics
until the inflaton sources the QFT out of equilibrium. (middle) The equivalent
figure for the inflaton. Initially it is dominated by the potential having
$\mathcal{P}=-\mathcal{E}$ after which it oscillates around the minimim.
(right) We show the total pressure over energy density, which is initially
dominated by the QFT, then by the inflaton and at late times again by the
reheated QFT.  Figure 5: We show in blue the temperature as measured by the
surface gravity of the apparent horizon (AH) and the event horizon (EH), which
are numerically indistinguishable. The dotted red line is the temperature of
the QFT as determined by the equation of state. The green dashed line shows
that the horizon temperatures are lower than the QFT temperature by exactly
the de Sitter temperature $T_{\rm ds}=H/2\pi$.

##  III Solution Method

Computing the time evolution of the scale factor $a(t)$, the inflaton
$\phi(t)$ and the energy-momentum tensor $T_{ij}(t)$ for a given set of
initial conditions, requires to solve the corresponding initial value problem
for Eqs. (12) to (14) together with the dual bulk initial-boundary value
problem in a self-consistent way. For this we follow essentially the same
procedure as in Ecker  _et al._ (2022), where we solve a similar system with
dynamical boundary metric, but with constant source $\Phi_{(0)}$ for the
scalar field operator. The only addition here is that we promote
$\Phi_{(0)}(t)=U(\phi(t))$ to a dynamical field whose time dependence is
determined by the inflaton equation of motion (14).

At the initial time $t=t_{\rm ini}$ we need to specify initial conditions for
the energy density $\mathcal{E}_{\rm QFT}(t_{\rm ini})=\mathcal{E}_{\rm
QFT}^{\rm ini}$, the inflaton $\phi(t_{\rm ini})=\phi_{\rm ini}$ and its time
derivative $\partial_{t}\phi(t_{\rm ini})=\phi^{\prime}_{\rm ini}$ as well as
a profile for the bulk scalar $\Phi(r,t_{\rm ini})=\Phi_{\rm ini}(r)$ along
the holographic coordinate $r$ and whose asymptotic value is consistent with
the inflaton $\lim_{r\to\infty}\,r\,\Phi_{\rm ini}(r)=U(\phi_{\rm ini})$. Eqs.
(12) to (14) then determine $\phi^{\prime\prime}_{\rm ini}$ as well as the
Hubble rate $H$ and its time derivative $H^{\prime}$. It is important to note
that $\mathcal{O}_{\rm QFT}$ depends on $H^{\prime}$ and also that
$\mathcal{E}_{\rm QFT}$ depends on $\partial_{t}^{2}\phi$. The equations are
hence coupled and lead to a sixth order polynomial equation which we solve
numerically 111We use _Mathematica_ to numerically solve for the six different
solutions for the Hubble rate and take the negative real solution with
smallest absolute magnitude to construct the initial data.. After the
initialisation we evolve $H$ and $\partial_{t}\phi$ using Eqs. (13) and (14).
As in Ecker  _et al._ (2022) for the boundary metric we replace
$\partial_{t}^{3}\phi$ and $\partial_{t}^{4}\phi$ derivatives that appear in
the regularized bulk equations by their solutions in terms of the near-
boundary expansion.

For the evolution presented in this work we set $\kappa_{5}=1/9$,
$\kappa_{4}=\frac{2\pi}{5625}$, $U(\phi)=\lambda\,\phi$ with $\lambda=1/30$
and use $\mathcal{E}_{\rm QFT}^{\rm ini}=13275$, $\phi_{\rm ini}=-30$,
$\phi^{\prime}_{\rm ini}=3/10$ and $\tilde{\Phi}(r)=-6+120/r-300/r^{3}$ as
initial conditions, where $\tilde{\Phi}(r)$ is defined by
$\Phi(r)\equiv\Phi_{NB}(r)+r^{-3}\tilde{\Phi}(r)$ and $\Phi_{NB}(r)$ contains
near-boundary terms up to $\mathcal{O}(r^{-2})$ and
$\mathcal{O}(r^{-4}\log(r))$. These parameters are tuned to get an evolution
that shows both an inflationary and a reheating phase. Key choices for the
evolution to be numerically stable are to start with a relatively high energy
density, which guarantees a large (and hence stable) bulk black hole horizon
for a relatively long time. The energy density should however not be so high
that it would dominate the inflaton dynamics for the entire evolution. In this
way the energy density at the end of the inflationary period is close to the
vacuum and significant effects of the inflaton coupling can be seen to heat up
the QFT in the reheating phase.

##  IV Results

Fig. 3 shows the resulting evolution of the inflaton (left), energy density
(middle) and Hubble rate (right) of the model. The early phase is dominated by
the high initial energy density of the QFT, but at $t=3.27$ the inflaton
energy density becomes dominant and the universe enters a phase of relatively
constant exponential expansion. Later at $t=14.3$ the inflaton reaches the
bottom of the potential and starts oscillating rapidly. These oscillations
form sources for the QFT energy, which then increases from a minimum of
$\mathcal{E}_{\rm QFT}=0.21$ at $t=13.5$ to a subsequent maximum of
$\mathcal{E}_{\rm QFT}=0.64$ at $t=17.3$. Crucially this reheating continues,
which is apparent from the relatively slow scaling $\mathcal{E}_{\rm
QFT}\propto t^{-1.17}$ of the QFT energy density. The universe then keeps
expanding at increasingly slower rates, thereby cooling down both the inflaton
and the QFT energy density. At late times the QFT energy density is dominant.

In Fig. 4 we show the evolution of the pressure of the QFT (left), the
inflaton (middle) as well as the total pressure (right). After a very short
far-from-equilibrium stage, we see that the QFT pressure is well described by
the equations as given by viscous hydrodynamics, much like what was found in
Ecker  _et al._ (2022). After the inflaton rolls down, however, we see that
the reheating pushes the QFT significantly out of equilibrium. After this the
system settles down to equilibrium rather quickly. At late times the evolution
is completely dominated by the QFT, which is now close to its conformal IR
fixed point where $\mathcal{P_{\rm QFT}}=\mathcal{E_{\rm QFT}}/3$ (Fig. 4
right).

In Fig. 5 we show in blue the temperature obtained from the surface gravity of
the apparent horizon $T_{\rm AH}$ (explicit formulas are given in Appendix D).
We verified that the event horizon location is numerically indistinguishable
from the apparent horizon throughout the evolution, which is expected for a
theory in thermal equilibrium but unlike the vacuum de Sitter case of
Casalderrey-Solana  _et al._ (2021). During the entire evolution the
temperature is dominated by the QFT energy $\mathcal{E}$. Since
$H^{2}\propto\mathcal{E}$ and $T_{\rm QFT}^{4}\propto\mathcal{E}$ at late
times, the temperature of the cosmological horizon $T_{\rm dS}=H/2\pi\propto
T_{\rm QFT}^{2}$ is negligible if $T_{\rm QFT}$ is small. At early times we
notice a significant difference between the apparent horizon temperature and
the temperature obtained from the QFT equation of state. This can be fully
explained by the fact that the universe is expanding. Indeed, subtracting
$T_{\rm QFT}\rightarrow T_{\rm QFT}-H/2\pi$ accurately describes the complete
evolution with the exception of a small off-equilibrium time-window where the
inflaton approaches the minimum of the potential for the first time.

It is hard to verify why this precise shift of $T_{\rm dS}$ in the temperature
arises, since we are not aware of any analytic solutions of a thermal plasma
in expanding de Sitter space. Consistently, for vacuum ($\mathcal{E}=0$) it is
possible to show that $T_{\rm AH}=-H/2\pi$ Buchel (2019); Casalderrey-Solana
_et al._ (2021), even though the physical interpretation is somewhat unclear.
We verified that the exact same subtraction describes the evolution in Fig. 9
of Casalderrey-Solana  _et al._ (2021) up to the point where $T_{\rm
QFT}\approx T_{\rm dS}$.

##  V Discussion

For simplicity, our work is restricted to a specific model which assumes a
holographic potential that realizes in the dual field theory a renormalization
group flow between UV- and IR-fixed points and leads to the thermodynamics of
a smooth cross over between two conformally symmetric phases.

Changing the potential would allow to study QFTs with different equilibrium
properties, like for example theories with phase transitions and confinement
Gursoy  _et al._ (2008, 2009), or one may vary the dimension $\Delta$ of the
scalar operator that couples to the inflaton. Choosing $\Delta<3$ for example,
makes the linear coupling to the inflaton relevant, as $\phi$ has a weak-
coupling dimension near $1$.

It would also be interesting to generalize the field content of our
construction, for example by adding the dynamics of a gauge field in the bulk
theory, which would allow to model the dynamics of conserved charges Folkestad
_et al._ (2019) and gauge fields Ecker  _et al._ (2018); Ahn  _et al._ (2023)
in the boundary theory.

One may also change the function $U(\phi)$ that controls the coupling of the
inflaton to the scalar QFT operator. Non-linear options for this function may
affect non-trvially the evolution, like for example a quadratic $U$ affects
the effective mass of the inflaton and may stop inflation if it becomes large
enough.

The most exciting avenue will be to make our model into a realistic
inflationary scenario for our own universe that satisfies all the constraints
known from cosmology. For this several steps are required, including a
realistic coupling of the QFT to fields of the Standard Model. With the
current numerical code it would be challenging to obtain the number of
e-foldings that are realistically required, but from Fig. 4 it can be seen
that accurate approximations using hydrodynamics may be feasible.

###### Acknowledgements.

We thank Valerie Domcke, David Mateos, Francesco Nitti and Tomislav Prokopec
for interesting discussions. C. E. acknowledges support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR
211 “Strong-interaction matter under extreme conditions”– project number
315477589 – TRR 211. E. K. was supported in part by CNRS contract IEA 199430.

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##  Appendix A Variational Principle with Mixed Boundary Conditions

In this appendix we review the variational principle in holography with
dynamical boundary conditions for the source fields Compere and Marolf (2008);
Ishibashi  _et al._ (2023); Ahn  _et al._ (2023). We will restrict to the case
relevant to this work, namely Einstein-dilaton gravity with dynamical boundary
conditions for the metric and the scalar field governed by four-dimensional
Einstein-inflaton equations of motion.

In the supergravity limit, the gauge/gravity duality allows to express the
partition function of a strongly coupled quantum field theory (QFT) as a path
integral of a higher dimensional gravity action

$Z_{\rm
QFT}[\gamma,\phi]=\int[\mathcal{D}g]_{\gamma}[\mathcal{D}\Phi]_{\phi}e^{-S_{\rm
hol}}\,,$ (18)

where $\gamma_{ij}$ and $\phi$ denote the background geometry of the dual QFT
and the inflaton field on the boundary, while
$\int[\mathcal{D}g]_{\gamma}[\mathcal{D}\Phi]_{\phi}$ means integration over
all bulk geometries and scalar fields with fixed boundary conditions
$\gamma_{ij}$ and $\phi$, respectively. Promoting $\gamma_{ij}$ and $\phi$ to
dynamical fields allows one to define an induced gravity partition function as
a path integral over all boundary fields

$\displaystyle Z_{\rm ind}=$
$\displaystyle\int\mathcal{D}\gamma\mathcal{D}\phi Z_{\rm QFT}[\gamma,\phi]$
$\displaystyle=$
$\displaystyle\int\mathcal{D}\gamma\mathcal{D}\phi\int[\mathcal{D}g]_{\gamma}[\mathcal{D}\Phi]_{\phi}e^{-S_{\rm
hol}}$ $\displaystyle=$ $\displaystyle\int\mathcal{D}g\mathcal{D}\Phi
e^{-S_{\rm hol}}\,.$ (19)

Because the boundary fields are dynamical, the variations of the action
result, in addition to the bulk equations of motion, also in some non-trivial
boundary contributions

$\displaystyle\delta_{g}S_{\rm hol}$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}\mathrm{d}x^{5}\sqrt{-g}\,{\rm
EOM_{bulk}^{(g)}}\,\delta g^{\mu\nu}$ (20) $\displaystyle+$
$\displaystyle\int_{\partial\mathcal{M}}\mathrm{d}x^{4}\sqrt{-\gamma}\frac{1}{2}\langle
T_{ij}^{\rm QFT}\rangle\delta\gamma^{ij}\,,$ $\displaystyle\delta_{\Phi}S_{\rm
hol}$ $\displaystyle=$ $\displaystyle\int\mathrm{d}x^{5}\sqrt{-g}\,{\rm
EOM_{bulk}^{(\Phi)}}\,\delta\Phi^{\mu\nu}$ (21) $\displaystyle+$
$\displaystyle\int_{\partial\mathcal{M}}\mathrm{d}x^{4}\sqrt{-\gamma}\langle\mathcal{O}\rangle\delta\phi\,.$

Since $S_{\rm hol}$ includes the counter terms, whose explicit form is derived
in the next section, the boundary contributions can be identified with the
expectation values of the renormalized holographic stress tensor $\langle
T_{ij}^{\rm QFT}\rangle$ and the scalar field operator
$\langle\mathcal{O}\rangle$ of the boundary theory.

There are different ways to make the actions (20) and (21) stationary Compere
and Marolf (2008). The simplest option is to impose Dirichlet boundary
conditions on $g_{\mu\nu}$ and $\Phi$, i.e., demanding $\delta\gamma_{ij}=0$
and $\delta\phi=0$, which makes the boundary geometry and the inflaton field
static. Another option is to impose Neuman boundary conditions, which is
demanding $\langle T_{ij}^{\rm QFT}\rangle=\langle\mathcal{O}\rangle=0$. In
this case $\gamma_{ij}$ and $\phi$ can remain dynamical and the bulk geometry
is fixed to empty AdS5. Combinations of these two options are of course also
possible. Finally, the most general possibility is to impose mixed boundary
conditions, that is to demand $\frac{1}{2}\langle T_{ij}^{\rm
QFT}\rangle+\frac{\delta S_{{\rm bdry,}\gamma}}{\delta\gamma^{ij}}=0$ and
$\langle\mathcal{O}\rangle+\frac{\delta S_{{\rm bdry,}\phi}}{\delta\phi}=0$
for some functionals of the boundary metric $S_{{\rm bdry,}\gamma}$ and the
scalar field $S_{{\rm bdry,}\phi}$ that can be added to the bulk action. In
this work we choose these boundary functionals to be given by the Einstein-
Hilbert plus inflaton action $S_{\rm EH+inf}[\phi,\gamma_{ij}]$ and the
interaction term $S_{\rm int}[\phi,\gamma_{ij}]$.

##  Appendix B Holographic Renormalization

In this appendix we derive the explicit form of the renormalized expectation
values of the holographic energy momentum tensor and the scalar field operator
Bianchi  _et al._ (2001, 2002); Skenderis (2002). In the following we assume
Fefferman–Graham (FG) gauge

$ds^{2}=L^{2}\frac{\mathrm{d}\rho^{2}}{4\rho^{2}}+\bar{g}_{ij}(\rho,x)\mathrm{d}x^{i}\mathrm{d}x^{j}\,,$
(22)

where the boundary is located at $\rho=0$ and is parametrized by the
coordinates $x^{i}$ with $i=0,\ldots,3$ and $L$ denotes the Anti-de Sitter
length scale which we set to unity. Near the boundary the metric and the
scalar field take the form

$\displaystyle\bar{g}_{ij}(\rho,x)$ $\displaystyle=$
$\displaystyle\frac{1}{\rho}\Big{[}\gamma_{ij}(x)+\rho\,\gamma_{(2)ij}(x)+\rho^{2}\gamma_{(4)ij}(x)$
(23) $\displaystyle+$
$\displaystyle\rho^{2}\log\rho\,h_{(4)ij}(x)+O(\rho^{3})\Big{]}\,,$
$\displaystyle\Phi(\rho,x)$ $\displaystyle=$
$\displaystyle\rho^{1/2}\Big{[}\Phi_{(0)}(x)+\rho\,\Phi_{(2)}(x)$ (24)
$\displaystyle+$
$\displaystyle\rho\,\log\rho\,\psi_{(2)}(x)+O(\rho^{2})\Big{]}\,.$

The first term $\gamma_{ij}(x)$ in the expansion of the metric is the boundary
metric and the term $\Phi_{(0)}(x)$ plays the role of the inflaton field in
the boundary theory. The Klein–Gordon equation for the scalar field fixes the
logarithmic coefficient $\psi_{(2)}$ in terms of $\gamma_{ij}$ and
$\Phi_{(0)}$ as

$\psi_{(2)}=\frac{1}{4}\left(\nabla^{2}\Phi_{(0)}-\frac{1}{6}\Phi_{(0)}R\right)\,.$
(25)

At leading order Einstein’s equations determine

$\gamma_{(2)ij}=-\frac{1}{2}\left(R_{ij}-\frac{1}{6}R\,\gamma_{ij}\right)-\frac{\Phi_{(0)}^{2}}{3}\gamma_{ij}\,.$
(26)

The logarithmic part at sub-leading order fixes

$\displaystyle h_{(4)ij}$ $\displaystyle=$ $\displaystyle
h_{(4)ij}^{\mbox{\tiny{grav}}}-\frac{1}{12}R_{ij}\Phi_{(0)}^{2}-\frac{1}{3}\nabla_{i}\Phi_{(0)}\nabla_{j}\Phi_{(0)}$
(27) $\displaystyle+$
$\displaystyle\frac{1}{12}\nabla_{i}\Phi_{(0)}\nabla^{i}\Phi_{(0)}\,\gamma_{ij}+\frac{1}{6}\Phi_{(0)}\nabla_{i}\nabla_{j}\Phi_{(0)}$
$\displaystyle+$
$\displaystyle\frac{1}{12}\Phi_{(0)}\,\square_{\gamma}\,\Phi_{(0)}\,\gamma_{ij}\,,$

where the pure gravitational part is given by

$\displaystyle h_{(4)ij}^{\mbox{\tiny{grav}}}$ $\displaystyle=$
$\displaystyle\frac{1}{8}R_{ikjl}R^{kl}-\frac{1}{48}\nabla_{i}\nabla_{j}R+\frac{1}{16}\nabla^{2}R_{ij}-\frac{1}{24}RR_{ij}$
(28) $\displaystyle+$
$\displaystyle\left(\frac{1}{96}R^{2}-\frac{1}{96}\nabla^{2}R-\frac{1}{32}R_{kl}R^{kl}\right)\gamma_{ij}\,.$

The expectation values of the holographic energy momentum tensor and the
scalar field operator follow from variations of the renormalized action of the
holographic model

$S_{\rm hol}=S_{\rm bulk}+S_{\rm GHY}+S_{\rm ct}\,.$ (29)

The bare bulk action $S_{\rm bulk}$ is defined in (5) and the second term is
the Gibbons–Hawking–York (GHY) boundary term

$S_{\rm GHY}=\frac{1}{\kappa_{5}}\int\mathrm{d}^{4}x\sqrt{-\gamma}K\,,$ (30)

where $K=\gamma^{ij}K_{ij}=\gamma^{ij}\nabla_{i}n_{j}$ denotes the trace of
the extrinsic curvature of a four-dimensional slice of the bulk geometry near
the boundary. The last contribution in (29) is a counter term that is defined
on a constant-$\rho$ hypersurface near the boundary, which is necessary to
render the on-shell action $S_{\rm hol}$ finite in the limit $\rho\to 0$

$\displaystyle S_{\mathrm{ct}}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa_{5}}\int\mathrm{d}^{4}x\sqrt{-\gamma}\Bigg{[}\left(-\frac{1}{8}R-\frac{3}{2}-\frac{1}{2}\Phi_{(0)}^{2}\right)$
(31) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(\log\rho\right)\mathcal{A}+\left(\alpha\mathcal{A}+\beta\Phi_{(0)}^{4}\right)\Bigg{]}\,,$

where the constants $\alpha$ and $\beta$ parametrize the residual
renomalization-scheme ambiguity of the model. The holographic conformal
anomaly Henningson and Skenderis (1998); Papadimitriou (2011)
$\mathcal{A}=\mathcal{A}_{g}+\mathcal{A}_{\phi}$ consists of a gravitational
part due to the curved boundary geometry

$\mathcal{A}_{g}=\frac{1}{16}(R^{ij}R_{ij}-\frac{1}{3}R^{2})\,,$ (32)

and a part due to scalar matter

$\mathcal{A}_{\Phi_{(0)}}=-\frac{1}{2}\left(\partial_{i}\Phi_{(0)}\partial^{i}\Phi_{(0)}+\frac{1}{6}R\Phi_{(0)}^{2}\right)\,.$
(33)

All this together results in the following expression for the holographic
stress tensor

$\displaystyle\langle T_{ij}^{\rm QFT}\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{-\gamma}}\frac{\delta S_{\rm
hol}}{\delta\gamma^{ij}}$ (34) $\displaystyle=$
$\displaystyle\frac{2}{\kappa_{5}}\Bigg{\\{}\gamma_{(4)ij}+\frac{1}{8}\left[\mathrm{Tr}\gamma_{(2)}^{2}-(\mathrm{Tr}\gamma_{(2)})^{2}\right]\gamma_{ij}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\gamma_{(2)}^{2}+\frac{1}{4}\gamma_{(2)ij}\mathrm{Tr}\gamma_{(2)}+\frac{1}{2}\partial_{i}\Phi_{(0)}\partial_{j}\Phi_{(0)}$
$\displaystyle+$
$\displaystyle\left(\Phi_{(0)}\Phi_{(2)}-\frac{1}{2}\Phi_{(0)}\psi_{(2)}-\frac{1}{4}\partial_{k}\Phi_{(0)}\partial^{k}\Phi_{(0)}\right)\gamma_{ij}$
$\displaystyle+$
$\displaystyle\alpha\left(\mathcal{T}^{\gamma}_{ij}+\mathcal{T}_{ij}^{\phi}\right)+\left(\frac{1}{18}+\beta\right)\Phi_{(0)}^{4}\gamma_{ij}\Bigg{\\}}\,.$

The anomalous contributions to the stress tensor are given by

$\displaystyle\mathcal{T}^{g}_{ij}$ $\displaystyle=$ $\displaystyle
2h_{(4)ij}\,,$ (35) $\displaystyle\mathcal{T}_{ij}^{\phi}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Phi_{(0)}^{2}R_{ij}-\frac{2}{3}\nabla_{i}\Phi_{(0)}\nabla_{j}\Phi_{(0)}+\frac{1}{6}\gamma^{kl}\nabla_{k}\nabla_{l}\Phi_{(0)}\gamma_{ij}$
(36) $\displaystyle+$
$\displaystyle\frac{1}{3}\Phi_{(0)}\nabla_{i}\nabla_{j}\Phi_{(0)}+\frac{1}{6}\Phi_{(0)}\square\Phi_{(0)}\gamma_{ij}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\gamma_{ij}\left(\Phi_{(0)}\square\Phi_{(0)}-\frac{1}{6}R\Phi_{(0)}^{2}\right)\,.$

The expectation value of the scalar operator in the field theory is then given
by

$\displaystyle\langle\mathcal{O}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-\gamma}}\frac{\delta S_{\rm
hol}}{\delta\Phi_{(0)}}$ (37) $\displaystyle=$
$\displaystyle\frac{2}{\kappa_{5}}\left[(1-4\alpha)\psi_{(2)}-2\Phi_{(2)}-4\beta\Phi_{(0)}^{3}\right]\,.$

The holographic stress tensor satisfies anomaly-corrected Ward identities

$\displaystyle\nabla^{i}\langle T_{ij}^{\rm QFT}\rangle$ $\displaystyle=$
$\displaystyle-\langle\mathcal{O}\rangle\nabla_{j}\Phi_{(0)}\,,$ (38)
$\displaystyle\gamma^{ij}\langle T_{ij}^{\rm QFT}\rangle$ $\displaystyle=$
$\displaystyle-\Phi_{(0)}\langle\mathcal{O}\rangle+\frac{1}{\kappa_{5}}\left(\mathcal{A}_{g}+\mathcal{A}_{\Phi_{(0)}}\right)\,.$
(39)

The inflaton field in the main text $\phi=\Phi_{(0)}/\lambda$ is related to
the source of the scalar operator $\Phi_{(0)}$ via the coupling constant
$\lambda$ and all QFT expectation values are given in units of the bulk
gravitational coupling

$\displaystyle\mathcal{T}_{ij}^{\rm QFT}$ $\displaystyle=$
$\displaystyle\kappa_{5}\langle T^{\rm QFT}_{ij}\rangle$ (40) $\displaystyle=$
$\displaystyle{\rm diag}\left(\mathcal{E}_{\rm QFT},\mathcal{P}_{\rm
QFT},\mathcal{P}_{\rm QFT},\mathcal{P}_{\rm QFT}\right)\,,$
$\displaystyle\mathcal{O}_{\rm QFT}$ $\displaystyle=$
$\displaystyle\kappa_{5}\langle\mathcal{O}\rangle\,.$ (41)

Finally, in our setup with dynamical boundary equations, $\alpha$ and $\beta$
renormalize the bare gravitational coupling and the cosmological constant in
the boundary theory Ecker  _et al._ (2022)

$\displaystyle\frac{1}{\kappa_{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa_{4,\rm bare}}+\frac{\alpha}{96\,\kappa_{5}}\,,$
(42) $\displaystyle\frac{\Lambda_{4}}{\kappa_{4}}$ $\displaystyle=$
$\displaystyle\frac{\Lambda_{4,\rm
bare}}{\kappa_{4}}-\frac{\beta}{1024\,\pi}\,.$ (43)

We fix $\alpha=0$ and $\beta=\frac{1}{16}$, because this choice leads to a
supersymmetric renormalisation scheme in which the full boundary stress tensor
vanishes identically if the boundary metric is flat.

##  Appendix C Properties of the holographic QFT

Here we review some basic properties of the holographic QFT used in this work
(see also Casalderrey-Solana  _et al._ (2021)). This theory has a relevant
scalar operator that is dual to a bulk scalar field $\Phi$. For convenience,
we repeat here the corresponding Einstein-dilaton type bulk action

$S_{\rm
bulk}=\frac{2}{\kappa_{5}}\int\mathrm{d}^{5}x\sqrt{-g}\left(\frac{1}{4}\mathcal{R}-\frac{1}{2}(\partial\Phi)^{2}-V_{\rm
bulk}(\Phi)\right)\,,$ (44)

where $\kappa_{5}$ denotes the bulk gravitational coupling, $\mathcal{R}$ is
the Ricci scalar associated to the bulk metric $g_{\mu\nu}$ and $\Phi$ is the
bulk scalar field with potential

$V_{\rm
bulk}(\Phi)=\frac{1}{L^{2}}\left(-3-\frac{3\Phi^{2}}{2}-\frac{\Phi^{4}}{3}+\frac{11\Phi^{6}}{96}-\frac{\Phi^{8}}{192}\right)\,.$
(45)

The bulk potential $V_{\rm bulk}(\Phi)$ has several extrema and is shown in
Fig. 6.

Figure 6:  Scalar field potential of the holographic model.

The most interesting for us is the maximum at $\Phi=0$, which corresponds to
an UV fixed point, or a CFTUV. On the two sides of this maximum are two
symmetric minima at $\Phi=\pm 2$ that correspond to two copies of an IR
conformal theory CFTIR. As the potential has a $\Phi\to-\Phi$ symmetry, the
two minima correspond to the same CFTIR. Around the maximum, the dimension of
the relevant scalar operator is $\Delta_{\rm UV}=3$, while at the minima the
dimension of the same scalar operator is $\Delta_{\rm IR}=25/6$. The relevant
coupling in the CFTUV has mass dimension one, and is therefore like a fermion
mass scale. Our QFT is a RG flow between CFTUV and CFTIR that is driven by the
source of the scalar operator with mass scale $m$. Moreover, the QFT has the
symmetry $m\to-m$. This QFT is therefore massless in the IR, with the massless
(and strongly coupled) degrees of freedom being those of CFTIR. Moreover,
although the QFT is strongly coupled at all scales, it is a non-confining QFT.

Figure 7: We show the thermodynamical entropy density (left), pressure
(middle) and bulk viscosity (right) as a function of temperature or energy
density. This figure is reproduced from Casalderrey-Solana  _et al._ (2021)
(see also Attems  _et al._ (2016)).

The thermodynamic and transport properties of the model were analyzed in
Attems  _et al._ (2016). At $T>0$ the theory is entering the black-hole phase
and remains in it, all the way to $T\to\infty$. In Fig. 7 (reproduced from
Casalderrey-Solana  _et al._ (2021)) we show from left to right the
dimensionless entropy ratio $s/T^{3}$ as a function of the (dimensionless)
temperature, the ratio $3{\cal P}/{\cal E}$ of the pressure to the energy
density and ratio of the bulk viscosity to the entropy density, $\zeta/s$, as
function of energy density. All these quantities asymptote to their conformal
values at small and large temperatures or energy densities.

Finally, we comment here on the evolution of the QFT when it is coupled to the
inflaton. The inflaton field is by construction proportional to the mass scale
of the QFT. When the inflaton slow-rolls in the potential on the left side of
Fig. 2, the mass scale of the QFT starts at large negative values and slowly
increases towards zero. This represents an inverse RG flow that is driven by
the cosmological evolution of the inflaton. Once the inflaton settles at the
minimum of the potentials at $\Phi=0$, the mass scale becomes zero and leaves
the QFT at its UV limit, namely CFTUV.

##  Appendix D Solving the Bulk Model Numerically

The action (5) results in a coupled set of equations of motion for the bulk
that are the five-dimensional Einstein–Klein–Gordon equations

$\displaystyle\mathcal{R}_{\mu\nu}-\frac{1}{2}\mathcal{R}\,g_{\mu\nu}$
$\displaystyle=$ $\displaystyle 2\partial_{\mu}\Phi\partial_{\nu}\Phi-
g_{\mu\nu}\left(2V_{\rm bulk}+(\partial\Phi)^{2}\right)\,,$ (46)
$\displaystyle\square_{g}\Phi$ $\displaystyle=$ $\displaystyle\frac{\partial
V_{\rm bulk}}{\partial\Phi}\,.$ (47)

The method we use to solve the corresponding initial value problem for fixed
boundary conditions was first presented in Chesler and Yaffe (2009) for pure
gravity and further reviewed in Chesler and Yaffe (2014); van der Schee
(2014). For the case of dynamical boundary conditions as studied here, the
method was first extended numerically in Ecker  _et al._ (2022). Here we give
a summary of this method.

For the numerical treatment of the initial value problem it is convenient to
use generalized Eddington–Finkelstein (EF) coordinates rather than FG gauge to
parametrize the bulk geometry and the scalar field

$\displaystyle\mathrm{d}s_{\rm bulk}^{2}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$ (48) $\displaystyle=$
$\displaystyle-A(r,t)\mathrm{d}t^{2}+2\mathrm{d}r\mathrm{d}t+S(r,t)^{2}\mathrm{d}\vec{x}^{2}\,,$
(49) $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\Phi(r,t)\,,$ (50)

where the asymptotic boundary is located at $r=\infty$. In this gauge the
coupled set of Einstein and scalar field equations result in the following
nested set of ODEs

$\displaystyle S^{\prime\prime}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}S\left(\Phi^{\prime}\right)^{2}\,,$ (51)
$\displaystyle\dot{S}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{2\dot{S}S^{\prime}}{S}-\frac{2SV}{3}\,,$ (52)
$\displaystyle\dot{\Phi}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{V^{\prime}}{2}-\frac{3\dot{S}\Phi^{\prime}}{2S}-\frac{3S^{\prime}\dot{\Phi}}{2S}\,,$
(53) $\displaystyle A^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{12\dot{S}S^{\prime}}{S^{2}}+\frac{4V}{3}-4\dot{\Phi}\Phi^{\prime}\,,$
(54) $\displaystyle\ddot{S}$ $\displaystyle=$
$\displaystyle\frac{\dot{S}A^{\prime}}{2}-\frac{2S\dot{\Phi}^{2}}{3}\,,$ (55)

where a prime denotes a radial derivative, $f^{\prime}\equiv\partial_{r}f$,
and an overdot is short-hand for the modified derivative
$\dot{f}\equiv\partial_{t}f+\frac{1}{2}A\partial_{r}f$. The beauty of the
scheme of this so-called characteristic formulation is that specifying
$\Phi(z)$ leads to $\partial_{t}\Phi(z)$ through this nested set of ODEs,
which is much simpler than the typical PDE system encountered in numerical
relativity. We solve the initial value problem using the procedure explained
in Casalderrey-Solana  _et al._ (2021) imposing the
Friedmann–Lemaître–Robertson–Walker metric

$\mathrm{d}s^{2}=\gamma_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}=-\mathrm{d}t^{2}+a(t)^{2}\mathrm{d}\vec{x}^{2}\,,$
(56)

which leads to the boundary condition $S(r)=r\,a(t)+\mathcal{O}(r^{0})$ for
Eq. (51). The boundary conditions for Eq. (52) and Eq. (53) is fixed by
demanding regularity near the boundary, while for Eq. 54 they are fixed by the
energy density $\mathcal{E}$. The energy density is evolved by using
conservation of the stress-energy tensor or alternatively by using Eq. (55).

In practice it is difficult to solve these equations directly. For numerical
efficiency it is better to switch coordinates to $z=1/r$ and to perform a
near-boundary (NB) expansion of both the equations and the functions $S$, $A$
and $\Phi$. One then redefines
$\Phi(r)\equiv\Phi_{NB}(r)+r^{-3}\tilde{\Phi}(r)$ with $\Phi_{NB}(r)$
containing near-boundary terms up to $\mathcal{O}(r^{-2})$ and
$\mathcal{O}(r^{-4}\log(r))$ (and analogously for $S$ and $A$). When using
spectral methods, it is especially important to subtract a high number of
logarithmic NB terms, as spectral methods rely on regularity of the functions
presented. Lastly, it is convenient to apply a gauge transformation
$r\rightarrow r+\xi(t)$ such that the apparent horizon (AH) remains at a
constant value of the $r$ coordinate. Since the condition for the location of
the AH equals $\dot{S}=0$ the equation for $\xi(t)$ can be obtained by solving
$\partial_{t}\dot{S}=0$ on the AH (note that in Fig. 1 we do not apply this
gauge transformation for clarity).

The methods to couple the equations with our boundary Friedmann+inflaton
equations are exactly the same as in Ecker  _et al._ (2022) with the only
exception that we now have a dynamical source for the scalar field as is
detailed in the main text. For completeness we note that we use a
pseudospectral grid with 5 domains each having 7 grid points. In the
simulations we use $\kappa_{5}=1$ and rescale to the chosen $\kappa_{5}$ only
when plotting results. We use timesteps of $\delta t=0.0005$ and filter then
numerical functions every $50$ timesteps by interpolating back and forth to a
spectral grid with 5 points (see also Chesler and Yaffe (2014)). We start with
a radial gauge transformation $r\rightarrow r+\xi$ with $\xi=1.709$. This
fixes the apparent horizon at $z=1/r=0.36$ and the evolution equations for
$\xi(t)$ guarantee that the horizon stays there (nevertheless, every 100
timesteps we perform a tiny gauge transformation to bring it back to
$z=0.36$). The full evolution then takes around 200 hours on a single core
using _Mathematica_ 11.

For reference we note that the temperature as derived from the surface gravity
$\kappa$ can be obtained from $T_{\rm
AH}=\kappa/2\pi=-\frac{z^{2}}{4\pi}\partial_{z}A$ evaluated at the location of
the apparent horizon. For the event horizon this requires solving the
differential equation $\partial_{t}r_{\rm EH}=-\tfrac{1}{2}A$ with boundary
condition $A(t,r_{\rm EH}(t=\infty))=0$. This reflects the fact that the event
horizon is teleological, e.g. it depends on the future spacetime. In our
simulations we go to a finite time where the geometry is sufficiently constant
such that a full solution of the event horizon can be obtained.

The full code can be downloaded at wilkevanderschee.nl.


# AI-Aided Integrated Terrestrial and Non-Terrestrial 6G Solutions for
Sustainable Maritime Networking

Salwa Saafi  Olga Vikhrova  Gábor Fodor  Jiri Hosek  and Sergey Andreev Salwa
Saafi (corresponding author, email<EMAIL_ADDRESS>is with Brno University of
Technology and Tampere University. Olga Vikhrova, and Sergey Andreev are with
Tampere University, Finland. Gábor Fodor is with Ericsson Research and KTH
Royal Institute of Technology. Jiri Hosek is with the Department of
Telecommunications, Brno University of Technology, Czech Republic.

###### Abstract

The maritime industry is experiencing a technological revolution that affects
shipbuilding, operation of both seagoing and inland vessels, cargo management,
and working practices in harbors. This ongoing transformation is driven by the
ambition to make the ecosystem more sustainable and cost-efficient.
Digitalization and automation help achieve these goals by transforming
shipping and cruising into a much more cost- and energy-efficient, and
decarbonized industry segment. The key enablers in these processes are always-
available connectivity and content delivery services, which can not only aid
shipping companies in improving their operational efficiency and reducing
carbon emissions but also contribute to enhanced crew welfare and passenger
experience. Due to recent advancements in integrating high-capacity and ultra-
reliable terrestrial and non-terrestrial networking technologies, ubiquitous
maritime connectivity is becoming a reality. To cope with the increased
complexity of managing these integrated systems, this article advocates the
use of artificial intelligence and machine learning-based approaches to meet
the service requirements and energy efficiency targets in various maritime
communications scenarios.

This article has been accepted for publication in a future issue of IEEE
Network, but has not been fully edited. Content may change prior to final
publication.
©2022 IEEE. Personal use of this material is permitted. Permission from IEEE
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##  I Introduction

The maritime industry anticipates a substantial increase in the number of
operating vessels, new harbors, and routes worldwide in response to the trade
facilitation initiatives supported by the World Trade Organization. These
initiatives aim to speed up international trade and unlock inclusive economic
development. As a fast-growing sector, this industry is also experiencing
external pressures due to environmental concerns. Today’s greenhouse gas
emissions from shipping are estimated as 2.6% of the total global emissions,
which is the equivalent of emissions from a large country [1]. To cope with
this escalation, environmental sustainability and digital inclusion practices
become fundamental and have to be incorporated across all maritime operations.

For sustainable industry, the maritime sector needs to go through an extensive
optimization and evolution toward fully autonomous, globally connected, and
digitalized operations with zero-emissions [2]. The success of automation of
the maritime industry relies heavily on dynamic networking and artificial
intelligence (AI) technologies that foster the emerging applications, such as
intelligent harbors, remote on-board maintenance, and autonomous docking.
These require unprecedentedly high data rates, large-scale connectivity
between a large number of dissimilar terminals, life-long learning and
inference at the smart end-points, and seamless operation of terrestrial and
non-terrestrial networks.

Recent findings showed that nearly 90% of data generated on-board never leaves
the deck, which means that operators are missing out on valuable insights and
analytics for improved logistics, cost of maintenance, and resource
utilization [3]. Owing to recent advances in satellite technology, the number
of connected vessels has doubled over the last 5 years, but only 75% of
vessels have on-board Internet access today. Satellite-based backhauling
remains extremely costly and inherently limited, thus presenting serious
challenges for the widespread technology adoption to meet the growing needs of
the maritime industry. However, existing maritime communication systems offer
dedicated services opportunistically (e.g., in proximity to coastal
infrastructure), rather than genuinely inter-connecting humans, vessels, and
ports into a holistic ecosystem.

While attempting to integrate several non-terrestrial networks into a unified
infrastructure to facilitate the demanding intelligent broadband services for
maritime operations, cellular Fifth Generation (5G) systems become
increasingly convoluted and energy consuming, which risks compromising the
fundamental need for sustainability [4]. By contrast, Sixth Generation (6G)
technologies are envisaged not only as those employing higher frequencies
(e.g., millimeter-wave (mmWave) and Terahertz bands) to achieve extreme
throughputs, but also as solutions capable of supporting AI-aided closed-loop
automation. Such systems are expected to enable the ultimate potential of the
zero-touch architecture proposed by the European Telecommunications Standards
Institute (ETSI) for fully automated networks [5].

In this article, we first examine the most important of the emerging maritime
use cases, which call for a careful design of a 6G maritime network (6G-MN)
that facilitates communication and computation applications. The key
components of 6G-MN s are dynamic networking – where communication links are
created and activated on-demand – and distributed intelligence, where learning
and inference are employed at different levels of the system. In our
supportive study, we demonstrate how machine learning (ML) can aid and
outperform traditional model-based approaches for energy-efficient topology
management and scheduling in dynamic maritime networks. We then identify
communication- and learning-related challenges in future AI-aided 6G-MN s.

The rest of this article is organized as follows. We first introduce the
rationale for building future maritime communication systems around cellular
networks and discuss the essential maritime use cases. We then review the
challenges of sustainable operation in 6G-MN s and highlight the key role of
AI for network-wide optimization in terms of topology management and resource
allocation. We also summarize the open issues related to the integration of
AI-based solutions in 6G-MN s and offer concluding remarks and future
perspectives on this work in the final section.

##  II Cellular-Enabled Maritime Networks

###  II-A Existing Maritime Communication Systems

Targeting navigation safety and maritime environment protection, the
International Maritime Organization (IMO) and the International
Telecommunication Union (ITU) cooperatively launched the global maritime
distress and safety system (GMDSS). The latter includes a set of terrestrial
and satellite radio technologies employed for people and vessel rescue in
distress. To further improve ship-to-ship and ship-to-shore navigation
accuracy, the IMO introduced the automatic identification system (AIS), which
complements marine radars with tracking information for vessel collision
avoidance and better situational awareness [6].

Despite being an effective technology for navigation assistance and maritime
emergency services, the AIS provides low data rate communications for the
exchange of basic navigation parameters such as speed, position, and direction
[4]. This limitation motivated the International Association of Marine Aids to
Navigation and Lighthouse Authorities (IALA) to develop their own very high
frequency data exchange system (VDES). Building upon AIS capabilities, VDES
encompasses several communication subsystems aiming to provide higher data
rates, enhance the operating ranges by the integration of satellite
components, augment security mechanisms, and support new maritime use cases
such as e-Navigation [4]. The concept was introduced by the IMO to harmonize
maritime navigation systems in offshore and coastal regions [6].

Although regulatory bodies continue to improve the systems discussed above,
the ongoing evolution toward a digitalized maritime industry poses new
challenges to maritime communication system design. With the increasing number
of vessels, growing level of ship autonomy, and widespread adoption of
Internet of Things (IoT) technologies, novel connectivity solutions need to
ensure cost-efficiency, scalability, and service availability [7]. Such
communication systems for modern maritime operations have to support not only
the existing e-Navigation and GMDSS services, but also the emerging broadband
and low-latency applications discussed in the sequel.

###  II-B Emerging 6G-MN Use Cases

Based on the IMO and AIS specifications and our understanding of market
trends, the emerging maritime use cases can be grouped into six categories as
illustrated in Fig. 1. This figure also maps the use cases onto connectivity
requirements in terms of 5G service classes, namely, enhanced mobile broadband
(eMBB), ultra-reliable low-latency communications (URLLC), and massive
machine-type communications (mMTC).

Figure 1: Use case categories in 6G-MNs

The navigation and fleet management applications facilitate the exchange of
telematic information for enhanced situational awareness and maritime fleet
management. They provide mission-critical services for vessels of different
types (i.e., cargo, law-enforcement, research, commercial, and leisure) and
shore-based traffic management organizations using high-speed broadband links.

Shipborne IoT aims to improve on-board operation and navigation by
communicating vessel’s speed, fuel consumption, and carbon dioxide emission
information to the on-board sensor fusion systems. Narrowband and massively
deployed sensors generate abundant machine-type raw data for subsequent
analysis and feature extraction. Vessel logistics is another shipborne use
case category where the crew utilizes on-board communications for staff
coordination and supply management and in the case of internal emergencies. On
top of that, on-board infotainment provides passengers with access to video
streaming, gaming, and interactive applications. These new shipborne use case
categories comprise the requirements of broadband, critical, and massive
machine-type communications, thus making the system highly heterogeneous in
terms of traffic patterns, quality of service (QoS) requirements, and device
capabilities.

As the name implies, the use case category of maritime search and rescue (SAR)
provides medical emergency and “Man Overboard” rescue services that entail
broadband and mission-critical applications to connect users in distress, on
vessels, and around shore-based facilities. To continue with shore-based
applications, harbor logistics offers a range of services for planning,
organization, and inspection of harbors and industrial port operations. Ship
loading/unloading coordination, asset tracking, warehouse management, short-
sea, and feeder shipping can be monitored via these services. Similarly to
vessel logistics use cases, harbor logistics may involve all three types of
communication regimes.

TABLE I: Overview of enabling solutions for maritime use cases Solution |  Related concepts |  Relevant 6G-MN use cases
---|---|---
5G service classes |  eMBB, URLLC, mMTC |  Broadband, critical, and massive machine-type communications for all 6G-MN use cases
NR NTN |  Satellite communication networks, unmanned aerial systems, HAPs |  Coverage extension, network access in offshore areas and NLOS scenarios
Mobile IAB |  Wireless backhaul, IAB-donor, IAB-nodes |  Capacity improvement within a vessel, coverage extension in offshore areas
MCX services |  MCPTT, MCData, MCVideo, off-network MCX |  Rescue services, response to shipborne emergencies
LTE/NR sidelink |  D2D communications, ProSe, UE-to-network relay |  Ship-to-ship communications for navigation and collision avoidance, sensor group communications for shipborne IoT, relaying for coverage extension
MEC |  Edge cloud servers, computation task offloading |  Low latency and low energy consumption in shore-based and offshore applications
Cellular LPWA |  LTE-M, NB-IoT, mMTC |  Shipborne cellular IoT
NR RedCap |  Industrial sensors, surveillance cameras, wearables |  Using data collected by RedCap devices in several 6G-MN use cases
5G XR |  AR, MR, VR |  XR for mission-critical maritime SAR, XR conferencing for vessel logistics
5G LAN |  5G LAN-type access, enterprise network communications |  On-board services including vessel logistics and infotainment
NPN |  Public network-integrated NPNs, standalone NPNs |  Smart ports/harbors, enhanced on-board connectivity
Positioning |  High-accuracy positioning, RAT-dependent, RAT-independent, hybrid solutions |  Vessel location awareness for navigation and fleet management, indoor positioning services for staff management
Platooning |  Cooperative platoons, autonomous vessels |  Short-sea shipping and feeder services within harbor logistics use cases

###  II-C Prospective Solutions for 6G-MNs

Several attempts to interconnect vessels in coastal waters and build a bridge
to the port have been successful by virtue of cellular coverage. However, due
to the limited capacity of communication links, existing systems for maritime
communications provided by, for example, Cellnex Telecom or Telenor Maritime,
fail to cover deep offshore areas and support delay-critical and bandwidth-
hungry use cases. By contrast, 5G and beyond networks can provide a flexible
and adaptive mobile communication platform for the modernization and long-term
support of the maritime industry not only in coastal but also in offshore
areas, as confirmed by the Third Generation Partnership Project (3GPP) in TR
22.819.

Initial 6G systems are to be primarily supported by the existing 5G
infrastructures, thus benefiting from the advancement of cellular technologies
that can foster the deployment of even more agile and intelligent applications
discussed above. Enabling solutions, related cellular concepts, and relevant
6G-MN use cases are offered in Table I.

Hybrid satellite–terrestrial networks have been utilized to complement the
high capacity of shore-based systems by the wide-area coverage of satellite
communications. In addition to satellite access, New Radio (NR) technologies
can support other non-terrestrial access components, such as high-altitude
platforms (HAPs) as identified by 3GPP in TR 38.811. By employing solutions
for NR to support non-terrestrial network (NTN), 6G-MN s can benefit from the
5-layer architecture for 6G setups as proposed in [8] to extend the coverage
of terrestrial systems and provide access to maritime services in offshore
areas and non-line-of-sight (NLOS) scenarios.

In addition to NR NTN, an attractive technology for coverage and capacity
extension in terrestrial 5G and beyond networks is integrated access and
backhaul (IAB), which was introduced by 3GPP TR 38.874 in Release 16. It is
based on using part of the wireless access spectrum for the backhaul
connections between remote base stations. While fixed IAB-nodes can extend the
capacity of coastal 6G-MN s, mobile IAB, as suggested among the 3GPP Release
17 enhancements, can be employed for dynamic backhaul solutions, thus shaping
mobile maritime mesh networks for both coverage and capacity extension. This
can be considered as a more flexible, scalable, and affordable broadband
access option for a wide range of cloud services.

Further, 3GPP public safety services, including mission-critical push-to-talk
(MCPTT), mission-critical data (MCData), and mission-critical video (MCVideo),
can be applied for emergency services. These services can also be provided
using the off-network mode in areas wherein cellular coverage is temporarily
unavailable or network performance is limited in terms of capacity or latency.

NR sidelink for device-to-device (D2D) communications and proximity services
(ProSe) can help avoid collisions as part of the navigation and fleet
management use case. Sidelink-based ship-to-ship communications allow vessels
to assist maritime coordination centers during SAR missions. User equipment
(UE)-to-network relay, as another form of D2D wherein an indirect network
connection is provided by a relay UE, can extend the coverage of terrestrial
systems for shipborne applications in coastal waters.

In 5G networks, computation resources are moved closer to the end devices.
Heavy computation tasks and raw data can be offloaded to proximate multi-
access edge computing (MEC) servers, thus minimizing end-to-end transmission
delays and energy consumption. Cellular low-power wide-area (LPWA)
technologies, namely, LTE machine-type communications (LTE-M) and narrowband
IoT (NB-IoT), can be deployed in 6G-MN s to support the use cases with mMTC
requirements. In a similar context, the need for NR devices with reduced
capabilities (RedCap), such as on-board industrial sensors, video surveillance
cameras, and wearables, has been addressed by 3GPP in TR 38.875. Future 6G-MN
s can benefit from the NR RedCap-enabled services for enhanced vessel
navigation and monitoring, harbor logistics, and remote on-board assistance.
The latter example refers to one of the extended reality (XR) use cases
defined in TR 26.928, which includes augmented reality-guided assistance in
remote locations, mixed reality-based sharing, and virtual reality-based
telepresence collaboration.

Non-public network (NPN) and local area network (LAN)-type access for
industrial IoT use cases has already been considered by 3GPP for maritime
scenarios in TR 22.819. With the planned enhancements in Release 17, NPN s can
be employed in smart harbors to build private networks with adequate QoS and
security guarantees. A range of positioning schemes, including radio access
technology (RAT)-dependent and RAT-independent solutions, are ratified in 3GPP
Release 16 specifications and can be used separately or in a hybrid manner to
meet the required positioning accuracy in 6G-MN use cases. Accurate
positioning can also enable future maritime platooning services. Aiming to
improve navigation safety and reduce fuel consumption, autonomous vessels can
move in cooperative platoons and be employed in feeder services to manage
short-sea shipping from hub ports to feeder ports in inland waterways [9].

Recently, recognizing the high interest from maritime stakeholders, 3GPP has
officially included the work on maritime communication (MARCOM) services over
cellular systems in its beyond Release 16 standardization efforts in TS
22.119. An important challenge that needs to be considered in these systems is
energy efficiency as part of the sustainability goal in maritime operations.
Even though 5G NR system design offers better bit-per-Joule energy efficiency
as compared to the previous generations of mobile technology, a typical 5G
site has nearly 70% higher energy consumption than a base station deploying a
mix of 2G, 3G, and 4G radios due to the use of additional power-hungry
components [10]. Further cell densification together with link heterogeneity
can make it difficult to optimize 5G and beyond deployments in real time,
which calls for a more adaptive and less energy consuming system design
discussed in the following section.

Figure 2: Our vision of a unified, scalable, reliable, and intelligent 6G
system with integrated features for sustainable maritime operations

##  III AI for Sustainable Maritime Networking

###  III-A Network Sustainability Challenges

The use cases identified in Fig. 1 are different from the scenarios behind the
operation of well-established mobile and vehicular ad-hoc networks. For
instance, they can create more stable multi-hop network formations over longer
time spans, may need to transmit heterogeneous data over larger communication
ranges, and might also include more complex and advanced on-board and in-port
communication scenarios similar to industrial IoT applications.

A large volume of data generated on-board the vessels in deep offshore areas
needs to be offloaded to the mainland for efficient port operation, or
distributed further within the network to facilitate optimal maritime
navigation and logistics. Real-time data collected along the navigation routes
are crucial for autonomous shipping, as ports and vessels can access this
information upon request. This can mitigate deviations from the optimal port
operation and failures of communication links. In contrast, infotainment and
extended reality-based applications for passengers and crew members of small
vessels and large cruisers can require rapid dissemination of heavy content
from the land to numerous destinations in different parts of the world. As
these examples suggest, maritime operation targets robust, scalable, high-
performance, and adaptive mechanisms for the orchestration of dissimilar
services over dynamically formed mobile networks.

In 6G-MN s, multi-hop communications and flexible mesh topologies can be
supported by IAB and D2D solutions. These technologies help overcome the
vulnerabilities of highly directional and fast fading links, tolerate
increased interference levels, and utilize radio resources more efficiently.
However, due to node mobility within the unique and challenging integrated
maritime infrastructure, 6G-MN s need to continuously adapt their resource
allocation and scheduling policies over dynamic topologies.

Our envisaged 6G-MN illustrated in Fig. 2 comprises a high-performance
terrestrial network segment that inherently supports operation and maintenance
of smart harbors and industrial ports. The non-terrestrial part encompasses a
dynamic IAB infrastructure, which ensures connectivity bridges between vessels
and the terrestrial segment. The former facilitates inland shipping and
ubiquitous monitoring of near-shore areas for improved human and marine life
safety. An important component of this system is the powerful edge and cloud
infrastructure for centralized and distributed learning to enable proactive
and resource-wise on-demand operation.

Traditional model-based approaches and optimization algorithms may not be
sufficient for satisfying the requirements of the above system. They typically
struggle with a lack of timely global information about the system (e.g.,
channel and buffer states, user mobility, or demand level) to provide optimal
control instructions and can thus yield impractical computations due to
excessive model dimensions. In turn, data-driven methods are known to
efficiently deal with both model and algorithm deficits, and with learning
functional relations between different system parameters that are difficult to
model. These parameters include (i) user profile data such as device position,
mobility, transmission, and energy consumption patterns, (ii) network
configuration data encompassing instantaneous link capacity and resource
utilization, and node capabilities, and (iii) service data covering quality of
experience, subscription to cooperative learning of a ML model, and
capabilities for execution of offloaded tasks.

Allowing to extract valuable information from these massive data, AI
techniques can be used to predict traffic peaks and system demands, detect
anomalies or near-overload conditions, and identify nodes or clusters with
high energy and resource consumption. This knowledge can alleviate the
topology management and scheduling problems and underpin the solution design
for self-optimized and automated 6G-MN s with energy-oriented optimization
goals. In what follows, we provide examples of AI applications for a more
energy-efficient maritime system operation.

###  III-B Energy-Centric Topology Management

Since both system topology and network load may evolve over time, storage,
compute, and spectrum resource allocations have to be provided on-demand and
thus periodically re-optimized. To manage on-demand topologies, joint link
activation and resource allocation problems need to be solved repeatedly, and
potentially compared with previous configurations. As traditional model-based
methods can be resource and time consuming, a more agile approach is required
to facilitate the repetitive optimization problems.

As an illustrative example, we consider a low-dynamic multi-hop wireless
network deployed over 100 km2, where access nodes are uniformly distributed
across the given area with the density of $0.5$ nodes per km2. Our goal is to
minimize the network energy consumption while delivering heterogeneous traffic
originating from cruisers or cargo vessels and rerouted to different
destinations (e.g., harbors or other vessels). Therefore, we are aiming at the
optimal allocation of time, spectrum, and power along the optimal routing
paths.

Given that the scale of the above optimization problem increases with the
number of nodes and potential routes, the performance of traditional
optimization approaches can degrade dramatically. For solving the joint power
allocation and link scheduling problem in this system to minimize its overall
energy consumption, one needs to know all the possible allocation patterns to
employ linear programming (LP). The number of patterns primarily depends on
the number of potential links, though not all of them may eventually be used
in the optimal solution [11].

Since the relation between the network flows and the set of active links in
the optimal configuration cannot be obtained analytically, it may be learned
by a data-driven method using information about only some of the optimal
configurations (e.g., by solving optimization problem analytically for a
number of system setups with fewer flows). Once the model is trained, it can
efficiently predict which links are critical for the optimal configuration
under any new traffic flows in the system. Therefore, by using a deep neural
network (DNN), particularly its sub-category deep belief network (DBN), we
reduce the number of links involved in the routing and scheduling decision in
a given multi-hop layout, and thereby alleviate the complexity and execution
time of finding the optimal configuration.

Hence, Fig. 3 demonstrates our system-level simulation results for the energy
efficiency of a maritime communication system operating at 3.5 GHz with
different traffic loads defined as the maximum system capacity share. These
results have been first obtained by using the DNN-aided LP optimization
framework described in [11], and then compared to the baseline system
operation.

Figure 3: Energy-efficient resource management for IAB-aided backhaul solution

Along these lines, other promising techniques such as long short-term memory
(LSTM) and auto-encoder can be applied to discover the temporal and spatial
correlations in traffic and support the formulation of multi-objective
optimization problems. For critical services with ultra-low latency
requirements, the challenge of routing is central. As shown in [12], a
supervised DNN-aided traffic routing scheme causes much lower overheads than
the traditional options such as open shortest path first (OSPF), while
guaranteeing acceptable delay.

###  III-C Energy-Efficient Scheduling

Due to large- and small-scale node mobility, as well as interference
fluctuations, the reported channel quality may become outdated, misleading, or
even lost at the network side. Systematic inaccurate or imperfect knowledge of
the channel state may cause significant QoS and energy efficiency degradation.
Hence, mechanisms for channel quality prediction can help combat different
types of link blockage effects and control radio interference while supporting
redundant on-demand topologies.

Deep learning (DL) methods can improve the accuracy of channel reporting and,
consequently, avoid inefficient resource utilization in the core network and
radio access parts. They are well-suited for capturing non-linear and dynamic
relationships between the model input and output data. They also have powerful
prediction, inference, and data analysis capabilities owing to the large
amounts of data generated by the environment and by the users. In particular,
LSTM can handle time series problems, which makes them attractive for channel
quality prediction and capable of alleviating the physical layer imperfections
[13].

The results of our system-level simulations summarized in Fig. 4 demonstrate a
significant improvement in energy efficiency by applying LSTM for resource
scheduling. We employ an open-source interface between network simulator-3 and
Python-based AI frameworks. The latter train the LSTM model using data
generated by the simulator and then return the data from the trained model
back to the simulator for testing. Communications over mmWave channels in a
single-cell network topology with mobile UE s ($10m/s$) are assumed. Unlike in
the baseline scenario (i.e., without ML), in the LSTM-aided case the base
station utilizes the predicted channel quality information when making
decisions about scheduling and radio resource allocation [13]. Not only the
system energy efficiency can be improved with better channel quality
predictions, but also the end-to-end packet delay may be significantly
decreased.

Figure 4: Use of LSTM for energy-efficient real-time scheduling

##  IV Open Issues in Employing AI for 6G-MNs

###  IV-A From Centralized to Distributed Learning

As the adoption of AI technologies accelerates, the integration of various
monitoring and control systems within a centralized cloud can limit the
scalability in such systems. Hence, today’s predominantly cloud-centric AI
solutions that rely on training and inference in the remote cloud have to be
complemented by more energy-efficient, partially distributed, and ultimately
fully distributed learning mechanisms where numerous devices collaboratively
train a part of a global model [14].

Pervasive system intelligence is vital for the evolution of maritime industry
and its sustainable operation. In particular, real-time decision-making vastly
improves port logistics and services. Through AI-assisted remote control, an
operator can digitally escort vessels safely to port. Smart fleet and asset
tracking features can improve load distribution in ports, which decreases the
volumes of carbon dioxide near the port areas. XR applications for field
engineers allow hands-on guidance from offsite support teams who can follow
the operator’s on-site view. All of these require lifelong learning where
autonomous edge nodes (on-board the vessels or in dock areas) can participate
in sensor data collection, processing, and sharing of resources for ML model
training.

Conventionally, DNN algorithms are executed in the cloud where training data
are preprocessed at the edge before being transferred to the cloud [14]. The
edge/fog computing infrastructures are intended to accommodate the needs of
multiple DNN models that require locality and persistent training. They also
prevent the transmission of massive raw data over the network. Federated
learning is a practical training mechanism wherein clients perform local ML
training and forward their results to an aggregator for further inference.
Devices, edge nodes, and cloud servers can be equivalently deemed as clients.
Under the risk of involving clients with poor channel conditions or limited
energy supply, ML model and client selection remains challenging in these
distributed learning-based systems.

###  IV-B UE Capabilities in Device-Level Solutions

Mobile RedCap devices such as wearables can assume a central role in real-time
monitoring and ubiquitous sensing of critical and highly dynamic processes in
maritime environments. Unmanned aerial vehicles help create situational
awareness for hinterland and smart fairway scenarios as well as provide remote
technical support for container handling equipment. Enhanced with additional
on-device learning and inference capabilities, these systems can utilize real-
time data to offer deeper insights into energy-efficient maritime operations.
Beam misalignment in dynamic maritime environments may lead to significant
data rate and energy efficiency degradation. DL-based proactive beam
management at the device side can help avoid this potential limitation.
Whenever the line-of-sight link is not available, reinforcement learning
allows the identification of optimal relay nodes in online fashion, even with
limited prior knowledge of the environment. However, selecting the optimal
relay nodes is non-trivial in dynamic environments and when both in-band and
out-of-band relaying options are available.

In systems relying on device-level solutions, UE capabilities are crucial when
selecting suitable ML models. For instance, if a moving IAB node is deployed
on a floating platform or on-board a vessel, it can be connected to a source
of renewable energy, while drones and wearable devices operate on battery.
Therefore, not all devices in 6G-MN s are capable of training complex DNN due
to their limited storage, processing, or power capacity. Standalone
compression techniques (such as pruning) have been optimized only for DNN
accuracy and without considering device energy consumption. By combining
multiple compression techniques, one may derive compressed DL models with
desired trade-offs between performance and resource utilization [15]. For
instance, AdaDeep can automatically select various compression techniques to
form a model according to not only device capability constraints but also
application-driven requirements.

###  IV-C Learning Delay and Network Reaction Time

In distributed learning under model or data split architecture, the involved
nodes need to periodically communicate ML model parameters over the network.
The time to synchronize their results can grow significantly due to in-network
transfer delays. This synchronization latency can become even higher when
using ML models such as DNN s with thousands of parameters. Although several
solutions were proposed to reduce the DNN training times, the latter still
depends on the used data samples and approximation functions. By properly
selecting the nodes (e.g., depending on channel conditions and energy
availability) and adjusting the ML model parameters (e.g., learning rates and
number of epochs in DNN s), one can reduce the learning delay.

Learning delay and ML convergence criteria are central in future AI-aided
6G-MN s. In learning-aided architectures, network reaction time can become a
new key performance indicator that tells how soon a new system configuration
can be enabled. It may be defined as the time between a parameter change
(e.g., number of active users or link quality) and the network response time
including ML (re-)training and inference. Due to the limited communication
ranges of high frequency radios, a potentially large number of hops may be
required to connect two nodes of interest, which may cause an increase in the
network reaction time under topology changes.

##  V Conclusions and Outlook

The convergence of AI and 6G allows to build sustainable AI-aided networks for
maritime communications. With the envisaged 6G-MN, the maritime industry can
benefit from the enabling effects of digitalization and virtualization in
reducing carbon dioxide emissions in ports and vessels. 6G-MN s permit the
integration of terrestrial and non-terrestrial network segments, applications,
and services in a holistic manner to accommodate the need for large-scale,
sustainable, and on-demand system infrastructures. Due to network complexity
and dynamics, AI-aided solutions are indispensable for prompt and customized
reactions of the network to demand fluctuations. In particular, deep learning
techniques discussed in this article can tackle 6G-MN optimization challenges
at different levels.

The challenges of efficient learning over 6G-MN s are shaped by the
distinctive features of the rapidly changing maritime environment, remote
operation with limited availability of energy and communication resources, and
considerable learning delays in distributed systems. However, several
approaches discussed in this work, such as reinforcement learning, can be
further developed and employed to address these issues. The insights offered
by this article motivate further research that can address the open questions
and challenges in intelligent 6G-MN s.

## Acknowledgments

The authors gratefully acknowledge funding from European Union’s Horizon 2020
Research and Innovation programme under the Marie Skłodowska-Curie grant
agreement No. 813278 (A-WEAR project). This work was also supported by the
Academy of Finland (projects Emc2-ML, RADIANT, and IDEA-MILL). G. Fodor was
partially supported by the European Celtic project 6G-SKY with project ID
C2021/1-9.

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## Biographies

Salwa Saafi<EMAIL_ADDRESS>is a Ph.D. student at the Department of
Telecommunications at Brno University of Technology, Czech Republic and the
Unit of Electrical Engineering at Tampere University, Finland. She received
her engineering degree (2017) in telecommunications from the Higher School of
Communication of Tunis, Tunisia. Her research interests include cellular radio
access technologies, future wireless architectures, and wearable applications.
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Olga Vikhrova<EMAIL_ADDRESS>is a researcher at Tampere University,
Finland. She received her Ph.D. (2021) in Information Engineering from
University Mediterranea of Reggio Calabria, Italy, and M.Sc. (2014) in
Information and Computer Science from Peoples’ Friendship University of Russia
(RUDN University), Russia. Her current research interests include distributed
edge learning and computing, integrated access and backhaul networks,
convergence of terrestrial and non-terrestrial networks.
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Gabor Fodor<EMAIL_ADDRESS>received his Ph.D. degree in electrical
engineering from the Budapest University of Technology and Economics in 1998,
his Docent degree from KTH Royal Institute of Technology, Stockholm, Sweden,
in 2019, and his D.Sc. degree from the Hungarian Academy of Sciences in 2019.
He is currently a master researcher with Ericsson Research and an adjunct
professor with KTH Royal Institute of Technology. He was a co-recipient of the
IEEE Communications Society Stephen O. Rice Prize in 2018. He is serving as
the Chair of the IEEE Communications Society Full Duplex Emerging Technologies
Initiative and as an Editor for IEEE Transactions on Wireless Communications
and IEEE Wireless Communications.
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Jiri Hosek<EMAIL_ADDRESS>received his M.S. and Ph.D. degrees in electrical
engineering from the Faculty of Electrical Engineering and Communication at
Brno University of Technology (BUT), Czech Republic, in 2007 and 2011,
respectively. He is currently an Associate Professor (2016) and Deputy Vice-
Head for R&D and International Relations at the Department of
Telecommunications (2018), BUT. Jiri is also coordinating the WISLAB research
group (2012), where he deals mostly with industry-oriented projects in the
area of IoT and home automation services. Jiri (co-)authored more than 130
research works on networking technologies, wireless communications, quality of
service, quality of experience, and IoT applications.
---
Sergey Andreev<EMAIL_ADDRESS>is an associate professor of
communications engineering and Academy Research Fellow at Tampere University,
Finland. He has been a Visiting Senior Research Fellow with King’s College
London, UK (2018-20) and a Visiting Postdoc with University of California, Los
Angeles, US (2016-17). He received his Ph.D. (2012) from TUT as well as his
Specialist (2006), Cand.Sc. (2009), and Dr.Habil. (2019) degrees from SUAI. He
served as lead series editor of the IoT Series (2018-21) for IEEE
Communications Magazine and as editor for IEEE Wireless Communications Letters
(2017-19).
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