# Finite Action Principle and Horava-Lifshitz Gravity: Early Universe, black-
holes and wormholes.

Jan Chojnacki1, Jan Kwapisz1,2 1Faculty of Physics, University of Warsaw, ul.
Pasteura 5, 02-093 Warsaw, Poland
2CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M,
Denmark

(January 2021)

###### Abstract

In this work, we elaborate on the finite action in the framework of Horava-
Lifshitz gravity. Assuming the Finite Action Principle we show that the
beginning of the universe is flat and homogeneous. Depending on the version of
the theory different cosmological scenarios are possible. Furthermore, we show
that the H-L gravity action selects only the regular black-hole spacetimes,
since the singular black holes possess infinite action. We also comment on the
possibility of traversable wormholes in theories with higher curvature
invariants. The possible cosmological solutions in Horava-Lifshitz and
quadratic gravity are similar, proving that the Finite Action Principle is not
model-sensitive.

##  I Introduction

The path integral approach yields a powerful framework in quantum theory since
it emphasises Lorentz covariance and allows for the description of non-
perturbative phenomena. In the path integral, one is supposed to sum over all
possible configurations of a field(s) $\Phi$ weighted by $e^{iS[\Phi]}$, where
$S[\Phi]$ is the classical action of the theory. In Wick rotated path
integral, where one takes $t\to i\tau$, the field configuration is weighted by
$e^{-S[\Phi]}$.
Following this notion, the Finite Action Principle (FAP) proposes that the
action should be fundamental entity instead of the field values. One can then
ask which of the field configurations results in the finite action and which
in the infinite one.
Recently, this principle has been applied to the study of black holes
Borissova and Eichhorn (2021). Since it is expected that the quantum gravity
should resolve the black-hole singularity problem, one may ask which of the
microscopic actions remain finite for non-singular black holes and conversely
interfere destructively for the singular ones. This we shall call the finite
action selection principle. Only after the inclusion of higher-curvature
operators, beyond the Einstein-Hilbert term, such selection principle can be
satisfied Borissova and Eichhorn (2021). Furthermore in asymptotic safety, the
quantum corrections to the Newtonian potential eliminate the classical-
singularity Bosma  _et al._ (2019).
On the other hand, requirement that an action of the Universe should be finite
Barrow and Tipler (1988), is well-motivated theoretically (see also a newly
proposed finite amplitudes principle Jonas  _et al._ (2021)). This principle
has a significant impact on the nature of quantum gravity and the evolution of
the Universe, once the higher-curvature terms are included Lehners and Stelle
(2019); Barrow (2020). Following this principle, unlike for the Einstein
action, in Stelle gravity Stelle (1977) the presence of the $R^{2}$ term
implies homogeneous and isotropic conditions for the early universe if
considering the off-shell action (for the on-shell action anisotropies are
supposed to be washed out by inflation Jonas  _et al._ (2021)). Furthermore,
the highly symmetric state yields a vanishing Weyl tensor Penrose (1989),
explaining the low entropy of the early universe.
These findings suggest that by taking into account the higher curvatures one
can resolve the singularities in the early universe and the black holes. Yet,
an issue with the higher-curvature theory of quantum gravity is the existence
of the particles with the negative mass-squared spectrum, known as ghosts,
which makes the theory non-unitary. It is the consequence of the Ostrogradsky
Theorem Woodard (2015) and the presence of higher than second-order time
derivatives in the terms beyond $R$ in the action. However this might be
resolved by additional symmetry Meissner and Veneziano (1991), giving up the
micro-causality, changing the propagator prescriptions Anselmi and Piva
(2018); Donoghue and Menezes (2019a) or taking into account infinitely many
derivatives Barnaby and Kamran (2008), see also the discussion Platania and
Wetterich (2020) on possible resolution in the context of asymptotic safety.
In this article, we explore yet another possibility, namely, we investigate
Horava-Lifshitz (H-L) gravity Hořava (2009), where the Lorentz Invariance (LI)
is broken at the fundamental level (see Wang (2017) for a comprehensive
progress report on this subject). Kinetic terms are first order in the time
derivatives, while higher spatial curvature scalars regulate the UV behavior
of the gravity. Furthermore, the lower-dimensional lattice studies of Causal
Dynamical Triangulations (CDT) give the same Hamiltonian as H-L gravity
Ambjorn  _et al._ (2010); Ambjørn  _et al._ (2013); Anderson  _et al._ (2012).
In the Euclidean path integral the notion of finite action principle is quite
natural, since the infinite action configurations clearly do not contribute.
On the other hand, the Wick rotation is not well defined in quantum gravity in
general, this become especially difficult for the theories with higher
curvature terms. In our investigation we follow Horava Hořava (2009) and
assume that Wick rotation is well defined. This can be motivated by the lack
of the higher time-like curvature invariants. In the usual higher derivative
constructions the existence of massive poles in the propagator makes the Wick
rotation troublesome Donoghue and Menezes (2019b). Conversely to the latter,
the new poles are massless and non-tachyonic for suitable choice of parameters
Bemfica and Gomes (2011). Furthermore, the existence of the foliation supports
that claim. For the discussion of Minkowski path integral and FAP, see
Borissova and Eichhorn (2021).
In this article, we show that the Finite Action arguments applied to the
projectable H-L gravity result in a flat, homogeneous, UV-complete, and ghost-
free beginning of the universe, supporting the topological phase conjecture
Agrawal  _et al._ (2020). We also show that the Finite Action selection
principle Borissova and Eichhorn (2021) works for H-L gravity in the context
of black holes (the action is finite for non-singular BH and conversely for
the singular). Furthermore, we have found that wormholes possess a Finite
Action and hence contribute to the path-integral of QG, therefore they are
consistent with ER=EPR hypothesis Maldacena and Susskind (2013). On the other
hand, the stable, traversable wormholes solutions are known only in the higher
derivative gravities Duplessis and Easson (2015) (without exotic matter), so
there seems to be a wormhole/non-singular BH trade-off after taking into
account the Finite Action Principle.

##  II Horava-Lifshitz gravity

In the Horava-Lifshitz gravity, space and time are scaled in a non-equivalent
way. Diffeomorphism invariance is broken by the foliation of the 4-dimensional
spacetime into 3-dimensional hypersurfaces of constant time, called leaves,
making the theory power-counting renormalizable (see also the renormalization
group studies of the subject D’Odorico  _et al._ (2014, 2015); Barvinsky  _et
al._ (2017)). The remaining symmetry respects transformations:

$\displaystyle t\xrightarrow{}\xi_{0}(t),\quad
x^{i}\xrightarrow{}\xi^{i}(t,x^{k}),$ (1)

and is often referred to as the foliation-preserving diffeomorphism, denoted
by Diff$(M,\mathcal{F})$. The diffeomorphism invariance is still present on
the leaves. The four-dimensional metric may be expressed in the Arnowitt-
Deser-Misner (ADM) Arnowitt  _et al._ (2008) variables:

$\displaystyle(N,N^{i},\,^{(3)}g_{ij}),$ (2)

where $N,\,N^{i},\,\,^{(3)}g_{ij}$ denote respectively the lapse function,
shift vector, and 3-dimensional induced metric on the leaves. The theory is
constructed from the following quantities:

$\,{}^{(3)}R_{ij},\quad K_{ij},\quad a_{i},\quad\,^{(3)}\nabla_{i},$ (3)

where $\,{}^{(3)}R_{ij}$ is the 3-dimensional Ricci curvature tensor,
$\,{}^{(3)}\nabla_{i}$ is the covariant derivative constructed from the
3-dimensional metric $\,{}^{(3)}g_{ij}$, and $a_{i}:=\frac{N,_{i}}{N}$.
Extrinsic curvature $K_{ij}$ is the only object, invariant under general
spatial diffeomorphisms containing exactly one time derivative of the metric
tensor $\,{}^{(3)}g_{ij}$:

$\displaystyle K_{ij}=\frac{1}{2N}\left(\frac{\partial\,^{(3)}g_{ij}}{\partial
t}-\,^{(3)}\nabla_{i}N_{j}-\,^{(3)}\nabla_{j}N_{i}\right).$ (4)

Quantities (2) are tensor/vectors with respect to Diff($M,\mathcal{F}$)
possessing the following mass dimensions:

$\displaystyle[\,^{(3)}R_{ij}]=2,\quad[K_{ij}]=3,\quad[a_{i}]=1,\quad[\,^{(3)}\nabla_{i}]=1.$
(5)

One may use (2) to construct, order by order, scalar terms appearing in the
Lagrangian of the theory. Following Wang (2017); Maier and Soares (2017) the
action of the Horava gravity takes the form:

$\displaystyle S_{g}=\zeta^{2}\int
dtdx^{3}N\sqrt{{}^{(3)}g}\left(\mathcal{K}-V\right),$ (6)

where $\mathcal{K}=K_{ij}K^{ij}-\lambda K^{2}$ with $K=K_{ij}\,^{(3)}g^{ij}$,
$\,{}^{(3)}g$ denotes the determinant of the 3-dimensional metric and
$\zeta^{2}=1/16\pi G$. It may be expressed as the difference of the kinetic
and potential part $\mathcal{L}=\mathcal{K}-V$ with
$\mathcal{K}=\left(K_{ij}K^{ij}-\lambda K^{2}\right)$. At the 6th order, the
potential part of the lagrangian contains over 100 terms Wang (2017). The
immense number of invariants is limited by imposing further symmetries. One
possible restriction for the potential comes from the projectability condition
$N=N(t)$, then terms proportional to $a_{i}\equiv 0$ vanish. Up to the sixth
order (compatible with power counting renormalizability), the potential $V$
restricted by the projectability condition is given by:

$\displaystyle
V=2\Lambda\zeta^{2}-\,^{(3)}R+\frac{1}{\zeta^{2}}\left(g_{2}\,^{(3)}R^{2}+g_{3}\,^{(3)}R^{ij}\,{}^{(3)}R_{ij}\right)$
$\displaystyle+\frac{1}{\zeta^{4}}\left(g_{4}{}^{(3)}R^{3}+g_{5}\,^{(3)}R\,^{(3)}R^{ij}\,{}^{(3)}R_{ij}+g_{6}\,^{(3)}R^{i}_{j}\,{}^{(3)}R^{j}_{k}\,{}^{(3)}R^{k}_{i}\right),$
$\displaystyle+\frac{1}{\zeta^{4}}\left(g_{7}{}^{(3)}R\nabla^{2}{}^{(3)}R+g_{8}(\nabla_{i}{}^{(3)}R_{jk})(\nabla^{i}{}^{(3)}R^{jk})\right),$
(7)

where $\Lambda$ is the cosmological constant and $\alpha_{ij}$ are the
coupling constants. For our purposes, we drop terms containing covariant
derivatives $\,{}^{(3)}\nabla_{i}$. One should also mention that this _minimal
theory_ Horava (2011) suffers from the existence of spin 0 graviton, which is
unstable in the IR, see Charmousis  _et al._ (2009); Li and Pang (2009); Blas
_et al._ (2009); Henneaux  _et al._ (2010). Various solutions to this problem
have been proposed. One can add the additional local $U(1)$ symmetry Wang
(2017); Horava and Melby-Thompson (2010). Then by the introduction of new
fields prevents the zero-mode from propagating. On the other hand, one can
drop the projectability condition $a_{i}=0$ and include the terms containing
$a_{i}$ in the potential term:

$\displaystyle
V=2\Lambda\zeta^{2}-\,^{(3)}R-\beta_{0}a_{i}a^{i}+\sum_{n=3}^{6}\mathcal{L}_{V}^{(n)},$
(8)

then for the spin-0 mode to be stable one requires $0<\beta_{0}<2$ Blas  _et
al._ (2011); Carloni  _et al._ (2011).
Note that we shall not be interested in the IR divergences stemming from
boundary terms in the action and hence the famous Gibbons-Hawking-York term in
context of black holes. Those divergences have nothing to do with resolution
of singularities, which is an UV issue, see also discussion in Borissova and
Eichhorn (2021). Nevertheless it might be useful for the reader to comment on
that issue. The variational principle with Dirichlet boundary condition
requires variation of the action to be zero, when we fix the boundary metric
Krishnan and Raju (2017). However this is not the case for the Einstein
Hilbert action and hence the famous GHY term York (1972); Gibbons and Hawking
(1977) has to be added, which is crucial for the finiteness of the action. On
the other hand for HL gravity the action possesses higher spatial derivatives.
The variational principle requires $\delta\gamma_{ij}$ along with its
derivatives to be zero at the spatial boundary, hence the variation is well
defined without the boundary term Lara  _et al._ (2021). We also comment
further that issue in the context of cosmological spacetimes in the Sec. III.
Furhermore absence of the boundary term has been recently proven for the
mimetic Horava gravity, see Chamseddine  _et al._ (2019); Malaeb and Saghir
(2020).

##  III Flatness, anisotropies and inhomogeneities in the early universe

#### Flatness

We begin our investigation of the early universe flatness by considering the
FLRW metric given by the formula Kiritsis and Kofinas (2009)

$\displaystyle N\to N(t),\quad N_{i}\to 0,\quad{}^{(3)}g_{ij}\to
a^{2}(t)\gamma_{ij},$ (9)

where $\gamma_{ij}$ is a maximally symmetric constant curvature metric, with
$k=+1$ for the metric on the sphere, $k=0$ for flat space time and $k=-1$ for
the hyperbolic metric. We have

$\displaystyle{}^{(3)}R_{ij}=2k\gamma_{ij},\quad{}^{(3)}R=\frac{6k}{a(t)^{2}},\quad\mathcal{K}=3(1-3\lambda)\left(\frac{\dot{a}}{a}\right)^{2},$
(10)

and $N\sqrt{{}^{(3)}g}=Na^{3}(t)$. For $a(t)=t^{s}$ the kinetic part of the
action gives us:

$\displaystyle N\sqrt{{}^{(3)}g}\mathcal{K}\sim t^{3s-2},$ (11)

since $N\sqrt{{}^{(3)}g}\mathcal{K}\sim t^{-1}$ leads to a logarithmic
divergence at $t\xrightarrow{}0$ after integrating over time, we impose that
the exponent of $t$ in the integrand should be greater than $-1$. Hence for
the action to remain finite as $t\to 0$, one requires $s>1/3$. In the
potential part we have exemplary terms

$\displaystyle N\sqrt{{}^{(3)}g}{}^{(3)}R\sim kt^{s},$ (12) $\displaystyle
N\sqrt{{}^{(3)}g}{}^{(3)}R^{2}\sim k^{2}t^{-s},$ (13) $\displaystyle
N\sqrt{{}^{(3)}g}{}^{(3)}R^{3}\sim k^{3}t^{-3s}.$ (14)

For $k\neq 0$ equations (11, 12, 13, 14) give rise to the following set of
contradicting inequalities:

$\displaystyle s>1/3,\quad s>-1,\quad s<1,\quad s<1/3,$ (15)

this shows that for $k\neq 0$ in there is no-FLRW like the beginning of the
Universe for the projectable action with potential given by (II). This means,
that in this version of H-L gravity, the Big Bang with power-law time
dependence of the scale factor cannot be realized (similar behavior has been
observed in Jonas  _et al._ (2021) for the LI gravity). Rejecting the cubic
$R^{3}$ terms from the potential, responsible for the contradictory
inequalities, yields the action to be finite. Below we also show (on the
example of Bianchi IX metric) that none of the anisotropic non-flat solutions
are allowed in the action with terms cubic in Ricci curvature.

#### Anisotropies

We consider Bianchi IX metric as a representative model of non-flat
anisotropic spacetimes (in this paragraph $k=1$):

$\displaystyle ds_{IX}^{2}=-N^{2}dt^{2}+h_{ij}\omega^{i}\omega^{j},$ (16)

where $h_{ij}=diag\left(M^{2},Q^{2},R^{2}\right)$ and $M,Q,R$ are functions of
the time only. The connection is

$\displaystyle
d\omega^{a}=\Gamma^{a}_{c}\wedge\omega^{c}=\Gamma^{a}_{cb}\omega^{b}\wedge\omega^{c}.$
(17)

The Bianchi IX one forms satisfy:

$\displaystyle
d\omega^{a}=\frac{1}{2}\epsilon^{abc}\omega^{b}\wedge\omega^{c}.$ (18)

Hence $\Gamma^{a}_{bc}=-\frac{1}{2}\epsilon^{abc}$. The usual closed FRLW
universe is obtained when $R(t)=M(t)=Q(t)=\frac{a(t)}{2}$, where $a(t)$ is the
scale factor. The explicit form of the curvature invariants was calculated in
Maier and Soares (2017):

$\,{}^{(3)}R$
$\displaystyle=\frac{-1}{2M^{2}Q^{2}R^{2}}\Big{(}M^{4}+Q^{4}+R^{4}-\left(R^{2}-Q^{2}\right)^{2}$
$\displaystyle-\left(R^{2}-M^{2}\right)^{2}-\left(M^{2}-Q^{2}\right)^{2}\Big{)},$
(19) $\,{}^{(3)}R^{i}_{j}\,{}^{(3)}R^{j}_{i}$
$\displaystyle=\frac{1}{4(MQR)^{4}}\Big{[}3M^{8}-4M^{6}\left(Q^{2}+R^{2}\right)$
$\displaystyle-4M^{2}\left(Q^{2}-R^{2}\right)^{2}\left(Q^{2}+R^{2}\right)$
$\displaystyle+2M^{4}\left(Q^{2}+R^{2}\right)^{2}+\left(Q^{2}-R^{2}\right)^{2}\left(3Q^{4}\right)$
$\displaystyle+\left(Q^{2}-R^{2}\right)^{2}\left(2Q^{2}R^{2}+3R^{4}\right)\Big{]},$
(20) $\,{}^{(3)}R^{i}_{j}\,{}^{(3)}R^{j}_{k}\,{}^{(3)}R^{k}_{i}$
$\displaystyle=\frac{1}{8(MQR)^{6}}\Big{(}\left[(M^{2}-Q^{2})^{2}-R^{4}\right]^{3}$
$\displaystyle+\left[(M^{2}-R^{2})^{2}-Q^{4}\right]^{3}$
$\displaystyle+\left[(Q^{2}-R^{2})^{2}-M^{4}\right]^{3}\Big{)}.$ (21)

The kinetic and the potential part are respectively:

$\displaystyle N\sqrt{{}^{(3)}g}\mathcal{K}$
$\displaystyle=\frac{MQR}{N}\Big{[}\left(1-\lambda\right)\left(\frac{\dot{M}^{2}}{M^{2}}+\frac{\dot{Q}^{2}}{Q^{2}}+\frac{\dot{R}^{2}}{R^{2}}\right)$
$\displaystyle-2\lambda\left(\frac{\dot{M}\dot{Q}}{MQ}+\frac{\dot{Q}\dot{R}}{QR}+\frac{\dot{M}\dot{R}}{MR}\right)\Big{]},$
(22) $\displaystyle N\sqrt{{}^{(3)}g}V$ $\displaystyle=-N\left(MQR\right)V.$
(23)

For the Bianchi IX metric we use the following ansatz:

$\displaystyle M(t)\sim t^{m},\quad Q(t)\sim t^{q},\quad R(t)\sim t^{r}.$ (24)

With such solutions, the kinetic term is proportional to

$\displaystyle N\sqrt{{}^{(3)}g}\mathcal{K}\sim t^{m+q+r-2}.$ (25)

This results in an inequality:

$\displaystyle m+q+r>1.$ (26)

Similar reasoning is applied to all of the curvature scalars in the potential.
Ricci scalar terms lead to conditions:

$\displaystyle 3m-q-r>-1,\quad 3q-m-r>-1,\quad 3r-m-q>-1,$ (27) $\displaystyle
r+q-m>1,\quad r+m-q>-1,\quad m+q-r>-1.$ (28)

Quadratic terms are numerous and we provide explicit conditions only for the
$R_{ij}R^{ij}$ terms:

$\displaystyle 5m-3q-3r>-1,\quad 3m-q-3r>-1,\quad 3m-3q-r>-1,$ $\displaystyle
3r-m-3q>-1,\quad r-m-q>-1,\quad q-m-r>-1,$ (29) $\displaystyle
3q-m-3r>-1,\quad m+q-3r>-1,\quad m-q-r>-1,$ (30) $\displaystyle
m+r-3q>-1,\quad 5q-3m-3r>-1,\quad 3q-3-r>-1,$ (31) $\displaystyle
q+r-3m>-1,\quad 3r-q-3m>-1,\quad-3m-3q+5r>-1.$ (32)

It is tedious to algebraically verify that the above set of conditions is not
contradictory. A geometrical interpretation brings more light to the problem:
each of the inequalities corresponds to a half of the $\mathbb{R}^{3}$ space
in $(q,m,r)$ coordinates. The common subspace restricted by a pair of such
inequalities vanishes if the planes corresponding to the boundary of the half-
spaces are parallel. This is easily verified by considering the vector normal
to each plane. For example the boundary plane obtained from inequality
$m+q+r>1$ is $m+q+r=1$ and the normal vector $(1,1,1)$. If two such normal
vectors are parallel (bearing in mind the correct inequality direction), the
half-spaces will be separate.
The kinetic part and scalars up to quadratic order in curvature do not lead to
contradictory conditions. However, including the $R^{3}$ term we have:

$\displaystyle N\sqrt{{}^{(3)}g}\,^{(3)}R^{3}\ni\frac{N}{MQR}\sim
t^{-m-q-r}\implies m+q+r<1.$ (33)

This is in clear contradiction with (26). This means, that also Bianchi IX
anisotropic spacetime leads to infinite action. Notice, that taking the
isotropic limit $m=q=r$ also leads to infinite action, as discussed in the
previous paragraph.
There are two ways of dealing with this - leaving the model unchanged and
considering other cosmological solutions, such as oscillating universe with
bounded $a(t)\in(a_{min},a_{max})$, see for example Fukushima  _et al._
(2019). On the other hand, this might be understood as indication that the
flatness problem is resolved via the Finite Action Principle without the need
of inlfation. In particular for flat, anisotropic Bianchi I spacetime, all of
the spatial invariants vanishes leaving us with the kinetic term condition

$\displaystyle m+q+r>1.$ (34)

Therefore, the action is finite for both isotropic and anisotropic flat
beginning of the universe in contradistinction to the Lorentz covariant
$R^{2}$ off-shell action Lehners and Stelle (2019), yet during the evolution
of the universe, the anisotropies might vanish dynamically Jonas  _et al._
(2021) if spacetime is accelerating. Furthermore, this is generically true for
$\Lambda>0$, see the dynamical systems studies of the matter Misonoh  _et al._
(2011). In our analysis we have neglected the boundary terms in the action. On
non-flat FLRW spacetime they could lead only to further contradictory
conditions. On the other hand for $k=0$ FLRW solution HL gravity those terms
will play no role, as it was investigated in Jonas  _et al._ (2021).

#### Inhomogeneities

Unlike the anisotropies, the finiteness of the action suppresses the
inhomogeneities already at the second-order of the spatial Ricci scalar
curvature. Investigation of the inhomogeneities concerns following isotropic
metric tensor:

$\displaystyle ds^{2}=-dt^{2}+\frac{A^{\prime
2}}{F^{2}}dr^{2}+A^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right),$ (35)

where $A=A(t,r)$, $F=F(r)$ and $A^{\prime}=\partial_{r}A$. The homogeneous
FRLW metric is retrieved, when $F\xrightarrow{}1$. The resulting Ricci scalar
and Ricci scalar squared contribution to the action are:

$\displaystyle\sqrt{\,{}^{(3)}g}\,^{(3)}R\sim
2AF^{\prime}+\frac{A^{\prime}\left(-1+F\right)^{2}}{F},$ (36)
$\displaystyle\sqrt{\,{}^{(3)}g}\,^{(3)}R^{2}\sim\frac{2AFF^{\prime}+A^{\prime}\left(F^{2}-1\right)}{A^{2}A^{\prime}F}.$
(37)

Again, we suppose that each term should be convergent as $t\xrightarrow{}0$.
By the ansatz $A(t)\sim t^{s}$, inequalities stemming from $\,{}^{(3)}R$ and
$\,{}^{(3)}R^{2}$ are contradictory. This means, that $F(r)\xrightarrow{}1$,
hence the metric of the early universe was homogeneous.

##  IV Black holes and wormholes

In this section, we show that H-L gravity satisfies the Finite Action
selection principle for the microscopic action of quantum gravity Borissova
and Eichhorn (2021). We study both the solutions of H-L gravity and the known,
off-shell BH spacetimes, due to the fact that there are no known regular BH
solutions in the HL gravity. Keep in mind that a metric does not need to be a
solution to the equations of motion to enter the path integral. We require
singular black-hole metrics to interfere destructively, while the regular ones
with finite action contribute to the probability amplitudes. We broaden this
analysis by studying the wormhole solutions.

###  IV.1 Singular black holes

Singularities may be categorised Ellis and Schmidt (1977) in the three main
groups: scalar, non-scalar and coordinate singularities. Scalar singularities
are the ones for which (some of) the curvature invariants, like Kretschmann
scalar, become divergent and hence they are the object of interest in our
considerations. Non-scalar singularities appear in physical quantities such as
the tidal forces. Finally, the coordinate singularities appear in the metric
tensor, however, one may get rid of the divergence with a proper coordinate
transformation. Yet, coordinate singularities of General Relativity may become
scalar singularities in the Horava-Lifshitz gravity Cai and Wang (2010). It is
due to the fact that the spacetime diffeomorphism of GR is a broader symmetry
than the foliation-preserving diffeomorphism of H-L gravity. As an example,
consider the Schwarzschild metric:

$\displaystyle
ds^{2}=-\left(1-\frac{2m}{r}\right)dt^{2}+\left(1-\frac{2m}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(38)

where $d\Omega^{2}=r^{2}(d\theta^{2}+\sin^{2}{\theta}\,d\phi^{2})$. The
singular points are $r=0$ and $r=r_{s}=2m$. In GR the singular point
$r_{s}=2m$ may be removed by the transformation

$\displaystyle dt_{PG}=dt+\frac{\sqrt{2mr}}{r-2m}dr.$ (39)

The resulting Painleve-Gullstrand metric is:

$\displaystyle
ds^{2}=-dt^{2}_{PG}+\left(dr-\sqrt{\frac{2m}{r}}dt_{PG}\right)^{2}+r^{2}d\Omega^{2}.$
(40)

For more details see e.g. Martel and Poisson (2001). In GR metric tensors (38)
and (40) describe the same spacetime with singularity at $r=0$. Notice,
however, that the coordinate transformation (39) does not preserve the
spacetime foliation, breaking the projectability condition. Hence, in the
framework of H-L gravity, metric tensors (38) and (40) describe distinct
spacetimes. Moreover, as we will show, Schwarzschild’s metric singularity at
$r=r_{s}$ becomes a spacetime singularity. Hence due to the unique nature of
the foliation-preserving diffeomorphism, investigating the singularities in
H-L gravity is a delicate matter.
We consider three representative solutions Cai and Wang (2010): (anti-) de
Sitter Schwarzschild, which is the simplest spacetime with the black hole and
the cosmological horizon, Kerr spacetime (see also the rotating H-L solution
Wang (2013)) and the H-L solution found by Lu, Mei and Pope (LMP) Lü  _et al._
(2009). In this section, we discuss the (anti-) de-Sitter Schwarzschild as the
clearest example. The other metrics have similar features and we explore them
in the Appendix. In the following, all of the curvature scalars are three-
dimensional, unless stated otherwise.

#### (Anti-) de Sitter Schwarzschild solution

The general static ADM metric with projectability condition takes the form:

$\displaystyle
ds^{2}=-dt^{2}+e^{2\nu}(dr+e^{\mu-\nu}dt)^{2}+r^{2}d\Omega^{2},$ (41)

where $\mu=\mu(r)$, $\nu=\nu(r)$. (Anti-) de Sitter Schwarzschild solutions
are be obtained for:
$\mu=\frac{1}{2}\ln{\left(\frac{M}{r}+\frac{\Lambda}{3}r^{2}\right)}$,
$\nu=0.$ The resulting kinetic terms and Ricci scalar are:

$\displaystyle{}^{(3)}R$ $\displaystyle=0,$ $\displaystyle K$
$\displaystyle=\left(\frac{3M+\Lambda
r^{3}}{12r^{3}}\right)^{\frac{1}{2}}\left(\frac{4}{r}-\frac{3M-2\Lambda
r^{3}}{3M+\Lambda r^{3}}\right),$ $\displaystyle K_{ij}K^{ij}$
$\displaystyle=\frac{3M+\Lambda r^{3}}{12r^{3}}\left[8+\left(\frac{3M-2\Lambda
r^{3}}{3M+\Lambda r^{3}}\right)^{2}\right].$ (42)

The kinetic part is divergent at $r=0$ and
$r=\left(\frac{3M}{|\Lambda|}\right)^{\frac{1}{3}}$ for the negative
cosmological constant. We investigate the finiteness of the function:

$\displaystyle
S_{s}(r_{UV},r_{IR}):=\int_{r_{UV}}^{r_{IR}}dr\,N\sqrt{g}\left(K_{ij}K^{ij}-\lambda
K^{2}+{}^{(3)}R\right),$ (43)

which is a part of the action qualitatively describing the singularities. The
$r_{IR}$ is chosen so that the volume integral is finite, hence we do not
consider singularities stemming from the IR behaviour (large distances) of the
spacetime boundary and time integration. In particular The $r_{UV}$ is the
minimal radius, which we take $r_{UV}\to 0$. For the scalars (IV.1) and value
of $\lambda\neq 1$, the function $S_{s}(r_{UV},r_{IR})$ is divergent at the
expected points $r_{s}=r_{UV}=0$ and
$r_{s}=\left(\frac{3M}{|\Lambda|}\right)^{\frac{1}{3}}$. However, for
$\lambda=1$, which is the value required for low energy Einstein-Hilbert
approximation, the terms divergent at
$r_{s}=\left(\frac{3M}{|\Lambda|}\right)^{\frac{1}{3}}$ remain finite as one
could expect, since $r_{s}$ corresponds to the cosmological horizon.
Explicitly we have:

$\displaystyle S_{s}(r_{UV},r_{IR})$ $\displaystyle=\frac{2}{9}\Lambda
r^{3}_{UV}-8\Lambda r^{2}_{UV}+\left(\frac{M}{4}-16\Lambda\right)\ln{r_{UV}}$
$\displaystyle-\frac{24M}{r_{UV}}+\frac{24}{r^{2}_{UV}}+\textrm{IR terms}.$
(44)

Here, only the spatial Ricci scalar is necessary for the singular solution to
be suppressed in the gravitational path integral.
As mentioned previously, different gauges of the same spacetime in GR,
correspond to distinct spacetimes in H-L gravity. Hence, we consider the
(anti-) de Sitter Schwarzschild metric in the orthogonal gauge, which is not a
solution to the projectable H-L theory, in contrast to the previous case:

$\displaystyle
ds^{2}=-e^{2\Psi(r)}d\tau^{2}+e^{2\Phi(r)}dr^{2}+r^{2}d\Omega^{2},$ (45)

here

$\displaystyle\Psi(r)=-\Phi(r)=\frac{1}{2}\ln\left(1-\frac{2M}{r}+\frac{1}{3}\Lambda
r^{2}\right).$ (46)

In the orthogonal gauge, the components of the metric tensor do not depend on
the time coordinate, hence the kinetic part vanishes $K_{ij}=0$. One finds,
that the Ricci scalar is constant ${}^{(3)}R=-2\Lambda$. However, the higher-
order curvature terms (see Appendix for the general form) are divergent at the
origin:

$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}$
$\displaystyle=\frac{4\Lambda^{2}}{3}+\frac{6M^{2}}{r^{6}},$
$\displaystyle{}^{(3)}R^{i}_{j}{}^{(3)}R^{j}_{k}{}^{(3)}R^{k}_{i}$
$\displaystyle=-\frac{8\Lambda^{3}}{9}-\frac{12\Lambda
M^{2}}{r^{6}}-\frac{6M^{3}}{r^{9}},$ (47)

yielding an infinite action and suppressing the singularity. The same
conclusions can be drawn for Kerr spacetime and singular Lu-Mei-Pope metric,
derived in the context of Horava gravity, see Appendix.

#### Regular black holes

Due to observation’s of the binary black holes mergers Abbott  _et al._ (2016)
and the Event Horizon Telescope observations Akiyama  _et al._ (2019a, b) the
structure of BH can be investigated on an unprecedented scale Dymnikova and
Kraav (2019). Furthermore, due to the expectation that the quantum gravity
shall resolve the BH singularity issue, the regular black holes have been of
interest recently, for discussions in various quantum gravity approaches
Bonanno and Reuter (2000); Ashtekar and Bojowald (2006); Modesto (2006);
Bonanno and Reuter (2006); Falls  _et al._ (2012); Held  _et al._ (2019);
Platania (2019); Faraoni and Giusti (2020) (see for more model independent
viewpoint Dymnikova (1992); Hayward (2006); Bambi and Modesto (2013); Frolov
(2016)). Following Borissova and Eichhorn (2021), we shall discuss Hayward
metric Hayward (2006) (Dymnikova spacetime Dymnikova (1992) is discussed in
the Appendix). The Hayward metric is an example of the regular black hole
solution in GR:

$\displaystyle ds^{2}$
$\displaystyle=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2},$ $\displaystyle
f(r)$ $\displaystyle=1-\frac{2Mr^{2}}{\left(r^{3}+2g^{3}\right)},$ (48)

where $g$ is an arbitrary positive parameter. The metric is non-singular in
$r\xrightarrow{}0$. It is not a solution to H-L theory, however, we consider
it as an off-shell metric present in the path integral.
The kinetic tensor vanishes $K_{ij}=0$, while the Ricci scalar and the second-
order curvature scalars are regular:

$\displaystyle{}^{(3)}R$ $\displaystyle=\frac{24g^{3}GM}{(2g^{3}+r^{3})^{2}},$
$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}$
$\displaystyle=\frac{6M^{2}\left(32g^{6}+r^{6}\right)}{\left(2g^{3}+r^{3}\right)^{4}},$
(49)

leading to finite action. A similar conclusion arises in the case of Dymnikova
spacetime, see Appendix. These two regular solutions to GR are also regular in
the off-shell H-L theory.

###  IV.2 Wormholes

Here, we take the first step in the direction of the investigations of the
consequences of the Finite Action Principle in the context of wormholes (WH).
The wormholes may be characterized in two classes: traversable and non-
traversable. The traversable WH, colloquially speaking, are such that one can
go through it to the other side, see Morris and Thorne (1988) for specific
conditions. The pioneering Einstein-Rosen bridge has been found originally as
a non-static, non-traversable solution to GR. The traversable solutions are
unstable, however, they might be stabilized by an exotic matter or inclusion
of the higher curvature scalar gravity Duplessis and Easson (2015). This is
important in the context of finite action since usually the divergences of
black holes do appear in the curvature squared terms. Hence, due to the
inclusion of the higher-order terms in the actions, the traversable wormholes
are solutions to the equations of motions without the exotic matter. The
exemplary wormhole spacetimes investigated here are the Einstein-Rosen bridge
proposed in Einstein and Rosen (1935), the Morris-Thorne (MT) wormhole Morris
and Thorne (1988), the traversable exponential metric wormhole Boonserm  _et
al._ (2018) and the wormhole solution discussed in the H-L gravity Botta
Cantcheff  _et al._ (2010). All of them have a finite action. Here, we shall
discuss the exponential metric WH. The conclusions for the other possible
wormholes are similar and we discuss them in the Appendix. For the exponential
metric WH, the line element is given by:

$\displaystyle
ds^{2}=-e^{-\frac{2M}{r}}dt^{2}+e^{\frac{2M}{r}}\left(dr^{2}+r^{2}d\Omega^{2}\right).$
(50)

This spacetime consists two regions: “our universe” with $r>M$ and the “other
universe” with $r<M$. $r=M$ corresponds to the wormhole’s throat. The spacial
volume of the “other universe” is infinite when $r\xrightarrow{}0$. Such
volume divergence is irrelevant to our discussion since it describes large
distances in the ”other universe”. Hence, we further consider only $r\geq M$.
The resulting Ricci and Kretschmann scalars calculated in Boonserm  _et al._
(2018) and the measure are non-singular everywhere:

$\displaystyle R$ $\displaystyle=-\frac{2M^{2}}{r^{4}}e^{\frac{-2M}{r}},$
$\displaystyle R_{\mu\nu\sigma\rho}R^{\mu\nu\sigma\rho}$
$\displaystyle=\frac{4M^{2}(12r^{2}-16Mr+7M^{2})}{r^{8}}e^{-4\frac{M}{r}},$
(51)

resulting in the finite action for the Stelle gravity. Similarly for the H-L
gravity:

$\displaystyle{}^{(3)}R=R,\quad K^{2}=K_{ij}K^{ij}=0,$
$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}=\frac{2M^{2}(M^{2}-2Mr+3r^{2})}{r^{8}}e^{-\frac{4M}{r}}.$
(52)

##  V Conclusions and discussion

The Finite Action Principle is a powerful tool to study quantum gravity
theories and also QFTs in general. In particular, we have shown that it can be
invoked to explain the flatness and homogeneity of the early universe and can
possible be a dynamic resolution of the singularities problem of black holes
in the context of Horava-Lifszyc gravity.
The conditions stemming from the Finite Action Principle justify the
Topological Phase hypothesis without the need for conversion of the degrees of
freedom in the early universe, which is assumed to take place in Agrawal  _et
al._ (2020). Furthermore, the anisotropic scaling of Horava gravity admits
only flat solutions for the cosmological metrics, see also the discussion on
the instanton “no-boundary”-like solution Bramberger  _et al._ (2018).
Moreover, the amplitude of the cosmological perturbations are scaling as:
$\delta\Phi=H^{\frac{3-z}{2z}},$ hence at $z=3$ they are almost scale-
invariant Mukohyama (2009). Finally, the Weyl anomalies structure in H-L
gravity does not lead to strong non-local effects during the radiation
domination epoch Godazgar  _et al._ (2017); Meissner and Nicolai (2017a). This
stems from the fact that these anomalies are of second order in derivatives in
the flat spacetimes Griffin  _et al._ (2013); Arav  _et al._ (2015, 2017,
2016), hence they are harmless and allow to avoid the vanishing of conformal
anomaly criteria Meissner and Nicolai (2017b, 2018). In particular, it would
be interesting to see whether the anisotropic Weyl anomalies can also give
departure from scale invariance, as it is discussed in Agrawal  _et al._
(2020). Yet we leave that for further investigation to be performed elsewhere.
Combined with earlier results, our investigation backs up fully the
topological phase conjecture, hence making inflation redundant. Furthermore,
it seems that this is in line with the swampland conjectures and the newly
proposed finite-amplitude principle Jonas  _et al._ (2021), making the
asymptotically safe quantum gravity to pick initial conditions such that
inflation ceases to be eternal Chojnacki  _et al._ (2021), see also Rudelius
(2019, 2021).
From the point of view of finite action selection principle Borissova and
Eichhorn (2021) they are equally good theories, resolving the black holes
singularities, assuming that ghost issue is resolved in the latter case. Yet
none of the regular B-H solutions have been found in the context of H-L
gravity Wang . Hence it is a strong suggestion that the wormholes may appear
in the UV regime of H-L gravity and can serve as a “cure” for singularities
Lobo (2016); Olmo  _et al._ (2017); Bambi  _et al._ (2016).
In the case of wormholes, both traversable and non-traversable wormholes are
on equal footing in the case of the Finite Action Principle. However, this
principle suggests that there is a trade-off between the resolution of black-
hole singularities and the appearance of wormhole spacetimes due to higher
curvature invariants. The wormhole solutions will remain in both the LI and
H-L path integrals. The higher-order curvature scalars, generically present in
the quantum gravity, stabilize the wormhole solutions without the need for an
exotic matter.
Finally, one should mention that there are many experiments to test the
Lorentz Invariance Violations (LIV) in the gravitational sector coming from
gravitational waves observations Yunes  _et al._ (2016); Emir Gümrükçüoğlu
_et al._ (2018); Abbott  _et al._ (2019); Ramos and Barausse (2019); Mewes
(2019); Frusciante and Benetti (2021), which could in principle validate
Horava’s proposal, yet we know much more about the LIV in the matter sector
(see for example Tasson (2014); Frusciante  _et al._ (2016)). Since these two
can be related Eichhorn  _et al._ (2020), one can speculate that H-L gravity
can be tested in the nearby future.

###  V.1 Acknowledgements

We thank J. N. Borrisova, A. Eichhorn, L. Giani, K. A. Meissner, A. G. A.
Pithis and A. Wang for inspiring discussions and careful reading of the
manuscript. J.H.K. was supported by the Polish National Science Centre grant
2018/29/N/ST2/01743. J.H.K. would like to thank the CP3-Origins for the
extended hospitality during this work. J.H.K acknowledges the NAWA Iwanowska
scholarship PPN/IWA/2019/1/00048.

##  Appendix A Appendix: Further black-holes and wormholes

Here we present further examples of interesting black-hole and wormhole
spacetimes in the context of the Finite Action Principle. We find the
restrictions on the LMP solutions necessary to resolve the singularity at the
origin. Similarly, the spatial Ricci scalar of Kerr’s spacetime yields
infinite action. We further give three examples of wormholes with finite
action: Einstein-Rosen bridge, Morris-Thorne wormhole, and a spatially
symmetric and traversable wormhole solution to H-L gravity.

###  A.1 Black-holes

#### LPM black hole

The popular LMP Lü  _et al._ (2009) metric is not a solution to the vacuum H-L
equations. However, the second class of the LMP solutions written in the ADM
frame with projectability condition satisfy the field equations of H-L gravity
coupled to anisotropic fluid with heat flow, see Cai and Wang (2010). The LMP
solutions were found in the orthogonal gauge (45), without the projectability.
There are two types of solutions. Class A solutions are:

$\displaystyle\Phi=-\frac{1}{2}\ln(1+x^{2}),\quad\Psi=\Psi(r).$ (A-1)

Class B solutions consist of:

$\displaystyle\Phi$ $\displaystyle=-\frac{1}{2}\ln\left(1+x^{2}-\alpha
x^{\alpha_{\pm}}\right),$ $\displaystyle\Psi$
$\displaystyle=-\beta_{\pm}\ln{x}+\frac{1}{2}\ln\left(1+x^{2}-\alpha
x^{\alpha_{\pm}}\right),$ (A-2)

where $x=\sqrt{|\Lambda_{W}|}r$, $\Lambda=\frac{3}{2}\Lambda_{W}$, $\alpha$ is
an arbitrary real constant, and $\alpha_{\pm}$ and
$\beta_{\pm}=2\alpha_{\pm}-1$ are parameters depending on $\lambda$. Their
explicit form may be found in Cai and Wang (2010). The LPM solutions (A-1) and
(A.1), have vanishing kinetic tensor $K_{ij}=0$, while the Ricci scalar and
the integral measure are given respectively by:

$\displaystyle{}^{(3)}R=\frac{2}{r^{2}}\left(\alpha(1+\alpha_{\pm})x^{\alpha_{\pm}}-3x^{2}\right),\quad
N\sqrt{g}=r^{2}x^{-\beta_{\pm}}.$ (A-3)

The $S_{s}$ function (43) stands:

$\displaystyle S_{s}(x_{UV},x_{IR})$
$\displaystyle=-\frac{2}{\sqrt{\Lambda_{W}}}\int_{x_{UV}/\sqrt{\Lambda_{W}}}^{x_{IR}/\sqrt{\Lambda_{W}}}dx\,(\alpha(1+\alpha_{\pm})x^{1-\alpha_{\pm}}$
$\displaystyle-3x^{3-2\alpha_{\pm}}),$ (A-4)

Where $x_{UV}=\sqrt{\Lambda_{W}}r_{UV}$ and $x_{IR}=\sqrt{\Lambda_{W}}r_{IR}$.
The necessary condition for the spatial Ricci scalar to be finite is
$2>\alpha_{\pm}$. We proceed in the ADM gauge, which describes an independent
theory in the H-L gravity. Then, the Class A solution is given by:

$\displaystyle\mu=-\infty,\quad\nu=-\frac{1}{2}\ln\left(1-\Lambda_{W}r^{2}\right)$
(A-5)

applied to (41), we get ${}^{(3)}R=6\Lambda_{W}$. The $S_{s}$ function is
given by:

$\displaystyle
S_{s}(r_{UV},r_{IR})=-\int_{r_{UV}}^{r_{IR}}\frac{6\Lambda_{W}r^{2}}{\sqrt{|1-\Lambda_{W}r^{2}|}}.$
(A-6)

The exact form of $S_{s}(r_{UV},r_{IR})$ depends on the sign of the scaled
cosmological constant $\Lambda_{W}$, nevertheless, it is always finite, when
$r_{UV}\xrightarrow[]{}0$. Indeed, for the negative $\Lambda_{W}<0$:

$\displaystyle S_{s}(r_{UV},r_{IR})$
$\displaystyle=-\frac{3}{\sqrt{-\Lambda_{W}}}\textrm{arcsinh}\left(\sqrt{-\Lambda_{W}}r_{UV}\right)$
$\displaystyle-3r_{UV}\sqrt{-\Lambda_{W}r^{2}_{UV}+1}.$ (A-7)

Positive cosmological constant splits the space in two regions. When
$r>\frac{1}{\sqrt{\Lambda_{W}}}$ we get:

$\displaystyle S_{s}(r_{UV},r_{IR})$
$\displaystyle=\frac{3}{\sqrt{\Lambda_{W}}}\textrm{arctanh}\left(\frac{\sqrt{\Lambda_{W}}r_{UV}}{\sqrt{\Lambda_{W}r^{2}_{UV}-1}}\right)$
$\displaystyle+3r_{UV}\sqrt{\Lambda_{W}r^{2}_{UV}-1},$ (A-8)

for a small, positive cosmological constant, above result is irrelevant for
our discussion, since it would describe large scales. When
$r<\frac{1}{\sqrt{\Lambda_{W}}}$:

$\displaystyle S_{s}(r_{UV},r_{IR})$
$\displaystyle=-\frac{3}{\sqrt{\Lambda_{W}}}\textrm{arcsin}\left(\sqrt{\Lambda_{W}}r_{UV}\right)$
$\displaystyle+3r_{UV}\sqrt{\Lambda_{W}r^{2}_{UV}-1}.$ (A-9)

The class B solution singularity at the origin, appearing when
$2\leq\alpha_{+}$ is suppressed by the Finite Action Principle. Class A
solutions are finite and contribute to the path integral, if the cosmological
constant is negative or small and positive when
$r_{UV}\xrightarrow{}\frac{1}{\sqrt{\Lambda_{W}}}$.

#### Kerr spacetime

Kerr spacetime corresponds to an axially symmetric, rotating black hole with
mass $M$ and angular momentum $J$. It is a solution to the Einstein Equations
in GR, however, it has been shown order by order in the parameter $a=J/M$,
that it is not a solution to the H-L field equations Lee and Myung (2012). Yet
it can still enter the path integral as an off-shell metric. The line element
in the Boyer-Lindquist coordinates is given by:

$\displaystyle
ds^{2}=-\frac{\rho^{2}\Delta_{r}}{\Sigma^{2}}dt^{2}+\frac{\rho^{2}}{\Delta_{r}}dr^{2}+\rho^{2}d\theta^{2}+\frac{\Sigma^{2}\sin^{2}{\theta}}{\rho^{2}}(d\phi-\xi
dt^{2})^{2},$ (A-10)

where

$\displaystyle\rho^{2}$ $\displaystyle=r^{2}+a^{2}\cos^{2}{\theta},$
$\displaystyle\Delta_{r}$ $\displaystyle=r^{2}+a^{2}-2Mr,$
$\displaystyle\Sigma^{2}$ $\displaystyle=(r^{2}+a^{2})^{2}-2Mr,$
$\displaystyle\xi$ $\displaystyle=\frac{2Mar}{\Sigma^{2}}.$ (A-11)

We are interested in the singularity on equator plane $\cos\theta=0$, $r=0$,
described in detail in Barrow (2020). For the explicit form of the extrinsic
curvature scalars and Ricci scalar refer to Lee and Myung (2012). Here, we
only show the form of the Ricci scalar on the $\cos{\theta}=0$ plane:

$\displaystyle{}^{(3)}R=-\frac{2a^{2}m^{2}(a^{2}+3r^{2})^{2}}{r^{4}(r^{3}+a^{2}(2M+r))^{2}.}$
(A-12)

It is singular at $r=0$. Integrating ${}^{(3)}R$ with the measure
$N\sqrt{g}=r^{2}$, results in the infinite action in the UV limit and the Kerr
spacetime does not contribute the path integral. The vanishing, 4-dimensional
Ricci scalar is restored in the LI limit $\lambda=1$. It is then necessary to
include the Kretschmann scalar to resolve the singularity as discussed in
Borissova and Eichhorn (2021).

#### Dymnikova spacetime

The Dymnikova spacetime is a regular solution in GR. It constructed with the
line element (IV.1) with:

$\displaystyle f(r)=1-\frac{2M(r)}{r},\quad
M(r)=M\left(1-e^{-\frac{r^{3}}{2g^{3}}}\right).$ (A-13)

The corresponding curvature scalars are non-singular:

$\displaystyle{}^{(3)}R$
$\displaystyle=\frac{6M}{g^{3}}e^{-\frac{r^{3}}{2g^{3}}},$
$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}$
$\displaystyle=\frac{3M^{2}}{2g^{6}r^{6}}e^{-\frac{r^{3}}{g^{3}}}\left(4g^{6}\left(e^{\frac{r^{3}}{2g^{3}}}-1\right)^{2}\right.$
$\displaystyle\left.-4g^{3}r^{3}\left(e^{\frac{r^{3}}{2g^{3}}}-1\right)+9r^{6}\right)$
(A-14)

and the action is finite in the limit $r_{UV}\xrightarrow{}0$. In particular,
in this limit we have
${}^{(3)}R_{ij}{}^{(3)}R^{ij}\xrightarrow{}12M^{2}/g^{6}$

#### Higher-order curvature scalars

Here we give a general expression for the higher-order scalars present in the
H-L potential for the projectable ADM and orthogonal gauge metric tensors. The
metric tensor in projectable ADM gauge (41) yields:

$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}$
$\displaystyle=\frac{2e^{-4\nu(r)}\left(2r^{2}\nu^{\prime}(r)^{2}+\left(r\nu^{\prime}(r)+e^{2\nu(r)}-1\right)^{2}\right)}{r^{4}},$
$\displaystyle{}^{(3)}R^{i}_{j}{}^{(3)}R^{j}_{k}{}^{(3)}R^{k}_{i}$
$\displaystyle=\frac{2e^{-6\nu(r)}\left(4r^{3}\nu^{\prime}(r)^{3}+\left(r\nu^{\prime}(r)+e^{2\nu(r)}-1\right)^{3}\right)}{r^{6}}.$
(A-15)

In the ortohonormal gauge they are given by

$\displaystyle{}^{(3)}R$
$\displaystyle=\frac{2e^{-2\Phi(r)}\left(2r\Phi^{\prime}(r)+e^{2\Phi(r)}-1\right)}{r^{2}},$
$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}$
$\displaystyle=\frac{2e^{-4\Phi(r)}\left(2r^{2}\Phi^{\prime}(r)^{2}+\left(r\Phi^{\prime}(r)+e^{2\Phi(r)}-1\right)^{2}\right)}{r^{4}},$
$\displaystyle{}^{(3)}R^{i}_{j}{}^{(3)}R^{j}_{k}{}^{(3)}R^{k}_{i}$
$\displaystyle=\frac{2e^{-6\Phi(r)}\left(4r^{3}\Phi^{\prime}(r)^{3}+\left(r\Phi^{\prime}(r)+e^{2\Phi(r)}-1\right)^{3}\right)}{r^{6}}.$
(A-16)

###  A.2 Wormholes

#### Einstein-Rosen bridge

The Einstein-Rosen (E-R) bridge smoothly glues together two copies of the
Schwarzschild spacetime: black-hole and the white-hole solutions corresponding
to the positive and negative coordinate $u$. Metric tensor of the Einstein-
Rosen wormhole proposed in Einstein and Rosen (1935) and discussed in e.g.
Katanaev (2014) is given by:

$\displaystyle
ds^{2}=\frac{-u^{2}}{u^{2}+4M}dt^{2}+(u^{2}+4M)du^{2}+\frac{1}{4}(u^{2}+4M)d\Omega^{2}.$
(B-1)

The E-R bridge is non-traversable and geodesically incomplete in $u=0$. This
fact, however, does not impact the regularity of the curvature scalars. The
4-dimensional Ricci scalar is:

$\displaystyle
R=\frac{2\left(64M^{2}+32Mu^{2}+4u^{4}+u^{2}\right)}{\left(4M+u^{2}\right)^{3}}.$
(B-2)

The second order curvature scalar $R_{\mu\nu}R^{\mu\nu}$ is:

$\displaystyle\frac{4\left(48M^{2}+8\left(4M+u^{2}\right)^{4}+(32M-1)\left(4M+u^{2}\right)^{2}\right)}{\left(4M+u^{2}\right)^{6}}.$
(B-3)

Both of which integrated with the measure are non-singular:

$\displaystyle\sqrt{g}=\frac{1}{4}u(4M+u^{2}).$ (B-4)

The wormhole solutions analyzed in this paper generally yield the finite
action in both GR and H-L. The finite Action Principle suggests, that in the
quantum UV regime, singular black-hole spacetimes may be replaced with the
regular wormhole solutions.

#### Morris-Thorne wormhole

The MT wormhole is defined in the spherically symmetric, Lorentzian spacetime
by the line element:

$\displaystyle
ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{dr^{2}}{1-\frac{b(r)}{r}}+r^{2}d\Omega^{2}$
(B-5)

where $\Phi(r)$ is known as the redshift and there are no horizons if it is
finite. Function $b(r)$ determines the wormhole’s shape. We choose
$\Phi(r),\,b(r)$ to be:

$\displaystyle\Phi(r)$ $\displaystyle=0,\quad
b(r)=2M\left(1-e^{r_{0}-r}\right)+r_{0}e^{r_{0}-r},$ (B-6)

where $r_{0}$ is the radius of the throat of the wormhole, such that
$b(r_{0})=r_{0}$. 4-dimensional curvature scalars for this spacetime have been
calculated in Mattingly  _et al._ (2020). The Ricci curvature scalar is
singular at $r=0$, however, the radial coordinate $r$ varies between $r_{0}>0$
and infinity:

$\displaystyle R=-2\left(2M-r_{0}\right)\frac{e^{r_{0}-r}}{r^{2}}.$ (B-7)

The resulting $S_{s}=\int_{r_{UV}=r_{0}}^{r_{IR}}\sqrt{g}R$ function is
divergent as $r_{UV}\xrightarrow{}r_{0}$ and cannot be expressed in terms of
simple functions:

$\displaystyle
2(2M-r_{0})\int_{r_{UV}=r_{0}}^{r_{IR}}\sqrt{\frac{r}{r-2M(1-e^{r_{0}-r})+r_{0}e^{r_{0}-r}}}e^{r_{0}-r}dr.$
(B-8)

However, this is only a coordinate singularity and one may get rid of it with
a proper transformation.
Higher-order curvature scalars for Morris-Thorne wormhole are:

$\displaystyle{}^{(3)}R$ $\displaystyle=\frac{2b^{\prime}(r)}{r^{2}}$
$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}$
$\displaystyle=\frac{3r^{2}b^{\prime}(r)^{2}-2rb(r)b^{\prime}(r)+3b(r)^{2}}{2r^{6}},$
$\displaystyle{}^{(3)}R^{i}_{j}{}^{(3)}R^{j}_{k}{}^{(3)}R^{k}_{i}$
$\displaystyle=\frac{-9r^{2}b(r)b^{\prime}(r)^{2}+5r^{3}b^{\prime}(r)^{3}+15rb(r)^{2}b^{\prime}(r)-3b(r)^{3}}{4r^{9}},$
(B-9)

and integrated give action that is finite.

#### H-L wormhole

Static spherically traversable symmetric wormholes have been constructed in
Botta Cantcheff  _et al._ (2010) in the H-L theory through the modification of
the Rosen-Einstein spacetime:

$\displaystyle
ds^{2}=-N^{2}(\rho)dt^{2}+\frac{1}{f(\rho)}d\rho^{2}+(r_{0}+\rho^{2})^{2}d\Omega^{2},$
(B-10)

with additional $\mathbf{Z}_{2}$ symmetry with respect to the wormhole’s
throat. There are solutions with $\lambda=1$ asymptotically corresponding to
the Minkowski vacuum. Explicitly we have:

$\displaystyle f=N^{2}$ $\displaystyle=1+\omega(r_{0}+\rho^{2})^{2}$
$\displaystyle-\sqrt{(r_{0}+\rho^{2})\left(\omega^{2}(r_{0}+\rho^{2})^{3}+4\omega
M\right)}.$ (B-11)

Radius of the wormhole’s throat is given by $r_{0}$. The parameters $\omega,$
and $M$ are connected to the coupling constants in H-L action. See Botta
Cantcheff  _et al._ (2010) for their explicit form. Ricci scalar of the H-L
wormhole invariants are given by

$\displaystyle{}^{(3)}R$
$\displaystyle=-\frac{1}{{(\rho^{2}+r_{0}){}^{2}}}\Big{[}2(-10\rho^{2}\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}$
$\displaystyle-4r_{0}\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}+16\rho^{6}\omega+8\rho^{2}$
$\displaystyle+36\rho^{4}r_{0}\omega+24\rho^{2}r_{0}^{2}\omega+4r_{0}^{3}\omega+4r_{0}-1)\Big{]},$
$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}$
$\displaystyle=\frac{1}{(\rho^{2}+r_{0})^{4}}\left\\{2(-7\rho^{2}\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}-2r_{0}\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}\right.$
$\displaystyle+10\rho^{6}\omega+6\rho^{2}+22\rho^{4}r_{0}\omega+14\rho^{2}r_{0}^{2}\omega+2r_{0}^{3}\omega+2r_{0}-1){}^{2}+4M$
$\displaystyle+\frac{4(\rho^{2}+r_{0})}{\omega(4M+\omega(r_{0}+\omega^{2}){}^{3})}\left[\rho^{2}\omega(-4(\rho^{2}+r_{0})\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}+\omega(r_{0}+\omega^{2}){}^{3})\right]$
$\displaystyle\left.-2\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}\left[-\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}+\omega(\rho^{2}+r_{0}){}^{2}+1)\right]{}^{2}\right\\}$
(B-12)

The kinetic terms with $K_{ij}=0$ are vanishing, while the spacial Ricci
scalar and higher curvature terms are finite.
From the point of view of the Finite Action Principle, all of the investigated
wormhole spacetimes are included in the gravitational path integral.

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# Team ÚFAL at CMCL 2022 Shared Task: Figuring out the correct recipe for
predicting Eye-Tracking features using Pretrained Language Models

Sunit Bhattacharya, Rishu Kumar  Ondřej Bojar
Charles University
Faculty Of Mathematics and Physics
Insititute of Formal and Applied Linguistics
<EMAIL_ADDRESS>

###### Abstract

Eye-Tracking data is a very useful source of information to study cognition
and especially language comprehension in humans. In this paper, we describe
our systems for the CMCL 2022 shared task on predicting eye-tracking
information. We describe our experiments with pretrained models like BERT and
XLM and the different ways in which we used those representations to predict
four eye-tracking features. Along with analysing the effect of using two
different kinds of pretrained multilingual language models and different ways
of pooling the token-level representations, we also explore how contextual
information affects the performance of the systems. Finally, we also explore
if factors like augmenting linguistic information affect the predictions. Our
submissions achieved an average MAE of 5.72 and ranked $5^{th}$ in the shared
task. The average MAE showed further reduction to 5.25 in post task
evaluation.

##  1 Introduction and Motivation

In the last decade that has seen rapid developments in AI research, the
emergence of the Transformer architecture Vaswani et al. (2017) marked a
pivotal point in Natural Language Processing (NLP). Fine-tuning pretrained
language models to work on various downstream tasks has become a dominant
method of obtaining state-of-the-art performance in different areas. Their
capability to capture linguistic knowledge and learn powerful contextual word
embeddings Liu et al. (2019) have made the transformer based models the work-
horses in many NLP tasks. Pretrained models like the multilingual BERT Devlin
et al. (2019) and XLM Conneau et al. (2020) have also shown state-of-the-art
performance on cross-lingual understanding tasks Wu and Dredze (2019); Artetxe
et al. (2019). In some cases like machine translation, there are even claims
that deep learning systems reach translation qualities that are comparable to
professional translators Popel et al. (2020).

Language processing and its links with cognition is a very old research
problem which has revealed how cognitive data (eg. gaze, fMRI) can be used to
investigate human cognition. Attempts at using computational methods for such
studies Mitchell et al. (2008); Dehghani et al. (2017) have also shown
encouraging results. However recently, there have been a number of works that
have tried to incorporate human cognitive data collected during reading for
improving the performance of NLP systems Hollenstein et al. (2019). The CMCL
2022 Shared Task of multilingual and cross-lingual prediction of human reading
behavior Hollenstein et al. (2022) explores how eye-gaze attributes can be
algorithmically predicted given reading data in multilingual settings.

Informed by the previous attempts at using pretrained multilingual language
models to predict human reading behavior Hollenstein et al. (2021) we
experiment with multilingual BERT and XLM based models to test which fares
better in this task. For the experiments with the pretrained models, we use
the trained weights from Huggingface Wolf et al. (2020) and perform the rest
of our experiments using PyTorch111https://pytorch.org/. Inspired by the
psycholinguistic research on investigating context length during processing
Wochna and Juhasz (2013), we experiment how different contexts affect model
performance. Finally, we merged the principles of the "classical" approach of
feature-based prediction with the pretrained-language model based prediction
for further analysis. In the following sections, we present our results from a
total of 48 different models.

##  2 Task Description

The CMCL 2022 Shared Task of Multilingual and Cross-lingual prediction of
human reading behavior frames the task of predicting eye-gaze attributes
associated with reading sentences as a regression task. The data for the task
was comprised of eye movements corresponding to reading sentences in six
languages (Chinese, Dutch, English, German, Hindi, Russian). The training data
for the task contained 1703 sentences while the development set and test set
contained 104 and 324 sentences respectively. The data was presented in a way
such that for each word in a sentence there were four associated eye-tracking
features in the form of the mean and standard deviation scores of the Total
Reading Time (TRT) and First Fixation Duration (FFD). The features in the data
were scaled in the range between 0 and 100 to facilitate evaluation via the
mean absolute average (MAE).

##  3 Experiments

A total of 48 models of different configurations were trained with the data
provided for the shared task. The different configurations used to construct
the models are based on intuition and literature survey.

Thee models were primarily categorized as System-1 (sys1) and System-2 (sys2)
models. For some word corresponding to a sentence in the dataset, System-1
models provided no additional context information. System-2 models on the
other hand, contained the information of all the words in the sentence that
preceded the current word, providing additional context. This setting was
inspired by works Khandelwal et al. (2018); Clark et al. (2019) on how context
is used by language models.

All systems under the System-1/2 labels were further trained as a BERT (bert)
based system or a XLM (xlm) based system. BERT embeddings were previously used
by Choudhary et al. (2021) for the eye-tracking feature prediction task in
CMCL 2021.

Corresponding to each such language models (bert and xlm), the impact of
different fine-tuning strategiesSun et al. (2019) on system performance was
studied. Hence, for one setting, only the contextualized word representation
(CWR) was utilized by freezing the model weights and putting a learnable
regression layer on top of the model output layer (classifier). Alternatively,
the models were fine-tuned with the regression layer on top of them (whole).
This setting is similar to the one used by Li and Rudzicz (2021). However in
our case, we experiment with a BERT and XLM pretrained model.

Additionally, we also performed experiments with pooling strategies for the
layer representations by either using the final hidden representation of the
first sub-word encoding of the input (first) or aggregating the
representations of all sub-words using mean-pooling (mean) or sum-pooling
(sum). The rationale behind using different pooling strategies was to have a
sentence-level representation of the input tokens. The impact of different
pooling strategies has previously been studied Shao et al. (2019); Lee et al.
(2019) for different problems. In this paper, we analyze the effect of pooling
feature-space embeddings in the context of eye-tracking feature prediction.

Finally, for the experiments where we augmented additional lexical features
(augmented) to the neural features for regression, we used word length and
word-frequency as the additional information following Vickers et al. (2021).

Constructing the experiments in this manner provided us with models with a
diverse set of properties and in turn provided insights into how well the
model behaves when all other things stay the same, and only one aspect of
learning is changed.

##  4 Results

The results corresponding to the top 10 systems based on the experiments
described above are shown in Table 1.

Model | MAE
---|---
bert_sys2_augmented_sum_classifier | 5.251
bert_sys2_unaugmented_first_classifier | 5.267
bert_sys2_augmented_mean_classifier | 5.272
bert_sys1_augmented_mean_classifier | 5.279
bert_sys2_augmented_first_classifier | 5.295
xlm_sys1_augmented_first_classifier | 5.341
xlm_sys2_augmented_first_whole | 5.346
bert_sys1_augmented_sum_classifier | 5.353
bert_sys2_augmented_sum_whole | 5.367
xlm_sys2_augmented_first_classifier | 5.373
Table 1: Top 10 best performing systems

It was observed that the maximum MAE scores (and the maximum variance of
scores) for all the models was obtained for the attribute "TRT_Avg". The
attribute wise variances corresponding to the test-data for all the models are
shown in Table 2. Similarly, the mean values of the attributes for all models
are shown in Table 3.

FFD_Avg | FFD_Std | TRT_Avg | TRT_Std
---|---|---|---
0.194 | 0.403 | 0.637 | 0.489
Table 2: Attribute wise variance of scores for all models FFD_Avg | FFD_Std | TRT_Avg | TRT_Std
---|---|---|---
5.691 | 2.646 | 8.633 | 5.806
Table 3: Attribute wise mean of scores for all models

An analysis of the models based on the different experimental configurations
are described in the following sections.

###  4.1 System-1 vs System-2

Table 4 shows the average model performance across System-1 and System-2
configurations for both BERT and XLM based models (based on the average MAE
values of the configurations). We see that for the BERT based models, the
average MAE for System-1 is lower than that of System-2. But for XLM-based
models, the difference is almost non-existent.

Model | Average MAE across models
---|---
Sys1_BERT | 5.66
Sys1_XLM | 5.70
Sys2_BERT | 5.72
Sys2_XLM | 5.69
Table 4: System-1 vs System-2 performance across models

However, it should be noted that 12 out of the first 20 best performing models
were System-2 models. Hence we posit that although the availability of the
full sentence context is a factor for having more efficient systems,
independently the factor does not seem to boost the overall performance much.

###  4.2 BERT vs XLM

Table 5 shows that there is only a tiny difference in average MAE for all four
attributes (FFD_$\mu$, FFD_$\sigma$, TRT_$\mu$, TRT_$\sigma$) for all BERT vs
XLM models . However, a brief look at Table 6 and Table 7 reveal that it was
the XLM models that were responsible for slightly decreased MAE scores for 3
of the 4 attributes that were being predicted.

Model | Average MAE across models
---|---
BERT | 5.6920
XLM | 5.6960
Table 5: BERT vs XLM performance across models Model | FFD_$\mu$ | FFD_$\sigma$ | TRT_$\mu$ | TRT_$\sigma$
---|---|---|---|---
BERT | 0.141 | 0.776 | 0.952 | 0.792
XLM | 0.236 | 0.045 | 0.349 | 0.204
Table 6: Attribute wise variance of scores for all BERT and XLM based models Model | FFD_$\mu$ | FFD_$\sigma$ | TRT_$\mu$ | TRT_$\sigma$
---|---|---|---|---
BERT | 5.592 | 2.679 | 8.645 | 5.852
XLM | 5.789 | 2.612 | 8.622 | 5.760
Table 7: Attribute wise mean of scores for all BERT and XLM based models

We also see that the amount of variance for XLM based models was also smaller
for 3 of the 4 attributes.

###  4.3 Augmented vs Un-Augmented models

Fig. 1 shows that augmented models. i.e. models that were fed information like
word-frequency and word-length along with the neural representation
information before being fed to the regression layer performed better than
models that used only contextual word embeddings resulting from pretrained
language models. Table 8 and Table 9 show the 5 best performing models of this
category sorted by their MAE.

Model | MAE
---|---
bert_sys2_unaugmented_first_classifier | 5.267
bert_sys2_unaugmented_mean_classifier | 5.405
xlm_sys1_unaugmented_mean_classifier | 5.5
xlm_sys2_unaugmented_mean_classifier | 5.55
xlm_sys1_unaugmented_mean_classifier | 5.557
Table 8: Performance of 5 best Un-Augmented models. Model | MAE
---|---
bert_sys2_augmented_sum_classifier | 5.251
bert_sys2_augmented_mean_classifier | 5.272
bert_sys1_augmented_mean_classifier | 5.279
bert_sys2_augmented_first_classifier | 5.295
xlm_sys1_augmented_first_classifier | 5.341
Table 9: Performance of 5 best Augmented models Model | FFD_$\mu$ | FFD_$\sigma$ | TRT_$\mu$ | TRT_$\sigma$
---|---|---|---|---
Aug | 5.502 | 2.511 | 8.181 | 5.436
Uaug | 5.88 | 2.78 | 9.086 | 6.176
Table 10: Attribute wise mean of scores for all Augmented and Un-augmented models Model | FFD_$\mu$ | FFD_$\sigma$ | TRT_$\mu$ | TRT_$\sigma$
---|---|---|---|---
Aug | 0.017 | 0.004 | 0.015 | 0.007
Uaug | 0.292 | 0.749 | 0.823 | 0.678
Table 11: Attribute wise variance of scores for all Augmented and Un-augmented
models

The mean and variance of attributes across models of these families presented
in Table 10 & 11 show that augmented models show way less variance in their
predictions in comparison with neural-representation only model families.

Figure 1: Augmented vs Un-augmented model performance. The x-axis represents
the 24 different models of each category. The y-axis shows the MAE
corresponding to each model.

###  4.4 Nature of representation of input tokens (Pooling strategies)

Fig. 2 shows that using the first sub-word token or the mean-pooled
representation of the entire input gives lesser MAE scores than the sum-pooled
representations. It was also observed that for System-2 family of models, the
mean-pooled representations were associated with lesser MAE scores in
comparison to the first sub-word representation. The attribute wise mean in
Table 15 and attribute wise variance of model MAEs shown in Table 16
illustrates this point. Table 12,Table 13 and Table 14 show the 5 best
performing models of this category sorted by their MAE.

Model | MAE
---|---
bert_sys2_unaugmented_first_classifier | 5.267
bert_sys2_augmented_first_classifier | 5.295
xlm_sys1_augmented_first_classifier | 5.341
xlm_sys2_augmented_first_whole | 5.346
xlm_sys2_augmented_first_classifier | 5.373
Table 12: Performance of 5 best first models Model | MAE
---|---
bert_sys2_augmented_mean_classifier | 5.272
bert_sys1_augmented_mean_classifier | 5.279
bert_sys2_augmented_mean_whole | 5.375
bert_sys2_unaugmented_mean_classifier | 5.405
xlm_sys1_augmented_mean_whole | 5.413
Table 13: Performance of 5 best Mean models Model | MAE
---|---
bert_sys2_augmented_sum_classifier | 5.251
bert_sys1_augmented_sum_classifier | 5.353
bert_sys2_augmented_sum_whole | 5.367
bert_sys1_augmented_sum_whole | 5.402
xlm_sys2_augmented_sum_classifier | 5.456
Table 14: Performance of 5 best Sum models Model | FFD_$\mu$ | FFD_$\sigma$ | TRT_$\mu$ | TRT_$\sigma$
---|---|---|---|---
first | 5.549 | 2.505 | 8.434 | 5.615
Mean | 5.57 | 2.538 | 8.416 | 5.636
Sum | 5.954 | 2.894 | 9.05 | 6.167
Table 15: Attribute wise mean of scores for models with different input token representations Model | FFD_$\mu$ | FFD_$\sigma$ | TRT_$\mu$ | TRT_$\sigma$
---|---|---|---|---
first | 0.036 | 0.004 | 0.118 | 0.054
Mean | 0.047 | 0.005 | 0.118 | 0.048
Sum | 0.383 | 1.082 | 1.374 | 1.139
Table 16: Attribute wise variance of scores for models with different input
token representations Figure 2: Model performance based on the nature of
representation of input tokens.The x-axis represents the 16 different models
of each category. The y-axis shows the MAE corresponding to each model.

###  4.5 Fine-tuning

Fine-tuning on large pretrained language models has become the standard way to
conduct NLP research after the widespread adoption of the transformer
architecture. And unsurprisingly, our experiments reveal (Fig. 3) that fine-
tuning of models give smaller MAE scores than training only the regression
layers. The stark difference in the variance for the predicted attributes
between fine-tuned models and regression only models (as illustrated in Table
17-18) further demonstrates the advantage of fine-tuning.

Model | FFD_$\mu$ | FFD_$\sigma$ | TRT_$\mu$ | TRT_$\sigma$
---|---|---|---|---
Aug | 5.502 | 2.511 | 8.181 | 5.436
Uaug | 5.88 | 2.78 | 9.086 | 6.176
Table 17: Attribute wise variance of scores for fine-tuned models vs regression-layer only models Model | FFD_$\mu$ | FFD_$\sigma$ | TRT_$\mu$ | TRT_$\sigma$
---|---|---|---|---
Aug | 0.017 | 0.004 | 0.015 | 0.007
Uaug | 0.292 | 0.749 | 0.823 | 0.678
Table 18: Attribute wise mean of scores for fine-tuned models vs regression-
layer only models Figure 3: Fine-tuning vs training only regression layer in
the models. The x-axis represents the 24 different models of each category.
The y-axis shows the MAE corresponding to each model.

##  5 Conclusion

In this paper, we have described our experiments with different kinds of
models that were trained on the data provided for this shared-task. We have
identified five ways in which we can make better systems to predict eye-
tracking features based on eye-tracking data from a multilingual corpus.
First, the experiments demonstrate that the inclusion of context (previous
words occurring in the sentence) helps the models to predict eye-tracking
attributes better. This reaffirms previous observations made with language
models that more context is always helpful. Second, we find that XLM based
models perform relatively better than the BERT based models. Third, our
experiments show the advantages of augmenting additional linguistic features
(word length and word frequency information in this case) to the contextual
word representations to make better systems. This is in agreement with the
findings from eye-tracking prediction tasks from last iterations of CMCL.
Fourth, we see how different pooling methods applied on the input token
representations affect the final performance of the systems. Finally, the
experiments re-validate the approach of fine-tuning pretrained language models
for specific tasks. Hence we conclude that contextualized word representations
from language models pretrained with many different languages, if carefully
augmented, engineered, and fine-tuned, can predict eye-tracking features quite
successfully.

##  6 Acknowledgement

This work has been funded from the grant 19-26934X (NEUREM3) of the Czech
Science Foundation.

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