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SEPLN-CEDI2024: Seminar of the Spanish Society for Natural Language Processing
at the 7th Spanish Conference on Informatics

[orcid=0000-0003-0752-8812<EMAIL_ADDRESS>url=https://www.grupolys.org/ cgomezr, ]

[orcid=0000-0002-4124-7929<EMAIL_ADDRESS>url=https://scholar.google.com/citations?user=cLXCYOcAAAAJ, ]

[orcid=0000-0002-1295-3840<EMAIL_ADDRESS>url=https://www.grupolys.org/ david.vilares/, ]

[orcid=0000-0003-3541-8303<EMAIL_ADDRESS>]

[orcid=0000-0001-8601-8305<EMAIL_ADDRESS>url=https://scholar.google.com/citations?user=XLj-Ol4AAAAJ, ]

# Dancing in the syntax forest: fast, accurate and explainable sentiment
analysis with SALSA

Carlos Gómez-Rodríguez  Universidade da Coruña, CITIC. Department of Computer
Science and Information Technologies. 15071 A Coruña, Spain  Universidade da
Coruña, Technology Transfer Unit (OTRI). 15071 A Coruña, Spain  Muhammad Imran
David Vilares  Elena Solera  Olga Kellert

(2024)

###### Abstract

Sentiment analysis is a key technology for companies and institutions to gauge
public opinion on products, services or events. However, for large-scale
sentiment analysis to be accessible to entities with modest computational
resources, it needs to be performed in a resource-efficient way. While some
efficient sentiment analysis systems exist, they tend to apply shallow
heuristics, which do not take into account syntactic phenomena that can
radically change sentiment. Conversely, alternatives that take syntax into
account are computationally expensive. The SALSA project, funded by the
European Research Council under a Proof-of-Concept Grant, aims to leverage
recently-developed fast syntactic parsing techniques to build sentiment
analysis systems that are lightweight and efficient, while still providing
accuracy and explainability through the explicit use of syntax. We intend our
approaches to be the backbone of a working product of interest for SMEs to use
in production.

###### keywords:

Sentiment analysis opinion mining syntax parsing

##  1 Introduction

We describe the project “efficient Syntactic Analysis for Large-scale
Sentiment Analysis (SALSA)”, which is currently being developed by members of
the LyS research group at the CITIC research center (Universidade da Coruña).
The project is funded by the European Research Council (ERC) with a budget of
150 000 €, under the Proof-of-Concept (PoC) grant scheme, with grant agreement
number 101100615. The Proof-of-Concept grant program intends to bridge the gap
between basic research results obtained in ERC projects and the early phases
of their commercialization, by exploring the innovation potential of the work
and bringing it closer to market. The SALSA project started on February 2023
and is scheduled to run until July 2024, with the possibility of an extension.

Given the nature of PoC grants, the project has a scientific research
component as well as an innovation and technology transfer component. In
Sections 2 and 3, we describe the motivation and methodology, respectively,
focusing on the scientific part of the project. Section 4 describes the
planning and the project team, with a more global focus that includes the
transfer parts as well.

##  2 Motivation

### The problem.

The Internet and social media are major platforms where people express their
views and share experiences about a variety of topics, including products,
services, and events. This results in a wealth of information that can be
leveraged to understand public perception, pinpoint product strengths and
weaknesses, discover and address people’s needs, or track political and market
trends. This is the goal of _opinion mining_ or _sentiment analysis_ systems,
which can analyze text and extract sentiment information from it. For
instance, a company that releases a new smartphone could gather user opinions
from social networks by collecting messages mentioning the product. Given each
such message, a sentiment analysis system can determine whether it contains a
positive, negative or neutral opinion, both towards the phone as a whole or
towards specific aspects like camera or battery.

While many approaches to sentiment analysis have been proposed, an extant
challenge lies in the absence of systems that can efficiently process opinions
while considering the complex structure of language. While efficient sentiment
analysis systems exist, like SentiStrength [1], they rely on shallow heuristic
methods that count sentiment words but may struggle with sentences where the
overall sentiment is shaped by the grammatical structure. For example,
SentiStrength produces the exact same result for “This phone is expensive, and
it’s not at all good” and “This phone is good, and it’s not at all expensive”:
both sentences actually have opposite sentiment polarities, but the heuristics
cannot correctly capture the scope of negation and its influence on how the
sentiment should be interpreted.

An alternative is to use approaches that take syntactic stucture into account.
Syntax can be injected into a sentiment analysis model either explicitly, by
using a parser [2, 3], or implicitly, by using pretrained language models [4]
which have been shown to internally represent syntax [5]. The advantage of the
former approach over the latter is that it can provide explainability, as an
explicit parse tree can be used to show how the polarity is obtained, while
the latter can be better in terms of raw accuracy. However, both of these
approaches have a high computational cost: pretrained models (especially newer
large language models) have considerable CPU and memory requirements [6],
while syntactic parsers have traditionally been slow [7, 8]. This makes
existing syntactically-guided sentiment analysis approaches infeasible for
small entities that have a constrained computational budget.

### The solution.

The SALSA project proposes to leverage a breakthrough achieved by the previous
ERC-funded FASTPARSE project [7] to build sentiment analysis systems that are
efficient and, at the same time, use explicit syntax to improve accuracy and
provide explainability. The FASTPARSE project had the goal to substantially
improve the speed of syntactic parsing, and was a resounding success,
advancing the state of the art in both parsing speed and accuracy several
times, and achieving accurate parsers that improved speed by an order or
magnitude over previous approaches.

In particular, within the FASTPARSE project, we defined a new syntactic
parsing paradigm that casts parsing as a sequence labeling task by defining
encodings that can represent syntax trees by means of discrete labels
associated with words. This was shown to work for the two main families of
grammatical formalisms, with encodings that proved viable for constituency
parsing [9] and dependency parsing [10], and new encodings have kept being
defined since then that further push parsing accuracy [11, 12].

For the purposes of this project, the approach to parsing as sequence labeling
has two especially relevant advantages:

  1. 1.

Pluggability: parse trees are represented with tags associated with words, a
standard representation that is easy to feed to downstream tasks, be it as
features [13] or by using multitask learning [14].

  2. 2.

Speed: these parsers can parse around a thousand sentences per second on a
standard consumer GPU [9, 10, 15].

Our proposal is, thus, to use this new parsing paradigm to create efficient
syntax-guided sentiment analysis systems that can operate at a large scale
with modest computational resources. This will democratize accurate large-
scale sentiment analysis technology, putting it within reach of smaller
institutions and companies that cannot afford to deploy slow parsers or
language models.

##  3 Methodology

Our project addresses two variants (or subtasks) of sentiment analysis:

  1. (a)

Polarity classification: a coarse-grained task where the input is a text, such
as a social network message, and the output is its polarity (i.e. a
representation, typically in a discrete scale, of how positive or negative the
global opinion expressed in the text is).

  2. (b)

Fine-grained sentiment analysis, often called aspect-based sentiment analysis
[16]: in this task, the goal is not to obtain a single value for the whole
text, but individual opinions about the different targets that may appear in
the message (for example, the text “I really like this phone’s camera, but the
battery life is not acceptable” contains both a highly positive opinion about
the camera and a negative one about the battery).

For this purpose, we will explore three approaches to integrate fast syntactic
parsing into solving these two sentiment analysis tasks:

  * •

Rule-based pipeline approach: we start by using a fast dependency parser to
obtain the syntactic tree for an input sentence. We then use syntax-based
rules and a polarity lexicon to navigate the tree, assigning polarity values
to its syntactic components (addressing subtask (b) above). These values can
then be integrated by the rules to obtain an overall polarity for the whole
sentence (addressing subtask (a)). The syntax-based rules are specifically
designed to address syntactic phenomena that influence opinion polarity, such
as negation (for instance, in the example above, "not acceptable" indicates a
negative sentiment because the negation alters the positive meaning of
"acceptable"), intensification (the phrase "really like" signifies a stronger
preference than merely "like"), and adversatives (where "but" juxtaposes a
positive statement against a negative one). It is worth noting that this kind
of approach have been applied in the past, particularly for subtask (a) (see
[2, 3]), but the use of slow parsers at the time rendered large-scale
application costly.

  * •

Multitask learning approach: thanks to the parsing-as-sequence-labeling
approach, we can train a multitask learning model to perform sentiment
analysis and syntactic parsing simultaneously. This allows each task to
benefit from the insights of the other implicitly, without relying on explicit
syntax-based rules. For subtask (a), this technique can be implemented by
formulating the sentiment analysis sequence labeling task in such a way that
the polarity of the whole sentence is represented by a tag associated with the
last word. For subtask (b), we will design a sequence labeling encoding to
represent fine-grained opinoins.

  * •

Integrated approach: this approach is specific to subtask (b). We represent
the sentiment structure of the sentence as a tree. This means that we can
apply the same algorithms that we use for syntactic parsing directly to
perform sentiment analysis. Furthermore, this strategy can be combined with
the multitask learning approach described above, enabling the training of a
sequence labeling model to simultaneously produce two types of trees: one for
syntactic parsing and another for sentiment analysis.

Each of the described approaches will be evaluated on freely-available
sentiment analysis corpora. We will take a multilingual focus, ensuring that
our systems work on a diverse range of languages (for the dependency parsing
part, this is ensured by the use of Universal Dependencies [17]). For the
initial value proposition, we will focus on Spanish and English, since our
main target is the local Spanish market. However, the core technology should
be available to easily adapt to more languages when needed.

In particular, a good starting point when it comes to datasets are the SemEval
2022 Task 10 corpora [18], as they cover five different languages and their
annotation includes individual polarity targets, thus supporting both our
subtasks (a) and (b). Given our focus on the Spanish market, we are also
working with a corpus of TripAdvisor reviews in Spanish, from Rest-Mex 2023
[19]. This also allows us to evaluate performance in a noisy context. Some
preliminary results of the first (rule-based pipeline) approach on the latter
corpus can be found in [20].

##  4 Planning and Team

The SALSA project is organized into three work packages. The first one
addresses the scientific research part of the project, and hence it is the one
we have addressed in Sections 2 and 3. The other two correspond to the
innovation and technology transfer parts of the project. In particular, the
work packages are as follows:

  * •

WP1: Technical and Performance Validation. This involves three tasks:

    * –

Validation of fast parsers for polarity classification: the main objective is
to adapt the parsers developed in the FASTPARSE project to work on polarity
classification tasks. To achieve this, we have created linguistic rules that,
when integrated with a parser’s output, allow us to determine the polarity of
sentences by analyzing the words and syntactic structures in the text [20].
Additionally, we are exploring a purely machine learning-based method that
does not rely on these rules. This method involves training a parser capable
of sequence labeling in conjunction with a sequence model that uses discrete
tags to represent sentence polarity, employing a multitask learning framework.
The success of this task is assessed through the validation of various models’
ability to classify polarity, as evidenced by performance metrics.

    * –

Validation of fast parsers for fine-grained sentiment analysis: in this task,
we aim to go beyond polarity classification by employing syntactic parsers for
more detailed sentiment analysis. The goal is to analyze sentiments at a more
granular level, capturing specific opinions within a sentence. This involves
recognizing how a single sentence might simultaneously provide positive
feedback on one aspect of a product while criticizing another. To achieve
this, we will adapt the approaches based on linguistic rules and on multitask
learning to process fine-grained information: the rules will be used to infer
sentiment information for smaller linguistic units apart from the whole
sentence, and a sequence labeling encoding will be used to represent fine-
grained sentiment information for the multitask setup. In addition, we will
try a third setup where the fine-grained sentiment information will be
represented as a tree, so that syntactic parsers can learn to output it
directly.

    * –

Cross-domain generalization. The performance of sentiment analysis models
trained on a given domain (e.g. restaurant reviews) often degrades when they
are used in a different domain (e.g. movie reviews). Since our project is
targeted primarily towards making the technology accessible to small entities,
which will often lack access to high-quality in-domain corpora, it is
important to ensure that our systems can adapt to different domains. In this
respect, rule-based approaches have been shown to be better than purely
supervised approaches [3]. In this task, we will evaluate the generalization
capabilities of the various models developed in the previous two tasks to
different domains, and explore specific domain adaptation approaches (like the
adaptation of sentiment dictionaries [2]) if needed.

  * •

WP2: Market Research and Validation. This involves three tasks:

    * –

Market Analysis. We have performed a study to identify current sentiment
analysis industry trends, market need and market size at the European level.
This study, which was subcontracted to an external consultancy company,
includes the potential market niches for our sentiment analysis models,
identification of comparative technologies and information on potential user
needs.

    * –

Validation of value proposition. This involves conducting interviews, as well
as Design Thinking workshops, to identify target user needs and validate our
proposition.

    * –

Commercial viability: product-market fit. This task consists in involving
potential users (identified in the previous tasks) in iterative testing and
validation of the developed models.

  * •

WP3: Pre-Technology Transfer. This includes two tasks:

    * –

Creating SALSA business model canvas. We will refine our business model canvas
through a process of continuous improvement. This will involve defining,
enhancing, and validating our value proposition, primary revenue streams, and
cost structures, alongside identifying the crucial partners, customers,
channels, and business relationships that form the core components of an open
source business model. We will assess the viability of establishing a spin-off
to offer service agreements related to SALSA.

    * –

Building a business environment. The aim of this task is to establish contact
with potential partners and stakeholders that can advance the TRL and scale
the technology, as well as with potential future investors. We will also
considering the possibility of preparing an EIC Transition proposal, depending
on the outcomes of previous tasks.

The project team is composed by the authors of this paper: a postdoctoral
researcher (O. Kellert), an MSCA predoctoral researcher (M. Imran), a
technical manager (D. Vilares) and a project manager (E. Solera), led by the
PI (C. Gómez-Rodríguez). In addition, some of the market-oriented tasks of the
project are being carried out with the help of Matical Innovation, which has
been subcontracted for that purpose.

###### Acknowledgements.

This project has received funding by the European Research Council (ERC),
under the Horizon Europe research and innovation programme (SALSA, grant
agreement No 101100615).

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# Constraining Barrow entropy-based Cosmology with power-law inflation

G. G. Luciano<EMAIL_ADDRESS>Applied Physics Section of
Environmental Science Department, Universitat de Lleida, Av. Jaume II, 69,
25001 Lleida, Spain

###### Abstract

We study the inflationary era of the Universe in a modified cosmological
scenario based on the gravity-thermodynamics conjecture with Barrow entropy
instead of the usual Bekenstein-Hawking one. The former arises from the effort
to account for quantum gravitational effects on the horizon surface of black
holes and, in a broader sense, of the Universe. First, we extract modified
Friedmann equations from the first law of thermodynamics applied to the
apparent horizon of a Friedmann-Robertson-Walker Universe in
$(n+1)$-dimensions. Assuming a power-law behavior for the scalar inflaton
field, we then investigate how the inflationary dynamics is affected in Barrow
cosmological setup. We find that the inflationary era may phenomenologically
consist of the slow-roll phase, while Barrow entropy is incompatible with
kinetic inflation. By demanding observationally consistency of the scalar
spectral index and tensor-to-scalar ratio with recent Planck data, we finally
constrain Barrow exponent to $\Delta\lesssim 10^{-4}$.

##  I Introduction

The effort to understand the statistical mechanics of black holes
Carlip:2014pma  has opened up new scenarios in modern theoretical physics,
including the study of the AdS/CFT correspondence Malda ; Aharony:1999ti  and
the investigation of the connection between gravity and thermodynamics. Beyond
their intrinsic interest, both these two lines of research might potentially
have a deep impact upon the development of quantum gravity, mainly because
they are the most successful realizations of the holographic principle tHooft
; Sussk . While the AdS/CFT correspondence is based on the description of the
background geometry in terms of anti-de Sitter vacuum solutions, the interplay
between gravity and thermodynamics finds its conceptualization in the so-
called _gravity-thermodynamics_ conjecture Padmanabhan:2003gd ; Eling:2006aw ;
Padmanabhan:2009vy , which states that Einstein field equations are nothing
but the gravitational counterpart of the laws of thermodynamics applied to
spacetime Jacobson:1995ab . Besides, in the cosmological context such a
conjecture allows to extract Friedmann equations by implementing the first law
of thermodynamics on the apparent horizon of the Universe Frolov:2002va ;
Cai:2005ra ; Akbar:2006kj ; Cai:2006rs .

In the original formulation the gravity-thermodynamic conjecture applies
Bekenstein-Hawking (BH) area law $S_{BH}=A/A_{0}$ to the Universe apparent
horizon of surface area $A=4\pi r_{hor}^{2}$ and radius $r_{hor}$.111Here and
henceforth we work in natural units $\hslash=c=G=k_{B}=1$. Accordingly, the
Planck area is $A_{0}=4G=4$. Nevertheless, generalized forms of BH entropy
have been discussed in recent literature, motivated by either nonextensive
Tsallis ; Tsallis2  or quantum gravity Barrow  arguments. To the latter class
belongs Barrow entropy, which deforms BH area-law to

$S\,=\,\left(\frac{A}{A_{0}}\right)^{1+\Delta/2}\,,\qquad 0\leq\Delta\leq
1\,,$ (1)

where Barrow exponent $\Delta$ embeds quantum gravitational corrections. In
particular, $\Delta=1$ corresponds to the maximal departure from BH entropy,
which is instead recovered for $\Delta=0$. Though being proposed for black
holes Barrow , Eq. (1) is also applied within the cosmological framework,
giving rise to modified Friedmann equations that predict a richer
phenomenology comparing to the standard one Saridakis:2020lrg . In addition,
one can rephrase the holographic principle in terms of Barrow entropy,
obtaining Barrow holographic dark energy (BHDE) (see, for instance
Saridakis:2020zol ; Dabrowski:2020atl ; Saridakis:2020cqq ; Sheykhi:2021fwh ;
Adhikary:2021xym ; Luciano ; Luciano2 ; Luciano3  for recent applications).
Comparison of the above constructions with observations sets upper limits on
Barrow exponent Anagnostopoulos:2020ctz ; Leon:2021wyx ; Barrow:2020kug ;
Jusufi:2021fek ; LucSar , which slightly deviates from zero, as expected.

In physical cosmology, inflation is supposed to be a crucial era in the
evolution of the Universe, consisting of a very short-lived, but extremely
accelerated expansion phase occurred right after the Big Bang. Originally
proposed in Inf1 ; Inf2 ; Inf3 ; Inf4 , it has been getting increasing
attention over the years, becoming one of the two pillars of the present
cosmological model along with the late time acceleration Perl ; Capoz ; Salu .
In spite of this, the origin of inflation has not been well understood yet.
The most commonly adopted scenario is that it has been driven by a particular
form of dark energy represented by a scalar field with slow rolling
assumptions LindeCosm . Alternative models have been recently proposed in Alt1
; Alt2 ; Alt3 ; Alt4 ; Alt5 ; Alt6 ; Alt7 ; Alt8 . The inflationary phase has
also been studied in connection with holographic dark energy InflatHDE1 ;
InflatHDE2 ; InflatHDE3 , motivated by the plausible role of the latter as a
mechanism responsible for the late time cosmic acceleration.

Starting from the above premises, in this work we study the evolution and
inflation of the Universe in the context of Barrow entropy-based Cosmology. In
this sense, our analysis should be regarded as a preliminary attempt to
explore the effects of quantum gravity on the dynamics of the Universe. In
particular, we apply Barrow formula (1) to the entropy associated with the
apparent horizon of a $(n+1)$-dimensional homogeneous and isotropic (Friedmann
Robertson Walker-like) Universe, assuming that the matter inside the horizon
is represented by a scalar field with a potential. In this setting, modified
Friedmann equations are derived from the first law of thermodynamics and
compared with the result of SheFr  for the specific case of $n=3$.
Furthermore, we investigate the early inflationary dynamics of Barrow
cosmology with the power-law potential function. Contrary to nonextensive
(Tsallis-like) scenario Keskin , where it has been shown that inflation may
consist of both slow roll- and kinetic-phases, here we find that only the
first stage is eligible, the kinetic energy era being incompatible with the
allowed values of Barrow exponent $\Delta$. After computing the characteristic
inflation parameters, we infer an upper bound on $\Delta$ in compliance with
recent observational constraints on the scalar spectral index and the tensor-
to-scalar ratio. We finally comment on the consistency of our results with
other approaches in literature aimed at exploring inflation driven by BHDE.

The remainder of the work is structured as follows: in the next Section, we
derive modified Friedmann equations from Barrow entropy. Sec. III is devoted
to to the study of the inflationary era in BHDE, while conclusions and outlook
are summarized in Sec. IV.

##  II Modified Friedmann equations in Barrow Cosmology

Let us consider a homogenous and isotropic Friedmann-Robertson-Walker (FRW)
Universe of spatial curvature $k$. We first set notation by following
Sheykhi:2021fwh  and focusing on $(3+1)$-dimensions. To be as general as
possible, the derivation of the modified Friedmann equations in Barrow
Cosmology is then performed for the $(n+1)$-dimensional case, with $n\geq 3$.

For a $(3+1)$-dimensional FRW Universe, the line element can be written as

$ds^{2}\,=\,h_{bc}\hskip
0.56905ptdx^{b}dx^{c}+\tilde{r}^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right),$
(2)

where we have denoted the metric of the $(1+1)$-dimensional subspace by
$h_{bc}=\mathrm{diag}[-1,a^{2}/(1-kr^{2})]$. Moreover, $x^{b}=(t,r)$,
$\tilde{r}=a(t)r$, $a(t)$ is the (time-dependent) scale factor and $r$ the
comoving radius.

Following Akbar2 , the dynamical apparent horizon is obtained from the
geometric condition

$h^{bc}\partial_{b}\tilde{r}\,\partial_{c}\tilde{r}=0\,.$ (3)

For the FRW Universe (2), explicit calculations yield

$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+{k}/{a^{2}}}}\,,$ (4)

where $H=\dot{a}(t)/a(t)$ is the Hubble parameter and the overhead dot
indicates derivative respect to the cosmic time $t$.

The apparent horizon has an associated temperature

$T=\,\frac{\kappa}{2\pi}=-\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right),$
(5)

where $\kappa$ represents the surface gravity. Clearly, for
$\dot{\tilde{r}}_{A}\leq 2H\tilde{r}_{A}$ we have $T\leq 0$. To avoid
meaningless negative temperatures, one can define $T=|\kappa|/2\pi$.
Furthermore, it is possible to assume that $\dot{\tilde{r}}_{A}\ll
2H\tilde{r}_{A}$ in an infinitesimal time interval $dt$, which amounts to
keeping the apparent horizon radius fixed. This implies the approximation
$T\simeq 1/{2\pi\tilde{r}_{A}}$ Cai:2005ra .

We now suppose that the matter content of the Universe is represented by a
scalar field $\phi$ characterized by a perfect fluid form. The corresponding
Lagrangian is given by $\mathcal{L}_{\phi}=X-V(\phi)$, where
$X=-\frac{1}{2}h^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ and $V(\phi)$
are the kinetic and (spatially homogenous) potential terms, respectively. In
turn, the stress-energy tensor is

$T_{\mu\nu}=(\rho_{\phi}+p_{\phi})u_{\mu}u_{\nu}+p_{\phi}\hskip
0.56905pth_{\mu\nu}\,,$ (6)

where $u_{\mu}$ is the four-velocity of the fluid and

$\rho_{\phi}=\frac{\dot{\phi}^{2}}{2}+V(\phi)\,,$ (7a)
$p_{\phi}=\frac{\dot{\phi}^{2}}{2}-V(\phi).$ (7b)

represent its energy density and pressure, respectively Alt8 . In turn, the
conservation equation $\nabla_{\mu}T^{\mu\nu}=0$ gives the continuity equation

$\dot{\rho}_{\phi}+3H(\rho_{\phi}+p_{\phi})=0\,.$ (8)

Combining Eqs. (7) and (8), we obtain the dynamics equation of the canonical
scalar field as

$\ddot{\phi}+3H\dot{\phi}+\partial_{\phi}V=0\,,$ (9)

where the term containing the Hubble constant serves as a kind of friction
term resulting from the expansion.

###  II.1 Modified Friedmann equations in $(n+1)$ dimensions

The above ingredients provide the basics to derive the modified Friedmann
equations in Barrow entropy-based Cosmology. Here, we extract such equations
from the first law of thermodynamics

$dE\,=\,T\hskip 0.56905ptdS+W\hskip 0.56905ptdV\,,$ (10)

applied to the apparent horizon of the FRW Universe in $(n+1)$-dimensions,
where

$W=(\rho_{\phi}-p_{\phi})/2\,,$ (11)

is the work density associated to the Universe expansion and

$S\,=\,\gamma\left(\frac{A}{A_{0}^{(n-1)/2}}\right)^{1+\Delta/2}\,,$ (12)

is the generalized Barrow entropy. We have denoted the $n$-dimensional horizon
surface by $A=n\Omega_{n}\tilde{r}_{A}^{n-1}$, where
$\Omega_{n}\,\equiv\,\frac{\pi^{n/2}}{\Gamma(n/2+1)}$ is the angular part of
the $n$-dimensional sphere and $\Gamma$ the Euler’s function. The
dimensionless constant $\gamma$ is such that $\gamma\rightarrow 1$ for $n=3$,
so that Eq. (1) is restored in this limit. Its explicit expression shall be
fixed later. In passing, we mention that an alternative derivation of modified
Friedmann equations can be built upon Padmanabhan’s paradigm of emergent
gravity Pad1 , which states that the spatial expansion of our Universe can be
understood as the consequence of the emergence of space with the progress of
cosmic time.

Now, by taking into account that the total energy of the Universe inside the
$n$-dimensional volume $V=\Omega_{n}\tilde{r}_{A}^{n}$ is $E=\rho_{\phi}V$, we
have

$dE=Vd\rho_{\phi}+\rho_{\phi}dV=\Omega_{n}\tilde{r}_{A}^{n}\hskip
0.56905pt\dot{\rho}_{\phi}\hskip 1.42262ptdt\,+\,\rho_{\phi}\hskip
1.13809pt\Omega_{n}n\tilde{r}_{A}^{n-1}d\tilde{r}_{A}\,.$ (13)

This relation can be further manipulated by resorting to the generalized
continuity equation

$\dot{\rho}_{\phi}+nH(\rho_{\phi}+p_{\phi})=0\,,$ (14)

to give

$dE\,=\,-\Omega_{n}\tilde{r}_{A}^{n}\hskip
0.56905ptnH\left(\rho_{\phi}+p_{\phi}\right)dt\,+\,\rho_{\phi}\hskip
1.13809pt\Omega_{n}n\tilde{r}_{A}^{n-1}d\tilde{r}_{A}\,.$ (15)

On the other hand, by differentiating the entropy (12) we get

$\displaystyle
dS=\gamma\left(\frac{1}{A_{0}^{(n-1)/2}}\right)^{1+\Delta/2}n\Omega_{n}\left(1+\frac{\Delta}{2}\right)\left(n-1\right)\left(n\Omega_{n}\hskip
0.56905pt\tilde{r}_{A}^{n-1}\right)^{\Delta/2}\tilde{r}_{A}^{n-2}\hskip
0.56905ptd\tilde{r}_{A}\,.$ (16)

By plugging Eqs. (13)-(16) into (10), we arrive to

$H\left(\rho_{\phi}+p_{\phi}\right)dt=\frac{\gamma\left(n-1\right)\left(1+\frac{\Delta}{2}\right)\left(n\Omega\tilde{r}_{A}^{n-1}\right)^{\Delta/2}}{2\pi\tilde{r}_{A}^{3}}\left(\frac{1}{A_{0}^{(n-1)/2}}\right)^{1+\Delta/2}\,dr\,.$
(17)

With the further use of the continuity equation (14), this becomes

$-\frac{2\pi\left(A_{0}^{(n-1)/2}\right)^{1+\Delta/2}}{\gamma\,n\left(n-1\right)(1+\frac{\Delta}{2})\left(n\Omega\right)^{\Delta/2}}\,d\rho_{\phi}\,=\,\tilde{r}_{A}^{(n-1)\Delta/2-3}d\tilde{r}_{A}\,.$
(18)

Integrating both sides, we are led to

$\tilde{r}_{A}^{(n-1)(1+\Delta/2)-n-1}=\frac{\pi\left[4-\left(n-1\right)\Delta\right]\left(A_{0}^{(n-1)/2}\right)^{1+\Delta/2}}{\gamma\hskip
1.13809ptn\left(n-1\right)\left(1+\frac{\Delta}{2}\right)\left(n\Omega\right)^{\Delta/2}}\,\rho_{\phi}\,,$
(19)

where the integration constant has been fixed by imposing the boundary
condition $8\pi\rho_{\phi}=\Lambda\simeq 0$. Finally, with the help of the
definition (4), we obtain

$\left(H^{2}+\frac{k}{a^{2}}\right)^{1-(n-1)\Delta/4}\,=\,\frac{8\pi\hskip
0.56905ptG_{eff}^{(n-1)/2}}{3}\hskip 0.56905pt\sigma\hskip
0.56905pt\rho_{\phi}\,,$ (20)

where we have defined

$\sigma\equiv\frac{3}{n-2}\,\frac{\left[n+1-\left(n-1\right)\left(1+\frac{\Delta}{2}\right)\right]}{\hskip
0.56905ptn\left(2-\Delta\right)}\,,$ (21)

and we have set

$\gamma\,=\,\frac{\pi^{(n-1)\Delta/4}}{2\left(n\Omega_{n}\right)^{\Delta/2}\,4^{(1+\Delta/2)(1-n)/2}}\,\frac{\left(n-2\right)}{\left(n-1\right)}\left(\frac{2-\Delta}{2+\Delta}\right)^{(3-n)/2}\,.$
(22)

Furthermore, we have introduced the effective gravitational constant
Sheykhi:2021fwh

$G_{eff}\,=\,\frac{A_{0}}{4}\left(\frac{2-\Delta}{2+\Delta}\right)\left(\frac{A_{0}}{4\pi}\right)^{\Delta/2}\,.$
(23)

Some comments are in order here: first, we notice that for $n=3$, we have
$\gamma\rightarrow 1$, consistently with the discussion below Eq. (12). The
same is true for $\sigma$, so that Eq. (20) for $n=3$ becomes

$\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}\,=\,\frac{8\pi
G_{eff}}{3}\hskip 0.56905pt\rho_{\phi}\,.$ (24)

This is nothing but the first modified Friedmann equation derived in
Sheykhi:2021fwh  when $\rho_{\phi}\equiv\rho$ (normal matter). Furthermore,
the limit $\Delta\rightarrow 0$ correctly reproduces the standard Friedmann
equation

$H^{2}+\frac{k}{a^{2}}\,=\,\frac{8\pi}{3}\hskip 0.56905pt\frac{A_{0}}{4}\hskip
0.56905pt\rho_{\phi}\,.$ (25)

As a final remark, it must be emphasized that, due to the positive definition
of the energy density, Eqs. (20) and (21) imply the upper bound

$n+1-\left(n-1\right)\left(1+\frac{\Delta}{2}\right)>0\,\,\,\,\Longrightarrow\,\,\,\,\Delta<\frac{4}{n-1}\,,$
(26)

which is obviously satisfied for any allowed value of $n$.

Now, from the time derivative of Eq. (24), one can easily obtain the second
modified Friedmann equation as follows

$2H\left[1-\left(n-1\right)\frac{\Delta}{4}\right]\left(H^{2}+\frac{k}{a^{2}}\right)^{-(n-1)\Delta/4}\left(\frac{\ddot{a}}{a}-H^{2}-\frac{k}{a^{2}}\right)=\frac{8\pi
G_{eff}^{(n-1)/2}}{3}\sigma\dot{\rho}_{\phi}\,.$ (27)

By use of the continuity equation (14), this gives

$\left[1-\left(n-1\right)\frac{\Delta}{4}\right]\left(H^{2}+\frac{k}{a^{2}}\right)^{-(n-1)\Delta/4}\left(\frac{\ddot{a}}{a}-H^{2}-\frac{k}{a^{2}}\right)=-\frac{4\pi
G_{eff}^{(n-1)/2}}{3}\,\sigma n\left(\rho_{\phi}+p_{\phi}\right).$ (28)

Replacing $\rho_{\phi}$ by the first Friedmann equation (20), we find after
some simplification

$\left[4-\left(n-1\right)\Delta\right]\frac{\ddot{a}}{a}\left(H^{2}+\frac{k}{a^{2}}\right)^{-(n-1)\Delta/4}+\left[2n-4+\Delta\left(n-1\right)\right]\left(H^{2}+\frac{k}{a^{2}}\right)^{1-(n-1)\Delta/4}=-\frac{16\pi
G_{eff}^{(n-1)/2}}{3}\,\sigma\hskip 0.56905ptn\hskip 0.56905ptp_{\phi}\,.$
(29)

This is the second modified Friedmann equation in Barrow Cosmology. Again, one
can check that $n=3$ gives back the result of Sheykhi:2021fwh

$\left(2-\Delta\right)\frac{\ddot{a}}{a}\left(H^{2}+\frac{k}{a^{2}}\right)^{-\Delta/2}+\left(1+\Delta\right)\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}\,=\,-8\pi
G_{eff}\,p_{\phi}\,.$ (30)

The further limit $\Delta\rightarrow 0$ reproduces the standard second
Friedmann equation, here rewritten as

$\dot{H}+H^{2}=-\frac{4\pi}{3}\left(\rho_{\phi}+3p_{\phi}\right),$ (31)

where we have used the relation

$\dot{H}=\frac{\ddot{a}}{a}-H^{2}\,.$ (32)

##  III Inflation in Barrow Cosmology

Let us now move onto the study of the inflationary era of the Universe. Within
the scalar theory framework considered above, the characteristic quantities to
compute are the inflation slow-roll parameters, which are defined by

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle-\frac{\dot{H}}{H^{2}}\,,$ (33) $\displaystyle\eta$
$\displaystyle=$ $\displaystyle-\frac{\ddot{H}}{2H\dot{H}}\,.$ (34)

Slow-roll conditions assert that both these two parameters take very small
values during inflation, i.e. $\epsilon,\eta\ll 1$. In the slow-roll
theoretical framework, the only requirement $\epsilon\ll 1$ is actually needed
to ensure the existence of an early inflationary era, Then, by imposing
$\dot{\phi}^{2},\ddot{\phi}\ll 1$ on the equation of motion of the theory, the
first Friedmann equation (20) under the slow-roll assumptions becomes

$H^{2}\simeq\left[\frac{8\pi G_{eff}}{3}\,V(\phi)\right]^{2/(2-\Delta)},$ (35)

where we have focused on the case $n=3$ and we have resorted to Eq. (7a).

On the other hand, from the second Friedmann equation (30) we get

$\dot{H}\simeq\frac{3\dot{\phi}^{2}}{2\left(\Delta-2\right)}\left(\frac{8\pi
G_{eff}}{3}\right)^{2/(2-\Delta)}V(\phi)^{\Delta/(2-\Delta)}\,.$ (36)

Combining Eqs. (35) and (36), the slow-roll parameters (33) and (34) take the
form

$\displaystyle\epsilon$ $\displaystyle\simeq$
$\displaystyle\frac{3\dot{\phi}^{2}}{2\left(2-\Delta\right)}V(\phi)^{-1}\,,$
(37) $\displaystyle\eta$ $\displaystyle\simeq$ $\displaystyle-\left(\frac{8\pi
G_{eff}}{3}V(\phi)\right)^{1/(\Delta-2)}\left[\frac{\ddot{\phi}}{\dot{\phi}}+\frac{\dot{\phi}\,\Delta}{4-2\Delta}\frac{\partial_{\phi}V(\phi)}{V(\phi)}\right].$
(38)

Let us now remark that the above parameters should be computed at horizon
crossing, where the fluctuations of the inflation field freeze Keskin .

The scalar spectral index of the primordial curvature perturbations and the
tensor-to-scalar ratio are defined by

$\displaystyle n_{s}$ $\displaystyle\simeq$ $\displaystyle
1-6\epsilon+2\eta\,,$ (39) $\displaystyle r$ $\displaystyle\simeq$
$\displaystyle 16\epsilon\,,$ (40)

respectively, which also need to be evaluated at the horizon crossing. For
later convenience, it is useful to introduce the e-folding time

$N=\int_{t_{i}}^{t_{f}}H(t)dt\,,$ (41)

where $t_{i}$ $(t_{f})$ represents the initial (final) time of the
inflationary era. Consistently with the above discussion, we consider
$t_{i}=t_{c}$ as the horizon crossing time, so that Eq. (41) can be rewritten
as $N=\int_{\phi_{c}}^{\phi_{f}}H{\dot{\phi}}^{-1}\,d\phi$, where we have used
the notation $\phi_{c}\equiv\phi(t_{c})$ and $\phi_{f}\equiv\phi(t_{f})$.

###  III.1 Slow-roll inflation with power-law potential

We now examine inflation from the dynamical point of view. Toward this end, we
assume a power-law behavior for the scalar potential $V(\phi)$ in the form

$V(\phi)\simeq\phi^{m}\,,$ (42)

where $m>0$ is the power-term. The latest observational data prefer models
with $m\sim\mathcal{O}(1)$ or $m\sim\mathcal{O}(10^{-1})$, while $m\geq 2$ is
disfavored in the minimally coupled scalar field. Henceforth, we shall focus
on such phenomenologically allowed values of $m$. We also remark that power
law inflation is a very useful model to assess approximation schemes for the
computation of scalar power spectra, since its spectrum is exactly
solvable222More generally, one can assume $V(\phi)=V_{0}\phi^{m}$, where
$V_{0}$ is a positive constant with dimensions of $[E]^{4-m}$. However, since
observational indices are shown to be independent of this quantity, we can set
$V_{0}=1$ in suitable units without loss of generality..

In order to extract analytical solutions of the inflationary observable
indices, we express $\dot{\phi}$ and $\ddot{\phi}$ in terms of the scalar
field by using the slow-roll conditions. In this regard, let us observe that
the evolution equation (9) can be rewritten as

$\dot{\phi}\simeq-\frac{1}{3H}\hskip 0.56905pt\partial_{\phi}V\,.$ (43)

By plugging into (35), we get

$\dot{\phi}=-\frac{m}{3}\hskip 0.56905pt\left(\frac{8\pi
G_{eff}}{3}\right)^{1/(\Delta-2)}\,\phi^{\left[\left(2-\Delta\right)\left(m-1\right)-m\right]/\left(2-\Delta\right)}\,.$
(44)

We can now derive the expression of $\phi_{f}$ by noticing that inflation is
supposed to end when $\epsilon(\phi_{f})\sim 1$. By inverting Eq. (37), we are
led to

$\phi_{f}=\left[\frac{6\left(2-\Delta\right)}{m^{2}}\hskip
0.56905pt\left(\frac{8\pi
G_{eff}}{3}\right)^{2/(2-\Delta)}\right]^{(2-\Delta)/[\Delta(2-m)-4]}\,.$ (45)

Similarly, insertion of Eqs. (35) and (44) into (41) allows to infer the
following expression for the scalar field at horizon crossing

$\phi_{c}=\Bigg{\\{}\frac{m}{3\left(2-\Delta\right)}\left(\frac{8\pi
G_{eff}}{3}\right)^{2/(\Delta-2)}\left\\{\frac{m}{2}+N\left[4+\Delta\left(m-2\right)\right]\right\\}\Bigg{\\}}^{(2-\Delta)/\left[4+\Delta(m-2)\right]}\,.$
(46)

The scalar spectral index (39) and the tensor-to-scalar ratio (40) can be cast
in terms of the power-term $m$ and the e-folding time $N$ as

$\displaystyle n_{s}$ $\displaystyle\simeq$ $\displaystyle
1-\frac{2\left[4+\Delta(m-2)+m\right]}{m\left\\{1+\frac{2N}{m}\left[4+\Delta(m-2)\right]\right\\}}\,,$
(47) $\displaystyle r$ $\displaystyle\simeq$
$\displaystyle\frac{16}{1+\frac{2N}{m}\left[4+\Delta(m-2)\right]}\,.$ (48)

Remarkably, we see that the slow-roll indices only depend on the power-term
$m$ and Barrow parameter $\Delta$. A similar result has been exhibited in the
context of Tsallis deformation of entropy-area law Keskin .

In order to constrain Barrow exponent $\Delta$, let us require consistency of
Eqs. (47) and (48) with observations. Specifically, we consider Planck 2018
measurements, which set the following bounds on $n_{s}$ and $r$ ConPlanck

$\displaystyle n_{s}$ $\displaystyle=$ $\displaystyle 0.9649\pm
0.0042\,\,\,\,(68\%\,\,\mathrm{CL})\,\,\,\,\mathrm{from\,\,Planck\,\,TT,TE,EE+lowE+lensing}\,,$
(49) $\displaystyle r$ $\displaystyle<$ $\displaystyle 0.064\hskip
51.21504pt(95\%\,\,\mathrm{CL})\,\,\,\,\mathrm{from\,\,Planck\,\,TT,TE,EE+lensing+lowEB}\,.$
(50)

For a sufficiently long inflation ($N\sim 30$ to $40$ e-folds) and
$m\sim\mathcal{O}(1)$, we obtain $\Delta\lesssim 10^{-4}$ (see also Fig. 1),
which improves the constraints $\Delta=0.0094^{+0.094}_{-0.101}$ and
$\Delta\lesssim 0.08$ obtained from Supernovae (SNIa) Pantheon sample
Anagnostopoulos:2020ctz ; Leon:2021wyx  and baryogenesis LucSar  measurements,
and is consistent with the most stringent bound $\Delta\lesssim 1.4\times
10^{-4}$ found so far through Big Bang Nucleosynthesis Barrow:2020kug . We
also observe that this bound is not improved by considering smaller power-
terms $m\sim\mathcal{O}(10^{-1})$.

Figure 1: Plot of $n_{s}$ (left panel) and $r$ (right panel) versus the power-
term $m$ and Barrow parameter $\Delta$ for $N\sim 30$.

###  III.2 Kinetic inflation with power-law potential

Above we have argued that the slow-roll inflation terminates when
$\epsilon\sim 1$. Two scenarios can then occur: either the scalar field
oscillates to the minimal value of the potential, leading the Universe into a
decelerated expansion phase, or the inflation goes on but with different
features. Here, we shall examine whether the latter possibility is allowed
within Barrow entropy-based Cosmology. In particular, a crucial assumption of
the slow-roll inflation is that the kinetic energy of the scalar field can be
neglected. However, if the volume of the Universe is large enough before the
field starts to oscillate, then a kinetic term might arise and drive a
transition from a vacuum state to quintessence. We assume the kinetic
contribute in the form

$\dot{\phi}^{2}=m\hskip 0.56905ptV(\phi)\,.$ (51)

The above expression can be actually deduced from the dynamics relation (9)
and the modified Friedmann equations (24) and (30), here rewritten for
convenience as

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\left(\frac{8\pi
G_{eff}}{3}\right)^{2/(2-\Delta)}\left[\frac{\dot{\phi}^{2}}{2}+V(\phi)\right]^{2/(2-\Delta)}\,,$
(52) $\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle\frac{3}{\Delta-2}\hskip 0.56905pt\dot{\phi}^{2}\left(\frac{8\pi
G_{eff}}{3}\right)^{2/(2-\Delta)}\left(\frac{\dot{\phi}^{2}}{2}+V\right)^{\Delta/(2-\Delta)}.$
(53)

These equations also allow us to express the slow-roll parameters as

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\frac{6m}{\left(2-\Delta\right)\left(m+2\right)}\,,$ (54)
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{m^{3/2}}{(\Delta-2)}\,\left(\frac{8\pi
G_{eff}}{3}\right)^{1/(\Delta-2)}\left(\frac{m+2}{2}\right)^{1/(\Delta-2)}\hskip
0.56905pt\phi^{\left[\Delta(2-m)-4\right]/\left[2(2-\Delta)\right]}\,.$ (55)

Now, the end of the kinetic inflation is set by the condition
$\eta(\phi_{f})\simeq 1$ Keskin , which gives from the definition (41)

$\phi_{c}=\left\\{\left(\frac{8\pi
G_{eff}}{3}\right)^{1/(\Delta-2)}\left(\frac{m+2}{2}\right)^{1/(\Delta-2)}\frac{m^{1/2}}{2(\Delta-2)}\left[2m+N\left[4+\Delta\left(m-2\right)\right]\right]\right\\}^{2(2-\Delta)/[4+\Delta(m-2)]}\,.$
(56)

From Eqs. (54) and (55), we then get (see also Fig. 2)

$\displaystyle n_{s}$ $\displaystyle=$
$\displaystyle\Bigg{\\{}1+4m\,\Bigg{\\{}\frac{9}{\left(m+2\right)\left(\Delta-2\right)}$
(57)
$\displaystyle+\,\frac{\left\\{\frac{m^{1/2}}{2(\Delta-2)}\left(\frac{m+2}{2}\right)^{1/(\Delta-2)}\left(\frac{8\pi
G_{eff}}{3}\right)^{1/(\Delta-2)}\left[2m+N\left[4+\Delta\left(m-2\right)\right]\right]\right\\}^{n\Delta/[2(\Delta-2)]}}{2m+N\left[4+\Delta\left(m-2\right)\right]}\Bigg{\\}}\Bigg{\\}}^{2(2-\Delta)/[4+\Delta(m-2)]},$
$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{96m}{\left(m+2\right)\left(2-\Delta\right)}\,.$ (58)

Unlike the previous scenario, we now find that observationally consistency is
obtained, provided that $\Delta$ assumes largely negative values. This occurs
for both $m\sim\mathcal{O}(1)$ and $m\sim\mathcal{O}(10^{-1})$, as it can be
easily seen from Eq. (58). However, such a condition is at odds with the
assumption (1), implying that a kinetic inflation could not be explained
within Barrow’s framework. This is a remarkable difference with the case of
inflation based on Tsallis entropy Keskin , which allows for kinetic phase
too. Specifically, in that case the kinetic inflation is associated to a
regime of decreasing horizon entropy and ensuing clumping of fluctuations in
particular regions of spacetime.

Figure 2: Plot of $r$ versus the power-term $m$ and Barrow parameter $\Delta$.

##  IV Discussion and Conclusions

Inspired by Covid-19 fractal structure, the modified entropy-area law (1) has
been proposed to take into account quantum gravitational effects on the black
hole horizon surface Barrow . In the lines of the gravity-thermodynamic
conjecture, this paradigm has been applied to the Universe horizon too, the
ensuing framework being known as Barrow Cosmology. Within this framework, we
have studied the evolution of FRW Universe, assuming the matter content to be
represented by a homogeneous scalar field in the form of a perfect fluid. As a
first step, by using the first law of thermodynamics applied to the horizon of
the FRW Universe, we have derived modified ($\Delta$-dependent) Friedmann
equations. The obtained result has been used to analyze the inflationary era.
Toward this end, we have supposed a power-law behavior for the scalar inflaton
field. We have found that inflation in Barrow Cosmology can consist of the
slow-roll phase only, the kinetic inflation being incompatible with the
allowed values of Barrow deformation parameter. We have finally constrained
Barrow exponent to $\Delta\lesssim 10^{-4}$ by demanding consistency of the
scalar spectral index and tensor-to-scalar ratio with recent observational
Planck data.

Other aspects deserve further analysis. Besides the background and
inflationary evolution, it would be interesting to study the growth rate of
matter density perturbations and structure formation. This is an important
testing ground to discriminate among existing modified cosmological models.
Preliminary investigation in this direction has been proposed in Perturb  in
the context of both Tsallis and Barrow entropies, showing that the entropic
deformation parameter significantly influences the growth of perturbations.
Moreover, one can attempt to extend the present considerations to Cosmology
based on Kaniadakis entropy Kan1 , which is a self-consistent relativistic
generalization of Boltzmann-Gibbs entropy with non-trivial cosmological
implications LucRev . In this way, a relationship between Barrow and
Kaniadakis formalisms can be established. Finally, since our models is an
effort to include quantum gravity corrections in the analysis of inflation, it
is essential to examine the obtained results in connection with predictions
from more fundamental theories of quantum gravity Addazi . Work along these
and other directions is under active consideration and will be presented
elsewhere.

Acknowledgements The author acknowledges the Spanish “Ministerio de
Universidades” for the awarded Maria Zambrano fellowship and funding received
from the European Union - NextGenerationEU. He is also grateful for
participation in the COST Action CA18108 “Quantum Gravity Phenomenology in the
Multimessenger Approach”.

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