# Reconstruction of $f(Q,T)$ Lagrangian for various cosmological scenario

Gaurav N. Gadbail 0000-0003-0684-9702<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Simran Arora 0000-0003-0326-8945
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of
Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

The variety of theories that can account for the dark energy phenomenon
encourages current research to concentrate on a more in-depth examination of
the potential impacts of modified gravity on both local and cosmic scales. We
discuss some cosmological reconstruction in $f(Q,T)$ cosmology (where $Q$ is
the non-metricity scalar, and $T$ is the trace of the energy-momentum tensor)
corresponding to the evolution background in Friedmann-Laîmatre-Robertson-
Walker (FLRW) universe. This helps us to determine how any FLRW cosmology can
arise from a specific $f(Q,T)$ theory. We use the reconstruction technique to
derive explicit forms of $f(Q,T)$ Lagrangian for the different kinds of matter
sources and Einstein’s static universe. We also formulate the models using
several ansatz forms of the $f(Q,T)$ function for $p=\omega\rho$. We
demonstrate that several classes of $f(Q,T)$ theories admit the power-law and
de-Sitter solutions in some ranges of $\omega$. Additionally, we reconstruct
the cosmological model for the scalar field with a specific form of $f(Q,T)$.
These new models with cosmological inspiration may impact gravitational
phenomena at other cosmological scales.

Keywords: $f(Q,T)$ gravity; Reconstruction; Perfect fluid; Power Law

##  I Introduction

General Relativity has been regarded as the geometry of spacetime since its
origin. This view has its foundation in the equivalence principle. Einstein
chose to define gravity as a result of the curvature of spacetime. Despite the
enormous success of Einstein’s General Relativity (GR), cosmic observations
introduced new challenges, such as dark energy and dark matter concerns. One
of the most important developments in cosmology over the last decade has been
the confrontation of observations with the standard $\Lambda$CDM, which led to
the discovery of an accelerating universe [1, 2, 3, 4, 5, 6]. Despite its
accomplishments, this motivated standard model has a severe flaw that stems
from the fact that there is a significant discrepancy between the value of
$\Lambda$ predicted by any quantum gravity theory and the observational value
[7, 8, 9, 10]. There are typically two ways to address these issues: modifying
the matter sector by including some additional dark components in the universe
energy budget and modifying general relativity. In addition to the
conventional curvature representation [11, 12], GR can be associated with
torsion or non-metricity. The former corresponds to the Teleparallel
Equivalent of GR (TEGR), whereas a Symmetric Teleparallel Equivalent of GR
(STEGR) can indeed be supported in flat and torsion-free spacetimes. Their
most straightforward modifications are the $f(\mathcal{T})$ theory [13, 14]
and $f(Q)$ theory [15, 16, 17, 18]. For instance, torsion is considered the
field, characterizing gravity in the so-called TEGR. This includes zero
curvature and non-metricity, with the Weitzenbock connection as the affine
connection. In this context, the virtual objects are tetrads from which the
affine connection, the torsion invariant, and, eventually, the field equations
can be derived. While geometry in STEGR has a non-metric connection with both
curvature and torsion at zero. Here, the non-metricity $Q$ geometrically
explains how a vector’s length changes in a parallel transport. Recently,
$f(Q)$ gravity has been thoughtfully considered mainly to explain the late-
time acceleration and dark energy concerns. For a review, one can check [19].
Furthermore, the $f(Q)$ gravity has been developed to incorporate a non-
minimal coupling in the Lagrangian, with the gravitational action generated by
an arbitrary function of the non-metricity $Q$ and the trace of energy-
momentum tensor $T$ [20]. Considerable work has been done in $f(Q,T)$ gravity
using simpler functional forms. Arora et al. demonstrated that $f(Q,T)$
gravity could explain the present cosmic acceleration and provide a feasible
solution to the dark energy issue [21, 22, 23]. Additionally, some quality
work based on various cosmological scenarios has been discussed in $f(Q,T)$
gravity theory [24].
On the other hand, the cosmological reconstruction approach has been designed
to precisely recover the properties of $\Lambda$CDM and know the expansion
history of the universe through modified theories of gravity. The complexity
of field equations, which makes it challenging to acquire exact and numerical
solutions that can be compared with observations, hinders studies of the
physics of such theories. In the reconstruction technique, it is believed that
the expansion history of the universe is understood precisely, and one inverts
the field equations to deduce which class of modified theory gives rise to a
given flat FRW model.
The cosmological reconstruction has been carried out in the framework of
$f(R)$ gravity under many scenarios [25, 26, 27, 28, 29, 30] to find realistic
cosmology that can represent the evolution of the matter-dominated era to the
DE phase. F. Esposito et al. [31] utilize reconstruction methods in $f(Q)$
gravity to examine precise isotropic and anisotropic cosmological solutions.
Exact power-law solutions in $f(G)$ gravity are found for a specific class of
models as demonstrated by Goheer et al. [32]. Furthermore, similar techniques
have been employed to model the cosmic evolution in accordance with the power-
law solutions, de Sitter universe, and phantom/non-phantom eras in different
extended theories [33, 34, 35, 36, 37].
We carry out several explicit reconstructions within the framework of $f(Q,T)$
gravity theory, yielding a variety of intriguing results. One noteworthy
approach here is to analyze known cosmic history and utilize field equations
to construct a specific type of Lagrangian that may reproduce the given
evolution background. First, we determine the real values of the Lagrangian
$f(Q,T)$ for various types of matter sources and Einstein’s static universe.
In fact, we discover the appropriate $f(Q,T)$ Lagrangian for modeling the
cosmic evolution by the power law and the de-Sitter solutions. We find the
gravity models that deviate from $\Lambda$CDM through the ansatz
$f(Q,T)=Q+F(T)$, with constant $Q$.
This paper is organized as follows. In section II, we present the general
$f(Q,T)$ gravity formalism with the Friedmann equations for the Friedman-
Lemaitre-Robertson-Walker universe. We reconstruct the $f(Q,T)$ Lagrangian for
different kinds of matter contributions in section III. By applying the
relation $p=\omega\rho$ and the specific ansatz forms of the $f(Q,T)$, we
reconstruct $f(Q,T)$ models in section IV. In section V, we go over the
prospect of obtaining gravitational Lagrangians in $f(Q,T)$ that are
appropriate for modeling the cosmic evolution implied by the power-law and the
de-sitter solutions. In section VI, we reconstruct the cosmological model for
the scalar field. Finally, the results are summarised in section VII.

##  II Field equations in $f(Q,T)$ Gravity

The $f(Q,T)$ theory of gravity which introduces an arbitrary function of
scalar non-metricity $Q$ and trace $T$ of the energy-momentum tensor, is an
intriguing modification to Einstein’s theory of gravity. The action of
$f(Q,T)$ theory coupled with matter Lagrangian $\mathcal{L}_{m}$ is given by
[20]

$S=\int\sqrt{-g}\left[\frac{1}{16\pi}f(Q,T)+\mathcal{L}_{m}\right]d^{4}x,$ (1)

where $g$ represents the determinant of $g_{\mu\nu}$. The non-metricity and
disformation tensor is defined as

$\displaystyle
Q\equiv-g^{\mu\nu}\left(L^{\alpha}_{\,\,\beta\mu}L^{\beta}_{\,\,\nu\alpha}-L^{\alpha}_{\,\,\beta\alpha}L^{\beta}_{\,\,\mu\nu}\right),$
(2) $\displaystyle
L^{\lambda}_{\,\,\,\,\mu\nu}=-\frac{1}{2}g^{\lambda\gamma}\left(\nabla_{\nu}g_{\mu\gamma}+\nabla_{\mu}g_{\gamma\nu}-\nabla_{\gamma}g_{\mu\nu}\right).$
(3)

The non-metricity tensor is defined as the covariant derivative of the metric
tensor, and its explicit form is

$Q_{\alpha\mu\nu}\equiv\nabla_{\alpha}g_{\mu\nu}.$ (4)

with the trace of a non-metricity tensor as

$Q_{\lambda}=Q_{\lambda\,\,\,\,\,\mu}^{\,\,\,\mu},\quad\quad\tilde{Q}_{\lambda}=Q^{\mu}_{\,\,\,\,\lambda\mu}.$

The Superpotential $P_{\,\,\mu\nu}^{\lambda}$ is defined as

$P_{\,\,\,\,\mu\nu}^{\lambda}=-\frac{1}{2}L^{\lambda}_{\,\,\,\,\mu\nu}+\frac{1}{4}\left(Q^{\lambda}-\tilde{Q^{\lambda}}\right)g_{\mu\nu}-\frac{1}{4}\delta^{\lambda}_{\,\,(\mu\,}Q_{\nu)},$
(5)

giving the relation of scalar nonmetricity as

$Q=-Q_{\lambda\mu\nu}P^{\lambda\mu\nu}.$ (6)

The field equations of $f(Q,T)$ theory by varying the action (1) with respect
to the metric tensor is obtained as

$-\frac{2}{\sqrt{-g}}\nabla_{\lambda}\left(f_{Q}\sqrt{-g}\,P^{\lambda}_{\,\,\,\,\mu\nu}\right)-\frac{1}{2}f\,g_{\mu\nu}+f_{T}\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)\\\
-f_{Q}\left(P_{\mu\lambda\alpha}Q_{\nu}^{\,\,\,\lambda\alpha}-2Q^{\lambda\alpha}_{\,\,\,\,\,\,\,\,\mu}\,P_{\lambda\alpha\nu}\right)=8\pi
T_{\mu\nu}.$ (7)

The terms used in the above are defined as

$\displaystyle\Theta_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g^{\alpha\beta}\frac{\delta T_{\alpha\beta}}{\delta g^{\mu\nu}},\quad
T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g^{\mu\nu}},$ (8) $\displaystyle f_{T}$ $\displaystyle=$
$\displaystyle\frac{\partial f(Q,T)}{\partial T},\quad f_{Q}=\frac{\partial
f(Q,T)}{\partial Q}.$ (9)

We are interested in exploring the cosmological consequences of the $f(Q,T)$
gravitational theory that is defined by a flat Friedmann-Lemaitre-Robertson-
Walker (FLRW) metric

$ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)\left(dx^{2}+dy^{2}+dz^{2}\right),$ (10)

where $N(t)$ is a Lapse function and $a(t)$ is a cosmic scale factor. Since
the coincident gauge is fixed during the diffeomorphism, we cannot choose a
specific Lapse function. However, the special case of $Q$ theories does allow
so because $Q$ retains a residual time reparameterization invariant. Hence,
one uses symmetry to set $N(t)=1$ [15, 16, 17]. By adopting the coincident
gauge, the covariant derivatives reduce to ordinary derivatives. If we adopt a
co-moving reference system, with $u^{\mu}=(1,0,0,0)$, then
${}^{\prime}=\frac{d}{d\tau}=\frac{d}{dt}$, $\tau$ is a proper time, giving
$H=\frac{\dot{a}(t)}{a(t)}$ as the Hubble parameter. For the above metric, the
non-metricity scalar is $Q=6H^{2}$. The trace of energy-momentum tensor is
given by $T=-\rho+3p$, where $p$ and $\rho$ are pressure and energy density,
respectively.
One can also mention that the dynamical evolution of massive particles is not
geodesics in $f(Q,T)$ [20], and an additional force does manifest itself as a
result of the coupling between $Q$ and $T$. The extra force is orthogonal to
matter four-velocity $u^{\mu}$ which is a prerequisite for a physical force.
The components that are orthogonal to the four-velocity of the particle can
contribute to the equation of motion which is a direct consequence of the co-
moving reference frame.

By substituting the above FLRW metric in Eq. (7) yields the two Friedmann
equations of $f(Q,T)$ theory

$\displaystyle\frac{f}{2}-6H^{2}f_{Q}=8\pi\rho+f_{T}(\rho+p),$ (11)
$\displaystyle\frac{f}{2}-2\left[\dot{f_{Q}}H+f_{Q}(\dot{H}+3H^{2})\right]=-8\pi
p.$ (12)

Here, dot (.) represent a derivative with respect to cosmic time $t$.
We can convert the cosmological evolution equations to a form that resembles
standard general relativity, by defining an effective pressure $p_{eff}$ and
an effective energy density $\rho_{eff}$ such that

$3H^{2}=8\pi\rho_{eff}=\frac{f}{4f_{Q}}-\frac{4\pi}{f_{Q}}\left[\left(1+\frac{f_{T}}{8\pi}\right)\rho+\frac{f_{T}}{8\pi}p\right],$
(13) $2\dot{H}+3H^{2}=-8\pi
p_{eff}=\frac{f}{4f_{Q}}-\frac{2\dot{f}_{Q}\,H}{f_{Q}}\\\
+\frac{4\pi}{f_{Q}}\left[\left(1+\frac{f_{T}}{8\pi}\right)\rho+\left(2+\frac{f_{T}}{8\pi}\right)p\right].$
(14)

Hence, follows the effective conservation equation [20]

$\dot{\rho}_{eff}=-3H(\rho_{eff}+p_{eff}).$ (15)

Using equations (13) and (14), we get the effective equation of state
parameter as

$\omega_{eff}=\frac{p_{eff}}{\rho_{eff}}=-1-\frac{\dot{H}}{H^{2}}$ (16)

##  III Reconstruction of $f(Q,T)$ cosmology

We shall now discuss the solutions of (11) and (12) pertinent from a
cosmological perspective. We aim to demonstrate that every cosmic epoch:
dominated by matter, radiation, or dark energy, may be developed in a model of
$f(Q,T)$ in the upcoming subsections. Adapting the approach of model
reconstruction, we consider the functional form of $f(Q,T)$ as

$f(Q,T)=f_{1}(Q)+f_{2}(T),$ (17)

where $f_{1}$ is a function of $Q$ and $f_{2}$ is a function of $T$.

The additive separable model above includes vastly different cosmological
limits, such as: STEGR ($f_{1}=Q$ and $f_{2}=0$), $\Lambda$CDM
($f_{1}+f_{2}=2\Lambda$), $f(Q)$ gravity ($f_{2}=0$), STEGR with a
modification ($f_{1}\neq Q$) allowing the $f_{1}(Q)$ and $f_{2}(T)$ function
to fully capture the behavior of the effective fluid component. This kind of
model has the advantage of yielding a decoupled system of ordinary
differential equations for the $f_{1}(Q)$ and $f_{2}(T)$ functions that are
simpler to solve.

###  III.1 Models for dust $(p=0)$

Here, the reconstruction for the dust fluid will be discussed. For this case,
the trace of energy-momentum tensor (EMT) is $T=-\rho$. Using Eq. (17) in Eq.
(11), we get

$\left(\frac{f_{1}}{2}-Qf_{1Q}\right)=-\left(\frac{f_{2}}{2}+8\pi
T+Tf_{2T}\right).$ (18)

One can observe that the left-hand side is entirely dependent on $Q$, whereas
the right-hand side is a function of $T$. So, due to in-dependency, both sides
must be equal to constant (say $\lambda$) using the separation of variables
approach. Hence, we have two ordinary differential equations

$\displaystyle\frac{f_{1}}{2}-Qf_{1Q}$ $\displaystyle=$
$\displaystyle\lambda,$ $\displaystyle\frac{f_{2}}{2}+8\pi T+Tf_{2T}$
$\displaystyle=$ $\displaystyle-\lambda.$

Solutions to the above differential equations are

$\displaystyle f_{1}(Q)$ $\displaystyle=$ $\displaystyle 2\lambda+c\sqrt{Q},$
(19) $\displaystyle f_{2}(T)$ $\displaystyle=$
$\displaystyle-2\lambda-\frac{16\pi T}{3}+\frac{c_{1}}{\sqrt{T}},$ (20)

where $c$ and $c_{1}$ are integrating constants. Substituting in the
functional form (17), we obtain

$f(Q,T)=c\sqrt{Q}-\frac{16\pi T}{3}+\frac{c_{1}}{\sqrt{T}}.$ (21)

One can deduce from the observation that the model in (21) is a real-valued
function of positive non-metricity scalar and the trace of EMT (and positive
energy density) if $c_{1}=0$. It implies the fact that there are classes of
real-valued functions $f(Q,T)=c\sqrt{Q}-\frac{16\pi T}{3}$ other than GR that
may simulate a dust-like expansion history.

###  III.2 Models for perfect fluid with EoS $p=-\frac{1}{3}\rho$

In this scenario, we reconstruct the $f(Q,T)$ Lagrangian for an accelerating
universe. The EoS value $\omega=-\frac{1}{3}$, which is near the limit of the
set of matter fields obeying the strong energy condition, is physically
intriguing. The trace of energy-momentum tensor (EMT) here becomes $T=-2\rho$.
Again Using Eq. (17) in Eq. (11), we have

$\left(\frac{f_{1}}{2}-Qf_{1Q}\right)=-\left(\frac{f_{2}}{2}+4\pi
T+\frac{T}{3}f_{2T}\right)$ (22)

The solution of the above differential equation (22) is

$\displaystyle f_{1}(Q)=2\mu+c\sqrt{Q},$ (23) $\displaystyle
f_{2}(T)=-2\mu-\frac{24\pi T}{5}+\frac{c_{2}}{T^{3/2}},$ (24)

where $c$ and $c_{2}$ are integrating constants. Here, $\mu$ is a separation
constant.
The obtained $f(Q,T)$ model is

$f(Q,T)=c\sqrt{Q}-\frac{24\pi T}{5}+\frac{c_{2}}{T^{3/2}}.$ (25)

One can deduce from the observation that the model in (25) is a real-valued
function of positive non-metricity scalar and the trace of EMT (and positive
energy density) if $c_{2}=0$. The obtained explicit form of the
$f(Q,T)=c\sqrt{Q}-\frac{24\pi T}{5}$ model is beneficial for studying the
acceleration scenario of the universe.

###  III.3 Models for $\Lambda$CDM $(p=-\rho)$

Let us now consider the standard case for an accelerating universe, i.e., the
$\Lambda$CDM. For this, we use a Friedmann equation (11), which reduces to

$\frac{f}{2}-Qf_{Q}=8\pi\rho_{\Lambda}.$ (26)

Further, one can rewrite the above equation in terms of density parameters,

$\frac{f}{2}-Qf_{Q}=4\pi Q\,\Omega_{\Lambda}.$ (27)

It can be observed that the solution of the differential equation (27) is
purely a function of $Q$.

$f(Q,T)=-8\pi Q\,\Omega_{\Lambda}+c_{3}\sqrt{Q},$ (28)

where $c_{3}$ is an integration constant. Consequently, the $\Lambda$CDM
expansion can be recreated in the modified gravity in the absence of a
cosmological constant. In this instance, the $f(Q,T)$ theory is limited to a
general relativity equivalent $f(Q)$ theory.

###  III.4 Einstein static universe in $f(Q,T)$ gravity

The Einstein static universe is described as $H=0$. Hence, by definition of
non-metricity, we have $Q=0$. We get the required Friedmann equation (11) as

$\frac{f}{2}-8\pi\rho-f_{T}(\rho+p)=0.$ (29)

The dust ($p=0$) results in the trace of EMT being $T=-\rho$, which simplifies
the above equation expressed as

$\frac{f}{2}+8\pi T+Tf_{T}=0.$ (30)

In general, we obtain the particular solution purely in $T$,

$f(Q,T)=-\frac{16\pi T}{3}+\frac{c_{4}}{\sqrt{T}}.$ (31)

where $c_{4}$ is an integration constant. For a real-valued solution, we have
$c_{4}=0$, i.e., $f(Q,T)=-\frac{16\pi T}{3}$.

##  IV Reconstruction of $f(Q,T)$ models for EoS $p=\omega\rho$

In this section, we reproduce $f(Q,T)$ models for the perfect fluid satisfying
$p=\omega\rho$, where $\omega$ is a constant. The trace of EMT is
$T=\rho(-1+3\omega)$. Here, we consider the particular ansatz forms of the
$f(Q,T)$ functional forms as follows:

  * A)

$g(Q)+h(T)$, B) $-Q+h(T)$,

  * C)

$Q\,h(T)$ , D) $T\,g(Q)$.

Model $B)$ is equivalent to general relativity theory as $f(Q,T)=-Q$ at
$h(T)=0$. The coupling possibilities in models $C)$ and $D)$ serve as
modification terms to the STEGR Lagrangian. Model $C)$ is fundamentally
different since it cannot recover STEGR and is non-trivial. The resultant
Lagrangian would therefore characterize cosmological processes without
requiring STEGR.

###  IV.1 $f(Q,T)=g(Q)+h(T)$

We apply additive separable models of $f(Q,T)$ to the Friedmann equation (11),
which yields a separable partial differential equation as

$\left(\frac{g}{2}-Q\,g_{Q}\right)=-\left(\frac{h}{2}-8\pi\rho-
h_{T}(\rho+p)\right).$ (32)

Here, $g_{Q}$ and $h_{T}$ represents derivatives with respect to $Q$ and $T$,
respectively.
Using the relation $\rho=\frac{T}{\left(-1+3\omega\right)}$, the above
equation can be written as

$\left(\frac{g}{2}-Q\,g_{Q}\right)=-\left(\frac{h}{2}-\frac{8\pi}{\left(-1+3\omega\right)}T-h_{T}\frac{1+\omega}{\left(-1+3\omega\right)}T\right)=\lambda.$
(33)

where $\lambda$ is a separating constant. The left-hand side and the right-
hand side are independent functions of $Q$ and $T$, respectively. Assuming
$\lambda\neq 0$, we note that we have a homogeneous differential equation with
the following analytical solutions:

$\displaystyle g(Q)$ $\displaystyle=$ $\displaystyle 2\lambda+c_{5}\sqrt{Q},$
(34) $\displaystyle h(T)$ $\displaystyle=$ $\displaystyle-\frac{16\pi
T}{3-\omega}-2\lambda+c_{6}\left[2T(1+\omega)\right]^{\frac{-1+3\omega}{2(1+\omega)}}.$
(35)

where $c_{5}$ and $c_{6}$ are integration constants and can be fixed by using
initial conditions on $g$ and $h$.
The obtained $f(Q,T)$ model is

$f(Q,T)=c_{5}\sqrt{Q}-\frac{16\pi
T}{3-\omega}+c_{6}\left[2T(1+\omega)\right]^{\frac{-1+3\omega}{2(1+\omega)}}.$
(36)

###  IV.2 $f(Q,T)=-Q+h(T)$

Now, we concentrate on a specific type of $f(Q,T)$ model for which the
Lagrangian is expressed as a minimal coupling between the non-metricity scalar
and a function of the trace of EMT. It is possible to reduce the Friedmann
equation (11) for this ansatz form of $f(Q,T)$ model.

$\frac{h(T)}{2}+\frac{Q}{2}=\frac{8\pi
T}{\left(-1+3\omega\right)}+\frac{1+\omega}{\left(-1+3\omega\right)}T\,h_{T}.$
(37)

Here, we treat $Q$ as a constant and obtain the solution of the above
differential equation.

$h(T)=-Q-\frac{16\pi
T}{3-\omega}+c_{7}\left[2T(1+\omega)\right]^{\frac{-1+3\omega}{2(1+\omega)}},$
(38)

where $c_{7}$ is an integration constant.
Consequently, the $f(Q,T)$ functional form is given by

$f(Q,T)=-2Q-\frac{16\pi
T}{3-\omega}+c_{7}\left[2T(1+\omega)\right]^{\frac{-1+3\omega}{2(1+\omega)}}.$
(39)

###  IV.3 $f(Q,T)=Qh(T)$

The following type of $f(Q,T)$ examines if a function $h(T)$ may satisfy a re-
scaling on $Q$. The equation for this paradigm using (11) is obtained as
below.

$-\frac{Q\,h(T)}{2}=\frac{8\pi
T}{\left(-1+3\omega\right)}+\frac{1+\omega}{\left(-1+3\omega\right)}Q\,T\,h_{T}$
(40)

In order to get the solution of equation (40), we treat $Q$ as a constant. The
analytical solution reads

$h(T)=-\frac{16\pi
T}{Q(1+5\omega)}+c_{8}\left[2T(1+\omega)\right]^{\frac{1-3\omega}{2(1+\omega)}},$
(41)

where $c_{8}$ is an integration constant. Therefore, the functional form of
$f(Q,T)$ becomes

$f(Q,T)=-\frac{16\pi
T}{(1+5\omega)}+c_{8}Q\left[2T(1+\omega)\right]^{\frac{1-3\omega}{2(1+\omega)}}$
(42)

###  IV.4 $f(Q,T)=T\,g(Q)$

We now explore the case of an arbitrary function of $Q$ fixing $T$ as a
scalar. In this case, we get the desired equation from (11) as

$\frac{T\,g(Q)}{2}-Q\,T\,g_{Q}=\frac{8\pi
T}{\left(-1+3\omega\right)}+\frac{1+\omega}{\left(-1+3\omega\right)}T\,g(Q).$
(43)

Here, considering $T\neq 0$ gives

$-Q\,g_{Q}+\frac{\omega-3}{2(-1+3\omega)}g(Q)=\frac{8\pi}{3\omega-1}.$ (44)

The general solution of (44) is obtained as

$g(Q)=\frac{16\pi}{\omega-3}+c_{9}\left[Q(2-6\omega)\right]^{\frac{\omega-3}{2-6\omega}},$
(45)

where $c_{9}$ is an integration constant.
The solution reproduces the following $f(Q,T)$ model.

$f(Q,T)=\frac{16\pi
T}{\omega-3}+c_{9}\,T\left[Q(2-6\omega)\right]^{\frac{\omega-3}{2-6\omega}}$
(46)

###  IV.5 Validity of the above models

We present two arguments $T(1+\omega)>0$ and $T(1+\omega)<0$ in order to
discuss the viability of Models (A), (B), and (C).
Here, $T(1+\omega)>0$ gives two conditions:

  * •

$T=\rho\left(-1+3\omega\right)>0$ and $1+\omega>0$ which gives
$\omega>\frac{1}{3}$ and $\omega>-1$, whose intersection gives
$\omega>\frac{1}{3}$ which may not be useful in investigating accelerating
scenario.

  * •

$T=\rho\left(-1+3\omega\right)<0$ and $1+\omega<0$ which gives
$\omega<\frac{1}{3}$ and $\omega>-1$, whose intersection gives $\omega<-1$,
which is viable to study the accelerated scenario.

Next, $T(1+\omega)<0$ gives two conditions:

  * •

$T=\rho\left(-1+3\omega\right)<0$ and $1+\omega>0$ which gives
$\omega<\frac{1}{3}$ and $\omega>-1$, whose intersection gives
$-1<\omega<\frac{1}{3}$ which can be studied for investigating accelerating
scenario only when $\frac{-1+3\omega}{2(1+\omega)}\in\mathcal{Z}$ .

  * •

$T=\rho\left(-1+3\omega\right)>0$ and $1+\omega<0$ gives $\omega>\frac{1}{3}$
and $\omega<-1$, which is not possible at the same time.

Model D), on the other hand, can be studied for cases:
$\left(2-6\omega\right)>0$ or $\left(2-6\omega\right)<0$ since $Q>0$. This
gives the model viability for either $\omega<\frac{1}{3}$ or
$\omega>\frac{1}{3}$ (with $\frac{\omega-3}{2-6\omega}\in\mathcal{Z}$, where
$\mathcal{Z}$ is the set of integer numbers), respectively.

##  V Cosmological solutions in $f(Q,T)$ gravity

In this section, we discuss the possibility of acquiring gravitational
Lagrangians $f(Q,T)$ that is suitable for simulating the cosmic evolution
presented by the power-law and the de-Sitter solutions.

###  V.1 Power law solutions

It would be fascinating to investigate the existence of exact power solutions
in $f(Q,T)$ gravity theory corresponding to various phases of cosmic
evolution. These solutions represent the decelerated and accelerated cosmic
eras, distinguished by the scale factor

$a(t)=a_{0}t^{m},\quad H(t)=\frac{m}{t}.\quad(m>0)$ (47)

It is known that the universe is viewed in its decelerated epoch for $0<m<1$.
Furthermore, the accelerated phase is experienced for $m>1$.

For the corresponding scale factor, the non-metricity scalar takes the form
$Q=6m^{2}t^{-2}$. Additionally, the conservation equation for
$p_{eff}=\omega_{eff}\,\,\rho_{eff}$ lead to

$\rho_{eff}(t)=\rho_{0}t^{-3m(1+\omega_{eff})}.$

Substituting Eq. (47) in (16), we get the effective EoS parameter as

$\omega_{eff}=\frac{(2-3\,m)}{3\,m}$

Then we may express the effective density $\rho_{eff}$ as a function of $Q$,

$\rho_{eff}(Q)=\rho_{0}\left(\frac{Q}{6\,m^{2}}\right).$ (48)

For the sake of simplicity, we suppose that the function can be written in the
following form

$f(Q,T)=f_{1}(Q)+f_{2}(T).$

In $f(Q,T)$ gravity, only effective thermodynamical quantities satisfy the
conservation equation, so we used the effective field equation (13). Using the
above effective quantities listed above, Eq. (13) becomes

$\displaystyle
8\pi\rho_{0}\left(\frac{Q}{6\,m^{2}}\right)f_{1Q}-\frac{f_{1}(Q)}{4}$
$\displaystyle=$ $\displaystyle K,$ (49)
$\displaystyle\frac{f_{2}(T)}{4}-4\pi\left[\left(1+\frac{f_{1T}}{8\pi}\right)\rho+\frac{f_{1T}}{8\pi}p\right]$
$\displaystyle=$ $\displaystyle K$ (50)

Here, $K\neq 0$ is a separating constant. The solution of the differential
equation (49) is

$f_{1}(Q)=-4\,K+\alpha_{0}\,Q^{\frac{3\,m^{2}}{16\pi\rho_{0}}}$ (51)

where $\alpha_{0}$ is an integration constant. In Eq. (50), real pressure $p$
and energy density $\rho$ are unknown quantities, so we cannot find the
analytical solution of differential equation (50). We investigate this
differential equation to obtain an analytical solution for three possible
cases: Dust, perfect fluid, and the $\Lambda$CDM.

  * •

For dust universe with $p=0$ and $\rho=-T$, equation (50) gives

$f_{2}(T)=4\,K-\frac{16\pi\,T}{3}+\frac{\alpha_{1}}{\sqrt{T}},$ (52)

where $\alpha_{1}$ is an integration constant.

  * •

For perfect fluid with pressure $p=-\frac{1}{3}\rho$ and $\rho=-T/2$, we get

$f_{2}(T)=4\,K-\frac{24\pi\,T}{5}+\frac{\alpha_{2}}{T^{3/2}},$ (53)

where $\alpha_{2}$ is an integration constant.

  * •

For $\Lambda$CDM case with $p=-\rho$ and $\rho=-T/4$, equation (50) gives the
solution

$f_{2}(T)=4\,K-4\pi\,T,$ (54)

Since the energy density in the present universe is positive, $T$ yields
negative values for models (52), (53) and (54). However, model (52) and (53)
take a complex value; therefore, it may be compatible with observations when
$\alpha_{1}$ and $\alpha_{2}$ vanish. The model (54), on the other hand, may
be consistent with cosmological observations.

###  V.2 de Sitter solutions

The interesting and well-known de-Sitter cosmic evolution effectively
describes the expansion of the universe. According to the idea, the universe
constantly expands as matter and radiation have a low energy density compared
to vacuum (energy density dark energy dominated epoch). The scale factor of
this evolutionary model increases exponentially with the constant Hubble
parameter $H(t)=H_{0}$ defined as

$a(t)=a_{0}\,e^{H_{0}t}.$

For the aforementioned scale factor, the non-metricity scalar is explicitly
written as $Q=6H_{0}^{2}$.
Again, using the same functional form $f(Q,T)=f_{1}(Q)+f_{2}(T)$, we have the
following expressions by using the Friedmann equation (11).

$\displaystyle\frac{f_{1}(Q)}{2}-Q_{0}f_{1Q}$ $\displaystyle=$
$\displaystyle\tilde{K},$
$\displaystyle\frac{f_{2}(T)}{2}-\frac{8\pi}{-1+3\omega}T-\left(\frac{1+\omega}{-1+3\omega}\right)Tf_{2T}$
$\displaystyle=$ $\displaystyle-\tilde{K}.$

We can directly solve the above equations for constant $\tilde{K}\neq 0$ to
get

$\displaystyle f_{1}(Q)$ $\displaystyle=$ $\displaystyle
2\tilde{K}+\alpha_{3}e^{\frac{Q}{2Q_{0}}},$ (55) $\displaystyle f_{2}(T)$
$\displaystyle=$ $\displaystyle-\frac{16\pi
T}{3-\omega}-2\tilde{K}+\alpha_{4}\left[2T(1+\omega)\right]^{\frac{-1+3\omega}{2(1+\omega)}}.$
(56)

The $f(Q,T)$ model becomes

$f(Q,T)=\alpha_{3}e^{\frac{Q}{2Q_{0}}}-\frac{16\pi
T}{3-\omega}+\alpha_{4}\left[2T(1+\omega)\right]^{\frac{-1+3\omega}{2(1+\omega)}}$
(57)

where $\alpha_{3}$ and $\alpha_{4}$ are integration constants. The model works
for the same conditions as discussed in section IV.

##  VI $f(Q,T)$ models for scalar field

The Lagrangian for a scalar field which is minimally coupled to the background
reads [38]

$L=-\frac{1}{2}\lambda\phi_{,\mu}\phi^{,\mu},$ (58)

Here $\lambda$ is a free parameter. The corresponding expression for the
stress-energy tensor is

$T_{\mu\nu}=-\frac{1}{2}\lambda\left(\phi_{,\mu}\phi_{,\mu}-\frac{1}{2}g_{\mu\nu}\phi_{,\alpha}\phi^{,\alpha}\right).$
(59)

Now, the expressions for $T$ and $\Theta$ read

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda\phi_{;\alpha}\phi^{;\alpha},$
$\displaystyle\Theta$ $\displaystyle=$
$\displaystyle-3\lambda\phi_{;\alpha}\phi^{;\alpha}.$

Eliminating the term $\phi_{;\alpha}\phi^{;\alpha}$ from the above two
equations, we have $\Theta=-6T$. Hence, we can write equation (7) as

$-\frac{2}{\sqrt{-g}}g^{\mu\nu}\nabla_{\lambda}\left(f_{Q}\sqrt{-g}\,P^{\lambda}_{\,\,\mu\nu}\right)-2f-5Tf_{T}\\\
-f_{Q}g^{\mu\nu}\left(P_{\mu\lambda\alpha}Q_{\nu}^{\,\,\,\lambda\alpha}-2Q_{\,\,\,\mu}^{\lambda\alpha}P_{\lambda\alpha\nu}\right)=8\pi
GT.$ (60)

We choose $f(Q,T)=Q+F(T)$ with a constant $Q$ to get a general solution

$f(Q,T)=Q-\frac{8\pi G}{7}T+\frac{C}{T^{\frac{2}{5}}},\quad T>0.$ (61)

where $C$ is an integrating constant.

##  VII Conclusion

Currently, one of the most considerable challenges is explaining late time
acceleration. The modified theory of gravity models is an intriguing approach
to describing such a late-time acceleration without inserting any exotic
matter component into the energy budget of the universe. Xu et al. [20]
proposed the generalization of the non-metric $f(Q)$ gravity theory by
introducing an arbitrary function of the non-metricity scalar $Q$ and the
trace of the energy-momentum tensor $T$. They examined the cosmological
evolution using three specific classes of the $f(Q,T)$ model. The specific
forms considered ignored the presence of specific mixed terms in the $QT$ or
its functions.
In this work, we presented the reconstruction of different classes of the
$f(Q,T)$ model for various cosmological scenarios. We obtained the $f(Q,T)$
Lagrangian for dust fluid, perfect fluid EoS $p=-\frac{1}{3}\rho$,
$\Lambda$CDM case and Einstein’s static universe using the additive ansatz
form of the $f(Q,T)$ Lagrangian. For consistent observational behavior, it is
noted that the explicit $f(Q,T)$ forms produced for all the cases in section
III contains the linear term $T$. In the $\Lambda$CDM scenario, the $f(Q,T)$
theory is reduced to the $f(Q)$ theory, while Einstein’s static universe is
reduced to a strictly $T$ dependent model.

It turns out that the barotropic EoS $p=\omega\rho$ follows the wider class of
theories using four different ansatzes $f(Q,T)$ forms as additive separable,
minimal coupling between the non-metricity scalar and a function of $T$, $Q$
re-scaling and $T$ re-scaling models. The additive separable model includes
vastly different cosmological models, such as STEGR ($g=Q$ and $h=0$),
$\Lambda$CDM ($g+h=2\Lambda$), $f(Q)$ gravity ($h=0$), STEGR with a
modification ($g\neq Q$) which enables the $g(Q)$ and $h(T)$ functions to
accurately represent the behavior of the effective fluid component. The second
model is an example of a successful theory of General Relativity that works
well when $h=0$. The $Q$ re-scaling is fundamentally different since it cannot
recover STEGR and is non-trivial. The resultant Lagrangian would therefore
characterize cosmological processes without requiring STEGR. The models in
section IV are viable for different ranges of $\omega$ as discussed.

In section (V), we proposed that possible cosmological trajectories for a
general $f(Q,T)$ may correspond to more complex precise solutions that
approximate scale like power-law. The expansion history of the universe is
believed to have experienced a stage of decelerated power-law expansion
followed by late-time acceleration. Therefore, power-law solutions play an
essential role in cosmology, as phases are dominated by matter and radiation
that eventually connect to an accelerating phase. The existence of the de-
Sitter solutions has also been investigated. Finally, we constructed the
$f(Q,T)$ Lagrangian for the scalar field using a specific form of $f(Q,T)$.
One can conclude that the reconstruction of the viable $f(Q,T)$ forms can be
tested through observational data and reproduce different cosmological
scenarios (like the background evolution of the universe and late-time
acceleration). Even one can study perturbation in future works that can behave
differently at the GR limit.

## Data Availability Statement

There are no new data associated with this article.

## Acknowledgments

GNG acknowledges University Grants Commission (UGC), New Delhi, India for
awarding Junior Research Fellowship (UGC-Ref. No.: 201610122060). SA
acknowledges BITS-Pilani, Hyderabad Campus, India for Institute Fellowship.
PKS thanks IUCAA, Pune, India for providing support through visiting
associateship program. We are very much grateful to the honorable referee and
to the editor for the illuminating suggestions that have significantly
improved our work in terms of research quality, and presentation.

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# Search for Neutrino Emission from GRB 221009A using the KM3NeT ARCA and ORCA
detectors

S. Aiello  A. Albert  M. Alshamsi  S. Alves Garre  A. Ambrosone  F. Ameli  M.
Andre  E. Androutsou  M. Anguita  L. Aphecetche  M. Ardid  S. Ardid  H. Atmani
J. Aublin  F. Badaracco  L. Bailly-Salins  Z. Bardačová  B. Baret  A. Bariego-
Quintana  S. Basegmez du Pree  Y. Becherini  M. Bendahman  F. Benfenati  M.
Benhassi  D. M. Benoit  E. Berbee  V. Bertin  S. Biagi  M. Boettcher  D.
Bonanno  J. Boumaaza  M. Bouta  M. Bouwhuis  C. Bozza  R. M. Bozza  H.Brânzaş
F. Bretaudeau  M. Breuhaus  R. Bruijn  J. Brunner  R. Bruno  E. Buis  R.
Buompane  J. Busto  B. Caiffi  D. Calvo  S. Campion  A. Capone  F. Carenini
V. Carretero  T. Cartraud  P. Castaldi  V. Cecchini  S. Celli  L. Cerisy  M.
Chabab  M. Chadolias  A. Chen  S. Cherubini  T. Chiarusi  M. Circella  R.
Cocimano  J. A. B. Coelho  A. Coleiro  A. Condorelli  R. Coniglione  P. Coyle
A. Creusot  G. Cuttone  R. Dallier  Y. Darras  A. De Benedittis  B. De Martino
V. Decoene  R. Del Burgo  I. Del Rosso  L. S. Di Mauro  I. Di Palma  A. F.
Díaz  C. Diaz  D. Diego-Tortosa  C. Distefano  A. Domi  C. Donzaud  D. Dornic
M. Dörr  E. Drakopoulou  D. Drouhin  J.-G. Ducoin  R. Dvornický  T. Eberl  E.
Eckerová  A. Eddymaoui  T. van Eeden  M. Eff  D. van Eijk  I. El Bojaddaini
S. El Hedri  A. Enzenhöfer  G. Ferrara  M. D. Filipović  F. Filippini  D.
Franciotti  L. A. Fusco  S. Gagliardini  T. Gal  J. García Méndez  A. Garcia
Soto  C. Gatius Oliver  N. Geißelbrecht  H. Ghaddari  L. Gialanella  B. K.
Gibson  E. Giorgio  I. Goos  P. Goswami  S. R. Gozzini  R. Gracia  K. Graf  C.
Guidi  B. Guillon  M. Gutiérrez  C. Haack  H. van Haren  A. Heijboer  A.
Hekalo  L. Hennig  J. J. Hernández-Rey  W. Idrissi Ibnsalih  G. Illuminati  D.
Joly  M. de Jong  P. de Jong  B. J. Jung  O. Kalekin  U. F. Katz  L.
Kharkhelauri  G. Kistauri  C. Kopper  A. Kouchner  V. Kueviakoe  V.
Kulikovskiy  R. Kvatadze  M. Labalme  R. Lahmann  G. Larosa  C. Lastoria  A.
Lazo  S. Le Stum  G. Lehaut  E. Leonora  N. Lessing  G. Levi  F. Longhitano
F. Magnani  J. Majumdar  L. Malerba  F. Mamedov  J. Mańczak  A. Manfreda  M.
Marconi  A. Margiotta  A. Marinelli  C. Markou  L. Martin  F. Marzaioli  M.
Mastrodicasa  S. Mastroianni  S. Miccichè  G. Miele  P. Migliozzi  E. Migneco
M. L. Mitsou  C. M. Mollo  L. Morales-Gallegos  G. Moretti  A. Moussa  I.
Mozun Mateo  R. Muller  M. R. Musone  M. Musumeci  S. Navas  A. Nayerhoda  C.
A. Nicolau  B. Nkosi  B. Ó Fearraigh  V. Oliviero  A. Orlando  E. Oukacha  D.
Paesani  J. Palacios González  G. Papalashvili  V. Parisi  E.J. Pastor Gomez
A. M. Păun  G. E. Păvălaş  I. Pelegris  S. Peña Martínez  M. Perrin-Terrin  J.
Perronnel  V. Pestel  R. Pestes  P. Piattelli  C. Poirè  V. Popa  T. Pradier
J. Prado  S. Pulvirenti  C.A. Quiroz-Rangel  U. Rahaman  N. Randazzo  S.
Razzaque  I. C. Rea  D. Real  G. Riccobene  J. Robinson  A. Romanov  A. Šaina
F. Salesa Greus  D. F. E. Samtleben  A. Sánchez Losa  S. Sanfilippo  M.
Sanguineti  C. Santonastaso  D. Santonocito  P. Sapienza  J. Schnabel  J.
Schumann  H. M. Schutte  J. Seneca  N. Sennan  B. Setter  I. Sgura  R.
Shanidze  A. Sharma  Y. Shitov  F. Šimkovic  A. Simonelli  A. Sinopoulou  M.V.
Smirnov  B. Spisso  M. Spurio  D. Stavropoulos  I. Štekl  M. Taiuti  R.
Tangorra-Cascione  Y. Tayalati  H. Thiersen  I. Tosta e Melo  E. Tragia  B.
Trocmé  V. Tsourapis  A.Tudorache  E. Tzamariudaki  A. Ukleja  A. Vacheret  A.
Valer Melchor  V. Valsecchi  V. Van Elewyck  G. Vannoye  G. Vasileiadis  F.
Vazquez de Sola  A. Veutro  S. Viola  D. Vivolo  J. Wilms  E. de Wolf  H.
Yepes-Ramirez  I. Yvon  G. Zarpapis  S. Zavatarelli  A. Zegarelli  D. Zito  J.
D. Zornoza  J. Zúñiga  N. Zywucka

###### Abstract

Gamma-ray bursts are promising candidate sources of high-energy astrophysical
neutrinos. The recent GRB 221009A event, identified as the brightest gamma-ray
burst ever detected, provides a unique opportunity to investigate hadronic
emissions involving neutrinos. The KM3NeT undersea neutrino detectors
participated in the worldwide follow-up effort triggered by the event,
searching for neutrino events. In this paper, we summarize subsequent
searches, in a wide energy range from MeV up to a few PeVs. No neutrino events
are found in any of the searches performed. Upper limits on the neutrino
emission associated with GRB 221009A are computed.

##  1 Introduction: GRB 221009A

On October 9, 2022, at 13:16:59.0 UT ($T_{0}$ hereafter) the Fermi satellite
detected an extraordinarily bright transient phenomenon by the Gamma-Ray Burst
Monitor (GBM) [1]. Shortly after that, at 14:10:17 UT, the Swift Burst Alert
Telescope (BAT) detected with better angular accuracy a transient event
consistent with the location of Fermi-GBM, with additional detection of
candidate counterparts by Swift-XRT and Swift-UVOT [2]. This transient
phenomenon was identified as a long Gamma-Ray Burst (GRB), whose most likely
origin is a supernova from the core collapse of a supermassive star [3].

The light curve measured by Fermi-GBM is composed of two consecutive emission
periods: a first single isolated peak followed by a longer multi-pulsed
episode in the 10–1000 keV energy range. The duration corresponding to 90% of
the central emission from this GRB ($T_{90}$ hereafter) is around $327$ s [4].
As the exceptional brightness of this event saturated the Fermi GBM and LAT
instruments during the main emission period, this value may be an
overestimation of the real $T_{90}$ [5].

The most accurate location for GRB 221009A has been provided by the Swift
spacecraft at R.A.(J2000) $=$ 288.26452∘ and DEC.(J2000) $=$ $+$ 19.77350∘,
with a $90\%$ confidence error radius of $0.61$ arcseconds [2]. This location
has been supported by the observations of multiple observatories [[6, 7, 8]].
The afterglow phase was detected and characterized in a wide frequency range,
with X-ray [9], optical [10, 11], and radio [12, 13] observations, in one of
the largest follow-up campaigns ever.

The LHAASO Water Cherenkov Detector Array detected photons from $200$ GeV up
to energies above $\,10$ TeV, the highest energy observed so far from a GRB
[14, 15]. Additionally, the Fermi-LAT satellite detected photons with energies
close to 99 GeV, which is to date the highest-energy detection by this
instrument for a GRB [16]. A GRB as extraordinary as GRB 221009A is expected
to occur only once every ten thousand years [17].

The exceptional intensity of this GRB may be explained by its proximity,
together with the fact that the jet emission is considered to be very
collimated [18]. A dedicated study suggests that the broadband afterglow
emission can indeed be explained by a broad structured jet [19]. The redshift
has been estimated to be $z=0.151$, following the optical observations of the
X-shooter VLT and the 10.4 m GTC telescopes [20, 3]. This corresponds to a
luminosity distance $d_{L}=745$ Mpc [21]. The isotropic-equivalent energy and
luminosity are estimated to be $10^{55}$ erg and $9.9\times 10^{53}$ erg s-1
respectively, which are among the highest values known for a GRB to date [5].

###  1.1 Neutrino emission from GRB 221009A

High-energy neutrinos (in the TeV–PeV energy range) are expected to be
generated through a wide variety of astrophysical processes in GRB events.
This is of particular interest in the case of GRB 221009A, as a recent study
[22] hinted towards a significant contribution of hadronic processes in its
multi-wavelength emission.

Current models of the prompt emission from a GRB describe it as a high-
temperature compacted plasma whose kinetic energy can be released through
internal shocks. These shocks can accelerate protons and other atomic nuclei
from the relativistic outflow, which may produce TeV–PeV neutrinos through
photo-meson production from the interaction with surrounding photon fields
[23]. Currently, this is expected to be the main mechanism for neutrino
production [24]. Nevertheless, there are still many other models proposed that
could explain neutrino emissions from GRBs such as subphotospheric neutrino
production [25], neutrino eruptions in jet propagation processes [26], regular
core-collapse processes [27] or even detectable neutrinos from neutron
collisions [28].

The KM3NeT Collaboration performed a real-time search for neutrino emission,
reporting a non-detection of candidate events during the [$T_{0}-50$ s,
$T_{0}+5000$ s] time window [29].

The IceCube Collaboration also reported a non-detection of neutrinos coming
from GRB 221009A [30]. The real-time analyses, conducted using the time
windows [$T_{0}$ $-$ $1$ hr, $T_{0}$ $+$ $2$ hr] and $T_{0}$ $\pm$ $1$ day,
were based on a Fast Response Analysis (FRA) framework optimized for TeV–PeV
neutrino detection. Refined upper limits in the time-integrated energy-scaled
neutrino emission from GRB 221009A were further reported [31]. The Baikal-GVD
Collaboration also performed an independent neutrino search with no
significant detection [32].

In this paper, the results of refined searches in the KM3NeT detectors for a
neutrino signal compatible with the location of GRB 221009A are shown,
covering various time windows. The data were analyzed with improved
calibrations with respect to the real-time search and considering systematic
effects such as the angular uncertainty in the reconstruction of the events.

##  2 The KM3NeT neutrino detectors

KM3NeT [33] is an international research collaboration currently deploying two
deep-sea neutrino telescopes in the Mediterranean Sea: ARCA (Astroparticle
Research with Cosmics in the Abyss) and ORCA (Oscillation Research with
Cosmics in the Abyss). These detectors consist of three-dimensional arrays
capable of detecting the Cherenkov light emission induced by the motion of
relativistic charged particles resulting from neutrino interactions.

The active component of the KM3NeT detectors is the Digital Optical Module
(DOM), a pressure-resistant glass sphere housing 31 Photomultiplier Tubes
(PMTs) [34]. The DOMs are embedded in vertical strings called Detection Units
(DUs), with 18 DOMs incorporated in each DU.

ORCA is located 40 km from Toulon (France) at a depth of 2.5 km. The distance
between DOMs is optimized for detecting neutrino events in the GeV energy
range. Although the main physics goal of ORCA is the study of neutrino
oscillation properties by the detection of atmospheric neutrinos, the energy
range covered makes it a well-suited instrument to perform GeV neutrino
astronomy as well.

ARCA, a larger array that will instrument about 1 km3 of deep-sea water, is
located 100 km offshore from Capo Passero (Sicily, Italy) at a depth of 3.5
km. The separation between DOMs makes this detector sensitive to neutrino
events from sub-TeV energies up to a few PeV. The main physics goal of ARCA is
the study of high-energy neutrinos originating from astrophysical sources. MeV
neutrinos can also be detected in ARCA and ORCA by looking for an increase in
the rate of PMT coincidences in each DOM [35, 36].

The high duty cycle of the detectors above 95%, the energy coverage from MeV
to PeV, the full-sky coverage and the angular resolution below one degree for
energies above 10 TeV for the searches performed in this work, turn the KM3NeT
detectors into suitable instruments to perform multi-messenger studies.

The final detector configurations will comprise 115 DUs for ORCA and 230 DUs
for ARCA. When GRB 221009A took place, ARCA was operated with 21 DUs, while
ORCA had 10 DUs in operation.

##  3 Search methods and results

The methods and results of the different refined searches performed using the
KM3NeT detectors are presented in this section. A description of the analysis
method employed in the GeV–PeV energy range is provided in Section 3.1. The
results for the TeV–PeV analyses, using ARCA data, are presented in Section
3.2. The GeV–TeV neutrino searches, using ORCA data, are detailed in Section
3.3. Finally, the method and results in the MeV range are presented in Section
3.4.

###  3.1 Search method: GeV–PeV neutrinos

At $T_{0}$, the location of GRB 221009A was above the horizon in the local sky
of the KM3NeT detectors, as shown in Figure 1. Therefore, neutrinos coming
from the direction of this source are reconstructed as downgoing events. The
downgoing data sample used in this analysis is composed of events
reconstructed with a cosine of the zenith angle larger than $0.1$. This part
of the sky is dominated by atmospheric muon events, and strict event
selections must be applied to reduce this background.

Figure 1: Skymap with the position of GRB 221009A in equatorial coordinates
(J2000). The green shadowed region denotes the instantaneous visibility for
upgoing events of ARCA at $T_{0}$. In the case of ORCA, the visibility region
is similar.

Two time windows, shorter than 24 hours, are considered: one for [$T_{0}$,
$T_{90}$], and another longer one for [$T_{0}$ $-$ $50$ s, $T_{0}$ $+$ $5000$
s], the latter covering the high-energy emission seen by Fermi-LAT [16] and
LHAASO [14]. In these two time windows, the GRB remains above the horizon.
Furthermore, two searches are conducted in the $T_{0}\pm 1$ day range: one
using only events reconstructed as upgoing, and another one employing events
reconstructed as downgoing. The search using upgoing events is motivated by
the fact that the location of GRB 221009A in the local detector sky is below
the horizon for about 45% of one day.

For events with reconstructed energy above a few GeV in ORCA and a few hundred
GeV in ARCA, two main topologies can be identified: track-like events, where
the Cherenkov PMT pulses (hits) are compatible with a straight line, and
cascade-like events, where the spatial distribution of the hits is compatible
with a quasi-spherical light emission. These topologies are associated with
the flavor of the interacting neutrino and with the kind of interaction.
Track-like signatures primarily arise from the charged-current interactions of
muon neutrinos and some charged-current interactions of tau neutrinos.
Cascade-like topologies arise from the interactions of the neutrinos of the
remaining flavors and neutral current muon neutrino interactions [37]. In the
analyses described in this paper, only track-like events are considered,
because of their superior angular resolution.

The search method is based on a binned ON/OFF technique [38]. The ON region is
defined as the region of the sky where a signal is expected. The OFF region
corresponds to a fraction of the sky where the background level is comparable
to the ON region but only background events are expected. As ON region, a
circular area centered on the GRB position, i.e. the Region of Interest (RoI)
is used. For the OFF region, declination bands are considered for searches
over time windows longer than one day, where the atmospheric background
depends only on the declination due to the Earth’s rotation. For shorter time
windows, elevation bands are used as the OFF region to account for the
dependence of the background on the local sky. In both cases, the number of
events in the OFF region is re-scaled to the solid angle and to the time
covered by the ON region.

The event selections are determined to obtain a background level such that the
detection of one event in the ON region is sufficient to obtain a $3\sigma$
excess above the background-only hypothesis. Using a $2$-sided convention,
this corresponds to an expected background lower or equal to $2.7\times
10^{-3}$ events. This optimization takes into account the performance of the
detector through two instrument response functions: the Point Spread Function
(PSF) and the detector effective area.

The PSF describes the dispersion of a signal coming from the direction of the
GRB due to the angular resolution of the detector. Systematic uncertainties
are included in the PSF, mainly coming from the determination of the absolute
orientation of the detectors and the geometrical shape of the detection units
as reconstructed through acoustic triangulation methodologies. The resulting
median of the PSF for the data sample considered in this work is estimated at
$0.8^{\circ}$ for ARCA and at 1.2∘ for ORCA, for an $E^{-2}$ neutrino
spectrum.

The detector effective area $A_{\rm eff}^{\delta}(E)$ at a declination
$\delta$ is defined as the quantity that provides the number of expected
signal events $N_{\rm s}$ when convoluted with the differential neutrino flux
$\Phi(E)$,

$N_{\rm s}=\int dt\int dE\cdot\Phi(E)\cdot A_{\rm eff}^{\delta}(E).$ (3.1)

The instrument response functions are computed using dedicated Monte Carlo
simulations of neutrino interactions generated with the gSeaGen software [39]
in the energy range between 1 GeV and 10 TeV for ORCA and from 100 GeV to 100
PeV for ARCA. The systematic uncertainty on the detector effective area is
conservatively estimated to be 30%, mainly due to the uncertainties on the
properties of the seawater medium and of the PMT, such as the light absorption
length or the PMT quantum efficiency [40].

A differential neutrino flux $\Phi(E)=\Phi_{0}(E/E_{0})^{-\gamma}$, with
spectral index $\gamma=2$ and $E_{0}=1$ GeV, is considered in all searches.
The normalization factor $\Phi_{0}$ is left as a free parameter in order to
determine its value or an upper limit in the non-detection case. The shape
assumed for the neutrino spectrum corresponds to the one expected from hadrons
accelerated by a second-order Fermi mechanism in strong shocks [41, 42].

###  3.2 Results of the TeV–PeV neutrino searches with the ARCA detector

A dataset with approximately 70 days of livetime is used to estimate the
expected background in the ARCA searches. The selection criteria are based on
the quality of the event reconstruction, the estimated angular uncertainty of
the event, the number of hits used in the reconstruction, and the estimated
length of the reconstructed track. In the analyses of downgoing tracks the
estimated energy of the events is also considered. The results for all
searches are summarized in Table 1.

SEARCH | KM3NeT $90\%$ CL upper limits on neutrino emission from GRB 221009A
---|---
ARCA (TeV–PeV) | Results for neutrino flux $\Phi(E)=\Phi_{0}(E/E_{0})^{-2}$ at $E_{0}=1$ GeV
| RoI
---
radius
| Expected
---
background
$(\times\,10^{-3})$
|  $\Phi_{0}$ UL
---
[GeV-1 cm-2 s-1]
| $E_{\rm min}$
---
[TeV]
| $E_{\rm max}$
---
[PeV]
| Fluence $\mathcal{F}$ UL
---
[GeV cm-2]
|  $E^{2}F(E)$ UL
---
[GeV cm-2]
$T_{90}$ | $2.1^{\circ}$ | $2.64\pm 0.02$ | $2.5\times 10^{-3}$ | $34$ | $13$ | 4.9 | $0.83$
$T_{0}$[$-50$ s, $+5000$ s] | $1.1^{\circ}$ | $2.53\pm 0.04$ | $2.8\times 10^{-4}$ | $110$ | $27$ | 7.9 | 1.4
$T_{0}\pm$1d downgoing | $1.0^{\circ}$ | $2.6\pm 0.1$ | $2.5\times 10^{-5}$ | $220$ | $36$ | 22 | 4.4
$T_{0}\pm$1d upgoing | $1.7^{\circ}$ | $2.7\pm 0.2$ | $6.2\times 10^{-6}$ | $8.1$ | $7.7$ | 7.4 | 1.1
ORCA (GeV–TeV) | Results for neutrino flux $\Phi(E)=\Phi_{0}(E/E_{0})^{-2}$ at $E_{0}=1$ GeV
| RoI
---
radius
| Expected
---
background
$(\times\,10^{-3})$
|  $\Phi_{0}$ UL
---
[GeV-1 cm-2 s-1]
| $E_{\rm min}$
---
[GeV]
| $E_{\rm max}$
---
[TeV]
| Fluence $\mathcal{F}$ UL
---
[GeV cm-2]
|  $E^{2}F(E)$ UL
---
[GeV cm-2]
$T_{90}$ | $2.0^{\circ}$ | $2.61\pm 0.04$ | $14$ | $150$ | $9.1$ | $1.9\times 10^{4}$ | $4.5\times 10^{3}$
$T_{0}$[$-50$ s, $+5000$ s] | $5.4^{\circ}$ | $2.6\pm 0.2$ | 1.9 | $54$ | $8.7$ | $4.9\times 10^{4}$ | $9.6\times 10^{3}$
$T_{0}\pm$1d downgoing | $1.0^{\circ}$ | $2.7\pm 0.3$ | $1.0\times 10^{-2}$ | $68$ | $8.8$ | $8.5\times 10^{3}$ | $1.7\times 10^{3}$
$T_{0}\pm$1d upgoing | $1.2^{\circ}$ | $2.7\pm 0.3$ | $4.7\times 10^{-4}$ | $130$ | $9.8$ | $3.5\times 10^{2}$ | $81$
MeV search | Results for quasi-thermal neutrino flux $F_{\bar{\nu}_{e}}(E)\propto E^{2}\text{exp}(-3E/\langle E\rangle)$ at $\langle E\rangle=15$ MeV
| Maximum
---
coincidence
level
| Expected
---
background
$p$-value |  | $E_{\rm min}$
---
[MeV]
| $E_{\rm max}$
---
[MeV]
| Total $\bar{\nu}_{e}$ flux
---
[cm-2]
| $E_{\rm tot,\nu}^{{\rm iso,}90\%}$
---
[erg]
$T_{90}$ | 27 | 29 | 0.99 | 5 | 30 | $2.5\times 10^{9}$ | $5.1\times 10^{62}$
$T_{0}$[$-50$ s, $+5000$ s] | 32 | 33 | 0.79 | $4.8\times 10^{9}$ | $9.7\times 10^{62}$
Table 1: $90\%$ CL upper limits on the neutrino emission from GRB 221009A for
the different time windows studied in each data sample. $E_{\rm min}$ and
$E_{\rm max}$ correspond respectively to the 5% and 95% quantiles in the
energy range of the expected neutrino flux for each search. For the GeV to PeV
analyses, the searches for the $T_{90}$ and $T_{0}$[$-50$ s, $+5000$ s] time
windows are only downgoing. The expected background level for each search is
also presented.

For the $T_{0}$ $\pm$ $1$ day searches, the RoI radius obtained for the
upgoing search is larger than in the case of the downgoing study, as expected
due to the lower atmospheric background. In the searches for downgoing
neutrino candidates, the RoI radius increases as the time window duration
decreases, which is expected since the use of shorter windows reduces the
background.

No candidate neutrino event coming from the ON region around GRB 221009A
direction is found in any of the time windows. Upper limits (UL) on the
neutrino emission from GRB 221009A are evaluated, taking into account the
instrument response functions of the current detector configurations.

The upper limit on the flux normalization factor $\Phi_{0}$ is determined as

$\Phi_{0}^{\rm UL}=\frac{\mu_{90}^{\rm FC}(n_{b})}{\int dt\int
dE\cdot\left(E/E_{0}\right)^{-\gamma}\cdot A_{\rm eff}^{\delta}(E)},$ (3.2)

where the quantity $\mu_{90}^{\rm FC}(n_{b})$ denotes the 90% confidence level
upper limit on the number of events according to the Feldman-Cousins method
[43], and $A_{\rm eff}^{\delta}(E)$ is the detector effective area at the
declination of GRB 221009A.

The upper limit on the radiant fluence, defined as the per-flavor neutrino
flux integrated in energy and time, over a given search time window is

$\mathcal{F}^{\rm UL}=\Delta T\int_{E_{\rm min}}^{E_{\rm max}}dE\cdot
E\cdot\Phi_{0}^{\rm UL}\cdot\left(\frac{E}{E_{0}}\right)^{-\gamma},$ (3.3)

where $\Delta T$ is the emission period considered, and $E_{\rm min}$ and
$E_{\rm max}$ correspond to the 5% and 95% quantiles of the energy range for
the neutrino flux [44].

The upper limit on the energy-scaled time-integrated neutrino flux, defined as
$E^{2}F(E)=\Delta T\cdot\Phi_{0}^{UL}\cdot E_{0}^{2}$, was computed in order
to perform a comparison with the upper limits reported by the IceCube
Collaboration [31].

###  3.3 Results of the GeV–TeV neutrino searches with the ORCA detector

A dataset with $\sim\,$41 days of livetime is used in the ORCA searches to
estimate the expected number of background events. The same four time windows
as for the ARCA analyses are inspected. The main difference comes from the use
of a machine learning classification algorithm [45] in the event selection,
which aims to reject atmospheric muons. The classifier is introduced
considering the larger impact of the atmospheric muon background in the ORCA
analyses, due to the lower depth of the site. The event selection optimization
procedure was applied to determine the best classification score and RoI
radius for each considered time window.

No candidate neutrino event is found in coincidence with the position of GRB
221009A in any of the searches. As in the case of ARCA, upper limits on the
neutrino flux normalization factor, time-integrated energy-scaled flux, and
time-integrated energy-integrated flux are determined. The results are
provided in Table 1.

###  3.4 Method and results for MeV neutrinos

The KM3NeT detectors can identify a burst of electron anti-neutrinos in the
5–30 MeV energy range through the observation of a global increase in the rate
of coincidences between PMTs in single DOMs. This method was developed to look
for neutrinos from core-collapse supernovae [35, 36]. A coincidence is defined
as at least four hits (PMT voltage above a given threshold) within 10 ns
inside single DOMs for PMTs within 90 degrees of each other. Coincidences are
aggregated in sliding windows of 500 ms computed every 100 ms independently
for each detector before being summed together. This number of coincidences is
referred to as the coincidence level.

The search for MeV neutrinos is made considering two time windows: [$T_{0}$,
$T_{90}$] and [$T_{0}-50$ s, $T_{0}+5000$ s]. To compute the significance and
associated p-value, the maximum coincidence level is evaluated for each time
window and compared to the expected background, obtained using 30 days of data
around $T_{0}$.

The 90% confidence level upper limits on the number of coincidences due to a
neutrino signal is computed following the Feldman-Cousins method [46]. This
quantity is used to compute the upper limit on the total time-integrated
neutrino flux and on the total energy emitted in isotropically distributed MeV
neutrinos by the source $E_{\rm tot,\nu}^{{\rm iso,}90\%}$ taking into account
the inferred GRB distance. These upper limits are computed assuming a quasi-
thermal emission of electron anti-neutrinos. The results are presented in
Table 1.

##  4 Discussion

In Figure 2 the main results of this work are presented and compared with the
upper limits derived by the IceCube Collaboration [31]. As a reference the
observations of the photon flux by Fermi-GBM [5], Fermi-LAT [16], and LHAASO
[14] are included.

Figure 2:  $90\%$ CL upper limits on $E^{2}F(E)$, the energy-scaled time-
integrated per flavor neutrino flux from GRB 221009A, for KM3NeT/ORCA (red
lines) and KM3NeT/ARCA (blue lines). The results from IceCube, taken from
[31], are also shown (green lines). Only the results for the $T_{0}\pm 1$ day
and $T_{90}$ searches are included, using a neutrino spectral index
$\gamma=2$, as they are the most relevant ones. For visualization purposes,
the gamma-ray observations are also included (gray dashed lines), from Fermi-
GBM [5] (section 8), Fermi-LAT [16], and LHAASO [14] (Table S2, assuming an
intrinsic spectrum and standard EBL). The right axis indicates the
differential isotropic equivalent energy.

The right $y$-axis of Figure 2 denotes the differential isotropic equivalent
energy $Ed\mathcal{E}_{\rm iso}/dE$. The upper limit on $E^{2}F(E)$ is
translated into an upper limit on $Ed\mathcal{E}_{\rm iso}/dE$ using the
relation $E^{2}F(E)=Ed\mathcal{E}_{\rm iso}/dE\times(1+z)/(4\pi d_{L}^{2})$,
accounting for the redshift $z$ and the luminosity distance $d_{L}$ [31].

In the case of the ARCA detector, the most restrictive upper limit on the
neutrino flux normalization $\Phi_{0}$ is provided by the searches for upgoing
neutrino candidates, see Table 1. This can be explained by considering the
lower atmospheric muon contamination in the upgoing sky. Instead, for the case
of $E^{2}F(E)$ the most restrictive value is provided by the $T_{90}$ time
window, as shown in Figure 2. This is expected since $E^{2}F(E)$ is a time-
integrated quantity, which favors shorter time window searches.

Although the same argument applies in ORCA, due to the larger atmospheric muon
contamination in the ORCA energy range, the performance of the downgoing
searches for this detector is a factor 20 worse than for the upgoing sky, as
can be seen in Table 1.

The limits presented in this paper are not as restrictive as the ones provided
by the IceCube Collaboration. This can be explained by considering the
position of the GRB in the local sky of the KM3NeT detectors, which happened
to be above the horizon at the time of the event, in contrast to the case of
IceCube. Furthermore, the current partial configuration ($\sim$ 10% of the
full configuration) of the ARCA and ORCA telescopes results in a limited
effective area.

##  5 Conclusions

Searches for a neutrino signal coming from GRB 221009A have been performed
with the KM3NeT ARCA and ORCA detectors. No candidate neutrino events were
found in coincidence with the GRB location. Taking into account the effective
area of the current detector configurations, upper limits on the neutrino
emission were obtained. Despite the reduced size of the detectors at the time
of GRB 221009A, about 10% of the final KM3NeT configuration, the results of
these analyses can be used to provide constraints on different neutrino
emission models for the case of GRB 221009A.

The ongoing construction of KM3NeT will increase the sensitivity for cosmic
neutrino detection in the coming years by one order of magnitude at least,
enhancing the relevant impact on forthcoming searches for neutrinos
originating from GRBs. For a significant detection it is crucial to
continuously monitor the sky with full coverage, using events coming from both
the upgoing and downgoing sky regions. This can be achieved by exploiting the
complementarity sky visibilities of the KM3NeT and the IceCube observatories,
together with the high duty cycle of the detectors.

## Acknowledgments

The authors acknowledge the financial support of the funding agencies: Czech
Science Foundation (GAČR 24-12702S); Agence Nationale de la Recherche
(contract ANR-15-CE31-0020), Centre National de la Recherche Scientifique
(CNRS), Commission Européenne (FEDER fund and Marie Curie Program), LabEx
UnivEarthS (ANR-10-LABX-0023 and ANR-18-IDEX-0001), Paris Île-de-France
Region, France; Shota Rustaveli National Science Foundation of Georgia
(SRNSFG, FR-22-13708), Georgia; The General Secretariat of Research and
Innovation (GSRI), Greece; Istituto Nazionale di Fisica Nucleare (INFN) and
Ministero dell’Università e della Ricerca (MUR), through PRIN 2022 program
(Grant PANTHEON 2022E2J4RK, Next Generation EU) and PON R&I program (Avviso n.
424 del 28 febbraio 2018, Progetto PACK-PIR01 00021), Italy; A. De Benedittis,
R. Del Burgo, W. Idrissi Ibnsalih, A. Nayerhoda, G. Papalashvili, I. C. Rea,
S. Santanastaso, A. Simonelli have been supported by the Italian Ministero
dell’Università e della Ricerca (MUR), Progetto CIR01 00021 (Avviso n. 2595
del 24 dicembre 2019); Ministry of Higher Education, Scientific Research and
Innovation, Morocco, and the Arab Fund for Economic and Social Development,
Kuwait; Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO), the
Netherlands; The National Science Centre, Poland (2021/41/N/ST2/01177); The
grant “AstroCeNT: Particle Astrophysics Science and Technology Centre”,
carried out within the International Research Agendas programme of the
Foundation for Polish Science financed by the European Union under the
European Regional Development Fund; National Authority for Scientific Research
(ANCS), Romania; Slovak Research and Development Agency under Contract No.
APVV-22-0413; Ministry of Education, Research, Development and Youth of the
Slovak Republic; MCIN for PID2021-124591NB-C41, -C42, -C43, funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, for
ASFAE/2022/014, ASFAE/2022 /023, with funding from the EU NextGenerationEU
(PRTR-C17.I01), Generalitat Valenciana, and for CSIC-INFRA23013, Generalitat
Valenciana for PROMETEO/2020/019, for Grant AST22_6.2 with funding from
Consejería de Universidad, Investigación e Innovación and Gobierno de España
and European Union - NextGenerationEU, for CIDEGENT/2018/034, /2019/043,
/2020/049, /2021/23 and for GRISOLIAP/2021/192 and EU for MSC/101025085,
Spain; The European Union’s Horizon 2020 Research and Innovation Programme
(ChETEC-INFRA - Project no. 101008324).

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