# Mimetic-$f(Q)$ gravity: cosmic reconstruction and energy conditions

Adam Z. Kaczmarek<EMAIL_ADDRESS><EMAIL_ADDRESS>Institute of Physics, Faculty of Science and
Technology, Jan Długosz University in Czȩstochowa, 13/15 Armii Krajowej Ave.,
42200 Czȩstochowa, Poland

###### Abstract

In this study, we present a novel approach to mimetic gravity incorporating a
non-zero nonmetricity tensor with vanishing torsion and curvature,
establishing a generalized mimetic-$f(Q)$ gravity framework. Using the
Lagrange multiplier method, we have obtained and discussed characteristics of
the theory’s field equations. In order to study cosmic evolution given by the
hybrid scale factor, we implemented the reconstruction method in two different
ways. In the first case, we have obtained corresponding Lagrange multiplier
$\eta$ and potential $U$ for the specific $f(Q)=f(Q)=Q-6\lambda
M^{2}\big{(}\frac{Q}{6M^{2}}\big{)}^{\alpha}$ function, while in the second
scenario we have recovered $f(Q)$ functional and mimetic potential for the
given Lagrange multiplier $\eta_{0}+\gamma H^{2}$. Subsequently, we explore
the fundamental properties of the $f(Q)=Q-6\lambda
M^{2}\big{(}\frac{Q}{6M^{2}}\big{)}^{\alpha}$ model and analyse the energy
conditions to establish its validity. Our findings indicate that the framework
introduced herein allows for the derivation of a wide range of viable
cosmological models that satisfy energy constraints that are necessary in
description of the accelerated expansion.

##  I Introduction

Recent experiments suggested that almost $95\%$ of the Universe is consisted
from the dark sector, indicating lack of completeness of the original
formulation of the Einstein’s work Clifton et al. (2012); Joyce et al. (2016);
Nojiri et al. (2017a); Dodelson and Schmidt (2020); Arbey and Mahmoudi (2021).
Those dark components are identified as dark energy, driving accelerated
expansion and dark matter, an entity clustering in galaxies as ’missing mass’
needed to properly explain large-scale cosmic structures Dodelson and Schmidt
(2020). However, current $\Lambda CDM$ model that combines (cold-) dark matter
and cosmological constant $\Lambda$ (dark energy) within formulation of GR,
undergoes serious pressure, facing new challenges Arbey and Mahmoudi (2021);
Perivolaropoulos and Skara (2022). In addition, many scientists are
unsatisfied with the reasoning behind dark sector, since if dark sector does
exist, only $5\%$ of matter-energy in the Universe is visible, and dark
components often require exotic and strange properties Clifton et al. (2012);
Joyce et al. (2015). In this context, thought that GR may not be the correct
proposal to describe gravity in large-scale structure led to the increased
interest in alternatives to the current $\Lambda CDM$ description. In such
context, modifications of Einstein’s formulation are proposed as a different
way of explaining cosmic phenomena and experimental issues without invoking
dark matter and dark energy Clifton et al. (2012); Joyce et al. (2016). It is
important to note, that some problems may raise during introduction of new
models. The resulting equations are often very complicated and hard to solve,
leading to technical difficulties in obtaining new solutions Clifton et al.
(2012); Nojiri et al. (2017a); Marciu (2020). The resulting issues can be
bypassed by employing the so-called reconstruction method, commonly used for
various, mainly cosmological models Nojiri et al. (2017a); Sardar and Debnath
(2022). The core of the reconstruction approach is to determine the desired
evolution first, and then recover (i.e. reconstruct) the corresponding model
of interests Bamba et al. (2012); Akarsu et al. (2014); Nojiri et al. (2017b);
Kaczmarek and Szczȩśniak (2020, 2021); Paul et al. (2022).

Note that, one cannot construct a specific cosmic model and ’call it a day’,
as there is no guarantee that the resulting theory is physically viable Koyama
(2016); Saridakis et al. (2021). Therefore, it is essential to employ various
tests and criteria to distinguish physically realizable models from those that
are not Clifton et al. (2012). One of such procedures is deeply rooted in the
mathematical formalism behind Einstein’s general relativity, by imposing
certain behaviour on the energy momentum tensor $T_{\mu\nu}$, in the form of
energy conditions Poisson (2004). In the conventional GR, these conditions
assist in selecting physically realistic matter content of spacetime. In the
context of the modified gravity, the energy conditions can be used to
constrain the degrees of freedom of the theory and test the physical viability
of cosmological scenarios Capozziello et al. (2014); Kaczmarek and Szczȩśniak
(2020). In order to achieve that goal, every alteration and modification to
the original GR can be treated as contribution from an effective energy-
momentum tensor Poisson (2004); Santos et al. (2007). By doing so, Universe is
filled with some kind of ’fictious’ (geometrical) fluids, that in principle
can play a role of dark matter and dark energy Capozziello et al. (2014);
Koussour et al. (2022). In general, energy conditions in the literature are
referred to and categorized based on four distinct criteria Poisson (2004);
Mandal et al. (2020):

  * •

The null energy condition (NEC) - guarantees positivity of matter density
along lightlike curves,

  * •

The weak energy condition (WEC) - states that the matter density is positive
along timelike curves,

  * •

The dominant energy condition (DEC) - velocity of the energy-momentum flow
cannot be greater than the speed of light,

  * •

The strong energy condition (SEC) - ensures attractive behaviour of gravity.

Those boundaries proved useful not only in proving BH theorems but also in the
context of cosmology Hawking and Ellis (2023); Mandal et al. (2020).
Noteworthy, violated SEC corresponds to the accelerated expansion of the
Universe Mandal et al. (2020); De and How (2022). Therefore, energy conditions
serve as crucial tools in defining the physical attributes of proposed
modified GR. This significance sparked substantial interest in various
modified theories of gravity Capozziello et al. (2014); Nojiri et al. (2017a);
Capozziello et al. (2018); Yousaf et al. (2018); Mandal et al. (2022).

Among the plethora of different models, one of the most interesting in recent
years is based on the old principles of differential geometry -
teleparallelism. The idea, considered even by Einstein himself, focusses on
the geometry, emphasizing the role of other notions than curvature in its
description. In fact, Levi-Civita connection is not the only possibility for
transporting vectors and tensors on the manifold Ortín (2004); Bahamonde et
al. (2018, 2023). From differential geometry, it is known that general affine
connection can be separated into the three parts:
$\Gamma^{\lambda}_{\;\;\mu\nu}=\\{^{\lambda}_{\;\;\mu\nu}\\}+K^{\lambda}_{\;\;\mu\nu}+L^{\lambda}_{\;\;\mu\nu}$,
with the Levi-Civita connection
($\\{^{\lambda}_{\;\;\mu\nu}\\}=\frac{1}{2}g^{\lambda\sigma}\left(\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right)$),
contorsion
($K^{\lambda}_{\;\;\mu\nu}=\frac{1}{2}g^{\lambda\beta}\Big{(}T_{\beta\mu\nu}+T_{\mu\beta\nu}+T_{\nu\beta\mu}\Big{)}$)
characterized by torsion
$T_{\;\;\mu\nu}^{\gamma}=\Gamma^{\gamma}_{\;\;\mu\nu}-\Gamma_{\;\;\nu\mu}^{\gamma}$
and disformation
$L^{\lambda}_{\;\;\mu\nu}=\frac{1}{2}g^{\lambda\beta}\left(Q_{\beta\mu\nu}-Q_{\mu\beta\nu}-Q_{\nu\beta\mu}\right)$
based on nonmetricity
$Q_{\gamma\mu\nu}=\nabla_{\gamma}g_{\mu\nu}=\partial_{\gamma}g_{\mu\nu}-\Gamma^{\beta}_{\;\;\gamma\mu}g_{\beta\nu}-\Gamma^{\beta}_{\;\;\gamma\nu}g_{\beta\mu}$
Bahamonde et al. (2023). Moreover, those different approaches merge into some
form of trinity referred to as ’geometrical trinity’, as they are different
representations of the same underlying theory (’three sides of the same coin’)
Beltrán Jiménez et al. (2019). In this aspect, one of the most intriguing
alternative descriptions of gravity is based on flat (curvatureless)
connection and vanishing torsion, i.e.
$\Gamma^{\lambda}_{\;\;\mu\nu}=L^{\lambda}_{\;\;\mu\nu}$, being fully
determined by nonmetricity tensor Jiménez et al. (2018); Bahamonde et al.
(2023). Theory of that kind is referred to as symmetric teleparallel
equivalent of GR (STEGR), where nonmetricity scalar ($Q$) plays a role that
usually is performed by Ricci curvature ($R$) Jiménez et al. (2018). The story
does not end here, as possible more general forms of nonmetricity theories are
grouped in the $f(Q)$ formulation, where gravity action is described by
general and analytical function of nonmetricity Jiménez et al. (2018, 2020);
Järv et al. (2018). This novel idea opens new possibilities and uncharted
territories in the study of the GR alternatives, in a manner similar to the
impact of the $f(R)$ theories on the curvature-based modified gravity Sotiriou
(2006); Capozziello et al. (2007); Clifton et al. (2012); Nojiri et al.
(2017a). Note that equations of motion from $f(R)$ gravity are higher order,
while remarkably $f(Q)$-theories lead to second order field equations,
avoiding Ostrogardski’s instabilities Jiménez et al. (2018); Järv et al.
(2018); Barros et al. (2020); Hu et al. (2022). Currently, $f(Q)$ gravity
achieved significant results despite being a few-years old Jiménez et al.
(2018); Mandal et al. (2020); Gakis et al. (2020); Frusciante (2021);
Albuquerque and Frusciante (2022). For example, it challenged the
aforementioned $\Lambda CDM$ model and provided an alternative that matches
with observations Beltrán Jiménez et al. (2019); Atayde and Frusciante (2021);
Anagnostopoulos et al. (2021). Thus, in the current state-of-the-art,
nonmetricity have been studied from different perspectives ranging from the
astrophysics and black holes Lazkoz et al. (2019); Bahamonde et al. (2022);
Calzá and Sebastiani (2023a); Errehymy et al. (2022); Mandal et al. (2022);
Bhar et al. (2023); Calzá and Sebastiani (2023b); Mishra et al. (2023), to the
applications in the theoretical cosmology Bajardi et al. (2020); Solanki et
al. (2021); D’Ambrosio et al. (2022); Koussour et al. (2022); Arora and Sahoo
(2022); Junior and Rodrigues (2023); Paliathanasis (2023); Subramaniam et al.
(2023) and often relying on reconstruction scheme Capozziello and Shokri
(2022); Myrzakulov et al. (2022); Singha et al. (2023). Moreover, energy
conditions have been proven useful in obtaining $f(Q)$ models compatible with
accelerated expansion of the Universe Mandal et al. (2020); Hu et al. (2022);
Mandal et al. (2022). For readers seeking more details, a comprehensive and
pedagogical introduction to $f(Q)$ gravity and related ’geometric trinity’ is
available in Heisenberg (2023).

Moreover, the growing interest in the generalized STEGR (i.e. $f(Q)$ gravity)
led to the various extensions and modifications of that theory, by looking for
additional degrees of freedom Harko et al. (2018). For example, the nontrivial
couplings between matter and nonmetricity tensor have been considered in
$f(Q,T)$ Xu et al. (2019) and $f(Q,L_{m})$ gravity Harko et al. (2018), for
trace of energy momentum tensor $T$ and matter Lagrangian, respectively.
Noteworthy, approaches of that kind can provide an alternative to the dark
energy Arora et al. (2020, 2021) or provide viable bounce cosmologies Agrawal
et al. (2021). From slightly different perspective, scalar tensor realizations
of $f(Q)$ gravity or metric-affine generalizations (i.e. curvature and torsion
is not-vanishing) have also been implemented in different contexts Järv et al.
(2018); Rünkla and Vilson (2018); Iosifidis et al. (2021); Bahamonde et al.
(2023). It is important to notice, that there are other different strategies
for changing GR. Instead of adding new terms to the action, one can focus on
the degrees of freedom that may not be visible upon the first look Sebastiani
et al. (2017); Nojiri et al. (2017a). This motivated Chamseddine and Mukhanov
to isolate the conformal degree of freedom for the metric, by introducing the
parametrization of the physical metric $g$ in terms of scalar field $\phi$ and
auxiliary metric $\hat{g}$ in the following way
$g_{\mu\nu}=\hat{g}_{\mu\nu}(g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi)\equiv
P\hat{g}_{\mu\nu}$. It was shown, that utilization of parametrization of that
kind leads to the inclusion of the scalar field that ’mimics’ the behaviour of
the dark matter Chamseddine and Mukhanov (2013); Chamseddine et al. (2014);
Mirza and Oboudiat (2019). Similarly to the theories build upon the
nonmetricity scalar, the mimetic theories with various extensions gained
significant interest over the recent years, providing another viable GR
alternative in the context of dark sector or inflation Nojiri and Odintsov
(2014); Astashenok et al. (2015); Myrzakulov et al. (2015); Nojiri et al.
(2016, 2017b); Casalino et al. (2018); Mansoori et al. (2021); Kaczmarek and
Szczȩśniak (2021). However, while torsion-mimetic-teleparallel
$f(\mathcal{T})$ gravity has seen some attention, the incorporation of the
mimetic degree of freedom into the broader scope of teleparallelism and
nonmetricity is still an underexplored area Mirza and Oboudiat (2019); Guo et
al. (2020); Izaurieta et al. (2020). Consequently, the general mimetic-STEGR
formulation remains uncharted territory, representing a key task of our
current work.

In the present paper, we follow the recent trend in the STEGR with extensions
and introduce mimetic generalization of the $f(Q)$ gravity. This is motivated
not only by the interesting and desired properties of both mimetic and STEGR
theories but also their flexibility in different physically crucial contexts
Astashenok et al. (2015); Odintsov and Oikonomou (2015, 2016); Jiménez et al.
(2018); Koussour et al. (2023); Heisenberg (2023). In details, after brief
overview of main characteristic of the STEGR and mimetic approaches, we
introduce the corresponding field equations and the main properties of the new
theory. The mimetic-$f(Q)$ theory introduced in sec. II can be interpreted as
extended STEGR that is supplied by the conformal degree of freedom arising
from the collisionless perfect fluid Chamseddine and Mukhanov (2013);
Sebastiani et al. (2017). In section III, we utilize a reconstruction scheme
to derive two models capable of describing FLRW cosmology at large scales by
employing the hybrid cosmic evolution Akarsu et al. (2014); Koussour et al.
(2022). This choice of scale factor allows us to obtain models suitable for
description of the different stages of the cosmic evolution in the concise and
unified manner Akarsu et al. (2014); Nojiri et al. (2017a). Firstly, we
reconstruct mimetic Lagrange multiplier and potential for a given $f(Q)$
cosmology. In the similar manner, we use inverted reconstruction scheme to
obtain mimetic potential and $f(Q)$ functional, that corresponds to the
assumed Lagrange multiplier. In section IV, the dynamical behaviour and energy
constraints are analysed in terms of redshift, to validate the reconstructed
mimetic-$f(Q)$ model for $f(Q)=Q-6\lambda
M^{2}\big{(}\frac{Q}{6M^{2}}\big{)}^{\alpha}$. By studying those properties,
we may verify whether the reconstructed model is viable or not, as realistic
behaviour is necessary for proper description of the current Universe’
expansion Arbey and Mahmoudi (2021). In the last part, we present the outline
and conclusion of our work, as well as presenting future perspectives that
arise from the mimetic approach within generalized mimetic-STEGR.

##  II Mimetic-$f(Q)$ Framework

In the following work, we concentrate on mimetic extension of theories built
upon nonmetricity alone. Hence, we consider torsionless and curvature free
geometry with connection Jiménez et al. (2018); Koussour et al. (2022);
Heisenberg (2023):

$\displaystyle\Gamma^{\lambda}_{\;\;\mu\nu}=L^{\lambda}_{\;\;\mu\nu}=\frac{1}{2}g^{\lambda\beta}\left(Q_{\beta\mu\nu}-Q_{\mu\beta\nu}-Q_{\nu\beta\mu}\right),$
(1)

characterized solely by the nonmetricity tensor:

$\displaystyle
Q_{\gamma\mu\nu}=\nabla_{\gamma}g_{\mu\nu}=\partial_{\gamma}g_{\mu\nu}-\Gamma^{\beta}_{\;\;\gamma\mu}g_{\beta\nu}-\Gamma^{\beta}_{\;\;\gamma\nu}g_{\beta\mu}.$
(2)

Thus, geometry build from nonmetricity is defined solely by changes of vectors
length upon parallelly transporting them on the manifold Beltrán Jiménez et
al. (2019). It is also important to add, that in the STEGR theories the
connection can be made vanishing by the appropriate coordinate transformation
in the form of so-called coincidental gauge Jiménez et al. (2018); Beltrán
Jiménez et al. (2019); Heisenberg (2023).

Before we go deeper into the formulation of theoretical framework, one should
note that nonmetricity has two independent traces:

$\displaystyle
Q_{\mu}=Q_{\mu\;\;\nu}^{\;\;\nu},\;\;\;\;\tilde{Q}_{\mu}=Q^{\nu}_{\;\;\mu\nu}.$
(3)

Moreover, one can introduce and define superpotential (conjugate) tensor in
terms of nonmetricity and its traces:

$\displaystyle
P^{\alpha}_{\;\;\mu\nu}=\frac{1}{2}Q_{(\mu\;\;\nu)}^{\;\;\alpha}-\frac{1}{4}\Big{(}Q^{\alpha}_{\;\;\mu\nu}-Q^{\alpha}g_{\mu\nu}-\tilde{Q}^{\alpha}g_{\mu\nu}-\delta^{\alpha}_{(\mu}Q_{\nu)}\Big{)}$
(4)

Therefore, nonmetricity scalar can be defined by:

$\displaystyle Q=-Q_{\alpha\mu\nu}P^{\alpha\mu\nu},$ (5)

and should be considered as a scalar trace of the nonmetricity tensor (i.e. no
free indices left). In what follows, we adopt the coicident gauge, where
$\nabla_{\mu}=\partial_{\mu}$ and
$Q_{\alpha\mu\nu}=\partial_{\alpha}g_{\mu\nu}$ Jiménez et al. (2018); Beltrán
Jiménez et al. (2019).

On the other hand, in the mimetic gravity, the Einstein-Hilbert metric
$g_{\mu\nu}$ can be written in terms of the auxiliary metric tensor
$\hat{g}_{\mu\nu}$ and the auxiliary scalar field $\phi$ Chamseddine and
Mukhanov (2013); Sebastiani et al. (2017):

$\displaystyle
g_{\mu\nu}=-\hat{g}^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi\hat{g}_{\mu\nu}.$
(6)

In this manner, the original metric $g_{\mu\nu}$ can be re-expressed with the
new degree of freedom linked to the invariance under conformal transformations
of the metric $\hat{g}_{\mu\nu}$. Consistency of the Eq.(1) leads to the
mimetic constraint equation imposed on the gradient of the scalar field $\phi$
Chamseddine and Mukhanov (2013):

$\displaystyle
g^{\mu\nu}(\phi,\hat{g}_{\mu\nu})\partial_{\mu}\phi\partial_{\nu}\phi=-1.$ (7)

By employing the mimetic framework, the resulting action can be seen as
conformally extended one and Weyl-invariant in terms of metric
$\hat{g}_{\mu\nu}$. For the detailed introduction to the mimetic theory,
please see Sebastiani et al. (2017).

As a way to impose given constraints, we use well-known Langrange multiplier
formalism, extensively used in the mimetic theories and extensions Odintsov
and Oikonomou (2015); Nojiri et al. (2016); Mansoori et al. (2021); Odintsov
and Oikonomou (2016); Cárdenas et al. (2021). Thus, action of the new
mimetic-$f(Q)$ gravity will take the following form:

$\displaystyle
S=\int\text{d}^{4}x\sqrt{-g}\frac{1}{2}\Big{[}f(Q)-U(\phi)+\eta(g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+1)\Big{]}+\int\text{d}^{4}x\sqrt{-g}\mathcal{L}_{m}$
(8)

with the mimetic field $\phi$ and the scalar (mimetic) potential $U(\phi)$. We
note that the presence of the potential in the action serves a minimal
extension of the original mimetic framework Chamseddine et al. (2014). This
augmentation allows for the portrayal of a wide array of models, since any
background spacetime can be obtained from apropiate potential $U(\phi)$
Chamseddine et al. (2014). We also note that the actions involving scalar
field constrained by Lagrange multiplier can be studied without referring to
the mimetic theory, see for instance Capozziello et al. (2010); Gao et al.
(2011); Nojiri et al. (2017b, 2019). For physical interpretation of the
mimetic-$f(Q)$ framework introduced herein, the mimetic degree of freedom (6)
arises from the collisionless perfect fluid, in the spirit of original
considerations of Chamseddine and Mukhanov Chamseddine and Mukhanov (2013);
Chamseddine et al. (2014).

Variation of the given mimetic-$f(Q)$ action in terms of $g$ yields the
following field equations:

$\displaystyle\frac{2}{\sqrt{-g}}\nabla_{\alpha}(\sqrt{-g}f_{Q}P^{\alpha}_{\;\;\mu\nu})+\frac{1}{2}g_{\mu\nu}\Big{(}f(Q)-U(\phi)+\eta(\partial_{\alpha}\phi\partial^{\alpha}\phi+1)\Big{)}-\eta\partial_{\mu}\phi\partial_{\nu}\phi+f_{Q}\big{(}P_{\mu\alpha\beta}Q_{\nu}^{\;\;\alpha\beta}-2Q_{\alpha\beta\mu}P^{\alpha\beta}_{\;\;\;\;\nu}\big{)}=T_{\mu\nu},$
(9)

or equivalently Heisenberg (2023):

$\displaystyle
f_{Q}G_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\Big{(}f(Q)-U(\phi)+\eta(\partial_{\alpha}\phi\partial^{\alpha}\phi+1)-Qf_{Q}\Big{)}+\eta\partial_{\mu}\phi\partial_{\nu}\phi+2f_{QQ}P^{\alpha}_{\;\;\mu\nu}\partial_{\alpha}Q=T_{\mu\nu}.$
(10)

In the above, the energy-momentum tensor is defined as:

$\displaystyle
T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g^{\mu\nu}}.$ (11)

On the other hand, variation with respect to the connection leads to the:

$\displaystyle\nabla_{\mu}\nabla_{\nu}(\sqrt{-g}P^{\mu\nu}_{\;\;\;\;\alpha})=0.$
(12)

The equations (9) or (10) for $f(Q)=Q$ and $\eta=U(\phi)=0$ reproduce standard
Einstein field equations of GR Jiménez et al. (2018); Beltrán Jiménez et al.
(2019). Note the sign convention in the definition of action (8). In
literature, there exist another possibility for extended STEGR with the extra
minus sign, for which $f(Q)=-Q$ restores Einstein’s formulation Myrzakulov et
al. (2022); Koussour et al. (2022); Heisenberg (2023).

Additionally, variation of the action with respect to the mimetic field $\phi$
results in the following scalar equation:

$\displaystyle 2\nabla^{\mu}(\eta\partial_{\mu}\phi)+U^{\prime}(\phi)=0,$ (13)

with $U^{\prime}(\phi)=\frac{\text{d}U(\phi)}{\text{d}\phi}$) and
$f_{Q}=\frac{dQ}{dt}$. It is important to remark, that variation of the action
with respect to Lagrange multiplier $\eta$ returns mimetic constraint
Chamseddine et al. (2014); Sebastiani et al. (2017).

Note that, from trace of Eq.(10) one can get:

$\displaystyle\eta=f_{Q}(G-2Q)-2U(\phi)+2f(Q)+2f_{QQ}P^{\alpha\;\;\mu}_{\;\;\mu}\partial_{\alpha}Q-T,$
(14)

where we have invoked relationship (7). Then, inserting it into the Eq.(10)
gives:

$\displaystyle
f_{Q}G_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\Big{(}f(Q)-U(\phi)+2f_{QQ}P^{\alpha}_{\;\;\mu\nu}\partial_{\alpha}Q+\eta(\partial_{\alpha}\phi\partial^{\alpha}\phi+1)-Qf_{Q}\Big{)}$
$\displaystyle+\Big{(}f_{Q}(G-2Q)-2U(\phi)+2f(Q)+2f_{QQ}P^{\alpha\;\;\mu}_{\;\;\mu}\partial_{\alpha}Q-T\Big{)}\partial_{\mu}\phi\partial_{\nu}\phi=T_{\mu\nu}.$
(15)

In that manner we have obtained equations that arise from the equivalent
action:

$\displaystyle
S=\int\text{d}^{4}x\sqrt{-g(\phi,\hat{g}_{\mu\nu})}\frac{1}{2}\Big{[}f(Q(\phi,\hat{g}_{\mu\nu}))-U(\phi)\Big{]}+\int\text{d}^{4}x\sqrt{-g(\phi,\hat{g}_{\mu\nu})}\mathcal{L}_{m},$
(16)

where variation of metric $\delta g_{\mu\nu}$ is treated as variation of
scalar field $\delta\phi$ and auxiliary metric $\delta\hat{g}_{\mu\nu}$, in
accordance with the Eq.(6) and original formulation of Chamseddine and
Mukhanov Chamseddine and Mukhanov (2013); Chamseddine et al. (2014);
Sebastiani et al. (2017); Mirza and Oboudiat (2019). By doing so, we have
linked two equivalent formulations of the mimetic-$f(Q)$ gravity. Namely, the
action (8) in terms of $g_{\mu\nu}$ with Lagrange multiplier and action (16)
written in terms of metric $\hat{g}_{\mu\nu}$.

Continuing our work, we will assume that geometry is described by the
isotropic and homogeneous FLRW spacetime. The line element for this spacetime
takes the following form:

$\displaystyle\text{d}s^{2}=-\text{d}t^{2}+a^{2}(t)\text{d}x_{i}\text{d}x^{i}=-\text{d}t^{2}+a^{2}(t)(\text{d}x^{2}+\text{d}y^{2}+\text{d}z^{2})$
(17)

in the usual Cartesian coordinates $(t,x,y,z)$ Dodelson and Schmidt (2020). We
also assume that scalar field depends only on time, i.e. $\phi=\phi(t)$. Then,
the nonmetricity in coincidence gauge will take the simple form Jiménez et al.
(2020); Koussour et al. (2022):

$\displaystyle Q=6H^{2},$ (18)

where ’dot’ ($\dot{}$) denotes differentiation w.r.t. cosmic time $t$ and
Hubble rate is given by $H=\frac{\dot{a}}{a}$. In addition, we assume perfect
fluid matter content of the Universe, described by the energy-momentum tensor
Hobson et al. (2006):

$\displaystyle T_{\mu\nu}=(\rho+p)v_{\mu}v_{\nu}+pg_{\mu\nu},$ (19)

for the four-velocity of the fluid $v_{\mu}$ that satisfy $v^{\mu}v_{\mu}=-1$.
The energy density and pressure are denoted as $\rho$ and $p$, respectively.
Furthermore, we can assume that perfect fluid satisfies standard isotropic
equation of state (EoS):

$\displaystyle w\rho=p,$ (20)

characterized by parameter $w$. Additionally, vanishing four-divergence of the
energy-momentum ($\nabla^{\mu}T_{\mu\nu}=0$) leads to the conservation
equation:

$\displaystyle\dot{\rho}+3H\rho(1+w)=0,$ (21)

leading to the general form of energy density for cosmic fluid:

$\displaystyle\rho(t)=\rho_{0}\Big{(}\frac{a}{a_{0}}\Big{)}^{-3(w+1)}.$ (22)

Then, corresponding nonmetricity field equations are:

$\displaystyle
6H^{2}f_{Q}-\frac{1}{2}\Big{(}f(Q)-U(\phi)+\eta(\dot{\phi}^{2}+1)\Big{)}=\rho$
(23)

and

$\displaystyle
2(\dot{H}+3H^{2})f_{Q}+2H\dot{f}_{Q}-\frac{1}{2}(f(Q)-U(\phi))=-p,$ (24)

for $00$th and $ii$th components respectively. Moreover, the scalar equation
take form:

$\displaystyle
6H\eta\dot{\phi}+2(\dot{\eta}\dot{\phi}+\eta\ddot{\phi})-U^{\prime}(\phi)=0.$
(25)

Mimetic constraint for the FLRW metric leads to the association of the scalar
field with a cosmic time $\phi=t$. We note that, this is an important result
in the mimetic gravities as in a wide class of the scalar-tensor theories, the
scalar field coupled to gravity often takes more complicated forms. By using
$\phi=t$, the field and scalar equations (23-25) reduce to:

$\displaystyle 6H^{2}f_{Q}-\frac{1}{2}\big{(}f(Q)-U(t)+2\eta\big{)}=\rho,$
(26) $\displaystyle
2(\dot{H}+3H^{2})f_{Q}+H\dot{f}_{Q}-\frac{1}{2}\big{(}f(Q)-U(t)\big{)}=-p,$
(27) $\displaystyle 6H\eta+2\dot{\eta}-\dot{U}=0.$ (28)

Combining equations (26) and (27) results in the expression for Lagrange
multiplier:

$\displaystyle\eta(t)=-\rho-p-H\dot{f}_{Q}-2f_{Q}\dot{H}.$ (29)

Note that for $f(Q)=Q$, standard mimetic gravity in Lagrange multiplier
formalism is obtained in Eqs. (21-24) Sebastiani et al. (2017). This confirms
that the construction of the mimetic-$f(Q)$ formalism is done properly, since
in the exact same manner $f(Q)=Q$ in standard $f(Q)$ gravity recovers
equations of GR: $3H^{2}=\rho$ and $2\dot{H}+3H^{2}=-p$ Jiménez et al. (2018);
Mussatayeva et al. (2023). For the particular form of $f(Q)$ gravity, by
solving Eq.(28), one can obtain the desired potential. Additionaly, the
Lagrange multiplier can be found from Eq.(29). This scheme will be covered in
details in the next section.

##  III Reconstruction of the Hybrid Expansion Law

Obtaining new solutions for field equations is not an easy task, due to the
inherent complexity and nonlinearity. Hence, studying cosmic scenarios is hard
to tackle as there is no guarantee that new model is solvable. That troubles
are even more evident and hard to bypass where GR modifications play a role
Clifton et al. (2012). However, in the reconstruction procedure, the field
equations are inverted to unveil the class of modified theories that give rise
to a specific flat Friedmann-Lemaître-Robertson-Walker (FLRW) model. In other
words, instead of solving field equations for a given model in order to obtain
scale factor, the equations are used to reconstruct a model that satisfy a
given scale factor Odintsov and Oikonomou (2015, 2016); Kaczmarek and
Szczȩśniak (2021); Gadbail et al. (2022); Gonçalves et al. (2022). It is
obvious, that ability to describe well-established cosmological models in the
GR modifications is desired Joyce et al. (2016); Nojiri et al. (2017a). The
hybrid expansion law (HEL) was introduced in Akarsu et al. (2014) as a product
of the power-law evolutions and the flat de-Sitter cosmology. Besides
describing these two phases, the hybrid scale factor allows for elegant and
nice transition between those stages in cosmic history, namely deceleration to
the cosmic acceleration Akarsu et al. (2014). This is particularly interesting
property for reconstruction procedure in the modified gravity, as one can at
hand reconstruct a model that describes cosmic history in the unified manner
Nojiri et al. (2017a). The corresponding scale factor and Hubble rate take the
following form Akarsu et al. (2014); Koussour et al. (2023):

$\displaystyle
a(t)=a_{0}\Big{(}\frac{t}{t_{0}}\Big{)}^{A}e^{B\big{(}\frac{t}{t_{0}}-1\big{)}},\;\;\;H=\frac{A}{t}+\frac{B}{t_{0}},$
(30)

where $A\geq 0$ and $B\geq 0$ and the present values of the scale factor and
Universe’ age are $a_{0}$ and $t_{0}$ accordingly Akarsu et al. (2014).

###  III.1 Reconstruction for $\eta(t)$ and $U(t)$

In this part, we will focus on obtaining Lagrange multiplier and potential for
the corresponding $f(Q)$ gravity model. For this purpose, we have to specify
the functional form of $f(Q)$. In this manner we will follow the steps of
Jimenez, Heisenberg and Koivisto in their pioneering work, by considering the
nonmetricity functional of Jiménez et al. (2018):

$\displaystyle f(Q)=Q-6\lambda M^{2}\big{(}\frac{Q}{6M^{2}}\big{)}^{\alpha},$
(31)

where $M$ should be treated as some scale, $\lambda$ is a dimensionless
parameter of order $1$. For the functional specified above, $\alpha=1/2$
recovers usual mimetic characteristics within standard GR for $M^{2}>>H^{2}$,
while modifications appear when $M^{2}\lesssim H^{2}$ Jiménez et al. (2018).
Thus, the combined equation (29) for $\eta$ is:

$\displaystyle\frac{2\alpha^{2}\dot{H}\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}-H^{2}\left(\eta+2\dot{H}+p+\rho\right)}{H^{2}}=0.$
(32)

Combining the Lgrange multiplier from (32) with the Klein-Gordon equation
(28), leads to:

$\displaystyle H^{3}\frac{dU}{dt}$ $\displaystyle=4\alpha(2\alpha-1)\lambda
M^{2}H\ddot{H}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}-2H^{3}\left(2\ddot{H}+\dot{p}+\dot{\rho}\right)+8\alpha\left(2\alpha^{2}-3\alpha+1\right)\lambda
M^{2}\dot{H}^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}$
$\displaystyle+12\alpha(2\alpha-1)\lambda
M^{4}\dot{H}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha+1}-6H^{4}\left(2\dot{H}+p+\rho\right).$
(33)

Thus, once the Lagrange multiplier is obtained, one can solve the differential
equation with respect to the mimetic potential $U$. Now, from applying scale
factor (30) in Eqs. (32) and (33) and keeping in mind the energy density (22),
the reconstructed Lagrange multiplier and potential are:

$\displaystyle\eta(t)$
$\displaystyle=\frac{2A\left(\text{t0}^{2}\left(A^{2}-\alpha(2\alpha-1)\lambda
M^{2}t^{2}\left(\frac{(A\text{t0}+Bt)^{2}}{M^{2}t^{2}\text{t0}^{2}}\right)^{\alpha}\right)+2ABt\text{t0}+B^{2}t^{2}\right)}{t^{2}(A\text{t0}+Bt)^{2}}$
$\displaystyle+\text{$\rho$0}(w+1)\left(\frac{t}{\text{t0}}\right)^{-3A}\left(-e^{B\left(3-\frac{3t}{\text{t0}}\right)}\right)\left(\left(\frac{t}{\text{t0}}\right)^{A}e^{B\left(\frac{t}{\text{t0}}-1\right)}\right)^{-3w}$
(34)

and

$\displaystyle U(t)$
$\displaystyle=\frac{2(2\alpha-1)\lambda\left(6ABt\text{t0}+A\text{t0}^{2}(3A-2\alpha)+3B^{2}t^{2}\right)\left(\frac{(A\text{t0}+Bt)^{2}}{M^{2}t^{2}\text{t0}^{2}}\right)^{\alpha-1}}{t^{2}\text{t0}^{2}}-\frac{2A((3A-2)\text{t0}+6Bt)}{t^{2}\text{t0}}$
$\displaystyle-2\text{$\rho$0}w\left(\frac{t}{\text{t0}}\right)^{-3A}e^{B\left(3-\frac{3t}{\text{t0}}\right)}\left(\left(\frac{t}{\text{t0}}\right)^{A}e^{B\left(\frac{t}{\text{t0}}-1\right)}\right)^{-3w}+U_{0}.$
(35)

Thus, it is possible to obtain hybrid-evolution law scale factor in the
mimetic framework of $f(Q)$, once the specific functional for extended STEGR
is specified. Moreover for the vacuum ($\rho=p=0$), in the limit of
$t\rightarrow\infty$ (i.e. $A=0$) corresponding to the de-Sitter evolution,
the resulting potential is constant $U(t)=6(2\alpha-1)\lambda
M^{2}\left(\frac{B^{2}}{M^{2}t_{0}^{2}}\right)^{\alpha}+U_{0}$, while $\eta$
vanishes. Thus, for the late time cosmic acceleration, the constant potential
will be responsible, mimicking the dark energy Sebastiani et al. (2017). We
note that this is a common characteristic for mimetic extensions of GR
modifications and alternatives Odintsov and Oikonomou (2016); Mirza and
Oboudiat (2019); Sebastiani et al. (2017); Kaczmarek and Szczȩśniak (2021);
Zhang et al. (2023).

###  III.2 Reconstruction for $f(Q)$ and $U$ from the Lagrange multiplier

In this scenario, instead of specifying a functional form of nonmetricity
$f(Q$ we will focus on obtaining a model that describes HEL for a specified
Lagrange multiplier. Herein we present the general procedure first, after
which we will use it for particular choice of $\eta$. In what follows, once
the multiplier $\eta$ is known, the Eq.(29) can be solved for $f_{Q}(t)$,
instead of $\eta$. In fact, the resulting differential equation can be easily
solved by integration with respect to the time $t$:

$\displaystyle
f_{Q}(t)=\frac{1}{H^{2}}\Big{[}\int-\Big{(}\rho(t)+p(t)+\eta(t)\Big{)}H\text{d}t+f_{0}\Big{]},$
(36)

where $f_{0}$ is the integration constant. Using the relationship between
nonmetricity scalar and Hubble rate $Q=6H^{2}$ (i.e. Eq.(18)) one can get an
explicit form of $t(Q)$ and obtain a function $f_{Q}(Q)$. Then, Eq.(36) can be
easily integrated in order to find $f(Q)$:

$\displaystyle f(Q)=\int f_{Q}(Q)\text{d}Q.$ (37)

On the other hand, scalar equation can be solved for $U$, leading to:

$\displaystyle
U(t)=U_{0}+\int_{1}^{t}2\left(3H(t_{1})\eta(t_{1})+\eta^{\prime}(t_{1})\right)dt_{1}.$
(38)

Note, that the inverted reconstruction is a standard approach in theories
built on the mimetic degree of freedom, mainly used to showcase the
flexibility of the introduced model Nojiri and Odintsov (2014); Astashenok et
al. (2015). Thus, one should treat this approach as a complement to the
procedure described in the previous subsection. Similarly to the mimetic
$f(R)$ gravity, one can also specify the potential $U(\phi)$ and then obtain
the Lagrange multiplier and corresponding action functional Nojiri and
Odintsov (2014).

In what follows, we will now reconstruct the $f(Q)$ functional for specified
Lagrange multiplier. The chosen multiplier will have simple quadratic form in
terms of Hubble factor:

$\displaystyle\eta(t)=\eta_{0}+\gamma H^{2},$ (39)

Now, due to the complexity of the equations, we will focus on the vacuum case
by taking $\rho=w=0$. This is a standard approach in mimetic gravity, as one
of the reason for introduction that kind of modification to Einsten’s theory
is that it can ’mimic’ matter Chamseddine and Mukhanov (2013); Sebastiani et
al. (2017). Hence, from $Q=6H^{2}$, the inverted function $t(Q)$ can be
obtained:

$\displaystyle
t=\frac{-6ABt_{0}-\sqrt{6}A\sqrt{Q}t_{0}^{2}}{6B^{2}-Qt_{0}^{2}}\;\;\;\;\text{or}\;\;\;\;t=\frac{\sqrt{6}A\sqrt{Q}t_{0}^{2}-6ABt_{0}}{6B^{2}-Qt_{0}^{2}}.$
(40)

Since we are dealing with the positive time, we restrict ourselves to the
second option of Eq.(40). The potential corresponding to the given Lagrange
multiplier, can be obtained immediately, by integrating Eq.(38) and takes
form:

$\displaystyle
U(t)=\frac{2B\left(-9A^{2}\gamma+2A\gamma+3\eta_{0}t^{2}\right)}{tt_{0}}+\frac{(2-3A)A^{2}\gamma}{t^{2}}+\frac{6A\log(t)\left(3B^{2}\gamma+\text{$\eta$0}t_{0}^{2}\right)}{t_{0}^{2}}+\frac{6B^{3}\gamma
t}{t_{0}^{3}}+U_{0}.$ (41)

On the other hand, from Eq.(36) one get the following form of $f_{Q}(t)$:

$\displaystyle f_{Q}(t)=\frac{A^{3}\gamma
t_{0}^{3}-2Btt_{0}^{2}\left(\eta_{0}t^{2}-3A^{2}\gamma\right)-2At^{2}t_{0}\log(t)\left(3B^{2}\gamma+\eta_{0}t_{0}^{2}\right)-2B^{3}\gamma
t^{3}+2c_{1}t^{2}t_{0}}{2t_{0}(At_{0}+Bt)^{2}}.$ (42)

Now, inserting the relation for $t(Q)$ of (40) leads to:

$\displaystyle f_{Q}(Q)$
$\displaystyle=\frac{t_{0}^{2}\left(\sqrt{6}\sqrt{Q}t_{0}-6B\right)^{2}}{\left(\sqrt{6}B\sqrt{Q}t_{0}^{2}-Qt_{0}^{3}\right)^{2}}\Big{[}-A\left(3B^{2}\gamma+\eta_{0}t_{0}^{2}\right)\log\left(\frac{At_{0}\left(\sqrt{6}\sqrt{Q}t_{0}-6B\right)}{6B^{2}-Qt_{0}^{2}}\right)-\frac{3AB\gamma\left(6B^{2}-Qt_{0}^{2}\right)}{6B-\sqrt{6}\sqrt{Q}t_{0}}$
$\displaystyle+\frac{A\gamma\left(Qt_{0}^{2}-6B^{2}\right)^{2}}{2\left(\sqrt{6}\sqrt{Q}t_{0}-6B\right)^{2}}+\frac{AB\eta_{0}t_{0}^{2}\left(6B-\sqrt{6}\sqrt{Q}t_{0}\right)}{6B^{2}-Qt_{0}^{2}}+\frac{AB^{3}\gamma\left(6B-\sqrt{6}\sqrt{Q}t_{0}\right)}{6B^{2}-Qt_{0}^{2}}+c_{1}\Big{]}.$
(43)

This lengthy expression can be integrated with respect to the $Q$, leading to
the explicit form of $f(Q)$ functional in the mimetic-nonmetricity regime
introduced herein. For the full form of the solution of Eq.(43), we refer a
reader to the Appendix. Thus, even the mathematically complicated forms of
scale factor can be obtained within mimetic-$f(Q)$, showing great flexibility
and adaptation of that framework. Noteworthy, the FLRW cosmology has been
reconstructed without invoking any matter content and as a consequence, full
hybrid cosmic evolution is characterized only by nonmetricity and mimetic
constraint from $f(Q)$ and $\eta$ with $U$, respectively. Note that from the
practical point of view, the standard approach with the specified $f(Q)$
functional is a more natural one, since the resulting models usually are
easier to handle and less mathematically complicated. In fact, this is a
common property of the theories twin to the mimetic-$f(Q)$ introduced herein,
such as mimetic-$f(R)$ or other higher-order models Nojiri and Odintsov
(2014); Astashenok et al. (2015); Sebastiani et al. (2017); Kaczmarek and
Szczȩśniak (2021).

##  IV Dynamical properties

###  IV.1 Redshift and equation of state

In this section, we will focus on main properties of the model reconstructed
in (A) part of the section $II$. In order to better relate to the current
observations and data, it is often more advantageous to use cosmic redshift
$z$ instead of time $t$. Redshift $z$ usually serves as a natural cosmic
measure, revealing not just distance but also the temporal context of
celestial phenomena. The higher $z$ value is, the corresponding cosmic epoch
is older and the distance to the remote object is greater Dodelson and Schmidt
(2020). For example, the beginning of the universe corresponds to the infinite
redshift ($z\rightarrow\infty$). As redshift is directly associated with the
geometry, the relation between $z$ and scale factor is:

$\displaystyle a(t)=\frac{a_{0}}{1+z},$ (44)

for the present value $a_{0}$ of a scale factor. In the case of the hybrid
expansion of (30), the age of Universe as a function of redshift reads
Koussour et al. (2023):

$\displaystyle t(z)=\frac{At_{0}}{B}f(z),$ (45)

for $f(z)=W[\frac{B}{A}\exp(\frac{B-\ln(1+z)}{A})]$, where $W$ is a Lambert
function (’product logarithm’) Corless et al. (1996); Koussour et al. (2023).
Moreover, Hubble factor and its time derivative in terms of redshift $z$ are
accordingly:

$\displaystyle H(z)=\frac{H_{0}B}{B+A}[1+\frac{1}{f(z)}]$ (46)

and

$\displaystyle\dot{H}=-(1+z)H(z)\frac{dH(z)}{dz},$ (47)

where $H_{0}=\frac{A+B}{t_{0}}$ stands for current value of Hubble parameter.
Moreover, the deccelaration parameter $q(z)$:

$\displaystyle q(z)=\frac{1}{H(z)^{2}}\left[(1+z)\frac{dH(z)}{dz}-1\right]$
(48)

can be introduced to address dynamics and behaviour of the expansion of
Universe. From its dynamics, one can easily deduce whether accelerated
expansion is slowing down Dodelson and Schmidt (2020); Koussour et al. (2023).
The deceleration as a function of redshift in the hybrid evolution scenario of
(30) has been plotted on panel (A) of Fig.(1) for different choices of $A$ and
$B$. In the present work, we will focus on the choice $A=B=1/2$. For general
details on dynamical behaviour of hybrid expansion law, we refer to Akarsu et
al. (2014); Koussour et al. (2023); Goswami et al. (2019), where other choices
of $A$ and $B$ have been studied.

Figure 1:  Evolution of the deceleration $q(z)$ (A) for different combinations
of hybrid scale factor parameters $A$ and $B$ in terms of redshift $z$.
Visualization of the energy density (B), effective pressure (C) and EoS
parameter (D) as a function of $z$ for different values of $\alpha$,where
$A=B=0.5$.

Now, we can address the dynamics of the reconstructed model for varying
redshift $z$. Firstly, we express the field equations in the form more
suitable for studying energy conditions and related properties. In order to
achieve this, we rewrite them in the Eisteinian form:

$\displaystyle 3H^{2}=\rho_{eff},\;\;\;\;-(\dot{H}+3H^{2})=p_{eff}$ (49)

where we have defined the effective energy density as:

$\displaystyle\rho_{eff}=\frac{1}{2f_{Q}}\Big{[}\rho+\frac{1}{2}\big{(}f(Q)-U(t)+2\eta\big{)}\Big{]},$
(50)

and effective pressure in the form of:

$\displaystyle
p_{eff}=\frac{1}{2f_{Q}}\Big{[}p+Hf_{QQ}\dot{Q}-\frac{1}{2}\big{(}f(Q)-U(t)\big{)}\Big{]}.$
(51)

The reader should treat the above reformulation as follows. Every term that is
not the ordinary left-hand side of FLRW equations in GR, can be treated as
effective fluid and component of the effective energy-momentum tensor
$T_{\mu\nu}^{eff}$ Nojiri et al. (2017a); Mandal et al. (2020). Moreover, any
geometrical quantity coming from the mimetic-$f(Q)$ action can be regarded as
the dark fluid that contributes to the matter content of the Universe. We
note, that this is standard association in modified gravity theories Clifton
et al. (2012); Nojiri et al. (2017a).

Additionally, in a manner similar to the usual matter content of the universe,
one can also define the effective equation of state parameter:

$\displaystyle
w_{eff}=\frac{p_{eff}}{\rho_{eff}}=\frac{2[H\dot{f}_{Q}+p+\eta+\rho]}{f(Q)+2(\eta+\rho)-U(t)}-1.$
(52)

As a check of consistency of the mimetic extension presented herein, we note
that for $f(Q)=Q$ and $\eta=U=0$, perfect fluid EoS of Einstein’s formulation
is recovered. For the function of $Q$ from Eq(31), we get:

$\displaystyle\rho_{eff}(z)=\frac{-3\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+3H^{2}+\eta(t)+\rho(t)-\frac{u(t)}{2}}{2\left(1-\alpha\lambda\left(\frac{H^{2}}{M^{2}}\right)^{\alpha-1}\right)},$
(53)

and

$\displaystyle p_{eff}(z)=\frac{4(\alpha-1)\alpha\lambda
M^{2}\dot{H}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}-H^{2}\left(6\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+2p+U\right)+6H^{4}}{4\alpha\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}-4H^{2}}.$ (54)

Dynamical nature of the $\rho_{eff}$ and $p_{eff}$ is visualized respectively
on the panels (B) and (C) of Fig.(1), where four different choices of
$\lambda$ have been considered. As can be seen, the effective energy density
stayed positive in the past. This is not the case for the effective pressure
that is currently negative at $z=0$, and except mimetic-$f(Q)$ model with
$\alpha=2$, it decreases as the redshift increase. Since negative pressure is
linked with the repulsive behaviour of gravity, the reconstructed model may
characterize accelerated expansion, since negative pressure is one of the main
characteristic of dark energy (cosmological constant) Arbey and Mahmoudi
(2021). Moreover, the corresponding EoS is given by:

$\displaystyle w_{eff}(z)=\frac{-4(\alpha-1)\alpha\lambda
M^{2}\dot{H}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+H^{2}\left(6\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+2p+U\right)-6H^{4}}{H^{2}\left(2\left(-3\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+\eta+\rho\right)+6H^{2}-U\right)}$
(55)

Having the effective EoS parameter, one can study and categorize behaviour of
the dark sector with its (effective) components. Currently, (i.e. $z=0$), the
observations suggest the equation of state parameter to be $w_{eff}<-1/3$, in
order to address accelerated expansion of the Universe. The behaviour of the
effective $w_{eff}$ for the model of Eq.(31) is presented on (D) plot of
Fig.(1), where different values of $\alpha$ has been taken. In current times,
EoS enters quintessence region of $w_{eff}\in(-1,1/3)$ and approaches $\Lambda
CDM$ value $w_{eff}=-1$ in the future as $z\rightarrow-1$. Hence, the model of
mimetic $f(Q)$ specified by Eq.(31) for the scale factor (30) leads to the
large acceleration phase for small redshift, resembling properties described
by the recent observations.

Figure 2: The energy conditions for the mimetic-$f(Q)$ gravity specified by
(31) for different values of parameter $\alpha$, where
$M=t_{0}=1,\lambda=\rho_{0}=a_{0}=2,w=C_{1}=0$.

###  IV.2 Energy conditions

After studying basic properties of the reconstructed mimetic-$f(Q)=Q-6\lambda
M^{2}\big{(}\frac{Q}{6M^{2}}\big{)}^{\alpha}$ gravity, one can determine the
energy constraints that have to be checked for the effective fluid
distribution of the Universe described by HEL. In order to study energy
conditions in general relativity, one should start with the Raychaudhury
equations: Poisson (2004):

$\displaystyle\frac{d\theta}{d\tau}=-\frac{1}{3}\theta^{2}+\omega_{\mu\nu}\omega^{\mu\nu}-\sigma_{\mu\nu}\sigma^{\mu\nu}-R_{\mu\nu}u^{\mu}u^{\nu},\;\;\;\frac{d\theta}{d\tau}=-\frac{1}{2}\theta^{2}+\omega_{\mu\nu}\omega^{\mu\nu}-\sigma_{\mu\nu}\sigma^{\mu\nu}-R_{\mu\nu}k^{\mu}k^{\nu}.$
(56)

where $\theta$, $\sigma_{\mu\nu}$, and $\omega_{\mu\nu}$ denote respectively
expansion factor, shear and rotation. the timelike and null tangent vectors
are denoted as $u^{\mu}$ and $k^{\mu}$. Those equations relate matter-geometry
and behaviour of congruences of curves in the spacetime manifold. Therefore,
for attractive gravity, Raychaudhury equations have to satisfy the presented
conditions:

$\displaystyle
R_{\mu\nu}u^{\mu}u^{\nu}=\Big{(}T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\Big{)}u^{\mu}u^{\nu}\geq
0,\;\;\;\;R_{\mu\nu}k^{\mu}k^{\nu}=\Big{(}T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\Big{)}k^{\mu}k^{\nu}\geq
0.$ (57)

Those geometric constraints (57) yield energy conditions for a perfect fluid
matter distribution:

  * •

NEC: $\rho+p\geq 0$,

  * •

WEC: $\rho\geq 0$, $\rho+p\geq 0$,

  * •

SEC: $\rho+p\geq 0$, $\rho+3p\geq 0$,

  * •

DEC: $\rho\geq 0$, $\rho\geq|p|$,

that obey $DEC\Longrightarrow WEC\Longrightarrow NEC\Longleftarrow SEC$.
Hence, if the NEC is violated, the remaining energy conditions will also be
violated Kaczmarek and Szczȩśniak (2020). For the modified theories of
gravity, energy constraints can be extended, due to the geometric character of
the Raychaudhuri equations. Nevertheless, certain ambiguities might arise,
such as alteration of the energy constraints upon transitioning between
conformally related frames Capozziello et al. (2014). For a comprehensive
discussion concerning energy conditions in modified gravity, we refer to
Capozziello et al. (2014, 2015, 2018).

Hence, for the hybrid cosmic evolution 41 and mimetic-$f(Q)$ gravity specified
by 31, the associated energy conditions are:

$\displaystyle\text{NEC}:\rho_{eff}+p_{eff}=\frac{H^{2}(p+\eta+\rho)-2(\alpha-1)\alpha\lambda
M^{2}\dot{H}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}}{2\left(H^{2}-\alpha\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}\right)}\geq,$ (58)
$\displaystyle\text{WEC}:\rho_{eff}(z)=\frac{-3\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+3H^{2}+\eta+\rho-\frac{U}{2}}{2\left(1-\alpha\lambda\left(\frac{H^{2}}{M^{2}}\right)^{\alpha-1}\right)}\geq
0,$ (59) $\displaystyle\text{DEC}:$
$\displaystyle\rho_{eff}-|p_{eff}|=\frac{-3\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+3H^{2}+\eta+\rho-\frac{U}{2}}{2\left(1-\alpha\lambda\left(\frac{H^{2}}{M^{2}}\right)^{\alpha-1}\right)}$
$\displaystyle-\bigg{|}\frac{4M^{2}(\alpha-1)\alpha\lambda\dot{H}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+6H^{4}-H^{2}\left(6M^{2}\lambda\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+2p+U\right)}{4M^{2}\alpha\lambda\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}-4H^{2}}\bigg{|}\geq
0,$ (60)

$\displaystyle\text{SEC}:\rho_{eff}+3p_{eff}=\frac{-6(\alpha-1)\alpha\lambda
M^{2}\dot{H}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+H^{2}\left(6\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}+3p+\eta+\rho+U\right)-6H^{4}}{2\left(H^{2}-\alpha\lambda
M^{2}\left(\frac{H^{2}}{M^{2}}\right)^{\alpha}\right)}\geq 0.$ (61)

Evolution of the given energy conditions in terms of redshift is presented on
panels (A)-(D) of the Fig.(2), where different possible values of $\alpha$
have been taken into account. In general, three of the four energy conditions
are satisfied, as contribution from effective energy density $\rho_{eff}$
dominates inequalities of NEC and DEC. Then, WEC is naturally satisfied, since
$\rho_{eff}$ stays positive, as it was discussed in the previous section. On
the other hand, SEC is generally violated in the current ($z=0$) and the
future ($z=-1$) stage, leading to the repulsive behaviour of gravity. In such
manner, acceleration of the universe can be currently predicted Visser and
Barcelo (2000); Koussour et al. (2022). Hence, the energy conditions for
mimetic-$f(Q)$ gravity, with the $f(Q)=Q-6\lambda
M^{2}\big{(}\frac{Q}{6M^{2}}\big{)}^{\alpha}$ are well-behaved in the context
of the cosmic dynamics. The resulting equation of state parameter $w_{eff}$ is
dynamical and ’mimics’ quintessence Joyce et al. (2016). Additionally, the
general behaviour of energy conditions for a broad range of parameters
$\alpha$ is presented on the Fig(3) in the Appendix.

##  V Summary and conclusions

The century after the Einstein original formulation of GR, there at three
equivalent descriptions of the gravity, based on the different construction of
the connection in the so-called geometrical trinity of gravity Beltrán Jiménez
et al. (2019); Heisenberg (2023). This intriguing fact led to the possible
existence of even more general framework, where formulations based on
nonmetricity, torsion or curvature theories are particular cases.
Additionally, can this geometrical trinity extend to the modifications of the
GR, merging different ideas and motivations Sebastiani et al. (2017)?

Within this context, the following work explores the new ground in the
modified GR theories by combines two interesting approaches within a single,
unified framework of the mimetic-$f(Q)$ gravity. The description introduced
herein can be understood and interpreted in two ways: i) as a mimetic gravity
with the gravitational action driven by the functional of the nonmetricity
tensor, similarly to the teleparallel mimetic $f(T)$ Chamseddine and Mukhanov
(2013); Chamseddine et al. (2014); Mirza and Oboudiat (2019); ii) as the
$f(Q)$ gravity with the degree of freedom introduced by the presence of the
presureless fluid Sebastiani et al. (2017); Jiménez et al. (2018, 2020).
Similarly to the original curvature-based formulation of mimetic gravity, the
extended $f(Q)$ approach can be presented in the two equivalent actions (8)
and (16), leading to the two equivalent sets of equations of motion. Moreover,
the obtained equations reduce to the usual mimetic gravity for
$f(Q)\rightarrow Q$, proving equivalence between two different constructions
of geometry, in accordance with the geometric trinity in gravity Beltrán
Jiménez et al. (2019); Bahamonde et al. (2023). As a next step, we have
obtained Friedmann equations within the mimetic-nonmetricity framework. In
this regard, extra terms present in the modified $f(Q)$ FLRW equations, can be
viewed as a contribution from the additional degree of freedom implemented via
Lagrange multiplier. After the discussion of the main properties of the
resulting theory, we have used the cosmological reconstruction method to study
the hybrid cosmology characterized by scale factor (30):
$a(t)=a_{0}\Big{(}\frac{t}{t_{0}}\Big{)}^{A}e^{B\big{(}\frac{t}{t_{0}}-1\big{)}}$.
By doing so, the question of finding models that can provide cosmic evolution
for distinct phases is answered, as the hybrid law describing two separate
stages of the universe history can be achieved, within the framework of
interest. Specifically, we have obtained two models from two different
approaches to the reconstruction method. In the first scenario, the Lagrange
multiplier $\eta(t)$ and potential $U(t)$ have been obtained for the assumed
functional of $f(Q)=Q-6\lambda M^{2}\big{(}\frac{Q}{6M^{2}}\big{)}^{\alpha}$.
In the second case, using the inverse reconstruction method for known Lagrange
multiplier $\eta_{0}+\gamma H^{2}$, we have obtained the corresponding $f(Q)$
function and potential $U(\phi)$ in the absence of the matter fields. Note,
that considerations of that type are an inherent feature of the mimetic
gravity, as it ’mimic’ evolution usually driven by some matter fields
Chamseddine and Mukhanov (2013). Then, we have studied the dynamics for the
model with $f(Q)=Q-6\lambda M^{2}\big{(}\frac{Q}{6M^{2}}\big{)}^{\alpha}$ in
terms of the redshift $z$. After analysing behaviour of the hybrid model for
different choices of constants $A$ and $B$, we have studied behaviour of the
effective energy density, pressure and effective EoS parameter where
$A=B=1/2$. The dynamical properties of the reconstructed mimetic gravity are
viable for a wide range of the constants $\alpha$ that correspond to the
different models within nonmetricity function of choice. Our work closes with
the discussion of the energy conditions associated with the analysed scenario.
In this aspect, SEC is generally violated in the present ($z=0$) for a broad
range of constants $\alpha$, a behaviour that aligns with the accelerated
expansion that is currently observed.

In what follows, mimetic-$f(Q)$ (nonmetricity) gravity should be treated as an
equivalent to the mimetic-$f(R)$ (curvature) and torsion mimetic-$f(T)$
(torsion) gravities Nojiri and Odintsov (2014); Mirza and Oboudiat (2019).
Thus, the geometrical trinity of mimetic gravity is completed and expanded to
the arbitrary functional of nonmetricity, torsion or curvature Beltrán Jiménez
et al. (2019). The overview and details of the new trinity will be done in the
future works. Moreover, since the main scope of our work was associated with
the introduction of the mimetic framework within STEGR along with the
cosmological reconstruction and energy conditions, the many aspects of the
theory introduced herein are yet to be studied. Specifically, in order to
better address current cosmology, the models within mimetic-$f(Q)$ theory
should be confronted with the current observations and $\Lambda CDM$, in order
to constrain parameters of the theory and unveil additional insight Lazkoz et
al. (2019); Xu et al. (2019).

Additionally, one can also extend the framework of mimetic-nonmetricity to the
case where coupling between matter and nonmetricty is present, by considering
$f(Q,T)$ extension of $f(Q)$ gravity Gadbail et al. (2021, 2023); Narawade et
al. (2023). Then, the starting action will have the form:

$\displaystyle
S=\int\text{d}^{4}x\sqrt{-g}\frac{1}{2}\Big{[}f(Q,T)-U(\phi)+\eta(g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+1)\Big{]}+\int\text{d}^{4}x\sqrt{-g}\mathcal{L}_{m},$
(62)

from which the field equations can be derived similarly to the steps (8)-(13).
We will focus on the mimetic $f(Q,T)$ gravity in one of the future works.

In summary, the integration of mimetic theories based on nonmetricity
introduces a novel and unexplored path toward further research avenues,
offering additional degrees of freedom. The dark matter and dark energy are
still mysterious, hence pursuing new models within modified gravity is still
viable and theoretically motivated. These secrets may be unravelled in the new
formulation of mimetic gravity. Moreover, since $f(Q)$ the theory is second
order in metric, the STEGR framework may be more suitable in approaching
various challenges normally present within higher-curvature formulations of
(mimetic) gravity, such as emergence of ghosts Sebastiani et al. (2017); Hu et
al. (2023). Therefore, within the domain of mimetic-$f(Q)$ gravity, numerous
implications—both theoretical and observational—remain uncharted, awaiting
further investigations and studies.

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##  VI Appendix

###  VI.1 Full form of $f(Q)$ function reconstructed from the Lagrange
multiplier

The integral of Eq.(43) w.r.t. nonmetricity scalar $Q$ takes the form:

$\displaystyle f(Q)=\frac{12}{t_{0}^{3}}\times$ (63)
$\displaystyle\Bigg{[}\frac{2}{\Big{(}\sqrt{6}\sqrt{q}t_{0}-6B\Big{)}^{3}\Big{(}A\Big{(}-108B^{3}\gamma-54\sqrt{6}B^{2}\gamma\sqrt{q}t_{0}-18Bt_{0}^{2}(\gamma
q-4\eta_{0})+\sqrt{6}\gamma
q^{3/2}t_{0}^{3}\Big{)}+12c_{1}\Big{(}6B+\sqrt{6}\sqrt{q}t_{0}\Big{)}\Big{)}}\times$
$\displaystyle\Big{(}A\Big{(}648B^{6}\gamma-108B^{4}t_{0}^{2}(4\eta_{0}+5\gamma
q)+12\sqrt{6}B^{3}\sqrt{q}t_{0}^{3}(18\eta_{0}+7\gamma
q)+18B^{2}qt_{0}^{4}(\gamma q-12\eta_{0})-6\sqrt{6}Bq^{3/2}t_{0}^{5}(\gamma
q-2\eta_{0})$ (64) $\displaystyle+\gamma
q^{3}t_{0}^{6}\Big{)}-12c_{1}\Big{(}6B^{2}-qt_{0}^{2}\Big{)}\Big{(}6B^{2}-2\sqrt{6}B\sqrt{q}t_{0}+qt_{0}^{2}\Big{)}\Big{)}\Big{(}36\sqrt{6}AB\log\Big{(}\sqrt{6}B+\sqrt{q}t_{0}\Big{)}\Big{(}B^{2}\gamma+\eta_{0}t_{0}^{2}\Big{)}$
$\displaystyle+\sqrt{q}t_{0}\Big{(}A\gamma\Big{(}-90B^{2}-6\sqrt{6}B\sqrt{q}t_{0}+qt_{0}^{2}\Big{)}+36c_{1}\Big{)}\Big{)}$
$\displaystyle+\frac{\sqrt{\frac{2}{3}}At_{0}\Big{(}3B^{2}\gamma+\eta_{0}t_{0}^{2}\Big{)}\Big{(}6B\Big{(}\sqrt{6}B-\sqrt{Q}t_{0}\Big{)}\log\Big{(}\frac{At_{0}\Big{(}\sqrt{6}\sqrt{Q}t_{0}-6B\Big{)}}{6B^{2}-Qt_{0}^{2}}\Big{)}+\Big{(}6\sqrt{6}B^{2}-12B\sqrt{Q}t_{0}+\sqrt{6}Qt_{0}^{2}\Big{)}\tanh^{-1}\Big{(}\frac{\sqrt{Q}t_{0}}{\sqrt{6}B}\Big{)}\Big{)}}{6B^{2}-2\sqrt{6}B\sqrt{Q}t_{0}+Qt_{0}^{2}}$
$\displaystyle-\frac{1}{6}At_{0}\Big{(}3B^{2}\gamma+\eta_{0}t_{0}^{2}\Big{)}\times$
$\displaystyle\Big{\\{}\frac{\Big{(}6B-\sqrt{6}\sqrt{Q}t_{0}\Big{)}\Big{(}\Big{(}6B-\sqrt{6}\sqrt{Q}t_{0}\Big{)}\log\Big{(}\sqrt{6}\sqrt{Q}t_{0}-6B\Big{)}+6B\Big{)}\Big{(}\log\Big{(}\frac{At_{0}\Big{(}\sqrt{6}\sqrt{Q}t_{0}-6B\Big{)}}{6B^{2}-Qt_{0}^{2}}\Big{)}+\log\Big{(}\frac{\sqrt{6}B}{t_{0}}+\sqrt{Q}\Big{)}\Big{)}}{6B^{2}-2\sqrt{6}B\sqrt{Q}t_{0}+Qt_{0}^{2}}$
$\displaystyle-6\Big{(}\text{Li}_{2}\Big{(}\frac{1}{12}\Big{(}\frac{\sqrt{6}\sqrt{Q}t_{0}}{B}+6\Big{)}\Big{)}+\log\Big{(}\frac{\sqrt{6}B}{t_{0}}+\sqrt{Q}\Big{)}\log\Big{(}\frac{1}{2}-\frac{\sqrt{Q}t_{0}}{2\sqrt{6}B}\Big{)}\Big{)}-\frac{6\sqrt{6}B\log\Big{(}\frac{\sqrt{6}B}{t_{0}}+\sqrt{Q}\Big{)}}{\sqrt{6}B-\sqrt{Q}t_{0}}+6\tanh^{-1}\Big{(}\frac{\sqrt{Q}t_{0}}{\sqrt{6}B}\Big{)}\Big{\\}}$
$\displaystyle-\frac{1}{6}At_{0}\Big{(}3B^{2}\gamma+\eta_{0}t_{0}^{2}\Big{)}\times$
$\displaystyle\Big{\\{}-6\Big{(}\text{Li}_{2}\Big{(}\frac{\sqrt{Q}t_{0}}{\sqrt{6}B}+1\Big{)}+\log\Big{(}\frac{\sqrt{6}B}{t_{0}}+\sqrt{Q}\Big{)}\log\Big{(}-\frac{\sqrt{Q}t_{0}}{\sqrt{6}B}\Big{)}\Big{)}+6\Big{(}\text{Li}_{2}\Big{(}\frac{1}{12}\Big{(}\frac{\sqrt{6}\sqrt{Q}t_{0}}{B}+6\Big{)}\Big{)}$
$\displaystyle+\log\Big{(}\frac{\sqrt{6}B}{t_{0}}+\sqrt{Q}\Big{)}\log\Big{(}\frac{1}{2}-\frac{\sqrt{Q}t_{0}}{2\sqrt{6}B}\Big{)}\Big{)}-\frac{6\sqrt{6}B\log\Big{(}\frac{\sqrt{6}B}{t_{0}}+\sqrt{Q}\Big{)}}{\sqrt{6}B-\sqrt{Q}t_{0}}+6\tanh^{-1}\Big{(}\frac{\sqrt{Q}t_{0}}{\sqrt{6}B}\Big{)}$
$\displaystyle+\frac{1}{6B^{2}-2\sqrt{6}B\sqrt{Q}t_{0}+Qt_{0}^{2}}\Big{[}\Big{(}6B-\sqrt{6}\sqrt{Q}t_{0}\Big{)}\Big{(}\Big{(}6B-\sqrt{6}\sqrt{Q}t_{0}\Big{)}\log\Big{(}\sqrt{6}\sqrt{Q}t_{0}\Big{)}$
$\displaystyle+\Big{(}\sqrt{6}\sqrt{Q}t_{0}-6B\Big{)}\log\Big{(}\sqrt{6}\sqrt{Q}t_{0}-6B\Big{)}+6B\Big{)}\Big{(}\log\Big{(}\frac{At_{0}\Big{(}\sqrt{6}\sqrt{Q}t_{0}-6B\Big{)}}{6B^{2}-Qt_{0}^{2}}\Big{)}+\log\Big{(}\frac{\sqrt{6}B}{t_{0}}+\sqrt{Q}\Big{)}\Big{)}\Big{]}\Big{\\}}\Bigg{]},$
(65)

where $Li_{2}(z)$ is Spence’s (dilogarithm) function Kirillov (1995).

###  VI.2 General behaviour of energy conditions

Figure 3: General behaviour of energy conditions across different values of
redshift $z$ and $\alpha$


# Breathing mode frequencies of a rotating Fermi gas in the BCS-BEC crossover
region

Theja N. De Silva  Department of Physics, Applied Physics and Astronomy, The
State University of New York at Binghamton, Binghamton, New York 13902, USA.

###### Abstract

We study the breathing mode frequencies of a rotating Fermi gas trapped in a
harmonic plus radial quartic potential. We find that as the radial
anharmonicity increases, the lowest order radial mode frequency increases
while the next lowest order radial mode frequency decreases. Then at a
critical anharmonicity, these two modes merge and beyond this merge the cloud
is unstable against the oscillations. The critical anharmonicity depends on
both rotational frequency and the chemical potential. As a result of the large
chemical potential in the BCS regime, even with a weak anharmonicity the
lowest order mode frequency increases with decreasing the attractive
interaction. For large enough anharmonicities in the weak coupling BCS limit,
we find that the excitation of the breathing mode frequencies make the atomic
cloud unstable.

##  I I. Introduction

The rapid progress of ultra-cold atomic gas experiments provides unique
opportunities for well controlled studies of quantum many body physics. For
the case of Fermi atomic systems, the possibility of controlling the s-wave
scattering length ($a$) between two different spin components allows to
control the interaction by using a magnetically tuned Feshbach resonance fb .
This unique capability allows one to investigate the cross-over between the
weakly interacting BCS regime (the regime where $a\rightarrow 0^{-}$) and
Bose-Einstein condensate of dimers (the regime where $a\rightarrow 0^{+}$) co
. These two regimes meet in strongly interacting limit where the scattering
length is divergent and at this unitarity limit, the physics is expected to be
universal uni .

The appearance of quantized vortices of a quantum fluid under rotation offers
direct evidence of superfluidity. For example, the observation of quantized
circulation in a rotating superfluid 4He vortexHe  and the observation of
vortex lattice in a rotating Fermi gas of 6Li vortexMIT  are two classic
demonstrations of phase coherence in a superfluid. These are analog to the
vortex lattice in type-II superconductors in the presence of a magnetic filed.
These vortices melt as the magnetic field increases and then the
superconductors turn into normal at sufficiently large magnetic fields. For
the case of rapidly rotating Fermi gasses, the force due to the trapping
frequency almost balances the centrifugal force and superfluid cloud spreads
in the plane perpendicular to the rotation axis. At the limit of very large
rotation, the theory predicts that the atomic system enters into the
fractional quantum Hall regime ho ; fqhe . However, the fractional quantum
Hall window is expected to be very small and inversely proportional to the
number of atoms in the trap. A possible way of stabilizing the fractional
quantum Hall regime is to add a positive quartic trapping potential.

In this paper, we study the collective breathing mode frequencies of a
rotating Fermi gas in the presence of a quartic trapping potential by using a
hydrodynamic approach. A negative, but small quartic term is always present
with the Gaussian optical potentials in current experimental setups while
added positive quartic term ensures the stability of the fast rotating regime.
Thermodynamic properties of a Bose gas confined in harmonic plus quartic
potential trap can be found in ref. quarticBose . As breathing mode
frequencies are very sensitive to the equation of state, these dynamical
quantities can be used as tests for various theories. The breathing mode
frequencies have been measured for non-rotating Fermi systems in the BCS-BEC
region modesEX1 ; modesEX2 . In most of the parameter regions, experimental
results agree well with the hydrodynamic approaches, variational approaches
and sum rule approaches modesTH . However, the measured finite temperature
axial and radial breathing mode frequencies show a striking increase in the
intermediate BCS regime modesEX1 ; modesEX2 , in contrast to zero temperature
theoretical calculations in a harmonic trap. Furthermore, experimentalists
were unable to measure the breathing mode frequencies in the weak coupling BCS
limit. The deviation of the experimental data from theoretical results and the
lack of experimental data in the weak coupling BCS limit were believed to be
due the large Landau damping when the superfluid energy gap is much smaller
than the collective oscillation energies. With inclusion of a positive quartic
term in the trapping potential, we find somewhat similar deviation of the
breathing mode frequencies in the intermediate regions of BCS regime. We find
that the breathing mode frequencies deviate significantly from the modes
frequencies calculated in a harmonic trap and the atomic cloud is unstable
against the breathing mode oscillations at larger chemical potentials. As the
chemical potential is larger in the weak coupling BCS limit, excitation of
breathing mode frequencies make the atomic cloud unstable. It should be noted
that the Gaussian optical trap potential provides a negative quartic term in
the trapping potential in experimental setups. We investigate the effect of
negative quartic term and find that the breathing mode frequencies tend to
decrease in the entire BCS-BEC crossover region.

This paper is organized as follows. In section II, we present the derivation
of breathing mode frequencies using a hydrodynamic approach. In section III,
we present our results together with a discussion. Finally in section IV, we
draw our conclusions.

##  II II. Formalism

We consider a rotating Fermi atomic system trapped in a harmonic plus radial
quartic potential in the BCS-BEC crossover region. The trapping potential is

$V_{ex}(r,z)=\frac{1}{2}M\omega_{r}^{2}r^{2}+\frac{1}{2}M\omega_{z}^{2}z^{2}+\frac{K}{4}r^{4}$
(1)

where $M$ is the atom mass, $\omega_{i}$’s are the harmonic trapping
frequencies, and $r^{2}=x^{2}+y^{2}$. The Fermi atomic system rotates about
the $z$-axis at frequency $\Omega$.

We restrict ourselves to the case of large number of vortices in the system
where the wavelength of the oscillation frequencies is much larger than the
inter-vortex distance. This condition always satisfies as the typical
wavelength of the lowest mode oscillations is of the order of the system size.
Further, we assume that the vortices are uniformly distributed in the
superfluid so that we do not have to consider microscopic details of the
single vortices. These assumptions are valid at the limit of large rotations
where the atomic cloud spreads in the plane perpendicular to the rotation.
Within this diffused vorticity approximation vorticity , diffuse vorticity is
given by $\mathbf{\nabla}\times\mathbf{v}=2\mathbf{\Omega}$, where superfluid
velocity is given by $\mathbf{v}=(\hbar/2M)\nabla\theta$. The local superfluid
density $n$ and the local phase $\theta$ are related through the wave function
$\psi=\sqrt{n}e^{i\theta}$. The uniform vortex density is given by
$n_{v}=2M\Omega/\hbar$.

Assuming local equilibrium, we start with the continuity and Euler equations
of rotational hydrodynamics,

$\frac{\partial n}{\partial t}=-\mathbf{\nabla}\cdot[n(r)\mathbf{v}]$ (2)

and

$\displaystyle M\frac{\partial\mathbf{v}}{\partial
t}=-\mathbf{\nabla}[\frac{1}{2}M\mathbf{v}^{2}+V_{ex}(r,z)-\frac{1}{2}\Omega^{2}r^{2}+\mu(n)]$
$\displaystyle+2M\mathbf{v}\times\mathbf{\Omega}+M\mathbf{v}\times\mathbf{\nabla}\times\mathbf{v}$
(3)

This hydrodynamic description is valid as long as the collisional relaxation
time $\tau$ is much smaller than the inverse of the oscillation frequencies;
$\omega\tau<<1$. The equation of state enters through the density dependent
local chemical potential $\mu(n)$. We fix the local chemical potential by
introducing the equation of state in the form of $\mu(n)\propto n^{\gamma}$.
As we will discuss in the next subsection, the polytropic index $\gamma$ is
calculated by the method proposed by Manini and Salasnich manini  in the
entire BCS-BEC crossover region. Linearizing the density $n$ and the
superfluid velocity $\mathbf{v}$ around their equilibrium values as
$n=n_{0}(r)+\delta n$, $\mathbf{v}=\mathbf{v}_{0}+\delta\mathbf{v}$ and
$\mu(n)=\mu(n_{0})+\delta\mu$ with $\delta\mu=(\partial\mu/\partial
n)|_{n=n_{0}}\delta n$, we obtain the linearized version of the hydrodynamic
equations.

$\frac{\partial\delta n}{\partial
t}=-\mathbf{\nabla}\cdot[n_{0}(r)\delta\mathbf{v}]$ (4)

and

$M\frac{\partial\delta\mathbf{v}}{\partial
t}=-\mathbf{\nabla}\delta\mu+2M\delta\mathbf{v}\times\mathbf{\Omega}$ (5)

Starting from these two linearized equations, collective breathing mode
frequencies have been calculated in ref.gosh  and ref.ant  for a harmonic
trap. As the authors have used two different ansatz for the velocity
fluctuation, they produce two different results for the breathing mode
frequencies. In this paper we closely follow the approach adopted in ref.ant ,
generalizing the theory to an anharmonic trap. The ansatz used in ref.ant
ensures the conservation of angular momentum properly. In order to solve the
linearized equations for the breathing mode frequencies, we take the
equilibrium density in the local density approximation as
$n_{0}(r)\propto[\mu_{0}-(1/2)M(\omega_{r}^{2}-\Omega_{0}^{2})r^{2}-(1/2)M\omega_{z}^{2}z^{2}-(K/4)r^{4}]^{1/\gamma}$
and use following variational ansatz for the density fluctuations and velocity
fluctuations.

$\displaystyle\delta\mathbf{v}=\\{\delta\mathbf{\Omega_{1}}\times\mathbf{r}+\delta\mathbf{\Omega_{2}}\times
r^{2}\mathbf{r}+\mathbf{\nabla}[\alpha_{\perp}r^{2}+\alpha_{z}z^{2}+\beta
r^{4}]\\}$ $\displaystyle\times\exp[-i\omega t]$ (6)

and

$\delta
n=n_{0}^{1-\gamma}\\{a_{0}+a_{\perp}r^{2}+a_{z}z^{2}+br^{4}\\}\exp[-i\omega
t]$ (7)

The first two terms $\delta\mathbf{\Omega_{1}}$ and
$\delta\mathbf{\Omega_{2}}$ in Eq. (II) are parallel to the axis of rotation
and guarantee that angular momentum is conserved during the oscillations.
Substituting these two ansatz into Eq. (4) and Eq. (5), we derive four linear
equations for the variational parameters $a_{0}$, $a_{\perp}$, $a_{z}$, and
$b$. These linear equations yield three non-zero solutions for the breathing
mode frequencies $\omega_{m}\equiv\omega/\omega_{r}$ as roots of the following
equation.

$A+B\omega_{m}^{2}+C\omega_{m}^{4}+\omega_{m}^{6}=0$ (8)

with,
$A\equiv-(1-\zeta^{2})^{2}\delta^{2}[64d\gamma(\gamma+1)+48\gamma^{2}+56\gamma+16]-(1-\zeta^{2})\zeta^{2}\delta^{2}(32\gamma^{2}+104\gamma+48)-16\zeta^{4}\delta^{2}(\gamma+2)$,
$B\equiv(1-\zeta^{2})^{2}[8(\gamma+1)(2\gamma+1)+16d\gamma(\gamma+2)]+(1-\zeta^{2})[\delta^{2}(8\gamma^{2}+26\gamma+12)+\zeta^{2}(40\gamma+24)]+8\zeta^{2}\delta^{2}(\gamma+2)+16\zeta^{4}$
and $C\equiv(1-\zeta^{2})(2-10\gamma)-\delta^{2}(\gamma+2)-8$. The constants,
$\zeta\equiv\Omega/\omega_{r}$, $\delta\equiv\omega_{z}/\omega_{r}$, and
$d\equiv[1/(1-\zeta^{2})^{2}](K\hbar/M^{2}\omega_{r}^{3})(\mu_{0}/\hbar\omega_{r})$
are a set of dimensionless parameters. The three solutions of Eq. (8) are the
lowest order axial breathing mode frequency $\omega_{1}$ and the lowest and
next lowest order radial breathing mode frequencies $\omega_{2}$ and
$\omega_{3}$.

###  II.1 The Effective polytropic index and the chemical potential in the
BCS-BEC crossover region

We use the proposal made by Manini and Salasnich manini  to calculate the
effective polytropic index $\gamma$ and the chemical potential $\mu$ in the
BCS-BEC crossover region. In the weak coupling BCS limit ($a\longrightarrow
0^{-}$) and the unitarity limit ($a\longrightarrow\infty$), the polytropic
index is $\gamma=2/3$. In the deep BEC limit ($a\longrightarrow 0^{+}$), the
polytropic index is $\gamma=1$. In the BCS-BEC crossover regime, the
scattering length’s dependence on $\gamma$ is given by manini

$\gamma=\frac{2/3-2y\epsilon^{\prime}(y)/5+y^{2}\epsilon^{\prime\prime}(y)/15}{\epsilon(y)-y\epsilon^{\prime}(y)/5}$
(9)

with the parameter $y=1/(k_{f}a)$ is the interaction parameter with $k_{f}$
being the Fermi wave vector. The function $\epsilon(y)$ is related to the
energy per atom given by $E=(3/5)E_{f}\epsilon(y)$, where
$E_{f}=\hbar^{2}k_{f}^{2}/2M$ is the Fermi atomic energy of a non-interacting
Fermi system in the trap. Above
$\epsilon^{\prime}(y)=\partial\epsilon(y)/\partial y$ and the double prime
indicates the second derivative of the function on its argument. Using the
data presented in reference astra , Manini and Salasnich manini  used a data
fitting scheme to derive an analytical form of the function $\epsilon(y)$ in
the entire BCS-BEC region,

$\epsilon(y)=\alpha_{1}-\alpha_{2}\arctan\biggr{[}\alpha_{3}y\frac{\beta_{1}+|y|}{\beta_{2}+|y|}\biggr{]}$
(10)

Two different sets of parameters are proposed for $\alpha_{i}$’s and
$\beta_{i}$’s in the BCS regime ($y<0$) and the BEC regime ($y>0$). In the BCS
regime, the parameters are $\alpha_{1}=0.42$, $\alpha_{2}=0.3692$,
$\alpha_{3}=1.044$, $\beta_{1}=1.4328$, and $\beta_{2}=0.5523$. In the BEC
regime, the parameters are $\alpha_{1}=0.42$, $\alpha_{2}=0.2674$,
$\alpha_{3}=5.04$, $\beta_{1}=0.1126$, and $\beta_{2}=0.4552$. The expression
for the chemical potential $\mu$ is given by manini

$\mu=E_{f}[\epsilon(y)-y\epsilon^{\prime}(y)/5]$ (11)

We determine the Fermi energy $E_{f}$ of a non-interacting Fermi system in a
harmonic plus radial quartic potential through the number equation.

$N=\frac{1}{15\delta}\sqrt{\frac{M^{2}\omega_{r}^{3}}{\hbar|K|}}\biggr{(}\frac{E_{f}}{\hbar\omega_{r}}\biggr{)}^{5/2}f(E_{f})$
(12)

We define the function $f(E_{f})$ as

$\displaystyle f(E_{f})=\pm 8\biggr{(}\pm
1+\frac{(1-\zeta^{2})^{2}}{4}\frac{1}{|\tilde{K}|\tilde{E_{f}}}\biggr{)}^{\frac{5}{2}}$
$\displaystyle\mp\sqrt{\frac{(1-\zeta^{2})^{2}}{4}\frac{1}{|\tilde{K}|\tilde{E_{f}}}}$
$\displaystyle\times\biggr{[}15+5(1-\zeta^{2})^{2}\frac{1}{\tilde{K}\tilde{E_{f}}}+\frac{1}{2}(1-\zeta^{2})^{4}\frac{1}{(\tilde{K}\tilde{E_{f}})^{2}}\biggr{]}$
(13)

where the scaled parameters are $\tilde{K}\equiv\hbar K/(M^{2}\omega_{r}^{3})$
and $\tilde{E_{f}}\equiv E_{f}/(\hbar\omega_{r})$. The upper and lower signs
are corresponding to $K>0$ and $K<0$ respectively. For the case of harmonic
trap, the Fermi energy is
$E_{f}=\hbar\omega_{r}[3N\delta(1-\zeta^{2})]^{1/3}$.

Figure 1: Central chemical potential ($\mu_{0}$) of a non-rotating Fermi gas
as a function of interaction parameter $1/(k_{f}a)$ for various values of
$\tilde{K}$. From top to bottom $\tilde{K}=0.05$ (long black dashed line), and
$0.01$ (short black dashed line), $0$ harmonic trap (black solid line),
$-0.005$ (short gray dashed line), and $-0.01$ (long gray dashed line). For
the calculation, we use $N=2.0\times 10^{6}$ and $\delta=0.045$.

##  III III. results and discussion

Figure 2: The lowest order radial breathing mode frequencies of a rotating
Fermi gas in a harmonic trap with aspect ratio $\delta=0.045$. The rotation
frequencies are $\zeta=0$ (solid line), $0.5$ (long dashed line) and $0.9$
(short dashed line).

In FIG. 1, we plot the central chemical potential of a non-rotating Fermi
system calculated from Eq. (11) as a function of inverse scattering length for
two different representative values of anharmonicity. We use $N=2.0\times
10^{6}$ number of atoms in the trap with $\delta=0.045$. Our calculation shows
a kink at unitarity limit due to the discontinuity in the function
$\epsilon(y)$ proposed by Manini and Salasnich manini  (This kink appears in
all the calculated macroscopic quantities).

Solving Eq. (8) for the case of harmonic potential trap ($K=0$) in a rotating
($\zeta\neq 0$) Fermi system, the lowest axial and radial breathing modes
frequencies are given by

$\displaystyle\omega_{m}^{2}=\frac{1}{2}\\{\gamma(2+\delta^{2}-2\zeta^{2})+2(1+\delta^{2}+\zeta^{2})$
$\displaystyle\pm\\{[\gamma(2+\delta^{2}-2\zeta^{2})+2(1+\delta^{2}+\zeta^{2})]^{2}-$
$\displaystyle 8\delta^{2}[\gamma(-3+\zeta^{2})-2(1+\zeta^{2})]\\}^{1/2}\\}.$
(14)

For the isotropic trap ($\delta=1$), at non-interacting limit and at unitarity
limit ($\gamma=2/3$), the mode frequencies are $\omega_{m}=2$ and
$\omega_{m}=\sqrt{2+2\zeta^{2}/3}$. In the deep BEC limit ($\gamma=1$), the
mode frequencies are $\omega_{m}=\sqrt{(1/2)(7\pm\sqrt{9-8\zeta^{2}})}$. For
the case of harmonic potential ($K=0$) in a non-rotating ($\zeta=0$) limit,
Eq. (III) reduces to

$\displaystyle\omega_{m}^{2}=\frac{1}{2}\\{2(1+\delta^{2})+\gamma(2+\delta^{2})$
$\displaystyle\pm\sqrt{[2(1+\delta^{2})+\gamma(2+\delta^{2})]^{2}-8(2+3\gamma)\delta^{2}}\\}.$
(15)

For the case of highly anisotropic limit ($\delta<<1$), the two mode
frequencies at $\gamma=2/3$ are $\omega_{1}/\omega_{z}=\sqrt{12/5}$ and
$\omega_{2}/\omega_{r}=\sqrt{10/3}$ as expected. For this case, the two mode
frequencies at $\gamma=1$ are $\omega_{1}/\omega_{z}=\sqrt{5/2}$ and
$\omega_{2}/\omega_{r}=2$. In the BCS-BEC crossover region, the lowest order
breathing modes frequencies are calculated from Eq. (III) by using the
$\gamma$ from Eq. (9). The results for several representative values of
$\zeta$ are given in FIG. 2 and FIG. 3.

Figure 3: The lowest order axial breathing mode frequencies of a rotating
Fermi gas in a harmonic trap with aspect ratio $\delta=0.045$. The rotation
frequencies are $\zeta=0$ (solid line), $0.5$ (long dashed line) and $0.9$
(short dashed line). Figure 4: The two lowest radial breathing mode
frequencies as a function of $d=[1/(1-\zeta^{2})^{2}]\tilde{K}\tilde{\mu}$.
Black (lowest mode) and gray (second lowest) solid lines are the mode
frequencies for weakly interacting limit and unitarity limit($\gamma=2/3$).
Black (lowest mode) and gray (second lowest) dashed lines are the mode
frequencies for deep BEC limit($\gamma=1$). The value of
$\delta=\omega_{z}/\omega_{r}=0.045$. The two modes merge at some critical $d$
and beyond this critical value, the atomic cloud is unstable against the
breathing mode oscillations.

We solve Eq. (8) for the breathing mode frequencies for various values of $K$
in both rotating and non-rotating Fermi systems. We calculate the central
chemical potential $\mu_{0}$ for fixed number of atoms $N=2.0\times 10^{6}$ in
the trap. We find that as
$d\equiv[1/(1-\zeta^{2})^{2}](K\hbar/M^{2}\omega_{r}^{3})(\mu_{0}/\hbar\omega_{r})=[1/(1-\zeta^{2})^{2}]\tilde{K}\tilde{\mu}$
increases, the lowest order radial breathing mode frequency increases while
the next lowest order breathing mode frequency decreases. Then at a critical
value of $d=d_{c}$, these two modes merge and beyond this critical $d_{c}$,
the atomic cloud is unstable against the oscillations. FIG. 4 shows the lowest
and the next lowest order radial mode frequencies at $\gamma=2/3$ (non-
interacting limit and unitarity limit) and $\gamma=1$ (deep BEC limit).

As evidenced by FIG. 5, the lowest order axial breathing mode frequencies are
almost insensitive to the radial anharmonicities. FIG. 5 shows the axial
breathing mode frequencies for various values of rotational frequencies as a
function radial anharmonicities at unitarity.

Figure 5: Axial breathing mode frequencies as a function of $\tilde{K}$ for
$\zeta=0$ (black), 0.3 (long dashed), 0.6 (short dashed) and, 0.9 (dotted) at
unitarity. We use the values $\delta=0.045$ and $N=2.0\times 10^{6}$.

In the BEC limit where $\gamma=1$, we calculate the lowest and next lowest
radial breathing mode frequencies as a function of radial anharmonicity
$\tilde{K}$ for three representative values of rotational frequencies $\zeta$.
We fixed the number of atoms to be $2.0\times 10^{6}$ and
$\delta=\omega_{Z}/\omega_{r}=0.045$. As shown in FIG. 6, as we increase
$\tilde{K}$ the lowest order radial breathing mode frequency increases, while
the next lowest order radial breathing mode frequency decreases. Further
increase of $\tilde{K}$ merges these two modes and beyond this merging point
the atomic cloud is unstable against the oscillations.

Figure 6: Radial breathing mode frequencies at BEC limit ($\gamma=1$) for
$\zeta=0$ (solid line), $0.3$ (long dashed line) and $0.5$ (short dashed
line). The black lines are lowest breathing mode and the gray lines are second
lowest breathing mode in an anharmonic trap. The value of
$\delta=\omega_{z}/\omega_{r}=0.045$ and the atom number is $N=2.0\times
10^{6}$.  Figure 7: The two lowest radial breathing mode frequencies at for
$\tilde{K}=0.005$ (solid line), $0.01$ (long dashed line) and $0.05$ (short
dashed line). The black lines are lowest breathing mode and the gray lines are
second lowest breathing mode in an anharmonic trap. The black dotted line is
the lowest breathing mode frequency in a harmonic trap. The gray dotted lines
are the breathing mode frequencies at $\tilde{K}=-0.005$. The value of
$\delta=\omega_{z}/\omega_{r}=0.045$ and the atom number is $N=2.0\times
10^{6}$.  Figure 8: The lowest order radial breathing mode frequencies at for
$\tilde{K}=0.005$, $0.025$ and $0.001$ (Top to bottom). The black lines is the
lowest order breathing mode in a harmonic trap. The value of
$\delta=\omega_{z}/\omega_{r}=0.045$ and the atom number is $N=2.0\times
10^{6}$. The dots are the experimental data for comparison modesEX2 .

The lowest order and next lowest order radial breathing mode frequencies as a
function of the interaction parameter [$1/(k_{f}a)$] are shown in FIG. 7. In
the presence of radial anharmonicity, the lowest order mode frequency tends to
increase in the BCS regime while the next lowest order breathing mode
frequency tends to decrease. This deviation becomes large as the anharmonicity
increases. As we have discussed before, at larger $K\mu_{0}$ values the two
lowest order modes merge and the cloud is unstable against the breathing mode
oscillations beyond this point. The data in the FIG. 8 shows the same
information as FIG. 7, but we plot only the lowest order radial breathing mode
frequencies for small quartic potentials together with experimental data from
ref. modesEX2 .

##  IV IV. Conclusions

We have discussed the breathing mode frequencies of a rotating Fermi gas
trapped in a harmonic plus radial quartic potential. We find that the radial
breathing mode frequencies strongly depend on the rotation and anharmonicity
through parameter
$d=[1/(1-\zeta^{2})^{2}](K\hbar/M^{2}\omega_{r}^{3})(\mu_{0}/\hbar\omega_{r})$.
As $d$ increases, the lowest order radial breathing mode’s frequency increases
and the next lowest order mode decreases. Beyond some critical $d_{c}$, these
two modes merge and the cloud is unstable against the oscillations.

As the chemical potential is large in the intermediate BCS regime, even with a
very weak quartic potential the parameter $d$ is large. As a result, the
lowest order breathing mode frequency increases in the intermediate BCS
regime. Even though the Gaussian optical trap potential provides a negative
anharmonic term in the trapping potential, this positive anharmonic behavior
has been seen in recent experiments modesEX1 ; modesEX2 . In the weak coupling
BCS limit, the chemical potential is even larger so that we find the atomic
cloud is unstable against the oscillations at large positive anharmonicities.
For negative quartic potentials, the breathing mode frequencies tend to
decrease in the BCS-BEC crossover region.

##  V V. Acknowledgements

This work was supported by the Binghamton University. We are very grateful to
Kaden Hazzard for very enlightening discussions and critical comments on the
manuscript.

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