11institutetext:  Grand AccÃlÃrateur National dâIons Lourds (GANIL), CEA/DRF-
CNRS/IN2P3, Caen, France 22institutetext: Physics Division, Argonne National
Laboratory, Lemont, USA 33institutetext: LP2IB, UniversitÃ de Bordeaux,
CNRS/IN2P3, Gradignan, France 44institutetext: Institut Laue-Langevin,
Grenoble, France

# Understanding the cosmic abundance of 22Na: lifetime measurements in 23Mg

C. FougÃres 1122<EMAIL_ADDRESS>F. de Oliveira Santos 11<EMAIL_ADDRESS>N. A. Smirnova 33 C. Michelagnoli 11 5 5 GANIL-E710 / AGATA collaborations

###### Abstract

Simulations of explosive nucleosynthesis in novae predict the production of
22Na, a key astronomical observable to constrain nova models. Its gamma-ray
line at 1.275 MeV has not yet been observed by the gamma-ray space telescopes.
The 20Ne/22Ne ratio in presolar grains, a possible tool to identify nova
grains, also depends on 22Na produced. Uncertainties on its yield in classical
novae currently originate from the rate of the 22Na(p, $\gamma$)23Mg reaction.
At peak novae temperatures, this reaction is dominated by a resonance at
E${}_{\text{R}}$=0.204 MeV, corresponding to the $E_{x}$=7.785 MeV excited
state in 23Mg. The resonance strengths measured so far disagree by one order
of magnitude. An experiment has been performed at GANIL to measure the
lifetime and the proton branching ratio of this key state, with a femtosecond
resolution for the former. The reactions populating states in 23Mg have been
studied with a high resolution detection set-up, i.e. the particle VAMOS,
SPIDER and gamma tracking AGATA spectrometers, allowing the measurements of
lifetimes and proton branchings. We present here a comparison between
experimental results and shell-model calculations, that allowed us to assign
the spin and parity of the key state. Rather small values obtained for reduced
$M1$ matrix elements, $|M(M1)|\lesssim 0.5$ $\mu_{N}$, and proton
spectroscopic factors, $C^{2}S_{\text{p}}$<10-2, seem to be beyond the
accuracy of the shell model. With the reevaluated 22Na(p, $\gamma$)23Mg rate,
the 22Na detectability limit and its observation frequency from novae are
found promising for the future space telescopes.

Many nuclei are synthesized in explosive stellar environments, like classical
novae, supernovae and neutron star mergers. Novae are transient astronomical
events and the most frequent explosive events in our galaxy, after X-ray
bursts. These explosions happen in a close binary stellar system consisting of
a white dwarf accreting hydrogen-rich matter from its companion. This matter
is progressively compressed at the white dwarf surface until the conditions
required for hydrogen combustion are reached. Nucleosynthesis takes place
during this explosive stage while a part of the new material is being ejected
in the interstellar medium. Beyond this well understood picture, large
uncertainties remain in our knowledge of novae, for example  the white dwarf
initial conditions (mass, luminosity), the accretion rate and its composition,
the amount of admixed white dwarf material with the accreta, the ejected mass,
to list a few bookJose . After the explosion, gas is crystallized, imprinting
the ejecta composition into grains. The isotopic composition of presolar
grains of a putative nova origin shed some lights into these stellar
explosions bookJose ; meteo ; meteoAl . Furthermore, novae are thought to end
as supernovae of type Ia SNIaref which are used as standard candles to
determine distances and, so, to estimate the cosmic expansion. Therefore,
astronomical observables are required to improve our understanding of novae:
22Na has revealed itself as a key candidate.

The lifetime of the 22Na radioisotope ($\tau$=2.6 yr) makes it a good
radioactive tracer of novae dhiel since: (i) the lifetime is longer than the
duration of the opaque phase following the explosion (on timescales of hours),
(ii) the decay time is short enough to ensure space-time correlation. Hence,
22Na is a promising candidate for observing $\gamma$-ray emissions from novae
refNalifetime . The emitted $\gamma$ rays at $E_{\gamma}$=1.275 MeV (and 0.511
MeV) are expected from hours to months after the explosion MHernanz . They
were searched for by the missions CGRO/COMPTEL refCOMPTEL and INTEGRAL/SPI
refINTEGRAL , leading only to upper limits for the ejected amount of 22Na
(left Fig. 1).

Figure 1: Left: The expected 22Na mass ejected from a nova is given as a
function of the strength of the E${}_{\text{R}}$=0.204 MeV resonance, from
simulations with the MESA code Paxton2013 . Colored lines mark the Integral
and Comptel space telescope lower sensitivity limits. Right: Levels scheme of
23Mg and resonance energies E${}_{\text{R}}$ of the proton capture reaction.
The Gamow window of novae is shown with the red zone.

New space telescopes are in preparation, namely COSI RefNextCSIastroGam and
e-ASTROGAM refeASTRO , with sensitivities 20 times higher than INTEGRAL. The
radioisotope 22Na can be also measured, indirectly, by the over-abundance of
the daughter nucleus 22Ne in presolar grains. Neon noble gas is not expected
to easily condense into dust grains and, so, the presence of 22Ne implies in-
situ $\beta^{+}$ decays of 22Na. A 22Ne excess was observed Black1972 .
Therefore, it is critical to well quantify 22Na nucleosynthesis in novae.

ONe novae are predicted to be the main site of 22Na production Jose1999 . In
these sites, the matter is composed of a mixture between the transferred
proton rich gas and the white dwarf envelope with mainly oxygen and neon. The
nucleosynthesis involves H to Ca isotopes, proton capture reactions and
$\beta^{+}$ decays bookJose . The production pathway of 22Na is
20Ne(p,$\gamma$)21Na(p,$\gamma$)22Mg($\beta^{+}$)22Na at high temperatures and
20Ne(p,$\gamma$)21Na($\beta^{+}$)21Ne(p,$\gamma$)22Na otherwise. The thermal
window of novae is [0.05, 0.5] GK. The destruction of 22Na proceeds via
$\beta^{+}$ decays and 22Na(p,$\gamma$)23Mg. The latter reaction is the least
known, resulting in a factor of 10 uncertainty in the predicted amount of 22Na
ejected (left Fig. 1). Since the direct capture contribution was found to be
negligible thesisFougeres ; Stegmuller1996 , the total reaction rate is
determined by several narrow resonances in the Gamow window $<\sigma
v>_{\text{tot}}=\sum_{i}(\frac{2\pi}{\mu_{(^{22}\text{Na,p})}\text{k}_{\text{BT}}})^{\frac{3}{2}}\hbar^{2}(\omega\gamma)_{i}\exp(-\frac{\text{E}_{\text{R}i}}{\text{k}_{\text{BT}}})$
(right Fig. 1). A direct measurement at astrophysical energies showed that a
resonance at E${}_{\text{R}}=0.204$ MeV dominates the 22Na(p,$\gamma$)23Mg
rate Direct . However, its contribution disagrees by a factor 3 or even more
with other measurements Stegmuller1996 ; Seuthe1990 . Such a direct
measurement is complex due to the radioactive nature of the target. The
resonance strength can be determined indirectly from the spectroscopic
properties of the $E_{x}$=7.785 MeV state in 23Mg, i.e.
$\omega\gamma=\frac{2\text{J}_{{}^{23}\text{Mg}}+1}{(2\text{J}_{{}^{22}\text{Na}}+1)(2\text{J}_{\text{p}}+1)}\frac{\Gamma_{\text{p}}\times\Gamma_{\gamma}}{\Gamma_{\text{tot}}}$.
The total and proton widths can be experimentally accessed via the state
lifetime $\tau=\hbar/\Gamma_{\text{tot}}$ and proton branching ratio
$BR_{p}=\Gamma_{\text{p}}/\Gamma_{\text{tot}}$. But the measured lifetime,
$\tau$=10(3) fs Jenkins2004 , is in contradiction with shell-model
predictions, as explained hereafter.

An experiment has been performed at GANIL facility, France, to strongly reduce
the rate uncertainties of 22Na(p,$\gamma$)23Mg through the measurements of
$\tau$ and BR${}_{\text{p}}$. It is briefly presented here. Deduced
spectroscopic properties are then discussed in relation with results from
shell-model calculations.

## Experimental approach

The state of interest has been populated with a 24Mg beam of E=4.6 MeV/u onto
a 3He-implanted gold target. The reactions 3He(24Mg, 4He)23Mg∗ have been
selected and measured with: the spectrometer VAMOS VAMOS1 for 4He
identification and momenta measurement, the $\gamma$-ray tracking spectrometer
AGATA AGATA2 , and the silicon telescope SPIDER for proton momenta
measurements. Momenta reconstruction has been improved thanks to a small gas
chamber placed in front of VAMOS. The $\gamma$-ray transitions and lifetimes
have been measured for about 20 identified states in 23Mg∗. The branching
ratios have been measured for the measured protons from states close to the
proton emission threshold. Details can be found in Ref. thesisFougeres and
results will be more extensively discussed in forthcoming publications.

## Nuclear structure considerations

Insights from theory are now discussed in relation with the performed
experimental study of 23Mg. The shell-model calculations are first briefly
explained. Then, the measured and theoretical reduced M1 matrix elements of
the $\gamma$-ray transitions are compared for several levels in 23Mg. We
finally focus on the spectroscopic properties of the $E_{x}$=7.785 and 7.803
MeV states, the latter being the Isobaric Analog State (IAS) of the 23Al
ground state ($J^{\pi}$=$\frac{5}{2}^{+}$). Among the calculated levels in
23Mg, the $E_{x}$=7.785 MeV state has not yet been clearly identified because
of its uncertain spin value Jenkins2004 ; Kwag2021 . While identifying the
best shell-model candidate for this state, its spin assignment and possible
isospin-mixing with the IAS are discussed.

### Shell-model calculations

We present here a theoretical analysis of the partial 23Mg level scheme, based
on the $sd$-shell calculations with charge-dependent versions of USDA and USDB
USD ; USDCAB (USDAcpn and USDBcpn) using the NushellX code NUSHELLX . In
general, those interactions are known to provide a very good description of
$sd$-shell nuclei. Assessment of shell-model states with observations is more
challenging at higher energies because of the increasing level density. This
has still been achieved for the $E_{x}$$\sim$7.5-8 MeV states in 23Mg (details
in thesisFougeres ). Excited states in 23Mg, including resonant ones, have
thus been identified by comparing experimental energies, gamma decay branching
ratios and 23Al $\beta^{+}$ feeding with theoretical quantities. Let us recall
that a partial electromagnetic decay width $\Gamma_{\gamma}$ is related to the
reduced $\gamma$-ray transition strength $B_{\gamma}$ via
$\Gamma_{\gamma,(\text{L,
i}\rightarrow\text{f})}=\frac{8\pi(\text{L}+1)}{\text{L}[(2\text{L}+1)\text{!!}]^{2}}(\frac{\text{E}_{\gamma,0}}{\hbar\text{c}})^{2\text{L}+1}B_{\gamma}$.
Then, the reduced M1 matrix elements $|M(M1)|$ can be derived with
$|M(M1)|=\sqrt{(2J_{i}+1)B(M1)}$. The proton decay width can be expressed as
$\Gamma_{\text{p}}=\sum_{\ell,j}C^{2}S_{p}(\ell
j)\times\Gamma_{\text{s.p.}}(\ell j)$ where $C^{2}S_{p}(\ell j)$ refer to
proton spectroscopic factors, $\Gamma_{\text{s.p.}}(\ell j)$ to the single-
particle proton decay widths which are calculated from proton scattering
cross-sections on a Woods-Saxon potential.

### Reduced M1 matrix elements of the 23Mg states

Experimental $B(M1)$ values have been determined from (i) measured ($\tau$,
Eγ,0), (ii) identified M1 transitions nndc2022 among observed $\gamma$ rays
and (iii) the assumption of small E2 contributions. The comparison of
experimental and theoretical values of reduced M1 matrix elements is presented
in Fig. 2 for transitions in 23Mg and 21,23Na BM1NaIso . Optimum effective
$g$-factors USDA(6) and USDB(6) from Table I of Ref. USDAB_em have been used
to parameterize the M1 operator. The results for the state of interest are
emphasized by the red contours. At the bottom, the ratios of shell-model to
experimental reduced matrix elements highlight differences by about a few
times for small values of $|M(M1)|$. Although the accuracy in reproduction of
$|M(M1)|$ for selected isotopes close to 23Mg is consistent with the rms
deviations of about 0.22 - 0.20 $\mu_{N}$ reported for USDA and USDB USDAB_em
, we see that the M1 transition from the key state of interest is among the
most challenging for theory.

Figure 2: Correlation between measured and theoretical reduced matrix elements
$|M(M1)|$ values for transitions in 23Mg and 21,23Na BM1NaIso . Theoretical
values were obtained from $sd$-shell calculations with USDAcpn and USDBcpn USD
; USDCAB , experimental ones from measured ($\tau$, Eγ,0), resulting in
rms=0.322 $\mu_{N}$ (USDAcpn) and rms=0.273 $\mu_{N}$ (USDBcpn). It follows
that the largest relative deviations are observed for the lowest $|M(M1)|$
values, as for the $E_{x}$=7.785 MeV state shown in the red contours.

### Spectroscopy of the $E_{x}$=7.785 and 7.803 MeV states

The two states measured at $E_{x}$=7.785 and 7.803 MeV (IAS) are now discussed
in more detail. The $\gamma$-ray width, never measured for the IAS, is 3.0(2)
eV for the mirror state, in agreement with the shell-model results of 2 eV
(USDAcpn) and 3 eV (USDBcpn). From the previous experiment on
BR${}_{\text{p}}$ Perjarvi , the proton width and spectroscopic factor are
deduced for this isospin-forbidden proton emission, i.e.
$\Gamma_{\text{p}}$=5(2) meV and $C^{2}S_{p}$=0.0056(20) versus
$C^{2}S_{p}$=0.0073 (USDAcpn) and 3$\times 10^{-5}$ (USDBcpn).

Figure 3: Comparison between $\gamma$-ray transitions from the $E_{x}$=7.785
state and from shell-model candidates: (n${}_{\text{SM}}$=7,
$J^{\pi}$=$\frac{7}{2}^{+}_{7}$) and (n${}_{\text{SM}}$=8,
$J^{\pi}$=$\frac{5}{2}^{+}_{8}$). A good agreement is observed for the blue
transitions, strengthening the $\text{n}_{\text{SM}}$=7 candidate as the
$E_{x}$=7.785 MeV state (intensities are given in %.)

Three different spins were proposed for the $E_{x}$=7.785 MeV state, either
$\frac{7}{2}^{+}$Jenkins2004 or ($\frac{3}{2}^{+}$, $\frac{5}{2}^{+}$)Kwag2021
. The value $\frac{3}{2}^{+}$Kwag2021 is unlikely since this state was well
observed in the beta-delayed proton emission of 23Al
($\frac{5}{2}^{+}$)Saastamoinen2011 ; Friedman2020 . The spin of 22Na ground
state being $3^{+}$, the transferred angular momentum should be at least
$\ell=2$, that is too high to be competitive with the $\gamma$ emission. As
shown in Fig. 3 (in red), the theoretical $\gamma$-decay pattern of the
$\frac{5}{2}^{+}_{8}$ state (eighth predicted state, $\text{n}_{\text{SM}}$=8)
strongly differs from the observed one. On the contrary, the
$J^{\pi}$=$\frac{7}{2}^{+}_{7}$ state (in blue) agrees well with the measured
decay pattern. This is true regardless of the interaction used. Moreover, a
spin assignment $\frac{5}{2}^{+}$ of the key state would imply a strong
isospin-mixing with the IAS (as predicted by the shell model) which is just 18
keV above. However, the measured log10(ft)=3.305(23) of the IAS zhai2007 is in
excellent agreement with no mixing hypothesis, and the one of the
$E_{x}$=7.785 MeV state agrees with the calculated $\frac{7}{2}^{+}_{7}$
state. Consequently, $J^{\pi}$=$\frac{7}{2}^{+}$ is assigned for the key
state.

This conclusion of no mixing is supported by the small gamma width of the key
state, $\Gamma_{\gamma}\lesssim 0.1$ eV, if compared to the gamma width of the
IAS. Regarding proton emission, from the measured $BR_{p}$, a small
spectroscopic factor $C^{2}S_{p}\approx 10^{-3}$ was obtained. Theoretical
predictions are $C^{2}S_{p}$=0.003 (USDAcpn) and 0.05 (USDBcpn), assuming
$7/2^{+}$ assignment. Although the former result captures the order of
magnitude, we remark that similar to M1 observable, very small values of
spectroscopic factors may not be perfectly described.

## Summary

Figure 4: Map of the Milky Way with the expected flux of 22Na, represented by
the red dots, from the ONe novae observed for the last 60 years. Circles
represent the limit distance of COSI and eASTROGAM to detect $\gamma$-rays
from 22Na. The former should detect such emission once every 20 yr.

In the present work, a detailed comparison has been drawn between
experimentally observed and shell-model states in 23Mg. It appears that shell-
model predictions may have large relative uncertainties, up to a factor of 10,
for very small (<10-2) spectroscopic factors and M1 transition probabilities.
This is apparently the case of the key state. Nevertheless, shell model has
brought helpful arguments favoring $\frac{7}{2}^{+}$ spin assignment of the
E${}_{\text{R}}$=0.204 MeV resonant state. The measured total and partial
widths of this state have led to a reevaluation of the 22Na(p,$\gamma$)23Mg
rate at novae temperatures. According to the future COSI and eASTROGAM
sensitivities, it allows us to estimate the maximum detectability distance and
a conservative observation frequency of 22Na based on the detected ONe novae
surveyONe ; bookJose . These predictions, illustrated in Fig. 4, are
encouraging the search for 22Na with the new space telescopes.

## Acknowledgments

The authors thank the GANIL accelerator staff for their beam delivery and
their support, the AGATA collaboration, and the Normandie region.

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# Metric-Affine Version of Myrzakulov $F(R,T,Q,{\cal T})$ Gravity and
Cosmological Applications

Damianos Iosifidis∗, Nurgissa Myrzakulov†,⋆, Ratbay Myrzakulov†,⋆ ∗Institute
of Theoretical Physics, Department of Physics, Aristotle University of
Thessaloniki, 54124 Thessaloniki, Greece.
† Ratbay Myrzakulov International Centre for Theoretical Physics, Nur-Sultan
010009, Kazakhstan.
⋆Eurasian National University, Nur-Sultan 010008, Kazakhstan.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We derive the full set of field equations for the Metric-Affine version of the
Myrzakulov gravity model and also extend this family of theories to a broader
one. More specifically, we consider theories whose gravitational Lagrangian is
given by $F(R,T,Q,{\cal T},{\cal D})$ where $T$, $Q$ are the torsion and non-
metricity scalars, ${\cal T}$ is the trace of the energy-momentum tensor and
${\cal D}$ the divergence of the dilation current. We then consider the linear
case of the aforementioned theory and assuming a cosmological setup we obtain
the modified Friedmann equations. In addition, focusing on the vanishing non-
metricity sector and considering matter coupled to torsion we obtain the
complete set of equations describing the cosmological behaviour of this model
along with solutions.

###### Contents

  1. I Introduction
  2. II Conventions/Notation
  3. III Cosmology with Torsion and Non-metricity
  4. IV MG-VIII model and Extension: The $F(R,T,Q,{\cal T},{\cal D})$ Theories
  5. V Cosmological Applications
    1. V.1 The Cosmology of $F=R+\beta T+\gamma Q+\mu{\cal T}+\nu{\cal D}$ Theory
    2. V.2 Scalar Field Coupled to Torsion
  6. VI Conclusions

##  I Introduction

Even though General Relativity (GR) is undeniably one of the most beautiful
and successful theories of physics, recent observational data have challenged
its statuswill2014confrontation . Probably the most important observations
that cannot be explained within the realm of GR are the early time as well as
the late time accelerated expansion of our Universe. This contradiction
between theory and observations have lead to the development of a fairly large
number of theories alternative to GR which collectively go by the name of
Modified Gravity saridakis2021modified . The search of a successful
alternative has been proven to be both fruitful as well as constructive in
regards with our understanding of gravity.

Among this plethora of modified gravities let us mention the metric $f(R)$
theories , the Metric-Affine (Palatini) $f(R)$ gravity sotiriou2008f ;
iosifidis2019torsion ; capozziello2010metric , the teleparallel $f(T)$
gravities aldrovandi2012teleparallel ; myrzakulov2011accelerating , the
symmetric teleparallel $f(Q)$ nester1998symmetric ; jimenez2018teleparallel ,
Scalar-Tensor theories bartolo1999scalar ; charmousis2012general , etc and
also certain extensions of them (see discussion on chapter $IV$). Of course,
the kind of modifications one chooses to adopt is highly a matter of personal
taste. In our point of view, interesting and well motivated alternatives are
those which extend the underlying geometry of spacetime by allowing a
connection more general than the usual Levi-Civita one. In generic settings,
when no a priori restriction is imposed on the connection and the latter is
regarded as another fundamental field on top of the metric the space will be
non-Riemannian eisenhart2012non  and possesses both torsion and non-metricity.
These last geometric quantities can then be computed once the affine
connection is found. The theories formulated on this non-Riemannian manifold
are known as Metric-Affine theories of gravity hehl1995metric ;
iosifidis2019metric .

In recent years there has been an ever increasing interest in the Metric-
Affine approach iosifidis2019exactly ; iosifidis2019scale ;
vitagliano2011dynamics ; sotiriou2007metric ; capozziello2010metric ;
percacci2020new ; Jimenez:2020dpn ; BeltranJimenez:2019acz ; Olmo:2011uz ;
aoki2019scalar ; Cabral:2020fax ; Ariwahjoedi:2020wmo ; Yang:2021fjy ;
helpin2020metric ; bahamonde2020new ; iosifidis2021parity  and especially to
its cosmological applications iosifidis2021riemann ; iosifidis2020cosmic ;
Iosifidis:2020gth ; Iosifidis:2021iuw ; jimenez2016spacetimes ;
beltran2017modified ; kranas2019friedmann ; barragan2009bouncing ;
shimada2019metric ; kubota2021cosmological ; Mikura:2020qhc ;
mikura2021minimal . This interest is possibly due to the fact that the
additional affects (compared to GR) that come into play in this framework have
a direct geometrical interpretation. That is, the modifications are solely due
to spacetime torsion and non-metricity. Furthermore, these geometric notions
are excited by matter that has intrinsic structurehehl1976hypermomentum ;
babourova1995variational ; obukhov1993hyperfluid ; Iosifidis:2020gth ;
Iosifidis:2021nra . This inner structure-generalized geometry interrelation
adds another positive characteristic to the MAG scheme. This is the framework
we are going to consider in this study.

In particular, the paper is organized as follows. Firstly, we fix conventions
and briefly review some of the basic elements of non-Riemannian geometry and
the physics of Metric-Affine gravity. We then consider an extended version of
the $F(R,T,Q,{\cal T},{\cal D})$ theory myrzakulov2012dark . To be more
specific, working in a Metric-Affine setup, we consider the class of theories
with gravitational Lagrangians of the form $F(R,T,Q,{\cal T},{\cal D})$, where
${\cal D}$ is the divergence of the dilation current, the new add-on we are
establishing here. Then, we obtain the field equations for this family of
theories by varying with respect to the metric and the independent affine
connection. Considering a linear function $F$ we then present a cosmological
application for this model and finally switching off non-metricity and
considering a scalar field coupled to torsion we obtain the modified Friedmann
equations and also provide solutions for this simple case.

##  II Conventions/Notation

Let us now briefly go over the basic geometric as well as physical setup we
are going to use and also fix notation. We consider a $4$-dim non-Riemannian
manifold endowed with a metric and an affine connection $({\cal M},g,\nabla)$.
Our definition for the covariant derivative, of a vector say, will be

$\nabla_{\alpha}u^{\lambda}=\partial_{\alpha}u^{\lambda}+\Gamma^{\lambda}_{\;\;\;\beta\alpha}u^{\beta}$
(1)

We also define the (Cartan) torsion tensor by

$S_{\mu\nu}^{\;\;\;\;\lambda}:=\Gamma^{\lambda}_{\;\;\;[\mu\nu]}$ (2)

and the non-metricity tensor as

$Q_{\alpha\mu\nu}:=-\nabla_{\alpha}g_{\alpha\beta}$ (3)

Contracting these with the metric tensor we obtain the associated torsion and
non-metricity vectors

$S_{\mu}:=S_{\mu\nu}^{\;\;\;\;\nu}$ (4)
$Q_{\mu}:=Q_{\mu\alpha\beta}g^{\alpha\beta}\;\;,\;\;q_{\mu}:=Q_{\alpha\beta\mu}g^{\alpha\beta}$
(5)

respectively. In addition, since we are in four dimensions, we can also form
the torsion pseudo-vector according to

$t^{\mu}:=\varepsilon^{\mu\alpha\beta\gamma}S_{\alpha\beta\gamma}$ (6)

Given the above definitions for torsion and non-metricity one can easily show
(see for instance iosifidis2019metric ) the affine connection
decomposition111From here onwards we shall use the tilde notation in order to
denote Riemannian objects, that is objects computed with respect to the Levi-
Civita connection $\tilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}$.

$\displaystyle\Gamma^{\lambda}_{\;\;\;\mu\nu}=N^{\lambda}_{\;\;\;\;\mu\nu}+\tilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}=\frac{1}{2}g^{\alpha\lambda}(Q_{\mu\nu\alpha}+Q_{\nu\alpha\mu}-Q_{\alpha\mu\nu})-g^{\alpha\lambda}(S_{\alpha\mu\nu}+S_{\alpha\nu\mu}-S_{\mu\nu\alpha})$
(7)

where $N^{\lambda}_{\;\;\mu\nu}$ is known as the distortion tensor. Continuing
we define the curvature tensor as usual

$R^{\mu}_{\;\;\;\nu\alpha\beta}:=2\partial_{[\alpha}\Gamma^{\mu}_{\;\;\;|\nu|\beta]}+2\Gamma^{\mu}_{\;\;\;\rho[\alpha}\Gamma^{\rho}_{\;\;\;|\nu|\beta]}$
(8)

and by a double contraction of the latter, we get the Ricci scalar

$R:=R^{\mu}_{\;\;\;\nu\mu\beta}g^{\nu\beta}$ (9)

Then by using decomposition (7) we obtain the post Riemannian expansion for
the Ricci scalar iosifidis2019metric

$R=\tilde{R}+T+Q+2Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+2S_{\mu}(q^{\mu}-Q^{\mu})+\tilde{\nabla}_{\mu}(q^{\mu}-Q^{\mu}-4S^{\mu})$
(10)

where $\tilde{R}$ is the Riemannian Ricci tensor (i.e. computed with respect
to the Levi-Civita connection) and we have also defined the torsion and non-
metricity scalars as

$T:=S_{\mu\nu\alpha}S^{\mu\nu\alpha}-2S_{\mu\nu\alpha}S^{\alpha\mu\nu}-4S_{\mu}S^{\mu}$
(11)

and

$Q:=\frac{1}{4}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}-\frac{1}{2}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}-\frac{1}{4}Q_{\mu}Q^{\mu}+\frac{1}{2}Q_{\mu}q^{\mu}$
(12)

respectively. Note that with the introduction of the superpotentials222Here we
are using the conventions of iosifidis2019scale .

$\Omega^{\alpha\mu\nu}:=\frac{1}{4}Q^{\alpha\mu\nu}-\frac{1}{2}Q^{\mu\nu\alpha}-\frac{1}{4}g^{\mu\nu}Q^{\alpha}+\frac{1}{2}g^{\alpha\mu}Q^{\nu}$
(13)
$\Sigma^{\alpha\mu\nu}:=S^{\alpha\mu\nu}-2S^{\mu\nu\alpha}-4g^{\mu\nu}S^{\alpha}$
(14)

these can more compactly be expressed as

$T=S_{\alpha\mu\nu}\Sigma^{\alpha\mu\nu}$ (15)
$Q=Q_{\alpha\mu\nu}\Omega^{\alpha\mu\nu}$ (16)

Equation (8) is of key importance in teleparallel formulations. For instance
by imposing vanishing curvature (which also implies $R=0$) and metric
compatibility ($Q_{\alpha\mu\nu}=0$) one obtains from $(\ref{N})$,

$\tilde{R}=-T+4\tilde{\nabla}_{\mu}S^{\mu}$ (17)

which is the basis of the metric teleparallel formulation. In a similar manner
the symmetric teleparallel (vanishing curvature and torsion) and also the
generalized teleparalelism (only vanishing curvature) are obtained
jimenez2019general .

Let us now turn our attention to the matter content. In Metric-Affine gravity
apart from the energy momentum tensor, which we define as usual

$T_{\mu\nu}:=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{M})}{\delta
g^{\mu\nu}}$ (18)

one also has to vary the matter part with respect to the affine-connection.
This new object, which is defined by

$\Delta_{\lambda}^{\;\;\;\mu\nu}:=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{M})}{\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}}$
(19)

is called hypermomentum hehl1976hypermomentum  and encodes the microscopic
characteristics of matter such as spin, dilation and shear. In the same way
that the energy momentum tensor sources spacetime curvature by means of the
metric field equations, the hypermomentum is the source of spacetime torsion
and non-metricity (through the connection field equations). Note that these
energy related tensors are not quite independent and are subject to the
conservation law

$\sqrt{-g}(2\tilde{\nabla}_{\mu}T^{\mu}_{\;\;\alpha}-\Delta^{\lambda\mu\nu}R_{\lambda\mu\nu\alpha})+\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}(\sqrt{-g}\Delta_{\alpha}^{\;\;\mu\nu})+2S_{\mu\alpha}^{\;\;\;\;\lambda}\hat{\nabla}_{\nu}(\sqrt{-g}\Delta_{\lambda}^{\;\;\;\mu\nu})=0\;,\;\;\hat{\nabla}_{\mu}:=2S_{\mu}-\nabla_{\mu}$
(20)

which comes from the diffeomorphism invariance of the matter sector of the
action (see Iosifidis:2020gth ). In the above discussion we have briefly
developed the geometric and physical setup needed for the rest of our study.
Let us focus on the cosmological aspects of theories with torsion and non-
metricity (i.e non-Riemannian extensions).

##  III Cosmology with Torsion and Non-metricity

Let us consider a homogeneous flat FLRW Cosmology, with the usual Robertson-
Walker line element

$ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}$ (21)

where $i,j=1,2,3$ and $a(t)$ is as usual the scale factor of the universe. As
usual the Hubble parameter is defined as $H:=\dot{a}/a$. Now, let $u^{\mu}$ be
the normalized $4$-velocity field and

$h_{\mu\nu}:=g_{\mu\nu}+u_{\mu}u_{\nu}$ (22)

be the projection tensor projecting objects on the space orthogonal to
$u^{\mu}$. The affine connection of the non-Riemannian FLRW spacetime reads
Iosifidis:2020gth

$\Gamma^{\lambda}_{\;\;\;\mu\nu}=\widetilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}+X(t)u^{\lambda}h_{\mu\nu}+Y(t)u_{\mu}h^{\lambda}_{\;\;\nu}+Z(t)u_{\nu}h^{\lambda}_{\;\;\mu}+V(t)u^{\lambda}u_{\mu}u_{\nu}+\epsilon^{\lambda}_{\;\;\mu\nu\rho}u^{\rho}W(t)\delta_{n,4}$
(23)

where the non-vanishing components of the Levi-Civita connection are in this
case

$\tilde{\Gamma}^{0}_{\;\;ij}=\tilde{\Gamma}^{0}_{\;\;ji}=\dot{a}a\delta_{ij}=Hg_{ij}\;,\;\;\tilde{\Gamma}^{i}_{\;\;j0}=\tilde{\Gamma}^{i}_{\;\;0j}=\frac{\dot{a}}{a}\delta^{i}_{\;\;j}=H\delta^{i}_{\;j}$
(24)

Continuing with the rest of the geometric objects, in this highly symmetric
spacetime, the torsion and non-metricity tensors take the forms
Iosifidis:2020gth

$S_{\mu\nu\alpha}^{(n)}=2u_{[\mu}h_{\nu]\alpha}\Phi(t)+\epsilon_{\mu\nu\alpha\rho}u^{\rho}P(t)$
(25)
$Q_{\alpha\mu\nu}=A(t)u_{\alpha}h_{\mu\nu}+B(t)h_{\alpha(\mu}u_{\nu)}+C(t)u_{\alpha}u_{\mu}u_{\nu}$
(26)

respectively. The five functions $\Phi,P,A,B,C$ describe the non-Riemannian
Cosmological effects. These, along with the scale factor, give the cosmic
evolution of non-Riemannian geometries. Let us note that using the relations
of the torsion and non-metricity tensors with the distortion tensor it is
trivial to show that the functions $X(t),Y(t),Z(t),V(t),W(t)$ are linearly
related to $\Phi(t),P(t),A(t),B(t),C(t)$ as Iosifidis:2020gth

$2(X+Y)=B\;,\;\;2Z=A\;,\;\;2V=C\;,\;\;2\Phi=Y-Z\;,\;\;P=W$ (27)

or inverting them

$W=P\;,\;\;V=C/2\;,\;\;Z=A/2$ (28)
$Y=2\Phi+\frac{A}{2}\;\;,\;\;\;X=\frac{B}{2}-2\Phi-\frac{A}{2}$ (29)

Now, using the definitions (11) and $(\ref{Qsc})$ for the torsion and non-
metricity scalars and the above cosmological forms for torsion and non-
metricity we find for the former

$T=24\Phi^{2}-6P^{2}$ (30) $\displaystyle
Q=\frac{3}{4}\left[2A^{2}+B(C-A)\right]$ (31)

respectively. These are the expressions for the torsion and non-metricity
scalars in a homogeneous cosmological setup when no teleparallelism is
imposed.

Finally, using the post Riemannian decomposition of the Ricci scalar and the
above forms of the torsion and non-metricity scalars we find

$\displaystyle
R=\tilde{R}+6\left[\frac{1}{4}A^{2}+4\Phi^{2}+\Phi(2A-B)\right]+\frac{3}{4}B(C-A)-6P^{2}$
$\displaystyle+3\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}u^{\mu}\left(\frac{B}{2}-A-4\Phi\right)\right]$
(32)

where

$\tilde{R}=6\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}\right]$
(33)

is the usual Riemannian part. The last decomposition will be very useful in
our subsequent discussion.

##  IV MG-VIII model and Extension: The $F(R,T,Q,{\cal T},{\cal D})$ Theories

In this paper, we are going to study the Myrzakulov gravity myrzakulov2012dark
VIII (MG-VIII)333See also anagnostopoulos2021observational ; Saridakis:2019qwt
for some observational implications of this theory.. Its action is given by
myrzakulov2012dark

$S[g,\Gamma,\phi]=S_{g}+S_{m}=\frac{1}{2\kappa}\int\sqrt{-g}d^{4}x\left[F(R,T,Q,{\cal
T})+2\kappa\mathcal{L}_{m}\right],$ (34)

where $R$ stands for the Ricci scalar (curvature scalar), $T$ is the torsion
scalar, $Q$ is the nonmetricity scalar and ${\cal T}$ is trace of the energy-
momentum tensor of matter Lagrangian $L_{m}$. The MG-VIII can be seen as some
kind of unification of $F(R),F(T),F(Q)$ or $F(R,{\cal T}),F(T,{\cal
T}),F(Q,{\cal T})$ theories (see Harko:2011kv ; harko2014f ; xu2019f
respectively). For instance, if one imposes flatness (i.e.
$R^{\lambda}_{\;\;\alpha\mu\nu}\equiv 0$) and metric compatibility
($Q_{\alpha\mu\nu}\equiv 0$) arrives at the $f(T)$ gravity
myrzakulov2011accelerating ; krvsvsak2016covariant . Demanding flatness and a
torsionless connection we get symmetric teleparallel $f(Q)$ gravity
nester1998symmetric ; jimenez2018teleparallel . More generally, imposing only
teleparallelism we arrive at the recently developed generalized teleparallel
scheme of f(G) jimenez2019general ; beltran2021accidental  theories. If no
restriction on the connection is assumed then $(\ref{2.1})$ serves as a
specific generalization of metric-affine $f(R)$ gravity where where the energy
momentum trace $\cal T$ and certain quadratic combinations of torsion and non-
metricity are added as well. In fact in this generalized metric-affine setup
one could consider also the presence of the hypermomentum analogue of the
(metrical) energy momentum trace. Giving it a little thought we observe that
similar to the trace $\cal T$ is the divergence of the dilation current as
they appear in the trace of the canonical444Here $t=t^{\mu\nu}g_{\mu\nu}$ is
the trace of the canonical energy momentum tensor $t^{\mu\nu}$. energy
momentum tensor (see for instance Iosifidis:2020gth )

$t={\cal
T}+\frac{1}{2\sqrt{-g}}\partial_{\nu}(\sqrt{-g}\Delta^{\nu})\;\;,\;\;\;\Delta^{\nu}:=\Delta_{\mu}^{\;\;\mu\nu}$
(35)

In this sense ${\cal T}$ and the divergence of $\Delta^{\nu}$ are placed on
equal footing as it is obvious from the above equation. Therefore, the scalar
obtained by the divergence of the dilation current

${\cal D}=\frac{1}{\sqrt{-g}}\partial_{\nu}(\sqrt{-g}\Delta^{\nu})$ (36)

would be trace analogue for the hypermomentum. With this inclusion we may
generalize the class of theories (34) to

$S[g,\Gamma,\phi]=S_{g}+S_{m}=\frac{1}{2\kappa}\int\sqrt{-g}d^{4}x\left[F(R,T,Q,{\cal
T},{\cal D})+2\kappa\mathcal{L}_{m}\right],$ (37)

The field equations of the family of theories given by the the above action
read as follows:

g-Variation:

$\displaystyle-\frac{1}{2}g_{\mu\nu}F+F_{R}R_{(\mu\nu)}+F_{T}\Big{(}2S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\alpha\beta}-S_{\alpha\beta\mu}S^{\alpha\beta}_{\;\;\;\;\nu}+2S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\beta\alpha}-4S_{\mu}S_{\nu}\Big{)}+F_{Q}L_{(\mu\nu)}$
$\displaystyle+\hat{\nabla}_{\lambda}(F_{Q}J^{\lambda}_{\;\;\;(\mu\nu)})+g_{\mu\nu}\hat{\nabla}_{\lambda}(F_{Q}\zeta^{\lambda})+F_{\cal
T}(\Theta_{\mu\nu}+T_{\mu\nu})+F_{D}M_{\mu\nu}=\kappa T_{\mu\nu}$ (38)

where

$\hat{\nabla}_{\lambda}:=\frac{1}{\sqrt{-g}}(2S_{\lambda}-\nabla_{\lambda})$
(39)

$\Omega^{\alpha\mu\nu}=\frac{1}{4}Q^{\alpha\mu\nu}-\frac{1}{2}Q^{\mu\nu\alpha}-\frac{1}{4}g^{\mu\nu}Q^{\alpha}+\frac{1}{2}g^{\alpha\mu}Q^{\nu}$
(40) $\displaystyle
4L_{\mu\nu}=(Q_{\mu\alpha\beta}-2Q_{\alpha\beta\mu})Q_{\nu}^{\;\;\;\alpha\beta}+(Q_{\mu}+2q_{\mu})Q_{\nu}+(2Q_{\mu\nu\alpha}-Q_{\alpha\mu\nu})Q^{\alpha})$
$\displaystyle-4\Omega^{\alpha\beta}_{\;\;\;\;\nu}Q_{\alpha\beta\mu}-4\Omega_{\alpha\mu\beta}Q^{\alpha\beta}_{\;\;\;\;\nu}$
(41) $\Theta_{\mu\nu}:=g^{\alpha\beta}\frac{\delta T_{\alpha\beta}}{\delta
g^{\mu\nu}}$ (42) $M_{\mu\nu}:=\frac{\delta D}{\delta g^{\mu\nu}}$ (43)

and we have also defined the densities

$J^{\lambda}_{\;\;\;\mu\nu}:=\sqrt{-g}\Big{(}\frac{1}{4}Q^{\lambda}_{\;\;\;\mu\nu}-\frac{1}{2}Q_{\mu\nu}^{\;\;\;\;\lambda}+\Omega^{\lambda}_{\;\;\;\mu\nu}\Big{)}$
(44)
$\zeta^{\lambda}=\sqrt{-g}\Big{(}-\frac{1}{4}Q^{\lambda}+\frac{1}{2}q^{\lambda}\Big{)}$
(45)

$\Gamma$-Variation:

$\displaystyle
P_{\lambda}^{\;\;\mu\nu}(F_{R})+2F_{T}\Big{(}S^{\mu\nu}_{\;\;\;\;\lambda}-2S_{\lambda}^{\;\;\;[\mu\nu]}-4S^{[\mu}\delta^{\nu]}_{\lambda}\Big{)}-M_{\lambda}^{\;\;\mu\nu\alpha}\partial_{\alpha}F_{D}$
$\displaystyle+F_{Q}\Big{(}2Q^{[\nu\mu]}_{\;\;\;\;\lambda}-Q_{\lambda}^{\;\;\mu\nu}+(q^{\nu}-Q^{\nu})\delta^{\mu}_{\lambda}+Q_{\lambda}g^{\mu\nu}+\frac{1}{2}Q^{\mu}\delta^{\nu}_{\lambda}\Big{)}=F_{\cal
T}\Theta_{\lambda}^{\;\;\mu\nu}+\kappa\Delta_{\lambda}^{\;\;\mu\nu}$ (46)

where

$\displaystyle
P_{\lambda}^{\;\;\;\mu\nu}(F_{R})=-\frac{\nabla_{\lambda}(\sqrt{-g}F_{R}g^{\mu\nu})}{\sqrt{-g}}+\frac{\nabla_{\alpha}(\sqrt{-g}F_{R}g^{\mu\alpha}\delta_{\lambda}^{\nu})}{\sqrt{-g}}+$
(47) $\displaystyle
2F_{R}(S_{\lambda}g^{\mu\nu}-S^{\mu}\delta_{\lambda}^{\nu}-S_{\lambda}^{\;\;\;\mu\nu})$

is the modified Palatini tensor and

$\Theta_{\lambda}^{\;\;\mu\nu}:=-\frac{\delta\cal
T}{\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}}\;\;,\;\;M_{\lambda}^{\;\;\mu\nu\alpha}:=\frac{\delta\Delta^{\alpha}}{\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}}$
(48)

Note: If matter does not couple to the connection (e.g. classical perfect
fluid with no inner structure) we have that $\Theta_{\lambda}^{\;\;\mu\nu}=0$
as well as $\Delta_{\lambda}^{\;\;\mu\nu}=0$ and
$M_{\lambda}^{\;\;\mu\nu\alpha}$. The above set of field equations constitutes
an extended (with the divergence of dilation included) Metric-Affine version
of the Myrzakulov gravities myrzakulov2012dark . Here we derived the field
equations with no restriction on the connection and also for the extended case
$F(R,T,Q,{\cal T},D)$. In the sequel we shall analyse further the linear case
$F=R+\beta T+\gamma Q+\mu{\cal T}+\nu{\cal D}$ and also touch upon
cosmological applications.

##  V Cosmological Applications

###  V.1 The Cosmology of $F=R+\beta T+\gamma Q+\mu{\cal T}+\nu{\cal D}$
Theory

Let us now analyse in more detail the linear case

$F=R+\beta T+\gamma Q+\mu{\cal T}+\nu{\cal D}$

and also obtain the associated cosmological equations. To start with let us
note that since $\sqrt{-g}{\cal D}$ is a total divergence, the dilation
current does not contribute to the field equations when included linearly.
Therefore we can safely set $\nu=0$ for the rest of our discussion. In
addition, in this linear case the metric field equations take the form

$\displaystyle-\frac{1}{2}g_{\mu\nu}F+R_{(\mu\nu)}+\beta\Big{(}2S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\alpha\beta}-S_{\alpha\beta\mu}S^{\alpha\beta}_{\;\;\;\;\nu}+2S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\beta\alpha}-4S_{\mu}S_{\nu}\Big{)}+\gamma
L_{(\mu\nu)}$ $\displaystyle+\hat{\nabla}_{\lambda}(\gamma
J^{\lambda}_{\;\;\;(\mu\nu)})+g_{\mu\nu}\hat{\nabla}_{\lambda}(\gamma\zeta^{\lambda})+\mu(\Theta_{\mu\nu}+T_{\mu\nu})=\kappa
T_{\mu\nu}$ (49)

Taking the trace of the last equation, using the post Riemannian expansion
(32) and also employing (25) along with (26) and after some long calculations
we finally arrive at

$\displaystyle\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}+(1+\beta)(4\Phi^{2}-P^{2})+\frac{1}{8}\Big{(}2A^{2}+B(C-A)\Big{)}+\Phi(2A-B)+\dot{f}+3Hf=-\mu(\Theta+{\cal
T})+\kappa{\cal T}$ (50)

where

$f:=\frac{1}{2}\left[(1-\gamma)\Big{(}\frac{B}{2}-A\Big{)}-4\Phi\right]\;,\;\;\Theta:=\Theta_{\mu\nu}g^{\mu\nu}$
(51)

which is a variant of the modified Friedmann equation. As for the second
Friedmann (acceleration) equation, its general form was derived in
iosifidis2020cosmic  for general non-Riemannian cosmological setups. It reads

$\displaystyle\frac{\ddot{a}}{a}=-\frac{1}{3}R_{\mu\nu}u^{\mu}u^{\nu}+2\left(\frac{\dot{a}}{a}\right)\Phi+2\dot{\Phi}+\left(\frac{\dot{a}}{a}\right)\left(A+\frac{C}{2}\right)+\frac{\dot{A}}{2}-\frac{A^{2}}{2}-\frac{1}{2}AC-2A\Phi-2C\Phi$
(52)

One could then proceed by contracting $(\ref{Pt})$ with $u^{\mu}u^{\nu}$ in
order to eliminate the first term ($R_{\mu\nu}u^{\mu}u^{\nu}$) and express
everything in terms of the scale factor and the torsion and non-metricity
variables. This results in a fairly complicated expression which we refrain
from presenting it here since it goes beyond the scope of the present
exposure. As a final note let us mention that in order to analyse in depth the
above cosmological model one should consider an appropriate form of matter for
which both the metrical energy momentum and hypermomentum tensors must respect
the cosmological principle. The fluid with such characteristics was
constructed in Iosifidis:2020gth  (see also for a generalized version
Iosifidis:2021nra ) and goes by the name Perfect Cosmological Hyperfluid. The
hypermomentum part of this fluid will then source the torsion and non-
metricity variables $\Phi,P,A,...$ etc. by virtue of the connection field
equations. We note that scalar fields coupled to the connection belong (are
certain subcases) to the aforementioned fluid description. For the sake of
illustration, below we present such an example with a scalar field non-
minimally coupled to the connection in the case of vanishing non-metricity and
also study some of the cosmological implications of this theory.

###  V.2 Scalar Field Coupled to Torsion

We shall now focus on the vanishing non-metricity sector and also set
$\gamma=0$ , that is we will concentrate on the case $F=R+\beta T$. As for the
matter part let us consider a scalar field. In the usual (i.e. purely
Riemannian) case one would have the usual Lagrangian

${\cal
L}^{(0)}_{m}=-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-V(\phi),$
(53)

for the scalar field $\phi$. However, in the presence of torsion nothing
prevent us to consider direct couplings of the scalar field with torsion. The
most straightforward form of such a coupling is a torsion vector-scalar field
derivative interaction of the form $\lambda_{0}S^{\mu}\nabla_{\mu}\phi$, where
$\lambda_{0}$ is the coupling constant measuring the strength of the
interaction. Including this term, our full matter Lagrangian now reads

${\cal
L}_{m}=-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-V(\phi)+\lambda_{0}S^{\mu}\nabla_{\mu}\phi$
(54)

Then, substituting this into $(\ref{2.1})$ and varying the latter with respect
to the scalar field, we obtain

$\frac{1}{\sqrt{-g}}\partial_{\mu}\Big{[}\sqrt{-g}(\partial^{\mu}\phi-\lambda_{0}S^{\mu})\Big{]}=\frac{\partial
V}{\partial\phi}$ (55)

which is the evolution equation for the scalar field under the influence of
torsion. In addition, the very presence of the interaction term
$\lambda_{0}S^{\mu}\nabla_{\mu}\phi$ produces a non-vanishing hypermomentum
which is trivially computed to be

$\Delta_{\lambda}^{\;\;\mu\nu}=2\lambda_{0}\delta^{[\mu}_{\lambda}\nabla^{\nu]}\phi$
(56)

With this result, starting from the connection field equations () which in our
case read

$\displaystyle
P_{\lambda}^{\;\;\mu\nu}+2\beta\Big{(}S^{\mu\nu}_{\;\;\;\;\lambda}-2S_{\lambda}^{\;\;\;[\mu\nu]}-4S^{[\mu}\delta^{\nu]}_{\lambda}\Big{)}=\kappa\Delta_{\lambda}^{\;\;\mu\nu}$
(57)

and contracting in $\mu=\lambda$ we find

$S^{\mu}=\frac{3\kappa\lambda_{0}}{8\beta}\partial^{\mu}\phi$ (58)

that is the presence of scalar field produces spacetime torsion555Of course
this is so because of the connection coupling which yields a non-vanishing
hypermomentum. If no such coupling is included the scalar field cannot either
feel or produce torsion.. In addition, contracting (57) with
$\varepsilon^{\lambda}_{\;\;\mu\nu\alpha}$ it follows that

$t_{\alpha}=0$ (59)

Note that we can now plug back to (54) the above form of the torsion tensor to
end up with

${\cal
L}_{m}=-\frac{1}{2}\left(1-\frac{3\kappa\lambda_{0}^{2}}{4\beta}\right)g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-V(\phi)$
(60)

Interestingly, from the last equation we conclude that the scalar-torsion
interaction changes the factor of the kinetic term for the scalar field. We
also see that the is a crucial value for the coupling
$|\lambda_{0}|=2\sqrt{\frac{\beta}{3\kappa}}$ above which the kinetic term
changes sign and for exactly this value vanishes identically. Since this last
case would require severe fine tuning we shall disregard it and we shall also
assume that $\lambda_{0}$ is under this bound so that the kinetic term keeps
its original sign.

Up to this point, the above considerations were general. Let us now focus on
the homogeneous FLRW cosmology of this theory. In this case, equation (59)
implies that $P=0$ and as a result, upon using $(\ref{sphi})$ the full torsion
tensor is given by

$S_{\mu\nu\alpha}=2u_{[\mu}h_{\nu]\alpha}\Phi(t)\;\;,\;\;\Phi=-\frac{\kappa\lambda_{0}}{8\beta}\dot{\phi}$
(61)

In the case of a free scalar field (i.e $V(\phi)=0$) inserting (58) into (55)
we obtain

$\left(1-\frac{3\kappa\lambda_{0}^{2}}{8\beta}\right)\partial_{\mu}\Big{[}\sqrt{-g}\partial^{\mu}\phi\Big{]}=0$
(62)

which for $|\lambda_{0}|\neq 2\sqrt{\frac{\beta}{3\kappa}}$ implies that

$\dot{\phi}=\frac{c_{0}}{a^{3}}$ (63)

On the other hand, the metric field equations in this case read

$\displaystyle-\frac{1}{2}g_{\mu\nu}F+R_{(\mu\nu)}+\beta\Big{(}2S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\alpha\beta}-S_{\alpha\beta\mu}S^{\alpha\beta}_{\;\;\;\;\nu}+2S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\beta\alpha}-4S_{\mu}S_{\nu}\Big{)}=\kappa
T_{\mu\nu}$ (64)

and by taking the trace, using the same procedure we outlined previously, we
finally obtain

$\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}=\left[-\frac{\kappa}{6}+(1-\beta)\left(\frac{\kappa\lambda_{0}}{4\beta}\right)^{2}\right]\dot{\phi}^{2}$
(65)

which is again a variant of the modified Friedmann equation. Let us now derive
the acceleration equation for this case. First, we contract the above field
equations with $u^{\mu}u^{\nu}$ to obtain

$R_{\mu\nu}u^{\mu}u^{\nu}=24\beta\Phi^{2}+\frac{\kappa}{2}(\rho+3p)$ (66)

which when substituted in $(\ref{accel})$ for vanishing non-metricity and the
given scalar matter results in the acceleration equation

$\frac{\ddot{a}}{a}=-8\beta\Phi^{2}-\frac{\kappa}{6}(\rho+3p)+2H\Phi+2\dot{\Phi}$
(67)

where $\rho$ and $p$ are the density and pressure associated to the scalar
field Lagrangian $(\ref{phi3})$. It is interesting to note that the first term
on the right hand side of the acceleration equation has a fixed sign depending
on the value of $\beta$. Intriguingly, for $\beta<0$ the contribution from
this term has always a fixed positive sign producing an accelerated expansion
regardless of the sign of $\Phi$ (or equivalently $\dot{\phi}$). As for the
last two terms, combining $(\ref{ss}b)$ and $(\ref{dotphi})$ we observe that
$\dot{\Phi}=-3H\Phi$ which when substituted to the above acceleration equation
yields

$\frac{\ddot{a}}{a}=-8\beta\Phi^{2}-\frac{\kappa}{6}(\rho+3p)+\frac{4}{3}\dot{\Phi}$
(68)

We can conclude therefore that the last term aids to to acceleration when
$\dot{\Phi}>0$ and slows it down whenever $\dot{\Phi}<0$. From the above
analysis we see that the non-Riemannian degrees of freedom play a crucial role
on the cosmological evolution providing new interesting phenomena. Now using
the latter form of the acceleration equation we can obtain the first Friedmann
equation from (65) by eliminating the double derivative of the scale factor.
For the simple case $V(\phi)=0$ we find

$\left(\frac{\dot{a}}{a}\right)^{2}=\left[\frac{\kappa}{6}+(1+\beta)\left(\frac{\kappa\lambda_{0}}{4\beta}\right)^{2}\right]\dot{\phi}^{2}-\frac{4}{3}\dot{\Phi}$
(69)

as the modified first Friedmann equation. Note that on substituting
($\ref{sphi}b$) in the above and completing the square in the resulting
expression we easily find the power-law solution

$a(t)\propto t^{1/3}$ (70)

which is the stiff matter solution. We see that in the simplified case of a
zero potential for the scalar we arrive at a known solution. However, we
should remark that the situation changes drastically when one considers a non-
vanishing potential. Note also that the torsion tensor in this case goes like
$1/t$ and therefore its effect diminishes with time.

Needless to say that when non-metricity is also included one gets more
complicated expressions with a much richer phenomenology. It would be quite
interesting to see exactly to what degree the simultaneous presence of torsion
and non-metricity alters the cosmological evolution in such models. This would
be the theme of a separate work.

##  VI Conclusions

By working in a Metric-Affine approach (i.e. considering the metric and the
connection as independent variables) we have considered a generalized version
of the theory proposed in myrzakulov2012dark . In particular, we derived the
full set of field equations of the class of theories whose gravitational part
of the Lagrangian is given by $F(R,T,Q,{\cal T},{\cal D})$, where $T$, $Q$ are
the torsion and non-metricity scalars, ${\cal T}$ is the trace of the energy-
momentum tensor and ${\cal D}$ is the divergence of the dilation current (one
of the hypermomentum sources). The family of theories contained in our
Lagrangian is fairly large since all, metric and Palatini $f(R)$ theories,
teleparallel $f(T)$, symmetric teleparallel $f(Q)$ or even generalized
teleparallel $f(G)$ and generalizations of them such as $f(R,{\cal T})$,
$f(T,{\cal T})$, $f(Q,{\cal T})$ can be seen as special cases of our theory.

Our contribution was two-fold. Firstly, we generalized the family of theories
to those including also the divergence of the dilation current (which is the
analogue of the energy-momentum trace for hypermomentum). Furthermore, as
already mentioned above, we worked in a Metric-Affine framework, considering
an independent affine connection as a fundamental variable along with the
metric. This allows one not only to study the aforementioned theories (by
restricting the connection one way or another), but also to analyse them in
this general Metric-Affine scheme. Having derived the complete set of Metric-
Affine $F(R,T,Q,{\cal T},{\cal D})$ theories we then concentrated our
attention on the linear case $F=R+\beta T+\gamma Q+\mu{\cal T}+\nu{\cal D}$
and obtained a variant version of the modified Friedmann equation. Finally, we
focused on the vanishing non-metricity sector and also considered a scalar
field coupled to torsion as our matter sector. In this case we derived both
the first and second (acceleration) Friedmann equations and examined under
what circumstances the presence of torsion can have an accelerating affect on
the cosmological evolution. For this simple case we were also able to provide
an exact power-law solution for the scale factor.

In closing let us note some further applications and additional developments
of our study here. Firstly, it would be interesting to study in more detail
the linear case especially in regard with its cosmological implications in the
presence of the cosmological hyperfluid Iosifidis:2020gth ; Iosifidis:2021nra
. In addition, as we have already mentioned, it would be worthy to elaborate
more on the coupled scalar field we presented when both torsion and non-
metricity are allowed and direct couplings of the latter with the scalar field
occur. Finally, it would be quite interesting to go beyond linear functions
$F$ of the new dilation current term we considered. In this way we will be
able to investigate what exactly is the effect of this new addition/extension
as well as its phenomenology especially with regards to its energy momentum
trace counterpart.

## Acknowledgements

The work was supported by the Ministry of Education and Science of the
Republic of Kazakhstan, Grant AP09058240.

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