#  $\gamma$ rays from in-flight positron annihilation as a probe of new
physics

Pedro De la Torre Luque<EMAIL_ADDRESS>Departamento de Física
Teórica, M-15, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, C/
Nicolás Cabrera, 13-15, 28049 Madrid, Spain  The Oskar Klein Centre,
Department of Physics, Stockholm University, Stockholm 106 91, Sweden  Shyam
Balaji<EMAIL_ADDRESS>Physics Department, King’s College London,
Strand, London, WC2R 2LS, United Kingdom  Pierluca Carenza
<EMAIL_ADDRESS>The Oskar Klein Centre, Department of Physics,
Stockholm University, Stockholm 106 91, Sweden  Leonardo Mastrototaro
<EMAIL_ADDRESS>Dipartimento di Fisica “E.R. Caianiello”, Università
degli Studi di Salerno, Via Giovanni Paolo II, 132 - 84084 Fisciano (SA),
Italy  INFN - Gruppo Collegato di Salerno, Via Giovanni Paolo II, 132 - 84084
Fisciano (SA), Italy.

###### Abstract

The $\gamma$ ray emission originating from in-flight annihilation (IA) of
positrons is a powerful observable for constraining high-energy positron
production from exotic sources. By comparing diffuse $\gamma$ ray observations
of INTEGRAL, COMPTEL and EGRET to theoretical predictions, we set the most
stringent constraints on electrophilic feebly interacting particles (FIPs),
thereby proving IA as a valuable probe of new physics. In particular, we
extensively discuss the case of MeV-scale sterile neutrinos, where IA sets the
most stringent constraints, excluding $|U_{\mu 4}|^{2}\gtrsim 10^{-13}$ and
$|U_{\tau 4}|^{2}\gtrsim 2\times 10^{-13}$ for sterile neutrinos mixed with
$\mu$ and $\tau$ neutrinos respectively. These constraints improve existing
limits by more than an order of magnitude. We briefly discuss the application
of these results to a host of exotic positron sources such as dark photons,
axion-like particles, primordial black holes (PBHs) and sub-GeV dark matter
(DM).

_Introduction—_ A steady production of positrons in the Milky Way is evidenced
by the long-standing observations of a diffuse $511$ keV $\gamma$ ray line
from electron-positron annihilation, whose emission seems constant over time
Siegert  _et al._ (2016); Kierans  _et al._ (2020). A full understanding of
the observed characteristics of this emission is still challenging, although
these observations allow us to constrain the properties of a variety of
Galactic positron emitters Prantzos  _et al._ (2011); Kierans  _et al._
(2019); Siegert (2023). As a direct consequence of this positron injection,
interactions of positrons with the interstellar medium (ISM) produce not only
a diffuse line signal but leads also to a continuum emission that extends
above and below $511$ keV. Here we list the processes responsible for a
$\gamma$ ray signal associated with high-energy positrons:

  * •

relativistic positrons lose energy by scattering on free and bounded electrons
in the ISM, leading to a continuum emission above $511$ keV (IA emission);

  * •

thermalized positrons produce a para-positronium (p-ps) bound state with free
electrons $25\%$ of the time, which annihilates in $\sim 0.12$ ns Badertscher
_et al._ (2007) into two $\gamma$ rays with $511$ keV energy each

  * •

in $75\%$ of the cases, thermalized positrons produce an ortho-positronium
(o-ps) bound state with free electrons, which decays in $\sim 140$ ns into
(predominantly) $3$ photons Badertscher  _et al._ (2007) with energy lower
than $511$ keV.

A variety of different positron sources are known, such as pulsar wind
nebulae, X-ray binaries (XRB), $\beta^{+}$ decay of unstable elements
synthesized in supernovae (SN) Prantzos  _et al._ (2011), and cosmic ray (CR)
interactions with the interstellar gas Gabici  _et al._ (2019). In addition,
“exotic” sources might also contribute to the positron flux. For instance,
feebly interacting particles (FIPs) produced in SN explosions, and decaying
into electrons and positrons (such as sterile neutrinos, dark photons or
axion-like particles) De la Torre Luque  _et al._ (2023a, b), low-mass
evaporating primordial black holes (PBHs) formed in the early Universe or sub-
GeV dark matter (DM) Koechler (2024); De la Torre Luque  _et al._ (2023c, d).
These exotic sources are expected to produce positrons in the MeV-GeV range
and are often not very well constrained at high energies, either by direct CR
measurements or from their secondary $\gamma$ ray emissions, such as via
inverse-Compton (IC) or bremsstrahlung.

In this letter, we propose the use of diffuse $\gamma$ ray emission produced
from IA to constrain the production of positrons from various exotic high-
energy emitters, extending the idea presented in Ref. Beacom and Yuksel (2006)
to a positron injection with an arbitrary new physics source and energy
spectrum. We will focus on electrophilic FIPs produced in SN explosions,
extensively discussing the case of sterile neutrinos. We also remark that IA
leads to very competitive constraints for evaporating light PBHs and sub-GeV
DM as well.

_Positron injection and propagation—_ Positrons injected in the Galaxy will
diffuse in the Milky Way and lose energy when interacting gas, the Galactic
magnetic field, and interstellar radiation fields, during their propagation,
leading to a secondary diffuse $\gamma$ ray flux. In order to evaluate the
effect of positron diffusion, we follow a similar procedure as in Refs. De la
Torre Luque  _et al._ (2023b, d). The spatial distribution and energy spectra
of the positrons are computed with a recent version De La Torre Luque (2023)
of the DRAGON2 code Evoli  _et al._ (2017, 2018), a dedicated CR propagation
code designed to numerically solve the full diffusion-advection-
reacceleration-loss equation for the transport of charged particles in the
Galactic environment Ginzburg and Syrovatskii (1969). The spatial distribution
of positron sources depends on the specific type of emitter that we consider.
In this letter, we will focus on FIPs produced in SN explosions and
subsequently decaying into electron-positron pairs. In this case, the spatial
distribution follows the SN one, taken to be the Ferriere distribution
Ferriere  _et al._ (2007).

The set of propagation parameters that we use, obtained through a combined fit
of different CR observables, allows us to reproduce the AMS-02 Aguilar  _et
al._ (2021, 2018, 2013) and Voyager-1 Webber  _et al._ (2002); Stone  _et al._
(2013) data in the range of some tens of MeV to the TeV scale De la Torre
Luque  _et al._ (2023d)111The input and outputs files from the used
propagation code are available at https://doi.org/10.5281/zenodo.10076728..

After injecting positrons in the Galaxy, they will reach a steady-state
distribution, with positrons annihilating once they reach the thermal energy
of a warm medium (T$\simeq 8000$ K), as suggested by the analysis of the width
of the $511$ keV line Knödlseder  _et al._ (2005). Therefore, the positron
distribution determines the spatial features of the 511 keV line emission De
la Torre Luque  _et al._ (2023c). The 511 keV flux, integrated over the line
of sight, is evaluated as Guessoum  _et al._ (1991)

$\frac{d\phi_{\gamma}^{511}}{d\Omega}=2k_{ps}\int
ds\,\frac{d\phi^{e^{+}}}{d\Omega}\cdot
n_{e}\cdot\sigma^{\textrm{ann}}(E_{\textrm{th}})\,,$ (1)

where $k_{ps}=1/4$ is the fraction of positronium decays contributing to the
511 keV line signal,
$\frac{d\phi^{e^{+}}}{d\Omega}=\int\frac{d\phi^{e^{+}}}{dEd\Omega}dE$ is the
energy-integrated flux of positrons, $\sigma^{\rm ann}$ is conservatively
approximated as the annihilation cross section of positrons with electrons at
rest Dirac (1930) and the factor $2$ accounts for the emission of two photons
per positron annihilation. Here, $d\Omega=dl\,db\cos b$ is the solid angle
element being $l$, $b$ and $s$ the Galactic longitude, latitude and distance
$s$ from the Earth.

Since we expect that the electron density follows a smooth distribution in the
disk of the Galaxy, we estimate the profile of the 511 keV line emission as
directly proportional to the spatial distribution of diffuse positrons, and
set the electron density to $n_{e}\simeq 1$ cm-3, and uniform. Given that the
electron density is suppressed out of the Galactic plane, we apply a scaling
relation to the 511 keV profile following the vertical distribution of free
electron density in the Galaxy De la Torre Luque  _et al._ (2023c), following
the NE2001 model Cordes and Lazio (2003, 2002). We have tested that this
approach leads to consistent results with previous evaluations Calore  _et
al._ (2021, 2022); De la Torre Luque  _et al._ (2023b, a); Carenza  _et al._
(2024).

The correct evaluation of this flux is of primary importance since the
predicted IA emission is directly proportional to the 511 keV line intensity.
Indeed, the IA emission is evaluated as Beacom and Yuksel (2006)

$\begin{split}&\frac{d\phi^{\textrm{IA}}}{d\Omega\,dE_{\gamma}}=\frac{d\phi^{511}}{d\Omega}\frac{n_{H}}{P(1-\frac{3}{4}f)}\times\\\
&\times\int^{E_{\textrm{max}}}_{E_{\gamma}}dE^{\prime}\,\frac{1}{N_{\rm
pos}}\frac{dN_{\rm
pos}}{dE^{\prime}}\int^{E^{\prime}}_{m_{e}}P_{E^{\prime}\rightarrow
E}\,\frac{d\sigma}{dE_{\gamma}}\frac{dE}{|dE/dx|}\,,\end{split}$ (2)

where $\frac{d\phi^{511}}{d\Omega}$ is the flux per solid angle of the 511 keV
line in the SPI region and the integral on $E^{\prime}$ accounts for the
energy-distribution of injected positrons determined by $dN_{\rm
pos}/dE^{\prime}$, being bounded between the highest energy used in the
positron propagation simulation, $E_{\rm max}=5$ GeV, to the minimum positron
energy able to produce $\gamma$ rays with energy $E_{\gamma}$. The term
$\frac{1}{N_{\rm pos}}\frac{dN_{\rm pos}}{dE^{\prime}}$ represents the
fraction of positrons at the energy $E^{\prime}$ contributing to the 511 keV
emission (when this fraction is $1$, we recover the case of monoenergetic
positrons generating the total 511 keV flux). Eq. (2) takes into account the
energy spectrum of injected positrons and it reduces to the expression used in
Ref. Beacom and Yuksel (2006) in the case of a monoenergetic injection. The
second integral takes into account the energy loss for a positron of energy
$E^{\prime}$ to the final energy before annihilating; and the number density
of hydrogen, $n_{H}$, counts the number of targets on which positrons scatter
and lose energy. Here, we defined $P_{E^{\prime}\to E}$ as the probability,
per unit of energy, for a positron with initial energy $E^{\prime}$ to produce
a $\gamma$ ray in flight before reaching the energy $E$, similarly
$P=P_{E^{\prime}\to m_{e}}$ is the probability for a positron with initial
energy $E^{\prime}$ to produce a $\gamma$ ray before thermalizing, and
$f=0.967\pm 0.022$ Jean  _et al._ (2006) is the fraction of positrons
annihilating at rest compared to the ones annihilating in-flight. We notice
that this expression eventually becomes independent of the gas density,
$n_{H}$, since $|dE/dx|$ (taken to be the ionization energy losses from the
DRAGON code, including H and He) is directly proportional to $n_{H}$.
Therefore, this factor does not involve additional uncertainties in the
calculation of the IA emission. The term $\frac{d\sigma}{dE_{\gamma}}$ is
taken from Eq. $(6)$ of Ref. Beacom and Yuksel (2006), which is normalized to
a $\gamma$ ray multiplicity of $2$ photons (i.e. two photons per positron
annihilated). Typically, the higher the energy of the positron, the lower the
probability of annihilating with an electron and, for a given positron energy,
the probability of producing a $\gamma$ ray is highest at the positron energy,
and non-zero below it.

On top of the positron-induced $\gamma$ ray continuum signals, we emphasize
that the Galactic IC and bremsstrahlung emission from CR electrons are the
dominant $\gamma$ ray production processes below energies of a few hundreds of
MeV. For this background, we use the electron model from Refs. de la Torre
Luque  _et al._ (2022); De la Torre Luque  _et al._ (2023e), which is
optimized to reproduce the electron and positron emission at Earth location,
as well as the local $\gamma$ ray emissivity, down to a few tens of MeV. We
have checked that this model shows strong agreement with the data down to $10$
keV. Some experiments employ simple power-laws fitted to data to characterize
this background, while the IA emission has a very different spectrum in
energy, helping to distinguish it from the background with high sensitivity.

_Feebly interacting particles as positron sources—_ Novel particles dubbed
FIPs are attracting increasingly more interest Lanfranchi  _et al._ (2021);
Agrawal  _et al._ (2021); Alekhin  _et al._ (2016); Antel  _et al._ (2023).
The most popular FIP models are axions Raffelt and Seckel (1988); Keil  _et
al._ (1997); Carenza  _et al._ (2019, 2021) and axion-like particles Caputo
_et al._ (2022); Lella  _et al._ (2023, 2024), light CP-even scalars Balaji
_et al._ (2022, 2023); Dev  _et al._ (2022, 2021, 2020), sterile neutrinos
Kolb  _et al._ (1996); Raffelt and Zhou (2011); Mastrototaro  _et al._ (2020)
and dark photons Chang  _et al._ (2017).

Sub-GeV FIPs may be produced in copious amounts during a SN explosion, the
final stage in the evolution of massive stars, thanks to the extreme
temperature and density conditions reached in this astrophysical environment.
After escaping the SN, FIPs might produce an energetic positron flux through
their decay. This scenario was considered in previous works De la Torre Luque
_et al._ (2023a, b), considering the observational consequences of both the
primary $e^{\pm}$ fluxes and the secondary photon fluxes. In this letter, we
will explore this possibility further, by considering the IA emission of FIP-
sourced positrons. Their injection spectrum is modeled with a modified
blackbody spectrum De la Torre Luque  _et al._ (2023b).

Once the positron injection spectrum is determined, we derive the $95\%$
confidence level bounds by comparing theoretical predictions to the available
diffuse $\gamma$ ray data in the MeV region, using mainly SPI Bouchet  _et
al._ (2008); Siegert  _et al._ (2016, 2022) (aboard INTEGRAL), COMPTEL
Kappadath (1998) and EGRET Strong  _et al._ (2004) (both, aboard CGRO). We
evaluate the bounds by calculating the best-fit value on the parameter that we
constrain and the $2\sigma$ statistical error is associated through a
$\chi^{2}$ test (in particular, we use the `curve_fit` function in the
`scipy.optimize` Python package).

Figure 1: Bounds at the $2\sigma$ level on the number of electrons and
positrons injected per SN ($N_{e}$) for the general FIP case. We show here the
limits derived from the local electron-positron flux by Voyager-1 (blue),
X-ray emission from IC scattering of FIP-induced electron fluxes (green), the
longitude profile of the $511$ keV line (red) and the ones derived comparing
IA emission with the diffuse $\gamma$ ray observations of COMPTEL in the
$|l|<30^{\circ}$ \- $|b|<15^{\circ}$ region (cyan). The hatched bars represent
the estimated uncertainties, as discussed in the text.

In order to show the power of this observable in constraining new physics, we
consider the generic FIP case, where we limit the number of $e^{+}$ injected
by FIP decays per SN explosion, $N_{e}$. In Fig. 1 we show the bounds on
$N_{e}$ for the FIP case. These results lead to limits that are stronger than
those previously found by considering the $e^{\pm}$ interstellar flux measured
by Voyager-1 Webber  _et al._ (2002), secondary X-ray emission in the MOS
detector energy range Foster  _et al._ (2021) and the 511 keV signal De la
Torre Luque  _et al._ (2023a, b). Notice that we have updated the bound from
the longitudinal profile of the 511 keV line emission (SPI data Siegert  _et
al._ (2016)) using the more accurate procedure detailed above, in very good
agreement with the literature Calore  _et al._ (2022); De la Torre Luque  _et
al._ (2023b). As discussed in Ref. De la Torre Luque  _et al._ (2023b), the
bound from the longitude profile of the 511 keV line is severely affected by
the systematic uncertainties at high longitudes, where the most constraining
data points lie. As a proxy to evaluate the impact of systematic uncertainties
at high longitudes, we repeat the analysis of the longitude profile
considering only data points below $20^{\circ}$, which leads to constraint
weaker by a factor $\simeq 2$ (hatched red region). Finally, we show the bound
obtained from IA emission using COMPTEL data in the $|l|<30^{\circ}$ \-
$|b|<15^{\circ}$ region Strong  _et al._ (1994a). We have explored other
regions of the sky, finding weaker constraints. In addition, we also add an
uncertainty band represented by the cyan hatched region, showing the
improvement in the bound when considering the background model as well. As we
see, the bound on $N_{e}$ becomes between a factor of a few, to more than one
order of magnitude better than the $511$ keV line constraint. Moreover, this
bound could be improved even further by optimizing the region of interest as
well as doing a more dedicated analysis of the raw data. This demonstrates
that IA emission leads to the strongest constraints so far on electrophilic
FIPs.

Figure 2: Predicted IA spectra for $|U_{\tau 4}|^{2}=10^{-11}$ and masses
between $10$ and $100$ MeV, in the $|l|<60^{\circ}$ \- $|b|<10^{\circ}$
region, compared to the COMPTEL and EGRET diffuse data.

_The case of sterile neutrinos—_ These results suggest that, for any FIP
produced in a SN and decaying into positrons, the IA is an excellent probe. In
the following, we consider the specific case of sterile neutrinos. In
particular, we will focus on a MeV-GeV scale sterile neutrino, $\nu_{4}$
(indicating the mass eigenstate), mostly a flavour-sterile one, $\nu_{s}$
(indicating a flavor eigenstate), with a small mixing to $\mu$ or $\tau$
neutrinos $\nu_{\alpha}$, $\alpha=\mu,\,\tau$. Sterile neutrinos have often
been introduced to explain the origin of neutrino masses Merle (2017);
Boyarsky  _et al._ (2019); Abazajian  _et al._ (2012); Abazajian and Kusenko
(2019), making them a well-motivated FIP. The positron spectrum injected by
sterile neutrino decays is calculated by following Ref. Carenza  _et al._
(2024).

In Fig. 2 we show the predicted IA spectrum for different sterile neutrino
masses in the $|l|<60^{\circ}$-$|b|<10^{\circ}$ region, up to a hundred MeV.
We compare these predictions with COMPTEL Strong  _et al._ (1994b) and EGRET
Strong  _et al._ (2004) diffuse data, with the former being the most
constraining dataset. We stress that one could also obtain constraints from
Fermi-LAT at the lowest edge of its energy range, where it’s accuracy is
lower. We leave this exercise for future work as a possibility to further
optimize our analysis.

Remarkably, we obtain the most stringent bound on MeV-scale sterile neutrinos,
more than an order of magnitude stronger than laboratory Bolton  _et al._
(2020) and astrophysical SN bounds Carenza  _et al._ (2024), as shown in red
in Fig. 3. The hatched band corresponds to systematic uncertainties, which
mainly consist of uncertainties on the electron density and distribution, the
gas phase and temperature where positrons thermalize and the propagation of
the positrons. To estimate the uncertainties in the predicted positron flux
related to our propagation setup, we create two extreme and opposite
scenarios, a pessimistic and optimistic one, that lead to the hatched region
shown in Fig. 3.

These scenarios mainly differ from our benchmark setup in the normalization of
the diffusion coefficient characterizing the diffusion process (i.e. $D_{0}$),
the volume where CRs are confined in the Galaxy (through the value of the halo
height $H$) and the level of reacceleration, parameterized in the Àlfven speed
$V_{A}$ Heinbach and Simon (1995); Strong and Moskalenko (1998). In the
benchmark case, we use $D_{0}=5.2\times 10^{28}$ cm2/s, $H=8$ kpc and
$V_{A}=13.4$ km/s (see Refs. De la Torre Luque  _et al._ (2023b, d); Seo and
Ptuskin (1994) for more details). The pessimistic setup features the scenario
with very slow diffusion, where we adopt a very small value of the halo
height, compared to typical values obtained in CR analyses, $H=3$ kpc. We keep
the ratio $H$/$D_{0}$ to the one that reproduces CR secondary ratios at Earth
(i.e. the same ratio as with our benchmark propagation parameters). In
addition, we assume no reacceleration (i.e. $V_{A}=0$ km/s), thus leading to
very conservative estimations of the positron flux produced from these
particles. In the optimistic case, we set $V_{A}=40$ km/s and $H=16$ kpc. We
notice that even using a very conservative estimation of the uncertainty
associated with propagation, the limits change by no more than a factor of
$4$-$5$ in the whole sterile neutrino mass range.

Figure 3: Overview of the astrophysical SN bounds Carenza  _et al._ (2024)
(green region), cosmology Sabti  _et al._ (2020); Boyarsky  _et al._ (2021);
Mastrototaro  _et al._ (2021) (blue region and hatched area for the out of
equilibrium extension) and laboratory experiments Bolton  _et al._ (2020)
(gray region) for sterile neutrinos mixed with $\nu_{\mu}$ (upper panel) or
$\nu_{\tau}$ (lower panel). The dashed lines represent the sensitivities of
the future experiments DUNE Ballett  _et al._ (2020); Krasnov (2019) (black)
and MATHUSLA Chou  _et al._ (2017) (brown). The IA bound discussed in this
work (red region, with hatched area corresponding to the uncertainty)
significantly probes a sizable portion of the parameter space for sterile
neutrinos with small mixing angle.

We stress that only the free-streaming regime, i.e. low mixing angle
$U_{\alpha 4}<\mathcal{O}(10^{-8})$, was considered because of the existing
bounds closing most of the parameter space in the large mixing parameter
range. Moreover, in this portion of parameter space, the SN evolution might be
significantly altered by strongly coupled sterile neutrinos.

The IA probes a portion of the parameter space constrained with cosmological
bounds, which strongly rely on modelling assumptions Sabti  _et al._ (2020);
Boyarsky  _et al._ (2021); Mastrototaro  _et al._ (2021). For even smaller
couplings, it is possible to extend these constraints but they are strongly
dependent on the cosmological assumptions Ovchynnikov (2021). In this context,
our astrophysical bound relies on different assumptions compared to
cosmological ones and it is therefore a valuable tool to independently probe
this portion of parameter space for such weakly coupled sterile neutrinos.

_Discussion and conclusion—_ In this work, we have presented a novel
observable, $\gamma$ ray emission from IA, which is able to provide very
strong constraints on the positron production from any exotic source of high
energy positrons. We investigate the case of electrophilic FIPs produced in
SNe, exemplifying the potential of IA emission to provide constraints from
diffuse $\gamma$ ray observations. The case of sterile neutrinos was discussed
in detail, and we expect similarly powerful results in the case of dark
photons and axion-like particles. In addition, this observable can also be
used to constrain the fraction of DM in the form of PBHs or the annihilation
or decay rate of DM producing electron-positron pairs as a final state, as we
will explore in a separate paper.

We remark that our evaluation should be taken as conservative, given that we
are only accounting for direct annihilation of positrons with electrons, as
pointed out in the main text. In addition, we did not add the internal
bremsstrahlung emission Beacom  _et al._ (2005), because it is subdominant
with respect to the IA emission, but that could improve the constraints by
$\mathcal{O}(10\%)$.

In conclusion, we demonstrate that this novel $\gamma$ ray emission mechanism
can provide very strong constraints in a variety of models for positron
emitters, regardless of the specific model, being thus a very powerful tool to
probe new physics.

_Acknowledgements—_ We would like to thank John Ellis for helpful discussions
at the beginning of this work and Tim Linden for feedback on the manuscript.
This publication is based upon work from COST Action COSMIC WISPers CA21106,
supported by COST (European Cooperation in Science and Technology). SB is
supported by the STFC under grant ST/X000753/1. The work of PC is supported by
the Swedish Research Council (VR) under grants 2018-03641 and 2019-02337. PDL
is supported by the Juan de la Cierva JDC2022-048916-I grant, funded by
MCIU/AEI/10.13039/501100011033 European Union ”NextGenerationEU”/PRTR. The
work of PDL is also supported by the grants PID2021-125331NB-I00 and
CEX2020-001007-S, both funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A
way of making Europe”. PDL also acknowledges the MultiDark Network, ref.
RED2022-134411-T. This project used computing resources from the Swedish
National Infrastructure for Computing (SNIC) under project Nos. 2021/3-42,
2021/6-326, 2021-1-24 and 2022/3-27 partially funded by the Swedish Research
Council through grant no. 2018-05973. The work of LM is supported by the
Italian Istituto Nazionale di Fisica Nucleare (INFN) through the “QGSKY”
project and by Ministero dell’Università e Ricerca (MUR).

## References

  * Siegert  _et al._ (2016) T. Siegert, R. Diehl, G. Khachatryan, M. G. H. Krause, F. Guglielmetti, J. Greiner, A. W. Strong, and X. Zhang, Astron. Astrophys. 586, A84 (2016), arXiv:1512.00325 [astro-ph.HE] .
  * Kierans  _et al._ (2020) C. A. Kierans _et al._ , Astrophys. J. 895, 44 (2020), arXiv:1912.00110 [astro-ph.HE] .
  * Prantzos  _et al._ (2011) N. Prantzos _et al._ , Rev. Mod. Phys. 83, 1001 (2011), arXiv:1009.4620 [astro-ph.HE] .
  * Kierans  _et al._ (2019) C. A. Kierans _et al._ , “Positron annihilation in the galaxy,” (2019), arXiv:1903.05569 [astro-ph.HE] .
  * Siegert (2023) T. Siegert, Astrophysics and Space Science 368 (2023), 10.1007/s10509-023-04184-4.
  * Badertscher  _et al._ (2007) A. Badertscher, P. Crivelli, W. Fetscher, U. Gendotti, S. N. Gninenko, V. Postoev, A. Rubbia, V. Samoylenko, and D. Sillou, Physical Review D 75 (2007), 10.1103/physrevd.75.032004.
  * Gabici  _et al._ (2019) S. Gabici, C. Evoli, D. Gaggero, P. Lipari, P. Mertsch, E. Orlando, A. Strong, and A. Vittino, Int. J. Mod. Phys. D 28, 1930022 (2019), arXiv:1903.11584 [astro-ph.HE] .
  * De la Torre Luque  _et al._ (2023a) P. De la Torre Luque, S. Balaji, and P. Carenza, To appear in PRD as a letter (2023a), arXiv:2307.13728 [hep-ph] .
  * De la Torre Luque  _et al._ (2023b) P. De la Torre Luque, S. Balaji, and P. Carenza, Accepted in PRD (2023b), arXiv:2307.13731 [hep-ph] .
  * Koechler (2024) J. Koechler, PoS TAUP2023, 044 (2024), arXiv:2309.10043 [hep-ph] .
  * De la Torre Luque  _et al._ (2023c) P. De la Torre Luque, S. Balaji, and J. Silk, (2023c), arXiv:2312.04907 [hep-ph] .
  * De la Torre Luque  _et al._ (2023d) P. De la Torre Luque, S. Balaji, and J. Koechler, Accepted in The Astrophysical Journal (2023d), arXiv:2311.04979 [hep-ph] .
  * Beacom and Yuksel (2006) J. F. Beacom and H. Yuksel, Phys. Rev. Lett. 97, 071102 (2006), arXiv:astro-ph/0512411 .
  * De La Torre Luque (2023) P. De La Torre Luque, “Dragon2_optimized_dm&antinuclei,”  (2023).
  * Evoli  _et al._ (2017) C. Evoli, D. Gaggero, A. Vittino, G. Di Bernardo, M. Di Mauro, A. Ligorini, P. Ullio, and D. Grasso, JCAP 02, 015 (2017), arXiv:1607.07886 [astro-ph.HE] .
  * Evoli  _et al._ (2018) C. Evoli, D. Gaggero, A. Vittino, M. Di Mauro, D. Grasso, and M. N. Mazziotta, JCAP 07, 006 (2018), arXiv:1711.09616 [astro-ph.HE] .
  * Ginzburg and Syrovatskii (1969) V. L. Ginzburg and S. I. Syrovatskii,  _The origin of cosmic rays_ (Gordon & Breach Publishing Group, 1969).
  * Ferriere  _et al._ (2007) K. Ferriere, W. Gillard, and P. Jean, Astronomy & Astrophysics 467, 611 (2007).
  * Aguilar  _et al._ (2021) M. Aguilar _et al._ , Physics Reports 894, 1 (2021), the Alpha Magnetic Spectrometer (AMS) on the International Space Station: Part II - Results from the First Seven Years.
  * Aguilar  _et al._ (2018) M. Aguilar _et al._ (AMS Collaboration), Phys. Rev. Lett. 120, 021101 (2018).
  * Aguilar  _et al._ (2013) M. Aguilar _et al._ (AMS Collaboration), Phys. Rev. Lett. 110, 141102 (2013).
  * Webber  _et al._ (2002) W. Webber, A. Lukasiak, and F. Mcdonald, ApJ 568, 210 (2002).
  * Stone  _et al._ (2013) E. C. Stone, A. C. Cummings, F. B. McDonald, B. C. Heikkila, N. Lal, and W. R. Webber, Science 341, 150 (2013), https://science.sciencemag.org/content/341/6142/150.full.pdf .
  * Knödlseder  _et al._ (2005) J. Knödlseder, P. Jean, V. Lonjou, G. Weidenspointner, N. Guessoum, W. Gillard, G. Skinner, P. von Ballmoos, G. Vedrenne, J.-P. Roques, S. Schanne, B. Teegarden, V. Schönfelder, and C. Winkler, Astronomy & Astrophysics 441, 513–532 (2005).
  * Guessoum  _et al._ (1991) N. Guessoum, R. Ramaty, and R. E. Lingenfelter, The astrophysical Journal 378, 170 (1991).
  * Dirac (1930) P. A. M. Dirac, Mathematical Proceedings of the Cambridge Philosophical Society 26, 361–375 (1930).
  * Cordes and Lazio (2003) J. M. Cordes and T. J. W. Lazio, (2003), arXiv:astro-ph/0301598 .
  * Cordes and Lazio (2002) J. M. Cordes and T. J. W. Lazio, (2002), arXiv:astro-ph/0207156 .
  * Calore  _et al._ (2021) F. Calore, P. Carenza, M. Giannotti, J. Jaeckel, G. Lucente, and A. Mirizzi, Phys. Rev. D 104, 043016 (2021), arXiv:2107.02186 [hep-ph] .
  * Calore  _et al._ (2022) F. Calore, P. Carenza, M. Giannotti, J. Jaeckel, G. Lucente, L. Mastrototaro, and A. Mirizzi, Phys. Rev. D 105, 063026 (2022), arXiv:2112.08382 [hep-ph] .
  * Carenza  _et al._ (2024) P. Carenza, G. Lucente, L. Mastrototaro, A. Mirizzi, and P. D. Serpico, Phys. Rev. D 109, 063010 (2024), arXiv:2311.00033 [hep-ph] .
  * Jean  _et al._ (2006) P. Jean, J. Knodlseder, W. Gillard, N. Guessoum, K. Ferriere, A. Marcowith, V. Lonjou, and J. P. Roques, Astron. Astrophys. 445, 579 (2006), arXiv:astro-ph/0509298 .
  * de la Torre Luque  _et al._ (2022) P. de la Torre Luque, M. N. Mazziotta, A. Ferrari, F. Loparco, P. Sala, and D. Serini, JCAP 07, 008 (2022), arXiv:2202.03559 [astro-ph.HE] .
  * De la Torre Luque  _et al._ (2023e) P. De la Torre Luque, F. Loparco, and M. N. Mazziotta, JCAP 10, 011 (2023e), arXiv:2305.02958 [astro-ph.HE] .
  * Lanfranchi  _et al._ (2021) G. Lanfranchi, M. Pospelov, and P. Schuster, Ann. Rev. Nucl. Part. Sci. 71, 279 (2021), arXiv:2011.02157 [hep-ph] .
  * Agrawal  _et al._ (2021) P. Agrawal _et al._ , Eur. Phys. J. C 81, 1015 (2021), arXiv:2102.12143 [hep-ph] .
  * Alekhin  _et al._ (2016) S. Alekhin _et al._ , Rept. Prog. Phys. 79, 124201 (2016), arXiv:1504.04855 [hep-ph] .
  * Antel  _et al._ (2023) C. Antel _et al._ , Eur. Phys. J. C 83, 1122 (2023), arXiv:2305.01715 [hep-ph] .
  * Raffelt and Seckel (1988) G. Raffelt and D. Seckel, Phys. Rev. Lett. 60, 1793 (1988).
  * Keil  _et al._ (1997) W. Keil, H.-T. Janka, D. N. Schramm, G. Sigl, M. S. Turner, and J. R. Ellis, Phys. Rev. D 56, 2419 (1997), arXiv:astro-ph/9612222 .
  * Carenza  _et al._ (2019) P. Carenza, T. Fischer, M. Giannotti, G. Guo, G. Martínez-Pinedo, and A. Mirizzi, JCAP 10, 016 (2019), [Erratum: JCAP 05, E01 (2020)], arXiv:1906.11844 [hep-ph] .
  * Carenza  _et al._ (2021) P. Carenza, B. Fore, M. Giannotti, A. Mirizzi, and S. Reddy, Phys. Rev. Lett. 126, 071102 (2021), arXiv:2010.02943 [hep-ph] .
  * Caputo  _et al._ (2022) A. Caputo, G. Raffelt, and E. Vitagliano, Phys. Rev. D 105, 035022 (2022), arXiv:2109.03244 [hep-ph] .
  * Lella  _et al._ (2023) A. Lella, P. Carenza, G. Lucente, M. Giannotti, and A. Mirizzi, Phys. Rev. D 107, 103017 (2023), arXiv:2211.13760 [hep-ph] .
  * Lella  _et al._ (2024) A. Lella, P. Carenza, G. Co’, G. Lucente, M. Giannotti, A. Mirizzi, and T. Rauscher, Phys. Rev. D 109, 023001 (2024), arXiv:2306.01048 [hep-ph] .
  * Balaji  _et al._ (2022) S. Balaji, P. S. B. Dev, J. Silk, and Y. Zhang, JCAP 12, 024 (2022), arXiv:2205.01669 [hep-ph] .
  * Balaji  _et al._ (2023) S. Balaji, M. E. Ramirez-Quezada, J. Silk, and Y. Zhang, Phys. Rev. D 107, 083038 (2023), arXiv:2301.02258 [hep-ph] .
  * Dev  _et al._ (2022) P. S. B. Dev, J.-F. Fortin, S. P. Harris, K. Sinha, and Y. Zhang, JCAP 01, 006 (2022), arXiv:2111.05852 [hep-ph] .
  * Dev  _et al._ (2021) P. S. B. Dev, R. N. Mohapatra, and Y. Zhang, JCAP 05, 014 (2021), arXiv:2010.01124 [hep-ph] .
  * Dev  _et al._ (2020) P. S. B. Dev, R. N. Mohapatra, and Y. Zhang, JCAP 08, 003 (2020), [Erratum: JCAP 11, E01 (2020)], arXiv:2005.00490 [hep-ph] .
  * Kolb  _et al._ (1996) E. W. Kolb, R. N. Mohapatra, and V. L. Teplitz, Phys. Rev. Lett. 77, 3066 (1996), arXiv:hep-ph/9605350 .
  * Raffelt and Zhou (2011) G. G. Raffelt and S. Zhou, Phys. Rev. D 83, 093014 (2011), arXiv:1102.5124 [hep-ph] .
  * Mastrototaro  _et al._ (2020) L. Mastrototaro, A. Mirizzi, P. D. Serpico, and A. Esmaili, JCAP 01, 010 (2020), arXiv:1910.10249 [hep-ph] .
  * Chang  _et al._ (2017) J. H. Chang, R. Essig, and S. D. McDermott, JHEP 01, 107 (2017), arXiv:1611.03864 [hep-ph] .
  * Bouchet  _et al._ (2008) L. Bouchet, E. Jourdain, J. P. Roques, A. Strong, R. Diehl, F. Lebrun, and R. Terrier, Astrophys. J. 679, 1315 (2008), arXiv:0801.2086 [astro-ph] .
  * Siegert  _et al._ (2022) T. Siegert, J. Berteaud, F. Calore, P. D. Serpico, and C. Weinberger, Astron. Astrophys. 660, A130 (2022), arXiv:2202.04574 [astro-ph.HE] .
  * Kappadath (1998) S. C. Kappadath, Ph.D. thesis, University of New Hampshire (1998).
  * Strong  _et al._ (2004) A. W. Strong, I. V. Moskalenko, and O. Reimer, The Astrophysical Journal 613, 962 (2004).
  * Foster  _et al._ (2021) J. W. Foster, M. Kongsore, C. Dessert, Y. Park, N. L. Rodd, K. Cranmer, and B. R. Safdi, Phys. Rev. Lett. 127, 051101 (2021), arXiv:2102.02207 [astro-ph.CO] .
  * Strong  _et al._ (1994a) A. W. Strong, K. Bennett, H. Bloemen, R. Diehl, W. Hermsen, D. Morris, V. Schoenfelder, J. G. Stacy, C. de Vries, M. Varendorff, C. Winkler, and G. Youssefi, A&A 292, 82 (1994a).
  * Merle (2017) A. Merle,  _Sterile Neutrino Dark Matter_ (IOP, 2017).
  * Boyarsky  _et al._ (2019) A. Boyarsky, M. Drewes, T. Lasserre, S. Mertens, and O. Ruchayskiy, Prog. Part. Nucl. Phys. 104, 1 (2019), arXiv:1807.07938 [hep-ph] .
  * Abazajian  _et al._ (2012) K. N. Abazajian _et al._ , (2012), arXiv:1204.5379 [hep-ph] .
  * Abazajian and Kusenko (2019) K. N. Abazajian and A. Kusenko, Phys. Rev. D 100, 103513 (2019), arXiv:1907.11696 [hep-ph] .
  * Strong  _et al._ (1994b) A. W. Strong, K. Bennett, H. Bloemen, R. Diehl, W. Hermsen, D. Morris, V. Schoenfelder, J. G. Stacy, C. de Vries, M. Varendorff, C. Winkler, and G. Youssefi, Astronomy & Astrophysics 292, 82 (1994b).
  * Bolton  _et al._ (2020) P. D. Bolton, F. F. Deppisch, and P. S. Bhupal Dev, JHEP 03, 170 (2020), arXiv:1912.03058 [hep-ph] .
  * Heinbach and Simon (1995) U. Heinbach and M. Simon, Astrophys. J. 441, 209 (1995).
  * Strong and Moskalenko (1998) A. W. Strong and I. V. Moskalenko, Astrophys. J. 509, 212 (1998), arXiv:astro-ph/9807150 [astro-ph] .
  * Seo and Ptuskin (1994) E.-S. Seo and V. S. Ptuskin, ApJ 431, 705 (1994).
  * Sabti  _et al._ (2020) N. Sabti, A. Magalich, and A. Filimonova, JCAP 11, 056 (2020), arXiv:2006.07387 [hep-ph] .
  * Boyarsky  _et al._ (2021) A. Boyarsky, M. Ovchynnikov, O. Ruchayskiy, and V. Syvolap, Phys. Rev. D 104, 023517 (2021), arXiv:2008.00749 [hep-ph] .
  * Mastrototaro  _et al._ (2021) L. Mastrototaro, P. D. Serpico, A. Mirizzi, and N. Saviano, Phys. Rev. D 104, 016026 (2021), arXiv:2104.11752 [hep-ph] .
  * Ballett  _et al._ (2020) P. Ballett, T. Boschi, and S. Pascoli, JHEP 03, 111 (2020), arXiv:1905.00284 [hep-ph] .
  * Krasnov (2019) I. Krasnov, Phys. Rev. D 100, 075023 (2019), arXiv:1902.06099 [hep-ph] .
  * Chou  _et al._ (2017) J. P. Chou, D. Curtin, and H. J. Lubatti, Phys. Lett. B 767, 29 (2017), arXiv:1606.06298 [hep-ph] .
  * Ovchynnikov (2021) M. Ovchynnikov, Ph.D. thesis, Leiden U. (2021).
  * Beacom  _et al._ (2005) J. F. Beacom, N. F. Bell, and G. Bertone, Phys. Rev. Lett. 94, 171301 (2005).


# Gravitational Slip parameter and Gravitational Waves in Einstein-Cartan
theory

Maryam Ranjbar<EMAIL_ADDRESS>Department of Physics, Faculty of
Sciences, Golestan University, Gorgan, Iran  Siamak Akhshabi
<EMAIL_ADDRESS>Department of Physics, Faculty of Sciences, Golestan
University, Gorgan, Iran  Mohsen Shadmehri<EMAIL_ADDRESS>Department of
Physics, Faculty of Sciences, Golestan University, Gorgan, Iran

###### Abstract

We study the evolution of scalar and tensor cosmological perturbations in the
framework of the Einstein-Cartan theory of gravity. The value of the
gravitational slip parameter which is defined as the ratio of the two scalar
potentials in the Newtonian gauge, can be used to determine whether or not the
gravity is modified. We calculate the value of slip parameter in the Einstein-
Cartan cosmology and show that it falls within the observed range. We also
discuss the evolution of the cosmic gravitational waves as another measure of
the modification of gravity.

††preprint: APS/123-QED

##  I Introduction

General Relativity (GR) has successfully predicted numerous phenomena, as
corroborated by various studies [1, 2, 3, 4, 5, 6, 7]. However, several
mysteries persist in the realm of physics. The nature of the cosmological
constant [8, 9], dark energy [8, 9, 10, 11, 12], and dark matter [13, 14, 15,
16] are among the phenomena yet to be comprehensively understood.

One potential reason for these uncertainties is that GR, in the context of
quantum mechanics, is fundamentally a classical theory. It is primarily
tailored for the analysis of macroscopic phenomena. To address this limitation
and extend the scope of GR to microphysical processes, various modified
gravity theories have been proposed. Notably among them are those that factor
in the torsion of spacetime, which is described as the non-symmetric part of
the affine connection.

The simplest of such theories, Einstein-Cartan theory (ECT) [17, 18, 19] has
the same Lagrangian as in GR but the geometry of space-time is no longer
Riemannian, but is a more general Riemann-Cartan space-time [17, 18, 19, 20,
21]. In Einstein-Cartan theory, torsion is coupled to the spin angular
momentum of the matter fields which acts as a source for gravitational
interactions. Einstein-Cartan theory is well studied in the literature and has
been used to study cosmological constant’s sign problem [22], Hubble tension
problem [23], Big Bang singularity [24, 25, 26, 27, 28], the emergent universe
scenario [29], and Dark matter [30].

The detection of gravitational waves by LIGO has provided a unique opportunity
to gain valuable insights into both astrophysics and cosmology, particularly
in relation to the early universe [6]. Within the framework of ECT, the
presence of torsion can introduce notable changes in the generation and
propagation of gravitational waves. Research presented in [31] has
demonstrated that neglecting the consideration of torsion may lead to
systematic errors when studying the propagation of gravitational waves.
Furthermore, as highlighted in [32], ECT can yield significantly different
descriptions of gravitational wave fluctuations compared to those derived from
General Relativity (GR). Given that the effects of torsion and spin during the
early stages of the universe are assumed to be significant [33], investigating
the behavior of GW through the ECT not only allows us to test the validity of
this model, but also has the potential to reveal novel understandings
regarding the nature of gravity.

Typically, modified gravity theories adjust the strength of gravity on large
scales and modify the weak gravitational lensing effect. Such alterations
permit us to construct a model-independent observational test to ascertain the
validity of the proposed theories [34]. Model-independent methods,
specifically at the linear perturbation level, were formulated in [35, 36, 37,
38, 39]. These studies employ parameterization and principal component
analysis to constrain the cosmic growth and expansion history. An alternative
strategy involves utilizing the constraints and known characteristics of dark
energy, along with its behavior, as a foundation for developing and evaluating
theories. This approach has been elaborated upon in [40, 41, 42].

Each of the methods mentioned above requires the use of parameterizations to
differentiate between various models based on data. However, it’s essential to
acknowledge that these parameterizations introduce biases that are contingent
on the specific parameterization chosen. These biases have the potential to
influence the accuracy of the results [43].

On the other hand, considering the perturbations level, there is an approach
that is fully model-independent. This measures properties like the anisotropic
stress without making any assumptions about the nature of the dark energy. The
existence of anisotropic stress results in the appearance of gravitational
slip i.e. the slip parameter $\eta$ defined as the ratio between two
gravitational potentials in a perturbed Friedmann-Robertson-Walker (FRW)
universe being different from unity. Given that the contribution of neutrinos’
free streaming to the anisotropic pressure in the late stages of the universe
becomes negligible in the presence of perfect fluid matter, the only source of
anisotropic stress could be a modification of gravity. In other words, the
existence of anisotropic stress leads to a difference in values between the
two scalar gravitational potentials that appear in the linear perturbation
equations of motion and can be described as the gravitational slip parameter
$\eta\equiv\frac{\Phi}{\Psi}\neq 1$ [43, 44].

The advantage of the gravitational slip parameter is that it can be
constructed based on model-independent observable quantities, which could
potentially confirm or rule out a particular model[45, 46]. Moreover, another
advantage of this method is the relation between gravitational slip parameter
and gravitational waves. The existence of anisotropic stress at the first
order in perturbations caused by perfect-fluid matter leads to modifications
in the propagation or behaviour of gravitational waves. In General Relativity,
gravitational waves are the only propagating degrees of freedom. Therefore, as
discussed earlier, it is logical to define modified gravity models as those in
which gravitational waves are altered. Imperfect fluids with anisotropic
stress may impact the two gravitational potentials but do not affect the
propagation of tensor modes which are associated with gravitational waves [47,
48]. So far, the relation between gravitational slip parameter and non-
standard gravitational waves in the context of scalar-tensor theories such as
Horndeski theory has been studied [49, 50].

Motivated by the above mentioned discussions, in this paper, we study the
gravitational slip parameter and the behaviour of gravitational waves in the
context of theories containing torsion, particularly the ECT as the simplest
model of this class. In section II we briefly review the definitions and field
equations in ECT, and obtain the scale factor and the spin density in three
different types of the universe including radiation-dominated era, matter-
dominated era, and dark energy-dominated era. In section III we investigate
linear scalar cosmological perturbations in ECT in order to find gravitational
slip parameter within this framework. In section IV we study the propagation
of tensor modes (i.e., gravitational waves) in EC cosmology and analyze the
behavior of gravitational waves in this model. Section V is devoted to our
concluding remarks.

##  II Einstein-Cartan cosmology

###  II.1  Definitions and Field Equations

Torsion tensor 111In this paper we assume a flat FRW universe with the metric
signature (+,-,-,-). is defined as the anti-symmetric part of the connection
[51]

$S_{\mu\nu}\,^{\alpha}\,=\,\Gamma_{[\mu\nu]}^{\alpha}\,=\,\frac{1}{2}\,(\Gamma_{\mu\nu}^{\alpha}\,-\,\Gamma_{\nu\mu}^{\alpha}).$
(1)

In order to compare torsion theories with general relativity, it is suitable
to divide the full connection in two parts

$\Gamma_{\mu\nu}^{\alpha}\,=\,\overset{\circ}{\Gamma}\,^{\alpha}_{\mu\nu}\,-\,K_{\mu\nu}\,^{\alpha},$
(2)

where $\overset{\circ}{\Gamma}\,^{\alpha}_{\mu\nu}$ is standard Christoffel
connection and $K_{\mu\nu}\,^{\alpha}$ is known as the contorsion tensor which
is related to Torsion tensor by the following relations

$K_{\mu\nu}\,^{\alpha}=-S_{\mu\nu}\,^{\alpha}+S_{\nu}\,^{\alpha}\,{}_{\mu}-S^{\alpha}\,_{\mu\nu},$
(3)
$S_{\mu\nu}\,^{\alpha}=-K_{[\mu\nu]}\,^{\alpha}=-\frac{1}{2}\,(K_{\mu\nu}\,^{\alpha}-K_{\nu\mu}\,^{\alpha}).$
(4)

For the remainder of this section we adopt the notation of reference [51].
Since the above mentioned connection 2 is not necessarily symmetric, the
Riemann tensor in ECT will have additional terms compared to GR 222Quantities
with an over-circle indicate GR quantities and are obtained from the
Christoffel connection

$\begin{split}\mathrm{R}_{\mu\nu\alpha}\,^{\beta}&=\overset{\circ}{\mathrm{R}}_{\mu\nu\alpha}\,^{\beta}\,-\nabla_{\mu}K_{\nu\alpha}\,^{\beta}\,-\nabla_{\nu}K_{\mu\alpha}\,^{\beta}\,\\\
&-K_{\nu\mu}\,^{\lambda}K_{\lambda\alpha}\,^{\beta}\,+K_{\mu\nu}\,^{\lambda}K_{\lambda\alpha}\,^{\beta}\,\\\
&-K_{\nu\alpha}\,^{\lambda}K_{\mu\lambda}\,^{\beta}\,+K_{\mu\alpha}\,^{\lambda}K_{\nu\lambda}\,^{\beta}\\\
&=\overset{\circ}{\mathrm{R}}_{\mu\nu\alpha}\,^{\beta}\,-2\nabla_{[\mu}K_{\nu]\alpha}\,^{\beta}\,+2K_{[\mu\nu]}\,^{\lambda}K_{\lambda\alpha}\,^{\beta}\,\\\
&+2K_{[\mu|\alpha}\,^{\lambda}K_{|\nu]\lambda}\,^{\beta},\end{split}$ (5)

Contracting the Riemann tensor, the Ricci tensor and Ricci scalar will be

$\begin{split}\mathrm{R}_{\mu\nu}&=\overset{\circ}{\mathrm{R}}_{\mu\nu}\,-\nabla_{\lambda}K_{\mu\nu}\,^{\lambda}\,+\nabla_{\mu}K_{\lambda\nu}\,^{\lambda}\,\\\
&+K_{\rho\mu}\,^{\lambda}K_{\lambda\nu}\,^{\rho}\,-K_{\mu\nu}\,^{\lambda}K_{\rho\lambda}\,^{\rho},\end{split}$
(6)

$\begin{split}\mathrm{R}&=\overset{\circ}{\mathrm{R}}\,+2\nabl
a_{\lambda}K_{\alpha}\,^{\lambda\alpha}\,+K_{\rho}\,^{\alpha\lambda}K_{\lambda\alpha}\,^{\rho}\,\\\
&-K_{\alpha}\,^{\alpha\lambda}K_{\rho\lambda}\,^{\rho}\\\
&=\overset{\circ}{\mathrm{R}}\,-4(\nabla_{\lambda}\,-S_{\alpha\lambda}\,^{\alpha})\,S_{\rho}\,^{\lambda\rho}\,\\\
&+2S_{\alpha\beta\gamma}S^{\gamma\beta\alpha}\,+S_{\alpha\beta\gamma}S^{\alpha\beta\gamma}.\end{split}$
(7)

The two field equations in Einstein-Cartan gravity are obtained by variation
of the Lagrangian with respect to the metric and connection fields. One of the
field equations gives an algebraic relation between the torsion and contorsion
and their source, the spin tensor of the matter fields $\tau_{\alpha\nu\nu}$
as [51]

$\begin{split}S_{\alpha\nu}\,^{\mu}=\kappa&(\tau_{\alpha\nu}\,^{\mu}\,+\frac{1}{2}\delta_{\alpha}^{\mu}\tau_{\nu\lambda}\,^{\lambda}\,-\frac{1}{2}\delta_{\nu}^{\mu}\tau_{\alpha\lambda}\,^{\lambda}),\\\
&-2S_{\alpha\lambda}\,^{\lambda}=\kappa\tau_{\alpha\lambda}\,^{\lambda}.\end{split}$
(8)

$K_{\mu\nu}\,^{\alpha}=\kappa(-\tau_{\mu\nu}\,^{\alpha}\,+\tau_{\nu}\,^{\alpha}\,{}_{\mu}\,-\tau^{\alpha}\,_{\mu\nu}\,-\delta_{\mu}^{\alpha}\tau_{\nu\lambda}\,^{\lambda}\,+g_{\mu\nu}\tau^{\alpha}\,_{\lambda}\,{}^{\lambda}).$
(9)

The other field equation of EC theory can be written as

$\begin{split}&\overset{\circ}{\mathrm{G}}_{\mu\nu}=\kappa\overset{\circ}{\mathrm{T}}_{\mu\nu}\,+\kappa^{2}\bigg{\\{}-4\tau_{\mu\lambda}\,^{\alpha}\tau_{\nu\alpha}\,^{\lambda]}\,-2\tau_{\mu\lambda\alpha}\tau_{\nu}\,^{\lambda\alpha}\,\\\
&+\tau_{\alpha\lambda\mu}\tau^{\alpha\lambda}\,_{\nu}+\frac{1}{2}g_{\mu\nu}\big{(}4\tau_{\lambda}\,^{\beta}\,{}_{[\alpha}\tau^{\lambda\alpha}\,_{\beta]}\,+\tau^{\alpha\lambda\beta}\tau_{\alpha\lambda\beta}\big{)}\bigg{\\}}\end{split}$
(10)

where $\overset{\circ}{\mathrm{G}}_{\mu\nu}$ and
$\overset{\circ}{\mathrm{T}}_{\mu\nu}$ refer to Einstein and Energy-Momentum
tensors in GR respectively, and $\tau_{\mu\lambda}\,^{\alpha}$ indicates spin
tensor.

###  II.2  Scale Factor and Spin Density

We now use the two field equation above to determine background cosmological
solutions for a single component universe. We begin by assuming the FRW metric
for the universe

$ds^{2}=dt^{2}\,-a^{2}(t)\ \gamma_{ij}dx^{i}dx^{j}.$ (11)

In EC cosmology, the symmetries of FRW space-time lead to the non-zero
component of spin and torsion tensors to be [52]

$\begin{split}&q(t)=\tau_{011}\ =\tau_{022}\ =\tau_{033}=-\tau_{A0A},\\\
&h(t)=S_{110}\ =S_{220}\ =S_{330}=-S_{A0A},\end{split}$ (12)

where spin density function $q(t)$ and torsion function $h(t)$ are functions
of cosmic time and are related to each other by the algebraic equation 8.
Substituting 11 and 12 into EC field equations, the Friedmann equations will
be

$3\bigg{(}\frac{\dot{a}(t)}{a(t)}\bigg{)}^{2}=\kappa\rho(t)\,+3\kappa^{2}a(t)^{-4}\
q(t)^{2},$ (13)

$\begin{split}&-\bigg{(}2\frac{\ddot{a}(t)}{a(t)}\,+(\frac{\dot{a}(t)}{a(t)})^{2}\bigg{)}a^{2}(t)\gamma_{ij}=\\\
&\bigg{(}\kappa a^{2}(t)\
p(t)\,-2\kappa^{2}a^{-4}(t)q(t)^{2}\bigg{)}\gamma_{ij}.\end{split}$ (14)

where $a(t)$ and $q(t)$ are scale factor and spin density respectively, and
$\kappa$ is Einstein gravitational constant $\kappa=4\pi$. Additionally,
pressure and energy density of a perfect fluid are indicated by $p(t)$ and
$\rho(t)$, respectively. For a single component universe

$\rho(t)=\frac{\rho_{0}}{a(t)^{3(1+\omega)}}.$ (15) $p(t)=\omega\rho(t)$ (16)

where $\omega$ is the equation of state parameter. In following subsection we
obtain the background scale factor and spin density as a function of time for
various different matter content for the universe.

####  II.2.1  Radiation dominated universe

By solving the system of equations 13 and 14 for $\omega=\frac{1}{3}$, we find
the scale factor and spin density as follows

$a(t)=Root\,of\Bigg{(}\int^{Z}\frac{3\ x}{\sqrt{x^{3}\,+\kappa\
\rho_{0}}}dx\,+t\Bigg{)},$ (17) $q(t)=\frac{\sqrt{9\ \dot{a}(t)^{2}\
a(t)^{2}\,-3\ \kappa\rho_{0}}}{3\ \kappa}.$ (18)

The integral has no analytical solution and should be solved numerically. In
order to obtain an analytical form for the background scale factor and spin
density for future use, we solve the integral in 17 numerically and try to fit
an analytical function to the numerical solution. Since the effects of torsion
on the cosmological dynamics in EC cosmology expected to be small, we assume
that the form the scale factor differ only slightly from its form in standard
GR cosmology. We assume that for a radiation dominated universe, $a(t)\propto
t^{\beta}$ where $\beta$ is a constant that differs only slightly from $1/2$.
First, we solved the integral 17 numerically by quadrature method using
Python. Then we test the ansatz by directly substituting it into the system of
equations 13 and 14. At the end, the value of constant $\beta$ can be deduced
by fitting an appropriate curve to the numerical solution. The result for
$a(t)$ would be

$a(t)\propto(t)^{\tfrac{1}{2}\,-0.001}.$ (19)

”Fig. 1”, shows the behavior of the fitted ansatz 19 to numerical solution of
the integral 17 333Note that in this paper we take $\rho_{0}=9.10^{-27}$ and
$\kappa=2.077.10^{-43}$.

Figure 1: The dashed line shows the ansatz (i.e.19) and the solid line
demonstrates the numerically obtained scale factor from the system of
equations [13,14]. The ansatz $a(t)=alphat^{\beta}$ with fixed parameters as
$\alpha=6,01$ and $\beta=\frac{1}{2}-0.001$ (i.e.19) behaves exactly same to
integral 17 which allows us to consider eq.19 as the scale factor of radiation
dominated universe for the rest of our calculations. Figure 2: Normalized
scale factor for a radiation dominated universe in Einstein-Cartan cosmology.
The present day scale factor $a_{0}$ was normalized to unity. Figure 3:  The
graph indicates the spin density’s value for the radiation dominated universe
changes regarding time (eq.18). The effects of torsion in very early universe
is considerable.

Figure 2 shows the behaviour of $a(t)$ in (eq.18) when the present day scale
factor $a_{0}$ was normalized to unity. The behavior of the spin density in a
radiation dominated universe, 18 is indicated in ”Fig. 3”.

####  II.2.2  Matter Dominated Universe

By solving the system of equations 13 and 14 for $\omega=0$, the scale factor
and spin density can be obtained for a matter dominated universe as

$a(t)=Root\,of\Bigg{(}\int^{Z}\frac{x\ \sqrt{3}}{\sqrt{x(x^{2}\,+\kappa\
\rho_{0})}}dx\,+t\Bigg{)},$ (20) $q(t)=\frac{-\sqrt{9\ \dot{a}(t)^{2}\
a(t)^{2}\,-3\ \kappa\rho_{0}a(t)}}{3\ \kappa}.$ (21)

Using the same method as in the previous sub-section II.2.1, we assume the
scale factor to be in the form $a(t)\propto t^{\lambda}$ where the constant
$\lambda$ differs only slightly from $2/3$. By fitting the ansatz to the
numerical solution of integral 20 we find

$a(t)\propto(t)^{\tfrac{2}{3}\,-0.13};$ (22)

”Fig. 4”, shows the behavior of the fitted ansatz 22 to numerical solution of
the integral 20.
The normalised scale factor of matter dominated universe is depicted in ”Fig.
5”. ”Fig. 6” shows the evolution of the spin density function $q(t)$ in a
matter dominated universe in EC cosmology. The effect of torsion in a matter
dominated universe is far less than in a radiation dominated universe.

Figure 4: The dashed line indicates the ansatz of scale factor (i.e.22) and
the solid line shows the obtained scale factor from the system of equations
[13-14]. The ansatz (i.e.22) behaves approximately the same as integral 20
which allows us to consider eq.22 as the scale factor for matter dominated
universe in the rest of our calculations. Figure 5:  Normalized scale factor
in a matter dominated universe in EC cosmology. The present day scale factor
($t=1*10^{10}$) was set to unity. Figure 6:  Evolution of spin density for the
matter dominated universe (eq.21) in EC cosmology. The torsion effects in this
universe is infinitesimal.

####  II.2.3  Dark Energy Dominated

At late times universe is dominated by the cosmological constant[53]. By
assuming $\omega=-1$ and solving the system of equations [13-14], we find the
scale factor and spin density of a the dark energy dominated universe as

$a(t)=\sqrt{2}\,+\,\sin\left(\frac{\sqrt{15}\ \sqrt{\kappa}\
\sqrt{\rho_{0}}}{3}\,t\right)\,+\,\cos\left(\frac{\sqrt{15}\ \sqrt{\kappa}\
\sqrt{\rho_{0}}}{3}\,t\right),$ (23) $q(t)=\frac{a(t)\sqrt{9\
\dot{a}(t)^{2}-3\ \kappa\rho_{0}\ a(t)^{2}}\,}{3\ \kappa}.$ (24)

Behavior of the spin density (i.e. eq. 24) is plotted in ”Fig. 7” while ”Fig.
8” represents the behavior of the normalized scale factor in a dark energy
dominated universe in EC cosmology.

Figure 7:  The graph demonstrates the evolution of spin density for the dark
energy dominated universe (eq.24). The torsion effect in Einstein-Cartan
theory for this universe is negligible. Figure 8:  Normalized scale factor for
a Dark Energy dominated universe with torsion.

For a better comparison, we plotted the scale factor and spin density of the
radiation, matter, and dark energy dominated universe in ”Fig. 9” and ”Fig.
10” respectively.

Figure 9:  The evolution of scale factor in Einstein-Cartan cosmology for dark
energy dominated universe 23 (green dashed line), matter dominated universe22
(pink dotted line) and radiation dominated universe 19 (orange solid line).
Figure 10:  Evolution of spin density in Einstein-Cartan cosmology for dark
Energy 24 (green dashed line), matter 21 (pink dotted line) and radiation 18
(orange solid line) dominated universes.

##  III  Slip Parameter

###  III.1  Theoretical aspect of Slip Parameter

The gravitational slip parameter $\eta$ is a direct measure of scalar (linear)
anisotropic stress. At the linear level, the anisotropic stress $\eta$ defined
as the ratio between the two gravitational potentials [44, 43]

$\eta(t,k)\equiv\Phi/\Psi.$ (25)

The existence of gravitational slip parameter ($\eta\neq 1$) in the presence
of perfect fluid matter is a clue of a modification in gravity. In addition,
this parameter is a model independent quantity which distinguishes the groups
of gravity models [44, 43, 54].

We consider only the scalar degrees of freedom of the perturbed metric around
a flat FRW background in the longitudinal (conformal Newtonian) gauge, where
the line-element is of the form

$ds^{2}=(1+2^{\prime}\Psi)dt^{2}-a(t)^{2}(1-2\Phi)\delta_{ij}dx^{i}dx^{j},$
(26)

where $\Psi\equiv\Psi(t,r)$ and $\Phi\equiv\Phi(t,r)$ are the gravitational
potentials and $a(t)$ is the scale factor. By plugging the perturbed metric 26
and non-zero components of the spin tensor 12 into the field equation of
Einstein-Cartan theory 10; the time-time component 27, time-space component 28
and space-space component 29 of perturbed field equation of ECT can be written
respectively as 444Here, and for the rest of calculations, an overdot denotes
a time derivative, i.e. $\dot{a(t)}=\frac{da(t)}{dt}$.

$\begin{split}&3\ H(t)^{2}\,+2\nabla^{2}\Phi(t,r)\,-6\ H(t)\
\dot{\Phi}(t,r)\\\ &\,=\kappa\Big{[}\rho(t)\,+\delta\rho(t,r)+2\rho(t)\
\Psi(t,r)\Big{]}\\\ &\,+\kappa^{2}\Big{[}3\ a(t)^{-4}\ q(t)^{2}\ +12\
a(t)^{-4}\ q(t)^{2}\ \Phi(t,r)\\\ &\,+6\ a(t)^{-4}\ q(t)\ Y(t,r)\ +6\
a(t)^{-4}\ q(t)^{2}\ \Psi(t,r)\Big{]},\end{split}$ (27)

$\begin{split}2\partial_{i}\Big{(}\dot{\Phi}\ (t,r)\,+H(t)\
\Psi(t,r)\Big{)}=0,\end{split}$ (28)

$\begin{split}&\Bigg{[}-2\Big{(}\frac{\ddot{a}(t)}{a(t)}+\big{(}\frac{\dot{a}(t)}{a(t)}\big{)}^{2}\Big{)}\
-\ \big{(}\frac{\dot{a}(t)}{a(t)}\big{)}^{2}+\nabla^{2}(\Psi(t,r)\\\
&-\Phi(t,r))+\bigg{(}4\
\frac{\ddot{a}(t)}{a(t)}+4\big{(}\frac{\dot{a}(t)}{a(t)}\big{)}^{2}+2\big{(}\frac{\dot{a}(t)}{a(t)}\big{)}^{2}\bigg{)}\\\
&\bigg{(}\Phi(t,r)+\Psi(t,r)\bigg{)}+2\ddot{\Phi}(t,r)\ +\
2H(t)\dot{\Psi}(t,r)\ \\\ &+\ 4H(t)\dot{\Phi}(t,r)\Bigg{]}\delta_{ij}+\
\partial_{i}\partial_{j}(\Phi(t,r)-\Psi(t,r))\\\ &=\Bigg{[}\kappa\
a(t)^{2}P(t)\ +\ \kappa\ a(t)^{2}\delta P(t,r)\\\ &-\ 2\kappa\
a(t)^{2}P(t)\Phi(t,r)-\ 25\kappa^{2}a(t)^{-2}q(t)Y(t,r)\\\
&-52\kappa^{2}a(t)^{-2}q(t)^{2}\Phi(t,r)+22\kappa^{2}a(t)^{-2}q(t)^{2}\Psi(t,r)\\\
&-\ 62\kappa^{2}a(t)^{-2}q(t)^{2}\Bigg{]}\delta_{ij}.\end{split}$ (29)

Note that we only consider scalar linear perturbation of the field equation
(i.e. $\Pi_{ij}=0$). Additionally, we only consider perturbation of the non-
zero component of the spin tensor which is represented by $Y(t,r)$.
$\delta\rho(t,r)$, $\delta P(t,r)$ and $H(t)$ refer to density perturbation,
pressure perturbation and Hubble parameter, respectively 555$\kappa$ is
Einstein gravitational constant

$H(t)=\frac{\dot{a}(t)}{a(t)},$ (30) $\begin{split}&P(t)=\omega\rho(t),\\\
&\delta P(t,r)=\omega\delta\rho(t,r).\end{split}$ (31)

In the above equations, any scalar field $\chi$ can be expanded into Fourier
modes

$\chi=\int\frac{d^{3}k}{(2\pi)^{3}}\
e^{\mathit{i}\mathbf{k}.\mathbf{x}}\chi(t),$ (32)

performing the expansion, we can write the equation for the fluctuations as
666$\mathbf{k}$ is wave-number

$\begin{split}&3\ H(t)^{2}\,-2\mathbf{k}^{2}\Phi(t)\,-6\ H(t)\
\dot{\Phi}(t)\\\ &=\kappa\Big{[}\rho(t)\,+\delta\rho(t)+2\rho(t)\
\Psi(t)\Big{]}\\\ &+\kappa^{2}\Big{[}3\ a(t)^{-4}\ q(t)^{2}+12\ a(t)^{-4}\
q(t)^{2}\ \Phi(t)\\\ &+6\ a(t)^{-4}\ q(t)\ Y(t)+6\ a(t)^{-4}\ q(t)^{2}\
\Psi(t)\Big{]},\end{split}$ (33)

$\begin{split}2\mathbf{k}\Big{(}\dot{\Phi}(t)\,+H(t)\
\Psi(t)\Big{)}=0,\end{split}$ (34)

$\begin{split}&\Bigg{[}-2\Big{(}\frac{\ddot{a}(t)}{a(t)}+\big{(}\frac{\dot{a}(t)}{a(t)}\big{)}^{2}\Big{)}\
-\ \big{(}\frac{\dot{a}(t)}{a(t)}\big{)}^{2}\ -\
\mathbf{k}^{2}(\Psi(t)+\Phi(t))\\\ &+\bigg{(}4\
\frac{\ddot{a}(t)}{a(t)}+4\big{(}\frac{\dot{a}(t)}{a(t)}\big{)}^{2}+2\big{(}\frac{\dot{a}(t)}{a(t)}\big{)}^{2}\bigg{)}\bigg{(}\Phi(t)+\Psi(t)\bigg{)}\\\
&+2\ddot{\Phi}(t)\ +\ 2H(t)\dot{\Psi}(t)\ +\
4H(t)\dot{\Phi}(t)\Bigg{]}\delta_{ij}\\\ &+\
\mathbf{k}^{2}(\Psi(t)-\Phi(t))=\Bigg{[}\kappa\ a(t)^{2}P(t)\\\ &+\ \kappa\
a(t)^{2}\delta P(t)\ -\ 2\kappa\ a(t)^{2}P(t)\Phi(t)\\\ &-\
25\kappa^{2}a(t)^{-2}q(t)Y(t)\ -\ 52\kappa^{2}a(t)^{-2}q(t)^{2}\Phi(t)\\\ &+\
22\kappa^{2}a(t)^{-2}q(t)^{2}\Psi(t)\ -\
62\kappa^{2}a(t)^{-2}q(t)^{2}\Bigg{]}\delta_{ij}.\end{split}$ (35)

In order to find the ratio of gravitational potentials, we shall insert the
background scale factor and spin density obtained in the previous section into
perturbed field equations and solve the system of equations (33, 34, 35).

####  III.1.1  Radiation Dominated

In a radiation dominated universe, the scale factor can be written in terms of
density as $a(t)=(\frac{\rho_{0}}{\rho(t)})^{1/4}$. As a result, spin density
[18] also can be written in terms of matter density

$q(t)=\frac{\sqrt{9\ (\frac{d}{dt}(\frac{\rho_{0}}{\rho(t)})^{1/4})^{2}\
(\frac{\rho_{0}}{\rho(t)})^{1/2}\,-3\ \kappa\rho_{0}}}{3\ \kappa}.$ (36)

Perturbation of previous equation 36 gives us a relation between $Y(t)$ and
$\delta\rho(t)$, where $Y(t)\propto(\frac{1}{\delta\rho(t)})^{1/14}$.
Therefore, $Y(t)$ in perturbed field equations for radiation dominated
universe is negligible.

In sec [II.2.1] we obtained background scale factor and spin density for
radiation dominated universe in EC cosmology. In order to find $\Psi(t)$,
$\Phi(t)$ and $\delta\rho(t)$ in this case, we substitute equations 18 and 19
into the system of equations [33, 34, 35] and by assuming $\omega=1/3$, solve
the system of equations numerically. The result for the evolution of
gravitational slip parameter over time is illustrated in ”Fig. 11”.

Figure 11:  The evolution of the ratio of two gravitational potentials
$\Phi(t)$ and $\Psi(t)$ (i.e. the Slip parameter) for a radiation dominated
universe in ECT.

####  III.1.2  Matter Dominated

Same as previous sub-section III.1.2, using eq.16, the scale factor can be
written in terms of density (i.e. $a(t)=(\frac{\rho_{0}}{\rho(t)})^{1/3}$). As
a result, spin density [21] can be written in terms of matter density as below

$q(t)=\frac{-\sqrt{9\ (\frac{d}{dt}(\frac{\rho_{0}}{\rho(t)})^{1/3})^{2}\
(\frac{\rho_{0}}{\rho(t)})^{2/3}\,-3\
\kappa\rho_{0}(\frac{\rho_{0}}{\rho(t)})^{1/3}}}{3\ \kappa}.$ (37)

By perturbing this equation, we found the relation between $Y(t)$ and
$\delta\rho(t)$ as
$Y(t)\propto(\frac{1}{\delta\rho(t)})^{2/22}-(\frac{1}{\delta\rho(t)})^{1/6}$.
Therefore, $Y(t)$ in perturbed field equations for matter dominated universe
will also be vanishingly small.

In the next step, we insert the scale factor 22, and the spin density 21 into
the system of equations [33-35] and by assuming $\omega=0$, we Solve this
system of equations numerically to find $\Psi(t)$, $\Phi(t)$ and
$\delta\rho(t)$. The result in the form of the ratio of gravitational
potentials (i.e. slip parameter $\eta$) is demonstrated in ”Fig. 12”.

Figure 12:  The evolution of the ratio of two gravitational potentials
$\Phi(t)$ and $\Psi(t)$ (i.e. the Slip parameter) for a matter dominated
universe in ECT.

####  III.1.3  Dark Energy Dominated

By the same analysis as the previous cases, we find that the perturbed spin
tensor $Y(t)$ is also negligible in dark energy dominated universe.

Consequently, we substituted the scale factor 23 and spin density 24 of dark
energy dominated universe into the system of equations [33\- 35] and solved it
numerically to find the slip parameter. The behavior of slip parameter over
time is demonstrated in ”Fig. 13”.

Figure 13:  The evolution of the slip parameter versus time for a dark energy
dominated universe ECT.

###  III.2  Slip Parameter from an observational viewpoint

In observational approaches, it is generally necessary to adopt a model or
make assumptions about the initial conditions of the Universe before
attempting to estimate a quantity[46]. Several studies [44, 43, 55, 56, 57]
demonstrate the possibility of obtaining three model-independent quantities,
denoted as $P1,P2,P3$, that effectively eliminate the influences of the
primordial power spectrum’s shape and galaxy bias, where $E=H/H_{0}$ and $z$
represents redshift. (see [46] for more explanation).

In [43], it has been shown that the ratio of the gravitational potentials and
therefore the gravitational slip parameter can be reconstructed in a model-
independent manner by pure observational quantities as following

$\eta_{obs}\equiv\frac{3\ P_{2}\ (1+z)^{3}}{2E^{2}\
(P_{3}+2+\frac{E^{\prime}}{E})}-1=\eta.$ (38)

$\eta_{obs}$ remains independent of any assumptions concerning the cosmic
expansion. In other words, it does not rely on a specific $\Lambda$CDM
background or any other particular model [46].

Using the above equation 38, the authors in reference [46], obtained the value
of the slip parameter $\eta_{obs}$ in different Redshift using three different
statistical methods. Their reported observational value of slip parameter is
summarized in Table 1. Note that the values generally have great amount of
uncertainties.

In order to compared our results for the slip parameter in Einstein-Cartan
cosmology summarized in figures ”Fig. 11”, ”Fig. 12” and ”Fig. 13”, with the
observational values presented in Table 1, we used the standard relation
between the redshift and the scale factor

$1+z=\frac{a(t_{0})}{a(t)},$ (39)

in order to convert cosmic times to redshifts and vice versa. As can be seen
from the values in the table, the results for Einstein-Cartan cosmology fall
within the observational values obtained by using GaPP and Linear Regression
methods.

Table 1: The first three rows show the observed value of slip parameter in
different Redshifts using three different statistical methods [46]. The last
row is the theoretically obtained value of slip parameter in each Redshift in
the context of the Einstein-Cartan theory obtained in this paper.  Observed
value of Slip parameter
---
Method |  Choice of $H_{0}$ | $\eta(z=0.294)$ | $\eta(z=0.58)$ | $\eta(z=0.86)$
Binning |  $H_{0}$ HST | $0.48\pm 0.45\,$ | $-0.03\pm 0.34$ | $\,-2.78\pm 6.84$
|  $H_{0}$ Planck | $0.56\pm 0.54\,$ | $-0.14\pm 0.32$ | $\,-6.75\pm 75.64$
GaPP |  $H_{0}$ HST | $\,0.49\pm 0.25$ | $0.94\pm 0.33$ | $0.27\pm 0.67\,$
|  $H_{0}$ Planck | $\,0.31\pm 0.22$ | $0.72\pm 0.33$ | $0.36\pm 0.79\,$
Linear Regression |  $H_{0}$ HST | $0.57\pm 1.05\,$ | $0.48\pm 0.96$ | $-0.11\pm 3.21\,$
|  $H_{0}$ Planck | $0.51\pm 1.07\,$ | $0.37\pm 0.93$ | $-0.18\pm 3.11\,$
Theoretically obtained value of Slip parameter in ECT
Einstein-Cartan cosmology | $z$ | $0.294$ | $0.58$ | $0.86$
| $\eta$ | $0.42$ | $0.40$ | $0.02$

##  IV  Gravitational Waves

In order to obtain the gravitational waves (GW), we only consider the tensor
perturbation of the metric[58]

$ds^{2}=dt^{2}\,-a^{2}(t)\ (\gamma_{ij}+h_{ij})dx^{i}dx^{j},$ (40)

where $h_{ij}$ decompose to scalar, vector and tensor perturbation

$h_{ij}=2C\delta_{ij}+\partial_{<i}\partial_{j>}E+2\partial_{(i}E_{j)}+2E_{ij}.$
(41)

focusing on the last term we have

$ds^{2}=dt^{2}\,-a^{2}(t)\ (\gamma_{ij}+2E_{ij})dx^{i}dx^{j}.$ (42)

Using the above metric 42 and finding the perturbed field equation of
Einstein-Cartan theory 10, the equation relevant to the propagation of
gravitational waves will be

$\begin{split}\Box E_{ij}(t,r)&=\kappa(a(t)^{2}P(t)+a(t)^{2}\delta
P(t,r))\delta_{ij}\\\ &+\kappa
a(t)^{2}\Pi_{ij}+\kappa^{2}18a(t)^{-2}q(t)Y(t,r)\gamma_{ij}\\\
&+\kappa^{2}2a(t)^{-2}q(t)^{2}E_{ij}(t,r),\end{split}$ (43)

where the d’Alembertian operator is

$\Box E_{ij}=\partial_{i}\partial^{j}E_{ij}.$ (44)

We aim to obtain gravitational waves in vacuum, consequently, the pressure and
its perturbation ($P(t)$ and $\delta P(t,r)$) vanish. Also, anistropic stress
($\Pi_{ij}$) and perturbation of spin tensor ($Y(t,r)$) are negligible in
cosmic scales. These considerations give us the equation for the Plane
gravitational waves in EC theory as

$\Box E_{ij}(t,r)=2\kappa^{2}a(t)^{-2}q(t)^{2}E_{ij}(t,r).$ (45)

GW for radiation dominated and matter dominated universe in the EC theory is
studied in detail in following sub-sections.

####  IV.0.1  Radiation dominated universe

By inserting scale factor 19 and spin density 18 into eq.45 and solving the
equation, the GWs in radiation dominated universe can be obtained as

$\begin{split}&E(t)=e^{-imt}t\bigg{(}C_{1}\,{}_{1}F_{1}[1-\frac{l}{2im},2,2imt]\\\
&+C_{2}\,U[1-\frac{l}{2im},2,2imt]\bigg{)},\end{split}$ (46)

where $l=17$, and $m$ is wave-number due to the Fourier transformation 32.
Additionally, ${}_{1}F_{1}(a,b,x)$ and $U(a,b,x)$ are Confluent Hypergeometric
Functions of first and second kinds, respectively [59]. The confluent
hypergeometric function can be expanded in hypergeometric series [60].

The behavior of gravitational waves for radiation dominated era in the EC
theory is illustrated in Fig[14].

Figure 14:  The graph shows the behavior of the normalized gravitational waves
regarding time in the context of EC theory for radiation dominated era. Note
that we consider m=10.

####  IV.0.2  Matter dominated universe

By substituting scale factor 22 and spin density 21 of matter dominated
universe into the GWs equation 45, the GWs in matter dominated universe will
be

$\begin{split}E(t)&=C_{1}\,M[-\frac{il_{2}}{2m},-\frac{1}{2}i\sqrt{4l_{1}-1},2imt]\\\
&+C_{2}\,W[-\frac{il_{2}}{2m},-\frac{1}{2}i\sqrt{4l_{1}-1},2imt],\end{split}$
(47)

where $l_{1}=4.8$ and $l_{2}=2.9$. The Whittaker functions (i.e. $M$ and $W$)
can be written in terms of confluent hypergeometric functions of first and
second kinds [61, 62] as below

$\begin{split}&E(t)=e^{-imt}(2imt)^{\frac{1}{2}\left(1-i\sqrt{4l_{1}-1}\right)}\\\
&\Bigg{[}C_{1}\,{}_{1}F_{1}\left(\frac{il_{2}}{2m}+\frac{1}{2}\left(1-i\sqrt{4l_{1}-1}\right),1-i\sqrt{4l_{1}-1},2imt\right)\\\
&+C_{2}\,U\left(\frac{il_{2}}{2m}+\frac{1}{2}\left(1-i\sqrt{4l_{1}-1}\right),1-i\sqrt{4l_{1}-1},2imt\right)\Bigg{]}.\end{split}$
(48)

Using power series expansion of hypergeometric functions, behavior of
gravitational waves for the matter dominated universe is depicted in Fig[15].

Figure 15: The graph shows the behavior of gravitational waves regarding time
through the EC theory for the matter dominated era. Note that we consider
m=10.

##  V  Conclusion

The presence of anisotropic stress introduces a disparity in the values of two
scalar gravitational potentials, $\phi$ and $\psi$, which appear in the first-
order perturbation equation of motion. The gravitational slip parameter,
measuring the anisotropic stress, is defined as the ratio of these two scalar
gravitational potentials $\eta=\frac{\phi}{\psi}$. Given that the existence of
anisotropic stress and gravitational slip parameter ($\eta\neq 1$) in the
presence of perfect fluid matter could signify a modification of gravity, the
gravitational slip parameter serves as a potential indicator of such
modifications.

Furthermore, the gravitational slip parameter can be constructed based on
model-independent observable quantities, providing a fully model-independent
approach to testing modified gravity theories. In this paper, we applied this
methodology to test Einstein-Cartan theory (ECT), a simple theory
incorporating torsion.

By deriving the equations of motion within the framework of ECT and
considering the Friedmann-Robertson-Walker metric for the universe, we
obtained the scale factor and spin density for three distinct single-component
universes: radiation-dominated, matter-dominated, and dark energy-dominated.
Studying the linear scalar perturbations with a non-zero spin tensor in the
ECT, we find the behavior of the slip parameter over time in the context of
ECT. The comparison of the theoretical values of the slip parameter within the
ECT framework with the observed values provides evidence supporting the
possibility of modified gravity in the sense of ECT.

Another test for the modification of gravity is the generation and propagation
of cosmic gravitational waves. As modified gravity theories alter the behavior
of gravitational waves, we can infer a connection between the modifications of
anisotropy constraints and gravitational wave propagation. Our analysis
demonstrates that introducing spin density to the matter part of the field
equations and modifying the source term influence the behavior and amplitude
of gravitational waves.

## References

  * Dyson  _et al._ [1920] F. W. Dyson, A. S. Eddington, and C. Davidson, Philosophical Transactions of the Royal Society of London Series A 220, 291 (1920).
  * Schneider  _et al._ [1992] P. Schneider, J. Ehlers, and E. E. Falco,  _Gravitational Lenses_ (1992).
  * Blandford and Narayan [1992] R. D. Blandford and R. Narayan, Annual Review of Astronomy and Astrophysics 30, 311 (1992).
  * Bartelmann and Schneider [2001] M. Bartelmann and P. Schneider, Physics Reports 340, 291 (2001).
  * Pound and Rebka [1960] R. V. Pound and G. A. Rebka, Apparent weight of photons, Phys. Rev. Lett. 4, 337 (1960).
  * Abbott  _et al._ [2016] B. P. Abbott _et al._ (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016).
  * Ghez  _et al._ [2008] A. M. Ghez, S. Salim, N. N. Weinberg, J. R. Lu, T. Do, J. K. Dunn, K. Matthews, M. R. Morris, S. Yelda, E. E. Becklin, T. Kremenek, M. Milosavljevic, and J. Naiman, The Astrophysical Journal 689, 1044 (2008).
  * Weinberg [1989] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
  * Peebles and Ratra [2003] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003).
  * Riess  _et al._ [1998] A. G. Riess _et al._ , The Astronomical Journal 116, 1009 (1998).
  * Sahni and Starobinsky [2006] V. Sahni and A. Starobinsky, International Journal of Modern Physics D 15, 2105 (2006).
  * Riess  _et al._ [2001] A. G. Riess _et al._ , apj 560, 49 (2001).
  * Navarro  _et al._ [1997] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 490, 493 (1997).
  * Sahni [2004] V. Sahni (Springer Berlin Heidelberg, 2004) pp. 141–179.
  * Bertone  _et al._ [2005] G. Bertone, D. Hooper, and J. Silk, Physics Reports 405, 279 (2005).
  * Clowe  _et al._ [2006] D. Clowe, M. Bradač, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, and D. Zaritsky, The Astrophysical Journal 648, L109 (2006).
  * Cartan [1923] E. Cartan, Annales scientifiques de l’École Normale Supérieure 3e série, 40, 325 (1923).
  * Cartan [1924] E. Cartan, Annales scientifiques de l’École Normale Supérieure 3e série, 41, 1 (1924).
  * Cartan [1925] E. Cartan, Annales scientifiques de l’École Normale Supérieure 3e série, 42, 17 (1925).
  * Hehl  _et al._ [1974] F. W. Hehl, P. von der Heyde, and G. D. Kerlick, Phys. Rev. D 10, 1066 (1974).
  * Hehl  _et al._ [1976] F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976).
  * Böhmer [2004] C. G. Böhmer, Classical and Quantum Gravity 21, 1119 (2004).
  * Akhshabi and Zamani [2023] S. Akhshabi and S. Zamani, (2023), arXiv:2305.00415 [gr-qc] .
  * Gasperini [1998] M. Gasperini, General Relativity and Gravitation 30, 1703 (1998).
  * Popławski [2012] N. J. Popławski, General Relativity and Gravitation 44, 1007 (2012).
  * Popławski [2012] N. Popławski, Phys. Rev. D 85, 107502 (2012).
  * Huang  _et al._ [2015] Q. Huang, P. Wu, and H. Yu, Phys. Rev. D 91, 103502 (2015).
  * Magueijo  _et al._ [2013] J. a. Magueijo, T. G. Zlosnik, and T. W. B. Kibble, Phys. Rev. D 87, 063504 (2013).
  * Hadi  _et al._ [2018] H. Hadi, Y. Heydarzade, M. Hashemi, and F. Darabi, The European Physical Journal C 78, 10.1140/epjc/s10052-017-5494-1 (2018).
  * Tilquin and Schücker [2011] A. Tilquin and T. Schücker, General Relativity and Gravitation 43, 2965 (2011).
  * Elizalde  _et al._ [2023] E. Elizalde _et al._ , Physics of the Dark Universe 40, 101197 (2023).
  * Alexander  _et al._ [2020] S. Alexander, L. Jenks, P. Jiroušek, J. a. Magueijo, and T. Złośnik, Phys. Rev. D 102, 044039 (2020).
  * de Sabbata [1994] V. de Sabbata, Nuovo Cimento A Serie 107, 363 (1994).
  * Amendola  _et al._ [2008] L. Amendola, M. Kunz, and D. Sapone, Journal of Cosmology and Astroparticle Physics 2008 (04), 013.
  * Caldwell and Doran [2005] R. R. Caldwell and M. Doran, Phys. Rev. D 72, 043527 (2005).
  * Linder [2005] E. V. Linder, Phys. Rev. D 72, 043529 (2005).
  * Hu and Sawicki [2007] W. Hu and I. Sawicki, Phys. Rev. D 76, 104043 (2007).
  * Bean and Tangmatitham [2010] R. Bean and M. Tangmatitham, Phys. Rev. D 81, 083534 (2010).
  * Pogosian  _et al._ [2010] L. Pogosian, A. Silvestri, K. Koyama, and G.-B. Zhao, Phys. Rev. D 81, 104023 (2010).
  * Bertschinger [2006] E. Bertschinger, Astrophys. J. 648, 797 (2006).
  * Baker  _et al._ [2011] T. Baker, P. G. Ferreira, C. Skordis, and J. Zuntz, Phys. Rev. D 84, 124018 (2011).
  * Battye and Pearson [2012] R. A. Battye and J. A. Pearson, Journal of Cosmology and Astroparticle Physics 2012 (07), 019.
  * Amendola  _et al._ [2013] L. Amendola, M. Kunz, M. Motta, I. D. Saltas, and I. Sawicki, Phys. Rev. D 87, 023501 (2013).
  * Motta  _et al._ [2013] M. Motta, I. Sawicki, I. D. Saltas, L. Amendola, and M. Kunz, Phys. Rev. D 88, 124035 (2013).
  * Amendola  _et al._ [2014] L. Amendola, S. Fogli, A. Guarnizo, M. Kunz, and A. Vollmer, Phys. Rev. D 89, 063538 (2014).
  * Pinho  _et al._ [2018] A. M. Pinho, S. Casas, and L. Amendola, Journal of Cosmology and Astroparticle Physics 2018 (11), 027.
  * Saltas  _et al._ [2014] I. D. Saltas, I. Sawicki, L. Amendola, and M. Kunz, Phys. Rev. Lett. 113, 191101 (2014).
  * Matos  _et al._ [2023] I. S. Matos, E. Bellini, M. O. Calvão, and M. Kunz, (2023), arXiv:2210.12174 [astro-ph.CO] .
  * Sawicki  _et al._ [2017] I. Sawicki, I. D. Saltas, M. Motta, L. Amendola, and M. Kunz, Phys. Rev. D 95, 083520 (2017).
  * Amendola  _et al._ [2018] L. Amendola, M. Kunz, I. D. Saltas, and I. Sawicki, Phys. Rev. Lett. 120, 131101 (2018).
  * Bravo Medina  _et al._ [2019] S. Bravo Medina, M. Nowakowski, and D. Batic, Annals of Physics 400, 64 (2019).
  * Goenner and Muller-Hoissen [1984] H. Goenner and F. Muller-Hoissen, Classical and Quantum Gravity 1, 651 (1984).
  * Riotto [2003] A. Riotto, ICTP Lect. Notes Ser. 14, 317 (2003), arXiv:hep-ph/0210162 .
  * Saltas  _et al._ [2018] I. D. Saltas, L. Amendola, M. Kunz, and I. Sawicki, (2018), arXiv:1812.03969 [astro-ph.CO] .
  * De Felice  _et al._ [2011] A. De Felice, T. Kobayashi, and S. Tsujikawa, Physics Letters B 706, 123 (2011).
  * Felice and Tsujikawa [2012] A. D. Felice and S. Tsujikawa, Journal of Cosmology and Astroparticle Physics 2012 (02), 007.
  * Nersisyan  _et al._ [2018] H. Nersisyan, N. A. Lima, and L. Amendola,  (2018).
  * Baumann [2022] D. Baumann,  _Cosmology, part III Mathematical Tripos_ (Cambridge University Press, 2022).
  * Sneddon [1961] I. N. Sneddon, Proceedings of the Edinburgh Mathematical Society 12, 169–169 (1961).
  * George B. Arfken [2005] H. J. W. George B. Arfken,  _Mathematical Methods for Physicists_ (Academic Press, 2005).
  * Milton Abramowitz [1972] I. A. S. Milton Abramowitz,  _Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables_ (Dover Publications, 1972).
  * E. T. Whittaker [1990] G. N. W. E. T. Whittaker,  _A Course of Modern Analysis_ (Cambridge University Press, 1990).
  * Roos [1994] M. Roos,  _Introduction to cosmology_ (Chichester: Wiley, 1994).
  * Zlatev  _et al._ [1999] I. Zlatev, L. Wang, and P. J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999).
  * Arkani-Hamed and Dimopoulos [2005] N. Arkani-Hamed and S. Dimopoulos, Journal of High Energy Physics 2005, 073 (2005).
  * Dvali  _et al._ [1994] G. Dvali, Q. Shafi, and R. Schaefer, Phys. Rev. Lett. 73, 1886 (1994).
  * Carroll [2001] S. M. Carroll, Living Reviews in Relativity 4, 10.12942/lrr-2001-1 (2001).
  * Faraoni and Capozziello [2011] V. Faraoni and S. Capozziello,  _Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics_ (Springer, Dordrecht, 2011).
  * Capozziello and Laurentis [2011] S. Capozziello and M. D. Laurentis, Physics Reports 509, 167 (2011).
  * Clifton  _et al._ [2012] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Physics Reports 513, 1 (2012).
  * Faraoni [2004] V. Faraoni,  _Cosmology in scalar tensor gravity_ (2004).
  * Müller-Hoissen [1990] F. Müller-Hoissen, Nuclear Physics B 346, 235 (1990).
  * Gasperini [1986] M. Gasperini, Phys. Rev. Lett. 56, 2873 (1986).
  * de Berredo-Peixoto and de Freitas [2009] G. de Berredo-Peixoto and E. A. de Freitas, Classical and Quantum Gravity 26, 175015 (2009).
  * Popławski [2010] N. J. Popławski, Physics Letters B 694, 181 (2010).
  * Ma  _et al._ [2014] M.-S. Ma, F. Liu, and R. Zhao, Classical and Quantum Gravity 31, 095001 (2014).
  * Solodukhin [1993] S. Solodukhin, Physics Letters B 319, 87 (1993).
  * Popławski [2016] N. Popławski, The Astrophysical Journal 832, 96 (2016).
  * Kalinowski [1981] M. W. Kalinowski, Letters in Mathematical Physics 5, 489 (1981).
  * Shafieloo and Clarkson [2010] A. Shafieloo and C. Clarkson, Phys. Rev. D 81, 083537 (2010).
  * Shafieloo and Linder [2011] A. Shafieloo and E. V. Linder, Phys. Rev. D 84, 063519 (2011).
  * Huterer and Starkman [2003] D. Huterer and G. Starkman, Phys. Rev. Lett. 90, 031301 (2003).
  * Adler Ronald and Schiffer [1975] M. B. Adler Ronald and M. Schiffer,  _Introduction to general relativity_ , Vol. 220 (McGraw-Hill, 1975).
  * Baudoin and O’Connell [2008] F. Baudoin and N. O’Connell,  (2008), arXiv:0809.2506 [math.PR] .