\makeatletter
\@ifundefined{HCode}
{\documentclass[CRPHYS,Unicode,screen,biblatex]{cedram}
\addbibresource{crphys20260373.bib}
\newenvironment{Table}{\begin{table}}{\end{table}}
\newenvironment{noXML}{}{}
\def\tsup#1{\textsuperscript{#1}}
\def\tsub#1{\textsubscript{#1}}
\def\tminus{$-$}
\def\ndash{\text{--}}
\usepackage[T1]{fontenc}
\def\thead{\noalign{\relax}\hline}
\def\tbody{\noalign{\relax}\hline}
\def\endthead{\noalign{\relax}\hline}
\def\tabnote#1{\vskip4pt\parbox{.96\linewidth}{#1}}
\def\hyphen{\text{-}}
\def\mathbi#1{\text{\textbf{\textit{#1}}}}
\RequirePackage{etoolbox}
\usepackage{amssymb} 
\def\sfrac#1#2{{#1}/{#2}}
\def\stfrac#1#2{({#1}/{#2})}
\def\sttfrac#1#2{({#1})/{#2}}
\def\stofrac#1#2{(({#1})/{#2})}
\def\bsigma{\boldsymbol{\sigma}}
\def\jobid{crphys20260373}
%\graphicspath{{/tmp/\jobid_figs/web/}}
\graphicspath{{./figures/}}
\newcounter{runlevel}
\let\MakeYrStrItalic\relax
\csdef{Seqnsplit}{\\}
\def\refinput#1{}
\def\back#1{}
\def\botline{\\\hline}
\def\Lbreak{\newline}
\def\dollar{\$}
\def\ubreak{\break}
\def\og{\guillemotleft}
\def\fg{\guillemotright}
\def\xmorerows#1#2{#2}
\def\mn{\phantom{$-$}}
\def\0{\phantom{0}}
\def\figType#1{}
\usepackage{multirow} 
\def\nrow#1{\@tempcnta #1\relax%
\advance\@tempcnta by 1\relax%
\xdef\lenrow{\the\@tempcnta}}
\def\morerows#1#2{\nrow{#1}\multirow{\lenrow}{*}{#2}}
\usepackage{hyperref}
\makeatletter
\g@addto@macro{\UrlBreaks}{\UrlOrds}
\gappto{\UrlBreaks}{\UrlOrds}
\DOI{10.5802/crphys.290}
\datereceived{2026-06-09}
\daterevised{2026-08-05}
\dateaccepted{2026-09-17}
\ItHasTeXPublished
}
{\documentclass[crphys]{article}
\usepackage[T1]{fontenc}
\def\CDRdoi{10.5802/crphys.290}
\let\refinput\input
\let\ubreak\relax
\makeatletter
\def\href#1#2{\url[#1]{#2}}
\def\tminus{\unient{2212}}
\def\CDRsupplementaryTwotypes#1#2{}
\let\splittabular\relax
\def\sfrac#1#2{{#1}/{#2}}
\def\stfrac#1#2{({#1}/{#2})}
\def\sttfrac#1#2{({#1})/{#2}}
\def\stofrac#1#2{(({#1})/{#2})}
\newcommand\@coi{}
\newcommand\COI[1]{\gdef\@coi{#1}}
\newcommand\printCOI{\ifx\@coi\@empty\else%
\section*{Declaration of interests}
\@coi\fi
}
}
\makeatother

\usepackage{upgreek}

\COI{The author does not work for, advise, own shares in, or receive
funds from any organization that could benefit from this article, and
has declared no affiliations other than their research organization.}

%\dateposted{2026-04-23}

\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Reconstructing dark matter phenomenology from noncommutative
geometry: a parametric and observational approach}

\alttitle{Reconstruction de la ph\'enom\'enologie de la mati\`ere noire \`a
partir de la g\'eom\'etrie non commutative : une approche param\'etrique et
observationnelle}

\author{\firstname{Berkane} \lastname{Amina}\CDRorcid{0009-0003-8039-5144}}
\address{Laboratory of Physics and Chemistry of Materials, Department
of Physics, University of M'Sila, Algeria}
\email{amina.berkane@univ-msila.dz}

\keywords{\kwd{Non-commutative geometry}
\kwd{Dark matter}
\kwd{Galactic rotation curves}
\kwd{Smeared mass distribution}
\kwd{\tralicstex{Λ}{${\Lambda}$}CDM model}
\kwd{NFW profile}
\kwd{NGC 3198}}

\altkeywords{\kwd{G\'eom\'etrie non commutative}
\kwd{Mati\`ere noire}
\kwd{Courbes de rotation galactique}
\kwd{Distribution de masse diffuse}
\kwd{Mod\`ele \tralicstex{Λ}{${\Lambda}$}CDM}
\kwd{Profil NFW}
\kwd{NGC 3198}}

\begin{abstract}
We investigate an alternative approach to the dark matter problem by
employing a non-commutative geometry (NCG) framework, wherein spacetime
coordinates exhibit intrinsic quantum uncertainty. In this model, the
classical point-mass source is replaced by a smeared energy density
characterized by a minimal length scale \tralicstex{\textit{β}}{$\beta$}. This
leads to a modified Schwarzschild-like solution with an effective mass
profile that yields flat galactic rotation curves---mimicking
the effects commonly attributed to dark matter. We derive the
corresponding gravitational potential and evaluate the velocity profile
analytically and numerically. A detailed comparison with the standard
\tralicstex{Λ}{${\Lambda}$}CDM model is performed using observational data from the
spiral galaxy NGC 3198. The results demonstrate that the NCG model can
successfully reproduce the flat rotation curve without invoking exotic
matter components. This suggests that quantum corrections to spacetime
geometry may provide a viable explanation for dark matter
phenomenology, motivating further investigation into geometric
extensions of gravity. The present work should be
interpreted as an effective non-commutative geometry-inspired model, in
which non-commutative effects are incorporated through a smeared matter
distribution while Einstein's field equations remain unchanged.
\end{abstract}

\begin{altabstract}
Nous \'etudions une approche alternative au probl\`eme de la mati\`ere noire
en utilisant un cadre de g\'eom\'etrie non commutative (NCG), dans lequel
les coordonn\'ees spatio-temporelles pr\'esentent une incertitude quantique
intrins\`eque. Dans ce mod\`ele, la source classique de masse ponctuelle
est remplac\'ee par une densit\'e d'\'energie \'etal\'ee caract\'eris\'ee par une
\'echelle de longueur minimale \tralicstex{\textit{β}}{$\beta$}. Cela conduit \`a une solution de type
Schwarzschild modifi\'ee, avec un profil de masse effective qui donne des
courbes de rotation galactique plates --- imitant les effets g\'en\'eralement
attribu\'es \`a la mati\`ere noire. Nous d\'erivons le potentiel gravitationnel
correspondant et \'evaluons le profil de vitesse de mani\`ere analytique et
num\'erique. Une comparaison d\'etaill\'ee avec le mod\`ele standard \tralicstex{Λ}{${\Lambda}$}CDM est
effectu\'ee \`a l'aide des donn\'ees d'observation de la galaxie spirale NGC
3198. Les r\'esultats d\'emontrent que le mod\`ele de la g\'eom\'etrie non
commutative parvient \`a reproduire la courbe de rotation plate sans
faire appel \`a des composants de mati\`ere exotique. Cela sugg\`ere que les
corrections quantiques apport\'ees \`a la g\'eom\'etrie de l'espace-temps
pourraient fournir une explication viable \`a la ph\'enom\'enologie de la
mati\`ere noire, ce qui justifie la poursuite des recherches sur les
extensions g\'eom\'etriques de la gravit\'e. Le pr\'esent travail doit \^etre
interpr\'et\'e comme un mod\`ele effectif inspir\'e de la g\'eom\'etrie non
commutative, dans lequel les effets non commutatifs sont int\'egr\'es par
le biais d'une distribution de mati\`ere \'etal\'ee, tandis que les \'equations
du champ d'Einstein restent inchang\'ees.
\end{altabstract}

%\input{CR-pagedemetas}

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}

One of the most significant open problems in modern astrophysics and
cosmology is the nature of dark matter, an invisible component that
constitutes approximately 85\% of the matter content of the Universe.
Multiple independent observations---including galactic rotation
curves~\cite{1}, gravitational lensing~\cite{2}, and fluctuations in
the cosmic microwave background (CMB)~\cite{3} indicate the presence of
a substantial, non-luminous form of matter that interacts
gravitationally but not electromagnetically.

The standard ${\Lambda}$CDM model (Lambda Cold Dark Matter)
successfully explains the large-scale structure of the Universe by
postulating a cold, non-relativistic form of dark matter. However, the
true microscopic identity of dark matter remains unknown, despite
extensive searches in underground laboratories, collider experiments,
and astrophysical observations~\cite{4,5}. This motivates alternative
approaches that attempt to explain the observed gravitational effects
without introducing new particles.

One such approach involves Non-Commutative Geometry (NCG), a
mathematical framework that modifies the structure of spacetime at very
small scales. In NCG, the usual commutative spacetime coordinates
$x_{\mu }$ satisfy a non-commutative relation:
{\begin{equation}\label{eq1}
\left[x^{\mu },x^{\upsilon }\right]=\mathrm{i}\theta ^{\mu \nu }
\end{equation}}\unskip
where $\theta ^{\mu \nu}$ is a real, constant antisymmetric matrix
characterizing the fundamental non-commutativity of
spacetime~\cite{6,7,8,9}. This idea, originally emerging from
string theory and quantum gravity, leads to the ``smearing'' of
point-like objects over minimal length scales, effectively replacing
the Dirac delta function with a Gaussian distribution. Such a smearing
naturally regularizes spacetime singularities and modifies the
energy--momentum tensor used in general relativity.

In the coordinate coherent state formulation adopted in this work,
spacetime non-commutativity is implemented through a smeared matter
distribution rather than through a direct modification of Einstein's
field equations. Consequently, the present model should be regarded as
an effective phenomenological framework inspired by non-commutative
geometry rather than a complete theory of non-commutative gravity.

In gravitational settings, these effects have been used to derive
non-singular black hole metrics, wormholes, and modifications to
Newtonian gravity~\cite{10,11,12,13}. Crucially, it has been shown that
these smeared mass distributions can reproduce flat galactic rotation
curves, which are one of the key pieces of evidence for dark
matter~\cite{14,15,16}. This opens the possibility that dark matter
phenomena might originate from the geometric properties of spacetime,
rather than requiring a new particle sector.

In this study, we propose a geometric explanation for dark matter
phenomena within the framework of non-commutative geometry (NCG). By
modeling galactic mass distributions as smeared sources characterized
by a minimal length scale $\sqrt{\beta}$, we derive the corresponding
gravitational field equations and obtain modified effective mass
profiles. These profiles naturally give rise to flat galactic rotation
curves, thereby reproducing key astrophysical observations without the
need to introduce exotic dark matter particles. This approach suggests
that the apparent dark matter effects may be an emergent consequence of
the underlying non-commutative structure of spacetime itself.

\section{Non-commutative geometry and modified gravitational field
equations}\label{sec2}

In classical general relativity, the gravitational field surrounding a
static and spherically symmetric mass distribution is described by the
Schwarzschild solution. The line element in geometrized units
${c}={G}=1$ is written as:
{\begin{equation}\label{eq2}
\mathrm{d}{s}^{2}=-\mathrm{e}^{{v}({r})}\mathrm{d}{t}^{2}+
\mathrm{e}^{\uplambda ({r})}\mathrm{d}{r}^{2}
+{r}^{2}\left(\mathrm{d}\theta ^{2}+\sin ^{2}\theta \mathrm{d}\phi
^{2}\right).
\end{equation}}\unskip
Here, ${v}(r)$ and $\uplambda (r)$ are functions of the radial
coordinate ${r}$, determined through Einstein's field equations:
{\begin{equation}\label{eq3}
G_{\mu \nu }=8\uppi T_{\mu \nu }.
\end{equation}}\unskip
In the present work, Einstein's field equations are not modified. The
effects of spacetime non-commutativity are introduced exclusively
through the effective matter source following the coordinate coherent
state approach proposed by Nicolini, Smailagic, and Spallucci~\cite{9,10}.
Consequently, the geometrical part of Einstein's equations remains
unchanged, while the matter sector is described by a non-local smeared
energy distribution.

In the context of non-commutative geometry (NCG), the classical concept
of a point-like mass is replaced by a \textit{smeared} energy
distribution~\cite{6,7}. This reflects the inherent quantum uncertainty
in spacetime coordinates, modeled by a non-zero commutator
$[x^{\mu },x^{\upsilon }]=\mathrm{i}\theta ^{\mu \nu }$. As a result,
the usual Dirac delta function is substituted with a Gaussian profile,
leading to the modified energy density. This
Gaussian profile should be viewed as an effective representation of the
matter source induced by the minimal length associated with
non-commutative geometry. It is not obtained by solving Moyal-deformed
Einstein equations but is postulated within the coordinate coherent
state formalism~\cite{10}:
{\begin{equation}\label{eq4}
\rho (r)=\frac{M}{(4\uppi \beta)^{3/2}}\exp \left(-\frac{r^{2}}{4\beta
}\right). 
\end{equation}}\unskip

Here, $M$ is the total mass, and $\beta $ is the non-commutative
parameter with dimensions of length squared, representing the minimal
length scale of spacetime uncertainty~\cite{17}.

The corresponding effective energy--momentum tensor is assumed to take
the anisotropic fluid form;
{\begin{equation}\label{eq5}
{T}_{\upsilon }^{\mu }=\mathrm{diag}(-\rho ,p_{r},p_{t},p_{t})
\end{equation}}\unskip
where the pressure components satisfy the conservation condition
{\begin{equation}\label{eq6}
\nabla _{\mu }T^{\mu \nu }=0.
\end{equation}}\unskip
The mass enclosed within a radius $r$, denoted $M(r)$, is then given by
the integral of the smeared energy density.
{\begin{equation}\label{eq7}
M(r)={\int }_{0}^{{r}}4\uppi {r}{'}^{2}\uprho (r')
\mathrm{d}r'=\frac{2M}{\sqrt{\uppi}}\gamma 
\left(\frac{3}{2},\frac{{r}^{2}}{4\beta }\right)
\end{equation}}\unskip
where $\gamma (s,\,x)$ is the lower incomplete gamma function. This mass
profile smoothly interpolates from zero at the origin to $M$ at large
$r$, thus regularizing the central singularity typical of point-mass
solutions~\cite{10,12}. Using this mass function, the modified
Schwarzschild-like metric becomes:
{\begin{equation}\label{eq8}
\mathrm{d}s^{2}=-\left(1-\frac{2M({r})}{{r}}\right)\mathrm{d}t^{2}+
\left(1-\frac{2M({r})}{{r}}\right)^{-1}\mathrm{d}r^{2}+{r}^{2}
\,\mathrm{d}\Omega ^{2}
\end{equation}}\unskip
with $\mathrm{d}\Omega ^{2}=\mathrm{d}\theta ^{2}+\sin ^{2}\theta
\mathrm{d}\phi ^{2}$. The above metric therefore represents the
solution of the classical Einstein equations sourced by the effective
smeared matter distribution, rather than the vacuum Schwarzschild
solution. In the limit $\beta \rightarrow 0$, the standard
Schwarzschild solution is recovered, and spacetime becomes commutative
again~\cite{10,18}.

The effective gravitational potential derived from this smeared mass
distribution leads to a modified radial acceleration. As will be shown
in Section~\ref{sec3}, this gives rise to \textit{flat galactic rotation
curves}, reproducing one of the key observational signatures of dark
matter---without invoking any new exotic matter
components~\cite{14,15,16,19,20,21}.

\section{Effective mass and galactic rotation curves}\label{sec3}

In galaxies, the rotational velocity of stars and gas as a function of
radial distance from the center provides a crucial probe of the
underlying mass distribution. In Newtonian dynamics, the circular
velocity $v(r)$ at $r$ due to a spherically symmetric mass distribution
$M(r)$ is given by:
{\begin{equation}\label{eq9}
v(r)=\sqrt{\frac{M(r)}{r}}
\end{equation}}\unskip
where we use geometrized units with ${c}={G}=1$. In the standard model,
this relation predicts a Keplerian decline $v(r)\propto
\sfrac{1}{\sqrt{r}}$ for large $r$, which contradicts observations
showing flat rotation curves. This discrepancy is one of the key
motivations for introducing dark matter.

Substituting Equations~(\ref{eq4}) and (\ref{eq5}) into the velocity
equation yields
{\begin{equation}\label{eq10}
v(r)=\sqrt{\frac{2M}{\sqrt{\uppi }r}\gamma \left(\frac{3}{2},
\frac{{r}^{2}}{4\beta }\right)}
\end{equation}}\unskip
to a flattened behavior at large $r$, consistent with observed rotation
curves. In the limit $\sqrt{\beta}\gg r$, the incomplete gamma
function approaches $\Gamma (3/2)=2/\sqrt{\uppi}$, and the
total mass saturates to $M$, making: $v(r)\rightarrow
\sqrt{\sfrac{M}{r}}$ (Newtonian~fall-off). However, if the smearing
parameter$\sqrt{ \beta }$ is large compared to the radial scale of the
galaxy, the transition to the Newtonian regime is delayed, and a flat
rotation curve persists over an extended region:
{\begin{equation}\label{eq11}
\sqrt{\beta }\succcurlyeq \mathrm{cost},\quad \mbox{for } v(r)\approx r.
\end{equation}}\unskip
This behavior is consistent with observational data~\cite{4,5,15} and
provides a natural explanation for galactic rotation curves
without the need for additional exotic dark
matter. The flattening arises purely from the quantum-gravitational
smearing encoded in the non-commutative structure of spacetime.
Furthermore, the effective gravitational potential derived from the
smeared mass function leads to a modification of the radial
acceleration:
{\begin{equation}\label{eq12}
a(r)=\frac{\mathrm{d}\Phi }{\mathrm{d}r}=\frac{M(r)}{r^{2}}-\frac{1}{r}
\frac{\mathrm{d}M(r)}{\mathrm{d}r}
\end{equation}}\unskip
which deviates from the Newtonian inverse-square law and becomes
scale-dependent due to the presence of $\beta $. These corrections can
be interpreted as mimicking the dynamical effects traditionally
attributed to a dark matter halo~\cite{13,14}.

\section{Numerical comparison: NGC 3198}\label{sec4}

To investigate the viability of the non-commutative geometry (NCG)
model in explaining galactic dynamics, we numerically evaluate the
rotational velocity profile using the expression previously derived in
Equation~(\ref{eq8}). This expression captures the influence of
spacetime smearing on the effective gravitational potential.

We compare the results to the standard ${\Lambda}$CDM model, which
employs the Navarro--Frenk--White (NFW) dark matter density profile given
by:
{\begin{equation}\label{eq13}
\rho _{\mathrm{NFW}}(r)=\frac{\rho _{0}}{\left(\frac{r}{r_{s}}\right)
\left(1+\frac{r}{r_{s}}\right)^{2}}.
\end{equation}}\unskip
The corresponding enclosed mass is expressed as:
{\begin{equation}\label{eq14}
M_{\mathrm{NFW}}(r)=4\uppi \rho _{0}{r_{s}}^{3}\left[\frac{r}{r_{s}+r}
-\mathrm{ln}\left(1+\frac{r}{r_{s}}\right)\right].
\end{equation}}\unskip
Our case study is the spiral galaxy NGC 3198, which is a benchmark
system due to its well-measured flat rotation curve extending to $r\sim
30$~kpc. 

We adopt the following parameter values for the numerical evaluation
(see Table~\ref{tab1} for a summary of the parameters used in the
simulation).

\begin{table*}
\caption{\label{tab1}Model parameters used in the numerical simulation}
\begin{tabular}{lccc}
\thead
Parameter & Symbol & Value & Model \\
\endthead
\raisebox{.8pc}{}
Total baryonic mass & $M$ & $5\times 10^{10}M_{\odot }$ & NCG \\
Smearing length scale & $\sqrt{\beta }$ & 10~kpc & NCG \\
Density scale & $\rho _{0}$ & $0.01\times 10^{9}{M_{\odot }}^{3}$ & ${\Lambda}$CDM(NFW) \\
Scale radius & $r_{s}$ & 15~kpc & ${\Lambda}$CDM(NFW)
\botline
\end{tabular} 
\end{table*}



\section{Discussion}\label{sec5}

The results obtained in this work highlight the potential of the
Non-Commutative Geometry (NCG) inspired effective model as an
alternative phenomenological framework capable of reproducing galactic
rotation curves without introducing additional dark matter particles
within the model itself. By smearing mass distributions through a
minimal length scale $\sqrt{\beta }$, the NCG model leads to a
gravitational potential that naturally flattens rotational velocity
profiles, consistent with astrophysical observations.

One of the key advantages of this approach is its basis in quantum
gravitational considerations. {It should be emphasized that the present
analysis does not derive a fundamental non-commutative theory of
gravity. Instead, it adopts the coordinate coherent state formalism, in
which non-commutative effects are incorporated through a smeared matter
distribution while Einstein's field equations remain unchanged.}
{Within the effective framework adopted here, the NCG-inspired model
reproduces the observed galactic dynamics through a smeared matter
distribution associated with a minimal length scale}. {This provides an
alternative phenomenological interpretation of galactic dynamics based
on a minimal-length-inspired matter distribution, while remaining fully
compatible with the classical Einstein field equations.}

The numerical comparison with the ${\Lambda}$CDM model using the NFW
profile shows that both models can reproduce the flat rotation curves
observed in galaxies such as NGC 3198. As illustrated in Figure~\ref{fig1}, the
predicted rotation curve obtained from the present NCG-inspired
effective model closely follows the observational data over the
considered radial range and exhibits a level of agreement comparable to
that of the NFW profile, despite the different physical assumptions
underlying the two approaches. However, within the effective framework
adopted in this work, the NCG-inspired model reproduces the observed
rotation curves through an effective smeared matter distribution
motivated by non-commutative geometry. suggesting that non-local
spacetime effects may contribute to an effective phenomenological
description of galactic dynamics. These results suggest that effective
minimal-length corrections inspired by non-commutative geometry may
play a role in describing large-scale astrophysical dynamics.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Comparison of Rotation Curves for NGC 3198 in NCG
and ${\Lambda}$CDM Models.}
\end{figure}

Nevertheless, the NCG model is not without limitations. The choice of
the smearing parameter $\beta $ significantly affects the
velocity profile, and a deeper understanding of its origin and scale
dependence is needed. Additionally, while the model works well in
spherically symmetric and static cases, more complex galactic
geometries and dynamical systems remain to be analyzed.

Future studies should also explore a rigorous derivation of the
present effective framework from a fundamental non-commutative theory
of gravity remains an open problem and represents an important
direction for future research.

\section{Conclusion and outlook}\label{sec6}

In this work, we have revisited the dark matter problem through the
lens of Non-Commutative Geometry (NCG). By modeling galactic mass
distributions as smeared sources characterized by a minimal length
scale $\beta $, we derived a modified Schwarzschild-like metric and
obtained an effective gravitational potential. The resulting circular
velocity profile, expressed in Equation~(\ref{eq10}), naturally leads to flat
rotation curves without the need for exotic dark matter particles.

To evaluate the physical relevance of the NCG model, we performed a
numerical comparison using observational data from the well-studied
galaxy NGC 3198. The predicted rotation curve matches the observed
profile remarkably well, reproducing the plateau behavior typically
attributed to dark matter halos. Compared to the standard
${\Lambda}$CDM model using the NFW profile (Equations~(\ref{eq13})
and~(\ref{eq14})), the NCG model offers a geometrical explanation
grounded in quantum gravitational effects, rather than invoking unknown
particles.

This approach suggests that non-commutative spacetime corrections could
be responsible for the apparent dark matter effects observed in
galactic dynamics. Although the present model provides an effective
phenomenological interpretation of galactic rotation curves, it does
not constitute a fundamental derivation of non-commutative gravity.
Establishing such a derivation remains an important objective for
future investigations, but rather an emergent phenomenon arising from
modifications to gravity at large scales.

The present work should therefore be viewed as an effective
non-commutative geometry-inspired model whose primary purpose is to
investigate the phenomenological consequences of minimal-length effects
in galactic dynamics rather than to formulate a complete theory of
non-commutative gravity.

Future work will explore:
\begin{itemize}
\item Applying the NCG model to a larger set of galaxies to assess its
universality.
\item Investigating cosmological implications and structure formation.
\item Extending the analysis to include gravitational lensing and
relativistic corrections.
\item Exploring connections with other quantum gravity frameworks such
as Loop Quantum Gravity and Causal Sets.
\end{itemize}
The results presented here provide compelling motivation for further
investigation of geometric alternatives to dark matter, particularly
those rooted in fundamental modifications of spacetime structure.

\printCOI

\back{}

\printbibliography
\refinput{crphys20260373-reference.tex}

\end{document}