contents=  Submitted to the Proceedings of the US Community Study on the
Future of Particle Physics (Snowmass 2021) ,
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# Dark Matter Complementarity - Summary

Antonio Boveia  Department of Physics and Center for Cosmology and
Astroparticle Physics, The Ohio State University, 191 W. Woodruff Avenue
Columbus, OH 43210, USA  Mohamed Berkat  Fysiska institutionen, Lunds
universitet, Professorsgatan 1, Lund, Sweden  Thomas Y. Chen  Fu Foundation
School of Engineering and Applied Science, Columbia University, New York, NY
10027, USA  Aman Desai  Caterina Doglioni  Fysiska institutionen, Lunds
universitet, Professorsgatan 1, Lund, Sweden  University of Manchester,
Department of Physics and Astronomy, Manchester M13 9PL, United Kingdom  Alex
Drlica-Wagner  Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL
60637, USA  Department of Astronomy and Astrophysics, University of Chicago,
Chicago IL 60637, USA  Susan Gardner  Department of Physics and Astronomy,
University of Kentucky, Lexington, KY 40506-0055  Stefania Gori  Physics
Department, University of California, Santa Cruz, CA 95064, USA  Joshua
Greaves  Fysiska institutionen, Lunds universitet, Professorsgatan 1, Lund,
Sweden  Patrick Harding  Physics Division, Los Alamos National Laboratory, Los
Alamos, NM 87545, USA  Philip C. Harris  Laboratory for Nuclear Science,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  W. Hugh
Lippincott  Department of Physics, University of California, Santa Barbara, CA
93106, USA  Maria Elena Monzani  SLAC National Accelerator Laboratory, Menlo
Park, CA 94025, USA  Kavli Institute for Particle Astrophysics and Cosmology,
Stanford University, Stanford CA, USA  Vatican Observatory, Castel Gandolfo,
V-00120, Vatican City State  Katherine Pachal  TRIUMF, Vancouver, BC V6T 2A3,
Canada  Chanda Prescod-Weinstein  Department of Physics and Astronomy,
University of New Hampshire, Durham, NH 03824, USA  Gray Rybka  Department of
Physics, University of Washington, Seattle WA 98195, USA  Bibhushan Shakya
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
Jessie Shelton  Department of Physics, University of Illinois Urbana-
Champaign, Urbana, IL 61801  Tracy R. Slatyer  Center for Theoretical Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Amanda
Steinhebel  NASA Goddard Space Flight Center, Greenbelt, MD, USA  Philip
Tanedo  Department of Physics and Astronomy, University of California
Riverside, Riverside, CA 92521, USA  Natalia Toro  SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA  Yun-Tse Tsai  SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA  Mike Williams  Laboratory for Nuclear
Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Lindley Winslow  Laboratory for Nuclear Science, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Jaehoon Yu  Physics Department,
University of Texas, Arlington, TX 76019, USA  Tien-Tien Yu  Department of
Physics and Institute for Fundamental Science, University of Oregon, Eugene,
OR 97403, USA

###### Abstract

The fundamental nature of Dark Matter is a central theme of the Snowmass 2021
process, extending across all Frontiers. In the last decade, advances in
detector technology, analysis techniques and theoretical modeling have enabled
a new generation of experiments and searches while broadening the types of
candidates we can pursue. Over the next decade, there is great potential for
discoveries that would transform our understanding of dark matter. In the
following, we outline a road map for discovery developed in collaboration
among the Frontiers. A strong portfolio of experiments that delves deep,
searches wide, and harnesses the complementarity between techniques is key to
tackling this complicated problem, requiring expertise, results, and planning
from all Frontiers of the Snowmass 2021 process.

## Executive Summary

The evidence for Dark Matter (DM) is overwhelming, yet the fundamental nature
of its constituents remains a mystery. Over the last decade, we have built a
powerful and diverse collection of tools to unlock this mystery, both by
refining established technologies and techniques and by harnessing new ones
including artificial intelligence/machine learning (AI/ML) and quantum
sensing/control. In parallel, we have continued to build our understanding of
how DM shapes our universe. We are well-positioned for a great discovery.

From its production to its interactions, DM is a major science driver across
all experimental Frontiers – Cosmic Frontier (CF), Energy Frontier (EF),
Neutrino Frontier (NF), and Rare Processes and Precision Frontier (RF) – as
well as the cross-cutting Frontiers: Accelerator Frontier (AF), Community
Engagement Frontier (CEF), Computational Frontier (CompF), Instrumentation
Frontier (IF), Underground Facilities (UF) and Theory Frontier (TF). Because
the science of DM does not respect Frontier boundaries, a unified strategy is
needed to maximize discovery potential.

Complementarity Within and Across Frontiers

Complementarity drives discovery in multiple ways. The space of viable DM
candidates and their properties is large, and a single experimental technique
or approach cannot be used to test all the possibilities; a diverse range of
techniques provides access to a much broader ensemble of DM scenarios. Where
different approaches have simultaneous sensitivity to a particular DM
candidate, they provide essential and complementary information and promote
healthy competition.

Some techniques can tell us whether a new particle constitutes the bulk of the
DM in the Galactic halo while others may better elucidate interactions of DM
with known particles, with a third category mapping the spectrum of new “dark
sector” particles related to the DM. In the event of a discovery, detection
and exclusion by complementary techniques will help triangulate the
fundamental nature of DM.

Maximize Opportunities for Discovery: Delve Deep, Search Wide

Embracing the role of complementarity, the DM community proposes a strategy to
_delve deep and search wide_ to maximize discovery potential. A range of
highly compelling theoretical targets arising from simple/minimal models are
accessible in the next decade via planned and proposed CF, EF, NF, and RF
experiments, colliders, and observatories. While discovering the fundamental
nature of DM is the ultimate prize, searching in these regions and _not_
finding DM would provide important information on the properties of DM.
Simultaneously, our strategy encompasses the development of new technologies
and techniques to explore new possibilities and complement the sensitivity of
existing searches.

Discovery Strategy
The community puts forth the following strategy for discovering the
fundamental nature of DM:

$\newmoon$ Build a portfolio of experiments of different scales: Experiments
at all scales are needed to untangle the mystery of DM and cover the very
broad range of theoretically motivated parameter space. Existing and planned
large-scale facilities across the HEP Frontiers have exceptional potential to
discover fundamental properties of DM. We should commit to scaling up mature
technologies that can promise significant sensitivity improvements, developing
potentially transformative new technologies to maturity, and supporting
efforts to maximize and make accessible large projects’ science output in the
search for DM. At smaller scales, execution of the existing Dark Matter New
Initiatives (DMNI) program and similar future calls are necessary to build the
most compelling DM portfolio, develop experience in project execution, and
accelerate the pace of discovery.

$\newmoon$ Leverage US expertise in international projects: The effort to
understand the fundamental nature of DM is a world-wide endeavor. Coordination
and cooperation across borders is critical for enabling this discovery. While
building a strong US-based program, we should pursue opportunities to leverage
key US expertise as a collaborative partner in international projects and play
a leadership role in this critical area.

$\newmoon$ Provide support to further strengthen the theory program: A strong
theory program is essential to make connections between experimental Frontiers
and take full advantage of new developments in analysis techniques. Theorists’
input has been and will be critical for developing innovative new approaches
to better understand and detect DM, and for determining how to predict and
relate signals across a range of experimental probes.

$\newmoon$ Support inter-disciplinary collaborations that enable discovery:
Many searches for DM benefit greatly from cross-disciplinary expertise, with
examples ranging from nuclear physics to metrology, and astrophysics to
condensed matter and atomic physics. Mechanisms to support such inter-
disciplinary collaborations should be established.

$\newmoon$ Targeted increase in the research budget: New research funding
targeted toward solving the DM problem is essential to enable new ideas, new
technologies, and new analyses. The number of active efforts exploring DM has
increased tremendously in the past decade, without a concomitant increase in
research funding. Across all Frontiers and project scales, research funding is
critical to enable discovery and build on new capabilities, both in projects
focused specifically on DM and to support DM analyses at multi-purpose
experiments. Without such support, the community will not be able to execute
the program described here, decreasing the chances of solving the mystery of
DM.

##  1 Introduction

Determining the fundamental nature of dark matter (DM) is one of the major
open questions that confronts our understanding of physics, and it has been
among the guiding themes of the Snowmass process in most HEP Frontiers. While
the microscopic properties of DM remain almost completely unknown, the
relatively similar energy densities of dark and visible matter in the Universe
— DM has five times more energy density than visible matter — is suggestive
that there may be non-gravitational interactions between DM and the Standard
Model (SM). However, the nature of these interactions is essentially
unconstrained, necessitating a broad and comprehensive approach to this
question to make progress in the next decade. This challenge requires
expertise, results, and planning from the full range of communities involved
in the Snowmass 2021 process, across all Frontiers.

The 2013 Snowmass process had a topical group (CF4) specifically devoted to
the complementarity of different DM studies. The white paper produced by that
group, “Dark Matter in the Coming Decade: Complementary Paths to Discovery and
Beyond,” [1] reviewed existing and planned DM efforts in direct detection,
indirect detection and collider experiments, as well as in astrophysical
probes.

The need for diverse and complementary approaches to the DM problem is even
more pressing now than it was in 2013. The DM search domain has broadened
significantly with promising new avenues now under development that will yield
results in the next decade. Our theoretical understanding of possible DM
parameter space has grown, and that space is being explored by a much larger
number of projects at different scales. Working in tandem, experiment,
observation, theory, and computation have the potential to identify key DM
properties while definitively excluding vast swathes of parameter space.

This document re-casts the scope and definition of complementary approaches to
DM identification to reflect the current state of the field (Section 2),
summarizes the needs of the different communities looking for DM and their
complementary strengths (Section 3), and supports these arguments with cross-
Frontier case studies (Section 4). An extended version of this report is also
available [2].

##  2 Dark matter complementarity in the coming decade

A complementary ensemble of DM searches is key to a comprehensive strategy,
with advantages including:

1\. Different approaches to DM searches allow us to probe different
fundamental properties of the DM. For this reason, complementary approaches
will be necessary to fully identify the DM and understand its physics. For
example, cosmological and astrophysical probes allow access to environments
not found on Earth and time/space scales dwarfing terrestrial experiments, and
consequently have unique sensitivity to a range of properties including the DM
lifetime, annihilation rate, and self-interaction cross section. Rare
interactions between DM and the SM are often most precisely probed in
terrestrial experiments, which enjoy controllable and clean environments. DM
production could be observed in extreme environments in the Cosmic Frontier,
or under controlled laboratory conditions at accelerators and colliders in the
Energy, Rare Processes and Precision, and Neutrino Frontiers; such
measurements can reveal DM interactions in energy domains beyond those of the
halo DM, including its early-universe behaviour and any dark sector particle
spectrum beyond DM. This kind of complementarity is showcased in the case
studies Minimal WIMP Dark Matter, involving the Energy and Cosmic Frontiers,
Sterile Neutrino Dark Matter, involving the Cosmic and Neutrino Frontiers, and
Wave-like Dark Matter: QCD Axion Discovery, involving the Cosmic, Neutrino,
and Rare Processes Frontiers.

2\. Different approaches to DM searches offer unique discovery sensitivity to
distinct scenarios and regions of parameter space. Only by performing a
variety of experiments, covering multiple Frontiers, can we span the wide DM
parameter space. For example, DM interactions with the SM may be either
suppressed or enhanced at low energies; to maximize the chances of a
discovery, it is important to support both experiments sensitive to non-
relativistic DM signals and those that explore the physics of DM at higher
(often relativistic) energies. This kind of complementarity is also explored
in the case study Generic BSM-mediated and Vector Portal Dark Matter,
involving the Energy and Rare Processes and Precision Frontiers.

3\. Results from any one class of searches can continuously inform the
interpretation of other measurements. A Cosmic Frontier detection of relic DM
would connect any observed candidate to the cosmological evidence that
motivates the entire DM program, and provide a target and motivation for
future efforts in other Frontiers. Combining results obtained with different
approaches can address otherwise-intractable uncertainties — e.g. DM
production probes are independent of the fraction of DM due to a particular
candidate, or the DM distribution. Complementary measurements can also reduce
systematic uncertainties directly: as two examples, cosmic measurements of the
density and velocity distribution of DM are essential for the interpretation
of direct and indirect searches, and accelerator experiments can constrain
cosmic ray physics relevant for indirect searches [3].

4\. Different DM experiments can be co-located and/or profit from the same or
similar technological infrastructure. Efficient use of shared resources
enables a wider exploration of DM. Examples include small Rare Processes and
Precision Frontier accelerator experiments that can be co-located with Energy
Frontier collider experiments, using the same high-energy beams to produce
different kinds of DM. Such experiments are discussed in the Rare Processes
and Precision Frontier and their connection to collider experiments is
discussed in the Energy Frontier report [4].

The community searching for DM has grown much more diversified in terms of
technologies, search targets, and project scales. Since the last iteration of
Snowmass in 2013, many new approaches to searching for DM, as well as new
theoretical hypotheses, have attained sufficient maturity to be part of the
toolkit that we will use to make progress in the quest for DM in the next
decade. The older approach to DM complementarity (as outlined in the previous
Snowmass whitepaper [1]) focused primarily on the Weakly Interacting Massive
Particle (WIMP) hypothesis and direct DM-SM interactions, and incorporated
high-mass new particle searches, direct and indirect detection, and
astrophysical probes. This remains an important hypothesis that should be
rigorously tested as part of a program that “delves deep.” However, the WIMP
is now joined by a greater diversity of alternative DM candidates. The QCD
axion has emerged as another key target for a focused suite of experiments
that will enable a definitive search for this candidate. Candidates and
possible signatures that inform the wider strategy include light particle-like
DM with masses in the MeV-GeV range, wave-like DM in addition to the QCD axion
(such as ultralight scalar or vector bosons), signatures of the greater dark
sector including long-lived particles, cosmological observations of DM
properties on large scales, and new signatures of DM in gravitational waves
and other multi-messenger observations.

The strategy presented above was developed from the bottom-up through the
communities represented by the Frontier Topical Groups. It is well-aligned
with the Basic Research Needs for Dark Matter Small Projects New Initiatives
report [5], which highlighted the growing landscape of smaller experiments to
produce and detect DM at accelerators, as well as the direct detection of
light and ultra-light DM. The goal of the following sections is to summarize
the needs of the individual Topical Groups and show that the needs are highly
complementary as are the techniques. A common strategy across HEP is needed to
enable the discoveries that will reveal the nature of DM.

##  3 Realizing dark matter complementarity across Frontiers

In this section, we briefly summarize the main approaches towards identifying
the fundamental nature of DM from each Snowmass Frontier, referring to the
Topical Group whitepapers for further information and needs.

### Cosmic Frontier 1 - Direct and Indirect Searches for Particle Dark Matter.

Direct detection experiments using mature technologies, like large liquid
noble detectors, probe some of the smallest cross sections ever measured in
the non-relativistic regime — favorable for models where interactions are
enhanced at low velocities — and provide leading sensitivity to heavy DM up to
ultraheavy mass scales. A new era of technological development has enabled
sensitivity to tiny energy depositions of eV-scale and below, opening up
sensitivity to unexplored parameter space for DM masses below 1 GeV. Indirect
detection provides a model-independent probe of the minimal thermal relic
scenario with $s$-wave annihilation, with expected sensitivity up to tens of
TeV masses in the next decade, and constraints of some form up to the Planck
mass. Such searches provide unique probes of the DM decay lifetime (and other
long-timescale processes), via their access to distance and time scales that
dwarf any terrestrial experiment, and cover enormous mass ranges.

The next decade offers a broad array of exciting opportunities in direct and
indirect detection, as discussed in Ref. [3]. To pursue these opportunities
requires a diverse, continuous portfolio of experiments that includes both
direct and indirect detection techniques at multiple scales. Moderate- and
large-scale experiments allow us to delve deep into high priority target
scenarios such as WIMPs, whereas an ensemble of small-scale experiments like
those supported by the Dark Matter New Initiatives (DMNI) program provides
versatility and the ability to test an expanded range of models. Support for
theory, simulations, calibration, background modeling and complementary
astrophysical measurements is essential to enabling discovery, as is R&D
towards improved detector technologies. Lastly, direct detection experiments,
particularly the next generation of WIMP searches, will require continued
investment in underground facilities.

### Cosmic Frontier 2 - Direct Searches for Axion Dark Matter.

Wave-like DM encompasses all candidates with masses less than 1 eV. Due to
their small masses, the detection principles are vastly different than those
traditionally used in high energy physics. It is here where quantum
measurement techniques become critical and advancements in this area have
opened up a broad horizon of new candidates to explore and many opportunities
for discovery. Within this group, the well-motivated QCD axion is an excellent
DM candidate that also solves the strong CP problem. Building on the success
of ADMX-G2, the new moderate-scale DMNI experiments ADMX-EFR and DMRadio-m3
are readying to start construction. In the next decade, a portfolio of
moderate-scale experiments are poised to explore significant QCD axion
parameter space. A concentrated effort coupling R&D, demonstrator-scale
experiments, and theory, would enable searches for a broader spectrum of
candidates.

### Cosmic Frontier 3 - Cosmic Probes of Dark Matter.

Cosmic probes provide the only direct, positive empirical measurements of the
existence and properties of DM. These probes complement terrestrial DM
searches by constraining the interaction strength between DM and the SM in
otherwise inaccessible regions of parameter space. In addition, cosmic probes
provide the only known way to directly study the fundamental properties of DM
through gravity, the only force to which DM is known to couple. Cosmic probes
are sensitive to the DM mass, lifetime, self-interaction cross section, and
other dark sector particles. In particular, cosmic probes are on the cusp of
detecting DM halos that are devoid of baryonic galaxies, providing a strong
test of the cold, collisionless DM paradigm.

The construction of future facilities spanning the electromagnetic spectrum,
as well as gravitational waves, can provide sensitivity to DM physics, as well
as the physics of dark energy and the early universe. Strategic HEP
investments in the construction and operation of Rubin LSST, CMB-S4, and
Spec-S5 should include DM physics as a core science driver to be considered
during the design and operation of these experiments. Cosmic probes provide
robust sensitivity to the microphysical properties of DM due to enormous
progress in theoretical modeling, numerical simulations, and astrophysical
observations. Theory, simulation, observation, and experiment must be
supported together to maximize the efficacy of cosmic probes of DM physics.

### Rare & Precision Frontier 6 - Dark Sectors at High Intensities.

Intensity Frontier experiments offer unique access to the physics of low-mass
DM by systematically probing a broad range of simple, well-motivated dark
sectors neutral under SM forces. Near-term searches for _DM production_ are
needed to thoroughly explore the coupling ranges motivated by MeV-to-GeV
thermal DM. Because the energies probed in Rare & Precision Frontier
experiments are similar to those relevant for light DM thermal freeze-out, the
range of production cross-sections expected for low-mass thermal relics is
compact—and accessible—regardless of the DM spin. In DM models where
interactions are suppressed at low velocities, cosmological and Galactic DM
signals can be suppressed by orders of magnitude. Thus, accelerator-based
production is the favored path to detection of these scenarios. Intensity
Frontier experiments can also discover and characterize _light dark-sector
particles that decay into SM particles_ , often with detectably long
lifetimes. Such particles arise in most generalized freeze-out scenarios such
as strongly interacting massive particles, forbidden DM, and secluded DM [6].
Their discovery can shed light on the interactions, nature, and origin of DM.
Both goals, searching for DM production and for visibly decaying particles
related to DM, were called out in the 2018 DM New Initiatives Basic Research
Needs report [5].

Achieving these goals over the next decade requires a four-pronged approach:
performing dark-sector analyses at multi-purpose experiments; realizing the
experiments selected through the competitive DMNI program; broadening the DMNI
experimental portfolio to achieve the goals laid out in the DMNI Report,
including a focus on signatures of long-lived dark sector particles decaying
(semi)visibly; and continued investment in dark-sector theory.

### Energy Frontier 10 - Dark Matter at Colliders.

Present and future colliders [7, 4] can search for particle DM and its
interactions, covering a broad swath of scenarios ranging from the canonical
WIMP to more general models of DM particles and the mediators of their
interaction or extended dark sectors. They can detect invisible particles
produced in collisions via missing transverse momentum, but their greatest
strengths are their ability to study how these invisible particles interact
with other particles and to search for additional particles involved in the DM
physics.

Over the next decade, the High-Luminosity LHC (HL-LHC) can explore whether DM
couples to the Higgs boson or other beyond the Standard Model (BSM) portal
particles, and test supersymmetric and other particle DM candidates that have
been long-term targets of the field. On longer timescales, an electron-
positron collider can push this sensitivity significantly further, as well
test DM models that favor couplings to leptons. A hadron or muon collider
would allow direct exploration of far-higher energy scales, with the greatest
prospects to discover WIMP multiplets, and kinetically-mixed thermal DM above
a GeV.

High-energy colliding beam facilities can also be used for special-purpose,
co-located DM experiments searching for long-lived, dark sector particles. An
example is the case of the proposed Forward Physics Facility [8] for the HL-
LHC, but similar facilities would provide similar capabilities at other future
colliders.

### Neutrino Frontier 3 - Dark Matter in Neutrino Experiments.

The high power beams, multi-kiloton scale far detectors underground, and near
detector systems needed for precision neutrino experiments make a broad
variety of BSM searches possible in the Neutrino Frontier. The high power
proton beams can produce dark sector particles, such as axion-like particles,
light DM and heavy neutral leptons, which can be detected in the precision
near detector complex. Dark sector particles with oscillatory behavior, e.g.
sterile neutrinos, can be probed in the far detector in combination with the
near detector complex in long baseline neutrino experiments. Dark sector
particles that alter neutrino fluxes from natural sources, or those with
cosmogenic origins such as boosted DM, can be detected in the underground,
massive far detectors.

Such rare dark sector interactions can be easily masked by backgrounds from a
variety of sources, such as cosmic rays, terrestrial radioactive sources, and
neutrino interactions. The current level of understanding of neutrino-nuclear
interactions as well as neutrino spectra from natural sources is insufficient
to effectively estimate or control them to the precision necessary for longer
term dark sector particle searches, necessitating a strong synergistic and
sustained collaboration between the nuclear physics and high energy physics
communities to perform measurements needed to improve the modeling of
neutrino-nuclear interactions, such as the $e4$nu collaboration.

### Theory Frontier.

The allowed mass ranges and interactions for DM are so vast that it is
impossible to undertake any meaningful exploration of this parameter space
without insight from theory. Theory is the language needed to make sense of
results from different experiments and Frontiers [9]. Broadly, theory
complements and enhances experimental DM searches in three ways:

Theory is essential to _define connections between experimental programs and
understand their complementarity_ in the context of specific DM models. This
role broadens the impact of null results, and becomes crucial in understanding
and verifying a potential signal.

Theory can motivate experimental programs for DM searches by tying DM to other
deep questions and insights about particle physics. Examples include the
hierarchy problem (a central problem of the Energy Frontier) inspiring WIMP
DM, neutrino mass generation (a central problem of the Neutrino Frontier)
inspiring sterile neutrino and neutrino-portal DM, and the strong CP problem
giving birth to axion DM.

Theory can also _enable new paths to DM detection_ , identifying DM scenarios
accessible to ongoing experiments or even leveraging new technologies,
experimental capabilities, and insights to define new experimental directions.
Recent examples of such interplay are found throughout the DMNI program across
Rare Processes and Cosmic Frontiers.

### Instrumentation Frontier.

Advances in instrumentation support every aspect of the hunt for DM, with new
technologies often opening new regions of parameter space for exploration. Two
examples in direct detection are in quantum sensors and noble element
detectors. Rapid progress in quantum sensors over the past decade has been key
in the success of the wave-like DM program. Such sensors come in a wide range
of technologies: atom interferometry, magnetometers, calorimeters, and
superconducting sensors to name a few. In searches for WIMP-like DM, the
development of liquid noble detectors has paved the way for the huge strides
in sensitivity seen since the last Snowmass, with LXe detectors leading the
way and LAr close behind. Of course, there are many more examples that enable
DM searches in all the Frontiers, including development in photon sensing,
timing, calorimetry, etc. Continued R&D into instrumentation, development of
the technical workforce, and tools to share common knowledge will be an
important component of the future DM program. More information can be found in
the Instrumentation Frontier report [10].

### Computational Frontier.

Computing and software are critical components of any search for DM, from the
collection and storage of raw data, through various stages of data movement,
data processing and data analysis, all the way to the interpretation of
results. Interestingly enough, the majority of DM searches are rapidly
converging towards exascale datasets that are not matched by technological
developments and budgets [11, 12, 13, 14]. Given this dramatic escalation in
data volume and complexity, common computing needs can be identified to enable
successful DM searches across all Frontiers, including: stronger partnerships
with the national supercomputing facilities, lowering the barrier of entry and
providing input on architecture evolution; scalable software infrastructure
tools across HEP, avoiding duplication of effort; simulation tools that are
efficient, well-maintained, well-understood, and thoroughly validated; and
industry collaborations on machine learning techniques, leveraging access to
external experts.

An additional challenge specific to complementarity is the need to exchange
data between different experiments and even Frontiers, which implies the
necessity to converge on data formats and analysis tools. The DM community
should take the lead in advocating for widely-adopted data and software
standards, in support of global analyses of experimental results.
Complementarity studies would be strengthened by an Open Science paradigm
including both data and software, with the added benefit of enhancing the
credibility of our potential discoveries, since progress in the field of DM
will require thorough scrutiny of data and results.

### Underground Facilities.

Most DM direct detection experiments must be sited in underground facilities
to evade cosmic ray backgrounds, and multiple new underground DM experiments
are expected and being planned (at both large and small scales). Currently,
underground facilities are largely subscribed by existing projects, with only
limited space available in the coming years. There is, then, a clear need for
additional underground space, tailored to the needs of DM experiments. This
underground space must accommodate experiments across scales, including large
liquid noble or freon experiments and smaller installations, for example mK
facilities. Assembly of future experiments will occur largely in the
underground environment, requiring underground radon-free clean rooms. Given
the volume of gas/cryogen, future liquid noble experiments also require
underground areas for staging (e.g., gas storage) and utilities (e.g. pumps,
distillation). These new suitable spaces must be available by the late 2020s
to meet the demand, which may be met in North America by proposed new
excavations at SURF or SNOLAB. More information can be found in the UF report
[15].

### Accelerator Frontier.

Current and future accelerator facilities are the underpinning of both
Intensity Frontier and Energy Frontier searches for DM and dark sectors.
Several beam facilities for axion and DM searches have been shown to have
great potential for construction in the 2030s in terms of scientific output,
cost and timeline, including PAR (a 1 GeV, 100 kW PIP-II Accumulator Ring). In
general, we should efficiently utilize existing and upcoming facilities to
explore dedicated or parasitic opportunities for rare process measurements —
examples include the SLAC SRF electron linac, MWs of proton beam power
potentially available after construction of the PIP-II SRF linac, spigots of
the future multi-MW FNAL complex upgrade, and at CERN, a Forward Physics
Facility at the LHC. At the Energy Frontier, an $e^{+}/e^{-}$ Higgs Factory
(e.g. FCC-ee, C3, etc) will likely be the next major accelerator facility;
interest in discovery machines such as O(10 TeV c.m.e.) muon colliders has
also gained significant momentum. These machines’ open-ended discovery
potential includes sensitivity to a range of particle DM scenarios. More
information can be found in the AF report [16].

### Community Engagement and Workforce Development for Dark Matter.

The last two decades have seen an explosion in the number of physicists
engaged in DM detection, and the enthusiasm for the topic has been palpable
throughout Snowmass 2021. In order to support this enthusiasm, the community
needs to enhance its engagement efforts at all levels of society, most notably
with the education system, industry partners, and policy makers. Outreach
widens access to information, and the exciting mystery of DM acts as a
powerful recruiting tool for HEP.

Career pipeline and development are crucial to sustaining such an expansion,
which in turn requires a renewed focus on the diversity, equity, inclusion,
and accessibility of the field. The historical exclusion of marginalized
people from high energy physics is fundamentally harmful to the humanity of
individuals who are excited and curious about science. Moreover, creating
equal opportunity and equality in particle physics is essential to
professional success in our field, ensuring breadth of perspectives and a deep
talent pool. Developing a broad talent pool sustains both scientific
advancements and the democratic principle of publicly supported and engaged
activities.

Equal opportunities in HEP go hand in hand with more traditional priorities.
The search for DM is highly interdisciplinary and is therefore an excellent
training ground for our next generation of scientists. DM searches that rely
on quantum sensing and AI/ML align with the need for training in these areas
both in HEP and beyond. The long gaps between design and commissioning of
large-scale HEP projects such as future colliders can lead to leaks in the
pipeline of key knowledge holders. Small- and medium-scale DM projects can
bridge these gaps, training scientists across all experimental phases, from
design to construction to commissioning and analysis. Thus, beyond their
scientific merits, these projects sustain expertise in hardware development,
construction, and their interface with science delivery, benefiting the field
overall. The Community Engagement Frontier report [17] addresses these and
other workforce questions.

##  4 Case Studies

Fig. 1 provides a graphical summary of the breadth of theoretical scenarios
that can provide DM candidates. The possibilities span enormous ranges in DM
mass and interaction strength.

In the case studies below, we briefly discuss several scenarios to illustrate
how complementarity between DM searches could enable discovery of the
fundamental nature of DM and allow triangulation of its properties. More
detailed versions of these scenarios may be found in Ref. [2]. Additional
relevant case studies may be found in Refs. [5, 18].

Figure 1: Summary of case studies presented in this document, shown in the
context of a sketch of the coupling-mass plane including the parameter space
typical of some of the rich variety of DM theories possible. The shaded colors
in this sketch are suggestive of the Frontiers with experiments represented in
the case studies in a given region, with color coding specified near the
rounded rectangles.

### Minimal WIMP Dark Matter.

One of the simplest possibilities for DM involves a new particle multiplet
that interacts with SM particles via the weak interaction [19]. Accidental or
imposed symmetries lead to the stability of the lightest particle of this
multiplet, providing a suitable DM candidate. The two most-widely studied
scenarios are the cases where the DM is part of a Dirac fermion doublet
(called the Higgsino) or of a Majorana fermion triplet (Wino); these
multiplets appear in supersymmetric (SUSY) theories as superpartners of the SM
Higgs/gauge bosons, and so the cases where the DM is close to “pure Higgsino”
or “pure Wino” can be realized in specific regions of the broader parameter
space of SUSY theories.111Setting aside the connection to supersymmetry, which
is not essential, many of the same general principles also apply to larger
electroweak multiplets; for a discussion of other options see e.g. [20, 21].
Such scenarios are very predictive, with the only free parameter being the DM
mass, and this can also be fixed (at 1.1 TeV for the Higgsino and 2.8 TeV for
the Wino) by matching the relic density under the assumption of a standard
cosmology with thermal freezeout of the DM [20, 21].

Indirect searches for gamma-rays and antiprotons already constrain the thermal
Wino (e.g. [22, 23]), albeit with significant systematic uncertainties
associated with the DM density and cosmic-ray propagation. Future cosmic
probes and complementary astrophysical measurements could reduce these
uncertainties and strengthen the bounds. In direct detection, the cross
section is very small [24], but large future direct detection experiments such
as DARWIN can probe the Wino scenario, with cross-sections still above the
neutrino fog [25]. The direct detection cross section for the thermal Higgsino
lies in the neutrino fog, but CTA should have sensitivity to the indirect-
detection signal [26]. Current colliders can only probe lighter non-thermal
Wino- and Higgsino-like particles, but a future multi-TeV lepton or hadron
collider could meet the thermal target for both Wino and Higgsino (and
eventually even for larger multiplets) [20, 21]. A direct or indirect
detection during the planning phase of such a collider would provide crucial
input to the design. A collider discovery would provide in-depth information
on the WIMP’s interactions with SM particles and its associated particle
spectra. Alternatively, null results at collider experiments could
significantly constrain the interpretation of a putative DM signal from direct
or indirect detection.

(a)

(b)

Figure 2: Indicative sketches depicting qualitatively how future collider,
indirect detection, and neutrino experiments
LABEL:sub@subfig:WIMP_ID_Colliders or collider and direct detection
experiments LABEL:sub@subfig:BSMMediation may complement each other during
discoveries of Wino/Higgsino DM or of BSM-mediated DM, respectively. The
$y$-axis indicates the annihilation cross section to example SM final states
in (a) and the spin-dependent scattering cross section on SM targets in (b).
Regions where “excluded” is mentioned in the figure have been covered by
published results, while other areas depict approximate regions of sensitivity
for current and future experiments. Regions of overlapping coverage, where
complementary observations in both types of experiments would be possible, are
indicated by saturated colors. Regions accessible by only one of the two types
of experiments are shown in muted colors or grayscale.

### Generic Beyond the Standard Model (BSM)-mediated and Vector Portal Dark
Matter.

If DM particles are discovered by either Cosmic Frontier or accelerator-based
(EF and/or RPF) experiments, the other technique will be essential to
understanding its properties. In the case of a CF detection of DM particles,
different types of cosmic probes and target materials can shed some light on
the nature of DM interactions. However, producing the same kind of DM in the
lab, with a known initial state, opens new windows to characterising DM
interactions and resolving the roles of related, cosmologically unstable,
particles. Likewise, while a signal of invisible particle production in a
fixed target or collider experiment can be related to DM models, a
simultaneous discovery in Cosmic Frontier experiments is needed to ascertain
the cosmological nature of the DM candidate.

This kind of complementarity is illustrated by theoretical scenarios that
extend the WIMP paradigm to include an additional particle beyond the SM that
mediates interactions between DM and SM. These mediator particles can decay
into both DM and SM particles; searches for each decay mode offer further
insight into the DM-SM interaction. A thermal history for the DM candidates in
the early universe can be attained depending on the coupling types and
strengths of the mediator (or portal) particle, as well as on the mediator and
DM particle masses. Specific realizations of these models are used as
benchmarks in e.g. Refs. [27, 4, 28, 29]. Figures 2
LABEL:sub@subfig:BSMMediation and 3 LABEL:sub@subfig:DarkPhoton illustrate
complementarity across CF, EF and RPF experiments in terms of both
opportunities for simultaneous discovery (CF1/EF10) and complementary
discovery sensitivity (EF10/RF6); we also refer the reader to Fig. 1-1 and
Case studies 1 and 2 of [5] for discussion of the CF1/RF6 complementarity for
low-mass DM. Further insights into these and related portal DM models can be
gained from MeV-energy indirect detection (CF1) and probes of DM self-
interaction (CF3).

(a)

(b)

Figure 3: LABEL:sub@subfig:DarkPhoton Sketch of how collider and accelerator
experiments together can reach sensitivity across many orders of magnitude of
DM mass to couplings expected for thermal-relic vector portal inelastic Dirac
DM production (note: lepton colliders not shown). $y$ parameterizes the DM-SM
interaction strength and depends on the kinetic mixing parameter $\epsilon$,
dark-sector coupling $\alpha$, and masses of the DM ($m_{\mathrm{DM}}$) and
mediator ($m_{\mathrm{med}}$). LABEL:sub@subfig:SterileNeutrino Sketch of
constraints on the mass and mixing angle $\theta$ of resonantly produced
sterile neutrino DM from indirect detection in X-rays, cosmic probes of small
scale structure, and projected sensitivity of tritium beta-decay neutrino
experiments.

### Sterile Neutrino Dark Matter.

Figure 3 LABEL:sub@subfig:SterileNeutrino is an illustration of the
complementarity between cosmological, astrophysical, and laboratory searches
for sterile neutrino DM (as compiled by [30]), with complementary reach
provided by a combination of indirect DM searches (CF1; [3]), cosmic probes of
structure formation (CF3; [18]), and laboratory neutrino facilities (NF3;
[31]). Indirect detection experiments searches for X-ray lines originating
from the decay of keV mass sterile neutrinos. Current constraints come from
Chandra, XMM-Newton, NuSTAR, INTEGRAL, and Fermi GBM observations of the
various astrophysical systems (e.g., the Milky Way, M31, dwarf galaxies, and
galaxy clusters) [32, 33, 34, 35, 36, 37, 38, 39]. Future X-ray facilities
such as XRISM, Athena, and the WFM instrument aboard the eXTP X-ray Telescope
could increase sensitivity to mixing angle by orders of magnitude [40, 41, 42,
43]. Cosmological constraints are due to the suppression of DM structure that
occurs for producing keV-mass sterile neutrinos with a “warm” initial momentum
distribution [44, 45]. These constraints will improve by orders of magnitude
as measurements of the least massive DM halos improves with DESI [46], Rubin
LSST [47], and future cosmological survey experiments [48]. Laboratory
searches (e.g., Katrin, TRISTAN, BeEST, and HUNTER) [49, 50, 51, 52] have
different dependencies on the model behavior of sterile neutrinos in the early
Universe (e.g., in cosmologies with large lepton asymmetry, low reheating
temperature and/or neutrino non-standard interactions) making them highly
complementary to indirect and cosmological searches [53, 54]. Below the lowest
solid line, achieving the correct relic density of sterile neutrino DM
requires other production mechanisms beyond active-sterile mixing; these
generally involve heavy BSM particles, which can be probed at the Energy
Frontier.

### Wave-like Dark Matter: QCD Axion Discovery.

The discovery of a QCD axion by a direct detection experiment such as
DMRadio-$m^{3}$ with a mass $<1\,\mu$eV can be used to illustrate of the wide-
ranging implication of a wave-like DM discovery. The detailed spectral
measurements from direct axion searches would almost instantaneously provide a
measurement of the velocity distribution of DM in the halo. These measurements
and subsequent measurements of the position distribution can then be compared
to the results from cosmic probes of DM. Since axions with masses below
$<1\,\mu$eV imply additional fields at the time of inflation, CMB B-modes
would then be out of range for next generation CMB experiments. A possible
discrepancy between such measurements would open the door to significant
changes in our understanding of particle physics and cosmology.

Most direct detection experiments use the axions’ coupling to photons. A
precision measurement of this coupling, or the coupling to other parts of the
SM, can disentangle which category of QCD axion or axion-like particle has
been discovered. There is complementarity here with other table-top precision
measurements. The discovery of a QCD axion implies additional particles, e.g.
KSVZ models predict additional quarks and DFSZ predict an expanded Higgs
sector. Either would be strong motivation for higher-energy colliders and
would guide the design of such efforts.

##  5 Conclusions

DM presents a fundamental puzzle to particle physics. The space of viable,
theoretically-motivated candidates is enormous and multi-dimensional, spanning
many orders of magnitude in mass and interaction strength. To make progress on
this challenging problem, maximize the chances of a transformative discovery,
and fully elucidate the properties of DM and related new physics in the event
of such a discovery, we advocate for a cross-Frontier effort incorporating
multiple complementary approaches to the problem. The suite of approaches
discussed in this summary, and in the various topical group and Frontier
reports, will allow us to delve deep into highly compelling, long-standing,
and well-studied scenarios for the nature of DM, and simultaneously to open up
our search to a wide and less-explored space of exciting and well-motivated
possibilities. A decade of coherent cross-Frontier DM exploration is an
opportunity that should not be missed.

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##  6 Acknowledgments

The authors would like to thank the following people for their input in the
discussions and cross-frontier meetings that led to this report, including
inputs to the figures: Suchita Kulkarni, Ben Loer, Bjoern Penning, Hai-Bo Yu.


# Late time phantom characteristic of the model in $f(R,T)$ gravity with
quadratic curvature term

Shaily<EMAIL_ADDRESS>Department of Mathematics, Netaji Subhas
University of Technology, New Delhi 110078, India  School of Computer Science
Engineering and Technology, Bennett University, Greater Noida 201310, India
Akanksha Singh<EMAIL_ADDRESS>Department of Mathematics, Netaji
Subhas University of Technology, New Delhi 110078, India  J. K. Singh
<EMAIL_ADDRESS>Department of Mathematics, Netaji Subhas University of
Technology, New Delhi 110078, India  Saibal Ray<EMAIL_ADDRESS>Centre
for Cosmology, Astrophysics and Space Science (CCASS), GLA University, Mathura
281406, Uttar Pradesh, India

###### Abstract

We propose a novel cosmological framework within the $f(R,T)$ type modified
gravity theory, incorporating a non-minimally coupled with the higher order of
the Ricci scalar ($R$) as well as the trace of the energy-momentum tensor
($T$). Therefore, our well-motivated chosen $f(R,T)$ expression is
$R+R^{m}+2\lambda T^{n}$, where $\lambda$, $m$, and $n$ are arbitrary
constants. Taking a constant jerk parameter ($j$), we derive expressions for
the deceleration parameter ($q$) and the Hubble parameter ($H$) as functions
of the redshift $z$. We constrained our model with the recent Observational
Hubble Dataset (OHD), $Pantheon$, and $Pantheon$ \+ OHD datasets by using the
analysis of Markov Chain Monte Carlo (MCMC). Our model shows early
deceleration followed by late-time acceleration, with the transition occurring
in the redshift range $1.10\leq z_{tr}\leq 1.15$. Our findings suggest that
this higher order model of $f(R,T)$ gravity theory can efficiently provide a
dark energy model for addressing the current scenario of cosmic acceleration.

PACS numbers: 04.20.-q, 04.50.Kd, 98.80.-k, 98.80.Es
Keywords: FLRW metric, $f(R,T)$ gravity, constant jerk parameter, phantom
model

##  I Introduction

Cosmological discoveries have revealed that astronomical structures in the
universe exhibit a systematic arrangement rather than a random distribution.
By studying this organized pattern and examining the physical characteristics
of interstellar objects, we can gain insights into the phenomenon of cosmic
accelerated expansion. While general relativity serves as the foundation of
modern physics and effectively explains black holes and gravitational
phenomena, it falls short of providing a comprehensive explanation for cosmic
acceleration. To shed light on this intriguing concept, researchers have
turned their attention towards investigating dark energy and exploring
modified theories of gravity. Various astronomical studies, which include the
Sloan Digital Sky Survey (SDSS), the 2-Degree Field Galactic Redshift Survey
(2DFGRS), and the Huge Synoptic Survey Telescope (HSST), have demonstrated the
significant influence that galaxies and stars have on the evolution of the
universe. Analyzing these fundamental components is crucial to deepening our
understanding of the formation and nature of the universe. Through the
examination of gravitating systems, we can uncover the underlying physics of
dark matter and dark energy. As theoretical cosmology confronts the challenges
posed by the existence of dark energy and compelling experimental evidence,
modified gravity theories emerge as encouraging avenues for refining our
theoretical understanding of late-time cosmic acceleration.

Prominent modified gravitational theories in astrophysics as well as cosmology
include $f(R)$ [1, 2, 3, 4], $f(R,T)$ [5, 6, 7], $f(T)$ [8, 9, 10], $f(G)$
[11, 12], $f(R,G)$ [13, 14], $f(G,T)$ [15, 16] and $f(R,\phi)$ [17] (however,
for interested readers, there are a few more references herein to consult the
alternative gravity theories [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32]). A cornerstone in the exploration of gravitational theories is
the $f(R)$ gravity theory, pioneered by Buchdahl [33], which involves
substituting the Ricci scalar $R$ in the Einstein-Hilbert (EH) action with an
arbitrary function $f(R)$. This theory has been examined and verified as a
viable explanation for late-time cosmic acceleration while also aligning with
local gravitational tests [3, 34, 35, 36, 37, 38, 39]. While $f(R)$ gravity
remains a focus of extensive study, recent decades have witnessed a surge of
interest in $f(T)$ gravity. This alternative theory has also provided
explanations regarding cosmic expansion and galactic dynamics, as evidenced by
references [40, 41, 42]. In their work, Harko et al. [43] first put forward
the $f(R,T)$ theory as a modification of general relativity (GR) that involves
altering the geometric part of the EH action, where $R$ is the Ricci scalar
and $T$ denotes the trace of the energy-momentum tensor (EMT). In this theory,
$R$ is replaced by an arbitrary $f(R,T)$ function, and due to the
incorporation of $T$, this theory encompasses novel gravitational aspects and
also satisfies the weak-field conditions of the solar system.

In the realm of cosmic dynamics, Houndjo [44] shed light on the transition
from a matter-dominated epoch to the late-time acceleration phase using the
minimally coupled $f(R,T)$ model. Within the vast expanse of scientific
literature, several distinct formulations emerge, each offering a unique lens
into the functional dependence of $f(R,T)$. Researchers have explored various
mathematical expressions to represent the dependence on $T$, including
polynomials, rational functions, and exponential expressions [45, 46, 47, 48,
49, 50, 51]. It is argued that the $T$-dependence can arise due to the
presence of quantum effects or imperfect fluids. Amidst the array of modified
gravitational theories, the $f(R,T)$ theory stands out as it incorporates an
energy transfer relationship between matter and geometry, thus potentially
exhibiting the observed phenomenon related to the accelerating phase of the
cosmic expansion. In connection to a few important features of $f(R,T)$
gravity theory, the following works are notable [52, 53, 54, 55, 56]. The
consequences of the $f(R,T)$ theory on cosmological and solar system models
have been investigated in [57]. In literature, discussions on bouncing models
within the $f(R,T)$ theory have also gained attention [58, 59, 60].

Motivated by the above-mentioned $f(R,T)$ theory-based works, our current
investigation starts with the general powers of curvature and trace of the
energy-momentum tensor. Specifically, we consider the form
$f(R,T)=R+R^{m}+2\lambda T^{n}$, subsequently delving into a detailed analysis
with distinct parameter values for $m$ and $n$. In this context, it is to be
noted that for wormhole-related investigations, Elizalde and Khurshudyan [61]
exploited a similar form of $f(R,T)$ with specific parametric values, such as
$m=2$ and $n=1$. We also notice that very recently Mahapatra and Das [62] have
employed a polytropic model under $f(R,T)$ in a restricted manner to study the
equilibrium configurations of relativistic neutron stars. However, in this
connection, we would like to emphasize that our approach is more general so
that the results can be expected to offer a wide range of flexibility to
explore the multifarious features of the late-time phantom behavior of the
universe.

The structured sequence of our research is: Section II serves as the starting
point, where we delve into the computation of the Hubble parameter by using
the jerk and deceleration parameter expressions. Next, we derive the Einstein
field equations for a specific form of the $f(R,T)$ function in Sec. III. In
our model, we constrain three model parameters using the OHD dataset, the
$Pantheon$ dataset, and their joint (OHD + $Pantheon$) dataset in Sec. IV.
Sec. V contains the explanation related to the dynamical behavior of the
universe for this model. At last, in Sec. VI, we discuss the results and
conclude our work.

##  II Formulation of the Hubble parameter

Let us first define here the jerk parameter $j$

$j(z)=q(z)+2q(z)^{2}+(1+z)\frac{dq(z)}{dz},$ (1)

where $q$ and $z$ are, respectively, the deceleration parameter and the
redshift.

Treating $j$ as a constant and subsequently solving the above differential
equation leads to

$\frac{\sqrt{8j+1}+4q+1}{\sqrt{8j+1}-4q-1}=\left(\alpha+\alpha
z\right)^{\sqrt{8j+1}},$ (2)

where $\alpha$ is an integration constant.

Rewriting Eq. (2) we get

$q=\frac{-1}{4}+\frac{\sqrt{8j+1}}{4}-\frac{\sqrt{8j+1}}{2\left(1+\left(\alpha+\alpha
z\right)^{\sqrt{8j+1}}\right)}.$ (3)

The equation representing $q$ as a function of the Hubble parameter $H$ and
redshift $z$ is

$q(z)=-1+\frac{(1+z)}{H(z)}\frac{dH}{dz}.$ (4)

Using Eq. (3) in Eq. (4), we get

$\frac{1}{1+z}\left[1-\frac{1}{4}+\frac{\sqrt{8j+1}}{4}-\frac{\sqrt{8j+1}}{2\left(1+\left(\alpha+\alpha
z\right)^{\sqrt{8j+1}}\right)}\right]dz=\frac{dH}{H}.$ (5)

By integrating both sides of Eq. (5), we obtain

$H=\beta\left(1+z\right)^{\frac{3+\sqrt{8j+1}}{4}}\sqrt{1+\left(\alpha+\alpha
z\right)^{-\sqrt{8j+1}}},$ (6)

where $\beta$ is an integration constant.

In order to work with observational data, we define $\beta$ as
$\frac{H_{0}}{\sqrt{1+\alpha^{-\sqrt{8j+1}}}}$, where $H_{0}=H(0)$.
Subsequently, one can find the $H$, which is as follows

$H=\frac{H_{0}}{\sqrt{1+\alpha^{-\sqrt{8j+1}}}}\left(1+z\right)^{\frac{3+\sqrt{8j+1}}{4}}\sqrt{1+\left(\alpha+\alpha
z\right)^{-\sqrt{8j+1}}}.$ (7)

##  III Formulation of $f(R,T)$ theory of gravity

The gravitational action corresponding to $f(R,T)$ gravity theory is

$\mathscr{S}=\int\Bigg{[}\frac{f(R,T)}{16\pi
G}+\mathscr{S}_{m}\Bigg{]}\sqrt{-g}d^{4}x,$ (8)

where $R$, $T$, $g$, $G$, and $\mathscr{S}_{m}$, respectively, denote the
Ricci scalar, the trace of the EMT, the determinant of the metric tensor, the
gravitational constant, and the matter Lagrangian. Here, the function $f(R,T)$
is explicitly defined as $f(R,T)=f_{1}(R)+2f_{2}(T)$ [43].

On varying the action (8) with respect to the metric tensor $g_{\mu\nu}$, one
can easily get the following gravitational equation [32]

$f_{1}^{R}(R)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}(f_{1}(R)+2f_{2}(T))+(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})f_{1}^{R}(R)=8\pi
GT_{\mu\nu}-2f_{2}^{T}(T)(T_{\mu\nu}+\Theta_{\mu\nu}),$ (9)

where $f_{1}^{R}(R)$ symbolizes the first order differentiation of $f_{1}$
with respect to $R$ and $f_{2}^{T}(T)$ means the first order differentiation
of $f_{2}$ with respect to $T$. The d’Alembert operator is denoted by $\Box$,
and $\nabla_{\mu}$ signifies the covariant derivative with respect to
$g_{\mu\nu}$. The expression $\Theta_{\mu\nu}$ can be written as

$\Theta_{\mu\nu}=g^{ij}\frac{\delta{T_{ij}}}{\delta{g_{\mu\nu}}}=-2T_{\mu\nu}+g_{\mu\nu}\mathscr{S}_{m}-2g^{ij}\frac{\delta^{2}\mathscr{S}_{m}}{\delta{g_{\mu\nu}}\delta{g_{ij}}}.$
(10)

Considering a perfect fluid, the matter Lagrangian is conventionally
represented as $\mathscr{S}_{m}=-p$. In this case, the EMT is given by
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu}$, where the physical parameters
have their usual meanings. Using these expressions, we get
$\Theta_{\mu\nu}=-2T_{\mu\nu}-pg_{\mu\nu}$. Therefore, Eq. (9) takes the
following form

$f_{1}^{R}(R)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}(f_{1}(R)+2f_{2}(T))+(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})f_{1}^{R}(R)=8\pi
GT_{\mu\nu}+2f_{2}^{T}(T)(T_{\mu\nu}+pg_{\mu\nu}).$ (11)

The form in which the Einstein field equations are expressed is given by

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi
G}{f_{1}^{R}(R)}(T_{\mu\nu}+T^{*}_{\mu\nu}),$ (12)

where
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T^{*}_{\mu\nu}=\frac{1}{8\pi
G}\Big{[}\frac{1}{2}g_{\mu\nu}((f_{1}(R)+2f_{2}(T))-Rf_{1}^{R}(R))+(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box)f_{1}^{R}(R)+2f_{2}^{T}(T)(T_{\mu\nu}+pg_{\mu\nu})\Big{]}.$

Now, the metric of the FLRW universe is expressed by

$ds^{2}=dt^{2}-a^{2}(t)\sum\limits_{i=1}^{3}dx_{i}^{2}.$ (13)

Here, $a(t)$ is the scale factor. Now, by considering $f_{1}(R)=R+R^{m}$ and
$f_{2}(T)=\lambda T^{n}$, where $T=\rho-3p$ and $R=-6(2H^{2}+\dot{H})$, we can
obtain the Einstein field equations (EFE) as

$3H^{2}=\frac{1}{1+mR^{m-1}}[8\pi\rho+\frac{1}{2}((1-m)R^{m}+2\lambda
T^{n})-3Hm(m-1)R^{m-2}\dot{R}+(\rho+p)2\lambda nT^{n-1}]$ (14)

or

$\displaystyle
3H^{2}=\frac{1}{1+mR^{m-1}}[8\pi\rho+\frac{1}{2}((1-m)(-6(2H^{2}+\dot{H}))^{m}+2\lambda
T^{n})]+~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\frac{1}{1+mR^{m-1}}[-3Hm(m-1)(-6(2H^{2}+\dot{H}))^{m-2}(-6(4H\dot{H}+\ddot{H}))+(\rho+p)2\lambda
nT^{n-1}],$

$2\dot{H}+3H^{2}=\frac{-1}{a^{2}(1+mR^{m-1})}[8\pi
pa^{2}+\frac{1}{2}(-a^{2})((1-m)R^{m}+2\lambda
T^{n})+am(m-1)R^{m-3}(a((m-2)\dot{R}^{2}+R\ddot{R})+3\dot{a}R\dot{R})]$ (15)

or

$\displaystyle 2\dot{H}+3H^{2}=\frac{-1}{a^{2}(1+mR^{m-1})}[8\pi
pa^{2}+\frac{1}{2}(-a^{2})((1-m)(-6(2H^{2}+\dot{H}))^{m}+2\lambda
T^{n})+~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle a\zeta
m(m-1)(-6(2H^{2}+\dot{H}))^{m-3}(a((m-2)(-6(4H\dot{H}+\ddot{H}))^{2}+(-6(2H^{2}+\dot{H}))(-6(4H\ddot{H}+4\dot{H}^{2}+\dddot{H})))+~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle
3\dot{a}(-6(2H^{2}+\dot{H}))(-6(4H\dot{H}+\ddot{H})))].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}`$

As the calculations involved in determining the physical parameters $\rho$ and
$p$ based on the above equations are complex, we simplify the analysis by
choosing $m=2$ and $n=1$. Now, we have taken $f_{1}(R)=R+R^{2}$, a choice
previously addressed by Starobinsky [63], and $f_{2}(T)=\lambda T$. The EFEs
for our chosen model are

$(8\pi+3\lambda)\rho-\lambda
p=3H^{2}+18(\dot{H}^{2}-6H^{2}\dot{H}-2H\ddot{H}),$ (16)
$(8\pi+3\lambda)p-\lambda\rho=-2\dot{H}-3H^{2}+6(26\dot{H}H^{2}+9\dot{H}^{2}+14H\ddot{H}+2\dddot{H}).$
(17)

##  IV Observational analysis

Observational analysis is an endeavoring field in cosmology where we calculate
the optimal value of model parameters with the help of some experiments. In
this analysis, the most accurate and strong persistence results are obtained
from the Planck satellite mission. In this model, we determine the optimal
values for model parameters by conducting a best-fit analysis with the help of
three distinct observational datasets, namely, the Hubble dataset (OHD) with
$77$ data points, the $Pantheon$ dataset with $1048$ data points, and the
combined dataset (OHD + $Pantheon$). The Hubble parameter value from Eq. (7),
which contains three model parameters named $\alpha$, $j$, and $H_{0}$, is
used for our analysis. In order to constrain the mentioned model parameters,
we basically execute the Markov Chain Monte Carlo (MCMC) method, implemented
through the $emcee$ library in Python.

###  IV.1 $H(z)$ Dataset

Exploring the dark division of the universe involves tapping into the
insightful $H(z)$ data, and this data is basically acquired from the cosmic
chronometers [64]. In our model, we use the OHD along with 77 data points
[65]. Thereafter, by using the $\chi^{2}$ test, we find the parametric best-
fit values of the present model.

Now, the formula for $\chi^{2}$ is drafted as

$\chi_{Hb}^{2}(\alpha,j,H_{0})=\sum\limits_{i=1}^{77}\frac{[H(\alpha,j,H_{0},z_{i})-H_{obs}(z_{i})]^{2}}{\sigma_{z_{i}}^{2}},$
(18)

where $H(\alpha,~{}j,~{}H_{0},~{}z_{i})$ and $H_{obs}$ are denoted for,
respectively, the theoretical and observed values of the Hubble parameter.
Additionally, $\sigma_{(z_{i})}$ stands for the standard deviation for every
value of $H(z_{i})$.

We determine that the analysis of only the Hubble dataset and the function
$\chi_{Hb}^{2}$ is not very authentic for the cosmological models. Therefore,
we take the $Pantheon$ dataset and their joint dataset into consideration.

Figure 1: Visualization of the likelihood contours using the $H(z)$ dataset. Table 1: Best-fit values of model parameters where $D_{1}=H(z)$, $D_{2}=Pantheon$, and $D_{3}=H(z)+Pantheon$. Dataset |  $\alpha$ |  $j$ |  $H_{0}$
---|---|---|---
$D_{1}$ |  $0.59905^{+0.00097}_{-0.00023}$ |  $0.9215^{+0.0076}_{-0.0210}$ |  $70.932^{+0.069}_{-0.020}$
$D_{2}$ |  $0.59998\pm 0.0001$ |  $0.949998\pm 0.000098$ |  $70.000013^{+0.000081}_{-0.000099}$
$D_{3}$ |  $0.598987^{+0.000094}_{-0.000110}$ |  $0.98999^{+0.000120}_{-0.000085}$ |  $70.3719\pm 0.0001$

###  IV.2 Pantheon Dataset

In the last two decades, Type Ia supernovae have emerged as pivotal tools in
the analysis of cosmological parameters, notably serving as the first evidence
for the late time accelerated expansion phase of the universe. Supernovae are
the stars, and as they burst, they release a lot of energy and enlarge their
outer shell. Thus, this part is related to examining the time evolution of
their brightness and their spectrum. In this model, the $Pantheon$ dataset
which comprises of $1048$ data points, is used to constrain the parameters of
the model [66]. The $Pantheon$ dataset contains the resulting compilation of
different surveys, including the CfA1-CfA4 surveys, the Sloan Digital Sky
Survey (SDSS), the Supernovae Legacy Survey (SNLS), the Carnegie Supernova
Project (CSP), the Pan-STARRS1 (PS1), and various Hubble Space Telescope (HST)
surveys, all of which span a redshift range from $0.01$ to $2.26$ [67, 68, 69,
70, 71]. In Type Ia supernovae analysis, we deal with redshift as well as
luminosity distance, which are closely related to standard candles in the
universe. The luminosity distance is defined as

$D_{L}(z)=(1+z)\int_{0}^{z}\frac{c}{H(z^{*})}dz^{*},$ (19)

with $c$ as the speed of light in a vacuum.

Now, $m$, the apparent magnitude can be provided as

$m(z)=M+5log_{10}\Bigg{[}\frac{D_{L}(z)}{1Mpc}\Bigg{]}+25,$ (20)

with $M$ denoting the absolute magnitude.

Here, a dissipation between $H_{0}$ and $M$ is observed, and given that the
distance modulus $\mu$ is equal to $m-M$, the $\chi^{2}$ formula is written as

$\chi_{Pn}^{2}(\alpha,j,H_{0})=\sum\limits_{i=1}^{1048}\left[\frac{\mu_{th}(\alpha,j,H_{0},z_{i})-\mu_{obs}(z_{i})}{\sigma_{\mu(z_{i})}}\right]^{2},$
(21)

where $\mu_{th}$ and $\mu_{obs}$ display the theoretical and observed distance
moduli, respectively.

Figure 2: Visualization of the likelihood contours for the $Pantheon$ dataset.

###  IV.3 Joint Dataset ($H(z)+Pantheon$)

For enhanced precision in constraining model parameters, we make use of the
joint dataset, and the results are calculated by the $\chi_{HP}^{2}$ function,
which is the sum of the functions $\chi_{Hb}^{2}$ and $\chi_{Pn}^{2}$ as

$\chi_{HP}^{2}(\alpha,j,H_{0})=\chi_{Hb}^{2}(\alpha,j,H_{0})+\chi_{Pn}^{2}(\alpha,j,H_{0}).$
(22)

Figure 3: Visualization of the likelihood contours for the $H(z)+Pantheon$
dataset.

The obtained values of $\alpha$, $j$, and $H_{0}$ from the above-mentioned
observational datasets are placed in Table 1. In Fig. 1, we notice that the
obtained contours are not properly in an oval shape, but we have tested using
the Gelman-Rubin convergence test that such contour patterns also converge
[72, 73].

For a better understanding of our model, we conduct a comparative analysis
with the $\Lambda$CDM model, illustrated through error bar plots. In Fig. 4,
there are three distinct trajectories concerning the best-fit parametric
values of the model. The Hubble dataset and the Type Ia Supernovae dataset
based error plots are presented in Figs. 4(a) and 4(b), respectively.

(a)

(b)

Figure 4: The error bar plots for $H(z)$ and Type Ia Supernovae datasets
exhibiting the similarities between the presented model and the $\Lambda$CDM.

##  V Physical parameters

This section involves the scrutinization of physical parameters,
systematically exploring their behavior across the best-fit values
corresponding to the Hubble dataset, the $Pantheon$ dataset, and their joint
dataset. To gain knowledge about phase-wise cosmic evolution, we examine the
features of the different model parameters, especially the dimensionless
deceleration parameter expressed in Eq. (3).

Fig. 5 demonstrates a trend in which the deceleration parameter $q$
experiences a declination from high to low redshift values. Also, this
parameter undergoes a sign change, indicating a phase shift from the
deceleration to the cosmic acceleration. It is observed that for all the
trajectories at the early epochs, the universe was in the deceleration mode,
whereas at present, as well as in late time eras, the universe is in the
accelerating mode. The values of redshift corresponding to the phase
transition phenomena are $1.14161$, $1.12361$, and $1.10797$ for the $H(z)$
dataset, the $Pantheon$ dataset, and their joint dataset, respectively.
Furthermore, the deceleration parameter assumes values of $-0.705673$,
$-0.715246$, and $-0.731007$ at the present for $H(z)$, $Pantheon$, and their
joint dataset, respectively.

Figure 5:  The graphical representation of $q$ vs. $z$.

For convenience,the jerk parameter $j$ can be articulated in relation to the
deceleration parameter $q$, as in Eq. (1). In the context of the $\Lambda$CDM
model, the jerk parameter maintains a constant value of $1$. In our model, we
determine the best-fit values for the jerk parameter to be $0.9215$,
$0.949998$, and $0.98999$ for the $H(z)$ dataset, the $Pantheon$ dataset, and
their joint dataset, respectively. Notably, all three values of $j$ are
remarkably close to unity, reflecting the consistency and agreement found
within the results.

By using Eqs. (16) and (17), we calculate the values of $\rho$ and $p$.
However, since their expressions are very large and complicated, we have
plotted their trajectories directly for every mentioned dataset. We have Fig.
6(a), in which it is observed that $\rho$ is decreasing monotonically from the
early to late times. For the isotropic pressure, Fig. 6(b) shows that all the
curves are in a highly positive region, but afterward, they decrease and enter
a negative region. At present, the negative value of $p$ signifies an
expanding and accelerating universe, while in late times, the model becomes
pressureless since $p$ tends to zero as $z$ approaches $-1$. In Fig. 6(c), the
progression of the Equation of State (EoS) parameter ($\omega=p/\rho$) has
been unfolded: starting from a perfect fluid, then it is entering the
quintessence region and eventually remaining in the phantom region. For all
observational datasets, the value of $\omega$ is less than $-1$, indicating
that our model represents a phantom scenario both at the present as well as in
the late times.

(a)

(b)

(c)

Figure 6: The graphical representations of the physical parameters, viz.,
$\rho$, $p$, and $\omega$ vs. $z$ for all the observational datasets.

Besides phenomenal fields, there are some major points to explore, such as the
classification of singularities, causal structure, energy conditions, etc. In
particular, energy conditions originated from the well-known Raychaudhuri
equation [74, 75, 76, 77, 78], serving as a valuable framework to assess the
influence of all the geometric terms within the stress-energy tensor. Stemming
from the Einstein field equations, energy conditions involve diminutions of
time-like or null vector fields with respect to the EMT and Einstein tensor.
These conditions stand out as prominent features in cosmological models,
defined as the weak energy condition (WEC: $\rho\geq 0,~{}\rho+p>0$), null
energy condition (NEC: $\rho+p\geq 0$), strong energy condition (SEC:
$\rho+3p\geq 0$), and dominant energy condition (DEC: $\rho\geq|p|$).

(a)

(b)

(c)

Figure 7: The representative graphical plots of the energy conditions.

In Fig. 7, we interestingly perceive that for all the observational datasets,
NEC and SEC behave alike. Both the energy conditions hold good in the early
universe but are violated in the present as well as in later eras. On the
other hand, DEC holds for all time periods. The violation of SEC hints at the
existence of dark energy in our model (see Fig. 7(b)). The breach of NEC
indicates the phantom behavior of the model in late times, indicating its
instability for an interval of redshift values (see Fig. 7(a)).

##  VI Conclusion

In the present investigation, we try to investigate a cosmological model in
the alternative gravity theory $f(R,T)$. The motivation for choosing this
gravity theory arises from its capacity to explore both the physical and
geometric behavior of the cosmos. We obtained the EFEs by taking a specific
form of the $f(R,T)$ function. The Hubble parameter has been calculated here
in terms of the redshift parameter along with three model parameters from the
jerk and deceleration parameter relation. Also, we enumerate the best-fit
values of the model parameters, namely, $j$, $H_{0}$, and $\alpha$, by
utilizing the observational datasets $H(z)$, $Pantheon$, and their joint data
(refer to Table 1). For this purpose, we used the MCMC method with the help of
the $emcee$ package in Python. Interestingly, one can find that our results of
$j$ and $H_{0}$ are consistent with the recently obtained Planck data. The
contours plotted with $1-\sigma$ and $2-\sigma$ confidence levels have been
discussed (see Figs. 1, 2, and 3). To assess the performance of our $f(R,T)$
model against the $\Lambda$CDM model, we have used error bar plots in Fig. 4.
We observe the cosmic phase transition from deceleration to acceleration mode,
marked by the shift of the $q$ from positive to negative values.

The top left panel of Fig. 6 provides a visual narrative of the evolution of
the energy density from the earlier to the later phase of the universe. For
all three datasets, the curves of energy density decrease monotonically from
high to low redshift. In Fig. 6(b), we notice that the isotropic pressure
shows positive behavior in the early cosmic epoch, whereas it has a negative
signature in the present stage and zero in the late epoch. This indicates an
accelerating as well as expanding universe at present and a pressureless
universe in the future. The trajectories of EoS parameter $\omega$ pass
through the various stages of evolution of the cosmos as it starts as perfect
fluid enters the quintessence dark energy region and then after remains in the
phantom region (see Fig. 6(c)).

In this model, we observe that the energy conditions, NEC and SEC, face
violations at low redshift values, whereas DEC holds across the entire value
of redshift (see Fig. 7).

Finally, based on the above-mentioned features, one can note that the
presented model of $f(R,T)$ gravity in the background of the flat FLRW metric
is quite different from the usual dark energy models. Undoubtedly, via the
plots and exhibited physical features, our obtained model in the $f(R,T)$
gravity theory emerges as a promising candidate among various modified gravity
theories and thus indicates the usefulness of further scope to work in the
modified theory of gravity for verification of observational signatures in a
more detailed and significant way.

## Acknowledgement

SR gratefully acknowledges support from the Inter-University Centre for
Astronomy and Astrophysics (IUCAA), Pune, India, under its Visiting Research
Associateship Programme as well as the facilities under ICARD, Pune, at CCASS,
GLA University, Mathura, India.

## Data Availability Statement

In this manuscript, we have used observational data as available in the
literature; as such, our work does not produce any form of new data.

## Conflicts of Interest

The authors assert that there are no conflicts of interest of the publication
of this work.

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