Institut Teknologi Bandung
Faculty of Mathematics and Natural Sciences
Astronomy Research Group1 and Bosscha Observatory2
1Jl. Ganesha No. 10 Bandung, West Java, Indonesia

2Jl. Peneropongan Bintang No. 45 Lembang, West Java, Indonesia

Can noncanonical domain walls play the role of halo dark
matter on the galactic scale?
Fargiza A. M. Mulki♠1♠Hesti Wulandari Taufiq Hidayat

fargiza@office.itb.ac.id

Introduction

Cosmological observations show that around 96% of the universe’s energy
budget is in the form of dark matter and dark energy, which are still poorlyunderstood. The nature of the two entities is different either at small orlarge scales. Notwithstanding that the empirical properties of the two arevery contradictive, several authors have attempted to unify dark matter anddark energy and perceive them as the same entity. Mulki et al. (2023a) showthat the noncanonical domain walls (NDWs) with Lagrangian)2
(1 µν
(1)
L = − g ∂µϕ∂ν ϕ − V (ϕ)
2can serve as dark energy and, in others, as cosmological dark matter withthe equation of state (EoS) parameter3 2wdw = − a .
(2)
2In addition, Mulki et al. (2023b, in preparation) examined the interaction
between domain walls with the Lagrangian (1). The authors found that twoNDWs separated at a distance smaller than a value l∗ will attract each other.
They also showed that the l∗ value is of the order of the cosmological scales.
Motivated by Mulki et al.’s results (2023a, 2023b), we can hypothesize thatthe network of NDWs may act as an attractive matter on the galactic scales.
Therefore, in this work, we test this hypothesis and examine the behavior
of NDWs at the galactic scales and their effect on the dynamics of a test
particle.
At galactic scales, many observations confirm the requirement of dark
matter to account for the flat rotation curve at the outer radii of galaxies. The
notion of dark matter in galaxies has recently become extensive, especiallysince astronomers discovered the so-called core-cusp, missing satellite, andtoo-big-to-fail problems. These challenges enrich the issues and repertoire ofdark matter in the galactic scales. Therefore, numerous models have been
put forward to address these challenges, such as fuzzy, self-interacting, andsolitonic dark matter (see Particle Data Group 2023 for more details).
Instead of examining these advanced problems, we address the noncanonicaldomain walls to the basic evidence of dark matter in the galactic scales, i.e.,
the flat rotation curve of galaxies.

• For an NDW with D−spatial dimensions:
The energy density and pressure:

4D2
ρϕ = V + (∂r ϕ)44(5)2D
44Pϕ = V + D(∂r ϕ) − (∂r ϕ)4(6)Polytropic NDWs

• We assume that the EoS of NDWs at the galactic scales obeys the polytropicrelation( )γρP = P∗(7)
ρ∗• Using Eq. (5), (6) and (4), it follows that γ = 1 is a polytropic index
solution for NDWs, regardless of its spatial dimension D.
5

Dynamics of a test particle• The pressure gradient inside an NDWs halo is given by the TOV equation)(()−112GMdPGM
2= − (ρ + P )+ 4πGr P
1−(8)
drrrr
• The EoS of NDWs is described by Eq. (7) with γ = 1• Approximate conditions:– Weak gravitational field limit (2GM /r ≪ 1)
– Pressure dominated dynamics (4πGr2P ≫ GM /r)• Solution:(P (r) ≈2πGρ∗ 2
12(r − r∗) +P∗P∗)−1(9)• Rotational velocity of a test particle in a distance r:
2r
8πGP∗2v ≈ρ∗ (r2 − r∗2)

(10)

8πGP∗
= constantv ≈ρ∗(11)• For large r
2

Objective

To examine whether NDWs with the Lagrangian (1) can play the role of halodark matter in the galactic scales, i.e., exhibiting a flat rotation curve at largeradii.
3Field equation• Spacetime background:)()−1(RsRs22
dt + 1 −dr2 + r2(dθ2 + sin2 θdφ2) (3)ds = − 1 −rr
where Rs ≡ 2GM• Assumptions for the field ϕ and its spacetime:– Weak gravitational field limit (Rs/r ≪ 1) (for the large radii cases)– Vacuum dominated phase (vacuum ≫ ϕ) −→ V (ϕ) = V ≈ constant(for a consistency with its cosmological solution (Mulki et al. 2023b))– Perfect fluid for general field ϕ• Equation of motion (EoM) of Lagrangian (1) in background (3):d2ϕ Rs dϕ+=0(4)2drr dr26Discussion

The NDWs embedded in the spherical symmetry background have apolytropic index γ = 1 so that the equation of state takes the simple form,i.e., P (r) ∝ ρ(r). The extensive presence of the NDWs in a galaxy canprovide an excess attractive pressure, forcing a test particle at large radii tomove faster than that of the Keplerian case. Thus, the NDWs can play therole of halo dark matter that explains the flat rotation curve at the outerradii of galaxies.
References
1. F. A. M. Mulki, H. Wulandari, and T. Hidayat. Noncanonical domain wall as a unified model of dark energy and dark matter:
I. Cosmic dynamics. Gravitation and Cosmology, accepted, 2023
2. F. A. M. Mulki, H. Wulandari, and T. Hidayat. Noncanonical domain wall as a unified model of dark energy and dark matter:
II. Physical properties. in preparation, 2023
3. Particle Data Group, Dark Matter, 2023 (https://pdg.lbl.gov/2023/reviews/astro-cosmo.html)

Acknowledgements
This research was funded by the PPMI (Penelitian dan Pengabdian pada Mayarakat) 2023 project of the Faculty of Mathematics
and Natural Sciences, Institut Teknologi Bandung (ITB).