# Strongly polarized narrow exciton lines from Ga2Se3/GaSe allotropic
nanostructures

Maxim V. Rakhlin  Ioffe Institute, 194021 St. Petersburg, Russia  Sergey V.
Sorokin  Ioffe Institute, 194021 St. Petersburg, Russia  Ilya A. Eliseyev
Ioffe Institute, 194021 St. Petersburg, Russia  Valery Yu. Davydov  Ioffe
Institute, 194021 St. Petersburg, Russia  Demid A. Kirilenko  Ioffe Institute,
194021 St. Petersburg, Russia  Alexey A. Toropov  Ioffe Institute, 194021 St.
Petersburg, Russia  Tatiana V. Shubina  Ioffe Institute, 194021 St.
Petersburg, Russia<EMAIL_ADDRESS>
###### Abstract

The ability to emit narrow exciton lines, preferably with a well-defined
polarization, is one of the key conditions for the use of nanostructures based
on III-VI monochalcogenides in nanophotonics and quantum technologies (for
single photon emitters). At present, the main method of their formation is the
exfoliation of layered crystals, supplemented by strain and defect
engineering. The factor limiting the use of epitaxy is the presence of
different polytypes and structural phases in the grown films. However, control
over their formation can make it possible to create structures with desired
properties. In this work, we propose a Ga2Se3/GaSe nanostructure fabricated by
van der Waals epitaxy with a high VI/III flux ratio as a source of strongly
polarized narrow lines of exciton emission. In fact, this nanostructure is
formed from allotropes: GaSe and Ga2Se3, which consist of the same atoms in
different arrangements. The energy position of the narrow lines is determined
by the quantum confinement in ultrathin Ga2Se3 inclusions of different sizes
in the GaSe matrix (barrier), and their linear polarization is due to the
ordering of Ga vacancies in a certain crystal direction in Ga2Se3. Using
transmission electron microscopy, Raman spectroscopy, and micro-
photoluminescence, the morphology of Ga2Se3/GaSe nanostructures, the
temperature and temporal behavior of the narrow lines have been established.
Our results open the way to the creation of allotropic nanostructures capable,
in particular, of providing narrow-line emission.

###### keywords:

American Chemical Society, LaTeX

IR,NMR,UV

##  1 Introduction

The last decades can be characterized as an era of two-dimensional (2D)
physics, when layered anisotropic semiconductors – transition metal
dichalcogenides and group III metal monochalcogenides, studied since the
middle of the last century, are being discovered from a new perspective.
Monochalcogenides such as GaSe, GaS, and InSe are attracting much attention as
parent materials for 2D and few-layer structures promising for applications in
nanoelectronics (field-effect transistors), photovoltaics, sensors, and
nonlinear optics 1, 2, 3, 4, 5. Strain and defect engineering has been used to
realize states in 2D structures, similar to quantum-dots, emitting narrow
exciton lines and acting as sources of single photons 6.

Gallium selenide is one of the most studied III–VI monochalcogenide
semiconductors (see reviews 7, 8). Unlike transition metal dichalcogenides,
bulk GaSe have almost degenerate indirect and direct band gaps, which greatly
simplifies the problem of forming effectively emitting structures 9, 10. The
published spectra of low-temperature photoluminescence (PL) of bulk GaSe are,
as a rule, inhomogeneous and contain a series of lines that were attributed to
free excitons and excitons bound at various defects 11. The presence of
different polytypes that differ from each other in the stacking sequence in
the unit cell 12, 13, 14, 15, 16 can also contribute to the complexity of PL.
As the number of monolayers decreases approximately to six, the emission from
GaSe disappears due to very weak absorption of the exciting light in indirect
gap structures. 17.

The so-called GaSe monolayer consists of four atomic layers, that is, a
tetralayer, where two layers of Ga atoms are located between two layers of
chalcogen atoms (Se-Ga-Ga-Se) (more details see in Supplementary). Such an
architecture allows sliding of the Se sublayer, which leads to the appearance
of intrinsic ferroelectricity 18. In addition, the formation of another
structural phase, Ga2Se3, is possible. Essentially, it is an allotrope of
GaSe, consisting of the same Ga and Se atoms, but in a different arrangement.
It is the so-called defective zinc blende, which, in the case of the ordering
of Ga vacancies, has either a monoclinic (mono) or orthorhombic (ortho)
crystal structure (see Fig. S3 in Supplementary). The calculated band gap
energy in mono-Ga2Se3 is somewhat higher than in ortho-Ga2Se3 19, 20. The
measured band gap values of Ga2Se3 are characterized by a large scatter and
depend both on the film preparation methods and on the spectroscopic technique
used 21, 22, 23, 24, 25. However, the difference in band gap between the mono-
and ortho- phases is always within 0.5 eV and the mono-phase has a higher
energy.

The ordered Ga2Se3 has unique structural and electronic properties, which are
almost independent on the particular crystal structure (mono or ortho), but
rather on the vacancies and their local environment 26. In contrast to GaSe
(Ga2Se2), one third of the gallium positions in this compound are not filled
and represents structural vacancies. Such vacancies are located not randomly,
but along every third line along a certain direction forming a “super-
structure”, that is, an extended unit cell 19 (see Fig. S4 in Supplementary).
As a result, the PL turns out to be strongly linearly polarized 22. Similar
phenomena, including optical anisotropy, were observed in Ga2S3 as well 27.
The nanostructures of these materials have not yet been studied.

So far, chalcogenide nanostructures have been fabricated predominantly using
exfoliation technology. However, this manual method does not provide
reproducible quality and sizes of monolayers suitable for chip technology. A
possible solution to the problem can be the use of epitaxial methods.
Molecular beam epitaxy (MBE) has certain advantages over other technologies
due to the use of high purity materials, ultrahigh vacuum in the chamber, as
well as the possibility of implementing the layer thickness at the atomic
level 28. MBE in the van der Waals growth mode was proposed back in the 1990s
29, 30 as a method for successful epitaxial growth even with a strong mismatch
between the layer and substrate lattices ($\Delta$a/a up to 30$\%$ or more)
and when their crystal structures are different. However, despite some
promising results 31, 32, 33, 34, van der Waals epitaxy of both mono- and
dichalcogenides is still under development 35.

In this paper, we report on the first successful formation of emitting
Ga2Se3/GaSe nanostructures by the van der Waals MBE. The technology for their
fabrication involves increasing the Se/Ga ratio by increasing the growth
temperature. Raman studies and transmission electron microscopy (TEM) have
shown that the grown samples contain ultrathin Ga2Se3 inclusions. Using micro-
PL spectroscopy ($\mu$-PL), highly linearly polarized narrow excitonic
emission lines ($\sim$1 meV wide) are observed, as expected from the ordering
of Ga vacancies in Ga2Se3. The short decay times ($\sim$400 ps) of the lines
are consistent with the direct band gap of Ga2Se3, and their quenching with
increasing temperature corresponds to the use of GaSe as a barrier.

##  2 Sample growth and characterization

###  2.1 Van der Waals epitaxy

Controlling the stoichiometry of van der Walls III-VI monochalcogenide films,
tending to spontaneous formation of a large number of polytypes, requires a
developed approach that takes into account relatively low growth temperatures
as well as the possibility of the formation of transitional submonolayers at
the heterointerface between the substrate and the layer 36, 33, 37.

The sample under study was grown on a GaAs(001) substrate using a two-chamber
MBE setup. First, a 200 nm thick GaAs buffer layer was grown in a separate
III-V chamber, then transferred to the III-VI chamber through vacuum, and the
growth of GaSe film was started at a substrate temperature of $T_{S}$=400∘C.
Such a relatively low value of $T_{S}$ provides the van der Waals growth mode
with the orientation of the c axis of the growing GaSe layer perpendicular to
the substrate surface, which ensures the formation of a sharp GaSe/GaAs(001)
interface 33. After the growth of the first $\sim$ 4 nm, the epitaxy
temperature was raised to $T_{S}$$\sim$500∘C and then remained unchanged until
the end of the growth run. One of the main reason for introducing this step is
that GaSe layers grown at high $T_{S}$ demonstrate efficient near band-edge PL
in contrast to the films grown at low $T_{S}$ 33. On the other hand, higher
$T_{S}$ requires an increase in the VI/III ratio to avoid the appearance of Ga
droplets on the growth surface. In this regard, it should be noted that the
conditions established for the MBE growth of single-crystal Ga2Se3 layers do
not differ much from that for GaSe: $T_{S}$ $\sim$ 450-550∘C (i.e., nearly the
same), VI/III ratio is 15 and higher (i.e, only a few times higher in contrast
to GaSe) if both standard Ga and Se sources as well as GaP(001) or GaAs(001)
substrates are used 38, 39, 40, 41. When using a valve cracking cell as a Se
source, the VI/III ratio required to maintain conditions with the high Se
content during GaSe growth is significantly higher and depends on the
substrate temperature due to the lower selenium sticking coefficient. Such a
high VI/III flux ratio, as well as relatively low growth temperatures
($T_{S}$$\sim$500∘C) can apparently promote the formation of a vacancy-ordered
Ga2Se3 phase inclusions in the growing film 20, 40. Proof of this assumption
is the existance of a peak associated with Ga2Se3 phase in the X-ray
diffraction curves of GaSe layers grown at high VI/III flux ratio ($\sim$
30-40) at $T_{S}$$\sim$400∘C using a Se valve cracking cell (see
Supplementary). Thus, one can expect the appearance of a small amount of
Ga2Se3 phase inclusions in GaSe layers grown in two-stage mode using the above
parameters.

Scanning electron microscopy (SEM) cross-sectional images show that the total
thickness of the grown film reached $\sim$ 0.36 $\mu$m (Fig. 1)a. The film
surface morphology is rough, enriched with trigonal or tetrahedral ridges,
which results from the increase in temperature at the second stage. Further
details of van der Waals epitaxy of GaSe can be found in the Methods section.

###  2.2 Raman studies

GaSe crystals consisting of Ga-Se-Se-Ga tetralayers exist as $\beta$,
$\gamma$, $\delta$, and $\epsilon$ 42, 43, 44 polytypes, which differ in
mutual arrangement of tetralayers in a unit cell. To determine the main
polytype of the sample grown by van der Waals epitaxy, we carried out Raman
studies in a wide frequency range. The $\epsilon$ polytype occurs most
frequently in epitaxial thin films. Within the framework of the D3h
$\epsilon$-GaSe symmetry group nomenclature, the high-frequency ($\omega>50$
cm-1) Raman spectra of bulk GaSe contain lines 60, 134, 214, 252 and 308 cm-1
corresponding to E${}^{{}^{\prime\prime}}$, A${}^{1}_{{}^{\prime}}$,
E${}^{{}^{\prime}}$( TO), E${}^{{}^{\prime}}$(LO) and A${}^{{}^{\prime}}_{1}$
respectively 45, 46. As seen in Figure 1b, the Raman spectrum measured in the
grown sample contains all these high-frequency lines. In addition, modes in
the low-frequency region ($\omega<50$ cm-1) make it possible to accurately
determine the polytype 46. Here we observe one peak at 19 cm-1, which
convincingly confirms the predominance of the $\epsilon$ polytype of GaSe in
the studied thin film.

Figure 1: SEM image of the cross section (a) and unpolarized Raman spectrum
(b) of a GaSe sample grown by van der Waals MBE on GaAs (001).

The vacancy-ordered Ga2Se3 can exist in both monoclinic and orthorhombic
modifications, whose electronic properties are similar 26. We denote these
modifications as “mono” and “ortho”, although some studies used the
designations $\beta$-Ga2Se3 for mono and $\gamma$-Ga2Se3 for ortho 47. In the
Raman spectrum of the sample under study, the 155 cm-1 and $\sim$250 cm-1
modes are observed, which were previously registered in Ga2Se3 48, 49, 50. The
peak at 155 cm-1 can be attributed to the A1 mode of the vacancy-ordered phase
48. The broadened shape of the peak indicates a change in the crystallinity of
the Ga2Se3 inclusions; a similar shape was observed in 50 for specimens grown
at relatively low VI/III flux ratios. In the region of 250 cm-1, a doublet is
observed, consisting of two peaks: a weak E${}^{{}^{\prime}}$(LO) peak of bulk
GaSe (252 cm-1) and a peak at 245 cm-1. It was suggested that the latter may
be due to short-range interactions in the Ga2Se3 unit cell 51. Thus, the Raman
spectrum is a combination of modes at frequencies characteristic of
$\epsilon$-GaSe and vacancy-ordered Ga2Se3.

In addition, several lines in the Raman spectrum in Figure 1b arise from
growth on the GaAs substrate and ridge morphology. For example, two additional
lines at 268 and 291 cm-1 correspond to transverse (TO) and longitudinal (LO)
optical phonons of the GaAs substrate. A distinct doublet at 200-220 cm-1 is a
sign of rough morphology 33, when the appearance of the low-frequency
E${}^{{}^{\prime\prime}}_{2}$ mode in GaSe became possible due to the
inclination of the ridge face depending on the incident light 52, 45.
Oxidation of GaSe in air can lead to the release of selenium 53 and the
appearance of weak of the Ga2O3 modes 54.

###  2.3 Transmission electron microscopy

The coexistence of the allotropic $\epsilon$-GaSe and Ga2Se3 phases in the
MBE-grown sample was confirmed by TEM studies of a thin flake (ridge)
mechanically separated from the substrate (Fig. 2) shows both the
$\epsilon$-GaSe hexagonal phase with a period of 0.8 nm and the Ga2Se3 phase
with a period of $\sim$0.6 nm at the top of this flake. These phases,
conjugated by their close-packed planes, have a small mismatch of about 2$\%$.
The Ga2Se3 inclusions have, on average, a thickness of several monolayers and
a transverse size of about 5 nm. In addition, hexagonal $\epsilon$-GaSe has a
fairly high stacking-fault density, which is clearly seen from electron
diffraction (not shown here).

According to previous theoretical studies 55, 19, only two types of short-
range ordering of vacancies are possible in Ga2Se3, which can occur either
along [$\bar{1}$10] in ortho–Ga2Se3 or a zigzag line along [$1\bar{1}$2] to
mono–Ga2S3 19, although some uncertainty in the possible directions is still
exists. In a orthorhombic cell $a$ = 5.477 Å, $c$ = 5.414 Å, while in a
monoclinic structure $a$ = 6.66 Å, $c$ = 11.65 Å, i.e. the value of $c$ is
twice as large 20. Therefore, in this flake we are most likely observing
ortho-Ga2Se3. Note that such characterization using TEM is possible only at
the thin tip of the trigonal flake, since the thicker part near the substrate
is opaque for this method.

Figure 2: High-resolution TEM images of Ga2Se3/GaSe nanostructures in cross-
sectional geometry: (a) general view with Ga2Se3 insertions within GaSe; (b)
enlarged image of the region with the Ga2Se3 phase.

However, in this part, the stresses caused by the mismatch between the lattice
parameters of the layer and the substrate are stronger, which can stimulate
the formation of a monoclinic phase with a zigzag arrangement of vacancies,
which is structurally more stable 20. Thus, the formation of both ortho- and
mono-Ga2Se3 in the film under study cannot be ruled out.

##  3 Narrow exciton lines

The possibility of quantum confinement in Ga2Se3 inclusions depends on their
characteristic dimensions relative to the exciton Bohr radius RB. In the first
approximation, we can assume that it is close to RB = 4.5 nm in bulk GaSe 56.
In this case, the exciton diameter of 9 nm significantly exceeds the
characteristic sizes of Ga2Se3 inclusions determined by the TEM method, and
quantum confinement can provide the formation of distinct exciton levels, as
in quantum dots. Therefore, the corresponding radiation must have a narrow
radiation line width and a higher energy than the bulk material. The minimum
energy of direct band gap transitions in bulk ortho-Ga2Se3 is about 1.75-1.85
eV 19. Although published data are strongly dispersed for mono-Ga2Se3, it is
reliably established that its band gap is noticeably higher.

The emission spectra measured in a grown film with a spatial resolution of
$\sim$1 $\mu$m at low temperature (10 K) and low excitation power (0.2 mW)
demonstrate ultra-narrow lines (Fig. 3). It should be emphasized that such
narrow lines are registered in local places. With a slight shift, narrow lines
quickly disappear (see Supplementary). The ultra-narrow lines have a width of
about 1 meV. Their intensity and number are increased as the excitation spot
is defocused up to several $\mu$m. During laser defocusing, the direction of
the incident light deviates slightly from the normal to the substrate, which
is obviously preferable for the effective excitation of Ga2Se3 inclusions
located inside inclined flakes (see the excitation diagram in Fig. 3). Under
this condition, a weak peak at 2.12 eV corresponding to GaSe emission is also
resolved. The region where ultranarrow lines appear (1.85-2.08 eV) indicates a
possible combination of ortho (in the lower part) and mono (in the upper part)
Ga2Se3 inclusions, for which the surrounding GaSe is a barrier.

Figure 3: Narrow-line PL spectra recorded in a Ga2Se3/GaSe film with different
excitation spot (focused to 1 $\mu$m and defocused). The excitation is
illustrated by the inset against the bird-view SEM. Excitation power 0.2 mW,
temperature 10 K.

At a higher excitation power (1 mW), a broad band appears centered around 1.73
eV (Fig. 4). This band is modulated by different peaks with a width of $\sim$3
meV. These peaks are probably due to the localization of excitons on
structural defects, such as stacking faults, which were diagnosed by TEM in
the film under study. Previously, we observed similar peaks in micro-PL
spectra measured directly at stacking faults crossing ZnSe quantum wells 57.
Such peaks and ultranarrow lines behave differently with increasing
temperature: ultranarrow lines disappear first in the upper region of the
spectrum, since the GaSe barrier is not too high for Ga2Se3 inclusions, and
broader peaks localized at deeper defect centers still exist. Interestingly,
in bulk GaSe, the free exciton peak dominates in the PL spectra at elevated
temperatures, although defect lines are present (see inset in Fig. 4).

Figure 4: Narrow-line PL spectra recorded at various temperatures in
Ga2Se3/GaSe. For comparison, the inset shows the PL spectra in bulk GaSe,
where the exciton peak begins to dominate with increasing temperature.

An important argument in favor of the correctness of our interpretation of
ultranarrow lines can be their linear polarization. The large linear
anisotropy of the transmission spectra in a vacancy-ordered bulk Ga2Se3 was
first discovered in 1994 by Okamoto et al. 22. Later this phenomenon was
theoretically considered by Nakayama and Ishikawa 19 and P$\acute{e}$ressy et
al. 26. It is shown that optical transitions in Ga2Se3 must be strictly
linearly polarized along certain crystal directions, where vacancies form a
one-dimensional super-structure. The ordering of vacancies occurs along a line
in the [$\bar{1}$10] direction in ortho–Ga2Se3 and in the form of a zigzag
along [$1\bar{1}$2] in mono–Ga2Se3. With a more complex zigzag ordering,
optical anisotropy may appear due to the off-diagonal component of the
dielectric function tensor, which changes the main axis of the crystal towards
the direction of vacancy ordering 19. As a result, a large optical anisotropy
should be observed in measurements along and perpendicular to the
corresponding direction in both mono- and ortho-Ga2Se3, and without a shift in
the PL line energy.

We have performed micro-PL polarization measurements within the full planar
angle from -180∘ to +180∘, which confirm our findings. A simultaneous change
in the polarization of narrow lines in Figure 5a indicates a strictly oriented
crystallographic direction of the radiating objects located inside the excited
spot. The difference in the angles of reaching the maximum and minimum of the
PL intensity, $I$, is exactly 90∘, as shown in Fig. 5c. The degree of
polarization, defined as $(I_{\bot}-I_{\parallel})/(I_{\bot}+I_{\parallel})$,
varies from $\sim$40$\%$ to 60$\%$ for different narrow lines. We emphasize
that such anisotropy is demonstrated by an ensemble of separate Ga2Se3/GaSe
nanostructures, whereas earlier the phenomenon was studied in bulk Ga2Se3 41,
22, 25.

At some points, it turns out to be possible to register simultaneously two
sets of lines, the polarization angle of which is shifted with relative to
each other by 60∘-90∘ (Fig. 5 (b,d)). In one of the set, the lines of the
strongest intensity are concentrated near 1.9 eV, and in the other, near 2.0
eV. Accordingly, we attribute these sets to the radiation of the orthorhombic
and monoclinic Ga2Se3 phases, which have the corresponding band gap energy for
the bulk $\sim$1.75 eV and $\sim$1.9 eV. The higher-energy lines arise from
mono-Ga2Se3 inclusions located closer to the substrate, where the stresses
caused by the lattice misfit is stronger. The shift between the angles of
polarization maxima in these two sets is due to the different crystal axes of
the vacancy ordering in these two modifications of the defective zinc blende
structure. Thus, this is an important observation that confirms the existence
of two structural phases of Ga2Se3/GaSe nanostructures implemented in the
sample under study.

Figure 5: (a, b) Mapping of the polarized spectra of narrow PL lines measured
over the full planar angle from -180∘ to +180∘ in two characteristic points
demonstrating the dominance of the ortho phase (a) and the combination of the
ortho and mono phases (b). The selected spectra demonstrate: (c) a difference
of 90∘ between the angles of maximum and minimum intensity in the ortho phase;
(d) shows the spectra characteristic of the ortho and mono phases.

Finally, we studied the type of optical transitions (direct or indirect)
corresponding to the narrow lines. Previously, by measuring the absorption and
cw PL, it was shown that bulk Ga2Se3 has a direct band gap 21, 27, 47. We
assume that the measurements of time-resolved $\mu$-PL (TRPL) at low
temperature is also one of the reliable methods, since in the case of a direct
band structure the characteristic decay times should be much shorter than
those of an indirect one. Figure 6 shows the measured decay curves of the
ultra-narrow lines possessing different wavelengths. The 629-nm (1.97 eV) line
probably comes from mono-Ga2Se3, and the following lines with wavelengths of
650 nm or more come from ortho-Ga2Se3. The characteristic decay times of the
dominant component in all narrow lines are about 400 ps. Such short decay
times are characteristic of direct-gap transitions. They cannot be attributed
to GaSe, where indirect transitions dominate at low temperatures and have much
longer decay times, on the order of tens of nanoseconds. For reference, we
present in Supplementary the data on the PL decay in bulk GaSe. The presence
of a weak and slowly decaying component in the decay curves is due to the
admixture of radiation from the surrounding GaSe. We cannot prevent this,
since the detection area of 1 $\mu$m significantly exceeds the dimensions of
the Ga2Se3/GaSe nanostructures. However, at a lower excitation power (0.1 mW),
when only the covering GaSe layer is effectively excited, the fast component
from the inserts disappears.

Figure 6: (a) Narrow-line PL spectrum and (b) PL decay curves of selected
narrow lines measured at the indicated wavelengths at 10 K using a 405-nm
pulsed-laser line. The excitation power is shown in the graphs.

Thus, these Ga2Se3/GaSe nanostructures have a direct type of transitions with
a corresponding alignment of the conduction and valence bands, which is very
important for the device application of the narrow lines.

##  4 Conclusions

Summarizing, this work presents Ga2Se3/GaSe nanostructures grown by van der
Waals epitaxy. The specificity of these nanostructures is that they can hardly
be called "hetero" structures, since they are formed from related materials,
consisting of the same atoms, but having different crystal structures. At a
certain extent, they are allotropic compounds. These nanostructures
demonstrate ultra-narrow lines of exciton radiation with a linewidth of
$\sim$1 meV. The presence of the narrow lines in the emission spectra is
possible as long as the energy of their quantum levels does not exceed the
height of the potential barrier, which is the surrounding GaSe matrix.
Importantly, the lines are strongly polarized with the degree of polarization
up to 60$\%$. Taking into account the energy position of the narrow lines and
their polarization, we assumed that the lines are associated with excitons
confined in the Ga2Se3 inclusions which have the vacancy-ordered orthorhombic
or monoclinic crystal structure. The existence of both crystal phases is
confirmed by the pattern of polarized PL mapping, in which the emission maxima
of high-energy and low-energy sets of narrow lines are shifted by a certain
angle relative each other. The time-resolved micro-PL spectroscopy established
the direct-gap nature of Ga2Se3/GaSe nanostructures. This combination of
optical properties is unique for monochalcogenide nanostructures. A similar
ordering of vacancies is possible in Ga2S3 and, possibly, in other
monochalcogenides such as In2Se3 and In2Te3, which paves the way for the use
of allotropic nanostructures in quantum photonics.

##  5 Experimental section

###  5.1 Van der Waals epitaxy

A (Ga,Se)-based layer was grown on epi-ready GaAs(001) substrate using a two-
chamber MBE setup (SemiTEq production). A standard Ga cell and Se valve
cracking cell (Veeco, USA) with the cracking zone temperature
$T_{Se}$(cr)=500∘C were used as molecular beam sources. To avoid problems with
thermal cleaning of the substrate to remove the native oxide layers in the
absence of an As flux, a GaAs buffer layer 200 nm thick was grown in a
separate III-V chamber and then transferred to the III-VI chamber through
vacuum. When the substrate temperature was stabilized at a targeted value of
$T_{S}$=400∘C, the growth of GaSe was initiated with simultaneous opening the
Ga and Se shutters. After the growth of first $\sim$ 4 nm, the epitaxy
temperature was increased up to the $T_{S}$$\sim$500∘C. The VI/III flux ratio
was controlled by measuring the beam equivalent pressures (BEPs) of the
corresponding elements at the substrate position by using a Bayard-Alpert ion
gauge and maintained as high as $\sim$ 20 and $\sim$ 30 for the $T_{S}$=4000C
and 500∘C, respectively. Taking into account that the stoichiometric
conditions on the growth surface correspond to $P_{Se}$/$P_{Ga}$ $\sim$ 12 and
$\sim$ 25 at these growth temperatures 33, the actual VI/III flux ratio can be
estimated as $\sim$ 1.2 and 1.6 for the first and second stages of growth,
respectively.

###  5.2 Structural characterization

SEM characterization was performed using JEOL JSM 7001F scanning electron
microscope. The crystal structure of mechanically separated flakes was
investigated by TEM studies using a Jeol JEM-2100F microscope.

###  5.3 Micro-Raman measurements

Raman spectra were obtained in backscattering geometry at room temperature.
For these investigations, we used a Horiba LabRAM HREvo UV-VIS-NIR-Open
spectrometer (Horiba, Lille, France) with confocal optics. Olympus MPLN100x
(Olympus, Tokyo, Japan) objective lens (NA = 0.9) focused the laser beam in a
spot $\sim$1 $\mu$m in diameter. As an excitation source, a Nd:YAG laser
(Torus Laser Quantum, Stockport, UK) with $\lambda$ = 532 nm was used. To
avoid damage and heating of the sample, the laser power on its surface was
limited to 80 $\mu$W. A set of Bragg reflection and transmission filters was
used to obtain Raman spectra in ultra-low-frequency range ($\omega$ < 50
cm-1).

###  5.4 Micro-photoluminescence measurements

$\mu$-PL setup was used for optical properties investigation of GaSe
structures. Non resonant optical excitation of a $cw$ laser (405 nm) was used
for the $\mu$-PL measurements. Incident radiation was focused in 2-3 $\mu$m
spot on the sample by an apoachromatic objective lens with NA of 0.42. The
power density was 4 W/cm2. The collected emission was dispersed by a 0.5m
monochromator with a 600/mm grating for detection $\mu$ PL spectra at selected
wavelengths. For time-resolved $\mu$-PL studies the 405 nm line of a pulsed
laser was used. Polarization control was achieved by means of a halfwave plate
and a film polarizer installed in the detection channel.

##  6 Conflict of Interest

The authors declare no conflict of interest.

##  7 Author Contributions

The manuscript was written using contributions of all authors. In particular,
M.V.R. and I.A.E. contributed to $\mu$-PL and Raman studies, respectively.
S.V.S. contributed to MBE growth. D.A.K. contributed to TEM measurements.
V.Yu.D., A.A.T. and T.V.S. proposed a general physical model and supervised
this work. All authors have given approval to the final version of the
manuscript.

The work of M.V.R. and I.A.E. was supported by a grant from the Russian
Science Foundation (no. 22-22-20049, https://rscf.ru/project/22-22-20049/) and
a grant from the St. Petersburg Science Foundation in accordance with
agreement no. 21/2022 dated April 14, 2022.

The data that support the findings of this study are available in the
supplementary material of this article.

## References

  * Bourdon et al. 1990 Bourdon, A.; Bringuier, E.; Portella, M.; Viviéres, M.; Piccioli, N. Angular properties of second-harmonic polarization due to high-order nonlinearities: Application to GaSe and InSe. _Physical Review Letters_ 1990, _65_ , 1925
  * Guo et al. 2015 Guo, J.; Xie, J.-J.; Li, D.-J.; Yang, G.-L.; Chen, F.; Wang, C.-R.; Zhang, L.-M.; Andreev, Y. M.; Kokh, K.; Lanskii, G.; Svetlichnyi, V. Doped GaSe crystals for laser frequency conversion. _Light: Science & Applications_ 2015, _4_ , e362
  * Sato et al. 2020 Sato, Y.; Tang, C.; Watanabe, K.; Ohsaki, J.; Yamamoto, T.; Tanabe, T.; Oyama, Y. InxGa1-xSe mixed crystals grown from an In flux by the traveling heater method for THz wave generation. _Journal of Physics Communications_ 2020, _4_ , 1065007
  * Cai et al. 2019 Cai, H.; Gu, Y.; Lin, Y.-C.; Yu, Y.; Geohegan, D.; Xiao, K. Synthesis and emerging properties of 2D layered III–VI metal chalcogenides. _Applied Physics Reviews_ 2019, _6_ , 041312
  * Wang et al. 2020 Wang, Y.; Gao, J.; Wei, B.; Han, Y.; Wang, C.; Gao, Y.; Liu, H.; Han, L.; Zhang, Y. Reduction of the ambient effect in multilayer InSe transistors and a strategy toward stable 2D-based optoelectronic applications. _Nanoscale_ 2020, _12_ , 18356–18362
  * Tonndorf et al. 2017 Tonndorf, P.; Schwarz, S.; Kern, J.; Niehues, I.; Pozo-Zamudio, O. D.; Dmitriev, A. I.; Bakhtinov, A. P.; Borisenko, D. N.; Kolesnikov, N. N.; Tartakovskii, A. I.; de Vasconcellos, S. M.; Bratschitsch, R. Single-photon emitters in GaSe. _2D Materials_ 2017, _4_ , 021010
  * Belen’kii and Stopachinskii 1983 Belen’kii, G. L.; Stopachinskii, V. B. Electronic and vibrational spectra of III-VI layered semiconductors. _Soviet Physics Uspekhi_ 1983, _26_ , 497–517
  * Sarkar and Stratakis 2020 Sarkar, A. S.; Stratakis, E. Recent Advances in 2D Metal Monochalcogenides. _Advanced Science_ 2020, _7_ , 2001655
  * Sun et al. 2018 Sun, Y.; Luo, S.; Zhao, X.-G.; Biswas, K.; Li, S.-L.; Zhang, L. InSe: a two-dimensional material with strong interlayer coupling. _Nanoscale_ 2018, _10_ , 7991–7998
  * Lai et al. 2022 Lai, K.; Ju, S.; Zhu, H.; Wang, H.; Wu, H.; Yang, B.; Zhang, E.; Yang, M.; Li, F.; Cui, S.; Deng, X.; Han, Z.; Zhu, M.; Dai, J. Strong bulk-surface interaction dominated in-plane anisotropy of electronic structure in GaTe. _Communications Physics_ 2022, _5_ , 143
  * Capozzi and Montagna 1989 Capozzi, V.; Montagna, M. Optical spectroscopy of extrinsic recombinations in gallium selenide. _Physical Review B_ 1989, _40_ , 3182
  * Brebner and Mooser 1967 Brebner, J. L.; Mooser, E. Excitons in GaSe polytypes. _Physics Letters A_ 1967, _24_ , 274–275
  * Blasi et al. 1989 Blasi, C. D.; Manno, D.; Rizzo, A. Convergent-beam electron diffraction study of melt-and vapour-grown single crystals of gallium chalcogenides. _Il Nuovo Cimento D_ 1989, _11_ , 1145–1163
  * Ueno et al. 1997 Ueno, K.; Takeda, N.; Sasaki, K.; Koma, A. Investigation of the growth mechanism of layered semiconductor GaSe. _Applied Surface Science_ 1997, _113_ , 38–42
  * Dai et al. 1999 Dai, Z.; Chegwidden, S. R.; Rumaner, L. E.; Ohuchi, F. S. Microstructure evolution of GaSe thin films grown on GaAs(100) by molecular beam epitaxy. _Journal of Applied Physics_ 1999, _85_ , 2603–2608
  * Liu et al. 2015 Liu, S.; Yuan, X.; Wang, P.; Chen, Z.-G.; Tang, L.; Zhang, E.; Zhang, C.; Liu, Y.; Wang, W.; Liu, C.; Chen, C.; Zou, J.; Hu, W.; Xiu, F. Controllable Growth of Vertical Heterostructure GaTexSe1–x/Si by Molecular Beam Epitaxy. _ACS Nano_ 2015, _9_ , 8592–8598
  * Pozo-Zamudio et al. 2015 Pozo-Zamudio, O. D.; Schwarz, S.; Sich, M.; Akimov, I. A.; Bayer, M.; Schofield, R.; Chekhovich, A.; Robinson, B. J.; Kay, N.; Kolosov, O. V.; Dmitriev, A. I.; Lashkarev, G.; Borisenko, D. N.; Kolesnikov, N. N.; Tartakovskii, A. I. Photoluminescence of two-dimensional GaTe and GaSe films. _2D Materials_ 2015, _2_ , 035010
  * Li et al. 2023 Li, W.; Zhang, X.; Yang, J.; Zhou, S.; Song, C.; Cheng, P.; Zhang, Y.-Q.; Feng, B.; Wang, Z.; Lu, Y.; Wu, K.; Chen, L. Emergence of ferroelectricity in a nonferroelectric monolayer. _Nature Communications_ 2023, _14_ , 2757
  * Nakayama and Ishikawa 1997 Nakayama, T.; Ishikawa, M. Bonding and Optical Anisotropy of Vacancy-Ordered Ga2Se3. _Journal of the Physical Society of Japan_ 1997, _66_ , 3887–3892
  * Huang et al. 2013 Huang, G.-Y.; Abdul-Jabbar, N.; Wirth, B. First-principles study of the structure and band structure of Ga2Se3. _Journal of Physics: Condensed Matter_ 2013, _25_ , 225503
  * Park et al. 1989 Park, K.-H.; Kim, H.-G.; Kim, W.-T.; Kim, C.-D.; Jeong, H.-M.; Lee, K.-J.; Lee, B.-H. Optical properties of Ga2Se3:Co2+ single crystals. _Solid State Communications_ 1989, _70_ , 971–974
  * Okamoto et al. 1994 Okamoto, T.; Yamada, A.; Konagai, M.; Takahashi, K. Polarized photoluminescence in vacancy-ordered Ga2Se3. _Journal of Crystal Growth_ 1994, _138_ , 204–207
  * Morley et al. 1996 Morley, S.; von der Emde, M.; Zahn, D. R. T.; Offermann, V.; Ng, T. L.; Maung, N.; Wright, A. C.; Fan, G. H.; Poole, I. B.; Williams, J. O. Optical spectroscopy of epitaxial Ga2Se3 layers from the far infrared to the ultraviolet. _Journal of Applied Physics_ 1996, _79_ , 3196–3199
  * von der Emde et al. 1996 von der Emde, M.; Zahn, D.; Ng, T.; Maung, N.; Fan, G.; Poole, I.; Williams, J.; Wright, A. MOCVD growth of Ga2Se3 on GaAs(100) and GaP(100): a Raman study. _Applied Surface Science_ 1996, _104-105_ , 575–579
  * Lovejoy et al. 2010 Lovejoy, T.; Yitamben, E.; Ohta, T.; Fain, S.; Ohuchi, F.; Olmstead, M. One-dimensional electronic states in Ga2Se3 on Si(001):As. _Physical Review B_ 2010, _81_ , 245313
  * Peressi and Baldereschi 1998 Peressi, M.; Baldereschi, A. Structural and electronic properties of Ga2Se3. _Journal of Applied Physics_ 1998, _83_ , 3092–3095
  * Ho and Chen 2014 Ho, C.-H.; Chen, H.-H. Optically decomposed near-band-edge structure and excitonic transitions in Ga2S3. _Scientific Reports_ 2014, _4_ , 6143
  * Toropov et al. 2020 Toropov, A.; Evropeitsev, E.; Nestoklon, M.; Smirnov, D.; Shubina, T.; Kaibyshev, V.; Budkin, G.; Jmerik, V.; Nechaev, D.; Rouvimov, S.; Ivanov, S.; Gil, B. Strongly Confined Excitons in GaN/AlN Nanostructures with Atomically Thin GaN Layers for Efficient Light Emission in Deep-Ultraviolet. _Nano Letteters_ 2020, _20_ , 158–165
  * Ohuchi et al. 1990 Ohuchi, F.; Parkinson, B.; Ueno, K.; Koma, A. van der Waals epitaxial growth and characterization of MoSe2 thin films on SnS2. _Journal of Applied Physics_ 1990, _68_ , 2168–2175
  * Koma 1999 Koma, A. van der Waals epitaxy for highly lattice-mismatched systems. _Journal of Crystal Growth_ 1999, _201_ , 236–241
  * Yuan et al. 2015 Yuan, X.; Tang, L.; Wang, P.; Chen, Z.; Zou, Y.; Su, X.; Zhang, C.; Liu, Y.; Wang, W.; Liu, C.; Chen, F.; Zou, J.; Zhou, P.; Hu, W.; Xiu, F. Wafer-scale arrayed p-n junctions based on few-layer epitaxial GaTe. _Nano Research_ 2015, _8_ , 3332
  * Sorokin et al. 2019 Sorokin, S.; Avdienko, P.; I.V. Sedova, D. K.; Yagovkina, M.; Smirnov, A.; Davydov, V.; Ivanov, S. Molecular-Beam Epitaxy of Two-Dimensional GaSe Layers on GaAs(001) and GaAs(112) Substrates: Structural and Optical Properties. _Semiconductors_ 2019, _53_ , 1131–1137
  * Sorokin et al. 2020 Sorokin, S. V.; Avdienko, P.; Sedova, I.; Kirilenko, D.; Davydov, V.; Komkov, O.; D.D. Firsov, D.; Ivanov, S. Molecular Beam Epitaxy of Layered Group III Metal Chalcogenides on GaAs(001) Substrates. _Materials_ 2020, _13_ , 3447
  * Ohtake and Sakuma 2021 Ohtake, A.; Sakuma, Y. Two-dimensional WSe2/MoSe2 heterostructures grown by molecular-beam epitaxy. _The Journal of Physical Chemistry C_ 2021, _125_ , 11257–11261
  * Singh and Gupta 2022 Singh, D. K.; Gupta, G. van der Waals epitaxy of transition metal dichalcogenides via molecular beam epitaxy: looking back and moving forward. _Materials Advances_ 2022, _3_ , 6142–6156
  * Komkov et al. 2020 Komkov, O.; Khakhulin, S.; Firsov, D.; Avdienko, P.; Sedova, I.; Sorokin, S. Investigation of Built-in Electric Fields at the GaSe/GaAs Interface by Photoreflectance Spectroscopy. _Semiconductors_ 2020, _54_ , 1198–1204
  * Dai and Ohuchi 1998 Dai, Z.; Ohuchi, F. Vacancy ordering of Ga2Se3 at GaSe/GaAs(100) interface. _Applied Physics Letters_ 1998, _73_ , 966–968
  * Teraguchi et al. 1991 Teraguchi, N.; Konagai, M.; Kato, F.; Takahashi, K. Growth and characterization of Ga2Se3 by molecular beam epitaxy. _Journal of Crystal Growth_ 1991, _115_ , 798–801
  * Teraguchi et al. 1991 Teraguchi, N.; Kato, F.; Konagai, M.; Takahashi, K.; Nakamura, Y.; Otuska, N. Vacancy ordering of Ga2Se3 films by molecular beam epitaxy. _Applied Physics Letters_ 1991, _59_ , 567–569
  * Okamoto et al. 1995 Okamoto, T.; Takegami, T.; Yamada, A.; Konagai, M. Control of the Arrangement of the Native Gallium Vacancies in Ga2Se3 on (100)GaAs by Molecular Beam Epitaxy. _Japanese Journal of Applied Physics_ 1995, _34_ , 5984–5988
  * Okamoto et al. 1993 Okamoto, T.; Konagai, M.; Kojima, N.; Yamada, A.; Takahashi, K.; Nakamura, Y.; Nittono, O. Anomalous anisotropy in the absorption coefficient of vacancy-ordered Ga2Se3. _Journal of Electronic Materials_ 1993, _22_ , 229–232
  * Nagel et al. 1979 Nagel, S.; Baldereschi, A.; Maschke, K. Tight-binding study of the electronic states in GaSe polytypes. _Journal of Physics C: Solid State Physics_ 1979, _12_ , 1625
  * Schlüter 1973 Schlüter, M. The Electronic Structure of GaSe. _Il Nuovo Cimento_ 1973, _3_ , 313
  * Segura 2018 Segura, A. Layered Indium Selenide under High Pressure: a Review. _Crystals_ 2018, _8_ , 206
  * Hoff 1975 Hoff, R. M. Raman scattering in GaSe. _The Journal of Physical Chemistry C_ 1975, _53_ , 17
  * Lei et al. 2013 Lei, S.; Ge, L.; Liu, Z.; Najmaei, S.; Shi, G.; You, G.; Lou, J.; Vajtai, R.; Ajayan, P. Synthesis and Photoresponse of Large GaSe Atomic Layers. _Nano Letters_ 2013, _13_ , 2777–2781
  * Ho 2020 Ho, C.-H. Ga2Se3 defect semiconductors: the study of direct band edge and optical properties. _ACS Omega_ 2020, _5_ , 18527–18534
  * Finkman et al. 1975 Finkman, E.; Tauc, J.; Kershaw, R.; Wold, A. Lattice dynamics of tetrahedrally bonded semiconductors containing ordered vacant sites. _Physical Review B_ 1975, _11_ , 3785
  * Kolodziejczyk et al. 92 Kolodziejczyk, M.; Filz, T.; Krost, A.; Richter, W.; Zahn, D. The likelihood of III$-2$–VI3 compound formation during epitaxial growth of II–VI on III–V semiconductors. _Journal of Crystal Growth_ 92, _117_ , 549–—553
  * Yamada et al. 1992 Yamada, A.; Kojima, N.; Takanashi, R. Raman study of epitaxial Ga2Se3 films grown by molecular beam epitaxy. _Japanese Journal of Applied Physics_ 1992, _31_ , L186
  * von der Emde et al. 1995 von der Emde, M.; Zahn, D.; Ng, T.; Maung, N.; Fan, G.; Poole, I.; Williams, J.; Wright, A. MOCVD growth of Ga2Se3 on GaAs(100) and GaP(100): a Raman study. _Applied Surface Science_ 1995, _104/105_ , 575–579
  * Irwin et al. 1973 Irwin, J.; Hoff, R.; Clayman, B.; Bromley, R. Long wavelength lattice vibrations in GaS and GaSe. _Solid State Communications_ 1973, _13_ , 1531–1536
  * Beechem et al. 2015 Beechem, T.; Kowalski, B.; Brumbach, M.; McDonald, A.; Spataru, C.; Howell, S.; Ohta, T.; Pask, J.; Kalugin, N. Oxidation of ultrathin GaSe. _Applied Physics Letters_ 2015, _107_ , 173103
  * Susoma et al. 2017 Susoma, J.; Lahtinen, J.; Kim, M.; Riikonen, J.; Lipsan, H. Crystal quality of two-dimensional gallium telluride and gallium selenide using Raman fingerprint. _AIP Advances_ 2017, _7_ , 015014
  * Newman 1962 Newman, P. Crystal structures of adamantine compounds. _Journal of Physics and Chemistry of Solids_ 1962, _23_ , 19
  * Budweg et al. 2019 Budweg, A.; Yadav, D.; Grupp, A.; Leitenstorfer, A.; Trushin, M.; Pauly, F.; Brida, D. Control of excitonic absorption by thickness variation in few-layer GaSe. _Physical Review B_ 2019, _100_ , 045404
  * Smirnov et al. 2018 Smirnov, D.; Belyaev, K.; Kirilenko, D.; Nestoklon, M. O.; Rakhlin, M. V.; Toropov, A.; Sedova, I.; Sorokin, S.; Ivanov, S.; Gil, B.; Shubina, T. Exciton Bound to 1D Intersections of Stacking Fault Planes with a ZnSe Quantum Well. _Physica Status Solidi-Rapid Research Letters_ 2018, _12_ , 1700410


# The Hubble Rate Trouble: An Effective Field Theory of Dark Matter

Alvaro S. de Jesus<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal do Rio Grande do Norte, 59078-970, Natal, RN, Brasil  Nelson Pinto-
Neto<EMAIL_ADDRESS>COSMO – Centro Brasileiro de Pesquisas Físicas, Xavier
Sigaud, 150, Urca, 22290-180, Rio de Janeiro, Brasil  Farinaldo S. Queiroz
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal do Rio
Grande do Norte, 59078-970, Natal, RN, Brasil  International Institute of
Physics, Universidade Federal do Rio Grande do Norte, 59078-970, Natal, RN,
Brasil  Millennium Institute for Subatomic Physics at High-Energy Frontier
(SAPHIR), Fernandez Concha 700, Santiago, Chile  Joseph Silk<EMAIL_ADDRESS>Institut d’Astrophysique de Paris (UMR7095: CNRS & UPMC- Sorbonne
Universities), F-75014, Paris, France  Department of Physics and Astronomy,
The Johns Hopkins University Homewood Campus, Baltimore, MD 21218, USA  BIPAC,
Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
Dêivid R. da Silva<EMAIL_ADDRESS>Departamento de Fisica,
Universidade Federal da Paraiba, 58051-970, João Pessoa, PB, Brasil

###### Abstract

The Hubble constant inferred from the 6-parameter fit to the CMB power
spectrum conflicts with the value obtained from direct measurements via type
Ia supernova and Cepheids observations. We write down effective operators
involving spin-0, spin-1/2, and spin-1 dark matter that lead to the
relativistic production of dark matter particles at early times, and
consequently lead to an increase in the number of relativistic degrees of
freedom. This mechanism which is amenable to CMB, BBN, and structure formation
observables can sufficiently raise the value of the Hubble constant derived
from CMB and reconcile local and CMB probes of the Hubble constant. This
mechanism alone increases $H_{0}$ up to $70\,{\rm kms^{-1}Mpc^{-1}}$, and with
the help of a Phantom-like cosmology, reach $H_{0}\simeq 71-73\,{\rm
kms^{-1}Mpc^{-1}}$. Lastly, we outline the region of parameter space which
reproduces $H_{0}\simeq 71-73\,{\rm kms^{-1}Mpc^{-1}}$ while obeying all
relevant constraints.

##  I  Introduction

The $\Lambda$CDM cosmological model is grounded in the idea that the universe
is nearly spatially flat, with its structures arising from quantum vacuum
fluctuations of cosmological perturbations from a highly homogeneous and
isotropic primordial era. Currently, the universe is dominated by dark energy
and cold dark matter Piattella (2018); Hobson  _et al._ (2006). This simple
description can nicely explain the abundance of light elements Cyburt  _et
al._ (2016), the CMB (Comic Microwave Background) power spectrum Aghanim  _et
al._ (2020a), the large scale structure as well as the ongoing accelerated
expansion era Collaboration (2020); Blanton  _et al._ (2017); Riess  _et al._
(1998); Perlmutter  _et al._ (1999), among others. Nevertheless, an important
discrepancy involving the Hubble constant surfaced. Considering the
$\Lambda$CDM model and CMB power spectrum, Planck data favors $H_{0}=67.27\pm
0.6$ km s-1 Mpc-1 Aghanim  _et al._ (2020a). However, adopting $\Lambda$CDM
model, quasar time-delay cosmography leads to $H_{0}=71.9^{+2.4}_{-3.0}$ Km
s-1 Mpc-1 Bonvin  _et al._ (2016). Parallax measurements of Cepheids provide
$H_{0}=73.24\pm 1.74$ km s-1 Mpc-1 Bernal  _et al._ (2016). In summary, early
measurements of the Hubble constant favor $H_{0}<69$ km s-1 Mpc-1, whereas
local measurements yield $H_{0}>71$ Km s-1 Mpc-1 Anchordoqui  _et al._ (2021).
The incompatibility found in the Hubble constant is known as the Hubble
tension, and its magnitude varies depending on the data set used. Table (1)
presents some of these measurements showing a discrepancy between late and
early universe data. Collectively speaking, it is clear that local
measurements do not agree with CMB inferred values for $H_{0}$.

EARLY UNIVERSE | Dataset
---|---
$H_{0}=70.0\pm 2.2kms^{-1}Mpc^{-1}$ | WMAP9 Hinshaw  _et al._ (2013)
$H_{0}=67.36\pm 0.54kms^{-1}Mpc^{-1}$ | CMB 2018 Aghanim  _et al._ (2020a)
$H_{0}==67.36\pm 0.54kms^{-1}Mpc^{-1}$ | SPT 2021 Dutcher  _et al._ (2021)
$H_{0}=69.72\pm 1.63kms^{-1}Mpc^{-1}$ | ACT 2019 Wang and Huang (2020)
$H_{0}=67.9\pm 1.1kms^{-1}Mpc^{-1}$ | BOSS data Ivanov  _et al._ (2020)
$H_{0}=69.6\pm 1.8kms^{-1}Mpc^{-1}$ | eBOSS Collab. Alam  _et al._ (2021)
LATE UNIVERSE | Dataset
$H_{0}=73.8\pm 2.1kms^{-1}Mpc^{-1}$ | SN1a 2021 Cardona  _et al._ (2017)
$H_{0}=75.4\pm 1.7kms^{-1}Mpc^{-1}$ | Pantheon 2019 Camarena and Marra (2020)
$H_{0}==72.8\pm 1.9kms^{-1}Mpc^{-1}$ | Gaia 2020 Breuval  _et al._ (2020)
$H_{0}=73.2\pm 1.3kms^{-1}Mpc^{-1}$ | Gaia and HST 2020 Riess  _et al._ (2021)
$H_{0}=69.8\pm 2.5kms^{-1}Mpc^{-1}$ | Red Giants 2019 Freedman  _et al._ (2019)
Table 1:  Early and late universe evaluations of the Hubble constant and their
respective data sets.

Studies raised systematic issues in the Planck analysis: One assumes the
$\Lambda$CDM model to infer $H_{0}$ from the CMB data. One first example is
the fact that Planck uses two different likelihood pipelines, Plik and
CamSpec, which consider different sky masks and could, in principle, shift by
$0.5\sigma$ the $H_{0}$ constraints coming from the CMB. More importantly, the
$A_{\rm lens}$ anomaly Calabrese  _et al._ (2008), which is a nonphysical
parameter equal to the unit if the gravitational lensing effects are the ones
predicted by the $\Lambda$CDM, and null if there is no lensing at all. Planck
collaboration sets $A_{\rm lens}>1$ at two standard deviations. As this
lensing anomaly has not been observed in the Planck trispectrum data, there is
still unknown small systematic error in the CMB data, which could reduce the
Hubble tension.

The late universe measurements of $H_{0}$ is direct, and come from measuring
the distance-redshift relation, in order words, the Hubble law. The most often
technique is parallax, i.e. use geometry to calibrate the luminosity of
pulsating Cepheid variables, for instance, which can be seen at great
distances and thus allow measurements of the cosmic expansion. We highlight
that this method treats such stars as standard candles.

Hence, once those stars are empirically standardized, the same type has the
same luminosity, without invoking any theoretical aspect. Though direct, this
type of measurement is much more subject to systematic errors than the early
universe measurements. Evidence of this fact is the last measurement presented
in table (1). It uses Red Giants instead of Cepheids or Supernovae to evaluate
the distance-redshift relation, and the value inferred for $H_{0}$ is a little
bigger but compatible with the early universe evaluations, with the error
bars. In many cases, the Red Giants and Cepheids used to obtain the different
data sets are located in the same galaxies, the discrepancy in the results
indicates the presence of a large systematic error in one or both of these
measurements. New observations using the James Webb telescope may solve this
issue in a couple of years, see a discussion of this conflict in Ref. Freedman
(2021).

This cosmological problem has triggered several solutions, see for recent
reviews Di Valentino  _et al._ (2021); Kamionkowski and Riess (2022). One
possibility to increase the Hubble constant inferred from CMB probes is to add
some amount of radiation at early times. A plausible way to accomplish this is
via the introduction of new light species that were in thermal equilibrium
much before CMB decoupling Anchordoqui and Goldberg (2012); Boehm  _et al._
(2012); Brust  _et al._ (2013). Such light species will contribute to the
number of degrees of relativistic degrees of freedom, $N_{eff}$, which is
positively correlated with the Hubble rate. An alternative way to increase
$N_{eff}$ is to introduce a relativistic production mechanism of dark matter
particles, which in turn mimic the effect of a neutrino species Hooper  _et
al._ (2012).

It is well known that dark matter cannot be relativistic at matter-radiation
equality for the sake of structure formation, and in this mechanism, dark
matter is not being converted into dark matter radiation after Big Bang
Nucleosynthesis Bringmann  _et al._ (2018). We are simply assuming that it was
produced relativistically, but later it became non-relativistic much before
Big Bang Nucleosynthesis or CMB decoupling. Therefore, this mechanism behaves
just like any other standard non-relativistic dark matter model at late times.
If a dark matter particle is produced relativisticaly it might be safely non-
relativistic depending on when it was produced and its initial kinetic energy,
as we will explain later. Anyway, this solution to the $H_{0}$ problem via
$N_{eff}$ has proven to be insufficient with the latest data from Planck and
new direct measurements of $H_{0}$. Within the $\Lambda$CDM one cannot find
$H_{0}>70kms^{-1}Mpc^{-1}$. Small deviations from the $\Lambda$CDM, however,
allow larger values Anchordoqui  _et al._ (2021).

In the $\Lambda$CDM, the dark energy equation of state, $p=w\rho$, has $w=-1$,
but in Phantom-like models $w<-1$ Caldwell (2002); Caldwell  _et al._ (2003);
Vikman (2005); Nojiri  _et al._ (2005); Shafer and Huterer (2014); Ludwick
(2017); Alestas  _et al._ (2020). This deviation in the equation of state
allows larger values of $H_{0}$ when global fits to the CMB spectrum are
performed Anchordoqui  _et al._ (2021). It is known that Phantom-like
cosmologies, that experience late times dark energy transitions at redshifts
$z\ll 0.1$ can raise the Hubble constant to values larger than $73{\rm
kms^{-1}Mpc^{-1}}$, while yielding equally good fit as $\Lambda$CDM at higher
redshift data, in particular from the cosmic microwave background and baryon
acoustic oscillations. Although, it faces some problems to raise $H_{0}$ to
large values when data from SHOES collaboration Riess  _et al._ (2022), which
consists in using Cepheid variables as intermediate calibrators, are accounted
for Benevento  _et al._ (2020). It has to do with the SNIa absolute magnitude
obtained in these data analyses, which disagree with the absolute magnitude
derived from SNIa, BAO and CMB data Alestas  _et al._ (2021); Camarena and
Marra (2021). Although, if a redshift dependence on the supernova absolute
magnitude is included, the Phantom-like solution to the $H_{0}$ problem
remains viable Alestas  _et al._ (2021); Camarena and Marra (2021). As
Phantom-like cosmology still stands as a plausible solution to the $H_{0}$
problem, we will consider it as our cosmological model, allowing us to connect
the increase in the Hubble rate with $N_{eff}$ in terms of dark radiation.

In this work, the dark radiation arises via the relativistic production of
dark matter that occurs through a decaying process, where a heavy particle
$(\chi^{\prime})$ decays into a dark matter particle $(\chi)$ plus a photon
$(\gamma)$. That decay adds an amount of hot dark matter which behaves as dark
radiation for a while but later becomes non-relativistic as its energy
decreases with the expansion. Hence, in this way, the production of dark
matter contributes to $N_{eff}$ and thus increasing $H_{0}$. As a side remark,
note that if the results of Refs. Freedman  _et al._ (2019); Freedman (2021)
discussed previously are correct, namely, that $H_{0}\approx
70kms^{-1}Mpc^{-1}$, this non-thermal production mechanism of dark matter
particles would be sufficient to reconcile CMB and late time measurements of
$H_{0}$ without appealing to any Phantom physics.

We will explain the mechanism in a model independent way, and later we write
down non-renormalizable operators encompassing spin 0, spin 1 and spin 1/2
dark matter particles that feature this non-thermal production of dark matter
particles. The important quantities are the masses of the particles and the
energy scale of the effective operator, $\Lambda$. With this at hand, we
delimit the region of parameter space iwhich offers a solution to the $H_{0}$
trouble.

This work is structured as follows: In section II we explain how the non-
thermal production of dark matter raises $H_{0}$; In section III, we present
effective operators that give rise to the $\chi^{\prime}\rightarrow\chi\gamma$
decay, and derive the corresponding decay width; In section IV we discuss the
results, before concluding in section V.

##  II Increase in relativistic energy density produced by dark matter

We are considering a radiation era where only photons and neutrinos are
relativistic. Therefore, the total energy density in this stage is,

$\rho=\rho_{rad}=\frac{\pi^{2}}{30}g_{*}T^{4},$ (1)

where $T$ is the temperature of the photons and $g_{*}$ is the total
relativistic degrees of freedom Piattella (2018). The factor $g_{*}$ gives,

$g_{*}=g_{\gamma}+\frac{7}{4}\,g_{\nu}N_{\nu}\left(\frac{T_{\nu}}{T_{\gamma}}\right)^{4}=2+\frac{}{}\frac{7}{4}\left(\frac{4}{11}\right)^{4/3}N_{\nu},$
(2)

where $g_{\gamma}=2$ indicates that photons have two polarization states,
$g_{\nu}=1$ informs that standard model neutrinos are only left-handed,
$T_{\nu}/T_{\gamma}=(4/11)^{1/3}$ is the ratio between the neutrinos and
photons temperature after the neutrinos decoupling Piattella (2018), and
$N_{\nu}$ is the number of neutrino flavors.

In the standard model, there are three neutrinos specie. Thus we expect
$N_{eff}$ to be close to three, not precisely three, because of some
temperature dependence. However, in non-standard cosmologies, we generally
write $N_{eff}=3+\Delta N_{eff}$, where $\Delta N_{eff}$ refers to the extra
number of relativistic degrees of freedom, which may come in sort of new light
species or other mechanisms that mimic this effect. Denoting the energy
density of a single standard model neutrino species as $\rho_{1\nu}$, we
define,

$\Delta N_{eff}=\frac{\rho_{extra}}{\rho_{1\nu}}\cdot$ (3)

Notice that the ratio between one neutrino species and cold dark matter energy
density in the matter-radiation equality is,

$\left.\frac{\rho_{1\nu}}{\rho_{CDM}}\right|_{t=t_{eq}}=\frac{\Omega_{\nu,0}\rho_{c}}{3a^{4}_{eq}}\times\left(\frac{\Omega_{CDM,0}}{a^{3}_{eq}}\right)^{-1}=0.16,$
(4)

where $\Omega_{\nu,0}=3.65\times 10^{-5}$, $\Omega_{CDM,0}=0.265$, and
$a_{eq}=3\times 10^{-4}$ Aghanim  _et al._ (2020b).

Consequently, one neutrino density energy is equivalent to $16\%$ of the cold
dark matter energy Alcaniz  _et al._ (2021) at the matter-radiation equality.
In other words, if a fraction of dark matter is relativistic at that time, it
can contribute to the energy density just like a neutrino species. That said,
we consider a heavy particle $\chi^{\prime}$ which decays in the radiation era
in two particles, dark matter $(\chi)$ and a photon $(\gamma)$. We also assume
that $m_{\chi^{\prime}}\gg m_{\chi}$, because we need dark matter to be
produced relativistically so it can mimic the effect of a neutrino species. We
avoid problems with structure formation by assuming that only a small fraction
of dark matter particles are produced in this way Alcaniz  _et al._ (2021). We
will now devote some time explaining how this decaying process can generate
$\Delta N_{eff}\neq 0$.

In the $\chi^{\prime}$ resting frame, the four-momentum of the particles are,

$\displaystyle p_{\chi^{\prime}}=(m_{\chi^{\prime}},\bm{0}),$ $\displaystyle
p_{\chi}=(E_{\chi}(\bm{p}),\bm{p}),$ $\displaystyle
p_{\gamma,\,\nu}=(|\bm{p}|,-\bm{p}).$

Imposing four-momentum conservation we obtain,

$\displaystyle\left|\bm{p}_{\chi}(\tau)\right|=\left|\bm{p}\right|=\frac{1}{2}m_{\chi^{\prime}}\left[1-\left(\frac{m_{\chi}}{m_{\chi^{\prime}}}\right)^{2}\right],$
(5) $\displaystyle
E_{\chi}(\tau)=m_{\chi}\left(\frac{m_{\chi^{\prime}}}{2m_{\chi}}+\frac{m_{\chi}}{2m_{\chi^{\prime}}}\right),$
(6)

where $\tau$ is the $\chi^{\prime}$ lifetime. Note that the equation above
refers to the energy and momentum at the moment immediately after the decay.
Hence, the Lorentz factor is,

$\gamma_{\chi}(\tau)=\left(\frac{m_{\chi^{\prime}}}{2m_{\chi}}+\frac{m_{\chi}}{2m_{\chi^{\prime}}}\right),$
(7)

where $E_{\chi}(\tau)=m_{\chi}\gamma_{\chi}(\tau)$.

The momentum of the particle is inversely proportional to the scale factor,
thus $\bm{p}^{2}_{\chi}\propto\frac{1}{a^{2}}$, which implies in,

$\begin{split}&E^{2}_{\chi}-m^{2}_{\chi}=\bm{p}^{2}_{\chi}\propto\frac{1}{a^{2}}\\\
&\Rightarrow\left(E^{2}_{\chi}(t)-m^{2}_{\chi}\right)a^{2}(t)=\left(E^{2}_{\chi}(\tau)-m^{2}_{\chi}\right)a^{2}(\tau)\\\
&\Rightarrow
E_{\chi}(t)=m_{\chi}\left[1+\left(\frac{a(\tau)}{a(t)}\right)^{2}\left(\gamma^{2}_{\chi}(\tau)-1\right)\right]^{1/2}.\end{split}$
(8)

From Eq.10 we can extract the Lorentz factor for the dark matter particles at
a given time $t$. Since we are considering a phase where the universe is
radiation dominated, we can substitute $a(\tau)/a(t)$ for $\sqrt{\tau/t}$ and
find the Lorentz factor Hobson  _et al._ (2006),

$\gamma_{\chi}(t)=\sqrt{\frac{(m^{2}_{\chi}-m^{2}_{\chi^{\prime}})^{2}}{4m^{2}_{\chi}m^{2}_{\chi^{\prime}}}\left(\frac{\tau}{t}\right)+1}.$
(9)

A particle in the non-relativistic regime has its mass as the mean
contribution of the total energy. Hence, the dark matter energy can be written
as,

$E_{\chi}=m_{\chi}(\gamma_{\chi}-1)+m_{\chi}.$ (10)

This equation provides a direct interpretation of $m_{\chi}(\gamma_{\chi}-1)$
as the mean contribution part of the particle energy in the ultrarelativistic
regime. Therefore, the total energy of dark matter particles is given by the
energy of the cold dark matter plus the energy of the relativistically
produced dark matter component,

$E_{DM}=N_{HDM}m_{\chi}(\gamma_{\chi}-1)+N_{CDM}m_{\chi},$ (11)

where $N_{HDM}$ and $N_{CDM}$ are the number of hot and cold dark matter
particles, respectively. To avoid conflicts with results from standard
cosmology, we enforce $N_{HDM}\ll N_{CDM}$, as will explain further.

The ratio between the hot and cold dark matter energy density is

$\frac{\rho_{HDM}}{\rho_{CDM}}=\frac{n_{HDM}m_{\chi}\left(\gamma_{\chi}-1\right)}{n_{CDM}m_{\chi}}\equiv
f\left(\gamma_{\chi}-1\right),$ (12)

where $n_{HDM}$ and $n_{CDM}$ are the number density of relativistic and
nonrelativistic produced dark matter particles, respectively. The factor $f$
is the ratio between these two number densities and it must be small. Here, we
consider $f=0.01$, which is an upper limit obtained from structure formation
Kelso  _et al._ (2013). Note that this relativistically produced dark matter
will be eventually cold, i.e. with $E_{\chi}\sim m_{\chi}$ shortly after
structure formation begins.

We assume that the extra source of radiation in (3) is the dark matter
particles in a hot stage. Using Eq.3, we get,

$\Delta
N_{eff}=\frac{\rho_{HDM}}{\rho_{1\nu}}=\frac{\rho_{CDM}f(\gamma_{\chi}-1)}{\rho_{1\nu}}\cdot$
(13)

Our next step is to calculate this expression at matter-radiation equality,
where $\rho_{CDM}/\rho_{1\nu}=1/0.16$,

$\Delta N_{eff}=\lim_{t\to
t_{eq}}\frac{f\left(\gamma_{\chi}-1\right)}{0.16}\cdot$ (14)

In the limit $m_{\chi^{\prime}}\gg m_{\chi}$, we can simplify Eq.(14) to,

$\gamma_{\chi}(t_{eq})-1\approx\gamma_{\chi}(t_{eq})\approx\frac{m_{\chi^{\prime}}}{2m_{\chi}}\sqrt{\frac{\tau}{t_{eq}}}\cdot$
(15)

which leads to,

$\Delta N_{eff}\approx 2.5\times 10^{-3}\sqrt{\frac{\tau}{10^{6}s}}\times
f\frac{m_{\chi^{\prime}}}{m_{\chi}},$ (16)

where we used $t_{eq}\approx 50000\leavevmode\nobreak\ \text{years}\approx
1.6\times 10^{12}\leavevmode\nobreak\ s$ Tong (2019).

The $\Delta N_{eff}$ is a function of four parameters: (i) the lifetime and
(ii) the mass of $\chi^{\prime}$; (iii) the mass of $\chi$; (iv) the fraction
of hot dark matter particles $(f)$, that we assume to be $0.01$. We will
address this assumption in the next section.

Figure 1: Allowed regions of parameters that connect our mechanism and the
value of Hubble constant in phantom-like cases. The first row corresponds to
the $\Lambda CDM$ model, and in the second and third rows a phantom-like
quintessence is introduced, first in a spatially flat model, then with non-
null spatial curvature. The second column presents the cases corresponding to
a non-zero $\Delta N_{eff}$. The data set that connects $\Delta N_{eff}$ and
$H_{0}$ showed in (a), (c), and (e) is taken from Anchordoqui  _et al._
(2021). In all figures, the lighter regions correspond to $99\%$ of CL, while
the darkest regions correspond to $68\%$ of CL. In (b), (d), and (f) the
orange, blue and gray regions correspond to the cases where $\chi^{\prime}$
lifetime is $10^{2}s$, $10^{3}s$ and $10^{4}s$ respectively. The bounds use
Planck 2018 CMB data, BAO, and type Ia data from the Pantheon sample.

As aforementioned, Planck collaboration has reported that $N_{eff}$ and
$H_{0}$ are positively correlated. Therefore, we can use Eq.(16) to connect
$H_{0}$ with $fm_{\chi^{\prime}}/m_{\chi}$ for a given lifetime.

Assuming that the Hubble constant measured locally should indeed be larger
than $70\,\text{Km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$, one can conclude that
the $\Lambda CDM$ model does not suffice Anchordoqui  _et al._ (2021). It is
necessary to consider non-standard cosmological scenarios Anchordoqui  _et
al._ (2021). Here we will consider Phantom-like models Yang  _et al._ (2018).
We use the positive correlation between $H_{0}$ and $N_{eff}$ found in
Anchordoqui  _et al._ (2021), and derive the allowed values of $H_{0}$ for
choices of the product $fm_{\chi^{\prime}}/m_{\chi}$ for $\tau=10^{2}$ s,
$10^{3}$ s, and $10^{4}$ s.

Phantom-like cosmologies alone allow $H_{0}$ values larger than
$70\,\text{Km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$, and consequently can solve
the $H_{0}$ trouble. This explains why in Fig.1 $fm_{\chi^{\prime}}/m_{\chi}$
can go to zero. However, if we adopted local measurements pointing to
$H_{0}\leq 70\,\text{Km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$, there would be no
need for a Phantom-like cosmology because our mechanism of non-thermal
production of dark matter particles can yield $N_{eff}=3.3$ and, consequently,
$H_{0}=70\,\text{Km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$ Anchordoqui  _et al._
(2021). In the middle panel of Fig.1 we assumed null curvature and in the
bottom a non-zero curvature. The difference in the parameter space is mild.
Thus, we can safely say that with or without curvature our work can solve the
$H_{0}$ trouble.

We stress that the advantage of our mechanism is the interplay between
particle physics and cosmology. Instead of relying simply on a cosmological
model such as Phantom-like cosmology, our idea invokes a connection to the
dark matter density and to the production mechanism of dark matter particles.
As far as typical direct detection searches go Aprile  _et al._ (2022), dark
matter particles with a non-thermal origin in the early universe produce no
effect on the scattering rate observed today. However, if dark matter
particles experienced in the early universe a non-thermal production, the
parameter space probed by direct detection experiments, in terms of mass and
coupling of given model, changes. Hence, If only a small fraction of dark
matter is produced non-thermally, this brings no impact to the typical direct
detection or accelerator searches for dark matter particles Duarte  _et al._
(2022). Be that as it may, this small fraction might serve an interesting
purpose in cosmology, a solution to the $H_{0}$ problem. We will now address
some important cosmological aspects of this non-thermal production mechanism
in the early universe. We start discussing structure formation and Big Bang
Nucleosynthesis.

##  III Relevant Bounds

###  III.1 Big Bang Nucleosynthesis

When electromagnetic energy is injected in to the universe through non-thermal
processes as the one we are considering Double Compton scattering ($\gamma
e^{-}\rightarrow\gamma\gamma e^{-}$), and bremsstrahlung ($e^{-}X\rightarrow
e^{-}X\gamma$) may alter the CMB spectrum Feng  _et al._ (2003, 2004),
relaxing it to a Bose-Einstein distribution function with chemical potential
different from zero. Given the existing upper limit on the chemical potential,
we can limit the energy injection at a given time. The bounds are rather
stringent, but for $\tau>10^{4}$ s. In our work, we will focus on the region
of parameter space in which $\tau<10^{4}$ s, to avoid conflicts with BBN
Hooper  _et al._ (2012); Alcaniz  _et al._ (2022).

###  III.2 Structure Formation

Galaxy cluster observations restrict the amount of hot dark matter in the
universe. Hot dark matter is typically treated as massive neutrinos. Those
studies limit the fraction of hot dark matter in the universe,
$\Omega_{HDM}/\Omega_{CDM}$ to be less than $0.01$. For this reason, we will
consider $f=0.01$. Notice that we are being very conservative by taking this
bound at face value, because the dark matter particles can be heavy,
conversely to neutrinos. Therefore, its free-streaming evolves differently. A
more robust calculation would have to be derive a more precise constraint.

###  III.3 Energy Evolution of Dark Matter

In our formalism, a fraction of dark matter particles are created in a hot
stage. But it is important that at matter-radiation equality time their
kinetic energy had been lost, due to the expansion of the universe. The
evolution of the dark matter particles are computed using Eq.(10). Therefore,
we can assess whether the dark matter particles produced this way are non-
relativistic, i.e, $E_{\chi}\sim m_{\chi}$, at the matter-radiation equality.
Focusing only on the region of parameter space which solves the $H_{0}$ we
compute the dark matter energy at the matter-radiation equality.

Taking $\tau\sim 10^{2}s-10^{4}s$ and $m_{\chi^{\prime}}/m_{\chi}\sim
10^{4}-10^{6}$, which is within the region of interest to solve the $H_{0}$
problem, we show in Fig. 2 that the dark matter particles become non-
relativistic at matter-radiation equality for
$m_{\chi^{\prime}}/m_{\chi}=10^{4}$. In Fig. 2 it is shown that for
$m_{\chi^{\prime}}/m_{\chi}=10^{6}$ dark matter particles are still
relativistic at matter-radiation equality. Enforcing the dark matter particles
to be cold at $t_{eq}$ we find a upper limit on the mass ration
$m_{\chi^{\prime}}/m_{\chi}$. We emphasize that this result is independent of
$f$. We highlight that the choices for the parameter in the figures solve the
$H_{0}$ discrepancy. Despite the energy of dark mater being independent of
$f$, we needed to assume $f$ to be small to reproduce the correct value of
$H_{0}$. Therefore, changing $f$ means changing the lifetime and mass ratio
that yields the correct $H_{0}$. That would consequently change the curves in
Fig. 2 at matter-radiation equality for $m_{\chi^{\prime}}/m_{\chi}=10^{4}$.
Anyway, in Fig. 2, our mechanism goes in the direction of a mixed cold+hot
dark matter scenario, which may solve some small scale problems appearing in
purely cold dark matter simulations Klypin  _et al._ (1993); Bose  _et al._
(2019); Lopez-Corredoira and Marmet (2022); Lahav and Liddle (2022); Workman
_et al._ (2022).

Figure 2: Time evolution of dark matter energy. We consider situations where
dark matter mother lifetime is $\tau=\\{10^{2}s,10^{3}s,10^{4}s\\}$ and the
ratio between dark matter mother and dark matter mass is
$m_{\chi^{\prime}}/m_{\chi}=\\{10^{4},10^{6}\\}$. In all situations of (a),
dark matter is cold at matter-radiation equality $(t_{eq})$, while in (b) dark
matter is hot in all scenarios.

In summary, our mechanism does not alter the CMB, BBN or structure formation
prediction for the region of interest. We now move to a more particle physics-
oriented section. Having in mind that this decay
$\chi^{\prime}\rightarrow\chi+\gamma$ can solve the $H_{0}$ problem, we write
down effective operators that feature this decay to determine the energy scale
$\Lambda$ at which the $H_{0}$ can be solved through non-thermal production of
dark matter particles.

##  IV Effective theory of Dark Matter

As the nature of dark matter is unknown, we will consider three effective
operators of dimension five covering spin-0, spin-1 and spin-1/2 dark matter
particles for the decay process $\chi^{\prime}\rightarrow\chi+\gamma$. The
corresponding Feynmann diagrams are displayed in Fig. 3.

$\chi^{\prime}$$\chi$$\gamma$

$\chi^{\prime}$$\chi$$\gamma$

$\chi^{\prime}$$\chi$$\gamma$

Figure 3: Diagrammatic representation of a heavy particle $(\chi^{\prime})$
that decay in hot dark matter $(\chi)$ and photon $(\gamma)$. Three cases are
considered: (a) $\chi^{\prime}$ is spin-1 and $\chi$ is a spin-0 particle; (b)
$\chi^{\prime}$ and $\chi$ are spin-1/2 particles; (c) $\chi^{\prime}$ is
spin-0 and $\chi$ is a spin-1 particle.

A two-body decaying rate is given by Griffiths (2020),

$\Gamma(\chi^{\prime}\rightarrow\chi+\gamma)=\frac{\left|\bm{p}_{\chi}(\tau)\right|}{8\pi
m^{2}_{\chi^{\prime}}}\left|\mathcal{M}\right|^{2},$ (17)

where $\mathcal{M}$ is the invariant amplitude. After plugging in the
kinematics given in Eq. (5), we get,

$\Gamma=\frac{1}{16\pi
m_{\chi^{\prime}}}\left[1-\left(\frac{m_{\chi}}{m_{\chi^{\prime}}}\right)^{2}\right]\left|\mathcal{M}\right|^{2}.$
(18)

We will use this general expression to calculate the lifetime $\tau=1/\Gamma$
for three different effective operators presented below.

###  IV.1 Decay in spin-0 dark matter and photon

In the first case we assume that $\chi^{\prime}$ is a spin-1, $\chi$ is a
spin-0 particle, and the effective Lagrangian describing this decay
$\chi^{\prime}\rightarrow\chi+\gamma$ is,

$\mathcal{L}_{eff}=\frac{1}{\Lambda}\phi_{\chi}\chi^{\prime}_{\mu\nu}F^{\mu\nu},$
(19)

where $\Lambda$ is an energy scale to be determined later. Note that
$\chi^{\prime}_{\mu\nu}\equiv\partial_{\mu}\chi^{\prime}_{\nu}-\partial_{\nu}\chi^{\prime}_{\mu}$.
The Feynman diagram for this process is shown in Fig. 3, which results in,

$\left|\mathcal{M}\right|^{2}=\frac{2m^{4}_{\chi^{\prime}}}{3\Lambda^{2}}\left[1-\left(\frac{m_{\chi}}{m_{\chi^{\prime}}}\right)^{2}\right].$
(20)

Substituting this result in Eq. (18) we obtain,

$\Gamma=\frac{m^{3}_{\chi^{\prime}}}{24\pi\Lambda^{2}}\left[1-\left(\frac{m_{\chi}}{m_{\chi^{\prime}}}\right)^{2}\right]^{3}\approx\frac{m^{3}_{\chi^{\prime}}}{24\pi\Lambda^{2}}\cdot$
(21)

Therefore, the lifetime is set by $\Lambda$ and $m_{\chi^{\prime}}$. We
exhibit this relation in Fig. 4 for $\tau=10^{2}\leavevmode\nobreak\
s,10^{3}\leavevmode\nobreak\ s,10^{4}$ s. As $\Delta N_{eff}$ is now a
function of $m_{\chi^{\prime}}$, $m_{\chi}$ and $\Lambda$, we can play with
those quantities to outline the region of parameter space that solves the
$H_{0}$ trouble exploiting its correlation with $\Delta N_{eff}$. In Fig. 4 we
set $m_{\chi}/m_{\chi}=10^{4}$ and show the values of $\Lambda$ which yield
$\Delta N_{eff}=0.1-0.6$ and lead to $H_{0}\sim 70-72{\rm kms^{-1}Mpc^{-1}}$
according to Fig. 1. We would like to stress once more that if local
measurement converge to $H_{0}\sim 70{\rm kms^{-1}Mpc^{-1}}$, our mechanism
alone is sufficient to solve the discrepancy on $H_{0}$, as can be seen in
Fig. 1.

Figure 4: Plot of $\Lambda$ as a function of the $\chi$ and $\chi^{\prime}$
masses for the case where $\chi^{\prime}$ is a spin-1, $\chi$ is a spin-0
particle. (a) $\Lambda\times m_{\chi^{\prime}}$ curves built from Eq. (21),
using $\tau=10^{2},10^{3}\,\text{and}\,10^{4}\,\text{s}$. (b) $\Lambda\times
m_{\chi}$ curves constructed from Eqs. (16) and (21). We consider the cases
where $\Delta N_{eff}=0.1-0.6$, with $f=0.01$ and
$m_{\chi^{\prime}}/m_{\chi}=10^{4}$.

###  IV.2 Decay in spin-1/2 dark matter and photon

In the second possibility, we consider that $\chi^{\prime}$ and $\chi$ are
spin-1/2 fermions, and the effective theory to describe the decay
$\chi^{\prime}\rightarrow\chi+\gamma$ is

$\mathcal{L}_{eff}=\frac{1}{\Lambda}\bar{\psi}_{\chi}\sigma^{\mu\nu}\psi_{\chi^{\prime}}F_{\mu\nu}+h.c.,$
(22)

where $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$. The
corresponding Feynman amplitude is,

$\left|\mathcal{M}\right|^{2}=\frac{8m_{\chi^{\prime}}^{4}}{\Lambda^{2}}\left[1-\left(\frac{m_{\chi}}{m_{\chi^{\prime}}}\right)^{2}\right]^{2}.$
(23)

Using Eq. (18) we find,

$\Gamma=\frac{m^{3}_{\chi^{\prime}}}{2\pi\Lambda^{2}}\left[1-\left(\frac{m_{\chi}}{m_{\chi^{\prime}}}\right)^{2}\right]^{3}\approx\frac{m^{3}_{\chi^{\prime}}}{2\pi\Lambda^{2}}\cdot$
(24)

In a similar vein, we use Eq.(24) to plot the relation between $\Lambda$ and
$m_{\chi}$ for $\tau=10^{2}\leavevmode\nobreak\ s,10^{3}\leavevmode\nobreak\
s,10^{4}\leavevmode\nobreak\ s$ in Fig. 5. Moreover, we derive the energy
scale $\Lambda$ that reproduces $\Delta_{eff}=0.1-0.6$ and can lead to a
solution to the $H_{0}$ trouble in Fig. 5, assuming
$m_{\chi^{\prime}}/m_{\chi}=10^{4}$, and $f=0.01$.

Figure 5: Plot of $\Lambda$ as a function of the $\chi$ and $\chi^{\prime}$
masses for the case where $\chi^{\prime}$ and $\chi$ are spin-1/2 particles.
(a) $\Lambda\times m_{\chi^{\prime}}$ curves built from Eq. (24), using
$\tau=10^{2},10^{3}\,\text{and}\,10^{4}\,\text{s}$. (b) $\Lambda\times
m_{\chi}$ curves constructed from Eqs. (16) and (24). We consider the cases
where $\Delta N_{eff}=0.1-0.6$, with $f=0.01$ and
$m_{\chi^{\prime}}/m_{\chi}=10^{4}$.

###  IV.3 Decay in spin-1 dark matter and photon

Lastly, we take $\chi^{\prime}$ to be a spin-0 particle, $\chi$ is a spin-1/2
fermion, which is described by the effective operator,

$\mathcal{L}_{eff}=\frac{1}{\Lambda}\phi_{\chi^{\prime}}\chi_{\mu\nu}F^{\mu\nu},$
(25)

where $\chi_{\mu\nu}\equiv\partial_{\mu}\chi_{\nu}-\partial_{\nu}\chi_{\mu}$,
which yields,

$\left|\mathcal{M}\right|^{2}=\frac{2m^{4}_{\chi^{\prime}}}{\Lambda^{2}}\left[1-\left(\frac{m_{\chi}}{m_{\chi^{\prime}}}\right)^{2}\right]^{2},$
(26)

and,

$\Gamma=\frac{m^{3}_{\chi^{\prime}}}{8\pi\Lambda^{2}}\left[1-\left(\frac{m_{\chi}}{m_{\chi^{\prime}}}\right)^{2}\right]^{3}\approx\frac{m^{3}_{\chi^{\prime}}}{8\pi\Lambda^{2}}\cdot$
(27)

The region of parameter that results in $\tau=10^{2},10^{3},10^{4}$ s are is
shown in Fig. 6, and the parameter space that may present a solution to the
$H_{0}$ problem is displayed in Fig. 6, setting
$m_{\chi^{\prime}}/m_{\chi}=10^{4}$, and $f=0.01$.

Figure 6: Plot of $\Lambda$ as a function of the $\chi$ and $\chi^{\prime}$
masses for the case where $\chi^{\prime}$ has spin-0 and $\chi$ is a spin-1
particle. (a) $\Lambda\times m_{\chi^{\prime}}$ curves built from Eq. (27),
using $\tau=10^{2},10^{3}\,\text{and}\,10^{4}\,\text{s}$. (b) $\Lambda\times
m_{\chi}$ curves constructed from Eqs. (16) and (27). We consider the cases
where $\Delta N_{eff}=0.1-0.6$, with $f=0.01$ and
$m_{\chi^{\prime}}/m_{\chi}=10^{4}$.

##  V Results and discussions

All the effective theories considered here feature
$\tau\propto\Lambda^{2}/m_{\chi^{\prime}}^{3}$. Hence, the larger the mass of
the mother particle the shorter the lifetime. This outcome is not new. Indeed,
searches for gamma-rays and x-rays resulted from long-lived particles have
been conducted exploiting Baring  _et al._ (2016). It allows us to place a
lower mass limit on its mass. To warrant a long-lived $\chi^{\prime}$, we need
to invoke a large $\Lambda$. This explains the large energy scale, $\Lambda$,
as shown in the figures 4, 5 and 6. It does not come as a surprise, because
long-lived particles are related to some suppression mechanism, either present
in the coupling constant or the energy scale. In our work, it is the latter
Mambrini  _et al._ (2016).

Noticed that the larger the effective energy scale, $\Lambda$ the longer the
lifetime. Consequently, larger values of $\Delta N_{eff}$ are found. Moreover,
the larger $m_{\chi^{\prime}}$ the smaller the lifetime. However, the larger
$m_{\chi^{\prime}}$ the larger $\Delta N_{eff}$. Hence, there are two
competing effects happening as we change $m_{\chi^{\prime}}$. Anyway, notice
that regardless of the spin of the particles involved, non-thermally produced
dark matter particles with masses at the electroweak scale can solve the
$H_{0}$ discrepancy in agreement with BBN, CMB, and structure formation
constraints.

Interestingly, these late-decaying particles producing dark matter appear in
UV complete models Feng  _et al._ (2003, 2004); Wang and Yang (2004);
Cembranos  _et al._ (2005); Kelso  _et al._ (2014). An exciting outcome of our
work is the correlation between particle physics, early, and late-time
cosmology.

##  VI  conclusions

In this work, we constructed non-renomalizable operators involving spin-0,
1/2, and 1 dark matter particles produced non-thermally via the decay of a
heavy companion through the process $\chi^{\prime}\rightarrow\chi+\gamma$.
These dark matter particles are produced relativistically at the decay time.
However, their energy decreases with the redshift, and they become essentially
non-relativistic at matter-radiation equality for the sake of structure
formation. This relativistic behavior of dark matter particles early on mimics
the effect of an extra degree of freedom that helps reconcile early and late
measurements of the Hubble constant. Depending on the local value adopted for
the $H_{0}$, our mechanism might solve the $H_{0}$ trouble within the
$\Lambda$CDM model, without evoking new dark energy densities. If $H_{0}$
turns out to be larger than $70kms^{-1}Mpc^{-1}$, then indeed a new equation
of state for the dark energy component is needed. Assuming that
$H_{0}>70kms^{-1}Mpc^{-1}$, under a Phantom-like cosmology, we showed that the
typical energy scale governing the decay $\chi^{\prime}\rightarrow\chi+\gamma$
should range from $10^{17}$ GeV to $10^{25}$ GeV depending on the spin nature
of the dark matter particle and the mass ratio $m_{\chi^{\prime}}/m_{\chi}$.
Such large energy scales are natural, as one needs to invoke a large
suppression mechanism to have a long-lived particle with a lifetime larger
than $10^{2}$ s. Our work shows that perhaps the solution to the $H_{0}$
trouble might reside in the production mechanism of dark matter particles or a
combination of both dark energy and dark matter components amenable to BBN,
CMB, and structure formation observables.

###### Acknowledgements.

ASJ acknowledges support from Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior (CAPES) under Grant No. 88887.497142/2020-00. D. R. da Silva
thanks for the support from CAPES under grant 88887.462084/2019-00. FSQ
acknowledges the Simons Foundation (Award Number:884966, AF), FAPESP grant
2021/01089-1, ICTP-SAIFR FAPESP grants 2021/14335-0, CNPq grant 307130/2021-5,
and ANID- Programa Milenio - code ICN2019_ 044. N.P.N. acknowledges the
support of CNPq of Brazil under grant PQ-IB 310121/2021-3.

## References

  * Piattella (2018) O. Piattella,  _Lecture notes in cosmology_ (Springer, 2018).
  * Hobson  _et al._ (2006) M. P. Hobson, G. P. Efstathiou, and A. N. Lasenby,  _General relativity: an introduction for physicists_ (Cambridge University Press, 2006).
  * Cyburt  _et al._ (2016) R. H. Cyburt, B. D. Fields, K. A. Olive, and T.-H. Yeh, Rev. Mod. Phys. 88, 015004 (2016), arXiv:1505.01076 [astro-ph.CO] .
  * Aghanim  _et al._ (2020a) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020a), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Collaboration (2020) D. E. S. Collaboration, Phys. Rev. D 102, 023509 (2020), arXiv:2002.11124 [astro-ph.CO] .
  * Blanton  _et al._ (2017) M. R. Blanton _et al._ , The Astronomical Journal 154, 28 (2017), arXiv:1703.00052 [astro-ph.GA] .
  * Riess  _et al._ (1998) A. G. Riess _et al._ , The Astronomical Journal 116, 1009 (1998), arXiv:astro-ph/9805201 .
  * Perlmutter  _et al._ (1999) S. Perlmutter _et al._ , Astrophys.J. 517, 565 (1999), arXiv:astro-ph/9812133 .
  * Bonvin  _et al._ (2016) V. Bonvin, F. Courbin, S. H. Suyu, P. J. Marshall, C. E. Rusu, D. Sluse, M. Tewes, K. C. Wong, T. Collett, C. D. Fassnacht, T. Treu, M. W. Auger, S. Hilbert, L. V. E. Koopmans, G. Meylan, N. Rumbaugh, A. Sonnenfeld, and C. Spiniello, Monthly Notices of the Royal Astronomical Society 465, 4914 (2016), https://academic.oup.com/mnras/article-pdf/465/4/4914/10254866/stw3006.pdf .
  * Bernal  _et al._ (2016) J. L. Bernal, L. Verde, and A. G. Riess, JCAP 10, 019 (2016), arXiv:1607.05617 [astro-ph.CO] .
  * Anchordoqui  _et al._ (2021) L. A. Anchordoqui, E. Di Valentino, S. Pan, and W. Yang, JHEAp 32, 28 (2021), arXiv:2107.13932 [astro-ph.CO] .
  * Hinshaw  _et al._ (2013) G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, and E. L. Wright, The Astrophysical Journal Supplement Series 208, 19 (2013).
  * Dutcher  _et al._ (2021) D. Dutcher _et al._ (SPT-3G), Phys. Rev. D 104, 022003 (2021), arXiv:2101.01684 [astro-ph.CO] .
  * Wang and Huang (2020) K. Wang and Q.-G. Huang, JCAP 06, 045 (2020), arXiv:1912.05491 [astro-ph.CO] .
  * Ivanov  _et al._ (2020) M. M. Ivanov, M. Simonović, and M. Zaldarriaga, JCAP 05, 042 (2020), arXiv:1909.05277 [astro-ph.CO] .
  * Alam  _et al._ (2021) S. Alam _et al._ (eBOSS), Phys. Rev. D 103, 083533 (2021), arXiv:2007.08991 [astro-ph.CO] .
  * Cardona  _et al._ (2017) W. Cardona, M. Kunz, and V. Pettorino, JCAP 03, 056 (2017), arXiv:1611.06088 [astro-ph.CO] .
  * Camarena and Marra (2020) D. Camarena and V. Marra, Phys. Rev. Res. 2, 013028 (2020), arXiv:1906.11814 [astro-ph.CO] .
  * Breuval  _et al._ (2020) L. Breuval _et al._ , Astron. Astrophys. 643, A115 (2020), arXiv:2006.08763 [astro-ph.SR] .
  * Riess  _et al._ (2021) A. G. Riess, S. Casertano, W. Yuan, J. B. Bowers, L. Macri, J. C. Zinn, and D. Scolnic, Astrophys. J. Lett. 908, L6 (2021), arXiv:2012.08534 [astro-ph.CO] .
  * Freedman  _et al._ (2019) W. L. Freedman _et al._ ,  (2019), 10.3847/1538-4357/ab2f73, arXiv:1907.05922 [astro-ph.CO] .
  * Calabrese  _et al._ (2008) E. Calabrese, A. Slosar, A. Melchiorri, G. F. Smoot, and O. Zahn, Phys. Rev. D 77, 123531 (2008), arXiv:0803.2309 [astro-ph] .
  * Freedman (2021) W. L. Freedman, Astrophys. J. 919, 16 (2021), arXiv:2106.15656 [astro-ph.CO] .
  * Di Valentino  _et al._ (2021) E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Class. Quant. Grav. 38, 153001 (2021), arXiv:2103.01183 [astro-ph.CO] .
  * Kamionkowski and Riess (2022) M. Kamionkowski and A. G. Riess, (2022), arXiv:2211.04492 [astro-ph.CO] .
  * Anchordoqui and Goldberg (2012) L. A. Anchordoqui and H. Goldberg, Phys. Rev. Lett. 108, 081805 (2012), arXiv:1111.7264 [hep-ph] .
  * Boehm  _et al._ (2012) C. Boehm, M. J. Dolan, and C. McCabe, JCAP 12, 027 (2012), arXiv:1207.0497 [astro-ph.CO] .
  * Brust  _et al._ (2013) C. Brust, D. E. Kaplan, and M. T. Walters, JHEP 12, 058 (2013), arXiv:1303.5379 [hep-ph] .
  * Hooper  _et al._ (2012) D. Hooper, F. S. Queiroz, and N. Y. Gnedin, Phys. Rev. D 85, 063513 (2012), arXiv:1111.6599 [astro-ph.CO] .
  * Bringmann  _et al._ (2018) T. Bringmann, F. Kahlhoefer, K. Schmidt-Hoberg, and P. Walia, Phys. Rev. D 98, 023543 (2018), arXiv:1803.03644 [astro-ph.CO] .
  * Caldwell (2002) R. R. Caldwell, Phys. Lett. B 545, 23 (2002), arXiv:astro-ph/9908168 .
  * Caldwell  _et al._ (2003) R. R. Caldwell, M. Kamionkowski, and N. N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003), arXiv:astro-ph/0302506 .
  * Vikman (2005) A. Vikman, Phys. Rev. D 71, 023515 (2005), arXiv:astro-ph/0407107 .
  * Nojiri  _et al._ (2005) S. Nojiri, S. D. Odintsov, and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005), arXiv:hep-th/0501025 .
  * Shafer and Huterer (2014) D. L. Shafer and D. Huterer, Phys. Rev. D 89, 063510 (2014), arXiv:1312.1688 [astro-ph.CO] .
  * Ludwick (2017) K. J. Ludwick, Mod. Phys. Lett. A 32, 1730025 (2017), arXiv:1708.06981 [astro-ph.CO] .
  * Alestas  _et al._ (2020) G. Alestas, L. Kazantzidis, and L. Perivolaropoulos, Phys. Rev. D 101, 123516 (2020), arXiv:2004.08363 [astro-ph.CO] .
  * Riess  _et al._ (2022) A. G. Riess _et al._ , Astrophys. J. Lett. 934, L7 (2022), arXiv:2112.04510 [astro-ph.CO] .
  * Benevento  _et al._ (2020) G. Benevento, W. Hu, and M. Raveri, Phys. Rev. D 101, 103517 (2020), arXiv:2002.11707 [astro-ph.CO] .
  * Alestas  _et al._ (2021) G. Alestas, L. Kazantzidis, and L. Perivolaropoulos, Phys. Rev. D 103, 083517 (2021), arXiv:2012.13932 [astro-ph.CO] .
  * Camarena and Marra (2021) D. Camarena and V. Marra, Mon. Not. Roy. Astron. Soc. 504, 5164 (2021), arXiv:2101.08641 [astro-ph.CO] .
  * Aghanim  _et al._ (2020b) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020b), arXiv:1807.06209 [astro-ph.CO] .
  * Alcaniz  _et al._ (2021) J. Alcaniz, N. Bernal, A. Masiero, and F. S. Queiroz, Phys. Lett. B 812, 136008 (2021), arXiv:1912.05563 [astro-ph.CO] .
  * Kelso  _et al._ (2013) C. Kelso, S. Profumo, and F. S. Queiroz, Phys. Rev. D 88, 023511 (2013), arXiv:1304.5243 [hep-ph] .
  * Tong (2019) D. Tong, Cambridge University (2019).
  * Yang  _et al._ (2018) W. Yang, S. Pan, E. Di Valentino, R. C. Nunes, S. Vagnozzi, and D. F. Mota, JCAP 09, 019 (2018), arXiv:1805.08252 [astro-ph.CO] .
  * Aprile  _et al._ (2022) E. Aprile _et al._ ((XENON Collaboration)††, XENON), Phys. Rev. Lett. 129, 161805 (2022), arXiv:2207.11330 [hep-ex] .
  * Duarte  _et al._ (2022) L. Duarte, L. Lin, M. Lindner, V. Kozhuharov, S. V. Kuleshov, A. S. de Jesus, F. S. Queiroz, Y. Villamizar, and H. Westfahl, (2022), arXiv:2206.05305 [hep-ph] .
  * Feng  _et al._ (2003) J. L. Feng, A. Rajaraman, and F. Takayama, Phys. Rev. D 68, 063504 (2003), arXiv:hep-ph/0306024 .
  * Feng  _et al._ (2004) J. L. Feng, S.-f. Su, and F. Takayama, Phys. Rev. D 70, 063514 (2004), arXiv:hep-ph/0404198 .
  * Alcaniz  _et al._ (2022) J. S. Alcaniz, J. P. Neto, F. S. Queiroz, D. R. da Silva, and R. Silva, Sci. Rep. 12, 20113 (2022), arXiv:2211.14345 [astro-ph.CO] .
  * Klypin  _et al._ (1993) A. Klypin, J. Holtzman, J. Primack, and E. Regos, Astrophys. J. 416, 1 (1993), arXiv:astro-ph/9305011 .
  * Bose  _et al._ (2019) S. Bose, M. Vogelsberger, J. Zavala, C. Pfrommer, F.-Y. Cyr-Racine, S. Bohr, and T. Bringmann, Mon. Not. Roy. Astron. Soc. 487, 522 (2019), arXiv:1811.10630 [astro-ph.CO] .
  * Lopez-Corredoira and Marmet (2022) M. Lopez-Corredoira and L. Marmet, Int. J. Mod. Phys. D 31, 2230014 (2022), arXiv:2202.12897 [astro-ph.CO] .
  * Lahav and Liddle (2022) O. Lahav and A. R. Liddle, (2022), arXiv:2201.08666 [astro-ph.CO] .
  * Workman  _et al._ (2022) R. L. Workman _et al._ (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Griffiths (2020) D. Griffiths,  _Introduction to elementary particles_ (John Wiley & Sons, 2020).
  * Baring  _et al._ (2016) M. G. Baring, T. Ghosh, F. S. Queiroz, and K. Sinha, Phys. Rev. D 93, 103009 (2016), arXiv:1510.00389 [hep-ph] .
  * Mambrini  _et al._ (2016) Y. Mambrini, S. Profumo, and F. S. Queiroz, Phys. Lett. B 760, 807 (2016), arXiv:1508.06635 [hep-ph] .
  * Wang and Yang (2004) F. Wang and J. M. Yang, Eur. Phys. J. C 38, 129 (2004), arXiv:hep-ph/0405186 .
  * Cembranos  _et al._ (2005) J. A. R. Cembranos, J. L. Feng, A. Rajaraman, and F. Takayama, Phys. Rev. Lett. 95, 181301 (2005), arXiv:hep-ph/0507150 .
  * Kelso  _et al._ (2014) C. Kelso, C. A. de S. Pires, S. Profumo, F. S. Queiroz, and P. S. Rodrigues da Silva, Eur. Phys. J. C 74, 2797 (2014), arXiv:1308.6630 [hep-ph] .