# Testing quantum gravity effects with global lepton number symmetry and dark
matter

Debasish Borah<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Guwahati, Assam 781039, India  Nayan Das<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Guwahati, Assam 781039,
India  Rishav Roshan<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Southampton, Southampton SO17 1BJ, United Kingdom

###### Abstract

We study the possibility of testing quantum gravity (QG) effects with a type-I
seesaw origin of light neutrino mass embedded in a global $U(1)_{L}$ symmetric
framework which also includes dark matter (DM). The spontaneous breaking of
$U(1)_{L}$ symmetry generates the seesaw scale as well as DM mass dynamically.
The (pseudo) Nambu-Goldstone boson, known as majoron, acquires non-zero mass
due to soft symmetry breaking terms of quadratic type in the scalar potential,
which eventually breaks $U(1)_{L}$ to its $Z_{2}$ subgroup. The spontaneous
symmetry breaking, which effectively breaks $Z_{2}$, leads to the formation of
domain walls (DW), posing a threat to successful cosmology, if allowed to
dominate. As gravity does not respect any global symmetries, we consider
higher dimensional operators suppressed by the scale of QG namely,
$\Lambda_{\rm QG}$ which introduces the required bias leading to DW
annihilation and emission of stochastic gravitational waves (GW) observable at
near future experiments. The same operators also lead to decay of DM bringing
interesting indirect detection aspects. While DM is produced non-thermally via
scalar portal interactions, light majoron can give rise to additional $\Delta
N_{\rm eff}$ within reach of future cosmic microwave background (CMB)
experiments.

##  I Introduction

Observations of non-zero neutrino mass and mixing, dark matter (DM) and baryon
asymmetry of the Universe (BAU) ParticleDataGroup:2020ssz ; Planck:2018vyg
have emerged as clear signs of beyond standard model (BSM) physics. While non-
zero neutrino mass and mixing can be explained within the framework of
canonical seesaw mechanisms like type-I seesaw Minkowski:1977sc ;
GellMann:1980vs ; Mohapatra:1979ia ; Sawada:1979dis ; Yanagida:1980xy ;
Schechter:1980gr , particle DM scenarios can be broadly categorised into
thermal and non-thermal DM depending upon their production mechanisms. Among
thermal and non-thermal DM scenarios, the weakly interacting massive particle
(WIMP) and feebly interacting massive particle (FIMP), respectively, have been
studied extensively in the literature. Recent reviews of WIMP and FIMP can be
found in Arcadi:2017kky  and Bernal:2017kxu  respectively. Similarly, popular
BSM scenarios to explain BAU can be categorized into baryogenesis
Weinberg:1979bt ; Kolb:1979qa  or leptogenesis Fukugita:1986hr .
Interestingly, leptogenesis can be realised within most of the seesaw
frameworks where one or more of the heavy particles introduced for seesaw
origin of neutrino mass can also be responsible for generating a non-zero
lepton asymmetry to be converted into BAU later by electroweak sphalerons
Kuzmin:1985mm .

In spite of having several motivations for BSM physics mentioned above, none
of the experiments across different frontiers have observed any convincing
signs of it so far. Canonical seesaw models consistent with neutrino mass and
leptogenesis typically have a very high scale for new physics, away from
direct experimental reach. On the other hand, WIMP DM has not shown up in
direct search experiments either LUX-ZEPLIN:2022qhg . This might be indicative
of the fact that DM is either feebly coupled or interact with the standard
model (SM) particles via superheavy mediators. Such null results at DM search
or other particle physics experiments searching for BSM physics have also
motivated the search for complementary probes at other experiments. For
example, some recent attempts have focused on probing high scale seesaw via
stochastic gravitational wave (GW) observations Dror:2019syi ; Blasi:2020wpy ;
Fornal:2020esl ; Samanta:2020cdk ; Barman:2022yos ; Huang:2022vkf ;
Dasgupta:2022isg ; Okada:2018xdh ; Hasegawa:2019amx ; Borah:2022cdx ;
Borah:2022vsu ; Barman:2023fad ; Borah:2023saq . The particle physics
scenarios explored in these works considered the presence of additional
symmetries whose spontaneous breaking not only lead to the dynamical
generation of seesaw scale but also leads to the generation of stochastic GW
from topological defects like cosmic strings Dror:2019syi ; Blasi:2020wpy ;
Fornal:2020esl ; Samanta:2020cdk ; Borah:2022vsu , domain walls Barman:2022yos
; Barman:2023fad ; Saikawa:2017hiv ; Roshan:2024qnv ; Bhattacharya:2023kws  or
from bubbles generated at first order phase transition Huang:2022vkf ;
Dasgupta:2022isg ; Okada:2018xdh ; Hasegawa:2019amx ; Borah:2022cdx ;
Borah:2023saq . GW observations has also been used to probe even higher scales
than seesaw, where quantum gravity (QG) effects can become important
King:2023ayw ; King:2023ztb ; Borah:2022wdy . In these works, the QG effects
on explicit global discrete symmetry breaking was considered in the context of
domain wall formation from spontaneous discrete symmetry breaking and
consequent GW emission. The authors of King:2023ayw ; King:2023ztb  also
considered the impact of such QG effects on decay of DM which is stabilised by
discrete symmetry at renormalisable level.

Motivated by these, we consider a global $U(1)$ symmetry related to the seesaw
scale and DM stability and study QG effects leading to stochastic GW and DM
decay. Connection to lepton sector and seesaw becomes natural if this Abelian
global symmetry is identified as the global lepton number symmetry $U(1)_{L}$.
In the simplest realisation of seesaw, the spontaneous breaking of $U(1)_{L}$
generates the right handed neutrino (RHN) masses or the seesaw scale
dynamically. The same breaking also results in a massless Nambu-Goldstone (NG)
boson, known as the majoron Chikashige:1980qk ; Chikashige:1980ui ;
Gelmini:1980re ; Manna:2022gwn . Majoron can acquire a non-zero mass from
explicit $U(1)_{L}$ breaking mass terms. When the singlet scalar responsible
for breaking $U(1)_{L}$ spontaneously acquires a non-zero vacuum expectation
value (VEV), it essentially breaks a $Z_{2}$ symmetry, remnant from $U(1)_{L}$
after explicit $U(1)_{L}$ breaking mass terms are included. Even without such
explicit symmetry breaking terms, $U(1)_{L}$ is anomalous in the SM and broken
to $Z_{3}$ by non-perturbative effects tHooft:1976rip ; Lazarides:2018aev
111See Lazarides:2018aev ; Brune:2022vzd  for non-minimal majoron models
without domain walls.. Spontaneous breaking of discrete symmetries like
$Z_{2}$ leads to the formation of domain walls (DW) Zeldovich:1974uw ;
Kibble:1976sj ; Vilenkin:1981zs ; Saikawa:2017hiv ; Roshan:2024qnv , which can
overclose the Universe, if allowed to dominate222Spontaneous breaking of
global $U(1)_{L}$ can also lead to formation of topological defects known as
cosmic strings. However, GW emission remains suppressed compared to particle
emission from global strings Baeza-Ballesteros:2023say .. We then consider
higher dimensional operators suppressed by QG scale as bias leading to
disappearance of DW thereby generating stochastic GW signal detectable at
ongoing and future experiments. We consider a Dirac fermion having chiral
transformation under $U(1)_{L}$ to be the DM which remains accidentally stable
even after $U(1)_{L}$ or $Z_{2}$ breaking, unlike earlier works King:2023ayw ;
King:2023ztb  where a separate $Z_{2}$ symmetry was introduced for DM
stability. The $U(1)_{L}$ global symmetry therefore, generates the scale of
seesaw, DM mass, ensures the stability of DM and also leads to DW formation in
the presence of $Z_{2}$-preserving explicit $U(1)_{L}$ breaking terms
generating majoron mass. The higher dimensional operators suppressed by QG
scale can however, lead to DM decay opening up some indirect detection
prospects. We discuss the details of DM which interacts with the SM via
singlet scalar portal, and the effects of QG scale suppressed operators on GW
emission from DW and on DM phenomenology. The RHNs not only generate light
neutrino masses via seesaw mechanism, but can also generate the baryon
asymmetry of the Universe via leptogenesis Fukugita:1986hr . The emergence of
$Z_{2}$ symmetry from a global $U(1)_{L}$, effects of QG on DW and DM decay,
the presence of the pseudo NG boson or majoron bring complementary detection
prospects at particle physics, gravitational waves as well as cosmology based
experiments.

This paper is organised as follows. In section II, we discuss the particle
physics setup followed by the details of DM and majoron cosmology in section
III. In section IV, and section V respectively, we discuss the details of DM
decay and domain wall disappearance emitting gravitational waves due to QG
scale suppressed operators. We finally conclude in section VI.

##  II The Setup

Fields |  $SU(2)_{L}\times U(1)_{Y}\times U(1)_{L}$ charge
---|---
$\chi_{L}$ | $(1,0,Q_{L})$
$\chi_{R}$ | $(1,0,Q_{L}+2)$
$N_{R}$ | $(1,0,1)$
$\ell_{L}$ | $(2,-\frac{1}{2},1)$
$\ell_{R}$ | $(1,-1,1)$
$H$ | $(2,\frac{1}{2},0)$
$\Phi$ | $(1,0,-2)$
Table 1: Relevant field content of the model.

The relevant field content and respective transformations under
$SU(2)_{L}\times U(1)_{Y}\times U(1)_{L}$ symmetry are shown in table 1 with
the BSM fields being the right handed neutrino $N_{R}$, fermion DM $\chi$ and
the singlet scalar $\Phi$ responsible for spontaneous symmetry breaking. The
relevant part of the Yukawa Lagrangian can be written as

$-\mathcal{L}_{Y}\supset\frac{f_{i}}{2}\Phi\overline{N^{c}_{Ri}}N_{Ri}+Y_{\alpha
i}\overline{\ell_{L_{\alpha}}}\tilde{H}N_{Ri}+y_{\chi}\overline{\chi_{L}}\Phi\chi_{R}+{\rm
h.c.}$ (1)

The scalar potential is

$V(H,\Phi)=\lambda_{H}\left(H^{\dagger}H-\dfrac{v^{2}}{2}\right)^{2}+\lambda_{\Phi}\left(|{\Phi}|^{2}-\dfrac{v_{\Phi}^{2}}{2}\right)^{2}+\lambda_{H\Phi}\left(H^{\dagger}H-\dfrac{v^{2}}{2}\right)\left(|{\Phi}|^{2}-\dfrac{v_{\Phi}^{2}}{2}\right)\,$
(2)

where $v(v_{\Phi})$ denotes the VEV of $H(\Phi)$. The spontaneous breaking of
$U(1)_{L}$ global symmetry after $\Phi$ acquires a VEV, leads to a massless
Goldstone boson or majoron, identified as the angular part of the field
$\Phi$. A non-zero majoron mass can be generated via explicit $U(1)_{L}$
breaking terms of the form

$V_{\not{L}}=-\frac{m^{2}}{4}(\Phi^{2}+{\rm h.c.})$ (3)

which breaks $U(1)_{L}$ to a remnant $Z_{2}$ symmetry in the scalar potential.

After the spontaneous breaking of global $U(1)_{L}$ symmetry, we can represent
$\Phi=(v_{\Phi}+\phi+i\eta)/\sqrt{2}$, where $\eta$ denotes the majoron. From
the potential terms in Eq. (2) and Eq. (3), the masses of Higgs ($h$), $\phi$
and majoron can be written as Gu:2010ys ; Queiroz:2014yna

$\displaystyle m^{2}_{h}\simeq 2\lambda_{H}v^{2},\quad
m^{2}_{\phi}=2\lambda_{\Phi}v^{2}_{\Phi},\quad m^{2}_{\eta}\simeq m^{2}.$ (4)

The coupling between the majoron and heavy right-handed neutrino can be
written as

$\displaystyle-\mathcal{L}\supset\frac{if_{i}}{2\sqrt{2}}\eta\overline{N_{i}}\gamma^{5}N_{i}=\frac{im_{N_{i}}}{2v_{\Phi}}\eta\overline{N_{i}}\gamma^{5}N_{i}.$
(5)

Similarly, the coupling between the majoron and DM can be written as

$\displaystyle-\mathcal{L}\supset\frac{iy_{\chi}}{\sqrt{2}}\eta\overline{\chi}\gamma^{5}\chi=\frac{im_{\chi}}{v_{\Phi}}\eta\overline{\chi}\gamma^{5}\chi.$
(6)

The details of derivation of coupling between CP-odd and CP-even scalars with
$N_{i}=N_{R_{i}}+N^{c}_{R_{i}}$ and $\chi=\chi_{L}+\chi_{R}$ are given in
appendix A. Light neutrino mass arises from the type-I seesaw mechanism as

$m_{\nu}=-M_{D}m^{-1}_{N}M^{T}_{D}$ (7)

where $M_{D}=Yv/\sqrt{2}$ is the Dirac mass. Here
$m_{N_{i}}=f_{i}v_{\Phi}/\sqrt{2}$, $m_{\phi}=\sqrt{2\lambda_{\Phi}}v_{\Phi}$,
$m_{\chi}=y_{\chi}v_{\Phi}/\sqrt{2}$. The free parameters of the model,
relevant for the phenomenology, are $f_{i},Y_{\alpha
i},y_{\chi},v_{\Phi},\lambda_{\Phi},\lambda_{H\Phi}$ and $m_{\eta}$. In terms
of masses of RHNs, $\phi$ and $\chi$, the free parameters are
$m_{N_{i}},m_{\chi},m_{\phi},v_{\Phi},Y_{\alpha i},\lambda_{H\Phi}$ and
$m_{\eta}$. We also denote the RHNs simply as $N$ hereafter.

Finally, there exists no exact (continuous or discrete) global symmetry in the
theory of QG Kallosh:1995hi . In other words, any global symmetry of a given
effective field theory (EFT) is at best an approximate symmetry emergent in
the IR Witten:2017hdv  and should be broken by a higher-dimensional operator.
As a result, the remnant $Z_{2}$ present in this setup will also be explicitly
broken by the higher dimensional operators Rai:1992xw ,

$\displaystyle\Delta V$ $\displaystyle=$ $\displaystyle\frac{1}{\Lambda_{\rm
QG}}(\alpha_{1}\Phi^{5}+\alpha_{2}\Phi^{3}H^{2}+\alpha_{3}\Phi H^{4})\;,$ (8)

where $\Lambda_{\rm QG}$ denotes the scale of QG. Here we assume a common
origin and therefore a common scale for the breaking of all global symmetries,
and for simplicity, we take all the dimensionless coefficients in Eq. (8) to
be of the same order, and we can make them of $\mathcal{O}(1)$ by redefining
$\Lambda_{\rm QG}$.

##  III Cosmology of majoron and dark matter

Majoron can have very interesting cosmology with the details being dependent
upon its mass and interactions. Very light majorons in eV or sub-eV scale with
negligible interactions can be approximated as a coherently oscillating scalar
field, produced via the misalignment mechanism similar to axions
Preskill:1982cy ; Abbott:1982af ; Dine:1982ah ; Co:2019jts . Depending upon
the initial displacement or kinetic energy of the scalar field, majoron can
play the role of cold dark matter in the Universe. In addition to that,
majoron can also lead to spontaneous baryogenesis via leptogenesis
Affleck:1984fy ; Cohen:1987vi ; Cohen:1988kt ; Co:2019wyp ; Chun:2023eqc ;
Ibe:2015nfa ; Domcke:2020kcp ; Co:2022aav ; Berbig:2023uzs ; Co:2020jtv ;
Chao:2023ojl ; Barnes:2024jap ; Barenboim:2024akt ; Barenboim:2024xxa ;
Datta:2024xhg ; Co:2024oek . In this work, we neither consider majoron DM nor
spontaneous leptogenesis. This allows more available parameter space in
singlet majoron model which otherwise gets constrained when majoron plays the
role of DM Akita:2023qiz ; deGiorgi:2023tvn . Also, majoron DM is not
perfectly stable but long-lived. However, one of our motivations is to study
the effect of QG scale on lifetime of DM, which otherwise is perfectly
stabilised by a global symmetry. Also considering thermal leptogenesis from
RHN decay makes the scenario independent of any initial conditions. While
leptogenesis can occur from CP violating out-of-equilibrium decays of right
handed neutrinos in type-I seesaw, DM is in the form a separate fermion field
which couples to the SM via radial and angular mode (majoron) of the singlet
scalar field responsible for spontaneous breaking of global lepton number
symmetry. We consider sizeable coupling of majoron with the SM bath such that
it can be thermally produced. Such majorons can not only act like a portal
between DM and SM but can also contribute to the effective relativistic
degrees of freedom, observable at cosmic microwave background (CMB)
experiments. Heavy majorons above the eV scale, which do not contribute to DM,
are likely to decay into the SM particles without leaving any relic in the
present Universe.

We first discuss the themalisation of majoron, $\eta$. For this, we have
considered the scattering process, $N\eta\to N\eta$ and compare its rate with
the Hubble expansion rate of the Universe $\mathcal{H}$. In the left panel
plot of Fig. 1, we show the ratio of interaction rate ($\Gamma=n^{\rm
eq}_{N}\langle\sigma v\rangle_{N\eta\to N\eta}$) to $\mathcal{H}$ as a
function of temperature for three different values of $v_{\Phi}$. For a larger
$m_{N}/v_{\Phi}$ ratio (in other words, a larger $f$), we can have thermalised
majoron. It is found that as long as $m_{\eta}\ll m_{N}$, the thermalisation
of majoron is independent of its mass. In the right panel plot of Fig. 1, we
show the parameter space in $m_{N}$ vs $v_{\Phi}$ plane where majoron
thermalises. We have not considered the upper triangular region where
$m_{N}>v_{\Phi}$, where the coupling can become non-perturbative.

Light majoron in sub-eV ballpark can contribute to the effective relativistic
degrees of freedom $\Delta N_{\rm eff}$ tightly constrained by CMB
observations as well as successful big bang nucleosynthesis (BBN). Majoron
contribution to $\Delta N_{\rm eff}$ can be calculated based on the thermal
history of majoron. The contribution to $\Delta N_{\rm eff}$ from light
thermal majoron at the epoch of recombination is given as

$\displaystyle\Delta N_{\rm eff}\equiv\frac{\rho_{\eta}(T_{\rm
CMB})}{\rho_{\nu,1}(T_{\rm CMB})}=0.027\times\left(\frac{106.75}{g_{*}(T_{\rm
dec})}\right)^{4/3}.$ (9)

Here $\rho_{\eta}$ and $\rho_{\nu,1}$ denotes the energy density of majoron
and a single species of active neutrino respectively and $g_{*}$ denotes the
effective degrees of freedom of SM particles. For all the blue shaded region
shown in the right panel plot of Fig. 1, majoron gets decoupled at a
temperature larger than $100$ GeV. This gives a constant contribution of
thermal majoron $\Delta N_{\rm eff}=0.027$ for all $v_{\Phi}$. Future CMB
experiment, specially, CMB-HD CMB-HD:2022bsz  can probe $\Delta N_{\rm eff}$
upto $0.014$ at $1\sigma$, keeping thermalised light majoron parameter space
verifiable.

Figure 1: Left panel: majoron interaction rate in comparison to the Hubble
expansion rate. Right panel: Parameter space in RHN mass $m_{N}$ and singlet
scalar VEV $v_{\Phi}$ showing the thermalisation region of majoron for
$m_{\eta}\ll m_{N}$.

DM couples to the SM bath or RHNs only via $\phi,\eta$ with a coupling
proportional to $m_{\chi}/v_{\Phi}$. We consider $v_{\Phi}\geq 10^{6}$ GeV
motivated from type-I seesaw, leptogenesis as well as domain wall related
gravitational wave signatures to be discussed below. For such a high scale
symmetry breaking, DM can not have large sizeable portal couplings required
for thermal freeze-out, for typical thermal DM masses around the electroweak
scale, below the unitarity limit Griest:1989wd . Therefore, it is more natural
to consider freeze-in dark matter Hall:2009bx . Depending upon the mass
spectrum of newly introduced particles, here we discuss two different
scenarios for freeze-in DM.

###  III.1 Dark matter with $m_{\phi}>2m_{N}$

To begin with, we consider the mass spectrum as
$m_{\phi}>2m_{N}>m_{\chi}>m_{\eta}$. We assume that RHNs are in thermal bath
via the Yukawa interaction with SM leptons and Higgs. Similarly, $\phi$ is
taken to be in thermal equilibrium initially. The DM is considered to be
feebly interacting, produced through two possible ways : from decay of CP even
scalar ($\phi\to\chi\bar{\chi}$) and from annihilation of RHNs
($N\bar{N}\to\chi\bar{\chi}$ with $\eta$ and $\phi$ as the mediator). Assuming
$N$ to be in thermal equilibrium, the Boltzmann equations (BE) for $\phi$ and
$\chi$ can be written as

$\displaystyle\frac{dY_{\phi}}{dx}$ $\displaystyle=$
$\displaystyle-\frac{\beta s}{\mathcal{H}x}\langle\sigma v\rangle_{\phi\phi\to
N\bar{N}}(Y^{2}_{\phi}-(Y^{\rm
eq}_{\phi})^{2})-\frac{\beta}{\mathcal{H}x}\Gamma_{\phi\to
N\bar{N}}\frac{K_{1}(m_{\phi}/T)}{K_{2}(m_{\phi}/T)}(Y_{\phi}-Y^{\rm
eq}_{\phi})$
$\displaystyle-\frac{\beta}{\mathcal{H}x}\Gamma_{\phi\to\chi\bar{\chi}}\frac{K_{1}(m_{\phi}/T)}{K_{2}(m_{\phi}/T)}Y_{\phi},$
$\displaystyle\frac{dY_{\chi}}{dx}$ $\displaystyle=$
$\displaystyle\frac{\beta}{\mathcal{H}x}\Gamma_{\phi\to\chi\bar{\chi}}\frac{K_{1}(m_{\phi}/T)}{K_{2}(m_{\phi}/T)}Y_{\phi}+\frac{\beta
s}{\mathcal{H}x}\langle\sigma v\rangle_{N\bar{N}\to\chi\bar{\chi}}(Y^{\rm
eq}_{N})^{2}.$ (11)

Here, $Y$ and $s$ represent the comoving number density and entropy density
respectively. The effect due to change of $g_{*,s}$ is incorporated into the
parameter $\beta=\left[1+\frac{Tdg_{*,s}/dT}{3g_{*,s}}\right]$ and $x=m_{0}/T$
where $m_{0}$ is some arbitrary reference mass. We consider $m_{0}=m_{\phi}$
for numerical calculations. $K_{i}$’s denote modified Bessel functions of the
second kind whereas $\mathcal{H}$ denotes the Hubble expansion rate. The
expression for relevant cross-sections are given in appendix B.

Figure 2: Evolution plot for $m_{\phi}>2m_{N}$ for two different $m_{\chi}=1$
GeV (left panel) and $m_{\chi}=10$ GeV (right panel). Due to the inverse decay
of $N$, $\phi$ follows equilibrium abundance throughout its lifetime. The blue
solid and dashed lines represent the respective contributions to DM production
from the decay of $\phi$ and the annihilation of $N$.

In Fig. 2, the evolution of $N$, $\phi$ and $\chi$ are shown. Here
$Y^{D}_{\chi}$ and $Y^{S}_{\chi}$ denote the contribution to the DM production
from the 1st and 2nd terms respectively in the BE for $Y_{\chi}$. Due to the
inverse decay and annihilation of $N$, $\phi$ remains in equilibrium during
its evolution. Also, the production of DM from annihilation is subdominant
compared to the production from decay of $\phi$, as expected. Increasing DM
mass from left to the right panel plots increases both $Y^{D}_{\chi}$ and
$Y^{S}_{\chi}$ as both $\Gamma_{\phi\to\chi\bar{\chi}}$ and $\langle\sigma
v\rangle_{N\bar{N}\to\chi\bar{\chi}}$ are proportional to $m^{2}_{\chi}$.

Figure 3: Parameter space in $v_{\Phi}$ vs $m_{N}$ plane with different
colored contours denoting relic satisfying regions for different DM masses.

Fig. 3 shows the parameter space in $v_{\Phi}$ vs $m_{N}$ plane with DM relic
satisfying regions are shown as colored contours for different DM masses.
Here, $\lambda_{\Phi}$ is taken as $10^{-3}$. This fixes $m_{\phi}$ for a
given $v_{\Phi}$. In the gray and magenta region $m_{N}>v_{\Phi}$ and
$m_{\phi}<2m_{N}$ respectively. The dashed magenta color line denotes
$m_{\phi}=2m_{N}$. As depicted in the evolution plots of Fig. 2, the
production of dark matter from decay of $\phi$ dominates over the production
from annihilation across the entire parameter space. The four different
colored contours correspond to different DM masses that satisfy correct relic
abundance. Staring from the dashed magenta line, a change in $m_{N}$ does not
change the DM relic for fixed $v_{\Phi}$ and $m_{\chi}$. This is because, in
the vertical portion, the coupling between $N$ and $\phi$ (which is
$\propto\frac{m_{N}}{v_{\Phi}}$) is larger enough so that $\phi$ remains in
equilibrium due to the inverse decay of $N$ ($N\bar{N}\to\phi$). However the
situation reverses for a sufficiently small values of $m_{N}$ where $\phi$
does not always follow the equilibrium distribution. In that region, a
different $v_{\Phi}$ (as well as $m_{\phi}$), does not alter DM abundances.
Hence, the horizontal portion of the curves are obtained. Moreover, to
indicate the parameter space that comes within the sensitivities of future
CMB-HD experiment, we show the blue colored region where majoron thermalises
(from Fig. 1).

###  III.2 Dark matter with $m_{\phi}<2m_{N}$

For the mass spectrum $2m_{N}>m_{\phi}>m_{\chi}>m_{\eta}$, i.e. the magenta
shaded region in Fig. 3, the decay of $\phi\to N\bar{N}$ ceases. Hence we have
only the 1st and last term in the right hand side of equation (III.1). Due to
the presence of only one decay channel of $\phi$, the comoving DM abundance
get enhanced. To see it, we consider a point in Fig. 3 denoted by $\times$.
The evolution plot for this point is shown in Fig. 4. In the plot,
$v_{\Phi}=10^{7}$ GeV and $\lambda_{\Phi}=10^{-3}$. This gives
$m_{\phi}=4.5\times 10^{5}$ GeV. Mass of $N$ is taken to be equal to
$m_{\phi}$. As shown in the plot, for these choice of parameters a higher
comoving abundance is obtained requiring DM mass to be as low as $9$ keV for
correct relic density. Also due to such low DM mass, the decay width of $\phi$
is small ($\propto m^{2}_{\chi}$) resulting $\phi$ decaying very close to the
BBN era. This brings two major problems namely, (i) the late decay of $\phi$
leads to a large the free-streaming length of DM (as shown in Decant:2021mhj ;
Biswas:2022vkq ) potentially giving hot DM that is already ruled out from
structure formation constraint; (ii) DM mass range lies in the boundary of
Tremaine-Gunn bound for fermionic dark matter PhysRevLett.42.407 . These
problems get worse for a higher $v_{\Phi}$ making the scenario
$m_{\phi}<2m_{N}$ for $\lambda_{\phi}=10^{-3}$ disfavoured. Hence, for our
current and future DM analysis, we only consider the region below the dashed
magenta line in Fig. 3.

Figure 4: Evolution plot for $m_{\phi}<2m_{N}$. Here, we take
$m_{N}=m_{\phi}$. The contribution from the $N\bar{N}\to\eta\eta$ is
negligible, hence not shown here.

###  III.3 Leptogenesis

Apart from producing correct relic abundance, the current set up also allows
to explain the observed baryon asymmetry of the Universe via thermal
leptogenesis. The CP violating out-of-equilibrium decay of RHN neutrinos
produce lepton asymmetry which is later converted to baryon asymmetry by SM
sphalerons. Assuming hierarchical RHN mass spectrum, the baryon-to-photon
ratio can be estimated as

$\displaystyle\eta_{B}\simeq 10^{-2}\epsilon_{1}\kappa_{1},$ (12)

where $\epsilon_{1}$ denotes the CP asymmetry and $\kappa_{1}$ represents the
efficiency factor. We use the maximum value of CP asymmetry parameter which is
given as

$|\epsilon_{1}|\leq\frac{3m_{N_{1}}\sqrt{(\Delta m_{\rm atm})^{2}}}{8\pi
v^{2}}.$ (13)

Here $(\Delta m_{\rm atm})^{2}\simeq 2.4\times 10^{-3}\rm eV^{2}$ denotes the
active neutrino atmospheric mass-squared difference. We consider efficiency
factor to be $0.01$ that corresponds to strong wash-out regime. Apart from
usual leptogenesis in type-I seesaw, in this setup we have extra annihilation
channel, $N\bar{N}\to\eta\eta$ that can keep RHNs in equilibrium for longer
and hence dilute the final baryon asymmetry. However for strong washout regime
and for equilibrium initial abundance of $N$, the effect from
$N\bar{N}\to\eta\eta$ can be neglected. With this, using Eq. (12), we get that
the mass of RHN should be larger than $m_{N_{1}}\gtrsim 6\times 10^{10}$ GeV.
As the DM analysis is valid for $m_{\phi}>2m_{N}$ as discussed in the previous
section, this requires $v_{\Phi}$ to be larger than $\sim 3\times 10^{12}$ GeV
for $\lambda_{\Phi}=10^{-3}$. Such large values of $v_{\Phi}$ are disfavoured
from DW overclosure and DM properties, as we discuss below. For a $v_{\Phi}$
smaller than $3\times 10^{12}$ GeV, required amount of lepton asymmetry can be
created via resonant leptogenesis Pilaftsis:2003gt . Resonant leptogenesis can
enhance CP asymmetry parameter upto $\mathcal{O}(0.1)$ where two RHNs have
nearly the same mass, with the mass splitting being nearly equal to the its
decay width. Therefore, correct baryon asymmetry is possible via resonant
scenario where $v_{\Phi}$ is as small as $10^{6}$ GeV.

##  IV Dark matter decay via quantum gravity effects

In the present setup, the QG effects also break the $U(1)_{L}$ symmetry
explicitly, and as a result, the DM decay can occur via dimension five
operators of the type

$-\mathcal{L}_{\rm decay}\supset\frac{Y_{D_{\alpha}}}{\Lambda_{\rm
QG}}\overline{\ell_{L\alpha}}\tilde{H}\chi_{R}\Phi+{\rm h.c.},$ (14)

where $\widetilde{H}=i\sigma_{2}H^{*}$. As discussed earlier, we assume that
all the global symmetries are broken by a common scale i.e. we simply consider
all the dimensionless coefficients in Eq. (8) and Eq. (29) to be of the same
order, and we set them to be ${\cal O}(1)$ by redefining $\Lambda_{\rm QG}$.
Looking at Eq. (29), one notices that, once the electroweak symmetry is
broken, the DM $\chi$ mixes with the SM neutrinos with a mixing angle
Datta:2021elq ; King:2023ztb ,

$\theta\simeq\sum_{\alpha=1,2,3}\left(\frac{Y_{D_{\alpha}}v_{\Phi}v}{\sqrt{2}\Lambda_{\text{QG}}m_{\chi}}\right)\;.$
(15)

As a result of this mixing, the DM decays to the SM particles. This provides
us a possibility to look for dark matter in the indirect search experiments.
Once such example is constraining the DM decay life time by the detection of
the galactic and extra-galactic diffuse X/$\gamma$-ray background. Due to the
mixing, the DM can have two-body (one-loop level mediated via $W^{\pm}$ and SM
charged leptons) and three-body (at tree-level mediated via $W^{\pm}$) decay
channels that can dominate if kinematically allowed. At one-loop level $\chi$
can decay to photons and active neutrinos via $\chi\to\nu\gamma$, the decay
rate of which is given by Shrock:1982sc ; Essig:2013goa

$\begin{split}\tau_{\chi\rightarrow\nu\gamma}&\simeq\left(\frac{9\alpha_{\mathrm{EM}}\sin^{2}\theta}{1024\pi^{4}}G_{F}^{2}m_{\chi}^{5}\right)^{-1}\\\
&\simeq 1.8\times 10^{17}~{}{\rm
s}\left(\frac{10\,\mathrm{MeV}}{m_{\chi}}\right)^{5}\left(\frac{\sin\theta}{10^{-8}}\right)^{-2}\;,\end{split}$
(16)

with $\alpha_{\rm EM}=1/137$ being the electromagnetic fine-structure constant
and $G_{F}$ denoting the Fermi constant. The three-body decay channel $\chi\to
e^{+}e^{-}\nu$, the decay rate of which can be approximately expressed as
Ruchayskiy:2011aa ; Essig:2013goa

$\begin{split}\tau_{\chi\to
e^{+}e^{-}\nu}&\simeq\left(\frac{c_{\alpha}\sin^{2}\theta}{96\pi^{3}}G^{2}_{F}m^{5}_{\chi}\right)^{-1}\\\
&\simeq 2.4\times 10^{15}~{}{\rm
s}\left(\frac{10\mathrm{MeV}}{m_{\chi}}\right)^{5}\left(\frac{\sin\theta}{10^{-8}}\right)^{-2}\;,\end{split}$
(17)

where $c_{\alpha}=(1+4\sin^{2}\theta_{W}+8\sin^{4}\theta_{W})/4\simeq 0.59$
with $\theta_{W}$ being the weak mixing angle. Contributions from the above
two channels to the X/$\gamma$-ray fluxes are roughly at similar levels
Essig:2013goa . The null detection of X-rays and $\gamma$-rays line sets a
lower bound on the lifetime of the decaying DM that can further be converted
into the constraints on the the DM mass and the mixing angle. The observed
diffuse photon spectra data obtained from the HEAO-1 Gruber:1999yr , INTEGRAL
Bouchet:2008rp , COMPTEL Sreekumar:1997yg  and EGRET Strong:2003ey  satellites
can restrict the parameter space of $m_{\chi}$ and $\theta$ within the range
$0.01~{}{\rm MeV}\lesssim m_{\chi}\lesssim 100~{}{\rm MeV}$, which can be
parameterized as $\theta^{2}\lesssim 2.8\times 10^{-18}({\rm
MeV}/m_{\chi})^{5}$ Boyarsky:2009ix . With the help of the expression of
$\theta$ in Eq. (15), we have

$\left(\frac{3v_{\Phi}v}{\sqrt{2}\Lambda_{\text{QG}}m_{\chi}}\right)^{2}\lesssim
2.8\times 10^{-18}\left(\frac{\text{MeV}}{m_{\chi}}\right)^{5}\;,$ (18)

where we consider that the DM interacts with different generations of
neutrinos with the identical ${\cal O}(1)$ strength. On top of this, the Fermi
Large Area Telescope (Fermi-LAT) Fermi-LAT:2012pls ; Fermi-LAT:2012ugx  has
presented a dedicated line search of the diffuse $\gamma$-ray background, the
null result of which also constrains the lifetime of the DM within the mass
range $1~{}{\rm GeV}\lesssim m_{\chi}\lesssim 1~{}{\rm TeV}$. Here we take the
95% C.L. limit on the lifetime of decaying DM given by ref. Fermi-LAT:2015kyq
, which assumes a Navarro-Frenk-White (NFW) profile for the DM distribution
Navarro:1995iw .

DM decay also faces stringent constraints from CMB, if the decay occurs after
or during the recombination. The DM decay can re-ionize the intergalactic
medium, and hence can modify the CMB power spectrum. Accurate measurements of
the CMB spectrum have been implemented by recent experiments including WMAP
WMAP:2012nax , ACT ACTPol:2014pbf , SPT Hou:2012xq  and Planck Planck:2018vyg
. The 95% C.L. lower bounds on the DM decay lifetime have been given in ref.
Slatyer:2016qyl , showing that $\tau_{\rm DM}\gtrsim 10^{25}~{}{\rm s}$ for
decays into both $e^{+}e^{-}$ pairs and photons. These DM decays can also
introduce radio signals originating inside the DM-dominated galaxies and
clusters, if the produced $e^{+}e^{-}$ pairs undergo energy loss via
electromagnetic interactions in the interstellar medium. Such radio waves can
be tested by radio telescopes like the Square Kilometer Array (SKA) radio
telescope Colafrancesco:2015ola . It is found that the DM decay width up to
$\Gamma_{\text{DM}}\gtrsim 10^{-30}~{}{\rm s}^{-1}$ is detectable at SKA
assuming 100-hour observation time Dutta:2022wuc . We will incorporate these
constraints while showing the final allowed parameter space, to be discussed
below.

##  V Domain walls and Gravitational Waves

Domain walls, a cosmological catastrophe, are 2-dimensional topological
defects resulting from a spontaneous symmetry breaking (SSB) of a discrete
symmetry like $Z_{2}$, as also considered in the present setup. Once formed,
their energy density varies inversely with the scale factor ($a^{-1}$) which
is much slower than that of matter ($a^{-3}$) or radiation ($a^{-4}$). As a
result, they can dominate the energy budget of the Universe at some later
stage and can alter the fate of CMB observations or successful light nuclei
synthesis during BBN. These cosmological catastrophes can be avoided by
introducing an energy bias in the scalar potential that makes the vacuum
unstable. In the present setup this can be achieved with the help of QG effect
as it is expected to break any global symmetry at high scale. Following this,
one can write the energy bias after replacing fields with their VEVs, this
leads to a bias contribution to the effective potential of the form

$V_{\rm bias}\simeq\frac{1}{\Lambda_{\rm
QG}}\left(v^{5}_{\Phi}+\frac{v^{3}_{\Phi}v^{2}}{2}+\frac{v_{\Phi}v^{4}}{4}\right)\;.$
(19)

Due to this uplifting of degeneracy, the DW start to collapse and annihilate,
resulting in the production of SGWB whose signals can be tested in the present
and future GW detectors. Under the assumption that $v_{\Phi}\gg v$, a
sufficiently large $V_{\rm bias}$ can be achieved rendering the production of
observable GW signals. Note that to have a DW, a very large $V_{\rm bias}$
must be avoided according to the prediction of percolation theory
Saikawa:2017hiv . Under such a hierarchy, the bias potential is dominated by,

$V_{\rm bias}\simeq\frac{v^{5}_{\Phi}}{\Lambda_{\rm QG}}\;.$ (20)

Before delving into the details of GW production from the annihilation of the
DW, we first like to discuss the constraints that appear on the $V_{\rm bias}$
as a result of DW annihilations to the SM particles. If the energy bias of the
DW is very small, they live for a very long time ($t_{\rm
ann}\propto\frac{1}{V_{\rm bias}}$, with $t_{\rm ann}$ being the annihilation
time of the DW). Requiring that their collapse happens before they dominate
the energy budget of the Universe puts a lower bound on the magnitude of
$V_{\rm bias}$ Saikawa:2017hiv ,

$V_{\rm bias}^{1/4}>2.18\times 10^{-5}{\rm GeV}~{}C_{\rm
ann}^{1/4}\mathcal{A}^{1/2}\left(\frac{\sigma^{1/3}}{10^{3}\rm
GeV}\right)^{3/2},$ (21)

where $C_{\rm ann}\simeq 2$ is a dimensionless constant, $\mathcal{A}\simeq
0.8$ denotes the area parameter and
$\sigma=\sqrt{\frac{8\lambda_{\Phi}}{9}}v_{\Phi}^{3}$ is the surface tension
of the DW. Even if Eq. (21) is satisfied and the DW are annihilated away
before their dominance, their decay products (assuming they are the SM
particles) can still destroy the light elements created at BBN epoch. This
demands $t_{\rm ann}\leq 0.01$ s and hence an additional constraint on the
magnitude of the energy bias can be obtained as,

$V_{\rm bias}^{1/4}>5.07\times 10^{-4}{\rm GeV}~{}C_{\rm
ann}^{1/4}\mathcal{A}^{1/4}\left(\frac{\sigma^{1/3}}{10^{3}\rm
GeV}\right)^{3/4}.$ (22)

The production of the GW from DW annihilation has been investigated in detail,
for example, see refs. Vilenkin:1981zs ; Gelmini:1988sf ; Larsson:1996sp ;
Hiramatsu:2013qaa ; Hiramatsu:2012sc ; Saikawa:2017hiv ; Roshan:2024qnv ;
Bhattacharya:2023kws . Assuming that the DW annihilate instantaneously
($t=t_{\text{ann}}$) during the radiation-dominated era, the peak frequency
$f_{p}$ and peak energy density spectrum $\Omega_{p}h^{2}$ of GW at present
can be expressed as Saikawa:2017hiv ; Chen:2020wvu

$\displaystyle f_{p}$ $\displaystyle\simeq 3.75\times
10^{-9}~{}\text{Hz}\times C_{\rm
ann}^{-1/2}\mathcal{A}^{-1/2}\bigg{(}\frac{10^{3}~{}\text{GeV}}{\mathcal{\sigma}^{1/3}}\bigg{)}^{3/2}\bigg{(}\frac{V_{\text{bias}}^{1/4}}{10^{-3}~{}\text{GeV}}\bigg{)}^{2}\,,$
$\displaystyle\Omega_{p}h^{2}$ $\displaystyle\simeq 5.3\times
10^{-20}\times\tilde{\epsilon}_{\text{GW}}~{}C_{\rm
ann}^{2}\mathcal{A}^{4}\bigg{(}\frac{\sigma^{1/3}}{10^{3}~{}\text{GeV}}\bigg{)}^{12}\bigg{(}\frac{10^{-3}~{}\text{GeV}}{V_{\text{bias}}^{1/4}}\bigg{)}^{8},$
(23)

where $\tilde{\epsilon}_{\text{GW}}\simeq 0.7$ Hiramatsu:2013qaa  denotes the
fraction of energy radiated into GW and can be regarded as a constant in the
scaling regime. The typical feature of the GW spectrum from DW is that it
follows a broken power law, where the breaking point has a frequency
determined by the annihilation time and the peak amplitude is determined by
the energy density in the domain walls as can be seen from Eq. (V). To depict
the GW spectrum, we adopt the following parametrization for a broken power-law
spectrum Caprini:2019egz ; NANOGrav:2023hvm

$\displaystyle\Omega_{\rm
GW}h^{2}=\Omega_{p}h^{2}\frac{(a+b)^{c}}{\left(bx^{-a/c}+ax^{b/c}\right)^{c}}\
,$ (24)

where $x\equiv f/f_{p}$, and $a$, $b$ and $c$ are real and positive
parameters. Here the low-frequency slope $a=3$ can be fixed by causality,
while numerical simulations suggest $b\simeq c\simeq 1$ Hiramatsu:2013qaa .

Figure 5: The orange and purple curve show the GW spectrum generated from the
annihilation of the DW. In both the panels, we set $\lambda_{\Phi}=10^{-3}$
just for the demonstrative purpose. While in the left Panel we fix
$\Lambda_{\rm QG}=10^{33}~{}\rm{GeV}$ and vary $v_{\Phi}$, in the right panel
we fix $v_{\Phi}=10^{6}~{}\rm{GeV}$ and vary $\Lambda_{\rm QG}$. The gray
color region denotes the portion excluded from $\Delta N_{\rm eff}$ bound.

Apart from the constraint on DW from equations (21) and (22), a large
amplitude of GW from DW annihilation can also contribute to extra effective
number of relativistic species, $\Delta N_{\rm eff}$. This gives an upper
bound on the GW energy denisty. Considering the Planck 2018 results on $\Delta
N_{\rm eff}$ at the time of BBN, the constraint on GW energy density is given
as Domenech:2020ssp

$\displaystyle\Omega_{\rm
GW,BBN}<\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\Delta N_{\rm eff}\sim
0.05.$ (25)

Hence, the maximum energy density of GW today which is allowed from $\Delta
N_{\rm eff}$ bound can be written as

$\displaystyle\Omega_{\rm GW}^{\rm\Delta N_{\rm
eff}}h^{2}=0.39\left(\frac{g_{*}(T_{\rm
BBN})}{106.75}\right)^{-1/3}\Omega_{r,0}h^{2}\Omega_{\rm GW,BBN}\simeq
1.75\times 10^{-6},$ (26)

where $\Omega_{r,0}h^{2}\sim 4.18\times 10^{-5}$ represents current radiation
energy density fraction. Therefore, to be consistent with this bound, the peak
energy density of GW from DW, $\Omega_{p}h^{2}$, should be less than
$\Omega_{\rm GW}^{\rm\Delta N_{\rm eff}}h^{2}$.

In Fig. 5, we present the GW spectrum for different choices of model
parameters. For both the plots we set $\lambda_{\Phi}=10^{-3}$. In Fig. 5, the
gray violins indicate the GW spectrum observed by the NANOGrav. We notice that
the spectra shown by the orange color lines in the Fig. 5 are in good
agreement with the NANOGrav 15-year result. Such domain wall interpretation of
the NANOGrav 15-year data was also discussed in earlier works, see
Barman:2023fad  and references therein. Using Eq. (24), we also evaluate the
sensitivity curves of the future GW detectors ET Punturo:2010zz , LISA
LISA:2017pwj , DECIGO Kawamura:2020pcg , $\mu$Ares Sesana:2019vho , SKA
Janssen:2014dka , IPTA Hobbs_2010  and THEIA Garcia-Bellido:2021zgu  by
calculating the signal-to-noise ratio (SNR) Maggiore:1999vm ; Allen:1997ad

$\varrho=\left[n_{\mathrm{det}}t_{\mathrm{obs}}\int_{f_{\min}}^{f_{\max}}df\left(\frac{\Omega_{\text{signal
}}(f)}{\Omega_{\text{noise }}(f)}\right)^{2}\right]^{1/2}\;,$ (27)

where $n_{\rm det}=1$ for auto-correlated detectors and $n_{\rm det}=2$ for
cross-correlated detectors, $t_{\rm obs}$ denotes the observational time, and
$\Omega_{\text{noise }}$ represents the noise spectrum expressed in terms of
the GW energy density spectrum Schmitz:2020syl . Integrating $(\Omega_{\rm
signal}/\Omega_{\rm noise})^{2}$ over the sensitive frequency range of
individual GW detectors, we obtain the SNRs for the GW spectra. Now, in the
left panel, we fix the QG scale at $\Lambda_{\rm QG}=10^{33}$ GeV while we
vary $v_{\Phi}$. Following Eq. (V), one finds that for dimension 5 operator,
$\Omega_{p}h^{2}\propto(\lambda_{\Phi}^{2}\Lambda_{\rm QG}^{2})/v_{\Phi}^{2}$
while $f_{p}\propto v_{\Phi}/(\lambda_{\Phi}^{1/4}\Lambda_{\rm QG}^{1/2})$. As
expected, a larger $v_{\Phi}$ results in a smaller $\Omega_{p}h^{2}$ and a
larger peak frequency. On the other hand, a larger $\Lambda_{\rm QG}$ scale
corresponds to a larger $\Omega_{p}h^{2}$ but a smaller peak frequency, this
behavior is visible in the right panel of Fig. 5.

BPs |  $\Lambda_{\rm QG}$ (GeV) |  $v_{\Phi}$ (GeV) |  $m_{N}$ (GeV) |  $m_{\chi}$ (MeV) | $\Delta N_{\rm eff}$
---|---|---|---|---|---
BP1 | $10^{33}$ | $10^{7}$ | $10^{5}$ | 40 | 0.027
BP2 | $10^{33}$ | $10^{6}$ | $10^{4}$ | 4 | 0.027
Table 2: Set of benchmark values of $\Lambda_{\rm QG}$ and $v_{\Phi}$.

In Fig. 6, we summarise our result. For the demonstrative purpose, we consider
different benchmark points (BPs) shown in Table 2. While in the left panel, we
show BP1, in the right panel, we show BP2. In both plots, the light gray and
the yellow-shaded regions are excluded from the null detection of X-ray
signals originating from the DM decay and CMB observations respectively, as
discussed in the previous section. The black dashed lines label the testing
capabilities of the upcoming SKA telescope. The pink and gray-shaded regions
are excluded from the constraints coming from the DW overclosure and BBN. The
BBN limit arises from the requirement of not disrupting light nuclei abundance
from DW annihilations at late epochs. We would also like to point out that the
$\Delta N_{\rm eff}$ bounds on the GW is much weaker than the one obtained
from the DW overclosure and hence we do not show it in the plots. The
parameter space below the horizontal dot-dashed line remains within the
sensitivity of future CMB experiment CMB-HD CMB-HD:2022bsz  which can measure
the enhanced $\Delta N_{\rm eff}$ due to light majoron. Next, we mark the
threshold SNR $\varrho=10$ for different detectors by the solid-colored
curves, which shows the parameter ranges detectable by these GW detectors.
From Fig. 6 and Fig. 5, it is clear that the BPs considered lie within the
detectable ranges of almost all the GW detectors except ET. Note that even
though the BP1 is allowed by the constraints that come from the indirect
searches of the DM and also lies within the reach of different GW experiments,
it still gets disallowed as it falls in the region of parameter space where
the DW can overclose the Universe. Finally, the predictability of the model is
also enhanced as one of the BPs considered in Table 2 is already disfavored
while other parameter space remain within reach of GW, CMB and DM indirect
detection experiments. As can be seen from the summary plots shown in Fig. 6,
the successful DM, DW and GW phenomenology requires the QG scale to be
$\Lambda_{\rm QG}\gtrsim 10^{21}$ GeV. The scale needs to be even higher for
the GW emitted by DW annihilation to remain within experimental sensitivities.
Such large QG scale can be natural in theories where the global symmetries are
broken by non-perturbative instanton effects for example, D-brane in string
theory Blumenhagen:2006xt ; Florea:2006si ; Blumenhagen:2009qh  or
gravitational instantons Giddings:1987cg ; Lee:1988ge ; Abbott:1989jw ;
Coleman:1989zu .

Figure 6: Combined constraints on $\Lambda_{\rm QG}$ and $v_{\Phi}$ from
indirect DM detection and GW observations with $\lambda_{\Phi}=10^{-3}$. The
light gray and yellow-shaded regions denote the excluded regions by X-ray and
CMB observations respectively. The black dashed lines label the testing
capabilities of the upcoming SKA telescope. The black dots correspond to the
BPs shown in Table 2. The red, blue, brown, cyan, purple, green, and orange
curves correspond to the sensitivities of THEIA, SKA, $\mu$Ares, DECIGO, LISA,
ET, and IPTA with SNR = 10. The gray-shaded regions are excluded by the
requirement of BBN. The pink-shaded region corresponds to the scenario where
DW may overclose the Universe at an early epoch. The region below the
horizontal dot-dashed line is within reach of CMB-HD experiment.

###  V.1 Analysis with dimension 7 operator

As seen from the above discussions, the lower bound on the QG scale from DM
and DW requirements remains high because explicit $Z_{2}$-breaking higher
dimensional operators arise at dimension five level in the minimal singlet
majoron model. It is possible to have successful DM and GW phenomenology with
a smaller QG scale if higher dimensional operators arise with dimensions more
than five. While in the minimal model discussed in this work, we can not
prevent dimension five operators suppressed by the QG scale, it is possible
with additional gauge symmetries that prevent such operators up to certain
dimensions. Similar ideas have already been adopted in the context of the
axion quality problem Barr:1992qq ; Dias:2002gg ; Carpenter:2009zs . Just for
the demonstrative purpose, we consider a dimension 7 operator (as it also
breaks the $Z_{2}$ symmetry explicitly) and write the energy bias after
replacing field $\Phi$ with its VEV, this leads to a bias contribution to the
effective potential of the form

$V_{\rm bias}\simeq\frac{v^{7}_{\Phi}}{\Lambda^{3}_{\rm QG}}\;.$ (28)

Following this, in Fig. 7, we set $\Lambda_{\rm QG}=M_{\rm Pl}$, where $M_{\rm
Pl}$ denotes Planck mass and then we show the GW spectrum for different
choices of the model parameters. Now from Eq. (V), we find that for the
dimension 7 operator, $\Omega_{p}h^{2}\propto(\lambda_{\Phi}^{2}M_{\rm
Pl}^{6})/v_{\Phi}^{2}$ and $f_{p}\propto
v_{\Phi}^{2}/(\lambda_{\Phi}^{1/4}M^{3/2}_{\rm Pl})$. Clearly, the presence of
$M_{\rm Pl}^{6}$ in numerator provides a huge enhancement in the peak
amplitude for the dimension 7 operator. To compensate for such enhancement, a
larger value of $v_{\Phi}$ together with a relatively smaller value of
$\lambda_{\Phi}$ are required. Fig. 7 shows that with $v_{\Phi}\sim 10^{12}$
GeV, $\lambda_{\Phi}\sim 10^{-9}$ and $\Lambda_{\rm QG}=M_{\rm Pl}$, the GW
spectrum generated by the DW annihilation remains within reach of different
near future GW experiments.

Similar to $Z_{2}$-breaking bias term, DM decay can also occur due to
dimension 7 operator of the form

$-\mathcal{L}_{\rm decay}\supset\frac{Y_{D_{\alpha}}}{\Lambda^{3}_{\rm
QG}}\overline{\ell_{L\alpha}}\tilde{H}\chi_{R}\Phi^{3}+{\rm h.c.},$ (29)

We choose a different benchmark point (BP3) to illustrate the combined result
for DM and GW phenomenology. The chosen values for BP3 are :
$v_{\Phi}=10^{12}$ GeV, $\lambda_{\Phi}=10^{-9}$, $m_{N}=200$ GeV,
$m_{\chi}=100$ MeV, $\Lambda_{\rm QG}=M_{\rm Pl}$ and majoron mass in sub-eV
scale. For BP3 the mass of $\phi$ is found to be $m_{\phi}=4.5\times 10^{7}$
GeV. Although to satisfy correct DM relic, it is sufficient to consider
$2m_{N}<m_{\phi}$ as discussed in sub-section III.1 and III.2, $m_{N}$ around
TeV scale requires a DM mass around GeV scale (e.g., $m_{N}=10^{4}$ GeV
requires $m_{\chi}\sim 1$ GeV). Now, a GeV scale DM mass along with
$v_{\Phi}=10^{12}$ GeV and $\Lambda_{\rm QG}=M_{\rm Pl}$ results in short DM
lifetime in tension with indirect detection limits. Therefore, we choose a
smaller $m_{N}=200$ GeV and hence a smaller $m_{\chi}=100$ MeV for BP3. The
evolution of relevant number densities together with DM number density for BP3
are shown in the left plot of Fig. 8. The final freeze-in abundance of DM,
dominantly produced from decay, is consistent with the observed DM abundance.
In the right panel of Fig. 8, we summarise our result for the dimension 7
operator. Here, we notice that the QG scale can be reduced to the Planck scale
but only for a relatively large value of $v_{\Phi}$ and a smaller value of
$\lambda_{\Phi}$. Note that the price we pay to achieve the desired DM and GW
phenomenology is a relatively small RHN mass (sub-TeV). As a result, its
interaction with majoron becomes very feeble and the majorons never
thermalises. This is the reason why we do not see the $\Delta N_{\rm eff}$
bounds from thermalised majoron in Fig. 8. Despite this, the parameter space
and BP3 in particular, remain within reach of near-future GW experiments like
LISA and DM indirect detection experiments.

Figure 7: The orange and purple curves show the GW spectrum generated from the
annihilation of the DW. In both the panels we set $\Lambda_{\rm QG}=M_{\rm
Pl}$ while fixing $\lambda_{\Phi}=10^{-9}$ (left panel) and
$v_{\Phi}=10^{12}~{}\rm{GeV}$ (right panel). The gray color region denotes the
portion excluded from $\Delta N_{\rm eff}$ bound on energy density of GW.

Figure 8: Left panel: evolution of comoving number densities for BP3:
$v_{\Phi}=10^{12}$ GeV, $\lambda_{\Phi}=10^{-9}$ (which gives
$m_{\phi}=4.5\times 10^{7}$ GeV), $m_{N}=200$ GeV and $m_{\chi}=100$ MeV.
Right panel: combined constraints on $\Lambda_{\rm QG}$ and $v_{\Phi}$ with
$\lambda_{\Phi}=10^{-9}$. The light gray and yellow-shaded regions denote the
excluded regions by $\gamma$-ray and CMB observations respectively with DM
mass 100 MeV. The black dashed line shows the sensitivity of upcoming SKA
telescope. The black dot corresponds to BP3. The red, blue, brown, cyan,
purple, and green curves represent the sensitivities of THEIA, SKA, DECIGO,
LISA, ET and IPTA with SNR = 10. The gray-shaded regions are excluded by the
requirement of BBN. The pink-shaded region corresponds to the scenario where
DW may overclose the Universe at an early epoch.

##  VI Conclusion

We have studied the possibility of testing quantum gravity effects in the
singlet majoron model with dark matter. The spontaneous breaking of global
lepton number symmetry generates right-handed neutrino mass or type-I seesaw
scale dynamically. Explicit $U(1)_{L}$ breaking of the quadratic type not only
leads to the mass of majoron, the pseudo-NG boson but also reduces the
$U(1)_{L}$ symmetry to a $Z_{2}$. The effective spontaneous breaking of
$Z_{2}$ symmetry then leads to the formation of domain walls. We consider
higher dimensional $Z_{2}$-breaking operators suppressed by the QG scale as
bias necessary for the domain walls to annihilate away while generating
stochastic gravitational waves. The same QG scale-suppressed operators also
lead to the decay of dark matter. In the minimal singlet majoron model, such
QG scale suppressed operators can arise at dimension five level and get
tightly constrained from DW, GW as well as DM lifetime criteria. These
constraints make the non-thermal production of DM via singlet scalar portal
couplings more realistic. We consider the majoron to be in the sub-eV regime
and produced thermally. This brings another detection prospect as light
majoron can lead to observable $\Delta N_{\rm eff}$ within reach of future CMB
experiments. While the scale of QG is required to be $\Lambda_{\rm QG}\gtrsim
10^{25}$ GeV for detectable GW from DW annihilation while satisfying indirect
detection bounds on decaying DM, it can be brought down to the Planck scale if
explicit global symmetry operators are considered at dimension 7 level instead
of dimension 5.

To conclude, the minimal singlet majoron model provides a dynamical origin of
the seesaw scale while also accommodating a stable DM without any additional
discrete symmetries. The heavy right handed neutrinos not only generate light
neutrino masses, but can also lead to successful leptogenesis with resonantly
enhanced CP asymmetry. Quantum gravity effects, providing an origin of
explicit global symmetry breaking, contribute to the generation of GW from
domain wall annihilations while also opening up possible indirect detection
aspects due to DM decay. Moreover, along with GW signatures, an observable
$\Delta N_{\rm eff}$ keeps the detection prospects of the model very promising
while providing a common solution to the neutrino mass, dark matter and baryon
asymmetry puzzles.

###### Acknowledgements.

The work of D.B. is supported by the Science and Engineering Research Board
(SERB), Government of India grants MTR/2022/000575 and CRG/2022/000603. D.B.
also acknowledges the support from the Simons Foundation (Award
Number:1023171-RC) to visit the International Institute of Physics, Natal,
Brazil in May 2024 when part of this work was completed. R.R. acknowledges
financial support from the STFC Consolidated Grant ST/T000775/1. The work of
N.D. is supported by the Ministry of Education, Government of India via the
Prime Minister’s Research Fellowship (PMRF) December 2021 scheme. N.D. would
like to thank Disha Bandyopadhyay for useful discussions. The authors would
also like to express special thanks to the organisers of the Workshop in High
Energy Physics Phenomenology (WHEPP XVII), IIT Gandhinagar, India where this
project was initiated.

##  Appendix A Coupling of CP-odd and CP-even scalars with $N$ and $\chi$

The interaction terms between $\Phi$ and $N_{R}$ are

$\frac{f_{i}}{2}\Phi\overline{N^{c}_{{R}_{i}}}N_{{R}_{i}}+\frac{f_{i}}{2}\Phi^{*}\overline{N_{{R}_{i}}}N^{c}_{{R}_{i}}.$
(30)

Writing $N_{i}=N_{{R}_{i}}+N^{c}_{{R}_{i}}=P_{R}N_{i}+P_{L}N_{i}$, we get-

$\frac{f_{i}}{2}\Phi\overline{P_{L}N_{i}}P_{R}N_{i}+\frac{f_{i}}{2}\Phi^{*}\overline{P_{R}N_{i}}P_{L}N_{i}\\\
=\frac{f_{i}}{2}\Phi\overline{N_{i}}P_{R}N_{i}+\frac{f_{i}}{2}\Phi^{*}\overline{N_{i}}P_{L}N_{i}.$

After SSB, we can write $\Phi=\frac{1}{\sqrt{2}}(\phi+v_{\Phi}+i\eta)$. This
gives -

$\displaystyle\frac{f_{i}}{2}\Phi\overline{N_{i}}P_{R}N_{i}+\frac{f_{i}}{2}\Phi^{*}\overline{N_{i}}P_{L}N_{i}$
(31) $\displaystyle=$
$\displaystyle\frac{f_{i}}{2}\frac{1}{\sqrt{2}}(\phi+v_{\Phi}+i\eta)\overline{N_{i}}P_{R}N_{i}+\frac{f_{i}}{2}\frac{1}{\sqrt{2}}(\phi+v_{\Phi}-i\eta)\overline{N_{i}}P_{L}N_{i}$
$\displaystyle=$
$\displaystyle\frac{f_{i}}{2\sqrt{2}}\phi\overline{N_{i}}N_{i}+\frac{f_{i}}{2\sqrt{2}}v_{\Phi}\overline{N_{i}}N_{i}+i\frac{f_{i}}{2\sqrt{2}}\eta\overline{N_{i}}\gamma^{5}N_{i}.$

Similarly the interaction terms between $\Phi$ and DM give -

$\displaystyle
y_{\chi}\Phi\overline{\chi_{L}}\chi_{R}+y_{\chi}\Phi^{*}\overline{\chi_{R}}\chi_{L}$
(32) $\displaystyle=$
$\displaystyle\frac{y_{\chi}}{\sqrt{2}}\phi\overline{\chi}\chi+\frac{y_{\chi}}{\sqrt{2}}v_{\Phi}\overline{\chi}\chi+i\frac{y_{\chi}}{\sqrt{2}}\eta\overline{\chi}\gamma^{5}\chi,$

where $\chi=\chi_{L}+\chi_{R}$.

##  Appendix B Cross-section

$\displaystyle(\sigma v)_{N\eta\to
N\eta}=\frac{f^{4}}{2048\pi\left(\left(m_{\eta}^{2}+m_{N}^{2}-s\right){}^{2}-4m_{\eta}^{2}m_{N}^{2}\right)}\Biggl{(}\frac{(-2m_{N}^{2}\left(m_{\eta}^{2}+s\right)+m_{N}^{4}+\left(s-m_{\eta}^{2}\right))}{s^{2}}\times$
$\displaystyle\Biggl{(}-2m_{\eta}^{2}+\frac{m_{\eta}^{4}\left(s-m_{N}^{2}\right){}^{2}\left(sm_{N}^{2}+m_{N}^{4}+2s^{2}\right)+2m_{\eta}^{10}\left(m_{N}^{2}+s\right)-m_{\eta}^{8}\left(m_{N}^{2}+s\right)\left(3m_{N}^{2}+s\right)}{\left(s-m_{N}^{2}\right){}^{2}\left(2m_{\eta}^{6}-m_{\eta}^{4}\left(3m_{N}^{2}+s\right)+\left(m_{N}^{3}-sm_{N}\right)^{2}\right)}$
$\displaystyle+m_{N}^{2}+5s\Biggl{)}-\frac{2\left(2m_{\eta}^{4}+\left(s-m_{N}^{2}\right){}^{2}\right)}{s-m_{N}^{2}}\log\Biggl{(}\frac{m_{\eta}^{4}-m_{N}^{2}\left(2m_{\eta}^{2}+s\right)+m_{N}^{4}}{s\left(2m_{\eta}^{2}+m_{N}^{2}-s\right)}\Biggl{)}\Biggl{)}.$
(33)

$\displaystyle(\sigma v)_{\phi\phi\to N\bar{N}}=-\frac{f^{4}}{512\pi
s\left(s-4m_{\phi}^{2}\right)}\times$
$\displaystyle\Biggl{(}s\sqrt{\frac{\left(s-4m_{N}^{2}\right)\left(s-4m_{\phi}^{2}\right)}{s^{2}}}+\left(8m_{N}^{2}-2m_{\phi}^{2}+s\right)\log\left(\frac{s\left(\sqrt{\frac{\left(s-4m_{N}^{2}\right)\left(s-4m_{\phi}^{2}\right)}{s^{2}}}-1\right)+2m_{\phi}^{2}}{2m_{\phi}^{2}-s\left(\sqrt{\frac{\left(s-4m_{N}^{2}\right)\left(s-4m_{\phi}^{2}\right)}{s^{2}}}+1\right)}\right)$
$\displaystyle+\frac{s\left(16m_{N}^{4}+m_{N}^{2}\left(s-12m_{\phi}^{2}\right)+2m_{\phi}^{4}\right)\sqrt{\frac{\left(s-4m_{N}^{2}\right)\left(s-4m_{\phi}^{2}\right)}{s^{2}}}}{m_{N}^{2}\left(s-4m_{\phi}^{2}\right)+m_{\phi}^{4}}$
$\displaystyle-\frac{2\left(-16m_{N}^{4}+4m_{N}^{2}s+m_{\phi}^{4}\right)\log\left(\frac{s\left(\sqrt{\frac{\left(s-4m_{N}^{2}\right)\left(s-4m_{\phi}^{2}\right)}{s^{2}}}-1\right)+2m_{\phi}^{2}}{2m_{\phi}^{2}-s\left(\sqrt{\frac{\left(s-4m_{N}^{2}\right)\left(s-4m_{\phi}^{2}\right)}{s^{2}}}+1\right)}\right)}{2m_{\phi}^{2}-s}\Biggl{)}$
(34)

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Chuuk | Fd. St. Micronesia | 1.01 | 3.0 | 4.8 | 3.2 | 4159.7 | 9376.6 | 10166.9 | 24077.9 | 35763.0 | 35998.8
Kapingamarangi | Fd. St. Micronesia | 0.67 | 765.7 | 1032.3 | 1298.7 | 32780.5 | 43647.5 | 49632.2 | 58552.4 | 68626.0 | 69991.1
Pohnpei-B | Fd. St. Micronesia | 0.72 | 556.1 | 786.6 | 761.2 | 25024.7 | 38395.8 | 43866.7 | 53733.5 | 64683.1 | 66171.7
Yap-B | Fd. St. Micronesia | 2.64 | 1.2 | 1.3 | 1.2 | 29.7 | 137.2 | 148.4 | 1287.5 | 1900.4 | 1612.4
Suva-C | Fiji | 0.73 | 1006.2 | 1301.6 | 1352.0 | 46453.7 | 50845.0 | 62450.7 | 73297.6 | 79385.0 | 77276.1
Noumea | France | 0.54 | 2716.1 | 3967.5 | 3795.2 | 43692.9 | 55365.7 | 64369.5 | 62814.9 | 74443.8 | 71836.0
Brest | France | 1.87 | 115.4 | 97.7 | 134.9 | 1041.6 | 1469.7 | 1587.8 | 4800.2 | 7394.6 | 5620.2
Marseille | France | 0.98 | 22.8 | 1545.3 | 32.2 | 2845.9 | 16949.3 | 4078.8 | 17836.1 | 31071.6 | 25817.0
Rikitea | French Polynesia | 0.34 | 17769.8 | 27627.1 | 27724.8 | 74777.4 | 80591.4 | 82129.7 | 79346.0 | 84219.9 | 82615.2
Papeete-B | French Polynesia | 0.94 | 2.9 | 3.9 | 3.8 | 7289.1 | 12427.6 | 14575.5 | 36066.2 | 50044.1 | 48578.7
Cuxhaven | Germany | 4.09 | 2.0 | 2.1 | 2.0 | 5.7 | 14.1 | 6.5 | 294.0 | 248.7 | 222.8
Malin Head | Ireland | 1.68 | 7.0 | 10.0 | 4.7 | 463.4 | 861.2 | 429.7 | 3062.6 | 2695.1 | 2733.3
Hakodate | Japan | 0.61 | 306.9 | 552.7 | 1445.1 | 13464.2 | 19964.6 | 27948.1 | 32966.9 | 45433.9 | 47047.5
Hamada | Japan | 1.30 | 2.8 | 3.3 | 4.5 | 1618.3 | 2299.6 | 3735.2 | 12026.5 | 20931.3 | 22583.2
Maisaka | Japan | 2.48 | 1.1 | 1.1 | 1.2 | 1.9 | 3.6 | 14.3 | 749.8 | 944.4 | 912.8
Ishigaki | Japan | 1.09 | 57.3 | 83.1 | 154.0 | 8209.6 | 11067.3 | 16926.9 | 31923.0 | 44837.0 | 46038.4
Naha | Japan | 1.17 | 13.7 | 22.7 | 56.1 | 4885.3 | 7073.3 | 12090.8 | 24776.0 | 37374.6 | 38516.4
Toyama | Japan | 0.75 | 196.6 | 415.9 | 1216.8 | 23548.6 | 33811.3 | 41563.2 | 51716.9 | 64782.3 | 63765.6
Hosojima | Japan | 1.56 | 2.0 | 2.3 | 2.7 | 589.9 | 748.6 | 1109.7 | 4699.7 | 7966.2 | 8917.5
Kushiro | Japan | 0.78 | 8512.9 | 14900.2 | 19000.6 | 90460.1 | 88966.1 | 90583.9 | 91285.0 | 90484.2 | 91212.0
Abashiri | Japan | 0.85 | 162.1 | 513.4 | 610.4 | 12710.8 | 20272.1 | 25672.6 | 39882.6 | 50971.5 | 52256.1
Mera | Japan | 1.20 | 9.5 | 11.6 | 22.8 | 7026.7 | 11955.7 | 16064.0 | 38523.0 | 52089.8 | 53518.6
Wakkanai | Japan | 0.99 | 15.1 | 43.1 | 106.8 | 9517.2 | 13334.5 | 20652.1 | 39038.1 | 51548.3 | 52521.4
Chichijima | Japan | 1.19 | 5.4 | 19.4 | 22.3 | 7002.6 | 9048.4 | 14424.1 | 29332.4 | 41111.9 | 42764.4
Nishinoomote | Japan | 1.04 | 46.7 | 73.9 | 181.8 | 8063.7 | 12476.5 | 17797.7 | 31876.0 | 45726.1 | 47144.8
Naze | Japan | 1.29 | 10.1 | 13.6 | 30.6 | 4535.2 | 6523.4 | 10243.3 | 24213.2 | 37680.9 | 38863.4
Hachinohe | Japan | 0.82 | 455.1 | 641.7 | 1679.5 | 38015.3 | 49823.7 | 55645.8 | 70360.0 | 78037.2 | 77295.2
Miyakejima | Japan | 1.68 | 8.7 | 9.5 | 14.5 | 5691.5 | 7779.2 | 11104.7 | 42015.4 | 55128.9 | 57206.1
Nakano Shima | Japan | 1.55 | 2.8 | 3.1 | 4.3 | 1595.5 | 2033.6 | 3230.7 | 12922.6 | 21883.8 | 23538.5
Ofunato | Japan | 0.74 | 5692.6 | 8106.9 | 14596.3 | 85635.4 | 88070.9 | 86722.2 | 91097.5 | 90633.4 | 90151.2
Nagasaki | Japan | 1.05 | 155.0 | 230.5 | 380.2 | 7696.2 | 11484.2 | 15893.4 | 27698.3 | 40991.7 | 41803.5
Aburatsu | Japan | 1.53 | 3.1 | 3.7 | 4.9 | 941.1 | 1229.0 | 1932.8 | 7726.6 | 13828.8 | 15228.5
Kushimoto | Japan | 0.92 | 365.5 | 502.3 | 973.7 | 26794.1 | 37762.8 | 44224.2 | 61662.8 | 72109.4 | 71444.6
Cendering | Malaysia | 1.29 | 12.8 | 17.3 | 22.4 | 2219.1 | 3586.5 | 4902.8 | 14357.5 | 22999.0 | 22493.2
Johor Baharu | Malaysia | 1.00 | 266.2 | 353.6 | 435.4 | 9829.7 | 14816.3 | 19296.0 | 32162.4 | 43810.1 | 43972.5
Kuantan | Malaysia | 1.29 | 33.8 | 45.8 | 58.8 | 3175.4 | 5016.7 | 6817.2 | 18037.4 | 27665.7 | 27214.0
Keling | Malaysia | 0.80 | 390.5 | 514.9 | 642.1 | 13908.9 | 19823.4 | 25568.4 | 35632.2 | 46594.1 | 46660.6
Lumut | Malaysia | 0.88 | 409.8 | 558.8 | 700.9 | 12477.2 | 18077.1 | 22858.0 | 33750.8 | 45350.8 | 44870.4
Kelang | Malaysia | 1.40 | 130.7 | 162.4 | 196.4 | 2893.0 | 3988.5 | 5357.5 | 14546.5 | 21481.1 | 20934.8
Langkawi | Malaysia | 0.97 | 385.7 | 542.3 | 685.2 | 9124.7 | 13723.9 | 17524.9 | 28361.7 | 39755.8 | 39224.2
Penang | Malaysia | 0.94 | 225.7 | 349.7 | 480.9 | 11125.0 | 16447.1 | 20902.1 | 32935.2 | 44464.6 | 44063.9
Port Louis-C | Mauritius | 0.49 | 1848.3 | 3257.4 | 5610.2 | 38976.8 | 53603.6 | 57972.4 | 58577.8 | 70815.7 | 71760.1
Rodrigues | Mauritius | 1.42 | 2.8 | 3.2 | 3.9 | 1623.6 | 2260.9 | 2897.7 | 12859.2 | 21911.1 | 21991.9
Manzanillo-A | Mexico | 0.72 | 1131.0 | 1937.6 | 2033.4 | 55303.8 | 64018.1 | 68917.4 | 79934.6 | 82282.4 | 83662.1
Ensenada | Mexico | 0.73 | 1267.7 | 1954.2 | 2094.0 | 24514.4 | 34709.0 | 40349.5 | 53663.9 | 63585.7 | 65056.7
Salina Cruz | Mexico | 0.89 | 50.1 | 106.9 | 101.5 | 14457.5 | 23775.3 | 27655.5 | 48136.9 | 60529.7 | 61396.3
Acapulco-A, Gro. | Mexico | 0.90 | 118.8 | 271.7 | 237.9 | 39725.2 | 50443.3 | 56119.2 | 77573.4 | 80842.2 | 82411.1
Cabo San Lucas | Mexico | 0.64 | 1468.5 | 2228.6 | 2324.3 | 29384.7 | 40279.6 | 45597.8 | 55212.5 | 65245.7 | 66217.7
Guaymas | Mexico | 0.72 | 807.9 | 1353.9 | 1916.9 | 45643.1 | 55248.7 | 60049.0 | 75171.2 | 79552.5 | 80992.7
Saipan-B | N. Mariana Islands | 3.85 | 1.1 | 1.1 | 1.1 | 1.2 | 1.3 | 1.3 | 282.0 | 385.1 | 316.3
Marsden Point | New Zealand | 5.38 | 1.1 | 1.1 | 1.1 | 1.2 | 1.2 | 1.2 | 1.3 | 21.3 | 1.4
Tauranga | New Zealand | 0.63 | 667.2 | 2631.1 | 1973.5 | 27072.0 | 46343.4 | 51139.3 | 52363.9 | 71123.0 | 71050.7
Taranaki | New Zealand | 3.99 | 1.2 | 1.2 | 1.2 | 1.4 | 1.5 | 1.6 | 183.3 | 200.9 | 173.9
Wellington | New Zealand | 0.65 | 857.0 | 3345.0 | 2400.7 | 34912.3 | 54956.9 | 59242.1 | 61140.5 | 78363.7 | 78370.3
Tregde | Norway | 1.18 | 9.3 | 28.4 | 7.7 | 1168.3 | 2385.1 | 1331.5 | 6488.8 | 6766.6 | 6707.0
Rorvik | Norway | 2.39 | 1.1 | 1.2 | 1.1 | 11.5 | 131.0 | 14.3 | 643.0 | 591.0 | 520.2
Ny-Alesund | Norway | 0.82 | 4.9 | 573.3 | 0.1 | 229.0 | 802.7 | 192.6 | 2165.0 | 961.3 | 1338.6
Vardo | Norway | 1.37 | 6.6 | 93.7 | 13.9 | 594.2 | 1953.8 | 972.7 | 3144.4 | 5343.2 | 3985.6
Balboa | Panama | 1.38 | 263.7 | 335.7 | 362.2 | 3156.8 | 5072.9 | 5620.0 | 14822.8 | 25219.9 | 24356.5
Cristobal | Panama | 0.40 | 9278.2 | 13596.8 | 16804.0 | 65421.6 | 75619.8 | 82205.6 | 73819.6 | 81834.0 | 83668.5
Rabaul | Papua New Guinea | 0.38 | 11014.3 | 14377.8 | 17497.4 | 64795.2 | 71141.3 | 77891.7 | 74111.8 | 80455.4 | 81473.9
Lobos de Afuera | Peru | 0.72 | 276.2 | 573.8 | 575.1 | 17694.5 | 26444.9 | 31047.5 | 42506.6 | 55264.3 | 55909.7
Callao-B | Peru | 0.80 | 8.2 | 21.4 | 24.4 | 5680.1 | 9493.2 | 11853.5 | 24474.1 | 37730.3 | 37804.3
Legaspi | Philippines | 0.77 | 672.4 | 1280.7 | 1393.6 | 39288.3 | 52445.8 | 60979.7 | 70550.7 | 77203.3 | 79735.2
Manila | Philippines | 0.98 | 1466.0 | 2092.1 | 2382.9 | 71211.7 | 78505.8 | 83330.9 | 90527.9 | 89315.4 | 90575.8
Cascais | Portugal | 1.11 | 157.1 | 133.1 | 230.1 | 3470.9 | 5448.2 | 7299.2 | 17329.6 | 27031.0 | 26251.7
Funchal-B | Portugal | 0.71 | 2580.3 | 2331.7 | 3677.8 | 27776.2 | 36911.4 | 43573.5 | 52316.2 | 62627.8 | 67887.2
Kanton-B | Rep. of Kiribati | 0.66 | 300.1 | 460.6 | 522.5 | 23931.7 | 34077.5 | 38042.5 | 50457.3 | 61644.6 | 63473.8
Christmas-B | Rep. of Kiribati | 0.52 | 2621.6 | 4558.7 | 4627.9 | 54577.5 | 61850.0 | 65740.1 | 69963.3 | 76608.6 | 78094.6
Majuro-A | Rep. of Marshall I | 0.65 | 3486.6 | 4463.4 | 4341.8 | 43283.8 | 53465.3 | 60070.9 | 65882.6 | 73996.2 | 76056.2
Kwajalein | Rep. of Marshall I | 0.55 | 7535.2 | 9514.0 | 9580.9 | 57710.2 | 68122.4 | 73093.5 | 73831.2 | 78943.6 | 81762.3
Malakal-B | Republic of Belau | 0.56 | 4020.4 | 5228.4 | 5829.8 | 46135.4 | 62482.2 | 67019.1 | 68355.4 | 76810.0 | 77964.2
Kaohsiung | Republic of China | 1.07 | 3.5 | 4.8 | 7.6 | 2745.8 | 4086.2 | 6102.2 | 14794.9 | 23462.8 | 23993.9
Keelung | Republic of China | 1.43 | 3.4 | 3.8 | 4.9 | 1656.4 | 2444.9 | 3500.2 | 13853.1 | 22131.6 | 23450.5
Luderitz | South Africa | 0.64 | 1614.5 | 2063.5 | 2531.6 | 34663.2 | 46826.7 | 54805.1 | 58991.6 | 70359.4 | 73323.0
Saldahna Bay | South Africa | 0.63 | 2972.8 | 3519.5 | 4297.4 | 36671.4 | 48734.4 | 54863.8 | 58856.5 | 69929.0 | 72186.4
Simon’s Town | South Africa | 0.80 | 495.8 | 681.2 | 872.5 | 20788.9 | 31289.8 | 38108.3 | 51010.2 | 64791.4 | 66982.6
Port Nolloth | South Africa | 0.72 | 1312.4 | 1631.3 | 1750.4 | 26278.7 | 37710.8 | 44505.3 | 53500.3 | 66525.6 | 69169.3
Port Elizabeth | South Africa | 0.94 | 188.3 | 296.3 | 315.1 | 8572.9 | 13846.3 | 18411.2 | 31228.4 | 46492.7 | 47493.2
La Coruna | Spain | 1.22 | 364.1 | 306.5 | 413.6 | 4292.4 | 6156.7 | 7399.9 | 19453.0 | 27394.6 | 26973.1
Ceuta | Spain | 0.60 | 339.5 | 731.1 | 1008.9 | 12885.0 | 19906.2 | 26779.3 | 37106.1 | 49493.7 | 52859.9
Vigo | Spain | 1.63 | 8.1 | 7.5 | 9.8 | 884.8 | 1383.2 | 1635.4 | 7279.0 | 11009.7 | 9524.1
Stockholm | Sweden | 1.14 | 3.1 | 6292.0 | 1.6 | 526.3 | 16766.7 | 528.8 | 1993.9 | 418.2 | 2212.6
Goteborg-Torsh. | Sweden | 1.69 | 2.7 | 51.3 | 2.2 | 296.8 | 2278.5 | 301.7 | 2120.0 | 3355.7 | 1873.8
Zanzibar | Tanzania | 1.13 | 960.4 | 1144.4 | 1236.9 | 7677.9 | 11082.1 | 14232.0 | 24700.0 | 37153.7 | 37779.2
Ko Lak | Thailand | 1.13 | 276.0 | 360.1 | 418.6 | 9199.4 | 14085.2 | 18055.8 | 33798.1 | 45554.8 | 45542.2
Stornoway | United Kingdom | 1.56 | 81.1 | 74.4 | 62.4 | 1003.9 | 1428.9 | 1030.0 | 5066.7 | 4250.9 | 4684.9
Lerwick | United Kingdom | 1.17 | 17.6 | 30.9 | 13.2 | 1336.2 | 2017.8 | 1346.9 | 6700.9 | 6205.4 | 6569.6
Faraday | United Kingdom | 1.77 | 1.3 | 1.4 | 1.3 | 1.8 | 2.0 | 2.6 | 279.2 | 1830.0 | 1590.9
Gibraltar-A | United Kingdom | 0.63 | 116.6 | 408.4 | 543.8 | 7593.1 | 12690.0 | 16945.0 | 26366.4 | 38980.4 | 41214.7
Bermuda-B | United Kingdom | 0.79 | 177.3 | 347.4 | 366.9 | 15488.1 | 21459.5 | 25739.3 | 43454.5 | 52731.1 | 49505.4
Newlyn, Cornwall | United Kingdom | 1.39 | 231.9 | 233.1 | 223.2 | 2478.0 | 3797.5 | 3131.7 | 10573.4 | 12929.0 | 13739.5
Seward-C, AK | USA | 1.78 | 0.6 | 0.8 | 0.7 | 133.1 | 186.8 | 135.2 | 1223.6 | 1455.2 | 1350.9
Ketchikan, AK | USA | 1.81 | 6.4 | 6.5 | 7.8 | 446.7 | 522.4 | 559.6 | 2715.9 | 2786.7 | 3318.2
Valdez, AK | USA | 1.69 | 0.3 | 0.6 | 0.4 | 88.2 | 169.3 | 92.6 | 1024.5 | 1258.7 | 1116.4
Yakutat, AK | USA | 1.42 | 0.0 | 0.0 | 0.0 | 57.4 | 109.3 | 64.6 | 694.6 | 660.5 | 677.3
Seldovia, AK | USA | 2.37 | 0.2 | 0.2 | 0.2 | 3.9 | 5.1 | 4.0 | 306.7 | 343.9 | 300.5
Sitka, AK | USA | 1.55 | 1.5 | 1.4 | 1.7 | 338.6 | 395.8 | 400.5 | 2107.0 | 2082.7 | 2409.4
Sand Point, AK | USA | 1.37 | 24.6 | 27.6 | 33.7 | 1609.8 | 1866.0 | 2447.9 | 9292.2 | 9977.8 | 14117.1
Dutch Harbor-B, AK | USA | 0.82 | 8.9 | 9.7 | 12.8 | 1408.5 | 1743.0 | 2193.2 | 5431.6 | 5939.9 | 7931.9
Cordova-B, AK | USA | 1.92 | 4.7 | 6.5 | 5.6 | 623.3 | 796.4 | 768.0 | 5696.0 | 6027.9 | 7050.1
Kodiak Isl., AK | USA | 2.09 | 0.5 | 0.5 | 0.5 | 1.0 | 1.8 | 1.2 | 356.5 | 343.7 | 322.7
Adak, AK | USA | 1.11 | 5.2 | 8.4 | 8.2 | 1383.2 | 1815.2 | 2132.2 | 6278.6 | 6874.7 | 9347.5
San Francisco, CA | USA | 1.06 | 15.5 | 29.9 | 31.8 | 4100.7 | 5853.1 | 8906.2 | 22800.8 | 35021.0 | 34393.6
San Diego, CA | USA | 0.82 | 685.6 | 1184.8 | 1254.6 | 18121.6 | 26799.8 | 32060.9 | 48584.5 | 59534.8 | 60938.9
Los Angeles, CA | USA | 0.76 | 500.0 | 966.2 | 996.9 | 12694.3 | 19522.0 | 24339.8 | 36256.4 | 48891.0 | 49459.8
Crescent City, CA | USA | 1.20 | 7.3 | 11.5 | 14.4 | 1169.1 | 1400.0 | 2108.9 | 6403.9 | 10043.0 | 9338.2
Monterey, CA | USA | 0.90 | 75.9 | 161.7 | 185.9 | 7473.5 | 11305.8 | 15807.4 | 29701.9 | 42331.8 | 42333.4
Port San Luis, CA | USA | 0.91 | 51.8 | 122.0 | 136.0 | 5535.8 | 8139.4 | 11871.8 | 23099.6 | 35399.1 | 34987.0
Santa Monica, CA | USA | 0.96 | 52.0 | 135.9 | 135.7 | 6182.5 | 9383.3 | 12858.6 | 26153.0 | 38887.2 | 38665.6
La Jolla, CA | USA | 0.76 | 1083.4 | 1715.4 | 1865.3 | 22921.9 | 33054.4 | 38521.9 | 53473.5 | 63448.4 | 65032.2
New London, CT | USA | 2.32 | 2.1 | 2.1 | 2.4 | 180.9 | 251.4 | 291.2 | 1676.3 | 3542.3 | 2172.2
Fernandina Beach, FL | USA | 1.47 | 7.1 | 8.4 | 10.1 | 1227.4 | 2412.2 | 2486.5 | 9862.5 | 19474.0 | 16696.3
St. Petersburg, FL | USA | 2.61 | 1.4 | 1.5 | 1.5 | 49.6 | 156.3 | 142.7 | 1325.6 | 2260.5 | 1578.4
Pensacola, FL | USA | 4.97 | 1.1 | 1.1 | 1.1 | 1.2 | 1.3 | 1.3 | 121.9 | 167.6 | 110.6
Mayport, FL | USA | 0.95 | 57.6 | 93.4 | 124.3 | 7375.9 | 15849.8 | 17493.3 | 33130.5 | 48435.1 | 50600.1
Limetree Bay, FL | USA | 2.46 | 1.2 | 1.1 | 1.2 | 108.3 | 139.3 | 138.7 | 1367.8 | 2040.6 | 1641.1
Key West, FL | USA | 0.96 | 7.9 | 31.9 | 16.6 | 6715.3 | 14048.9 | 15782.9 | 33432.6 | 48509.5 | 49892.8
Fort Pulaski, GA | USA | 1.17 | 33.9 | 43.5 | 65.2 | 4763.4 | 9952.9 | 11180.8 | 27439.7 | 43026.5 | 43911.6
Hilo, HI | USA | 0.67 | 1101.0 | 1398.0 | 2054.7 | 48909.9 | 58668.3 | 64368.1 | 71676.9 | 78764.7 | 78545.2
French Frigate, HI | USA | 0.47 | 5112.1 | 9501.6 | 9810.6 | 60720.3 | 71033.6 | 74442.4 | 71625.4 | 79126.1 | 78932.7
Kahului, HI | USA | 0.52 | 4370.1 | 5496.7 | 7507.0 | 63429.8 | 70423.9 | 74299.0 | 75574.3 | 81667.8 | 81311.8
Mokuoloe, HI | USA | 0.44 | 7846.7 | 10190.2 | 13258.2 | 67323.2 | 73246.4 | 76634.2 | 75038.4 | 81544.9 | 81232.8
Honolulu-B, HI | USA | 0.45 | 6664.0 | 8728.3 | 11408.8 | 65557.3 | 71928.3 | 75504.8 | 74255.5 | 81007.5 | 80723.4
Nawiliwili, HI | USA | 0.71 | 224.1 | 262.8 | 392.7 | 23574.0 | 33271.0 | 38319.2 | 49882.4 | 62393.3 | 62933.5
Grand Isle, LA | USA | 10.00 | 1.3 | 1.2 | 1.3 | 1.4 | 1.4 | 1.4 | 1.5 | 1.6 | 1.6
Woods Hole, MA | USA | 1.65 | 5.2 | 6.5 | 7.2 | 890.1 | 1945.4 | 1658.5 | 5841.4 | 15392.8 | 9538.4
Nantucket, MA | USA | 3.74 | 1.2 | 1.3 | 1.3 | 1.7 | 1.9 | 2.0 | 390.1 | 651.2 | 417.8
Boston, MA | USA | 1.75 | 5.1 | 6.2 | 7.0 | 782.7 | 1499.7 | 1401.0 | 5036.4 | 12481.3 | 7633.6
Portland, ME | USA | 1.33 | 26.3 | 41.8 | 44.3 | 2054.5 | 4493.9 | 4260.2 | 9109.6 | 22304.1 | 15760.2
Eastport, ME | USA | 1.72 | 19.3 | 33.7 | 34.5 | 1154.2 | 2723.6 | 2423.1 | 5694.6 | 14587.3 | 8648.3
Duck Pier, NC | USA | 1.40 | 17.9 | 29.9 | 35.3 | 2492.5 | 5948.8 | 6505.1 | 18241.9 | 34590.0 | 31860.1
Wilmington, NC | USA | 1.24 | 3.9 | 7.7 | 6.0 | 1782.8 | 4022.9 | 4379.2 | 13403.3 | 27017.5 | 23543.8
Atlantic City, NJ | USA | 1.62 | 8.0 | 10.9 | 12.6 | 1407.0 | 3299.1 | 3205.9 | 10439.6 | 24855.3 | 19048.4
Cape May, NJ | USA | 1.43 | 14.5 | 19.6 | 24.3 | 2118.9 | 5730.6 | 5395.8 | 14979.9 | 32360.6 | 28499.7
Montauk, NY | USA | 1.53 | 10.6 | 13.7 | 16.0 | 1350.4 | 2788.1 | 2665.4 | 8412.9 | 21327.4 | 15755.3
New York, NY | USA | 2.61 | 1.8 | 1.9 | 2.0 | 93.3 | 172.3 | 183.6 | 1300.5 | 2547.7 | 1560.4
Charleston, OR | USA | 1.18 | 37.7 | 53.8 | 70.5 | 2505.9 | 3108.4 | 4800.4 | 13168.6 | 20908.5 | 19984.6
South Beach, OR | USA | 1.37 | 22.2 | 31.1 | 38.2 | 2037.2 | 2469.1 | 3720.7 | 12483.4 | 19973.4 | 18772.7
Astoria, OR | USA | 1.25 | 9.9 | 15.3 | 20.0 | 1179.6 | 1421.2 | 2048.2 | 6321.3 | 10045.6 | 9001.4
Newport, RI | USA | 1.46 | 8.3 | 11.7 | 12.4 | 1572.1 | 3427.3 | 3138.9 | 8996.1 | 22403.0 | 16090.1
Charleston, SC | USA | 1.61 | 3.0 | 3.2 | 3.6 | 835.1 | 1508.4 | 1703.2 | 7925.2 | 15785.5 | 13080.1
Galveston (Pier 21), TX | USA | 2.96 | 1.9 | 1.9 | 1.9 | 33.4 | 155.6 | 138.6 | 1648.4 | 3005.7 | 1985.6
Rockport, TX | USA | 1.97 | 2.1 | 2.2 | 2.3 | 510.5 | 923.3 | 913.8 | 7270.5 | 15475.8 | 11903.8
Port Isabel, TX | USA | 2.76 | 1.3 | 1.3 | 1.3 | 7.5 | 126.3 | 115.7 | 1251.6 | 2014.6 | 1396.5
Galveston (P. Pier), TX | USA | 5.26 | 1.3 | 1.2 | 1.3 | 1.6 | 1.7 | 1.7 | 152.5 | 211.6 | 145.1
Chesapeake BBT, VA | USA | 2.02 | 3.4 | 3.7 | 4.2 | 461.4 | 909.9 | 895.4 | 4740.5 | 10662.9 | 7203.3
Neah Bay, WA | USA | 1.31 | 4.2 | 7.1 | 10.8 | 686.8 | 818.8 | 1061.3 | 3497.6 | 5115.0 | 4501.7
Willapa Bay, WA | USA | 2.05 | 2.2 | 2.6 | 2.8 | 251.3 | 269.6 | 382.2 | 2185.3 | 3014.1 | 2590.3
Lewes, DE | USA | 2.18 | 2.5 | 2.6 | 2.9 | 269.9 | 551.6 | 557.6 | 2789.4 | 6303.5 | 4022.3
Apra Harbor, Guam | USA Trust | 0.50 | 2417.4 | 3602.8 | 3793.9 | 51797.9 | 62775.5 | 70531.0 | 69295.5 | 75780.4 | 78992.1
Wake | USA Trust | 1.82 | 1.5 | 1.6 | 1.7 | 618.0 | 927.2 | 813.2 | 5042.8 | 8596.8 | 8543.8
Johnston | USA Trust | 0.89 | 45.7 | 171.4 | 76.8 | 11344.6 | 20089.5 | 22332.8 | 35174.6 | 47531.8 | 49323.9
Midway | USA Trust | 1.12 | 7.5 | 16.1 | 14.8 | 3714.3 | 6376.9 | 9526.7 | 22321.2 | 34381.6 | 34488.4
Pago Pago | USA Trust | 0.48 | 7167.3 | 11780.5 | 12692.2 | 75498.9 | 81710.1 | 82693.4 | 82939.5 | 86495.5 | 86097.5
Charlotte Amalie, VI | USA Trust | 1.24 | 1.8 | 1.7 | 1.8 | 1590.1 | 2712.5 | 2494.8 | 13142.3 | 23022.3 | 22474.6
San Juan, PR | USA Trust | 0.93 | 4.4 | 8.7 | 5.7 | 5947.9 | 11394.3 | 11413.4 | 31020.1 | 44208.2 | 46298.4
Magueyes Island, PR | USA Trust | 1.09 | 1.9 | 1.9 | 2.0 | 2508.2 | 4807.1 | 4466.8 | 18458.3 | 30861.5 | 31125.9