# Catastrogenesis with unstable ALPs as the origin of the NANOGrav 15 yr
gravitational wave signal

Graciela B. Gelmini<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of California, Los Angeles
Los Angeles, California, 90095-1547, USA  Jonah Hyman<EMAIL_ADDRESS>Department of Physics and Astronomy, University of California, Los Angeles
Los Angeles, California, 90095-1547, USA

(November 30, 2023)

###### Abstract

In post-inflation axion-like particle (ALP) models, a stable domain wall
network forms if the model’s potential has multiple minima. This system must
annihilate before dominating the Universe’s energy density, producing ALPs and
gravitational waves (a process we dub “catastrogenesis,” or “creation via
annihilation”). We examine the possibility that the gravitational wave
background recently reported by NANOGrav is due to catastrogenesis. For the
case of ALP decay into two photons, we identify the region of ALP mass and
coupling, just outside current limits, compatible with the NANOGrav signal.

##  I Introduction

The fundamental importance of gravitational waves (GWs) as messengers of the
pre-Big Bang Nucleosynthesis (BBN) era, a yet unknown epoch of the Universe
from which we do not yet have any other remnants, cannot be overestimated. The
NANOGrav pulsar timing array collaboration has recently reported the
observation of a stochastic gravitational wave background (NANOGrav:2023gor, )
in 15 years of data, and has examined its possible origin in terms of new
physics (NANOGrav:2023hvm, ). They showed that the pre-BBN annihilation of
cosmic walls provides a good fit to their signal, both as the sole source and
in combination with a background from a population of inspiraling supermassive
black hole binaries (SMBHBs), which is expected to be its primary conventional
physics origin (NANOGrav:2023hvm, ). The annihilation produces a peaked
spectrum, whose peak frequency $f_{\rm peak}$ is given by the inverse of the
cosmic horizon $\simeq t_{\rm ann}$ at annihilation redshifted to the present.
In their fit to the wall annihilation model NANOGrav finds (NANOGrav:2023hvm,
) a peak frequency

$f_{\rm peak}=c_{f}10^{-8}\leavevmode\nobreak\ {\rm Hz}\leavevmode\nobreak\ ,$
(1)

and a peak energy density

$\left.\Omega_{\rm GW}h^{2}\right|_{\rm
peak}=c_{\Omega}10^{-8}\leavevmode\nobreak\ ,$ (2)

with coefficients $c_{f}$ and $c_{\Omega}$ of order 1. In particular
$c_{\Omega}\simeq 1$, while $c_{f}$ can have larger values.

Here we consider the annihilation of a $U(1)$ pseudo Nambu-Goldstone boson
stable string-wall system as the origin of the NANOGrav signal, based on our
previous recent work (Gelmini:2021yzu, ; Gelmini:2022nim, ; Gelmini:2023ngs,
), to which we refer often in the following.

Many extensions of the Standard Model (SM) of elementary particles assume an
approximate global $U(1)$ symmetry spontaneously broken at an energy scale
$V$. The symmetry is not exact, but explicitly broken at another scale $v\ll
V$. Thus the model has a pseudo Nambu-Goldstone boson we denote with $a$, with
mass $m_{a}\simeq v^{2}/V$. These models, include the original axion
(Peccei:1977hh, ; Weinberg:1977ma, ; Wilczek:1977pj, ), invisible axions (also
called “QCD axions”) (Kim:1979if, ; Shifman:1979if, ; Zhitnitsky:1980tq, ;
Dine:1981rt, ), majoron models (Chikashige:1980ui, ; Gelmini:1980re, ;
Rothstein:1992rh, ; Gu:2010ys, ; Lazarides:2018aev, ; Reig:2019sok, ;
Abe:2020dut, ; Bansal:2022zpi, ), familon models (Wilczek:1982rv, ;
Reiss:1982sq, ; Gelmini:1982zz, ), and axion-like particles (ALPs) (e.g.
Svrcek:2006yi ; Arvanitaki:2009fg ; Acharya:2010zx ; Dine:2010cr ;
Jaeckel:2010ni ). Many models predict a large mass for the QCD axion
(Fukuda:2015ana, ; Dimopoulos:2016lvn, ; Agrawal:2017ksf, ; Gaillard:2018xgk,
), including the “high-quality QCD axion” (Hook:2019qoh, ) and previous models
(see e.g. Section 6.7 of DiLuzio:2020wdo ). Heavy majorons, which could get a
mass from soft breaking terms or from gravitational effects (see e.g.
Rothstein:1992rh ; Gu:2010ys ; Lazarides:2018aev ; Reig:2019sok ; Abe:2020dut
), have been considered as well (see e.g. Gu:2010ys ; Abe:2020dut ), even of
mass in the TeV range. Since we need a specific type of model to take into
account existing experimental bounds, we concentrate on ALPs coupled to
photons. ALPs are one of the most studied types of dark matter candidates.
They are extensively searched for in a variety of laboratory experiments and
astrophysical observations, their coupling to photons being one of the most
studied as well.

We assume that the spontaneous symmetry breaking happens after inflation, in
which case cosmic strings appear during the spontaneous breaking transition,
and a system of cosmic walls bounded by strings is produced when the explicit
breaking becomes dynamically relevant, when $t\simeq m_{a}^{-1}$. The cosmic
strings then enter into a “scaling” regime, in which the number of strings per
Hubble volume remains of order 1 (see e.g. Vilenkin:1984ib  and references
therein). The subsequent evolution of the string-wall system depends crucially
on the number of minima of the potential after the explicit breaking, which
may be just one minimum, $N=1$, or several, $N>1$. With $N=1$, “ribbons” of
walls bounded by strings surrounded by true vacuum form, which shrink very
fast due to the pull of the walls on the strings, leading to the immediate
annihilation of the string-wall system (see e.g. Gorghetto:2021fsn ).

We concentrate on the $N>1$ case, where the $U(1)$ symmetry is broken into a
discrete $Z_{N}$ symmetry, in which each string connects to $N$ walls forming
a stable string-wall system. A short time after walls form, when friction of
the walls with the surrounding medium is negligible, the string-wall system
enters into another scaling regime in which the linear size of the walls is
the cosmic horizon size $\simeq t$. Thus its energy density is $\rho_{\rm
walls}\simeq\sigma/t$, where $\sigma$ is the energy density per unit area of
the walls. The energy density in this system grows faster with time than the
radiation density, and would come to dominate the energy density of the
Universe, leading to an unacceptable cosmology (Zeldovich:1974uw, ), unless it
annihilates earlier.

If the $Z_{N}$ is also an approximate symmetry, then there is a “bias,” a
small energy difference between the $N$ minima, which chooses one of them to
be that with minimum energy. The energy difference between two vacua at both
sides of each wall accelerates each wall toward its adjacent higher-energy
vacuum, which drives the domain walls to their annihilation (Zeldovich:1974uw,
) (see also e.g. Gelmini:1988sf ). As in our previous recent work
(Gelmini:2021yzu, ; Gelmini:2022nim, ; Gelmini:2023ngs, ), we adopt the
$Z_{N}$ explicit breaking term in the scalar potential originally proposed for
QCD axions (Sikivie:1982qv, ; Chang:1998bq, ), and parameterized as $V_{\rm
bias}\simeq\epsilon_{b}v^{4}$, with a dimensionless positive coefficient
$\epsilon_{b}\ll 1$. For small enough $\epsilon_{b}$ values, ALPs are
dominantly produced when the string-wall system annihilates, together with
GWs, a process that we named “catastrogenesis” (Gelmini:2022nim, ), after the
Greek word katastrofh́, for “overturn” or “annihilation.”

The emission of GWs by the initial system of cosmic strings ends when walls
appear. Thus, there is a low-frequency cutoff of 82 $(m_{a}/{\rm GeV})^{1/2}$
Hz (Chang:2019mza, ; Gouttenoire:2019kij, ; Gorghetto:2021fsn, ),
corresponding to the inverse of the cosmic horizon when walls appear,
redshifted to the present. This is much higher than the relevant frequencies
for $m_{a}\simeq$ GeV, so strings do not contribute to the NANOGrav signal in
this model.

We assume radiation domination during the times of interest. In this case, the
present peak GW density is related to the temperature at annihilation $T_{\rm
ann}$ by

$f_{\rm peak}\simeq 0.76\times 10^{-7}\text{Hz}\leavevmode\nobreak\
\frac{T_{\rm ann}}{\rm GeV}\leavevmode\nobreak\ \frac{\left[g_{\star}(T_{\rm
ann})\right]^{1/2}}{\left[g_{s\star}(T_{\rm
ann})\right]^{1/3}}\leavevmode\nobreak\ ,$ (3)

where $g_{\star}$ and $g_{s\star}$ are the energy and entropy density numbers
of degrees of freedom. Thus, Eq. (1) also gives $T_{\rm ann}$ in terms of
$c_{f}$

$T_{\rm ann}\simeq 82.5\leavevmode\nobreak\ c_{f}\leavevmode\nobreak\ {\rm
MeV}\left[\frac{16.5}{g_{\star}(T_{\rm ann})}\right]^{1/2}\leavevmode\nobreak\
\left[\frac{g_{s\star}(T_{\rm ann})}{16.5}\right]^{1/3},$ (4)

while in terms of the parameters of our model it is

$T_{\rm ann}\simeq\frac{2.2\times 10^{9}\leavevmode\nobreak\ {\rm
GeV}}{[g_{\star}(T_{\rm ann})]^{1/4}}\leavevmode\nobreak\
\sqrt{\frac{\epsilon_{b}\leavevmode\nobreak\
m_{a}}{f_{\sigma}\leavevmode\nobreak\ {\rm GeV}}}\leavevmode\nobreak\ .$ (5)

The peak energy density is

$\left.\Omega_{\rm GW}h^{2}\right|_{\rm peak}\simeq\frac{1.2\times
10^{-79}\epsilon_{\rm GW}\leavevmode\nobreak\ g_{\star}(T_{\rm
ann})}{\epsilon_{b}^{2}\leavevmode\nobreak\ \left[g_{s\star}(T_{\rm
ann})\right]^{4/3}}\left(\frac{f_{\sigma}V}{N{\text{GeV}}}\right)^{4}$ (6)

where $f_{\sigma}$ is a parameter entering into the definition of the energy
per unit area of the walls, $\sigma\simeq f_{\sigma}v^{2}V/N$, and
$f_{\sigma}\simeq 6$ for most assumed potentials. We include in Eq. (6) a
dimensionless factor $\epsilon_{\rm GW}$ found in numerical simulations (e.g.
Hiramatsu:2012sc ). When needing to fix its value we use $\epsilon_{\rm
GW}=0.7$ as adopted in the NANOGrav fit (NANOGrav:2023hvm, ) following
Hiramatsu:2013qaa  (in our earlier work we took instead $\epsilon_{\rm
GW}=10$, using Fig. 8 of Hiramatsu:2012sc ). Since $g_{\star}=g_{s\star}$ for
$T>1$ MeV, we set them equal in the following. We address the reader to
Gelmini:2021yzu ; Gelmini:2022nim  for the derivation of these equations.

Our previous results (Gelmini:2021yzu, ; Gelmini:2022nim, ) show that the
requirement that the ALP density not exceed that of dark matter,
$\Omega_{a}h^{2}\lesssim 0.12$, implies

$\frac{\left.\Omega_{\rm GW}h^{2}\right|_{\rm
peak}}{10^{-17}}\left(\frac{f_{\rm peak}}{10^{-9}{\rm
Hz}}\right)^{2}<10^{-2}\leavevmode\nobreak\ ,$ (7)

so the model cannot produce the NANOGrav signal with stable ALPs. Thus we
concentrate on ALPs that are unstable and decay into SM products that
thermalize early enough to leave no trace by the time of Big Bang
Nucleosynthesis (BBN), such as we considered in Gelmini:2023ngs . To escape
existing laboratory, astrophysical, and cosmological limits on ALP decays into
SM products, these ALPs must have a mass $m_{a}$ in the GeV range or higher,
depending on the decay mode (see e.g. Gelmini:2023ngs  and references
therein).

Similar or related models have been studied recently in relation to pulsar
timing array data, e.g. Guo:2023hyp ; Madge:2023cak ; Kitajima:2023cek ;
Gouttenoire:2023ftk ; Blasi:2023sej ; Servant:2023mwt ; Ge:2023rce .
Servant:2023mwt  considered the same type of models we study here, but with
the purpose of excluding parameter regions disfavored by the NANOGrav 15 yr
data, which they analyzed independently. Our purpose is instead to try to
explain the signal, and thus we stay away from the disfavored region (shown in
gray in the lower left panel of Fig. 12 of NANOGrav:2023hvm  and the right
panel of Fig. 2 of Servant:2023mwt ).

##  II Unstable ALP models that can produce the NANOGrav signal

In Gelmini:2023ngs  we assumed $m_{a}$ was sufficiently larger than 1 GeV for
ALPs decaying into SM particles to comfortably escape existing experimental
limits. However, we need to be more nuanced here and explore the viability of
somewhat lighter ALPs. The reason is that requiring the walls to form at least
one order of magnitude in temperature after strings appear, combined with
upper limits on $T_{\rm ann}$ determined by NANOGrav to explain the signal,
impose $m_{a}\lesssim 1.8$ GeV, as we are going to show now.

Walls appear when the Hubble parameter is $H(T_{\rm w})\simeq m_{a}/3$, i.e.
when the temperature is

$T_{\rm w}\simeq\frac{1.6\times 10^{9}\leavevmode\nobreak\ {\rm
GeV}}{\left[g_{\star}(T_{\rm w})\right]^{1/4}}\left(\frac{m_{a}}{\rm
GeV}\right)^{1/2}.$ (8)

Thus $T_{\rm w}$ depends only on $m_{a}$ ($g_{\star}(T_{\rm w})\simeq 105$,
since $T_{\rm w}>100$ GeV). As in our previous papers, we consider $m_{a}$ to
be temperature independent. A temperature dependence would not affect the
annihilation process, which happens late enough for $m_{a}$ to have reached
its present constant value in any case, but could affect $T_{\rm w}$.

Combining Eqs. (2) and (6) fixes the ratio $V^{2}/\epsilon_{b}$, and Eqs. (4)
and (5) fix the product $\epsilon_{b}m_{a}$. Thus, given Eqs. (1) and (2), we
obtain $V$ as a function of $m_{a}$,

$V\simeq\frac{5.0\times 10^{7}\leavevmode\nobreak\ \rm{GeV}}{\epsilon_{\rm
GW}^{1/4}}\frac{N}{f_{\sigma}^{1/2}}\left(\frac{\rm
GeV}{m_{a}}\right)^{1/2}c_{f}c_{\Omega}^{1/4}\left[\frac{g_{\star}(T_{\rm
ann})}{16.5}\right]^{1/6},$ (9)

and consequently $m_{a}$ in terms of the ratio $T_{\rm w}/V$,

$m_{a}\simeq\frac{(T_{\rm
w}/V)}{0.1}c_{f}c_{\Omega}^{1/4}\frac{N}{f_{\sigma}^{1/2}}\frac{10.3\leavevmode\nobreak\
{\rm MeV}}{\epsilon_{\rm GW}^{1/4}}\left[\frac{g_{\star}(T_{\rm
ann})}{16.5}\right]^{1/6}.$ (10)

We require $T_{\rm w}/V\lesssim 0.1$ so walls form at least one order of
magnitude in temperature after strings appear. Larger values of $N$ are
favorable to allow larger $m_{a}$. To our knowledge, upper limits on $N$ have
been studied only in QCD axion models (DiLuzio:2017pfr, ) in which $N=20$ is
possible. For axions coupled to gluons, $N$ is given by the color anomaly
coefficient. Similarly, non-perturbative effects in a dark sector
(Arvanitaki:2009fg, ; Jaeckel:2010ni, ), generically lead to $N>1$ for ALPs,
thus possibly to similarly large $N$ values. We will thus adopt $N=20$.
Replacing also $f_{\sigma}=6$ and $\epsilon_{\rm GW}=0.7$,

$m_{a}\simeq\frac{(T_{\rm w}/V)}{0.1}c_{f}c_{\Omega}^{1/4}\leavevmode\nobreak\
92\leavevmode\nobreak\ {\rm MeV}\left[\frac{g_{\star}(T_{\rm
ann})}{16.5}\right]^{1/6}.$ (11)

An upper limit on $c_{f}$ thus provides an upper limit on $m_{a}$ and vice
versa (since the NANOGrav fit prefers $c_{\Omega}\simeq$ 1 (NANOGrav:2023hvm,
)). Looking in Fig. 12 of NANOGrav:2023hvm , the range of annihilation
temperatures (called $T_{\star}$ in that paper) where the NANOGrav signal can
be explained by the annihilation of domain walls into SM products (DW-SM, the
model most similar to ours), we can see that $T_{\rm ann}\lesssim$ 1 GeV
(close to the upper boundary of the red region in the figure). By Eq. (4),
this corresponds to $c_{f}\lesssim 15$ (taking into account the rapid change
of $g_{\star}$ values for temperatures in the 100 MeV range,
$g_{\star}(1\leavevmode\nobreak\ \rm{GeV})\simeq 70$). Through Eq. (11), this
implies $m_{a}\lesssim 1.8$ GeV. A more conservative upper limit on the
annihilation temperature is the upper boundary of the 95% credible interval
including a SMBHB contribution quoted in the text of NANOGrav:2023hvm , 843
MeV. This implies through Eq. (4) (with $g_{\star}(0.84\leavevmode\nobreak\
{\rm GeV})\simeq 68$) $c_{f}\lesssim 13$, and through Eq. (11) $m_{a}\lesssim
1.5\leavevmode\nobreak\ \rm{GeV}$.

Let us now consider the experimental limits on ALPs coupled to photons through
a Lagrangian term

$\mathcal{L}_{a\gamma\gamma}=\frac{c_{\gamma\gamma}}{f_{a}}aF_{\mu\nu}\tilde{F}^{\mu\nu}\leavevmode\nobreak\
,$ (12)

where $F_{\mu\nu}$ is the electromagnetic field tensor and
$\tilde{F}^{\mu\nu}$ its dual, $|c_{\gamma\gamma}|$ is a dimensionless
coupling constant, and $f_{a}=V/N$ is given by Eq. (9) divided by $N$, and is
independent of $N$, thus

$\frac{1}{f_{a}}\simeq\frac{1.4\times 10^{-8}}{c_{f}c_{\Omega}^{1/4}{\rm
GeV}}\left(\frac{m_{a}}{100\leavevmode\nobreak\ {\rm
MeV}}\right)^{1/2}\left[\frac{16.5}{g_{\star}(T_{\rm ann})}\right]^{1/6}.$
(13)

Or using Eq. (11) in Eq. (13),

$\frac{1}{f_{a}}\lesssim\frac{1.4\times
10^{-8}}{(c_{f}c_{\Omega}^{1/4})^{1/2}\leavevmode\nobreak\
\rm{GeV}}\left[\frac{16.5}{g_{\star}(T_{\rm ann})}\right]^{1/12}.$ (14)

So a larger $c_{f}$ (thus also a larger $T_{\rm ann}$) makes the coupling
smaller.

Requiring the upper limit on the ALP mass in Eq. (11) to reach $m_{a}\simeq$
300 MeV (to avoid experimental limits on lighter ALPs shown in Fig. 1), we
obtain $c_{f}\gtrsim 2.9$ and $T_{\rm ann}\gtrsim$ 200 MeV (thus
$g_{\star}\simeq 42$), which implies $1/f_{a}<7.5\times 10^{-9}/{\rm GeV}$.
Requiring instead the upper limit to be $m_{a}\simeq 1.8$ GeV, which
corresponds to $T_{\rm ann}\simeq$ 1 GeV since as mentioned above, $c_{f}=15$,
with $g_{\star}(1\leavevmode\nobreak\ {\rm GeV})\simeq 70$, Eq. (13) or (14)
implies $1/f_{a}<3.1\times 10^{-9}/{\rm GeV}$.

Assuming $|c_{\gamma\gamma}|\lesssim$ 1, these upper limits on $1/f_{a}$
translate into upper limits on the ALP coupling to photons as a function of
$m_{a}$. These limits constitute the upper boundary of the gray region in Fig.
1. Fig. 1 also shows relevant regions rejected by the most up-to-date limits
on ALPs. Notice that if $|c_{\gamma\gamma}|>1$, the region extends upward (as
indicated by the dashed lines) where experimental limits (not only
astrophysical limits) become important.

The value of $|c_{\gamma\gamma}|$ depends on the completion of the ALP model.
It has been extensively studied only for the QCD axion, where
$|c_{\gamma\gamma}|\simeq\alpha_{\rm EM}/8\pi\simeq 2.9\times 10^{-4}$ in the
simplest models. However, $|c_{\gamma\gamma}|$ can be many orders of
magnitude, even exponentially, larger in some models (see e.g. the “Axions and
Other Similar Particles” review in ParticleDataGroup:2022pth  or
Farina:2016tgd ; Agrawal:2017cmd ; Daido:2018dmu ; DiLuzio:2020wdo ;
Plakkot:2021xyx ). Notice that with the $|c_{\gamma\gamma}|$ value in the
simplest QCD axion models, the upper boundary of our region of compatibility
(gray) in Fig. 1 would move to the dot-dashed line in the region excluded by
BBN limits (yellow), i.e. the region of compatibility would not exist.

Figure 1: Region (in gray) of ALP coupling to two photons versus ALP mass
$m_{a}$ for models which could explain the NANOGrav 15 yr signal, where 300
MeV $<m_{a}<$ 1.8 GeV, together with (colored) relevant regions excluded by:
SN 1987A cooling (Caputo:2022mah, ; Caputo_2022b, ), SN 1987A ALP decay (Solar
Maximum Mission) (müller2023investigating, ), SN 1987A ALP decay (Pioneer
Venus Orbiter) (Diamond_2023, ), supernovae (SN) explosion energy
(Caputo:2022mah, ), GW170817 (diamond2023multimessenger, ), BBN +
$N_{\rm{eff}}$ limits (Depta_2020, ), and experimental limits (Dolan:2017osp,
; Dobrich:2019dxc, ). This figure reproduces a portion of Fig. 9 of
Antel:2023hkf  with additions from AxionLimits .

We will now check the lifetime and the fraction of the density constituted by
ALPs at the time of decay. The decay rate (see e.g. Eq. (138) of Antel:2023hkf
)

$\Gamma(a\to\gamma\gamma)=\frac{|c_{\gamma\gamma}|^{2}m_{a}^{3}}{4\pi
f_{a}^{2}}$ (15)

corresponds to a lifetime (using Eq. (13))

$\tau=\frac{c_{f}^{2}c_{\Omega}^{1/2}}{|c_{\gamma\gamma}|^{2}}4.2\times
10^{-5}{\rm\leavevmode\nobreak\ sec}\left(\frac{100\leavevmode\nobreak\ {\rm
MeV}}{m_{a}}\right)^{4}\left[\frac{g_{\star}(T_{\rm
ann})}{16.5}\right]^{1/3}\leavevmode\nobreak\ .$ (16)

With $|c_{\gamma\gamma}|$ in the range 0.1 to 1, we can have $\tau\simeq
t_{\rm ann}$, i.e. the decay can happen at annihilation. Requiring
$\tau\lesssim 0.1$ sec, so that the decay happens early enough not to affect
BBN, translates through Eq. (15) into $|c_{\gamma\gamma}|/f_{a}\gtrsim
0.9\times 10^{-11}({\rm GeV}/m_{a})^{3/2}/{\rm GeV}$. This determines the
lower boundary of the gray region shown in Fig. 1 (where $|c_{\gamma\gamma}|$
goes from $\simeq 10^{-3}$ for $m_{a}\simeq 1.8$ GeV to $\simeq 10^{-2}$ for
$m_{a}\simeq 0.3$ GeV).

To compute the density of the string-wall system with respect to that of
radiation at annihilation, we consider that, had the system not annihilated,
its energy density $\rho_{\rm{walls}}\simeq\sigma/t$ would have continued to
grow until the moment we call wall-domination $t_{\rm{wd}}$, at which it
becomes as large as the radiation energy,
$\rho_{\rm{walls}}(t_{\rm{wd}})\simeq\rho_{\rm{rad}}(t_{\rm{wd}})$. The
temperature of wall-domination is (see Gelmini:2022nim ; Gelmini:2023ngs )

$T_{\rm wd}\simeq\frac{3.4\leavevmode\nobreak\ {\rm GeV}}{[g_{\star}(T_{\rm
wd})]^{1/4}}\frac{f_{\sigma}^{1/2}}{N}\left(\frac{V}{\rm
10^{9}\leavevmode\nobreak\ GeV}\right)\left(\frac{m_{a}}{\rm
10\leavevmode\nobreak\ GeV}\right)^{1/2}.$ (17)

Besides, $\rho_{\rm{walls}}(t_{\rm ann})/\rho_{\rm{walls}}(t_{\rm{wd}})\simeq
t_{\rm{wd}}/t_{\rm{ann}}$, and the ratio of radiation densities at wall-
domination and annihilation is given by the ratio of $g_{\star}T^{4}$ at each
temperature. Combining these equations we find

$\frac{\rho_{\rm{walls}}(T_{\rm ann})}{\rho_{\rm{rad}}(T_{\rm
ann})}\simeq\left(\frac{g_{\star}(T_{\rm{wd}})}{g_{\star}(T_{\rm
ann})}\right)^{1/2}\left(\frac{T_{\rm{wd}}}{T_{\rm
ann}}\right)^{2}\leavevmode\nobreak\ .$ (18)

Using Eq. (4) and Eq. (9) in Eq. (17), we find

$\frac{\rho_{\rm{walls}}(T_{\rm ann})}{\rho_{\rm{rad}}(T_{\rm ann})}\simeq
0.13\leavevmode\nobreak\ c_{\Omega}^{1/2}\left(\frac{g_{\star}(T_{\rm
ann})}{16.5}\right)^{1/6},$ (19)

which shows that this ratio is always $<1$ for the annihilation temperatures
we consider. Since practically all the density in the string-wall system goes
into nonrelativistic (or quasi-nonrelativistic) ALPs at annihilation,
considering the redshift of the ALP and radiation densities until ALPs decay
at temperature $T_{\rm decay}$,

$\frac{\rho_{\rm ALPs}(T_{\rm decay})}{\rho_{\rm rad}(T_{\rm
decay})}\simeq\left(\frac{T_{\rm ann}}{T_{\rm decay}}\right)\frac{\rho_{\rm
walls}(T_{\rm ann})}{\rho_{\rm rad}(T_{\rm ann})}.$ (20)

As we mentioned above, $T_{\rm decay}$ can be very close to $T_{\rm ann}$, so
ALPs do not get to matter dominate in our model and the decays happen early
enough for the products to thermalize long before BBN. Otherwise, there would
be a period of ALP matter domination before ALPs decay, which is in principle
not problematic since the decays happen much before BBN, but would be a
scenario deserving further study.

The range of $c_{f}$ values, 2.9 to 15, that we have found above corresponds
to a peak frequency range through Eq. (1). In Fig. 2 we indicate two
approximate spectra, with the maximum and minimum $f_{\rm peak}$ in the
mentioned range. Frequencies $f<f_{\rm peak}$ correspond to super-horizon
wavelengths at annihilation, so causality requires a $\sim f^{3}$ dependence
(Caprini:2009fx, ) for wavelengths that enter into the horizon during
radiation domination, see e.g. Barenboim:2016mjm ; Cai:2019cdl ; Hook:2020phx
. For $f>f_{\rm peak}$ the spectrum depends instead on the particular
production model. Hiramatsu:2012sc  finds a roughly $1/f$ dependence (although
the approximate slope slightly depends on $N$), which we use for Fig. 2. In
Fig. 2 the rough signal region of NANOGrav, as well as the limits and reach of
other GW observatories, is shown.

Figure 2: Approximate spectra which could account for the NANOGrav signal in
our catastrogenesis model, with peak amplitude
$\left.\Omega_{\rm{GW}}h^{2}\right|_{\rm{peak}}=10^{-8}$ and peak frequencies
$f_{\rm peak}=2.9\times 10^{-8}$ Hz and $1.5\times 10^{-7}$ Hz, which are the
minimum and maximum frequencies we found (see text). Allowed spectra would
have peak frequencies in between these two. Also shown are the approximate
NANOGrav 15 yr signal (NANOGrav:2023hvm, ) (in purple) and limits (solid line
boundaries) or reach (dashed line boundaries) of other GW detectors: the
European Pulsar Timing Array (EPTA) (Antoniadis:2023rey, ) and the Square
Kilometre Array (SKA) (Janssen:2014dka, ) in purple; the space-based
experiments TianQin (TianQin:2015yph, ), Taiji (Ruan:2018tsw, ), and the Laser
Interferometer Space Antenna (LISA) (LISA:2017pwj, ) in green; the Atom
Interferometer Observatory and Network (AION) (Badurina:2019hst, ), the Atomic
Experiment for Dark Matter and Gravity Exploration in Space (AEDGE)
(AEDGE:2019nxb, ), the Deci-hertz Interferometer Gravitational wave
Observatory (DECIGO) (Seto:2001qf, ), and the Big Bang Observer (BBO)
(Corbin:2005ny, ) in blue; the ground-based Einstein Telescope (ET) in red
(Sathyaprakash:2012jk, ); and the Laser Interferometer Gravitational-wave
Observatory (LIGO) in gray (LIGOScientific:2019vic, ). The cyan band
corresponds to the 95% C.L. upper limit on the effective number of degrees of
freedom during CMB emission from Planck and other data (Pagano:2015hma, ),
which imposes $\Omega_{\rm GW}h^{2}<10^{-6}$.

##  III Possibility of primordial black hole formation

The formation of primordial black holes (PBHs) during the process of
annihilation of the string-wall system is an exciting possible aspect of ALP
models with $N>1$. We recently dealt with the possibility of producing
“asteroid-mass” PBHs, in the range in which they could constitute all of the
dark matter, in Gelmini:2023ngs . If formed, the PBH mass in the models in the
present paper would be in the range of $0.1$ to a few solar masses, but PBH
abundance would be too large to be allowed, and this would reject these
models.

However, the formation of PBHs is uncertain. The argument for PBH formation,
first presented in Ferrer:2018uiu  for QCD axions, is that in the latest
stages of wall annihilation in $N>1$ models ($t>t_{\rm ann}$) closed walls
could arise and collapse in an approximately spherically symmetric way. In
this case, if the characteristic linear size of the walls continues to grow
with time after annihilation starts, some fraction of the closed walls could
reach their Schwarzschild radius $R_{\rm Sch}$ and collapse into PBHs. The
figure of merit used is $p(t)=R_{\rm Sch}/t={2M(t)}/({t\,M_{\rm P}^{2}})$,
where $M_{\rm P}$ is the Planck mass and $M(t)$ is the mass within the
collapsing closed wall at time $t$. Reaching $p(t)=1$ would indicate the
formation of PBHs. This definition is based on the fact that while walls are
in the scaling regime, the linear size of the walls $L$ is close to the
horizon size ($L\simeq t$).

Annihilation starts when surface tension of the walls, which produces a
pressure $p_{T}\simeq\sigma/t$ that decreases with time (which tends to
rapidly straighten out curved walls to the horizon scale $t$), is compensated
by the volume pressure $p_{V}\simeq V_{\rm bias}$ (which tends instead to
accelerate the walls toward their higher-energy adjacent vacuum). In our
model, $p_{V}\ll p_{T}$ when walls form. At a later time, when $p_{T}\simeq
p_{V}$, the bias drives the walls (and the strings bounding them) to
annihilate within a Hubble time. This defines $t_{\rm ann}\simeq\sigma/V_{\rm
bias}$, after which $V_{\rm bias}$ dominates the energy density. After
annihilation starts, $L\simeq t$ is no longer guaranteed.

We have checked that for the models in this paper, we always have $p(t_{\rm
ann})<1$. If $L$ continues being close to $t$ for $t>t_{\rm ann}$, then
$p(t>t_{\rm ann})\simeq V_{\rm bias}L^{3}/L$ grows with time as $t^{2}$ and
eventually reaches 1. However, if $L$ decreases with time at some point after
annihilation starts, the figure of merit may never reach 1. Based on the
simple power-law parameterization we used in our previous recent work
(Gelmini:2022nim, ; Gelmini:2023ngs, ) for the evolution of the energy density
after annihilation starts, namely ${\rho_{\rm walls}(T)}/{\rho_{\rm
walls}(T_{\rm ann})}\simeq\left({T}/{T_{\rm ann}}\right)^{\alpha}$ (with a
parameter $\alpha$ that needs to be extracted from simulations), we can make a
naive estimate of how the characteristic linear wall size $L$ within a Hubble
volume $t^{3}$ evolves with time. In Appendix A we show how this naive
estimate requires $\alpha<6$ for $L$ to ever become larger than $t_{\rm ann}$.
The only simulations of the annihilation process available (Kawasaki:2014sqa,
) find $\alpha\geq 7$ (Gelmini:2022nim, ; Gelmini:2023ngs, ). On the other
hand, they also seem to indicate that the evolution of the string-wall system
continues being close to that in the scaling regime for some time. Therefore,
more detailed simulations of the annihilation process are required to
elucidate the appearance of PBHs.

In addition, a large enough departure from spherical symmetry due to angular
momentum or vacua with different energy on different sides of the collapsing
closed wall could prevent the formation of PBHs. Since the formation of PBHs
is such an uncertain consequence of ALP models with $N>1$, we do not use this
feature to reject any of these models.

##  IV Conclusions

We pointed out that the recently confirmed stochastic gravitational wave
background could be due to pseudo Nambu-Goldstone bosons, whose existence
could only be revealed through their decays and this background. In
particular, we examined unstable ALP models which can produce the recent
NANOGrav 15 yr signal. ALP models have a complex cosmology in which a stable
system of walls bounded by strings develops (for $N>1$), and nonrelativistic
ALPs and gravitational waves are produced when the cosmic string-wall system
annihilates (a process we dubbed “catastrogenesis” in our recent work on these
models). The annihilation produces a distinctive peaked spectrum, at a
frequency corresponding to the inverse of the cosmic horizon at annihilation.
Thus, this peak frequency is related to the annihilation temperature.

We require ALPs to decay into Standard Model (SM) products which thermalize
much before BBN. In particular, we have shown that ALPs decaying into two
photons in the region of masses and couplings necessary to explain the signal
can evade existing observational limits, the most relevant of which are
derived from supernova data (see Fig. 1), for ALP masses from about 300 MeV to
1.8 GeV. The model closest to ours that NANOGrav fitted to their signal is
that of domain walls decaying into SM products (DW-SM). Our model is very
similar to this one if the ALP decay happens very shortly after string-wall
system annihilation, which we showed is possible. Thus we use the NANOGrav
fits to this model to select a range of annihilation temperatures and thus
peak frequencies.

We have found a range of $c_{f}$ values (as defined in Eq. (1)) which
corresponds to the range of peak frequencies from $f_{\rm peak}=2.9\times
10^{-8}$ Hz to $f_{\rm peak}=1.5\times 10^{-7}$ Hz. This corresponds to
annihilation temperatures in the range 200 MeV to 1 GeV. This temperature
range overlaps with the upper portion of the 68% credible interval (which goes
to 275 MeV) and the 95% credible interval (which goes to 505 MeV) quoted by
NANOGrav (NANOGrav:2023hvm, ) if their DW-SM model is the sole origin of the
signal. Considering their fit done with the addition of a SMBHB contribution,
our temperature range overlaps with a larger portion of both the 68% credible
interval (which goes to 309 MeV) and the 95% credible interval (which goes to
843 MeV) quoted in the text, and is included within the red region in the
lower left corner of Fig. 12 of NANOGrav:2023hvm  (for its DW-SM + SMBHB fit).

The upper portion of the region of ALP-photon coupling and mass necessary to
explain the NANOGrav signal (shown in Fig. 1), for couplings above $10^{-7}$
GeV-1, could be tested in the future by DarkQuest, HIKE-dump and SHiP, as
shown e.g. in Fig. 133 of Antel:2023hkf  (see references therein). The lower
portion would be tested if a new supernova is observed by the Supernova Early
Warning System (SNEWS) (SNEWS:2020tbu, ), which would allow to extend
considerably all the limits derived from SN1987A.

## Acknowledgements

We thank E. Vitagliano for useful comments. The work of GG was supported in
part by the U.S. Department of Energy (DOE) Grant No. DE-SC0009937.

##  Appendix A Appendix A

Before annihilation starts, the energy density of the walls in the scaling
regime is $\rho_{\rm walls}\simeq\sigma/t\gg V_{\rm bias}$. The annihilation
of the string-wall system starts when the bias volume energy density, or
magnitude of volume pressure, $V_{\rm bias}$ becomes of the same order as the
energy density, or surface tension, of the walls $\sigma/t$ ($t_{\rm
ann}\simeq\sigma/V_{\rm bias}$), after which $V_{\rm bias}$ dominates and
accelerates walls towards the higher-energy vacuum adjacent to each wall. If
PBHs do not form at annihilation, i.e. $p(t_{\rm ann})<1$, the energy
contained in a closed wall will need to increase with time for PBHs to form
later, at a time $t_{\star}$ such that $p(t_{\star})=1$. Since the energy
density $V_{\rm bias}$ is constant, this requires that the dimensions of the
closed walls keep growing for $t>t_{\rm ann}$. In fact, if the characteristic
linear dimension $L$ of walls continues being close to $t$, $L\simeq t$, then
$p(t>t_{\rm ann})\sim V_{\rm bias}L^{3}/L$ grows with time as $t^{2}$ and
eventually reaches 1. However, if $L$ decreases with time and never becomes
larger than $t_{\rm ann}$, the figure of merit $p(t)$ decreases after
annihilation starts and never reaches 1.

Based on the simple power-law parameterization we used in our previous recent
work (Gelmini:2022nim, ; Gelmini:2023ngs, ) for the evolution of the energy
density after annihilation starts, for $T<T_{\rm ann}$,

$\frac{\rho_{\rm walls}(T)}{\rho_{\rm walls}(T_{\rm
ann})}\simeq\left(\frac{T}{T_{\rm
ann}}\right)^{\alpha}\simeq\left(\frac{t_{\rm ann}}{t}\right)^{\alpha/2},$
(21)

with a real positive power $\alpha$ that needs to be extracted from
simulations of the annihilation process, we can make a naive estimate of how
the characteristic linear wall size $L$ within a Hubble volume $t^{3}$ evolves
with time. It is easy to do it either assuming that walls dominate the energy
density, or that volume density dominates. In both cases we find the same
condition on $\alpha$ for $L$ to continue growing with time, i.e. $L>t_{\rm
ann}$ for $t>t_{\rm ann}$. Therefore it is reasonable to assume that the same
condition holds in the transition period, when both volume and walls
contribute significantly to the energy density of the annihilating string-wall
system.

If the energy in walls still dominates

$\left(\frac{t_{\rm ann}}{t}\right)^{\alpha/2}\simeq\frac{\rho_{\rm
walls}(T)}{\rho_{\rm walls}(T_{\rm ann})}\simeq\frac{\sigma
L^{2}}{t^{3}}\frac{t_{\rm ann}^{3}}{\sigma t_{\rm
ann}^{2}}\simeq\frac{L^{2}}{t^{3}}t_{\rm ann}.$ (22)

Thus

$L=t\left(\frac{t_{\rm ann}}{t}\right)^{(\alpha-2)/4}$ (23)

and requiring $L/t_{\rm ann}>1$ for $t>t_{\rm ann}$, means that $(t/t_{\rm
ann})^{(6-\alpha)/4}>1$, i.e. ${(6-\alpha)/4}>0$, thus $\alpha<6$.

A similar calculation can be done assuming volume energy dominates, $\rho_{\rm
walls}(t)\simeq V_{\rm bias}L^{3}/t^{3}$ to find

$L=t\left(\frac{t_{\rm ann}}{t}\right)^{\alpha/6}$ (24)

and requiring $L/t_{\rm ann}>1$ for $t>t_{\rm ann}$, means that $(t/t_{\rm
ann})^{(6-\alpha)/6}>$ 1, with ${(6-\alpha)/6}>0$, i.e. $\alpha<6$ again.

In both cases we find the condition $\alpha<6$ for $L$ to become larger than
$t_{\rm ann}$ after annihilation stars. However the only simulations available
to estimate values of $\alpha$ (Kawasaki:2014sqa, ) lead to $\alpha\geq 7$
(see Gelmini:2022nim ; Gelmini:2023ngs  for details). In this case, with our
naive estimates the linear size of walls would decrease with time after
annihilation starts and PBHs would not form if $p(t_{\rm ann})<1$.

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<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
# Dirac and Majorana Fermions in the Anti-de Sitter spacetime with Tachyonic
approaches

Chia-Li Hsieh<EMAIL_ADDRESS>Vahideh Memari  [  Mustafa Halilsoy  [
Department of Physics, Faculty of Arts and Sciences, Eastern Mediterranean
University, Famagusta, North Cyprus via Mersin 10, Turkey

###### Abstract

We analyze the Dirac equation in Anti-de Sitter (AdS) spacetime upon the
comformally flat spacetime. We also apply the analysis to the Majorana
condition. The analytical solutions are neat. We observe that Dirac fermions
and Majorana fermions behave differently in AdS background. The wavefunction
of tachyonic neutrinos leaves an track, clumping in a special flatland,
embedded in the curved 3+1-D AdS spacetime. Besides, there emerges a boundary,
and neutrinos also oscillate.

.

Suggested keywords

##  I Introduction

The task to find solutions of Dirac equations in curved spacetime relies on
decoupling the equations, especially in curved spacetime. The well-known
examples of Dirac equations in Kerr geometry and de Sitter spacetime can be
seen from Chandrasekhar[1] and Dirac[2]. In this paper, we will study the
Dirac equation in Anti-de Sitter (AdS) spacetime. Instead of going directly to
the curved spacetime [3], we take advantage of the conformally flat coordinate
in AdS spacetime, which is rarely discussed. Then with Newman-Penrose (NP)
formalism [4] and tricks in section II, we decouple the equations neatly and
have profound results.

Physically, AdS spacetime is a solution of Einstein’s field equation with a
negative constant $\Lambda$, giving an attracting geometry. And the metric can
be read as
$ds^{2}=(1+\frac{\Lambda}{3}r^{2})d\bar{t}^{2}-\frac{dr^{2}}{1+\frac{\Lambda}{3}r^{2}}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$, with cosmological constant $\Lambda$. We can transform it into a
conformally flat spacetime without changing physical meaning (just seen from
another frame)

$ds^{2}=\frac{1}{z^{2}}(dt^{2}-dz^{2}-dx^{2}-dy^{2})$ (1)

by the recipe[5]

$\begin{split}&\sqrt{r^{2}+a^{2}}\cos\bar{t}=\frac{1}{2z}(a^{2}+x^{2}+y^{2}+z^{2}-t^{2})\\\
&\sqrt{r^{2}+a^{2}}\sin\bar{t}=\frac{at}{z}\\\
&r\sin\theta\cos\phi=\frac{ax}{z}\\\ &-r\sin\theta\sin\phi=\frac{ay}{z}\\\
&-r\cos\theta=\frac{1}{2z}(-a^{2}+x^{2}+y^{2}+z^{2}-t^{2}),\end{split}$ (2)

where $a^{2}=\frac{3}{|\Lambda|}$ and the $\Lambda$ is absorbed.
Alternatively, the AdS metric (1) can be obtained as a result of reflecting
null-shells from a boundary [6]. we can see that for constant z the AdS
spacetime can be reduced to a $2+1-D$ flat spacetime, that is, every slice of
constant z corresponds to a lower/flat spacetime.

In addition to solving the Dirac equations, a wavefunction of tachyon is also
introduced. We resemble the wavefunctions as neutrinos, which inspires some
interesting phenomena. Besides, we also derive the Majorana fermions in the
AdS spacetime. Since gravity distinguishes Dirac and Majorana particles [7],
then, we may ask the question: Can the attractive AdS geometry also
distinguish between Dirac and Majorana?

Our paper is organized as follows. In section II, we discuss the differential
equations and derive the wavefunctions and the boundary. We consider the
tachyonic fields in the AdS background in section III. In Section IV, th
Majorana condition is imposed. We complete the paper with our discussion and
conclusion in section V.

##  II The Method

Consider the Dirac equation in AdS spacetime by using the Newman-Penrose (NP)
formalism [4] refering to Chandrasekhar [1].

$\begin{split}&(D+\epsilon-\rho)F_{1}+(\bar{\delta}+\pi-\alpha)F_{2}=imG_{1}\\\
&(\Delta+\mu-\gamma)F_{2}+(\delta+\beta-\tau)F_{1}=imG_{2}\\\
&(D+\bar{\epsilon}-\bar{\rho})G_{2}-(\delta+\bar{\pi}-\bar{\alpha})G_{1}=imF_{2}\\\
&(\Delta+\bar{\mu}-\bar{\gamma})G_{1}-(\bar{\delta}+\bar{\beta}-\bar{\tau})G_{2}=imF_{1},\end{split}$
(3)

where $\alpha,\beta,\gamma,\rho,\epsilon,\mu,\pi,\tau$ are spin coefficients,
$D,\Delta,\delta$, the directional derivatives, m the mass and
$F_{1},F_{2},G_{1},G_{2}$ the spinor components. From the Eqn (1), we may
choose the null tetrad basis 1-forms as

$\begin{split}&\sqrt{2}l=\frac{1}{z}(dt-dz)\\\
&\sqrt{2}n=\frac{1}{z}(dt+dz)\\\ &\sqrt{2}m=\frac{1}{z}(dx+idy)\\\
&\sqrt{2}\bar{m}=\frac{1}{z}(dx-idy),\end{split}$ (4)

which gives us the only non-zero spin coefficients
$\epsilon=\gamma=-\frac{1}{2\sqrt{2}},\mu=\rho=\frac{1}{\sqrt{2}}$ ,Ricci
scalar $R=-12$ and the directional derivatives

$\begin{split}&D=\frac{z}{\sqrt{2}}(\partial_{t}+\partial_{z})\\\
&\Delta=\frac{z}{\sqrt{2}}(\partial_{t}-\partial_{z})\\\
&\delta=-\frac{z}{\sqrt{2}}(\partial_{x}+i\partial_{y})\\\
&\bar{\delta}=-\frac{z}{\sqrt{2}}(\partial_{x}-i\partial_{y}).\end{split}$ (5)

Therefore, we have the Dirac equations

$\begin{split}&(D-\frac{3}{2\sqrt{2}})F_{1}+\bar{\delta}F_{2}=imG_{1}\\\
&(\Delta+\frac{3}{2\sqrt{2}})F_{2}+\delta F_{1}=imG_{2}\\\
&(D-\frac{3}{2\sqrt{2}})G_{2}-\delta G_{1}=imF_{2}\\\
&(\Delta+\frac{3}{2\sqrt{2}})G_{1}-\bar{\delta}G_{2}=imF_{1}.\end{split}$ (6)

To solve these equations, we consider the ansatz

$\begin{split}&F_{i}=f_{i}(z)e^{i\omega t+ip_{1}x+ip_{2}y}\\\
&G_{i}=g_{i}(z)e^{i\omega t+ip_{1}x+ip_{2}y}.\end{split}$ (7)

Therefore, we derive

$\begin{split}&Pf_{1}-(p_{2}+ip_{1})f_{2}=\frac{i\sqrt{2}mg_{1}}{z}\\\
&\bar{P}f_{2}-(p_{2}-ip_{1})f_{1}=-\frac{i\sqrt{2}mg_{2}}{z}\\\
&Pg_{2}-(p_{2}-ip_{1})g_{1}=\frac{i\sqrt{2}mf_{2}}{z}\\\
&\bar{P}g_{1}-(p_{2}+ip_{1})g_{2}=-\frac{i\sqrt{2}mf_{1}}{z},\end{split}$ (8)

with the definition of operator, $P=\partial_{z}-\frac{3}{2z}+i\omega$ and its
complex conjugate $\bar{P}$. By multiplying all the equations with its complex
conjugate operator, we obtain

$\begin{split}&f^{\prime\prime}_{1}-\frac{3}{z}f^{\prime}_{1}+(\frac{15-8m^{2}}{4z^{2}}+\omega^{2}-p^{2}_{1}-p^{2}_{2})f_{1}=-\frac{i\sqrt{2}m}{z^{2}}g_{1}\\\
&f^{\prime\prime}_{2}-\frac{3}{z}f^{\prime}_{2}+(\frac{15-8m^{2}}{4z^{2}}+\omega^{2}-p^{2}_{1}-p^{2}_{2})f_{2}=\frac{i\sqrt{2}m}{z^{2}}g_{2}\\\
&g^{\prime\prime}_{2}-\frac{3}{z}g^{\prime}_{2}+(\frac{15-8m^{2}}{4z^{2}}+\omega^{2}-p^{2}_{1}-p^{2}_{2})g_{2}=-\frac{i\sqrt{2}m}{z^{2}}f_{2}\\\
&g^{\prime\prime}_{1}-\frac{3}{z}g^{\prime}_{1}+(\frac{15-8m^{2}}{4z^{2}}+\omega^{2}-p^{2}_{1}-p^{2}_{2})g_{1}=\frac{i\sqrt{2}m}{z^{2}}f_{1},\end{split}$
(9)

Now consider the on-shell condition and set
$\omega^{2}-p^{2}_{1}-p^{2}_{2}=p^{2}_{3}+m^{2}=k^{2}$. One observes that from
the equations the choices $f_{1}=ig_{1}$ and $f_{2}=ig_{2}$ decouple the
equations and the Dirac equations are in consistent with each other. To solve
the Dirac equations, we consider

$\begin{split}&f_{1}=A+iB\\\ &f_{2}=C+iD\\\ &g_{1}=E+iF\\\ &g_{2}=G+iH\\\
\end{split}$ (10)

where A, B,… are real functions and set $\Box
f=f^{\prime\prime}-\frac{3}{z}f^{\prime}+(\frac{15-8m^{2}}{4z^{2}}+k^{2})f$.
We then derive

$\begin{split}&\Box A=\frac{\sqrt{2}m}{z^{2}}F,\Box
B=-\frac{\sqrt{2}m}{z^{2}}E\\\ &\Box C=-\frac{\sqrt{2}m}{z^{2}}H,\Box
D=\frac{\sqrt{2}m}{z^{2}}G\\\ &\Box G=\frac{\sqrt{2}m}{z^{2}}D,\Box
H=-\frac{\sqrt{2}m}{z^{2}}C\\\ &\Box E=-\frac{\sqrt{2}m}{z^{2}}B,\Box
F=\frac{\sqrt{2}m}{z^{2}}A.\\\ \end{split}$ (11)

Equation (11) plays the crucial to get the consistent Dirac equaitons.
However, to decouple the equations, one may choose

$\begin{split}&A=-F,C=-H\\\ &B=E,D=G.\\\ \end{split}$ (12)

Thus,

$\begin{split}&f_{1}=A+iB,f_{2}=C+iD\\\ &g_{1}=B-iA,g_{2}=D-iC,\\\
\end{split}$ (13)

and $f_{1}=ig_{1}$, $f_{2}=ig_{2}$ are satisfied. The equation (11) should be
reduced to $\Box A=\frac{\sqrt{2}m}{z^{2}}A$, $\Box
B=-\frac{\sqrt{2}m}{z^{2}}B$ or

$\begin{split}&A^{\prime\prime}-\frac{3}{z}A^{\prime}+(\frac{15-8m^{2}-4\sqrt{2}m}{4z^{2}}+k^{2})A=0\\\
&B^{\prime\prime}-\frac{3}{z}B^{\prime}+(\frac{15-8m^{2}+4\sqrt{2}m}{4z^{2}}+k^{2})B=0.\\\
\end{split}$ (14)

To solve A and B, we refer to Lommel [8] that

$z^{2}f^{\prime\prime}+(2\alpha-2\beta\nu+1)zf^{\prime}+[\alpha(\alpha-2\beta\nu)+\beta^{2}\gamma^{2}z^{2\beta}]f=0$
(15)

with a solution

$f=z^{\beta\nu-\alpha}[a_{0}J_{\nu}(\gamma z^{\beta})+b_{0}J_{-\nu}(\gamma
z^{\beta})],$ (16)

where $a_{0}$ and $b_{0}$ are constants. Therefore,

$A(z)=z^{2}[a_{0}J_{\nu}(kz)+b_{0}J_{-\nu}(kz)],$ (17)

where $\nu=\pm\sqrt{\frac{1}{4}+\sqrt{2}m(\sqrt{2}m+1)}$, and

$B(z)=z^{2}[c_{0}J_{\nu}(kz)+d_{0}J_{-\nu}(kz)]$ (18)

with $\nu=\pm\sqrt{\frac{1}{4}+\sqrt{2}m(\sqrt{2}m-1)}$ and ($c_{0},d_{0}$)
constants. For massless case, $\nu=\pm\frac{1}{2}$, we got
$z^{2}J_{\pm\frac{1}{2}}$. The Bessel function performs as a periodic function
with decay. We may consider that asymptotic approach, $z>>1$, the differential
equation reduces to

$f^{\prime\prime}-\frac{3}{z}f^{\prime}+k^{2}f=0,$ (19)

which gives a neat solution, $z^{2}J_{2}(kz)$ or $z^{2}N_{2}(kz)$, where
$N(kz)$ is the Neumann function. Therefore, from our ansatz (7), we can build
the spinor.

In the kinetic energy dominated regime, $k>>1$ , we can see from the
differential equations that

$f^{\prime\prime}+k^{2}f\approx 0,$ (20)

which brings us the solution, $f=A_{0}sin(kz)+B_{0}cos(kz)$. One may regard
these solutions as neutrino because we cannot bring it to rest. For
simplicity, we let 2 mass states to be

$\begin{split}&f_{\nu 1}=A_{0}sin(k_{1}z)\\\ &f_{\nu
2}=A_{0}sin(k_{2}z),\end{split}$ (21)

where $k_{1}=\sqrt{p^{2}_{3}+m^{2}_{1}}\approx p_{3}+\frac{m^{2}_{1}}{2p_{3}}$
and $k_{2}\approx p_{3}+\frac{m^{2}_{2}}{2p_{3}}$. Set
$k=\frac{k_{1}+k_{2}}{2}$ and $\Delta k=\frac{m^{2}_{1}-m^{2}_{2}}{4p_{3}}$,
then $k_{1}=k+\Delta k$ and $k_{2}=k-\Delta k$. The combination of neutrinos
becomes

$\nu_{1}+\nu_{2}=2A_{0}sin(kz)cos(\Delta kz).$ (22)

The simple version of neutrino beat or neutrino oscillation is recovered in
the AdS background.

The AdS spacetime is an attractive geometry as it is well-known from AdS/CFT
correspondence [9]. Particles tend to deviate inward. We can observe that
$z^{2}$ accompanies all the Bessel functions, which makes all the
wavefunctions become zero as $z\to$ 0\. There emerges a boundary at $z=0$. In
this way all the information would be lost in the boundary, the wavefunction
would disappear. However, after examining the geodesic (in the Appendix), we
find that the wavefunction can only diminish and fall into a $z=z_{0}$ plane.
We may choose the $\xi$ to be very large, so that $z_{0}=\frac{1}{\xi}$ would
be in the edge of boundary $z=0$ (in the Appendix). Therefore,
$z^{2}_{0}J(kz_{0})$ becomes constant, the other components still propagate in
the $z_{0}$ plane, the $2+1-D$ lower and flat spacetime, see Figure 1. As a
result, we find that higher dimension may collapse into lower dimension and
the wavefunctions are special signals to mediate between higher and lower
dimensions in the AdS geometry,.

##  III Tachyonic condition

The equation of motion includes not only usual Dirac fermions but tachyons.
The motivation comes from Chodos [10] that neutrinos may be in the tachyonic
form. Therefore, in this section we assume the tachyonic condition, that is,

$\omega^{2}-p^{2}_{1}-p^{2}_{2}=p^{2}_{3}+m^{2}=-\kappa^{2}.$ (23)

Obviously, $p^{2}_{3}>0$. So $m$ has to be replaced by $im$ for
$\kappa^{2}>0$. The equations of motion now become

$\begin{split}&\bar{\Box}f_{1}=\frac{\sqrt{2}m}{z^{2}}g_{1},\bar{\Box}f_{2}=-\frac{\sqrt{2}m}{z^{2}}g_{2}\\\
&\bar{\Box}g_{1}=-\frac{\sqrt{2}m}{z^{2}}f_{1},\bar{\Box}g_{2}=\frac{\sqrt{2}m}{z^{2}}f_{2}\\\
,\end{split}$ (24)

where
$\bar{\Box}f=f^{\prime\prime}-\frac{3}{z}f^{\prime}+(\frac{15+8m^{2}}{4z^{2}}-\kappa^{2})f$.
The condition $f_{1}=ig_{1}$, $f_{2}=ig_{2}$ also applies here, but the
function can not be decomposed into real and imaginary part easily. We have to
directly solve equations

$\bar{\Box}f_{1}=-i\frac{\sqrt{2}m}{z^{2}}f_{1},\bar{\Box}f_{2}=i\frac{\sqrt{2}m}{z^{2}}f_{2},$
(25)

The solutions are $z^{2}K_{\nu}(\kappa z)$, where $K_{\nu}(\kappa z)$ is
modified Bessel function, and $\nu=\frac{1}{2}-i\sqrt{2}m$ for $f_{1}$, and
$\nu={\frac{1}{2}+i\sqrt{2}m}$ for $f_{2}$.

If the neutrinos are tachyons [10], we see that the index $\nu=\frac{1}{2}$
dominates the modified Bessel function due to negligible mass of neutrinos. As
one can see from Figure 2, before the wavefunction diminishes to the boundary,
it has a higher chance to be seen at constant slice $z_{d}$. The particles
here are fermions, therefore they are more likely to clump in a $z=z_{d}$
slice, that is, a $2+1-D$ spacetime. There arises a phenomenon: In AdS
spacetime, we have found a special shell that all 3 types of neutrinos
inhabit. In other words, we find a ”Flatland” for neutrinos, since the shell
is a $2+1-D$ flat spacetime. It is important to find a flat frame embedded in
the curved spacetime. Thus, in curved AdS background, there may exist a flat
shell, and in this shell, one could discuss the propagation with no curvature
effects.

Figure 1: The $3+1-D$ wavefunction diminishes and collapses into a $2+1-D$
plane wave at $z=z_{0}$. Figure 2: The wavefunction of tachyon has a maximum
at $z=z_{d}$.

##  IV Majorana condition

In Dirac equations, we may impose the Majorana condition [11],

$\Psi_{M}=\begin{bmatrix}\phi\\\ i\sigma_{2}\phi^{*}\end{bmatrix},$ (26)

such that $\Psi_{M}$ would be its own antiparticle, where
$\sigma_{2}=\begin{bmatrix}0&-i\\\ i&0\end{bmatrix}$. Set
$\phi=\begin{bmatrix}A\\\ B\end{bmatrix}$.

$\therefore\Psi_{M}=\begin{bmatrix}A\\\ B\\\ B^{*}\\\ -A^{*}\end{bmatrix}.$
(27)

Comparing Chandrasekhar’s assumption [1] with the Majorana condition leads to

$\begin{bmatrix}P^{A}\\\
\bar{Q}^{A^{\prime}}\end{bmatrix}=\begin{bmatrix}P^{0}\\\ P^{1}\\\
\bar{Q}^{0^{\prime}}\\\
\bar{Q}^{1^{\prime}}\end{bmatrix}=\begin{bmatrix}F_{1}\\\ F_{2}\\\ -G_{2}\\\
G_{1}\end{bmatrix}=\begin{bmatrix}A\\\ B\\\ B^{*}\\\ -A^{*}.\end{bmatrix}$
(28)

In the equation, $A\rightarrow F_{1},B\rightarrow F_{2},B^{*}\rightarrow-
G_{2},-A^{*}\rightarrow G_{1}$, we therefore find the Majorana condition for
Dirac equation in AdS spacetime,

$F_{1}=-G_{1}^{*},F_{2}=-G_{2}^{*}.$ (29)

Since the fields are Majorana, we may suggest that each component is complex
conjugate to itself,

$F_{i}=F_{i}^{*},G_{i}=G_{i}^{*}$ (30)

and cast the ansatz as

$\begin{split}&F_{i}=f_{i}(z)e^{-\omega t-p_{1}x-p_{2}y}\\\
&G_{i}=g_{i}(z)e^{-\omega t-p_{1}x-p_{2}y}.\end{split}$ (31)

Bringing (31) into (3) gives

$\begin{split}&Mf_{1}+(p_{1}-ip_{2})f_{2}=\frac{i\sqrt{2}mg_{1}}{z}\\\
&\bar{M}f_{2}+(p_{1}+ip_{2})f_{1}=\frac{i\sqrt{2}mg_{2}}{z}\\\
&Mg_{2}+(-p_{1}-ip_{2})g_{1}=\frac{i\sqrt{2}mf_{2}}{z}\\\
&\bar{M}g_{1}+(-p_{1}+ip_{2})g_{2}=\frac{i\sqrt{2}mf_{1}}{z},\end{split}$ (32)

where we define $M=-\omega+\partial_{z}-\frac{3}{2z}$ and
$\bar{M}=-\omega-\partial_{z}-\frac{3}{2z}$, and multiply $M$ and $\bar{M}$
for each as we did previously. The differential equations read

$\begin{split}&f^{\prime\prime}_{1}-\frac{3\omega}{z}f_{1}-(\frac{3+8m^{2}}{4z^{2}}+k^{2})f_{1}=\frac{i\sqrt{2}m}{z^{2}}g_{1}\\\
&f^{\prime\prime}_{2}-\frac{3\omega}{z}f_{2}-(\frac{15+8m^{2}}{4z^{2}}+k^{2})f_{2}=\frac{i\sqrt{2}m}{z^{2}}g_{2}\\\
&g^{\prime\prime}_{1}-\frac{3\omega}{z}g_{1}-(\frac{15+8m^{2}}{4z^{2}}+k^{2})g_{1}=\frac{i\sqrt{2}m}{z^{2}}f_{1}\\\
&g^{\prime\prime}_{2}-\frac{3\omega}{z}g_{2}-(\frac{3+8m^{2}}{4z^{2}}+k^{2})g_{2}=\frac{i\sqrt{2}m}{z^{2}}f_{2},\end{split}$
(33)

and also the on-shell condition applies here. The difference between $f_{i}$
and $g_{i}$ are due to that $M$ and $\bar{M}$ do not commute. Then considering
(29) and (30), $g_{1}\rightarrow-f_{1}^{*}\rightarrow-f_{1}$, we obtain the
Whittaker equation (see [12])

$\begin{split}f^{\prime\prime}_{1}-\frac{3\omega}{z}f_{1}-(\frac{3+8m^{2}-i4\sqrt{2}m}{4z^{2}}+k^{2})f_{1}=0\end{split},$
(34)

and similar for the rest equations.

For heavily sterile Majorana particles in seesaw mechanism [13], $m>>1$ and
$z<<1$, eqn (34) can be reduced to

$\begin{split}f^{\prime\prime}_{1}-\frac{2m^{2}}{z^{2}}f_{1}\approx
0.\end{split}$ (35)

The wavefunction of Majorana particles is

$f_{1}=a_{0}e^{k_{1}\ln z}+b_{0}e^{k_{2}\ln z},$ (36)

where $k_{1,2}=\frac{1\pm\sqrt{1+8m^{2}}}{2}$ with ($a_{0},b_{0}$) constants,
and similarly for the $g_{i}$ functions but picking up a minus sign. Then
$f_{i}=-g_{i}^{*}$ and $f_{i}=f_{i}^{*}$ fit the criteria of our assumptions
(29) and (30).

If we loosely regard condition (29) as a boundary only involved in solving
differential equations, we got some interesting results. From the ansatz (31),
we may focus on $z<<1$, therefore

$f_{1}^{{}^{\prime\prime}}-\frac{3+8m^{2}+i4\sqrt{2}m}{4z^{2}}f_{1}\approx 0.$
(37)

We can see from the massless case, the solution is $f_{1}$ =$z^{\frac{-1}{2}}$
or $z^{\frac{3}{2}}$, and $f^{*}_{1}=f_{1}$ is satisfied. And the solution of
massive case gives

$f_{1}=e^{\frac{1}{2}(1+c+id)\ln z}+e^{\frac{1}{2}(1-c-id)\ln z},$ (38)

where
$c^{2}=\frac{1+4a\pm\sqrt{(1+4a)^{2}+16b^{2}}}{2},d=2b/c,a=\frac{3+8m^{2}}{4},b=-\sqrt{2}m$.
The massive $f_{1}$ is obviously not complex conjugate to itself, unless
$c=0$. However, $c=0$ implies $m=0$. To fix this, we may replace $m$ by $im$.
$f_{1}$ becomes

$f_{1}=a_{0}e^{k_{1}\ln z}+b_{0}e^{k_{2}\ln z},$ (39)

where $k_{1,2}=\frac{1\pm\sqrt{4-8m^{2}+4\sqrt{2}m}}{2}$ with ($a_{0},b_{0}$)
constants. We may choose $a_{0}=b_{0}$. Therefore, $f^{*}_{1}=f_{1}$.

We could also consider that $f_{1}$ is a wavefunction consisted of a pair of
electron and hole. Then one may observe that this self-complex conjugated
wavefunction evanesce or grow in z direction but with limitation, $z<<1$, if
we can create a AdS-like potential in condensed matter.

##  V Discussion and Conclusion

We would like to promote a concept here. Spacetime like AdS can be projected
to the Ginzburg-Landau theory [14]. Vice versa, quantum mechanical materials
can simulate spacetime [15]. One may consider the electromagnetic interaction
and gravitational effect in the derivative of the spinor $\Psi$[16, 17],
$D_{\mu}\Psi=\partial_{\mu}\Psi+iq_{1}A_{\mu}\Psi+q_{2}\Gamma_{\mu}\Psi$,
where $A_{\mu}$ and $\Gamma_{\mu}$ are vector potential and Riemannian
connection with coupling constant $q_{1}$ and $q_{2}$, respectively. We can
find a proper $A_{\mu}$ to replace $\Gamma_{\mu}$, or we can say, we mimic the
non-flat space by $A_{\mu}$. Then we may try the AdS-like potential to test
the wavefunction $z^{2}J_{\nu}(kz)$ of electrons and finally applying to find
the Majorana particles in condensed matter [18].

By exploiting the conformally flat spacetime, we find a general form of Dirac
fermions in AdS spacetime, which can be applied to free neutrinos, electrons,
quarks and so on. The Majorana fermions are also discussed, and we derive the
wavefunction of sterile Majorana particles. Now we can respond to the
question. Can AdS geometry distinguish between Dirac and Majorana particles?
Upon our calculation, we may support that Dirac and Majorana wavefunctions can
be distinguished in AdS spacetime. For theoretical interest, we introduce
tachyons in Dirac and Majorana equations. A ”Flatland” for tachyonic neutrinos
and boundaries are naturally built. In addition, the wavefunctions we found
here play the role to bridge higher and lower dimensions. Finally, we hope our
finding can deepen the understanding of Dirac equations in non-flat spacetime
and inspire the searching of Majorana particles as well.

## Appendix

Since we have

$ds^{2}=\frac{1}{z^{2}}(dt^{2}-dz^{2}-dx^{2}-dy^{2}),$ (40)

the geodesic lagrangian is

$L=\frac{1}{2z^{2}}(\dot{t}^{2}-\dot{z}^{2}-\dot{x}^{2}-\dot{y}^{2}).$ (41)

Because x, y, and t parts are cyclic, we have

$\begin{split}&\frac{\partial
L}{\partial\dot{t}}=\frac{\dot{t}^{2}}{z^{2}}=\alpha_{0}\\\ &\frac{\partial
L}{\partial\dot{x}}=\frac{\dot{x}^{2}}{z^{2}}=\beta_{0}\\\ &\frac{\partial
L}{\partial\dot{y}}=\frac{\dot{y}^{2}}{z^{2}}=\gamma_{0},\end{split}$ (42)

where a ’dot’ denotes derivative with respect to the proper time $\tau$. As
for the z part, we should consider
$1=\frac{1}{z^{2}}(\alpha_{0}^{2}z^{4}-\dot{z}^{2}-\beta_{0}^{2}z^{4}-\gamma_{0}^{2}z^{4})$,
therefore

$\dot{z}^{2}=-z^{2}+(\alpha_{0}^{2}-\beta_{0}^{2}-\gamma_{0}^{2})z^{4}=\xi^{2}z^{4}-z^{2},$
(43)

where $\xi^{2}=\alpha_{0}^{2}-\beta_{0}^{2}-\gamma_{0}^{2}$. Also,
$\frac{dz}{d\tau}=\frac{dz}{dt}\frac{dt}{d\tau}=\frac{dz}{dt}\alpha_{0}z^{2}$.
As a result,
$\frac{dz}{dt}=\frac{1}{\alpha_{0}}\sqrt{\frac{\xi^{2}z^{2}-1}{z^{2}}}$, and
brings us

$z=\sqrt{\frac{(t-t_{l})^{2}\xi^{2}}{\alpha_{0}^{2}}+\frac{1}{\xi^{2}}}.$ (44)

We define the constant $t_{l}$ for life time in z-axis.

Data Availability Statement: No Data associated in the manuscript.

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