Axion and the Supermassive Black Holes at High $z$

Takeshi Fukuyamaa,111E-mail<EMAIL_ADDRESS>
aResearch Center for Nuclear Physics (RCNP), Osaka University,
Ibaraki, Osaka, 567-0047, Japan

Axion dark matter (DM) is studied on the formation of supermassive black holes
(SMBHs) at high red shift $z$. It is shown that the attractive self
interaction of this DM may resolve the tension between the large mass with
high angular momentum of SMBH and its early time formation. We consider the
ultra-light axion DM around $O(10^{-21})$ eV as its origin, which may also
cause nano-Hz stochastic gravitational wave background recently observed.

##  1 Introduction

It is well known that supermassive black holes (SMBHs) ($M_{SMBH}\geq
O(10^{6}M_{\odot})$ with solar mass $M_{\odot}$) reside at the center of most
galaxies [1, 2, 3]222Also the supermassive galaxies at $z\geq 10$ is observed
by James Webb Space Telescope [4].. The accretion processes of both baryonic
and collisionless DM are rather slow. So there appears a problem why the SMBHs
appear at high redshift $z\geq 5$. Furthermore, BHs have in general large
angular momenta, which makes it more difficult to form BHs at such high $z$.
In this paper, we consider axion DM [5] and show that its attractive self-
interaction may solve the above mentioned problem. QCD axion appears in
connection with the chiral condensate, which is the main origin of baryonic
masses, and it is natural to consider that axion is the main constituent of
DM. Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) axion [6, 7] is naturally
formulated in the framework of the minimal SO(10) GUT [8]. Indeed, in the
minimal SO(10) model, there are two Higgs doublets, $H_{u}$ and $H_{d}$,
coming from ${\bf 10}$-plet and $\overline{126}$-plet Higgs in GUT. They are
coupled with ${\bf 210}$-plet Higgs [9, 10], which induces the
$H_{u}-H_{d}-\phi\mbox{(axion)}^{2}$ coupling in the Standard Model (SM)
phase. The fundamental spinor representation is ${\bf 16}$-plet, including
just all SM matters of quarks and leptons, and no room for the additional
heavy fermions unlike Kim-Shifman-Vainshtein-Zakharov (KSVZ) axion [11, 12].
Thus, the DFSZ axion model can be arranged with the minimal SO(10) model by
adjusting global $U(1)_{PQ}$ charges. That is, ${\bf 16},~{}{\bf
10},~{}{\overline{\bf 126}},~{}{\bf 210}$ fields have $U(1)_{PQ}$ charge
$-1,~{}+2,~{}+2,~{}0$, respectively. As will be shown, string-inspired axion
may play the essential role in the formation of the SMBH and it is very
suggestive that the axion decay constant $f_{a}$ is of $O(10^{16})$GeV,
coincident with GUT scale (See (38)). It is also shown that axion (-like) mass
is related with the unified strong coupling ((41)). However, DFSZ model
involves the domain wall problem since $N_{DW}=2N_{g}=6(N_{g}\mbox{:generation
number}$) [10, 13, 14], whereas KSVZ has $N_{DW}=1$ and no problem.
Fortunately enough, axion field acts as $\phi=const.$ in the $m_{a}t\ll
1~{}(m_{a}\mbox{:axion mass})$ era and induces the inflation, which dilutes
away such topological defect as is shown below [15, 16]. In this letter, we
show that the ultra-light axion induces the rapid formation of SMBH and may
also be the origin of the stochastic gravitational wave backgroud (SGWB) found
by the North American Nanoherz Observatory for Gravitational Waves (NANOGrav)
[17, 18] and Pulsar Timing Array Collaborations (PTAs) [19, 20].

This letter is organized as follows. In section 2, we review the axion model
in the expanding universe, where axion is divided into fast and slow
oscillating parts. The slow oscillating part forms overdense part, whose
evolution is discussed in section 3. In section 4, we discuss about the
problem on the axion model before or during inflation and about SGWB. We use
$\hbar=c=1$ units in this letter.

##  2 The review of axion cosmology

In this section, we review the background of the scenario [16]. We introduce
the Peccei-Quinn scalar field $\phi$. The axion field $\varphi$ is introduced
as its phase

$\phi=|\phi|e^{i\theta}=|\phi|e^{i\varphi/f_{a}}$ (1)

$\phi$ in the expanding universe is described as

$\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{\mu}}\left(\sqrt{-g}g^{\mu\nu}\frac{\partial\phi}{\partial
x^{\nu}}\right)+m^{2}\phi=0.$ (2)

The metric $g_{\mu\nu}$ is

$ds^{2}=(1+2\Phi)dt^{2}-a^{2}(1-2\Phi)(dx^{2}+dy^{2}+dz^{2}).$ (3)

Here $\Phi$ is the gravitational potential whose meaning will be discussed
later and $a$ is the Friedmann-Robartson-Walker (FRW) metric. We have
neglected the interaction with gauge bosons and axion self-coupling since we
are concerned with the global behaviours of axion in the expanding universe
here. Substituting (3) into (2), we obtain

$\ddot{\phi}+3H\dot{\phi}-4\dot{\Phi}\dot{\phi}-\frac{1+4\Phi}{a^{2}}\Delta\phi+m_{a}^{2}(1+2\Phi)\phi=0,$
(4)

where $\dot{\phi}\equiv\frac{\partial\phi}{\partial t}$ and
$H=\frac{\dot{a}}{a}$. Firstly we want to consider the global behaviours of
axion and make an approximation

$\ddot{\phi}+3H\dot{\phi}+m_{a}^{2}\phi=0$ (5)

with $H=3p/t$ when $a\propto t^{p}$. The general solution of (5) is

$\phi=a(t)^{-3/2}(m_{a}t)^{1/2}\left(AJ_{n}(m_{a}t)+BY_{n}(m_{a}t)\right),$
(6)

where $J_{n}$ and $Y_{n}$ are Bessel and Neumann functions, respectively, and
$n=\frac{1}{2}\sqrt{9p^{2}-6p+1}$. Imposing the regularity at $m_{a}t\to 0$,
we obtain the integral constant $B=0$.

When $m_{a}t\ll 1$,

$J_{n}(m_{a}t)\approx\frac{1}{\Gamma(n+1)}\left(\frac{m_{a}t}{2}\right)^{n},$
(7)

and

$\phi(t)\propto t^{-\frac{3}{2}p+\frac{1}{2}+n}.$ (8)

Then $\phi(t)=const.$ either radiation ($p=\frac{1}{2}$) or matter
($p=\frac{2}{3}$) dominant universe. Thus axion acts as dark energy (DE).

When $m_{a}t\gg 1$,

$J_{n}(m_{a}t)\approx\left(\frac{2}{\pi
m_{a}t}\right)^{1/2}\cos\left(m_{a}t-\frac{n\pi}{2}-\frac{\pi}{4}\right)$ (9)

and

$\phi(t)=A\sqrt{\frac{2}{\pi}}t^{\frac{3}{2}p}\cos\left(m_{a}t-\frac{n\pi}{2}-\frac{\pi}{4}\right).$
(10)

Thus the energy density of axion

$\displaystyle\rho_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}m_{a}^{2}\phi^{2}$ (11)
$\displaystyle=$
$\displaystyle\frac{A^{2}m_{a}^{2}}{\pi}\frac{1}{a^{3}(t)}\left[1+\frac{9p^{2}}{4}\frac{1}{(m_{a}t)^{2}}\cos^{2}\left(m_{a}t-\frac{n\pi}{2}-\frac{\pi}{4}\right)\right]$
$\displaystyle\approx$
$\displaystyle\frac{A^{2}m_{a}^{2}}{\pi}\frac{1}{a^{3}(t)}.$ (12)

Then $\phi$ acts as dark matter (DM) for $m_{a}t\gg 1$

Thus $\phi$ acts as DE for the early $m_{a}t\ll 1$ era and does as DM for the
later $m_{a}t\gg 1$ era. However, we must consider another aspect of axion as
a quantum field and the result of Bose Einstein Condensate (BEC). Aa we
mentioned, at $m_{a}t\gg 1$, axion oscillates coherently and indistinguishable
with BEC. However, this conclusion comes from (5) with neither gravitational
nor axion self interactions. Indeed, the BEC proceeds in a Bose gas of mass
$m$ and number density $n$, when the thermal de Broglie wavelength
$\lambda_{dB}\equiv\sqrt{2\pi^{2}/(mkT_{m})}$ exceeds the mean interparticle
distance $n^{-1/3}$, and the wavepacket percolates in space [21]:

$kT_{m}<\frac{2\pi^{2}n^{2/3}}{m},$ (13)

where $n$ is the number density, $n\equiv N/V$. On the other hand, cosmic
evolution has the same temperature dependence [21] since the matter-dominant
universe behaves, in an adiabatic process, as

$\rho\propto T_{m}^{3/2}.$ (14)

Hence, if the boson temperature is equal to radiation temperature at $z=1000$,
for example, we have the critical temperature at present $T_{critical}=0.0027$
K, since $T_{m}\propto a^{-2}$ and therefore $T_{\gamma}/T_{m}\propto a$ in an
adiabatic evolution. Using the present energy density of the universe
$\rho=9.44\times 10^{-30}$ g cm-3, the BEC takes place provided that the boson
mass satisfies

$m<1.87\text{eV}.$ (15)

However, axion has the self attraction which makes BEC unstable and axion
oscillates between BEC and DM gass. The difference between BEC and DM occurs
on length scales smaller than the de Brogile length. So for the usual QCD
axion it is too small to consider [22]. However, it may not be negligible for
the string-inspired axion which will be discussed later.

Axion field potential in the classic dilute gas approximation becomes

$V(\phi)=m_{u}\Lambda_{QCD}^{3}\left[1-\cos\left(\frac{\varphi}{f_{a}}\right)\right]\equiv\Lambda^{4}\left[1-\cos\left(\frac{\varphi}{f_{a}}\right)\right].$
(16)

Using the Gellmann-Oakes-Renner relation [23],

$\Lambda_{QCD}^{3}=\frac{F_{\pi}^{2}m_{\pi}^{2}}{m_{u}+m_{d}}.$ (17)

Here $F_{\pi}=93$ MeV, and $\frac{m_{u}}{m_{d}}\approx 0.47$, and we obtain
axion mass

$m_{a}=5.7\times
10^{-6}\left(\frac{10^{12}\mbox{GeV}}{f_{a}}\right)\text{eV}.$ (18)

Then, the axion self coupling constant $\lambda$ in
$\frac{\lambda}{4!}\varphi^{4}$ becomes

$\lambda=-0.47\frac{F_{\pi}^{2}m_{\pi}^{2}}{f_{a}^{4}}<0.$ (19)

Thus, the self coupling is attraction, and not repulsion unlike the usual
Gross-Pitaevskii equation [24, 25]. As is well known, there are wide ranges of
axion mass $m_{a}$ and breaking energy scale $f_{a}$, or equivalently
$\Lambda$ and $f_{a}$ but potential form of (16) is not altered. In the
subsequent sections, we will show that this attractive force may solve the
above mentioned problem of the SMBH at high $z$ and confine the ambiguities of
axion models.

##  3 Slow oscillation part

Axion field, $\phi$, is divided into fast oscillation ($e^{im_{a}t}$) part and
slow one ($\psi$), as

$\phi=\frac{1}{\sqrt{2m_{a}}}\left(\psi
e^{-im_{a}t}+\psi^{*}e^{im_{a}t}\right).$ (20)

This is also the process from the relativistic to nonrelativistic transition.

Substituting (20) into (4), we obtain

$i\dot{\psi}+i\frac{3}{2}H\psi+\frac{1}{2m_{a}^{2}}\Delta\psi-
m_{a}\Phi\psi=0.$ (21)

Here we consider the flucutuation (the overdense region) of the Peccei-Quinn
field decoupled from the cosmological expansion and use Gaussian approximation
[26] with angular momentum,

$|\psi\left(t,x\right)|=\frac{1}{\sqrt{2\pi(l+1)!\sigma^{3}}}\left(\frac{r}{\sigma}\right)^{l}e^{-r^{2}/(2\sigma^{2})}Y_{lm}(\theta,\varphi).$
(22)

From (21) with axion self coupling (16), the axion Lagrangian density becomes

$\mathcal{L}=\frac{i}{2}\left(\psi^{*}\frac{\partial\psi}{\partial
t}-\psi\frac{\partial\psi^{*}}{\partial
t}\right)-\frac{1}{2m_{a}}\nabla\psi^{*}\cdot\nabla\psi-\frac{gN}{2}|\psi|^{4}+N|\psi(x)|^{2}\int\frac{Gm_{a}^{2}}{|{\bf
x}-{\bf y}|}|\psi(y)|^{2}d^{3}y.$ (23)

Here $N$ is the total number of axion particles in the overdense region.
Dimensional coupling $g$ is related with the above $\lambda$ by

$g=\frac{\lambda}{m_{a}^{2}}=\frac{4\pi a_{s}}{m_{a}},$ (24)

where $|a_{s}|$ is the scattering length. Then we obtain the effective
potential,

$V_{eff}=\frac{1}{2m_{a}\sigma^{2}}-\frac{\sqrt{2}}{3\pi}\frac{GNm_{a}^{2}}{\sigma}+\frac{l(l+1)}{2m_{a}\sigma^{2}}+\frac{gN}{6\sqrt{2}\pi^{3/2}\sigma^{3}}.$
(25)

So far we have discussed axion field as DM gass. However, I mentioned this DM
acts soon as BEC. However, BEC is not stable also: For the negative scattering
length, the energy of axion DM is given by [27, 28]

$\frac{E}{N}=\frac{2\pi
a_{s}n}{m_{a}}\left[1+\frac{128}{15}\sqrt{\frac{a_{s}^{3}n}{\pi}}\right].$
(26)

That is, if there is a self-attraction ($a_{s}<0$), BEC decays to DM gass at
the rate

$\Gamma=\frac{128\sqrt{\pi}|a_{s}|^{5/2}n^{3/2}}{15m_{a}}.$ (27)

Thus

$\mathcal{O}(\Gamma)=\mathcal{O}(|a_{s}|^{3}n)^{3/2}\times\frac{1}{m_{a}a_{s}^{2}}.$
(28)

So the BEC half-life time $1/\Gamma$ is very short in comparison with the
cosmological time scale over the possible wide ranges of $|a_{s}|^{3}n<1$, and
DM and BEC are indistinguishable. This conclusion is analogous to that by
Sikivie-Yang [22] except their $a_{s}^{3}n$ in place of our
$(a_{s}^{3}n)^{3/2}$ in (28). See also [29].

Thus we have obtained the main features of axion DM and DE, and let us proceed
to the main problem of Black Hole formation of axion DM taking its self-
coupling into consideration. The effective potential (25) has the stable
orbits at [30, 31]

$\sigma_{min}=\frac{\frac{\tilde{l}^{2}}{2m_{a}}\pm\sqrt{\left(\frac{\tilde{l}^{2}}{2m_{a}}\right)^{2}+\frac{gGN^{2}m_{a}^{2}}{6\pi^{5/2}}}}{\frac{\sqrt{2}}{3\pi}GNm_{a}^{2}}$
(29)

with

$\tilde{l}^{2}\equiv l(l+1)+1.$ (30)

Here it is very important that axion DM attracts to each others since $g<0$,
which allows the two extrema in (29). The minus (plus) case corresponds to
stable (unstable) orbit. These maximum and minimum points coalesce at
$\sqrt{{...}}=0$ in (29) and the stable orbit disappears, leading to so called
Kaup radius,

$\sigma_{Kaup}=\frac{\sqrt{3}}{2\pi^{1/4}}\frac{\sqrt{-g}}{\sqrt{G}m_{a}}=\sqrt{-6\pi^{1/2}g}\frac{M_{Pl}^{*}}{m_{a}}$
(31)

with the reduced Planck mass $M_{Pl}^{*}=\sqrt{\frac{1}{8\pi G}}=2.4\times
10^{18}$ Gev and Kaup mass,

$M_{Kaup}=Nm_{a}=\frac{\sqrt{6\pi^{5/2}}}{2}\frac{\tilde{l}^{2}}{\sqrt{G|\lambda|}}=25\tilde{l}^{2}\frac{M_{Pl}^{*}}{\sqrt{|\lambda|}}.$
(32)

This mass is eventually reduced to BH mass since there is no repulsive force
to prevent the collapse,

$M_{Kaup}=M_{BH}.$ (33)

$\lambda\equiv gm_{a}^{2}$ is estimated from (16) as

$\lambda=-\left(\frac{\Lambda}{f_{a}}\right)^{4}.$ (34)

Then

$\lambda=-0.47\frac{(0.093\times 0.140)^{2}}{10^{48}}=-7.9\times 10^{-54}$
(35)

for a dilute gass approximation ((16)). Substituting this value into (32), we
obtain $M_{BH}=2.1\tilde{l}^{2}\times 10^{-10}M_{\odot}\ll M_{\odot}$. If we
consider the string-inspired axion [16, 32, 33, 34], the axion potential
roughly becomes

$V(\phi)=\Lambda_{string}^{4}\left[1-\cos\left(\frac{\phi}{f_{a}}\right)\right]$
(36)

with

$\Lambda_{string}^{4}=M_{SUSY}^{2}M_{Pl}^{*2}e^{-S_{instanton}}.$ (37)

Here $f_{a}$ and $m_{a}$ become independent parameters and are given as
follow: Firstly, $f_{a}$ is [32]

$f_{a}=\frac{\alpha_{GUT}M_{Pl}^{*}}{\sqrt{2}2\pi}\approx 1.1\times
10^{16}~{}\mbox{GeV}.$ (38)

Here we have set the strong coupling constant $\alpha_{GUT}$ as

$\alpha_{GUT}=\frac{1}{25}.$ (39)

The string-inspired axion mass is

$m_{a}=\frac{M_{SUSY}M_{Pl}^{*}}{f_{a}}e^{-\frac{S_{instanton}}{2}}=1.2\times
10^{-14}~{}\mbox{eV}$ (40)

with supersymmetry breaking scale $M_{SUSY}$. Here we have used

$M_{SUSY}=10^{9}~{}\mbox{GeV},~{}~{}\frac{S_{instanton}}{2}=\frac{\pi}{\alpha_{GUT}}\approx
79$ (41)

and the instanton number $N_{instanton}$,

$N_{instanton}=\frac{1}{64\pi^{2}}\int
d^{4}x\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}^{a}F_{\rho\sigma}^{a}=1.$ (42)

Thus we obtain $f_{a}\approx 10^{16}$ GeV, $m_{a}\approx 10^{-14}$ eV, and
$M_{BH}=50\tilde{l}^{2}M_{\odot}$. However, there are many parameters in
axiverse [33], $M_{SUSY},~{}S_{instanton},~{}\mbox{misaligment
parameter}~{}\theta_{i},$ primordial isocurvature fraction $\alpha$((49)).
Visinelli and Vagnozzi obtained

$S_{instanton}=198\pm 28~{}~{}\mbox{and}~{}~{}M_{SUSY}=10^{11}\mbox{GeV}$ (43)

by Bayesian parameter inference in light of many cosmological data [34]
444This SUSY breaking scale seems to be too large in comparison with that
obtained from the gauge coupling unification [8]. However, there are many
ambiguities on the intermediate states appearing from GUT to SM scales.. Then,
if we adopt the center value $198$ for $S_{instanton}$, we obtain

$\mbox{log}_{10}(m_{a}/\mbox{eV})=-21.5^{+1.3}_{-2.3}$ (44)

and

$M_{BH}=2.5~{}\tilde{l}^{2}\times 10^{8}M_{\odot}.$ (45)

This may explain the origin of SMBH [1, 2, 3]. It is very interesting that
these values lead to the observed magnitude of DM

$\Omega_{a}\equiv\frac{\rho_{a}}{\rho_{critical}}=\frac{1}{3H_{0}^{2}M_{Pl}^{*2}}\rho_{a}=\frac{1}{3H_{0}^{2}M_{Pl}^{*2}}\times
a_{osc}^{3}\times\frac{1}{2}m_{a}^{2}f_{a}^{2}=\sqrt{\frac{m_{a}}{10^{-27}\text{eV}}}\left(\frac{f_{a}}{M_{Pl}^{*}}\right)^{2}$
(46)

with the present Hubble constant $H_{0}(=10^{-33}$ eV). Here in the last
equality we have used axion with $m_{a}=10^{-21}$ eV begins to oscillate at
$z=10^{7}$. Thus $\Omega_{a}$ is within the observed value $\Omega_{a}=O(1)$.
See [35] on the effect of the reaction of gauge fields.

##  4 Discussion

We have studied the two scenarios of axion models, QCD axion with
$f_{a}=10^{12}$ GeV and the string-inspired axion with $f_{a}=10^{16}$ GeV.
$f_{a}=10^{12}$ GeV and $f_{a}=10^{16}$ GeV correspond to canonical seesaw (or
Pati-Salam) and GUT energy scales, respectively in SO(10) GUT theory. The
early formation of SMBH prefers the latter scenario, which may reduce the
ambiguities of axiverse parameters. In either model, axion appears before or
during inflation, and is free from the domain wall problem [13] but induces
the isocurvature perturbation [36]. The latter is constrained by the cosmic
microwave background (CMB) [37]. Axion acquires fluctuations during inflation,

$\delta\varphi=f_{a}\delta\theta=\frac{H_{inf}}{2\pi},$ (47)

which leads to isocurvature perturabation $\delta\rho_{a}$ after it got mass,

$\frac{\delta\rho_{a}}{\rho_{a}}\approx\frac{H_{inf}}{f_{a}}.$ (48)

and the observed perturbation is constrained as

$\alpha\equiv\frac{(\mbox{isocurvature perturbation})^{2}}{(\mbox{adiabatic
perturbation})^{2}+(\mbox{isocurvature perturbation})^{2}}<0.077~{}(95\%CL).$
(49)

There is an indication that $\alpha=0.05$ is the best fit to the observed
power spectrum $P(k)$ [38]. However, there is also an indication that CDM
isocurvature perturbation is not preferable at a statistically significant
level [39],

$\alpha<0.038~{}~{}(95\%~{}CL).$ (50)

Axion also produces non-Gaussianity in general. However, if
$\Omega_{a}h^{2}=0.2$, the non-Gaussianity is negligible. Indeed, it is
described as [40, 41]

$\Omega_{a}h^{2}=10^{-3}\left(\frac{f_{a}}{10^{10}\mbox{GeV}}\right)^{1.2}\left(\frac{H_{inf}/2\pi}{10^{10}\mbox{GeV}}\right)^{2}.$
(51)

Therefore, $f_{a}=10^{16}$ GeV and $H_{inf}=O(10^{9})$GeV give almost full
value of DM and we may neglect non-Gaussian contribution. Concerning with
this, we give a comment on the relation among DM and DE. As we have said,
axion acts as DM at $m_{a}t\gg 1$ and have not explained the present value of
$\Omega_{\Lambda}\approx 0.7$. If axion dominates DM and DE, DE may remain as
BH if

$M_{BH}(a)=M_{BH}(a_{i})\left(\frac{a}{a_{i}}\right)^{3}.$ (52)

Here $M_{BH}(a)$ is the total mass of BHs within the FRW scale factor $a$
[42]. There is a positive observational indication on it [43] though also an
alternative result [44] and a critical comment on the interpretation of (52)
[45]. There is an argument that DE appears as a cosmological constant coming
from de-Sitter invariant gauge theory of gravitation [46].

Finally, we add some comments on the discovery of the SGWB by the NANOGrav and
PTAs. The purpose of the present paper was to find the reason of the early
formation of SMBHs. However, our model also gives the possibility of causing
SGWB. Indeed, the metric in the Einstein equation is generalized from (3) to

$ds^{2}=(1+2\Phi(\bf{x},t))dt^{2}-(1+2\Psi(\bf{x},t))\delta_{ij}dx^{i}dx^{j},$
(53)

and $\Phi$ and $\Psi$ are divided into the static and oscillation parts [47],

$\Phi(\bf{x},t)=\Phi_{0}(\bf{x})+\Phi_{c}(\bf{x})\cos(\omega
t+2\alpha(\bf{x}))+\Phi_{s}(\bf{x})\sin(\omega
t+2\alpha(\bf{x}))~{}~{}\mbox{etc.}$ (54)

with

$\Delta\Phi_{0}=4\pi G\rho_{DM}~{}\mbox{and}~{}\Phi_{0}=-\Psi_{0}$ (55)

etc. The oscillation part satisfies

$6\ddot{\Psi}-2\Delta(\Psi+\Phi)=8\pi G\rho_{DM}(1+3\cos 2m_{a}t).$ (56)

and from the geodesic equation of photon, we obtain

$\omega_{obs}=\omega_{0}+\omega_{0}\left(\Phi(x_{obs})-\Phi(x_{s})\right),$
(57)

where $x_{s}$ and $x_{obs}$ are the positions of the pulsar and observation,
respectively. Thus, we observe the monochromatic oscillation of gravitational
wave having the amplitude $A$,

$A=2\times
10^{-15}\left(\frac{\rho_{DM}}{0.3\mbox{GeV}/cm^{3}}\right)\left(\frac{10^{-23}\mbox{eV}}{m_{a}}\right),$
(58)

and the frequency $f$,

$f=5\times 10^{-9}~{}\mbox{Hz}\left(\frac{m_{a}}{10^{-23}\mbox{eV}}\right).$
(59)

This may be within 2 $\sigma$ of observations [17]. We need further
measurements [48] towards more definitive conclusions.

Thus the ultralight axion required for the early formation of SMBHs is one of
candidates of origin of SGWB among inspiralling binary SMBH models and cosmic
string etc. [49].

We have considered axion DM models. One is the QCD axion whose potential is
given by (17) and another is string-inspired axion given by (36). It has been
shown that the string-insired axion may resove the tension of the early
formation of primordial SMBHs and also may be the origin of SGWB recently
observed.

## Acknowledgments

We would express our sincere thanks to Dr. J. Soda for useful discussions and
comments. This work is supported by JSPS KAKENHI Grant Numbers JP22H01237.

## References

  * [1] D. Richstone et al., Nature 395, A14 (1998).
  * [2] J. Kormendy and L.C. Ho, Ann. Rev. Astron. Astrophys. 51, 511 (2013).
  * [3] E. Banados, Nature 553, 473 (2018).
  * [4] N.J. Adams et al., MNRAS, 518, 4755 (2023).
  * [5] R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).
  * [6] A.R. Zhitnicki, Soviet J. Nucl. Phys. 31, 260 (1980).
  * [7] M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. B 104, 199 (1981).
  * [8] T. Fukuyama, Int. J. Mod. Phys. A28, 1330008 (2013).
  * [9] T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac, and N. Okada, Eur. Phys. J. C42, 191 (2005).
  * [10] K.S. Babu, T. Fukuyama, S. Khan, and S. Saad, JHEP 06, 045 (2019).
  * [11] J.E. Kim, Phys. Rev. Lett. 43, 103 (1979).
  * [12] M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov, Nucl. Phys. B166, 493 (1980).
  * [13] P. Sikivie, Phys. Rev. Lett. 48, 1156 (1982).
  * [14] K.A. Beyer and S. Sarker, SciPost Phys. 15, 003 (2023).
  * [15] D.J.E. Marsh, Phys. Rep. 643, 1-79 (2016).
  * [16] J. Soda, Lecture on Axion Cosmology and Gravitation at Rikkyo University Dec.1-4 (in Japanese).
  * [17] G. Agazie et al. (NANOGrav Collaboration), arXiv:2306.16213 (2023).
  * [18] A. Afzal et al. (NANOGrav Collaboration), arXiv:2306.16219 (2023).
  * [19] J. Antoniadis et al. (EPTA+InPTA Collaboration), A&A 678, A50 (2023) arXiv:2306.16214.
  * [20] D.J. Reardon et al. (PPTA Collaboration), Astrophys. J. Lett. 951, L6 (2023).
  * [21] T. Fukuyama, M. Morikawa, and T. Tatekawa, JCAP 06, 033 (2008).
  * [22] P. Sikivie and Q. Yang, Phys. Rev. Lett. 103, 111301 (2009).
  * [23] M. Gell-Mann, R. Oakes, and B. Renner, Phys.Rev. 175, 2195 (1968).
  * [24] E.P. Gross, Nuovo Cimento 20, 454 (1961).
  * [25] L.P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961).
  * [26] P. Das Gupta and E. Thareja, Class. Quantum Grav. 34, 035006 (2017).
  * [27] T.D. Lee and C.N. Yang, 105, 1119 (1957).
  * [28] E.M. Lifshitz and L.P. Pitaevskii, Section 25 in Statistical Physics Part II (Pergamon Press 1980).
  * [29] A.H. Guth, M.P. Hertzberg, and C. Prescod-Weinstein, Phys. Rev. D 92, 103513 (2015).
  * [30] J. Eby, C. Kouvavaris, N.G. Nielsen, and L.C.R. Wijewardhana, JHEP 02, 028 (2016).
  * [31] M.P. Hertzberg and E.D. Sciappacasse, JCAP 08, 028 (2018).
  * [32] P. Svrcek and E. Witten, JHEP 06, 051 (2006).
  * [33] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. March-Russell, Phys. Rev. D81, 123530 (2010).
  * [34] L. Visinelli and S. Vagnozzi, Phys. Rev. D99, 063517 (2019).
  * [35] N. Kitajima, J. Soda, and Y. Ukawa, Phys. Rev. Lett. 126, 121301 (2021).
  * [36] D. Lyth, Phys.Rev.D 45 3394 (1992) and references therin.
  * [37] G. Hinshaw et al. (WMAP Collaboration), Astrophys. J. Suppl. 208, 19 (2013).
  * [38] E. Komatsu et al. (WMAP Collaboration), Astrophys. J. Suppl. 192, 18 (2011).
  * [39] P.Ade et al. (Planck Collaboration),1502.02114 (2015).
  * [40] M.S. Turner, Phys. Rev. D33, 889 (1986).
  * [41] M. Kawasaki, K. Nakayama, and F. Takahashi, JCAP 11, 019 (2008).
  * [42] K.S. Croker, M. Zevin, D. Farrah, K.A. Nishimura, and G. Tarle, Astrophys. J. Letteres, 921, L22 (2021).
  * [43] D. Farrah et al., Astrophys. J. Letters, 944, L31 (2023).
  * [44] L. Lei et al., arXiv:2305.03408[astro-ph.CO].
  * [45] T. Mistele, Research Notes on the AAS, 7, 101 (2023). arXiv:2304.09817[gr-qc].
  * [46] T. Fukuyama, Ann. Phys. 157, 321 (1984).
  * [47] A. Khmelnitsky and V. Rubakov, JCAP 02, 019 (2014).
  * [48] F. Combes et al. (Square Kilometer Array Collaboration), arXiv:2107.03915[astro-ph.CO].
  * [49] J. Ellis et al., arXiv:2308.08546[astro-ph.CO].