TU-1199

KEK-QUP-2023-0015

Nanohertz gravitational waves from cosmic strings
and dark photon dark matter

Naoya Kitajima(a,b) and Kazunori Nakayama(b,c)

(a)Frontier Research Institute for Interdisciplinary Sciences, Tohoku
University, Sendai 980-8578, Japan

(b)Department of Physics, Tohoku University, Sendai 980-8578, Japan

(c)International Center for Quantum-field Measurement Systems for Studies of
the Universe and Particles (QUP), KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801,
Japan

The recent observations by pulsar timing array (PTA) experiments suggest the
existence of stochastic gravitational wave background in the nano-Hz range. It
can be a hint for the new physics and cosmic string is one of the promising
candidate. In this paper, we study the implication of the PTA result for
cosmic strings and dark photon dark matter produced by the decay of cosmic
string loops. It can simultaneously explain the PTA result and present dark
matter abundance for the dark photon mass $m\sim 10^{-6}$–$10^{-4}\,{\rm eV}$.
Implications for the gravitational wave detection with multi-frequency bands
are also discussed.

Recently pulsar timing array (PTA) collaborations, including the North
American Nanohertz Observatory for Gravitational Waves (NANOGrav) [1],
Eulopean Pulsar Timing Array (EPTA) [2], Parkes Pulsar Timing Array (PPTA) [3]
and Chiniese Pulsar Timing Array (CPTA) [4], reported a detection of
stochastic gravitational waves (GW) around the frequency of $\sim 10^{-8}$ Hz
at the level of GW energy density parameter $\Omega_{\rm GW}\sim 10^{-9}$. It
may be explained by the super massive black hole binaries, while
interpretations in terms of new physics phenomena are also possible [5]. In
this letter we consider the cosmic strings that are source of the dark photon
dark matter, as the origin of stochastic GWs detected by the NANOGrav.111 See
Refs. [6, 7, 8] for the cosmic string explanation of the NANOGrav signal
reported in 2020 [9].  The possibility of cosmic strings have been already
discussed in detail in Ref. [5], where it has been concluded that the stable
cosmic strings are not suitable for explaining the PTA data, since the
spectrum is too soft around the PTA frequency range. We point out that, for
the case of cosmic strings in association with the light dark photon dark
matter, the spectrum fits well with the PTA result.

There are many production mechanisms of dark photon dark matter, such as the
gravitational production during inflation or reheating [10, 11, 12, 13, 14,
15, 16, 17, 18], resonant production through the oscillation of the axion or
dark-Higgs [19, 20, 21, 22, 23], production from the cosmic string network
[24, 25], misalignment production during inflation [26, 27, 28]. Among them,
in this letter we focus on the production from the cosmic strings [24, 25]. We
consider the following Abelian-Higgs model with the dark U(1) gauge symmetry:

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-|D_{\mu}\Phi|^{2}-V(|\Phi|),$
(1)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ with $A_{\mu}$
being the dark photon field and $D_{\mu}\Phi=\partial_{\mu}\Phi-igA_{\mu}\Phi$
with $\Phi$ being the Higgs field in association with the dark U(1) gauge
symmetry and $g$ is the gauge coupling constant. The potential $V$ is chosen
as $V=\frac{\lambda}{4}\left(|\Phi|^{2}-v^{2}\right)^{2}$ with the Higgs self
coupling constant $\lambda$ and the vacuum expectation value $v$. After the
symmetry breaking, the dark photon obtains a mass of $m=\sqrt{2}gv$.

In this model, cosmic string networks are formed in the universe if the
symmetry breaking happens after inflation. Even in the case of gauged U(1)
symmetry, the actual string dynamics is almost the same as global strings for
small enough gauge coupling $g$ [25]. The dynamics of global strings, often in
the context of axion, have been studied in Refs. [29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54].
The distribution of cosmic string networks approaches to the so-called scaling
regime, in which there are $\xi\sim\mathcal{O}(1)$ number of infinite strings
in one Hubble volume. To maintain the scaling, the network continuously should
produce small string loops in each Hubble time through reconnections.
Recently, violation of the scaling law has been reported based on highly-
resolved long-term field theoretic simulations [46, 47, 51]. Numerical results
suggest that $\xi$ is not constant but rather evolves logarithmically with
time. Physical interpretation of such logarithmic time dependence is still
unclear but it can be partially explained by the logarithmic growth of the
string tension with time, i.e. $\mu\propto\log(t/\delta)$ with $\delta$ being
the string width. It implies that the energy loss rate of the (infinite)
string network decreases inversely proportional to $\log(t/\delta)$ due to the
emission of axions/dark photons and thus the string density increases
accordingly [47].222Another group reported that their simulation does not show
the scaling violation [49, 52]. More refined simulation is necessary to
confirm whether the scaling violation occurs or not. String loops produced in
this way will shrink by emitting either dark photons or GWs. Loops with a
length $\ell$ shorter than $m^{-1}$ loses their energy by emitting dark photon
within one Hubble time, while loops with $\ell\gtrsim m^{-1}$ loses their
energy only through the GW emission. The dark photon number density produced
in one Hubble time is estimated as $n\sim\xi\mu/t$ with $\mu$ being the string
tension, by assuming that the typical loop length is comparable to the Hubble
radius $t$. Assuming the radiation-dominated universe when the dark photon is
produced, the final energy density of the dark photon is dominated by those
produced around $t\sim m^{-1}$, after which loops cannot emit vector boson.
The resulting dark photon abundance is estimated as [24, 25]

$\displaystyle\Omega_{\gamma^{\prime}}h^{2}\simeq
0.091\bigg{(}\frac{\xi}{12}\bigg{)}\bigg{(}\frac{m}{10^{-13}\,{\rm
eV}}\bigg{)}^{1/2}\bigg{(}\frac{v}{10^{14}\,{\rm GeV}}\bigg{)}^{2},$ (2)

in terms of the density parameter, where $\xi\simeq
0.15\log(\sqrt{\lambda}v/m)$. It can easily explain the present dark matter
abundance for wide parameter ranges. Thus dark photon produced by cosmic
string loops is one of the dark matter candidates.

Cosmic strings also produce GWs that form the stochastic GW background in the
present universe [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
70, 71, 72]. In the scenario of light dark photon/axion, one should take
account of the effect of dark photon/axion emission by the loops, which
significantly suppress the GW emission efficiency for small loops. The
resulting GW spectrum is very different from both the local and global string
cases. A detailed analysis of GW spectrum in such a case is performed in Ref.
[25]. The differential present GW background energy density, $d\rho_{\rm
GW}/df$ with $f$ being the present GW frequency, produced by the cosmic string
loops is given by

$\displaystyle\frac{d\rho_{\rm GW}}{df}=\int dt\int
d\ell\,G\mu^{2}(t)\,S\left(\frac{\ell
fa_{0}}{a(t)}\right)\,\frac{dn_{\ell}(t)}{d\ln\ell}\left(\frac{a(t)}{a_{0}}\right)^{3}.$
(3)

where the function $S(x)$ is given by

$\displaystyle S(x)=(q-1)2^{q-1}\Gamma_{\rm GW}\frac{\theta(x-2)}{x^{q}}.$ (4)

Here $\Gamma_{\rm GW}\sim 50$ and $q=4/3$ $(5/3)$ if the cusps (kinks) on the
loop dominate the GW emission [55, 56, 60, 64]. The loop distribution function
$dn_{\ell}(t)/d\ln\ell$ is given by

$\displaystyle\frac{dn_{\ell}(t)}{d\ln\ell}\simeq\frac{\tau(\ell)}{\tau(\ell)+t_{i}}\frac{C\xi(t_{i})}{\alpha
t_{i}^{3}}\,\left(\frac{a(t_{i})}{a(t)}\right)^{3}\theta\left(t-\frac{\ell_{i}}{\alpha}\right),$
(5)

by assuming that newly produced loops at the cosmic time $t=t_{i}$ has a fixed
size $\ell_{i}=\alpha t_{i}$. In a numerical calculation below we tale
$\alpha=0.1$. The loop lifetime $\tau(\ell)$ is determined by the GW emission
for $\ell\gtrsim m^{-1}$ and by the vector boson emission for $\ell\lesssim
m^{-1}$:

$\displaystyle\tau(\ell)\sim\begin{cases}\displaystyle\frac{\ell}{\Gamma_{\rm
GW}G\mu}&{\rm for~{}~{}~{}}\ell\gtrsim m^{-1}\\\
\displaystyle\frac{2\pi\log(\sqrt{\lambda}v/M)}{\Gamma_{\rm vec}}\ell&{\rm
for~{}~{}~{}}\ell\lesssim m^{-1}\end{cases}.$ (6)

where $\Gamma_{\rm vec}\sim 65$ [73, 31, 72] and $M\equiv{\rm max}[m,H]$. We
will express the present GW spectrum in terms of the density parameter:

$\displaystyle\Omega_{\rm GW}(f)=\frac{1}{\rho_{\rm crit}}\frac{d\rho_{\rm
GW}(t_{0})}{d\ln f},$ (7)

where $\rho_{\rm crit}$ denotes the critical energy density of the present
universe.

Figure 1: The predicted GW spectrum from cosmic strings for $v=2\times
10^{12}\,{\rm GeV}$ and $m=(10^{-4},10^{-12},10^{-20})$ eV. The upper (lower)
figure corresponds to the case where cusps (kinks) dominate the GW emission
from string loops. Note that the present dark matter abundance is explained by
the dark photon produced by string loops for $m=10^{-4}$ eV. Together shown
are current/future sensitivities of various experiments.

Fig. 1 shows the predicted GW spectrum from cosmic strings for $v=2\times
10^{12}\,{\rm GeV}$ and $m=(10^{-4},10^{-12},10^{-20})$ eV. We also take
$\lambda=1$. The upper (lower) figure corresponds to the case where cusps
(kinks) dominate the GW emission from string loops. Note that the present dark
matter abundance is explained by the dark photon produced by string loops for
$m=10^{-4}$ eV (see Eq. (2)). Together shown are future sensitivities of
LIGO/Virgo [74, 75, 76], Einstein Telescope (ET) [77], LISA [78] and Square
Kilometer Array (SKA) [79]. The current limits from pulsar timing arrays,
European Pulsar Timing Array (EPTA) [80] and NANOGrav 12.5 year data [9] are
also shown. These sensitivities are taken from Refs. [81, 82].

It is seen that the PTA result can be explained for $v\sim 10^{12}\,{\rm GeV}$
for wide dark photon mass range $m\lesssim 10^{-4}\,{\rm eV}$. Heavier dark
photon predicts too much dark matter abundance unless thermal history in the
early universe is modified. The GW amplitude at the PTA frequency range is
almost independent of the dark photon mass up to the small logarithmic
dependence of $\mu$ and $\xi$. On the other hand, the GW amplitude at the
LIGO/Virgo frequency range significantly depends on the dark photon mass.
Interestingly, only for the case of $m\sim 10^{-4}\,{\rm eV}$, the GW is
detectable in future LIGO/Virgo detectors and it is exactly the dark photon
mass that explains the present dark matter abundance. For lower dark photon
mass, GW detection at the LIGO/Virgo may not be possible, but they are
detectable at future space laser interferometers, LISA or DECIGO, and also
ground-based detector (ET). In any case, it is the high frequency tail of the
GW spectrum produced by cusps on the loop (4) that might be detectable at the
LIGO/Virgo. However, one should note that the cusp formation on the string
loops may not be guaranteed so far for the case of (nearly) global strings. On
the other hand, kinks are formed in the process of string reconnection, and
hence their existence on the loops may be more promising. For the case of
kink-dominated GW emission, as shown in the lower panel of Fig. 1, the GW
amplitude is much suppressed for higher frequency. Therefore, by combining the
information of the pulsar timing arrays and future space/ground-based
detectors, we can obtain information on the dark photon mass and the
fundamental nature of cosmic string loops.

Fig. 2 shows the constraint on the model parameter. The gray- and orange-
shaded regions are ruled out from the dark matter abundance and the black hole
superradiance [83] respectively. It should be emphasized that PTA result (blue
hached region) implies $m\sim 10^{-5}$-$10^{-6}$ eV to explain the dark matter
abundance by the dark photon matter and it can be tested by future LIGO/Virgo
[74, 75, 76], as shown by a red solid line. Region above the red dashed line
is excluded by the current LIGO/Virgo constraint [81].

Let us comment on the difference between the cosmic string scenario discussed
in Ref. [5] and ours. For the case of stable local strings, by fixing the
string tension $\mu$ so that it reproduces the PTA signal, the peak frequency
is also fixed: $f_{\rm peak}\sim(\Gamma_{\rm GW}G\mu t_{0})^{-1}$ and
$\Omega_{\rm GW}(f_{\rm peak})\sim G\mu$. The spectral index $n$ of
$\Omega_{\rm GW}$, defined by $n=d\ln\Omega_{\rm GW}(f)/d\ln f$, is much
softer than the observationally favored value. In our case, the cosmic strings
are rather close to global strings, in which case the typical number of
strings in one Hubble volume, $\xi$, is likely to be logarithmically enhanced
[43, 44, 45, 46, 47, 48, 50, 51, 53, 54, 25], although the discussion is not
settled yet [49, 52]. Thus the GW amplitude is enhance by this factor, which
makes it possible that the hard part of the GW spectrum reaches to the PTA
level. In this sense, our scenario is phenomenologically similar to the cosmic
superstrings with small reconnection probability mentioned in Ref. [5], which
shows good agreement with the observation. For the superstring case, the
current result from the LIGO/Virgo experiment severely constrains the
parameter region unless some modified cosmological history is assumed, while
in our case the GW spectrum is naturally suppressed at the LIGO/Virgo
frequency. This is because small loops decays to dark photons and the
efficiency for the GW emission is suppressed at high frequency.

It should be noticed that theoretical prediction of the GW spectrum suffers
from various uncertainties due to incomplete understandings of long term
cosmic string dynamics. One should consider that our estimation may have at
least $\mathcal{O}(1)$ numerical uncertainty. Although we have used $v=2\times
10^{12}\,{\rm GeV}$ as a reference value, $v$ may have a bit large or smaller
values to explain the PTA result taking account of theoretical uncertainty.
Fig. 3 shows the spectral index $n$ in our scenario for $v=2\times
10^{12}\,{\rm GeV}$ and $v=10^{12}\,{\rm GeV}$. Around the PTA frequency range
it is around $0.5$–$0.8$. Accidentally it is close to the case of supermassive
blackhole binaries [1], which typically predicts $n=2/3$. Therefore our model
fits the data at the same level as the supermassive blackhole binaries.

To summarize, the PTA result for the detection of GWs may be explained by the
cosmic strings interacting with light dark photon. The dark photon mass is
predicted to be $m\sim 10^{-6}$–$10^{-4}$ eV if the dark photon produced by
strings accounts for the present dark matter density of the universe. Note
that GWs with higher frequency band (e.g. $\mathcal{O}(10\,{\rm Hz})$) emitted
by the same string network is within the sensitivity of future LIGO/Virgo
observation, while the current constraint is evaded. Dark photon with this
mass range may be proved by several experiments [84] such as haloscope, if the
dark photon has a small but finite kinetic mixing with the Standard Model
photon. Such a dark photon dark matter scenario is likely to predict
observable level of GW amplitude at the LIGO/Virgo frequency range. On the
other hand, we also need more robust theoretical understandings for the long
term cosmological dynamics of strings in order to predict the GW spectrum
precisely.

Figure 2:  Constraint on the model parameter. NANOGrav result is shown by the
blue hached region (factor 2 uncertainty is assumed by hand). Future designed
sensitivity of LIGO/Virgo experiment is shown by the red line. Thick and thin
lines represent respectively the cusp and kink contributions. Dashed-red line
shows the current sensitivity.  Figure 3: The spectral index $n$ of
$\Omega_{\rm GW}(f)$ for $v=2\times 10^{12}\,{\rm GeV}$ and $v=10^{12}\,{\rm
GeV}$.

## Acknowledgment

This work was supported by JSPS KAKENHI Grant Nos. 19H01894 (N.K.), 20H01894
(N.K.), 20H05851 (N.K.), 21H01078 (N.K.), 21KK0050 (N.K.). This work was
supported by World Premier International Research Center Initiative (WPI),
MEXT, Japan.

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