# Gravitational waves induced by the local-type non-Gaussian curvature
perturbations

Chen Yuan<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
China  School of Physical Sciences, University of Chinese Academy of Sciences,
No. 19A Yuquan Road, Beijing 100049, China  Qing-Guo Huang<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China  School of Physical
Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road,
Beijing 100049, China  School of Fundamental Physics and Mathematical Sciences
Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China  Center
for Gravitation and Cosmology, College of Physical Science and Technology,
Yangzhou University, Yangzhou 225009, China  Synergetic Innovation Center for
Quantum Effects and Applications, Hunan Normal University, Changsha 410081,
China

###### Abstract

The current observational constraints still leave a substantial mass window
$\sim[10^{-16},10^{-14}]\cup[10^{-13},10^{-12}]M_{\odot}$ for primordial black
holes (PBHs) representing all of dark matter (DM) in our Universe. The
gravitational waves (GWs) induced by the curvature perturbations are
inevitably generated during the formation of these PBHs, and fall in the
frequency band of LISA. Such scalar induced gravitational waves (SIGWs) are
supposed to be definitely detected by LISA even when the second-order local-
type non-Gaussianity characterized by the parameter $F_{\rm NL}$ is taken into
account. In this letter, we give a comprehensive analysis of the GWs induced
by the local-type non-Gaussian curvature perturbations up to the third-order
denoted by the non-linear parameter $G_{\rm NL}$, and find that a log-
dependent slope of SIGWs in the infrared region is generically predicted and
the amplitude of SIGWs can be further suppressed by several orders of
magnitude. Therefore, the null-detection of SIGWs by LISA cannot rule out the
possibility of PBHs making up all of DM.

Introduction.  The nature of dark matter (DM) remains one of the most
important problems in fundamental physics. The detection of gravitational
waves (GWs) generated by the coalescence of two $\sim 30M_{\odot}$ black holes
Abbott  _et al._ (2016) has not only marked the beginning of GW astronomy but
also renewed the postulation that primordial black holes (PBHs) may consist of
all or some of DM in our Universe. PBHs are supposed to form from the collapse
of over-densed regions generated by the enhanced scalar curvature
perturbations in the radiation dominated (RD) era Hawking (1971); Carr and
Hawking (1974). PBHs lighter than $\sim 10^{-18}M_{\odot}$ or heavier than
$\sim 10^{3}M_{\odot}$ have been drastically constrained through Hawking
radiation Carr  _et al._ (2010) and the accretion of primordial gas onto PBHs
on CMB scales Chen  _et al._ (2016); Ali-Haimoud and Kamionkowski (2017);
Aloni  _et al._ (2017); Horowitz (2016) respectively. Although various
investigations have put constraints on the fraction of PBHs in DM in the mass
range $[10^{-12},10^{3}]M_{\odot}$ to no more than a few parts in hundred Bird
_et al._ (2016); García-Bellido (2017); Sasaki  _et al._ (2018); Barack  _et
al._ (2019); Chen and Huang (2018); Chen  _et al._ (2019a); Chen and Huang
(2019); Barnacka  _et al._ (2012); Graham  _et al._ (2015); Niikura  _et al._
(2019a); Griest  _et al._ (2013); Tisserand  _et al._ (2007); Brandt (2016);
Gaggero  _et al._ (2017); Niikura  _et al._ (2019b); Wang  _et al._ (2018);
Abbott  _et al._ (2018); Magee  _et al._ (2018); Chen and Huang (2019);
Montero-Camacho  _et al._ (2019); Laha (2019); Chen  _et al._ (2019b), the
mass window $[10^{-16},10^{-14}]\cup[10^{-13},10^{-12}]M_{\odot}$ still opens
for the possibility that all of DM are in the form of PBHs Defillon  _et al._
(2014); Katz  _et al._ (2018). See a recent review in Carr  _et al._ (2020).

It is known that the scalar perturbations would source the gravitational waves
at second-order Tomita (1967); Matarrese  _et al._ (1993, 1994, 1998); Noh and
Hwang (2004); Carbone and Matarrese (2005); Nakamura (2007), namely the scalar
induced gravitational waves (SIGWs). SIGW inevitably generated during the
formation of PBHs provides a new powerful tool to search for the PBHs (in
particular, those in the mass window
$[10^{-16},10^{-14}]\cup[10^{-13},10^{-12}]M_{\odot}$ for constituting all of
DM) Saito and Yokoyama (2009); Yuan  _et al._ (2019a), and the log-dependent
slope of SIGWs in the infrared region near the peak is supposed to be a
distinguishing feature from other astrophysical sources Yuan  _et al._
(2019b). See some recent relevant works about SIGWs in Ananda  _et al._
(2007); Baumann  _et al._ (2007); Assadullahi and Wands (2010); Bugaev and
Klimai (2010); Saito and Yokoyama (2010); Bugaev and Klimai (2011); Nakama and
Suyama (2016); Garcia-Bellido  _et al._ (2017); Espinosa  _et al._ (2018);
Kohri and Terada (2018); Bartolo  _et al._ (2019a, b); Inomata and Nakama
(2019); Clesse  _et al._ (2018); Cai  _et al._ (2019a); Inomata  _et al._
(2019a, b); Inomata and Terada (2019); Domènech (2019); Cai  _et al._ (2019b,
c); Tomikawa and Kobayashi (2019); De Luca  _et al._ (2019); Fu  _et al._
(2019); Kawasaki  _et al._ (2020); Lin  _et al._ (2020); Ota (2020); Inomata
_et al._ (2020); Giovannini (2020); Lu  _et al._ (2020).

Usually, the scalar curvature perturbation is naively assumed to be Gaussian.
However, PBHs are significantly generated by the enhanced scalar curvature
perturbation, and the non-negligible non-Gaussianity is also expected to be
generated. Since the formation of PBHs takes place at the tail of the PDF, the
non-Gaussianities would drastically alter the PBH abundance Byrnes  _et al._
(2012), and can suppress the SIGWs by several orders of magnitude Nakama  _et
al._ (2017). Recently, the SIGWs induced by scalar curvature perturbation with
a second-order (i.e. $F_{\mathrm{NL}}$-order) local-type non-Gaussianity were
studied in Cai  _et al._ (2019d); Unal (2019). However, $F_{\mathrm{NL}}$ is
just the coefficient of even order curvature perturbation, and it is necessary
to extend to the odd order curvature perturbation (i.e.
$G_{\mathrm{NL}}$-order) for achieving a general analysis of GWs induced by
the local-type non-Gaussian scalar curvature perturbations.

In this letter, we provide a comprehensive analysis of GWs induced by the
local-type non-Gaussian curvature perturbations up to the third-order (i.e.
$G_{\mathrm{NL}}$-order). However, different from the main results in Cai  _et
al._ (2019d), we find a universal $k^{3}\ln^{2}(4k_{*}^{2}/3k^{2})$ behavior
of non-Gaussian contribution to SIGWs in the infrared region instead of
$k^{3}$ behavior claimed in Cai  _et al._ (2019d), and we do not observe
multiple-peak structures around a cutoff like those showed in Fig. 1 of Cai
_et al._ (2019d). Furthermore, although the SIGWs are claimed to be definitely
detected by LISA if PBHs with mass of $10^{20}$ to $10^{22}$g are identified
as all of DM in our Universe in Cai  _et al._ (2019d), we find that the SIGWs
can be undetectable by LISA once the $G_{\mathrm{NL}}$-order non-Gaussianity
is taken into account.

SIGWs induced by general non-Gaussian scalar curvature perturbations.  Let’s
start with the perturbed metric in Newton gauge

$ds^{2}=a^{2}\left\\{-(1+2\phi)d\eta^{2}+[(1-2\phi)\delta_{ij}+{1\over
2}h_{ij}]dx^{i}dx^{j}\right\\},$ (1)

where $\phi$ is the linear scalar perturbation and $h_{ij}$ is the second-
order tensor perturbation. The equation of motion for $h_{ij}$ is obtained by
the second-order Einstein equation

$h_{ij}^{\prime\prime}+2\mathcal{H}h_{ij}^{\prime}-\nabla^{2}h_{ij}=-4\mathcal{T}_{ij}^{\ell
m}S_{\ell m},$ (2)

where $\mathcal{H}\equiv a^{\prime}/a$ is the conformal Hubble parameter and
the prime denotes derivative with respect to the conformal time $\eta$. The
transverse and traceless projection operator in Fourier space is defined by

$\mathcal{T}_{ij}^{\ell m}(\bm{k})\equiv
e_{ij}^{(+)}(\bm{k})e^{(+)lm}(\bm{k})+e_{ij}^{(\times)}(\bm{k})e^{(\times)lm}(\bm{k}),$
(3)

where the polarization tensors are
$(e_{i}e_{j}-\bar{e}_{i}\bar{e}_{j})/\sqrt{2}$ and
$(e_{i}\bar{e}_{j}+\bar{e}_{i}e_{j})/\sqrt{2}$ for $+$ and $\times$ mode,
respectively. From now on, we choose $e=(1,0,0)$ and $\bar{e}=(0,1,0)$. In the
RD era, the source term reads

$\displaystyle
S_{ij}\\!\\!=\\!3\phi\partial_{i}\partial_{j}\phi\\!-\\!{1\over\mathcal{H}}(\partial_{i}\phi^{\prime}\partial_{j}\phi+\partial_{i}\phi\partial_{j}\phi^{\prime})\\!-\\!{1\over\mathcal{H}^{2}}\partial_{i}\phi^{\prime}\partial_{j}\phi^{\prime}\\!.\quad$
(4)

The observational quantity, $\Omega_{\mathrm{GW}}(k)$, is defined as the
energy density of GWs per logarithm wavelength normalized by the critical
energy, $\rho_{c}(\eta)$, namely

$\Omega_{\mathrm{GW}}(k,\eta)\equiv{\rho_{\mathrm{GW}}(k,\eta)\over\rho_{c}(\eta)}={k^{3}\over
48\pi^{2}}\left(k\over\mathcal{H}\right)^{2}\overline{\left\langle|h_{\bm{k}}(\eta)|^{2}\right\rangle}.$
(5)

The GWs will be induced by curvature perturbations throughout the RD era, and
the density parameter till the matter-radiation equality is
$\Omega_{\mathrm{GW}}(k)=\Omega_{\mathrm{GW}}(k,\eta\rightarrow\infty)$. After
solving Eq. (2) by Green’s function in Fourier space,
$\Omega_{\mathrm{GW}}(k)$ can be expressed by, Kohri and Terada (2018),

$\Omega_{\mathrm{GW}}(k)=\int_{0}^{\infty}\mathrm{d}u\int_{|1-u|}^{1+u}\mathrm{d}v~{}P_{\zeta}(uk)P_{\zeta}(vk)I(u,v),$
(6)

where the dimensionless curvature power spectrum $P_{\zeta}(k)$ is defined as

$\displaystyle\left\langle\zeta(\bm{k})\zeta(\bm{k^{\prime}})\right\rangle$
$\displaystyle\equiv{2\pi^{2}\over
k^{3}}P_{\zeta}(k)\delta(\bm{k}+\bm{k^{\prime}}).$ (7)

$\Omega_{\mathrm{GW}}$ in Eq. (6) is valid during the RD era and the nowadays
density parameter of SIGWs is given by
$\Omega_{\mathrm{GW},0}(k)=\Omega_{r}\times\Omega_{\mathrm{GW}}(k)$ where
$\Omega_{r}$ is the density parameter for radiation. Here we use the the
comoving curvature perturbation $\zeta$ which is related to $\phi$ by
$\phi=(2/3)\zeta$, and $I(u,v)$ takes the form, Kohri and Terada (2018),

$\displaystyle I(u,v)$ $\displaystyle={1\over
12}\left({4v^{2}-(1+v^{2}-u^{2})^{2}\over
4uv}\right)^{2}\left(3(u^{2}+v^{2}-3)\over 4u^{3}v^{3}\right)^{2}$ (8)
$\displaystyle\times\Bigg{(}\Big{(}-4uv+(u^{2}+v^{2}-3)\ln\Big{|}{3-(u+v)^{2}\over
3-(u-v)^{2}}\Big{|}\Big{)}^{2}$
$\displaystyle+\pi^{2}\left(u^{2}+v^{2}-3\right)^{2}\Theta(u+v-\sqrt{3})\Bigg{)}.$

For the local-type non-Gaussian curvature perturbation, in real space, the
curvature curvature perturbation $\zeta$ can be expanded in terms of the
Gaussian part, $\zeta_{g}$, as follows

$\zeta(\zeta_{g})=\zeta_{g}+F_{\mathrm{NL}}\left(\zeta_{g}^{2}-\left\langle\zeta_{g}^{2}\right\rangle\right)+G_{\mathrm{NL}}\zeta_{g}^{3},$
(9)

where $\left\langle\zeta_{g}^{2}\right\rangle=\int P_{g}(k)\mathrm{d}\ln k$ is
the variance of the Gaussian part. For convenience, we define $P_{g}(k)\equiv
A_{g}\tilde{P}_{g}(k)$ with $A_{g}$ standing for the dimensionless amplitude
of the Gaussian spectrum and $\tilde{P}_{g}(k)$ being normalized by
$\int\tilde{P}_{g}(k)\mathrm{d}\ln k=1$. Therefore, we have

$\displaystyle{P}_{\zeta}(k)=\alpha_{1}{\tilde{P}}_{1}(k)+\alpha_{2}{\tilde{P}}_{2}(k)+\alpha_{3}{\tilde{P}}_{3}(k),$
(10)

where $\alpha_{1}\equiv A_{g}\left(1+3G_{\mathrm{NL}}A_{g}\right)^{2}$,
$\alpha_{2}\equiv F_{\mathrm{NL}}^{2}A_{g}^{2}$, $\alpha_{3}\equiv
3G_{\mathrm{NL}}^{2}A_{g}^{3}$, and ${\tilde{P}}_{1}\equiv{\tilde{P}}_{g}$ and
the non-Gaussian power spectra $\tilde{P}_{2}(k)$ and $\tilde{P}_{3}(k)$ read

$\displaystyle{\tilde{P}}_{2}(k)$
$\displaystyle\equiv\int\mathrm{d}^{3}p{k^{3}\tilde{P}_{g}(p)\tilde{P}_{g}(|\bm{k}-\bm{p}|)\over
2\pi p^{3}|\bm{k}-\bm{p}|^{3}}$ (11)
$\displaystyle=\int_{0}^{+\infty}\\!\\!\\!\mathrm{d}v\\!\int_{|1-v|}^{1+v}\mathrm{d}u~{}\frac{\tilde{P}_{g}(uk)\tilde{P}_{g}(vk)}{u^{2}v^{2}},$
$\displaystyle{\tilde{P}}_{3}(k)$
$\displaystyle\equiv\int\mathrm{d}^{3}p\mathrm{d}^{3}q{k^{3}\tilde{P}_{g}(p)\tilde{P}_{g}(q)\tilde{P}_{g}(|\bm{k}-\bm{p}-\bm{q}|)\over
8\pi^{2}p^{3}q^{3}|\bm{k}-\bm{p}-\bm{q}|^{3}}$ (12)
$\displaystyle=\int_{0}^{+\infty}\\!\\!\\!\mathrm{d}v\\!\int_{|1-v|}^{1+v}\mathrm{d}u~{}\frac{\tilde{P}_{g}(uk)\tilde{P}_{2}(vk)}{2u^{2}v^{2}}.$

Here we change the variables to $|\bm{k}-\bm{p}|/k=u$ and $p/k=v$. And then
the density parameter $\Omega_{\mathrm{GW}}(k)$ can be written by

$\displaystyle\Omega_{\mathrm{GW}}(k)=\sum_{i,j=1}^{3}\alpha_{i}\alpha_{j}{\tilde{\Omega}}_{ij}(k),$
(13)

where

$\displaystyle{\tilde{\Omega}}_{ij}(k)=\int_{0}^{\infty}\mathrm{d}u\int_{|1-u|}^{1+u}\\!\\!\mathrm{d}v{\tilde{P}}_{i}(uk){\tilde{P}}_{j}(vk)I(u,v).$
(14)

Roughly speaking, the amplitudes of shape functions ${\tilde{\Omega}}_{ij}$
are ${\cal O}(1)$ because ${\tilde{P}}_{g}$ is normalized to be unity, and
thus the amplitude of $\Omega_{\mathrm{GW}}(k)$ can be estimated by the
dominated coefficient among $\alpha_{i}\alpha_{j}$.

It is difficult to obtain an analytic result of ${\tilde{\Omega}}_{ij}$ for
the general curvature power spectrum. Here we consider a log-normal Gaussian
part of power spectrum, namely

${\tilde{P}}_{1}(k)={A_{g}\over\sqrt{2\pi\sigma_{*}^{2}}}\exp\left(-\frac{\ln(k/k_{*})^{2}}{2\sigma_{*}^{2}}\right),$
(15)

to illustrate the main features of SIGWs. In the limit of
$\sigma_{*}\rightarrow 0$, $\tilde{P}_{1}(k)$ approaches to a $\delta$ power
spectrum, namely $k_{*}\delta(k-k_{*})$. In such limit, the higher order power
spectra become

$\displaystyle\tilde{P}_{2}(k)$
$\displaystyle=\tilde{k}^{2}\Theta(2-\tilde{k})$ (16)
$\displaystyle\tilde{P}_{3}(k)$
$\displaystyle=M(\tilde{k})\Theta(3-\tilde{k}),$ (17)

where $\tilde{k}\equiv k/k_{*}$ and
$M(\tilde{k})\equiv\min(2\tilde{k}^{3},(3-\tilde{k})\tilde{k}^{2})$, and then
${\tilde{\Omega}}_{ij}$ becomes

$\displaystyle\tilde{\Omega}_{11}$
$\displaystyle={\tilde{k}^{-2}}I\left({1\over\tilde{k}},{1\over\tilde{k}}\right)\Theta(2-\tilde{k}),$
$\displaystyle\tilde{\Omega}_{12}$
$\displaystyle=\tilde{k}\int_{\big{|}1-{1\over\tilde{k}}\big{|}}^{\min({2\over\tilde{k}},1+{1\over\tilde{k}})}\mathrm{d}v~{}v^{2}I\left({1\over\tilde{k}},v\right)\Theta(3-\tilde{k}),$
$\displaystyle\tilde{\Omega}_{22}$
$\displaystyle=\tilde{k}^{4}\int_{0}^{{2\over\tilde{k}}}\mathrm{d}v\int_{|1-v|}^{1+v}\mathrm{d}u~{}u^{2}v^{2}I(u,v)\Theta(2-u\tilde{k}),$
$\displaystyle\tilde{\Omega}_{13}$
$\displaystyle=\tilde{k}^{-1}\int_{\big{|}1-{1\over\tilde{k}}\big{|}}^{\min({3\over\tilde{k}},1+{1\over\tilde{k}})}\mathrm{d}v~{}M(v\tilde{k})I\left({1\over\tilde{k}},v\right)\Theta(4-\tilde{k}),$
$\displaystyle\tilde{\Omega}_{23}$
$\displaystyle=\tilde{k}^{2}\int_{0}^{3\over\tilde{k}}\mathrm{d}v\int_{|1-v|}^{1+v}\mathrm{d}u~{}u^{2}I(u,v)M(v\tilde{k})\Theta(2-u\tilde{k}),$
$\displaystyle\tilde{\Omega}_{33}$
$\displaystyle=\int_{0}^{{3\over\tilde{k}}}\mathrm{d}v\int_{|1-v|}^{1+v}\mathrm{d}u~{}M(u\tilde{k})M(v\tilde{k})I(u,v)\Theta(3-u\tilde{k}).$

For a more general investigation, we plot the shape functions
${\tilde{\Omega}}_{ij}(k)$ in Fig. 1 for the log-normal power spectra with
$\sigma_{*}\rightarrow 0$ (solid lines) and $\sigma_{*}=0.5$ (dashed lines),
respectively. Here ${\tilde{\Omega}}_{11}(k)$ is contributed by the Gaussian
power spectrum and the others ${\tilde{\Omega}}_{ij}(k)$ are related to the
non-Gaussian parts. From Fig. 1, the amplitudes of ${\tilde{\Omega}}_{ij}(k)$
is ${\cal O}(1)$ which is consistent with our previous estimation. In the
infrared region $(k<k_{*})$, the Gaussian contribution goes like
$\tilde{\Omega}_{11}(f)\propto f^{2}\ln^{2}(4f_{*}^{2}/3f^{2})$ for a very
narrow power spectrum and $\tilde{\Omega}_{11}(f)\propto
f^{3}\ln^{2}(4f_{*}^{2}/3f^{2})$ for a wide power spectrum, and the others
${\tilde{\Omega}}_{ij}(k)$ contributed by non-Gaussian parts goes like
$\tilde{\Omega}_{ij}(f)\propto f^{3}\ln^{2}(4f_{*}^{2}/3f^{2})$ for both the
narrow and wide power spectra. It indicates that the non-Gaussian
contributions to the shape functions have a universal behavior with slope
$n_{\rm GW}\equiv\frac{{\rm d}\ln\tilde{\Omega}_{ij}}{{\rm d}\ln
f}=3-2/\ln(2f_{*}/\sqrt{3}f)$ which are significantly different from $3$
claimed in Cai  _et al._ (2019d), and such a difference can be probably
distinguished by LISA at high confidence level Yuan  _et al._ (2019b). Here
$k=2\pi f$ and $k_{*}=2\pi f_{*}$. Even though $n_{\rm GW}\rightarrow 3$ in
the far infrared limit ($k\ll k_{*}$) except ${\tilde{\Omega}}_{11}$ for a
very narrow power spectrum, the SIGWs in the far infrared limit should be
extremely suppressed and have much less opportunities to be detected. In this
sense, such a log-dependent slope given in Yuan  _et al._ (2019b) is also
reliable even for the non-Gaussian case, and can be taken as a generic feature
for SIGWs accompanying the formation of PBHs.

Figure 1:  The shapes of $\tilde{\Omega}_{ij}$ for the log-normal power
spectrum. The solid lines correspond to $\sigma_{*}\rightarrow 0$ and dashed
lines correspond to $\sigma_{*}=0.5$. Two reference lines of
$f^{2}\ln^{2}(4f_{*}^{2}/3f^{2})$ and $f^{2}\ln^{2}(4f_{*}^{2}/3f^{2})$ are
also illustrated, corresponding to slopes of $2-2/\ln(2f_{*}/\sqrt{3}f)$ and
$3-2/\ln(2f_{*}/\sqrt{3}f)$ respectively.

Here we only consider PBHs formed during the radiation dominant era, and the
PBH mass is estimated by the horizon mass when the corresponding curvature
perturbation exceeding the threshold (for example, $\zeta>\zeta_{c}\simeq 1$
Musco  _et al._ (2009, 2005); Musco and Miller (2013); Harada  _et al._
(2013)) re-entered the horizon. The PBH mass, corresponding to the peak
perturbation mode $k_{*}$, is given by, Hawking (1971); Carr and Hawking
(1974),

${m_{\mathrm{pbh}}^{*}}\approx 2.3\times
10^{18}M_{\odot}\left(\frac{3.91}{g_{*}^{\mathrm{form}}}\right)^{1/6}\left(\frac{H_{0}}{f_{*}}\right)^{2},$
(18)

where $g_{*}^{\mathrm{form}}$ is the degrees of freedom at formation and
$H_{0}$ is the Hubble constant by today.

Keeping $A_{g}=10^{-3}$ fixed, for $m_{\rm pbh}^{*}=10^{-12}M_{\odot}$, the
density parameter $\Omega_{\mathrm{GW},0}(f)$ with non-Gaussianities is
illustrated in Fig. 2 for some typical values of $F_{\mathrm{NL}}$ and
$G_{\mathrm{NL}}$. Since $F_{\mathrm{NL}}$ contributes to
$\Omega_{\mathrm{GW},0}(f)$ in form of
$\alpha_{2}=F_{\mathrm{NL}}^{2}A_{g}^{2}$, $\Omega_{\mathrm{GW},0}(f)$ is only
related to the absolute value of $F_{\mathrm{NL}}$. In contrast, the
contribution to $\Omega_{\mathrm{GW},0}(f)$ from $G_{\mathrm{NL}}$ encodes in
terms of $\alpha_{1}=A_{g}\left(1+3G_{\mathrm{NL}}A_{g}\right)^{2}$ and
$\alpha_{3}=3G_{\mathrm{NL}}^{2}A_{g}^{3}$, and then a negative value of
$G_{\mathrm{NL}}$ can suppress $\Omega_{\mathrm{GW},0}(f)$ compared to a
positive $G_{\mathrm{NL}}$. From Fig. 2, despite the main peak locating at
$f=2f_{*}/\sqrt{3}$, we do not observe a multiple-peak structure around the
cut-off frequency like that shown in Cai  _et al._ (2019d) for both narrow and
wide curvature power spectra.

Figure 2:  The density parameter $\Omega_{\mathrm{GW},0}(f)$ induced by non-
Gaussian curvature perturbations. Here we fix the amplitude of Gaussian
spectrum to be $A_{g}=10^{-3}$ for $m_{\rm{pbh}}^{*}=10^{-12}M_{\odot}$. The
solid lines correspond to $\sigma_{*}\rightarrow 0$ and dashed lines
correspond to $\sigma_{*}=0.5$.

Testing the postulation of PBH DM by LISA.  Similar to Cai  _et al._ (2019d),
we also consider a narrow curvature power spectrum, and then the PBHs are
roughly monochromatic. According to Byrnes  _et al._ (2012), the formation
probability $\beta$ of PBHs can be evaluated by

$\displaystyle\beta=\int_{\zeta(\zeta_{g})>\zeta_{c}}{1\over\sqrt{2\pi
A_{g}}}\exp\left(-{\zeta_{g}^{2}\over 2A_{g}}\right)\mathrm{d}\zeta_{g}.$ (19)

Here, the integrating regions include all of regions satisfying
$\zeta(\zeta_{g})>\zeta_{c}$. And the formation probability $\beta$ is related
to the abundance of PBHs in DM by, Nakama  _et al._ (2017),

$\displaystyle f_{\mathrm{pbh}}\simeq 2.5\times
10^{8}\beta\left(\frac{g_{*}^{\mathrm{form}}}{10.75}\right)^{-\frac{1}{4}}\left(\frac{m_{\mathrm{pbh}}}{M_{\odot}}\right)^{-\frac{1}{2}}.$
(20)

To quantitatively estimate the detectability of the SIGW signals by LISA, we
evaluate the expected signal-to-noise ratio (SNR) ($\rho$). The SNR of
detecting a GW background is given by, Thrane and Romano (2013); Allen and
Romano (1999),

$\displaystyle\rho^{2}\\!=\\!T\\!\\!\int\\!\\!\mathrm{d}f\frac{R(f)^{2}S_{h}(f)^{2}}{[1+R(f)^{2}]S_{h}(f)^{2}+P_{n}(f)^{2}+2S_{h}(f)P_{n}(f)},$

where $S_{h}(f)=3H_{0}^{2}\Omega_{\mathrm{GW},0}/(2\pi^{2}f^{3})$. $R(f)$ and
$P_{n}(f)$ are the signal transfer function and the total noise spectrum
respectively Robson  _et al._ (2019). In order to test the postulation of PBH
DM, we assume that PBHs constitute all of DM in our Universe, i.e. $f_{\rm
pbh}=1$, and plot the expected SNR for a different mass of PBHs in Fig. 3
where a dotted line corresponds to $\rho=5$ beyond which the GW signals are
supposed to be distinguished from the noise. Here we focus on PBHs with mass
heavier than $10^{-18}M_{\odot}$, otherwise, they would have been evaporated
completely by Hawking radiations. For the Gaussian curvature perturbations,
LISA can detect SIGWs accompanying the formation of PBHs in the mass range
$\sim[10^{-18},1.4\times 10^{-8}]M_{\odot}$. Taking into account the non-
Gaussianity at the $F_{\mathrm{NL}}$-order, the amplitude of
$\Omega_{\mathrm{GW},0}$ will be mostly suppressed in the limit of
$F_{\mathrm{NL}}=+\infty$. Although $\Omega_{\mathrm{GW},0}$ is suppressed by
several orders of magnitude in such a limit, it is still optimistic that LISA
can detect the SIGWs for PBHs in the mass range $\sim[4.1\times
10^{-15},6.0\times 10^{-10}]M_{\odot}$. However, from Fig. 3, we see that the
SNR is always less than 5 for PBHs with arbitrary mass if, for example,
$F_{\mathrm{NL}}=0$ and $G_{\mathrm{NL}}=-2000$. Therefore, a null detection
of SIGWs can not rule out the possibility of PBH constituting all of DM in the
remaining mass window
$\sim[10^{-16},10^{-14}]\cup[10^{-13},10^{-12}]M_{\odot}$, but may leave a
possibility for large primordial non-Gaussianities at the third order.

Figure 3:  The expected SNR of SIGWs generated by monochromatic PBHs as a
function of $m_{\rm{pbh}}$. We keep $f_{\mathrm{pbh}}=1$ fixed and assume a
four year observation time for LISA. A reference line, $\rho=5$, is drawn for
comparison.

Conclusion.  In this letter, we provide a comprehensive analysis of the GWs
induced by the local-type non-Gaussian curvature perturbations up to the
third-order (i.e., the $G_{\mathrm{NL}}$ order). We find that in general the
SIGWs can be written in the form of Eq. (13) where the amplitude of
$\Omega_{\mathrm{GW},0}(f)$ is mainly determined by the dominated coefficient
among $\alpha_{i}\alpha_{j}$, and the shape functions from the non-Gaussian
parts have a universal behavior with slope $n_{\rm
GW}=3-2/\ln(2f_{*}/\sqrt{3}f)$ in the infrared region. Even though a log-
dependent slope of SIGW is proposed for a Gaussian curvature perturbation in
Yuan  _et al._ (2019b), we find such a log-dependent slope is quite generic
even for the non-Gaussian curvature perturbations and can be taken as a
generic feature for the SIGWs accompanying formation of PBHs.

In addition, we also explore the detectability of the GWs induced by the
local-type non-Gaussian curvature perturbation for PBHs serving as all of DM
by LISA in detail. Since the third-order non-Gaussianity can further suppress
the amplitude of $\Omega_{\mathrm{GW},0}(f)$, the SIGWs may not be detected by
LISA even for PBHs in the mass range
$\sim[10^{-16},10^{-14}]\cup[10^{-13},10^{-12}]M_{\odot}$ making up all of the
DM in our Universe. In this sense, a more sensitive GW detector than LISA is
needed for testing the postulation of PBH DM.

Acknowledgments. We would like to thank Zu-Cheng Chen and Shi Pi for useful
conversations. Cosmological perturbations are derived using the xPand Pitrou
_et al._ (2013) package. We also acknowledge the use of HPC Cluster of ITP-
CAS. This work is supported by grants from NSFC (grant No. 11690021, 11975019,
11847612, 11991053), the Strategic Priority Research Program of Chinese
Academy of Sciences (Grant No. XDB23000000, XDA15020701), and Key Research
Program of Frontier Sciences, CAS, Grant NO. ZDBS-LY-7009.

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