# Enhancing gravitational wave anisotropies with peaked scalar sources

Ema Dimastrogiovanni  Matteo Fasiello  Ameek Malhotra  Gianmassimo Tasinato

###### Abstract

Gravitational wave (GW) backgrounds of cosmological origin are expected to be
nearly isotropic, with small anisotropies resembling those of the cosmic
microwave background. We analyse the case of a scalar-induced GW background
and clarify in the process the relation between two different approaches to
calculating GW anisotropies. We focus on GW scenarios sourced by a
significantly peaked scalar spectrum, which are frequently considered in the
context of primordial black holes production. We show that the resulting GW
anisotropies are characterised by a distinct frequency dependence. We explore
the observational consequences concentrating on a GW background enhanced in
the frequency band of space-based GW detectors. We study the detectability of
the signal through both cross-correlations among different space-based GW
detectors, and among GW and CMB experiments.

##  1 Introduction

The detection of a stochastic gravitational wave background (SGWB) is one of
the key targets of gravitational wave astronomy. Given the increasing
sensitivity of ground based detectors [1, 2], and the tantalising hints of a
signal at NANOGrav [3], the detection of a background of astrophysical origin
is eagerly anticipated. In addition to the astrophysical background [4], one
should also expect the existence of a cosmological GW background (CGWB). Since
the universe is, to a good approximation, transparent to gravitational waves
below the Planck scale, the CGWB provides an excellent handle on the physics
of the primordial universe (see for example the reviews in [5, 6]).
The CGWB may arise from a variety of processes in the early universe, from
inflation to later sources including (p)re-heating dynamics, phase
transitions, cosmic strings, and primordial black holes (PBH) (see [5] for a
review). Remarkably, the GW signal associated to several such sources may be
detected in the near future. Indeed, the corresponding frequencies are the one
accessed by ground-based interferometers such as ET [7], CE [8]
$(1\text{--}10^{3}\,\rm Hz)$ and space-based experiments including LISA [9],
Taiji [10] $(10^{-4}\text{--}10^{-1}\,\rm Hz)$, all of which are expected to
become operational by 2035.
The GW background generated during inflation stands out in that (i) it is
universal in nature and (ii) it may span a wide range of frequencies, from the
CMB up to (and above) laser interferometer scales. In single-field slow-roll
scenarios, the amplitude of this background is directly related to the energy
scale at which inflation occurs111One should stress that this relation is not
necessarily one-to-one. There are indeed very interesting cases that break
this simple correspondence. Such classes of models include so-called
$P(X,\phi)$ theories (also known as k-essence), and the more general EFT of
(single-field) inflation. One should also mention specific constructions with
non-minimal couplings, such as those considered in [11] and [12]. Quite
interesting in this context is also the setup known as “Galileon inflation”
[13], which has received considerable attention. As to the origin of, for
example, a subluminal sound speed in the (scalar) sector, one may point to the
integrating out of additional fields in the inflationary Lagrangian, such as
those kinetically coupled via the so-called gelaton mechanism of [14]..
Efforts to detect this background using the B-mode polarisation of the CMB
have so far led to increasingly stringent constraints on the tensor-to-scalar
ratio [15, 16, 17], the most recent one being $r<0.032$ [18]. As a result of
the ever-improving experimental bounds, various inflationary models have
fallen by the wayside. Future CMB experiments like CMB-S4 [19], Simons
Observatory [20] and LiteBIRD [21] will be able to probe this background for
values of the tensor-to-scalar ratio down to $r\sim 10^{-3}$ allowing us to
further rule in(out) several classes of inflationary models.

The CMB-S4/LiteBIRD threshold on $r$ is particularly significant in that it
will put to the test celebrated models of inflation such as the Starobinsky
model [22] and (in the large field limit [23]) Higgs inflation. We stress
therefore that, although direct detection of a SGWB from single-field slow-
roll models is beyond the reach of most PTAs and interferometers (with the
possible exception of the proposed BBO [24] in the deciHz range), the lack of
detection itself will be almost equally as informative. Crucially, GW
detection by upcoming experiments would point to a number of interesting
inflationary scenarios, including multi-field models (see e.g. [25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35]), theories featuring alternative symmetry
breaking patterns [36, 37, 38, 39, 40], and non-attractor phases [41, 42] (see
[6] for a comprehensive review).
A SGWB detection will also provide the opportunity to test its dependence on
direction. Most cosmological sources of GWs are known to be predominantly
isotropic, with a small level of anisotropy that may arise from i) the
specific production mechanism and ii) propagation in a perturbed universe.
While the former anisotropies are typically model-dependent222For example,
anisotropies can be induced by primordial squeezed non-Gaussianity in the
tensor sector [43, 44, 45, 46, 47]. the latter are fairly universal in nature,
being a consequence of the fact that the universe is not perfectly homogeneous
and isotropic [48, 49, 50, 51, 52]. The anisotropies of the CGWB may therefore
hold precious information on the early universe333SGWB anisotropies have also
been studied in the context of phase transitions [53, 54, 55], cosmic strings
[56, 57, 58, 59], as probes of $\Delta N_{\rm eff}$ [60], and several pre-
recombination scenarios [61]. A detailed analysis of their cross-correlation
with the CMB temperature and E-mode polarisation can be found in [62]..
Understanding their behaviour is also a critical step towards being able to
tell them apart from anisotropies of astrophysical origin [63, 64, 65, 66, 67,
68, 69, 70, 71, 72, 73], which act as a foreground for the CGWB.
Propagation anisotropies [48] are similar in origin to those of the CMB and
have been studied via a Boltzmann approach in [49, 50, 51]. We will point out
how the two approaches (in [48] and [49, 50, 51] respectively) are related to
each other under certain assumptions about the behaviour of the primordial
perturbations as well as the time when GW are generated.
An important, ever-present, source of cosmological GW is the background
sourced at second order in the primordial curvature perturbations [74, 75, 76,
77]. At linear order in perturbation theory tensor and scalar modes are
decoupled from each other. However, this is no longer the case at second
order: as the primordial scalar perturbations re-enter the horizon, they
source GW via their second order anisotropic stress. Whenever these scalar
perturbations have spectra peaked at scales which re-enter the horizon during
the radiation domination epoch, the resulting scalar induced gravitational
waves (SIGW) may have an amplitude large enough to be detected at
interferometer scales. For sufficiently peaked spectra, possible in
inflationary models with significant primordial black hole (PBH) production
(see e.g. [78] for a review), this small-scale enhancement imparts a distinct
spectral shape to the SIGW and its anisotropies. We will show how this also
leads to a significant enhancement in the anisotropy spectrum at certain
frequencies, with potentially observable consequences. The frequency
dependence of the anisotropies is present not just for the SIGW, but also for
any other SGWB whose spectral shape deviates from a power law (e.g.
(p)reheating, phase transitions, cosmic strings [5]). See also [79] for a
general review.

This paper is organised as follows. In Sec. 2 we first review the Boltzmann
approach to the SGWB anisotropies and discuss the role of the GW initial
conditions. We then demonstrate how and under what assumptions the result of
[48] can be related to those of the Boltzmann approach. In Sec. 3 we discuss
the frequency dependence of the anisotropy spectrum for the SIGW. We show that
if the primordial scalar power spectrum is sharply peaked at certain scales,
the resulting SIGW anisotropies can be enhanced at certain frequencies by a
factor $\mathcal{O}(10\text{--}100)$ relative to the anisotropies for a
standard power-law spectrum. We then discuss the implications of this
enhancement for the detection of GW anisotropies using LISA-Taiji and BBO.
Finally, we present our conclusions in Sec. 4.

##  2 Anisotropies of SGWB

In this section we start by reviewing the propagation anisotropies of the
SGWB, both via the Boltzmann approach of refs. [49, 50, 51] and via the
calculation in [48]. As discussed in ref. [49], the Boltzmann formalism relies
on the geometrical optics approach to GW propagation which is valid for GW
whose wavelengths are much smaller than the length scales over which the
background curvature varies [80, 81]. The same holds for the approach of [48].
Later in this section we discuss the relation between these results and the
role of the GW initial conditions. The results we develop here will be applied
in Sec. 3 to study GW anisotropies from scalar induced gravitational waves.

###  2.1 Boltzmann approach to SGWB anisotropies

We begin with a brief review of the main results of [49, 50, 51]. The starting
point is the distribution function for gravitons $f(x^{\mu},p^{\mu})$444Note
that, as a consequence of equivalence principle, the stress-energy carried by
gravitational waves cannot be localised within their wavelength. Thus, the
energy density should be understood as obtained by averaging over
length(time)-scales much larger than the wavelength(time)-scales associated to
the GW [80, 81].. Here $x^{\mu},p^{\mu}$ denote the position and momentum of
the gravitons with $p^{\mu}=dx^{\mu}/d\lambda$, $\lambda$ being the affine
parameter along the graviton trajectory which, in the geometric optics limit,
is given by the null geodesics of the perturbed spacetime. The evolution of
the distribution function is governed by the Liouville equation

$\displaystyle\frac{df}{d\lambda}=C[f(\lambda)]+I[f(\lambda)],$ (2.1)

where $C$ and $I$ are the collision and injection terms. The collision term is
generally absent since gravitons are decoupled below the Planck scale. For
stochastic backgrounds of cosmological origin, the injection term can be
treated as an initial condition of the distribution [50, 51]. Thus, one must
solve the free Boltzmann equation $df/d\lambda=0$ in the perturbed universe.
One can write it as

$\displaystyle\frac{df}{d\eta}=\frac{\partial f}{\partial\eta}+\frac{\partial
f}{\partial x^{i}}\frac{dx^{i}}{d\eta}+\frac{\partial f}{\partial
q}\frac{dq}{d\eta}=0.$ (2.2)

Here,

$q\,\equiv\,|\vec{p}|a$ (2.3)

is the comoving momentum of the gravitons, $a$ is the cosmological scale
factor, and $\hat{n}=\hat{p}$ is the direction of propagation. In the
Newtonian gauge with $\Phi$ and $\Psi$ as the large-scale scalar potentials we
have,

$\displaystyle
ds^{2}=a^{2}(\eta)\left[-(1+2\Phi)d\eta^{2}+(1-2\Psi)\delta_{ij}dx^{i}dx^{j}\right],$
(2.4)

and the Boltzmann equation at first order in $\Phi,\Psi$ can be written as
[50, 51],

$\displaystyle\frac{\partial f}{\partial\eta}+\frac{\partial f}{\partial
x^{i}}n^{i}+q\frac{\partial f}{\partial
q}\left[\frac{\partial\Psi}{\partial\eta}-\frac{\partial\Phi}{\partial
x^{i}}n^{i}\right]=0.$ (2.5)

The total SGWB energy density observed today is given by

$\displaystyle\rho_{\rm GW}(\eta_{0},\vec{x}_{0})=\int
d^{3}p\,pf(\eta_{0},\vec{x_{0}},q,\hat{n})\,,$ (2.6)

and the more commonly used fractional density parameter $\Omega_{\rm GW}(q)$
is defined as [82]

$\displaystyle\rho_{\rm GW}(\eta_{0},\vec{x}_{0})=\rho_{cr}\int d\ln
q\,\Omega_{\rm GW}(\eta_{0},\vec{x}_{0},q),$ (2.7)

where $\rho_{cr}$ is the critical energy density of the universe. Upon
expanding the distribution function into a homogeneous and isotropic part
$\bar{f}(q)$ and a perturbation $\delta
f=-q\Gamma(\eta,\vec{x},q,\hat{n})\partial{\bar{f}}/\partial{q}$ one obtains
the following (very reminiscent of the analogous one in the CMB context)
equation [50, 51],

$\displaystyle\Gamma(\eta_{0},\vec{x}_{0},\hat{n},q)=\underbrace{\Gamma(\eta_{\rm
i},\vec{x}_{i},q)}_{\Gamma_{I}}+\underbrace{\Phi(\eta_{\rm
i},\vec{x}_{i})+\int_{\eta_{\rm
i}}^{\eta_{0}}d\eta(\Phi^{\prime}+\Psi^{\prime})}_{\Gamma_{S}},$ (2.8)

where $\eta_{0},\vec{x}_{0}$ correspond to the time and position of
observation and one has $\vec{x}_{i}=\vec{x}_{0}-(\eta_{0}-\eta_{i})\hat{n}$.
The first term on the right-hand side (R.H.S). denotes the initial
perturbation at the time of emission $\eta_{i}$ while the second and third
terms arise due to propagation in an inhomogeneous universe, analogous to the
Sachs-Wolfe (SW) and integrated Sachs-Wolfe (ISW) effects for the CMB. As
pointed out in [51], for adiabatic initial conditions the term $\Gamma_{I}$ is
also correlated with the scalar term and contributes to the SW effect for
gravitons. We explicitly evaluate this term in Sec. 2.1.1 for the case of
single clock inflation.

The GW anisotropy observed today is defined as [51, 50],

$\displaystyle\delta_{\rm
GW}(\eta_{0},\vec{x}_{0},\hat{n},q)\equiv\frac{\omega_{\rm
GW}(\eta_{0},\vec{x}_{0},\hat{n},q)}{\overline{\Omega}_{\rm
GW}(\eta_{0},q)}-1=\left[4-\frac{\partial\ln\overline{\Omega}_{\rm
GW}(\eta_{0},q)}{\partial\ln q}\right]\Gamma(\eta_{0},\vec{x}_{0},\hat{n},q),$
(2.9)

where the quantity $\omega_{\rm GW}$ is given by

$\displaystyle\Omega_{\rm GW}(\eta,\vec{x},q)=\frac{1}{4\pi}\int
d^{2}\hat{n}\,\omega_{\rm GW}(\eta,\vec{x},q,\hat{n}),$ (2.10)

with $\overline{\Omega}_{\rm GW}$ denoting the spatial average of $\Omega_{\rm
GW}(q,\vec{x})$. In terms of the distribution function we have [51, 50],

$\displaystyle\overline{\Omega}_{\rm GW}(q,\eta)=\frac{4\pi}{\rho_{\rm
cr}}\left(\frac{q}{a}\right)^{4}\,\bar{f}(q).$ (2.11)

We can now calculate how the initial condition $\Gamma_{I}$ is related to the
initial density fluctuation $\delta\rho_{\rm GW}$, with $\rho_{\rm
GW}(\eta_{i},\mathbf{x})\equiv\bar{\rho}_{\rm GW}(\eta_{i})+\delta\rho_{\rm
GW}(\eta_{i},\mathbf{x})$. This will then be used in Sec. 2.1.1 to obtain
$\Gamma_{I}$ in terms of the primordial potential $\Phi$. One finds:

$\displaystyle\rho_{\rm GW}(\eta_{i},\vec{x})$ $\displaystyle=\int
d^{3}p\,p\bar{f}(q)\left[1-\frac{\partial\ln\bar{f}}{\partial\ln
q}\Gamma(\eta_{i},\vec{x},q,\hat{n})\right]$
$\displaystyle=\frac{4\pi}{a^{4}}\int dq\,\bar{f}(q)q^{3}-\frac{1}{a^{4}}\int
dq\frac{\partial\bar{f}}{\partial q}q^{4}\int
d^{2}\hat{n}\,\Gamma_{I},\quad\text{with
}\Gamma_{I}\equiv\Gamma(\eta_{i},\vec{x},q,\hat{n})\,.$ (2.12)

If the only contribution to $\Gamma_{I}$ arises from the adiabatic primordial
perturbations, which is the scenario we are interested in here, $\Gamma_{I}$
can be safely assumed to be $q$-independent555In general $\Gamma_{I}$ can
indeed be $q$-dependent, the specific form of this dependence arises from the
production mechanism of the SGWB. An example of this $q$-dependence is
discussed in ref. [51]. In case such a $q$-dependence is present, the
correlators of the observed anisotropy $\Gamma$ are no longer frequency
dependent and one has $\Gamma(q,\hat{n})=\sum_{\ell m}\Gamma_{\ell
m}(q)Y_{\ell m}(\hat{n})$.. In this case, upon integrating by parts the
integral over $q$ on the R.H.S., one arrives at

$\displaystyle\rho_{\rm GW}(\eta_{i},\vec{x})=\frac{4\pi}{a^{4}}\int
dq\,\bar{f}(q)q^{3}\left(1+4\int\frac{d^{2}\hat{n}}{4\pi}\,\Gamma_{I}\right)\equiv\bar{\rho}_{\rm
GW}+\delta\rho_{\rm GW}.$ (2.13)

From here one can see that the intrinsic density perturbation in the Newtonian
gauge at the initial time $\eta_{i}$ is related to the monopole of
$\Gamma_{I}$ by

$\displaystyle\frac{\delta\rho_{\rm GW}}{\bar{\rho}_{\rm
GW}}=4\int\frac{d^{2}\hat{n}}{4\pi}\,\Gamma_{I}\equiv 4\Gamma_{I}^{(0)}.$
(2.14)

We are already familiar with this result from the CMB
($\Gamma=\Theta\equiv\delta T/T$), where we have
$\delta\rho_{\gamma}/\bar{\rho}_{\gamma}\equiv\delta_{\gamma}=4\Theta_{0}$,
$\Theta_{0}$ being the intrinsic temperature fluctuation at recombination.
Note that in our analysis we will neglect any higher order multipole terms in
$\Gamma_{I}$ and therefore from now on we will simply identify
$\Gamma_{I}=\Gamma_{I}^{(0)}=\delta\rho_{\rm GW}/(4\bar{\rho}_{\rm GW})$.

When to evaluate initial conditions. For GW detectable at interferometer
scales, the initial conditions are set early during radiation domination (RD)
since that is when these GW are generated (e.g. from 1st order phase
transitions, second order GW from scalar perturbations). Similarly, for
inflationary GW to be tested e.g. at interferometers, RD is when the GW modes
re-enter the horizon and start to propagate freely. It is only once the GW
modes become sub-horizon ($q\ll\mathcal{H}$ ) that $\rho_{\rm GW}\propto
a^{-4}$ and one can correspondingly think of them as behaving like
relativistic particles. Instead, for superhorizon modes ($q\gg\mathcal{H}$)
the energy density scales as $\rho_{\rm GW}\propto a^{-2}$ [83].

####  2.1.1 Initial conditions for GW anisotropies

We now calculate the initial GW overdensity $\delta\rho_{\rm
GW}(\eta_{i},\mathbf{x})$ at the time of emission $\eta_{i}$, assuming that
the primordial perturbations from inflation are adiabatic. These perturbations
correspond to common, local time shifts in all background quantities as a
result of which they have the property that for two different species $i$ and
$j$,

$\displaystyle\frac{\delta\rho_{i}}{(1+\mathsf{w}_{i})\bar{\rho}_{i}}=\frac{\delta\rho_{j}}{(1+\mathsf{w}_{j})\bar{\rho}_{j}}\,,$
(2.15)

where $\mathsf{w}_{i}$ denotes the equation of state parameter for species
$i$. Keeping this in mind, we can now easily relate the initial GW density
perturbation to the potential $\Phi$ (we neglect any anisotropic stresses so
that $\Phi=\Psi$). Firstly, note that the super-horizon solution for the
photon density contrast during radiation domination is given by [84],

$\displaystyle\delta_{\gamma}=-2\Phi\,.$ (2.16)

Therefore, by adiabaticity (neglecting sub-leading slow-roll corrections) and
using the fact that $\mathsf{w}_{\rm GW}=\mathsf{w}_{\gamma}=1/3$, we
have666We should stress here that, in our notation, ${\delta\rho_{\rm
GW}}/{\bar{\rho}_{\rm GW}}$ is not the same as the quantity $\delta_{\rm GW}$
defined in Eq. (2.9). $\delta\rho_{\rm GW}$ denotes the perturbation to the GW
energy density $\bar{\rho}_{\rm GW}$ whereas $\delta_{\rm GW}$ is the
anisotropy in the fractional energy density parameter $\overline{\Omega}_{\rm
GW}$.,

$\displaystyle\frac{\delta\rho_{\rm GW}}{\bar{\rho}_{\rm
GW}}=\delta_{\gamma}=-2\Phi\
\implies\Gamma_{I}=\frac{1}{4}\frac{\delta\rho_{\rm GW}}{\bar{\rho}_{\rm
GW}}=-\frac{1}{2}\Phi\,,$ (2.17)

with all the quantities evaluated at the initial time $\eta_{i}$.

Alternatively, one can obtain the result of Eq. (2.17) for the initial density
perturbation in the Newtonian gauge by first converting the metric element to
an unperturbed form

$\displaystyle
ds^{2}=a^{2}(\tilde{\eta})\left[-d\tilde{\eta}^{2}+\delta_{ij}d\tilde{x}^{i}d\tilde{x}^{j}\right]$
(2.18)

via the coordinate transformation $x^{\mu}\to\tilde{x}^{\mu}\equiv
x^{\mu}+\xi^{\mu}$ with [85]

$\displaystyle\xi^{0}=\begin{dcases}-\frac{1}{5}\zeta\eta,&\text{M.D.}\\\
-\frac{1}{3}\zeta\eta,&\text{R.D.}\end{dcases}$ (2.19)

and

$\displaystyle\xi^{i}$ $\displaystyle=-\zeta x^{i}\,.$ (2.20)

In the unperturbed coordinates we have $\delta\tilde{\rho}_{\rm GW}=0$. Thus,
the density perturbation $\delta\rho_{\rm GW}$ in the Newtonian gauge is,

$\displaystyle\delta\rho_{\rm GW}$ $\displaystyle=\delta\tilde{\rho}_{\rm
GW}+\bar{\rho}_{\rm GW}^{\prime}\xi^{0}$
$\displaystyle=-4\mathcal{H}\bar{\rho}_{\rm GW}\xi^{0}\,,$ (2.21)

where $\bar{\rho}_{\rm GW}^{\prime}=-3\mathcal{H}(1+\mathsf{w}_{\rm
GW})\bar{\rho}_{\rm GW}$. During radiation domination one then finds,

$\displaystyle\frac{\delta\rho_{\rm GW}}{\bar{\rho}_{\rm GW}}=-2\Phi\,,$
(2.22)

where we used the radiation domination expression for $\xi^{0}$ given in Eq.
(2.19) and the relations $a\propto\eta$, and $\Phi=-2\zeta/3$. In contrast,
for the CMB the relevant epoch for the initial conditions is that of matter
domination and one obtains the standard result [85]

$\displaystyle\delta_{\gamma}=-4\mathcal{H}\xi^{0}=-\frac{8}{3}\Phi,$ (2.23)

where the matter domination relations $a\propto\eta^{2}$ and $\Phi=-3\zeta/5$
have been used.

Putting together Eqs. (2.8) and (2.17), the total anisotropy observed today
for GW produced during the radiation dominated epoch, under the assumption of
adiabatic initial conditions, is given by

$\displaystyle\Gamma(\eta_{0},\vec{x}_{0},\hat{n})$
$\displaystyle=\underbrace{\Gamma(\eta_{i},\vec{x}_{i})}_{\Gamma_{I}}+\Phi(\eta_{i},\vec{x}_{i})+\int_{\eta_{\rm
i}}^{\eta_{0}}d\eta(\Phi^{\prime}+\Psi^{\prime})$
$\displaystyle=\frac{1}{2}\Phi(\eta_{i},\vec{x}_{i})+\int_{\eta_{\rm
i}}^{\eta_{0}}d\eta(\Phi^{\prime}+\Psi^{\prime})$ (2.24)
$\displaystyle=-\frac{1}{3}\zeta(\eta_{i},\vec{x}_{i})+\int_{\eta_{\rm
i}}^{\eta_{0}}d\eta(\Phi^{\prime}+\Psi^{\prime})\,,$

where we remind the reader that
$\vec{x}_{0}=\vec{x}_{i}+(\eta_{0}-\eta_{i})\hat{n}$. Thus, although the large
scale GW anisotropies have the same origin as those of the CMB, the
coefficient appearing in front of the potential $\Phi$ is $1/2$ instead of
$1/3$ due to the fact that the gravitons of interest (for direct detection at
intermediate and small scales) start propagating during radiation domination.
Indeed, if instead recombination took place during radiation domination, then
the CMB SW effect would also be modified accordingly [86].

The effect of primordial isocurvature perturbations on the GW anisotropy has
also been studied recently in [54, 87]. The isocurvature perturbations modify
the GW anisotropy as [54],

$\displaystyle\Gamma_{\rm iso}=\Gamma_{\rm ad}+\frac{1}{3}(1-f_{\rm GW})S_{\rm
GW}\,.$ (2.25)

Here $f_{\rm GW}\equiv\rho_{\text{GW}}/\rho_{\text{tot}}$, with
$\rho_{\text{tot}}$ the total energy density, and $\Gamma_{\rm ad}$ is the GW
anisotropy, assuming adiabatic initial conditions, given in Eq. (2.1.1).
$S_{\rm GW}$ is the GW isocurvature perturbation w.r.t photons defined as
$S_{{\rm GW}}\equiv 3(\zeta_{{\rm GW}}-\zeta_{\gamma})$, with
$\zeta_{i}=-\Psi-H{\delta\rho_{i}}/{\dot{\bar{\rho}}_{i}}$ the curvature
perturbation on the hyper-surfaces of uniform energy of the fluid $i$.

###  2.2 Uniform density gauge calculation

We now compute the propagation anisotropy following ref. [48]. The starting
point for the calculation of [48] is the uniform density gauge during matter
domination for which the metric is given by [88],

$\displaystyle
ds^{2}=a^{2}(\eta)\left[-d\eta^{2}+(1+2\zeta)\delta_{ij}dx^{i}dx^{j}-\frac{4}{5aH}\partial_{i}\zeta
d\eta dx^{i}\right]\,.$ (2.26)

To facilitate the comparison with the results of the previous section, we
compute here the GW anisotropy via the Boltzmann approach working in the
uniform density gauge given by Eq. (2.26). Although ref. [48] proceeds in a
slightly different manner, the results are equivalent. For completeness we
also report the original calculation of ref. [48] as well as some details
related to the Boltzmann approach in Appendix A.

To solve the Boltzmann equation (2.2), we express more conveniently its
various terms. We write the graviton 4-momentum $P^{\mu}$ in terms of the
magnitude of the physical momentum $p$ and the direction of propagation
$\hat{n}$ as (see Appendix A for the details)

$\displaystyle P^{\mu}=\frac{p}{a}\left(1-\frac{2}{5aH}\partial_{i}\zeta
n^{i},(1-\zeta)\hat{n}\right)\,.$ (2.27)

From this one can simply read the derivative

$\displaystyle\frac{dx^{i}}{d\eta}=\frac{P^{i}}{P^{0}}=n^{i}\,.$ (2.28)

Note that we only need this term at zeroth order, since it multiplies
$\partial f/\partial x^{i}$ which is a first order quantity. Next, we need
$dq/d\eta$. After some manipulations (see Appendix A), one finds

$\displaystyle\frac{dq}{d\eta}=\frac{1}{5}q\,\partial_{i}\zeta n^{i}\,.$
(2.29)

Using the above results, one can finally write

$\displaystyle\frac{\partial f}{\partial\eta}+\frac{\partial f}{\partial
x^{i}}n^{i}+\frac{1}{5}q\partial_{i}\zeta n^{i}\frac{\partial f}{\partial
q}=0\,.$ (2.30)

In terms of the zeroth order distribution $\bar{f}$ and the first order
perturbation $\delta
f=-q\Gamma(\eta,\vec{x},q,\hat{n})\partial{\bar{f}}/\partial{q}$ one obtains,

$\displaystyle\frac{\partial\Gamma}{\partial\eta}+n^{i}\frac{\partial\Gamma}{\partial
x^{i}}$ $\displaystyle=\frac{1}{5}\frac{\partial\zeta}{\partial
x^{i}}n^{i}\,.$ (2.31)

This equation can be integrated along the line of sight to give,

$\displaystyle\Gamma(\eta,\vec{x}_{0},q,\hat{n})=\underbrace{\Gamma(\eta_{\rm
in},\vec{x}_{i},q,\hat{n})}_{\Gamma_{I}}-\underbrace{\frac{1}{5}\zeta(\eta_{\rm
in},x_{i})}_{\Gamma_{S}}\,,$ (2.32)

where $\vec{x}_{0}=\vec{x}_{i}+(\eta-\eta_{\rm in})\hat{n}$. The curvature
perturbation at the observer’s position can be ignored since it does not have
any direction dependence. Note that the Authors of ref. [48] consider the case
of single-field inflation and assume adiabatic primordial perturbations. Since
the gauge of Eq. (2.26) is the uniform matter density gauge, it must also be
the uniform GW density gauge, i.e. by adiabaticity one finds

$\displaystyle\frac{\delta\rho_{\rm GW}}{\bar{\rho}_{\rm
GW}}=\frac{4\delta\rho_{\rm m}}{3\bar{\rho}_{\rm m}}=0\implies\Gamma_{I}=0,$
(2.33)

where we have used Eq. (2.14). Using this result in Eq. (2.32), the relation
in Eq. (2.9) between the GW anisotropy $\delta_{\rm GW}$ and $\Gamma$, and Eq.
(2.11), one obtains

$\displaystyle\delta_{\rm GW}(q,\eta_{0},\hat{n})$
$\displaystyle=-\frac{1}{5}\zeta(\eta_{i},\vec{x}_{i})\left[4-\frac{\partial\ln\overline{\Omega}_{\rm
GW}}{\partial\ln
q}\right]=\frac{1}{5}\zeta(\eta_{i},\vec{x}_{i})\frac{\partial\ln\bar{f}(q)}{\partial\ln
q}\,,$ (2.34)

which is the result of ref. [48]. Let us now see how to relate this to the
Newtonian gauge result of Eq. (2.1.1). Firstly, note that under the assumption
of matter domination one can simply disregard the ISW term from that equation
and write

$\displaystyle\Gamma=\frac{1}{4}\frac{\delta\rho_{\rm GW}}{\bar{\rho}_{\rm
GW}}+\Phi\,.$ (2.35)

In this equation, we now use the matter domination relation, Eq. (2.23),
together with the adiabaticity condition, to get
$\delta_{\gamma}=\delta\rho_{\rm GW}/\bar{\rho}_{\rm GW}=-8\Phi/3$. Thus Eq.
(2.35) becomes,

$\displaystyle\Gamma=\frac{1}{3}\Phi=-\frac{1}{5}\zeta$ (2.36)

and

$\displaystyle\delta_{\rm GW}$
$\displaystyle=-\left[4-\frac{\partial\ln\overline{\Omega}_{\rm
GW}}{\partial\ln q}\right]\frac{\zeta}{5}\,,$ (2.37)

which matches Eq. (2.34). However, we stress once again that given the choice
of the metric in Eq. (2.26), the result of ref. [48] is valid only if we
assume matter domination throughout. As mentioned in Sec. 2.1, for most
cosmological SGWB the initial conditions are instead evaluated during
radiation domination and the corresponding expression is that of Eq. (2.1.1).

##  3 Anisotropies for peaked spectra

We now use the results of Sec. 2 to calculate the propagation anisotropy
spectrum associated to a cosmological background of gravitational waves
sourced at second order in the curvature perturbations which re-enter the
horizon during the epoch of radiation domination [75, 76]. We are interested
in scenarios where the primordial curvature power spectrum is sharply peaked
on small scales. We parametrize these primordial spectra around the peak
wavenumber $k_{*}$ by a log-normal, i.e. we take

$\displaystyle\mathcal{P_{\mathcal{R}}}(k)|_{k\gg k_{\rm
CMB}}=\frac{A_{\mathcal{R}}}{\sqrt{2\pi}\Delta}\exp\left[-\frac{\ln^{2}(k/k_{*})}{2\Delta^{2}}\right]\,.$
(3.1)

The parameter $\Delta$ then controls the width of the spectrum and
$A_{\mathcal{R}}$ represents its amplitude. In the limit $\Delta\to 0$ one
recovers the Dirac-delta power spectrum
$\mathcal{P}_{\mathcal{R}}=A_{\mathcal{R}}\delta(\ln(k/k_{*}))$. The log-
normal serves as a useful representative of a peaked spectrum which can arise
in several inflationary models producing PBH (see [89] and references
therein). Note that the quantity $\mathcal{R}$ is the comoving gauge curvature
perturbation which on super-Hubble scales is equivalent to the uniform density
gauge curvature perturbation $\zeta$.

###  3.1 Scalar induced GW background and its anisotropies

The GW energy density spectrum observed today for a log-normal power spectrum
was calculated in [89] and can be written as

$\displaystyle\Omega_{\rm GW}(k,\eta_{0})h^{2}\simeq 1.6\times
10^{-5}\left(\frac{g_{*s}(\eta_{k})}{106.75}\right)^{-1/3}\left(\frac{\Omega_{r,0}h^{2}}{4.1\times
10^{-5}}\right)\Omega_{\rm GW,r}(k)\,,$ (3.2)

where $\Omega_{\rm GW,r}$ is the GW energy density at matter radiation
equality and is given by

$\displaystyle\Omega_{\rm
GW,r}(k)=3\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du\frac{\mathcal{T}(u,v)}{u^{2}v^{2}}\mathcal{P}_{\mathcal{R}}(vk)\mathcal{P}_{\mathcal{R}}(uk)\,,$
(3.3)

with the function $\mathcal{T}(u,v)$ defined in Eq.(10) of [89]. A generic GW
background of cosmological origin will have anisotropies in the angular
distribution of its energy density. These anisotropies777In this section we
focus only on the propagation anisotropies of the SGWB and therefore the
quantity $C_{\ell}^{\Gamma}$ is frequency independent [51]. We neglect any
other sources of anisotropies that might be relevant for scalar induced GW,
e.g. those arising from primordial non-Gaussianity due to a local
$\langle\zeta^{3}\rangle$ bispectrum [90]. are given by

$\displaystyle\delta_{\rm
GW}(k,\hat{n})=\left(4-n_{\Omega}(k)\right)\Gamma(k,\hat{n}),\quad
n_{\Omega}(k)\equiv{\partial\ln\overline{\Omega}_{\rm GW}}/{\partial\ln k}\,,$
(3.4)

where $\Gamma$ was defined in Eq. (2.8). Thus, if the spectrum is sharply
peaked ($|n_{\Omega}(k)|\gg 1$) one can expect enhancement of $C_{\ell}^{\rm
GW}$ relative to $C_{\ell}^{\Gamma}$ (see Eqs. 3.11, 3.12) by a factor
$\sim\mathcal{O}(10\text{--}1000)$ at certain scales.

|
---|---
|
Figure 1: The dashed curves show the spectral tilt factor $4-n_{\Omega}$, the
solid curves show the GW amplitude $\Omega_{\rm GW,r}/A_{\mathcal{R}}^{2}$.

The GW spectrum and this spectral tilt factor $n_{\Omega}$ as a function of
$k$ are plotted in Fig. 1 for different choices of $\Delta$. The frequency
profile of the stochastic gravitational wave spectrum induced at second order
from scalar fluctuations can be understood analytically, at least for simple
ansätze of the sourcing scalar spectrum. For example, for a delta-like scalar
source, peaked at momentum $k\equiv k_{*}$, the resulting induced SGWB
spectrum has a dip and goes to zero for $k/k_{\star}=\sqrt{2/3}$, and a
pronounced resonant peak at $k/k_{\star}=\sqrt{4/3}$.

If the source scalar spectrum profile is narrow but not a delta-function, then
the previous features are smoothed. Fig. 1 shows the SGWB induced by a log-
normal peak, for different choices of the characteristic width. The technical
reason for these features can be found in the convolution integrals that give
the SGWB at second order: see e.g. [75, 76, 91] for important early works on
the subject, as well as [77] for a recent systematic review. The features are
due to interference and resonant effects among the scalar modes that source
the GW background.

The spectrum is normalised against $A_{\mathcal{R}}^{2}$ because the latter
quantity always results as overall coefficient in front of the induced
$\Omega_{\rm GW}$ at second order (see also [75, 76, 91, 77]). It is customary
to plot the GW frequency profile singling out such overall factor.

####  3.1.1 Inflationary models with peaked spectra

A number of inflationary models can produce a sufficiently narrow peak in the
GW spectrum which can enhance the SGWB anisotropies by a factor of
$\mathcal{O}(100)$ or more. Within single-field models, the curvature power
spectrum has bounds in its growth rate (see e.g. [92, 93, 94, 95]) which
restricts the steepness of the induced GW spectrum [74, 75, 76], and –
consequently – how large $n_{\Omega}$ can be. In fact, we find that single-
field scenarios can in general be well-represented by the $\Delta=10^{-1}$
case of Fig. 1. On the other hand, models with modified gravity [96], multiple
fields [97, 98, 99, 100, 101], parametric resonance [102, 103, 104, 105] or
particle production [106] may produce the narrow spectra corresponding to
scenarios with $\Delta\leq 10^{-2}$. One should also mention models with non-
standard kinetic terms, these too can support an enhanced GW spectrum, see
e.g. [41, 42].

####  3.1.2 Other sources of peaked GW spectra

GW from non-inflationary sources, e.g. phase transitions [107], cosmic strings
[108], can also produce peaked spectra although the enhancement from the tilt
is typically of the same order as for a power-law spectrum. As an example, let
us consider the SGWB produced from cosmological phase transitions. For these
backgrounds, the sound wave contribution is generally larger than the other
contributions and has a steeper spectral shape given by [109]

$\displaystyle
S(k,k_{*})=\left(\frac{k}{k_{*}}\right)^{3}\left(\frac{7}{4+3(k/k_{*})^{2}}\right)^{7/2}\,,$
(3.5)

with

$\displaystyle\Omega_{\rm GW}(k)=\Omega_{\rm GW}(k_{*})S(k,k_{*})\,,$ (3.6)

where $\Omega_{\rm GW}(k_{*})$ is the GW amplitude at the peak wavenumber
$k_{*}$. From Eq. (3.5) one can infer that the factor $(4-n_{\Omega}(k))$
appearing in the R.H.S. of Eq. (3.4) is $\mathcal{O}$(1). Thus, the
enhancement in this case is much smaller than what GW sourced from peaks in
the curvature power spectrum allow for. In the next section, we focus on the
latter scenario and explore the observational consequences through an example
of SIGW detectable at mHz scales.

###  3.2 LISA GW-PBH scenario

As a representative example, we consider a scenario in which the scalar
induced GW spectrum can be detected by LISA [9]. For this, we take the peak of
the log-normal power spectrum of Eq. (3.1) to correspond to a frequency within
the LISA band $10^{-4}\text{ Hz}\leq f\leq 10^{-1}\text{ Hz}$. The relation
between frequency and wavenumber is given by

$\displaystyle\frac{k}{\text{Mpc}^{-1}}\simeq 6.5\times
10^{14}\frac{f}{\text{Hz}}\,.$ (3.7)

The induced GW spectrum is plotted in the left panel of Fig. 2 for the
parameter choice of table 1. Note that to avoid numerical artifacts arising in
the calculations we have also smoothed the $\Omega_{\rm GW}$ spectrum near the
peak using the peak width as the smoothing scale. The smoothed spectrum
$\widetilde{\Omega}_{\rm GW}$ is defined as

$\displaystyle\widetilde{\Omega}_{\rm
GW}(f)=\frac{1}{2\Delta}\int_{fe^{-\Delta}}^{fe^{\Delta}}\Omega_{\rm
GW}(f^{\prime})\,d\ln f^{\prime}\,.$ (3.8)

$\Delta$ | $10^{-2}$
---|---
$f_{*}$ | $5\times 10^{-3}\,\rm Hz$
$k_{*}$ | $3\times 10^{12}\,\rm Mpc^{-1}$
$A_{\mathcal{R}}$ | $7.5\times 10^{-3}$
Table 1: Parameter choice for the LISA GW scenario.

The enhancement of the curvature power spectrum can also lead to the
production of PBH and one ought to ensure that the scenario considered here is
well within the region allowed by current constraints on the PBH abundance.
The relation between PBH mass at formation and the comoving scale $k$ re-
entering the horizon (during the radiation era) can be written as [110, 78],

$\displaystyle M_{\rm PBH}\simeq
30M_{\odot}\left(\frac{\gamma}{0.2}\right)\left(\frac{g_{*}}{10.75}\right)^{-1/6}\left(\frac{2.9\times
10^{5}{\rm Mpc^{-1}}}{k}\right)^{2}\,.$ (3.9)

For peak frequencies/wavenumbers relevant for LISA this leads to PBH with
masses in the range $M_{\rm PBH}\sim\mathcal{O}(10^{-15}\text{ --
}10^{-12})M_{\odot}$ (see [111, 112] for a detailed analysis of the PBH-GW
scenario for LISA). In our case the corresponding PBH abundance is shown in
the right panel of Fig. 2 for $\Delta=10^{-2}$ along with the corresponding
constraints, compiled using the tool provided in [113]. To calculate the mass
function of PBH we have followed the method of ref. [114] which uses a simple
Press-Schechter approach and accounts for the effects of critical collapse888A
detailed comparison of different methodologies for calculating the PBH
abundances and mass distribution can be found in [115].. The PBH mass function
$f_{\rm PBH}(M)$ is defined as,

$\displaystyle\frac{\Omega_{\rm PBH}}{\Omega_{\rm DM}}=\int f_{\rm
PBH}(M)\,d\ln M\,,$ (3.10)

where the L.H.S. denotes the total fraction of the dark matter density
constituted by PBH. Note that the constraints shown in Fig. 2 are strictly
valid only for a monochromatic mass function and for extended mass functions
the constraints need to be calculated differently [116].

Figure 2: Left - The induced GW spectrum at LISA scales for the parameter
choice given in table 1 and the smoothed spectrum defined in Eq. (3.8). The
green dashed curve shows power-law integrated sensitivity curve for LISA [117]
(see [118] for the updated sensitivity curves due to the impact of foregrounds
and resolvable binaries). Right - The PBH abundance for the same values of the
parameters plotted along with the constraints from microlensing [119, 120,
121, 122, 123, 124, 125], accretion [126, 127, 128, 129], GW [130],
evaporation [131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141] and
dynamical constraints [142, 143].

####  3.2.1 Angular power spectrum of GW anisotropies

We now calculate the angular power spectrum of the GW anisotropies for the
LISA GW scenario of table 1. The angular power spectrum is defined as

$\displaystyle\langle\Gamma_{\ell
m}\Gamma_{\ell^{\prime}m^{\prime}}\rangle=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}C_{\ell}^{\Gamma}\,,$
(3.11)

where we have assumed statistical isotropy. We denote by $C_{\ell}^{\rm GW}$
the angular power spectrum of $\delta_{\rm GW}$ which is related to the above
quantity as follows,

$\displaystyle C_{\ell}^{\rm
GW}(k)=\left(4-n_{\Omega}(k)\right)^{2}C_{\ell}^{\Gamma}\,,$ (3.12)

with $n_{\Omega}$ given by Eq. (3.4). In the left panel of Fig. 3 we plot the
autocorrelation of the GW anisotropies and their cross-correlation with CMB
temperature and E-mode polarisation anisotropies. We obtained the T and E
transfer functions from CAMB [144]. Note that the propagation anisotropies of
the CGWB quantity $\Gamma$ (Eq. (2.1.1)), and consequently
$C_{\ell}^{\Gamma}$, are frequency independent. In the right panel, we show
the frequency dependence of $C_{\ell}^{{\rm GW}}$ (for $\ell=2$) for the LISA
GW scenario and for a flat spectrum of GW for comparison. For simplicity, we
have only considered the SW contribution for GW anisotropies, Eq. (2.1.1),
since this is the dominant contribution on large angular scales, and ignored
the ISW term. These scales are the most relevant for the GW detectors under
consideration here since these are limited by their angular resolution
$\ell_{\rm max}\sim 15\text{--}30$ [145, 146, 147]. It is worth noting that
this frequency dependence of the anisotropies arises for any SGWB that is
different from a power-law ($n_{\Omega}=\rm const.$), whereas
$C_{\ell}^{\Gamma}$ for the anisotropies from propagation is expected to be
the same for all CGWB.

|
---|---
Figure 3: Left - Angular power spectrum of the GW anisotropies and their
cross-correlation with CMB-T, E. Right - Frequency dependence of the angular
power spectrum, for $\ell=2$, for the LISA model of table 1 (blue line) and
for a flat spectrum (orange line) plotted in a frequency range where LISA has
the highest sensitivity.

####  3.2.2 SNR of the anisotropies

In this section we show that the enhancement of anisotropies for a peaked
spectrum can make these anisotropies easier to detect compared to the case of
a standard power-law spectrum. A related phenomenon was recently pointed out
in [148] for the case of kinematic anisotropies. For the purpose of this
comparison, we consider a flat power-law spectrum with an amplitude such that
the resulting SNR of the monopole is the same in the LISA range. We have
verified that the results for a power law with a small non-zero spectral tilt
$n_{\Omega}\sim\mathcal{O}(1)$ are similar so we do not show them separately.
We plot the SNR for the individual multipoles in Fig. 4. Our results indicate
that detecting the anisotropy is easier for a peaked spectrum than for a flat
spectrum. We see that, although the $\ell=2$ is detectable with a LISA-Taiji
network for the peaked spectra, it is not detectable for the flat spectra.

|
---|---
Figure 4: Left - $\Omega_{\rm GW}(f)$ plotted for the two spectra. Right - SNR
of the individual multipoles with $T_{\rm obs}=3\rm\,years$.

The multipole SNR is defined as [149]

$\displaystyle\text{SNR}_{\ell}^{2}=\int df\,C_{\ell}^{{\rm
GW}}(f)\left(\frac{\Omega_{\rm GW}(f)}{\Omega^{\ell}_{{\rm
GW},n}(f)}\right)^{2}\,,$ (3.13)

where the quantity $\Omega^{\ell}_{{\rm GW},n}(f)$ is the effective angular
sensitivity of the detector network to the $\ell-$th multipole. The quantity
$\Omega_{\rm GW,n}^{\ell}(f)$ is defined as

$\displaystyle\Omega_{\rm GW,n}^{\ell}(f)^{-1}\equiv T_{\rm
obs}\sum_{AB}\left(\frac{2}{5}\right)^{2}\left(\frac{4\pi^{2}f^{3}}{3H_{0}^{2}}\right)^{-2}\frac{1}{N^{A}_{f}N^{B}_{f}}\frac{\sum_{m}|\mathcal{A}_{AB}^{\ell
m}(f)|^{2}}{(2\ell+1)},$ (3.14)

where $T_{\rm obs}$ is the time of observation, $N_{f}^{A}$ denotes the noise
Power Spectral Density for detector $A$ and $\mathcal{A}^{\ell m}_{AB}$ the
spherical harmonic transform of the antenna pattern for the detector pair
$AB$.

We use the code schNell999https://github.com/damonge/schNell (see also the
companion work in [146]). to compute $\Omega_{\rm GW,n}^{\ell}(f)$. Fig. 5
contains a plot of the quantity $\Omega_{\rm GW,n}^{\ell}(f)$ for LISA-Taiji
[150] and for the BBO configuration with 4 constellations [24]. We include all
LISA-Taiji cross-correlations as well correlations internal to both LISA and
Taiji while for BBO we only consider cross-correlations among the different
vertices. The addition of multiple detectors separated by large distances
improves angular sensitivity compared to a single LISA-like constellation,
e.g. see Fig. 9 of ref. [149] or Fig. 6 of ref. [145]. The noise curves for
the various detectors can be found in [117] for LISA, [150] for Taiji and
[151] for BBO.

Figure 5: Sensitivity of LISA-Taiji and BBO to the different multipoles of the
SGWB as a function of frequency. The time of observation is taken to be 3
years for both plots.

#### SNR of the cross-correlation

Figure 6: SNR of the cross-correlation between the GW anisotropies and the
CMB-T, E mode anisotropies.

The SNR of the cross-correlation with the CMB-T, E anisotropies is defined
as101010Note that incomplete sky coverage can lead to correlations between
nearby $\ell$ thus degrading the sensitivity to the individual multipoles. As
in ref. [62], we have assumed for simplicity that the GW and CMB maps used to
calculate the cross-correlation are full sky which makes our estimate an
optimistic one. A reduction of the SNR by a factor $\sqrt{f_{\mathrm{sky}}}<1$
would be expected in a more realistic case [152, 153].

$\displaystyle\text{SNR}^{2}=\sum_{\ell=2}^{\ell_{\rm max}}\sum_{\rm
X=T,E}(2\ell+1)\frac{\left(C^{\rm{X}\Gamma}\right)^{2}}{\left(C^{\rm{X}\Gamma}\right)^{2}+\left(C_{\ell}^{\Gamma}+N_{\ell}^{\Gamma}\right)C_{\ell}^{\rm
X}}\,,$ (3.15)

where the quantity $N_{\ell}^{\Gamma}$ is (for $\ell>0$)

$\displaystyle N_{\ell}^{\Gamma}\equiv\left[\int
df\frac{(4-n_{\Omega}(f))^{2}\overline{\Omega}_{\rm GW}(f)^{2}}{\Omega_{\rm
GW,n}^{\ell}(f)^{2}}\right]^{-1}\,.$ (3.16)

We plot the SNR in Fig. 6 as a function of $\ell_{\rm max}$ for observation by
LISA-Taiji and BBO. For the representative example considered here, this
cross-correlation will be detectable by a BBO-level GW detector but not with
LISA-Taiji. The GW-T cross-correlation provides the entirety of the cross-
correlation SNR while the GW-E contribution is negligible.

##  4 Conclusion

These are very exciting times for GW astronomy: observations of compact object
mergers by LIGO-Virgo and KAGRA have already demonstrated its tremendous
potential to shed light on stellar and black hole physics, late-time cosmology
as well as general relativity itself.
The detection of a background resulting from the superposition of a large
number of these astrophysical sources is also expected to happen in the near
future. When it comes to a cosmological background, such as the one from
inflation, detection will likely be more challenging and require space-based
and future ground-based interferometers. The importance of detecting
cosmological backgrounds cannot be understated. These are generated well
before the CMB and thus provide access to the primordial universe at much
earlier times than what is possible by electromagnetic observations alone.

Much like the CMB, the CGWB is predicted to be nearly isotropic with small
fluctuations that may be inherent to the production mechanism itself or may
arise due to propagation in a universe that is not perfectly homogeneous and
isotropic.

In the first part of this paper we have reviewed two different approaches to
derive these “propagation” anisotropies and shown how, under some simplifying
assumptions, they can be related to each other once a careful choice of the GW
initial conditions is performed.
We also pointed out how the adiabatic or isocurvature nature of the primordial
perturbations affects this anisotropy through the intrinsic density
fluctuation at the time when the GW start propagating. Whether or not a
significant amplitude of isocurvature perturbations could be produced to
noticeably alter the SGWB anisotropies while leaving the CMB unaffected is a
subject that merits further investigation. We plan to address such matters in
an upcoming work.

In the second part of this paper, we analysed the propagation anisotropy
spectrum of SGWB sourced at second order in the curvature perturbation. We saw
that if the primordial curvature power spectrum is sharply peaked around some
scales, as is possible in scenarios of PBH production, the SGWB anisotropies
can be enhanced by a factor $\sim\mathcal{O}(10\text{--}100)$ relative to
those of a power-law SGWB spectrum. This enhancement arises due to the fact
that the multiplicative factor $(4-n_{\Omega})$ in the definition of
$\delta_{\rm GW}$ in Eq. (3.4) becomes large if the SGWB spectrum is sharply
peaked.
Through the representative example of a scalar induced SGWB detectable at mHz
frequencies, we showed that even though this enhancement only affects certain
scales, it can nevertheless make these anisotropies easier to detect relative
to those of a standard power-law spectrum. For this representative example, we
demonstrated the capability of a LISA-Taiji network to detect the $\ell=2$
multipole as well as that of BBO to detect the cross-correlation with CMB
anisotropies. Larger values of the SGWB monopole would naturally increase the
detectability of these anisotropies, as would the improved sensitivity and
angular resolution of futuristic detector networks like BBO/DECIGO.
We also considered SGWB from other sources which may exhibit a peaked spectral
shape. We find that in a representative case (see section 3.1.2) the
enhancement of the anisotropies is small compared to the PBH scenario
mentioned above. Irrespective of whether the enhancement is significant or
not, the distinct frequency dependence of the SGWB anisotropies for such
spectra could possibly be exploited to separate them from the anisotropies
associated with other CGWB or even SGWB of astrophysical origin. This idea has
been previously used for the CMB to subtract the contribution from foreground
sources. A dedicated analysis of its feasibility for GW is also worth pursuing
and would bring us one step closer to the detection of these GW anisotropies.

### Acknowledgments

M.F. would like to acknowledge support from the “Atracción de Talento” grant
2019-T1/TIC15784, his work is partially supported by the Spanish Research
Agency (Agencia Estatal de Investigación) through the Grant IFT Centro de
Excelencia Severo Ochoa No CEX2020-001007-S, funded by
MCIN/AEI/10.13039/501100011033. GT is partially funded by the STFC grant
ST/T000813/1. For the purpose of open access, the authors have applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising.

##  Appendix A Boltzmann equation in $\zeta$-gauge

In this Appendix, we provide additional details relevant to the calculations
of Sec. 2.2 and then briefly review the original computation of ref. [48].

To begin with, let $\lambda$ be the affine parameter along the graviton
geodesic. The graviton 4-momentum is then $P^{\mu}=dx/d\lambda$. Let us
express this in terms of the quantity $p$ defined as $p^{2}\equiv
g_{ij}P^{i}P^{j}$, and $\hat{n}$, the unit vector along the GW direction. To
this end, we first write $P^{i}=C\hat{n}$, where $C$ can be determined in
terms of $p$ as,

$\displaystyle p^{2}$
$\displaystyle=g_{ij}P^{i}P^{j}=a^{2}(1+2\zeta)C^{2}\delta_{ij}n^{i}n^{j}$
$\displaystyle\implies C=\frac{p}{a}(1-\zeta)\,.$ (A.1)

Next, using $g_{\mu\nu}P^{\mu}P^{\nu}=0$ we find,

$\displaystyle P^{0}=\frac{p}{a}\left(1-\frac{2}{5aH}\partial_{i}\zeta
n^{i}\right)\,.$ (A.2)

To solve the Boltzmann equation (2.2) we need to calculate $dq/d\eta$ with
$q=|\vec{p}|a$. Using Eqs. (A) and (A.2) one finds $P_{0}=-q$. Thus, it only
remains to solve the geodesic equation for $P_{0}$. This is given by [154],

$\displaystyle\frac{dP_{0}}{d\lambda}=\frac{1}{2}\partial_{0}g_{\alpha\beta}P^{\alpha}P^{\beta}\,.$
(A.3)

Using the fact that $d/d\lambda=P^{0}d/d\eta$ one can re-write the above
equation as

$\displaystyle
P^{0}\frac{dP_{0}}{d\eta}=\frac{1}{2}\partial_{0}g_{\alpha\beta}P^{\alpha}P^{\beta}.$
(A.4)

To simplify things further, we can work with the re-scaled metric without the
scale factor which we denote by $\tilde{g}_{\mu\nu}$, i.e.
$g_{\mu\nu}=a^{2}\tilde{g}_{\mu\nu}$. The null geodesics remain the same; the
affine parameters and the tangent vectors in the rescaled metric, on the other
hand, are respectively given by $d\tilde{\lambda}=a^{-2}d\lambda$ and
$\tilde{P}_{\mu}=a^{2}P^{\mu}$ [155]. One also finds
$P_{\mu}=\tilde{P}_{\mu}$. Using these relations the geodesic equation reads,

$\displaystyle\frac{d\tilde{P}_{0}}{d\eta}=-\frac{2}{5}\tilde{P}^{0}\left(\frac{a}{a^{\prime}}\right)^{\prime}\partial_{i}\zeta
n^{i}\,,$ (A.5)

where the prime denotes a partial derivative with respect to conformal time.
Using $P_{0}=\tilde{P_{0}}$ and re-writing the above expression using the
quantities defined in the original metric $g_{\mu\nu}$, one obtains at linear
order in $\zeta$,

$\displaystyle\frac{dP_{0}}{d\eta}$
$\displaystyle=-\frac{1}{5}a^{2}P^{0}\,\partial_{i}\zeta n^{i}$
$\displaystyle=-\frac{1}{5}q\,\partial_{i}\zeta n^{i}\,.$ (A.6)

The Boltzmann equation then becomes

$\displaystyle\frac{df}{d\eta}=\frac{\partial f}{\partial\eta}+\frac{\partial
f}{\partial x^{i}}n^{i}+\frac{1}{5}q\partial_{i}\zeta n^{i}\frac{\partial
f}{\partial q}$ (A.7)

(Eq. (2.30) of the main text). Upon expanding the distribution function into a
zeroth order term plus perturbations,

$\displaystyle
f(\eta,\vec{x},q,\hat{n})=\bar{f}(q)-q\frac{\partial\bar{f}}{\partial
q}\Gamma(\eta,\vec{x},q,\hat{n})\,,$ (A.8)

the Boltzmann equation at first order reads,

$\displaystyle\frac{\partial\Gamma}{\partial\eta}+n^{i}\frac{\partial\Gamma}{\partial
x^{i}}=\frac{1}{5}\frac{\partial\zeta}{\partial x^{i}}n^{i}\,.$ (A.9)

In Fourier space, one finds

$\displaystyle\frac{\partial\Gamma_{k}}{\partial\eta}+ik\mu\Gamma_{k}=\frac{ik\mu}{5}\zeta_{k}\,,$
(A.10)

with $\mu\equiv\hat{k}\cdot\hat{n}$. This can be solved to give

$\displaystyle\Gamma_{k}(\eta)=e^{ik\mu(\eta_{\rm
in}-\eta)}\left[\Gamma_{k}(\eta_{\rm
in},q,\hat{n})-\frac{1}{5}\zeta_{k}(\eta_{\rm in})\right]\,,$ (A.11)

which is Eq. (2.32) in Fourier space.

We also provide here the original derivation of ref. [48]. Starting from Eq.
(A.5) and using the fact that the derivative along the geodesic can be written
as

$\displaystyle\frac{d}{d\eta}=\frac{\partial}{\partial\eta}+n^{i}\partial_{i}\,,$
(A.12)

one obtains [156]

$\displaystyle\frac{dP_{0}}{d\eta}=-\frac{2}{5}P^{0}\left(\frac{a}{a^{\prime}}\right)^{\prime}\frac{d}{d\eta}\zeta\,.$
(A.13)

At zeroth order $P_{0}=-P^{0}$ and one finds,

$\displaystyle P_{0}(\eta)=P_{0}(\eta_{\rm
i})\left[1+\frac{1}{5}\left(\zeta(\eta,\mathbf{x})-\zeta(\eta_{\rm
i},\mathbf{x}_{i})\right)\right]\,,$ (A.14)

where $\vec{x}_{0}=\vec{x}_{i}+(\eta_{0}-\eta_{i})\hat{n}$. Equivalently, we
can write for the comoving momentum $q$,

$\displaystyle\frac{q}{q_{i}}=\left[1+\frac{1}{5}\left(\zeta(\eta,\vec{x})-\zeta(\eta_{\rm
i},\vec{x}_{i})\right)\right].$ (A.15)

In this case one obtains a SGWB anisotropy equal to:

$\displaystyle\delta_{\rm GW}(q,\hat{n})$
$\displaystyle=\frac{1}{5}\zeta(\eta_{i},\vec{x}_{i})\frac{\partial\ln\bar{f}(q)}{\partial\ln
q}$ (A.16)

(Eq. (2.34) of the main text). Note the implicit assumption here that there is
no initial perturbation to the distribution function, i.e.
$f(\eta_{i},\vec{x}_{i})=\bar{f}(\eta_{i})$, which is valid for adiabatic
primordial perturbations when working in the uniform matter density gauge (see
the discussion in Sec. 2.2.). The curvature perturbation at the observer’s
position can also be ignored since it only contributes to the monopole. Ref.
[48] considers the case of GW from single-field slow-roll inflation, with a
distribution function
$\bar{f}(q)\propto(q_{0}/q)^{2(\nu+1+\epsilon)}$.111111Note that the quantity
$\langle N\rangle$ of [48] is the same as $\bar{f}$ up to some constant
factors. Here $q_{0}$ is a reference frequency, $\epsilon$ the standard slow-
roll parameter during inflation and $\nu\equiv 2/(1+3\mathsf{w})$ with
$\mathsf{w}$ evaluated at the time when the GW mode re-enters the horizon.
Evaluating the SGWB anisotropy in terms of these quantities one arrives at
[48],

$\displaystyle\delta_{\rm
GW}(q,\hat{n})=-\frac{2}{5}(\nu+1+\epsilon)\zeta(\eta_{i},\vec{x}_{i})\,.$
(A.17)

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