Kinematic anisotropies and pulsar timing arrays

Gianmassimo Tasinato1,2

1 Dipartimento di Fisica e Astronomia, Università di Bologna, Italia

2 Physics Department, Swansea University, SA28PP, United Kingdom
email: g.tasinato2208 at gmail.com

###### Abstract

Doppler anisotropies, induced by our relative motion with respect to the
source rest frame, are a guaranteed property of stochastic gravitational wave
backgrounds of cosmological origin. If detected by future pulsar timing array
measurements, they will provide interesting information on the physics
sourcing gravitational waves, which is hard or even impossible to extract from
measurements of the isotropic part of the background only. We analytically
determine the pulsar response function to kinematic anisotropies, including
possible effects due to parity violation, to features in the frequency
dependence of the isotropic part of the spectrum, as well as to the presence
of extra scalar and vector polarizations. For the first time, we show how the
sensitivity to different effects crucially depends on the pulsar configuration
with respect to the relative motion among frames. Correspondingly, we propose
examples of strategies of detection, each aimed at exploiting future
measurements of kinematic anisotropies for characterizing distinct features of
the cosmological gravitational wave background.

##  1 Introduction

Four pulsar timing array (PTA) collaborations [1, 2, 3, 4] recently detected a
signal compatible with a stochastic gravitational wave background (SGWB). At
the moment, various open questions remain on the origin of the gravitational
wave (GW) signal, since diverse sources, from astrophysical to cosmological,
are compatible with current observations (see e.g. [5] for a recent multi-
model assessment). Properties of the SGWB can in principle allow us to
distinguish astrophysical from cosmological sources of GW: parity violation
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], non-Gaussianities (see e.g.
[19, 20, 21, 22]), or anisotropies (see [23] for a recent survey). If the SGWB
has cosmological origin, besides its intrinsic anisotropies, the signal is
expected to be characterised by a Doppler anisotropy due to our relative
motion with respect to the source rest frame. As found by cosmic microwave
background (CMB) experiments [24, 25, 26, 27], our velocity with respect to
the cosmic rest frame has an amplitude of size $\beta\,=\,v/c\,=\,1.23\times
10^{-3}$ with respect to the velocity of light, and points in the direction
$(l,b)\,=\,(264^{\rm o},48^{\rm o})$. Correspondigly, the size of the dipolar
kinematic anisotropy of a cosmological SGWB should be one-thousandth smaller
than the amplitude of the isotropic part of the GW spectrum. This is smaller
in size than the expected anisotropies of the astrophysical SGWB in the PTA
band (see e.g. [28, 29, 30, 31, 32] and in particular [33] for a recent
study). But, as for the CMB, it is potentially well larger than intrinsic
cosmological anisotropies (see, e.g., the studies [34, 35, 36, 37, 38, 39,
40]). Kinematic anisotropies is a topic of active research in the context of
GW interferometers [23, 41, 42, 43, 44, 45]. It is then worth asking what
information we can obtain from a possible future detection of kinematic
anisotropies with PTA experiments.

This is the scope of this work. The development and refinement of detection
techniques of anisotropies of SGWB from PTA observations have a long history –
see e.g. [46, 47, 48, 49, 50, 51, 52, 53] – often borrowing and elaborating
techniques developed in the context of ground-based [54, 55, 56, 57, 58] and
space-based [59, 60, 61, 62, 63] interferometers. Current PTA measurements do
not find yet indications of SGWB anisotropies [64, 65], but as data are
becoming more and more accurate, a detection might be forthcoming. Since in
this work we specialize to kinematic anisotropies, we express our findings in
the most convenient way to extract information from a detection of Doppler
effects in the SGWB. We stress the geometrical aspects of the problem, and we
make manifest how the sensitivity to different GW effects depends on the
position of pulsars with respect to the relative velocity vector among frames.
These findings – of which many are new while others are known but we set them
in a new perspective – can be useful to plan future PTA observations. In fact,
as we can plan the location of ground based interferometers in the Earth in
order to extract most physics from observations (for example, see the recent
study [66] in the context of the Einstein Telescope), we can also plan which
pulsars are worth monitoring to learn new physics from GW measurements. This
is an invaluable opportunity for forthcoming observations with SKA facilities,
which will monitor and detect signals from a large number of pulsars [67] with
important opportunities for GW physics: see e.g. [68]. The physical effects we
explore using SGWB kinematic anisotropies in PTA band are:

  * $\triangleright$

Parity violation: Well motivated early-universe scenarios predict the
existence of parity-violating effects in gravitational and gauge-field
interactions, with interesting cosmological and GW consequences, see e.g. [6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. These effects manifest as
circular polarization of the SGWB. This quantity is hard or even impossible to
detect with single interferometers, when focussing on the isotropic part of
the background only: see e.g. the clear analysis of [69]. Opportunities of
detection arise when cross correlating different experiments, see e.g. [70,
71, 72], or by probing anisotropies of the SGWB: [73, 74, 75, 76]. For the
case of PTA measurements, [77] shown for the first time that one needs to
measure SGWB anisotropies for being sensitive to circular polarization (see
also [78]). In our work, we show that kinematic anisotropies are in fact
potentially able to probe circular polarization with PTA detections, and the
PTA sensitivity depends on the pulsar location with respect to the relative
motion along frames. Hence, a careful plan of the pulsars to monitor is needed
for increasing opportunities to detection (see sections 2, 3, and 5).

  * $\triangleright$

Frequency features in GW spectrum: Kinematic anisotropies are obtained from
Doppler boosting the isotropic part of the SGWB: they depend on derivatives
along frequency of this quantity [41]. Hence, Doppler effects can be a
complementary probe of the frequency dependence of the SGWB, alternative to
more direct methods (see e.g. [79]). Kinematic anisotropies are more
pronounced if the SGWB has features in frequency: suitable combinations of PTA
anisotropy measurements allow us to specifically detect the slope of the
background (see sections 3, and 5).

  * $\triangleright$

Presence of extra scalar and vector polarizations: Alternative theories of
gravity predict the existence of additional scalar and vector polarizations
with respect to the spin-2 ones of General Relativity (see e.g. [80] for a
general discussion, and [81] for a comprehensive review). Previous works
investigated how PTA measurements can probe the GW polarization content, see
e.g. [82, 83, 84, 85, 86, 87, 88]. Here we point out that kinematic
anisotropies, which depend on the slope of the spectrum, can be sensitive to
extra polarizations also when the latter give a small contribution to the
isotropic part of the spectrum (see sections 4, and 5).

We discuss the motivations and theoretical formulation of these topics in
sections 2, 4 and 3, while in section 5 we elaborate strategies of detection
of the effects listed above. Two Appendixes develop technical tools needed in
the main text.

##  2 Set-up

In this section we investigate the response of a PTA experiment to an
anisotropic SGWB characterized by the GW intensity and GW circular
polarization. We will start appreciating how the PTA response to GW depend on
the pulsar configuration. In the next sections, the resulting formulas will
then be applied to the specific case of kinematic anisotropies.

The GW is expressed in terms of fluctuations of the Minkowski metric

$ds^{2}\,=\,-dt^{2}+\left(\delta_{ij}+h_{ij}(t,\vec{x})\right)\,dx^{i}dx^{j}\,.$
(2.1)

We decompose the GW in Fourier modes as

$h_{ij}(t,\vec{x})\,=\,\sum_{\lambda}\,\int_{-\infty}^{+\infty}df\,\int
d^{2}\hat{n}\,{e}^{-2\pi if\,\hat{n}\vec{x}}\,e^{2\pi ift}\,{\bf
e}_{ij}^{\lambda}(\hat{n})\,h_{\lambda}(f,\hat{n})\,,$ (2.2)

imposing the condition

$h_{\lambda}(-f,\hat{n})\,=\,h^{*}_{\lambda}(f,\hat{n})$ (2.3)

which ensures that $h_{ij}(t,\vec{x})$ is real. We also assume that the
polarization tensors ${\bf e}_{ij}^{\lambda}$ are real quantities. See
Appendix A for more details on our conventions. The presence of a GW deforms
the geodesics of light, and produces a time delay $\Delta T_{a}(t)$ on the
period of a pulsar. We denote with $\tau_{a}$ the time travelled from a pulsar
to the Earth, setting from now on $c=1$. The pulsar is situated at the
position (from now on, hat quantities correspond to unit vectors)

$\vec{x}_{a}\,=\,\tau_{a}\,\hat{x}_{a}$ (2.4)

with respect to the Earth, located at $\vec{x}\,=\,0$. The direction of the
vector $\hat{x}_{a}$ of the pulsar with respect to the Earth plays an
important role for our arguments.

The time delay of the light geodesics reads

$\displaystyle z_{a}(t)$ $\displaystyle=$ $\displaystyle\frac{\Delta
T_{a}(t)}{T_{a}(t)}\,=\,\int_{-\infty}^{+\infty}df\,e^{2\pi ift}\,z_{a}(f)$
(2.5) $\displaystyle=$ $\displaystyle\int_{-\infty}^{+\infty}df\,e^{2\pi
ift}\,\left(\sum_{\lambda}\,\int d^{2}\hat{n}\,D_{a}^{ij}(\hat{n})\,{\bf
e}_{ij}^{\lambda}(\hat{n})\,h_{\lambda}(f,\hat{n})\right)\,.$ (2.6)

$D^{ij}_{a}$ is the so-called detector tensor, controlling the connection
between the light delay 111 In writing eq (2.6), we neglect as usual the
‘pulsar terms’ and focus on the ‘Earth terms’. We refer the reader to Chapter
23 of [89] and to the recent [90] for textbook discussions on the quantities
we are introducing, their physical properties, and more general information on
how PTA respond to GW physics. and the GW:

$D^{ij}_{a}\equiv\frac{\hat{x}_{a}^{i}\,\hat{x}_{a}^{j}}{2(1+\hat{n}\cdot\hat{x}_{a})}\,.$
(2.7)

Starting from the time delay $z_{a}(t)$, it is convenient to compute the time
residual

$R_{a}(t)\,\equiv\,\int_{0}^{t}\,dt^{\prime}z_{a}(t^{\prime})\,,$ (2.8)

which is easier to handle when Fourier transforming the signal. We assume that
the correlators of Fourier modes of GW fluctuations can be expressed as

$\langle
h_{\lambda}(f,\hat{n})\,h^{*}_{\lambda^{\prime}}(f^{\prime},\hat{n}^{\prime})\rangle\,=\,\frac{1}{2}\,S_{\lambda\lambda^{\prime}}(f,\hat{n})\,\delta(f-f^{\prime})\,\frac{\delta^{(2)}(\hat{n}-\hat{n}^{\prime})}{4\pi}\,,$
(2.9)

where the tensor $S_{\lambda\lambda^{\prime}}(f,\hat{n})$ defines the
properties of the SGWB, and $\lambda$ is the polarization index (we adopt a
$(+,\times)$ basis for the polarization tensors, see Appendix A). The quantity
$S_{\lambda\lambda^{\prime}}(f,\hat{n})$ can be decomposed in intensity and
circular polarization as

$S_{\lambda\lambda^{\prime}}(f,\hat{n})\,=\,I(f,\hat{n})\delta_{\lambda\lambda^{\prime}}-iV(f,\hat{n})\,\epsilon_{\lambda\lambda^{\prime}}\,,$
(2.10)

where the $2\times 2$ tensor $\epsilon_{\lambda\lambda^{\prime}}$ is defined
as $\epsilon_{+\times}\,=\,1\,=\,-\epsilon_{\times+}$, while
$\epsilon_{++}\,=\,0\,=\,\epsilon_{\times\times}$. The SGWB intensity
$I(f,\hat{n})$ is real and positive, and the circular polarization
$V(f,\hat{n})$ is a real quantity. Both quantities can depend on the GW
frequency and direction, and behave as scalars under boosts. Our aim is to
compute the response of a PTA system to the presence of a SGWB, whose spectrum
is characterized by (possibly anisotropic) intensity $I$ and circular
polarization $V$ parameters.

In order to do so, we compute the correlation among the time residuals of a
pair of pulsars, denoted respectively by the letters $a$ and $b$, which is
induced by the presence of GW. The correlations among time-residuals is
essential for detecting and characterizing the SGWB [91]. We introduce the
short-hand notation

$D_{a}^{\lambda}(\hat{n})\,\equiv\,D_{a}^{ij}(\hat{n})\,\,{\bf
e}_{ij}^{\lambda}(\hat{n})\,,\hskip 28.45274pt;\hskip 28.45274pt\Delta
t_{12}\,=\,t_{1}-t_{2}\,,$ (2.11)

and compute the two-point correlators of the the two pulsar time delays (we
sum over repeated polarization indexes):

$\displaystyle\langle z_{a}(t_{1})z_{b}(t_{2})\rangle$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}df\,df^{\prime}\,\int
d^{2}\hat{n}\,d^{2}\hat{n}^{\prime}D_{a}^{\lambda}(\hat{n})D_{b}^{\lambda^{\prime}}(\hat{n}^{\prime})\,e^{2\pi
i\left(ft_{1}+f^{\prime}t_{2}\right)}\langle
h_{\lambda}(f,\hat{n})\,h_{\lambda^{\prime}}(f^{\prime},\hat{n}^{\prime})\rangle\,.$

Making appropriate changes of variable, $f\to-f$ and
$f^{\prime}\to-f^{\prime}$, we can use relation (2.3) to recast the previous
expression in a convenient form, which allows us to apply eq (2.9):

$\displaystyle\langle z_{a}(t_{1})z_{b}(t_{2})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\int
df\,df^{\prime}\,d^{2}\hat{n}\,d^{2}\hat{n}^{\prime}D_{a}^{\lambda}(\hat{n})D_{b}^{\lambda^{\prime}}(\hat{n}^{\prime})$
$\displaystyle\left(e^{2\pi i\left(ft_{1}-f^{\prime}t_{2}\right)}\langle
h_{\lambda}(f,\hat{n})\,h^{*}_{\lambda^{\prime}}(f^{\prime},\hat{n}^{\prime})\rangle+e^{-2\pi
i\left(ft_{1}-f^{\prime}t_{2}\right)}\langle
h^{*}_{\lambda}(f,\hat{n})\,h_{\lambda^{\prime}}(f^{\prime},\hat{n}^{\prime})\rangle\right)\,,$
$\displaystyle=$ $\displaystyle\frac{1}{4}\,\int
df\,d^{2}\hat{n}\,D_{a}^{\lambda}(\hat{n})D_{b}^{\lambda^{\prime}}(\hat{n})\left(e^{2\pi
if\Delta t_{12}}\,S_{\lambda\lambda^{\prime}}(f,\hat{n})+e^{-2\pi if\Delta
t_{12}}\,S^{*}_{\lambda\lambda^{\prime}}(f,\hat{n})\right)\,,$
$\displaystyle=$ $\displaystyle\frac{1}{4}\,\int
df\,d^{2}\hat{n}\,D_{a}^{\lambda}(\hat{n})D_{b}^{\lambda^{\prime}}(\hat{n})$
$\displaystyle\left[\cos{(2\pi f\Delta
t_{12})}\,\left(S_{\lambda\lambda^{\prime}}(f,\hat{n})+S^{*}_{\lambda\lambda^{\prime}}(f,\hat{n})\right)+i\sin{(2\pi
f\Delta
t_{12})}\,\left(S_{\lambda\lambda^{\prime}}(f,\hat{n})-S^{*}_{\lambda\lambda^{\prime}}(f,\hat{n})\right)\right]\,,$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int{df}\,d^{2}\hat{n}\,D_{a}^{\lambda}(\hat{n})D_{b}^{\lambda^{\prime}}(\hat{n})\left[\cos{(2\pi
f\Delta t_{12})}\,I(f,\hat{n})\delta_{\lambda\lambda^{\prime}}+\sin{(2\pi
f\Delta t_{12})}\,V(f,\hat{n})\epsilon_{\lambda\lambda^{\prime}}\right]\,.$

The previous result can be plugged in eq (2.8) to compute the two-point
function of time residuals:

$\displaystyle\langle R_{a}(t_{A})R_{b}(t_{B})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{t_{A}}\int_{0}^{t_{B}}\,dt_{1}dt_{2}\,\int
df\,d^{2}\hat{n}$ (2.14) $\displaystyle
D_{a}^{\lambda}(\hat{n})D_{b}^{\lambda^{\prime}}(\hat{n})\,\Big{[}\cos{\left(2\pi
f\Delta
t_{12}\right)}\,I(f,\hat{n})\delta_{\lambda\lambda^{\prime}}+\sin{\left(2\pi
f\Delta t_{12}\right)}\,V(f,\hat{n})\epsilon_{\lambda\lambda^{\prime}}\Big{]}$
$\displaystyle=$ $\displaystyle\int\frac{df\,\sin{\left(\pi
ft_{A}\right)}\sin{\left(\pi ft_{B}\right)}}{\pi\,f^{2}}$
$\displaystyle\times\left[\bar{I}(f)\,\Gamma^{I}_{ab}(f)\,\cos{\left(2\pi
f\Delta t_{AB}\right)}\,+\bar{V}(f)\,\Gamma^{V}_{ab}(f)\,\sin{\left(2\pi
f\Delta t_{AB}\right)}\right]\,,$

where $\bar{I}(f)$ is the isotropic value of the intensity integrated over all
directions, while $\bar{V}(f)$ is its analog for circular polarization.

The response functions of a pulsar pair to the GW intensity and circular
polarization are obtained integrating over $\hat{n}$, and read

$\displaystyle\Gamma^{I}_{ab}(f)$ $\displaystyle=$
$\displaystyle\frac{1}{{2\pi\,\bar{I}(f)}}\int
d^{2}\hat{n}\left(D_{a}^{\lambda}(\hat{n})D_{b}^{\lambda^{\prime}}(\hat{n})\,\delta_{\lambda\lambda^{\prime}}\right)\,{I(f,\hat{n})}\,,$
(2.15) $\displaystyle\Gamma^{V}_{ab}(f)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\,{\bar{V}(f)}}\int
d^{2}\hat{n}\left(D_{a}^{\lambda}(\hat{n})D_{b}^{\lambda^{\prime}}(\hat{n})\,\epsilon_{\lambda\lambda^{\prime}}\right)\,{V(f,\hat{n})}\,.$
(2.16)

These quantities depend on the relative position of the pulsars in the sky
(given the dependence of the quantities $D_{a,b}^{\lambda}$ on the pulsar
location) and on the properties of the SGWB. Hence, the response of a PTA
system to GW depends on the pulsar configuration. Notice that only correlators
(2.14) evaluated at different times $t_{A}$ and $t_{B}$ are sensitive to
circular polarization. In actual measurements though, correlators are weighted
by suitable filters to better extract the signal. We discuss in Appendix B and
section 5 the relation between correlators as above and measurable GW signals,
using a a match-filtering technique.

Our task 222While in this section and the next we focus on spin-2 GW modes,
the same procedure can be applied to study alternative gravity models in which
gravitation is mediated by a mixture of spin-2 and spin-0 or spin-1 fields. We
examine this possibility in section 4. is to compute $\Gamma^{I}_{ab}(f)$ and
$\Gamma^{V}_{ab}(f)$. The results depend also on the theory of gravity one
considers. For the case GW are carried by spin-2 fields, as in General
Relativity, we can make use of eqs (A.14) and (A.15). The quantities within
parenthesis 333Formulas similar to (2.17) have been used in [52, 53] to
characterize scenarios with anisotropic GW intensity. See also the treatment
in [46]. As far as we are aware, our formulas for circular polarization are
instead new in this context. in eqs (2.15) and (2.16) result:

$\displaystyle
D_{a}^{\lambda}(\hat{n})D_{b}^{\lambda^{\prime}}(\hat{n})\,\delta_{\lambda\lambda^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{(\hat{x}_{a}\cdot\hat{n})^{2}+(\hat{x}_{b}\cdot\hat{n})^{2}+(\hat{x}_{a}\cdot\hat{n})^{2}(\hat{x}_{b}\cdot\hat{n})^{2}-1}{8(1+\hat{x}_{a}\cdot\hat{n})(1+\hat{x}_{b}\cdot\hat{n})}$
(2.17) $\displaystyle+$
$\displaystyle\frac{(\hat{x}_{a}\cdot\hat{x}_{b})^{2}-2(\hat{x}_{a}\cdot\hat{x}_{b})(\hat{x}_{a}\cdot\hat{n})(\hat{x}_{b}\cdot\hat{n})}{4(1+\hat{x}_{a}\cdot\hat{n})(1+\hat{x}_{b}\cdot\hat{n})}\,,$

and

$\displaystyle
D_{a}^{\lambda}(\hat{n})D_{b}^{\lambda^{\prime}}(\hat{n})\,\epsilon_{\lambda\lambda^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{\left[\hat{x}_{a}\cdot\hat{x}_{b}-(\hat{x}_{a}\cdot\hat{n})(\hat{x}_{b}\cdot\hat{n})\right]\left[\hat{n}\cdot(\hat{x}_{a}\times\hat{x}_{b})\right]}{{4(1+\hat{x}_{a}\cdot\hat{n})(1+\hat{x}_{b}\cdot\hat{n})}}\,.$
(2.18)

where $\times$ denotes cross product among vectors. Plugging these results in
eqs (2.15), (2.16), we are left with angular integrals to carry on, which
depend on $I(f,\hat{n})$ and $V(f,\hat{n})$. The computation gives quantities
depending on the GW frequency. We can already notice that, being the quantity
in eq (2.18) an odd function of the vector directions, the integral over all
directions give zero, unless the circular polarization function $V(f,\hat{n})$
depend explicitly on the direction $\hat{n}$. Hence we need an anisotropic
signal to measure circular polarization [77, 78], completely analogously to
what happens for planar interferometers (see e.g. [69]).

Starting from these basic formulas, in the next section we apply them to the
specific case of spin-2 GW (including circular polarization), and study
anisotropies induced by the motion of our reference frame with respect to the
rest frame of the SGWB source.

##  3 PTA response to kinematic anisotropies: the spin-2 case

In the previous section we developed a general treatment to investigate the
PTA response to anisotropic GW signals. We now specialise our attention to the
case of Doppler anisotropies, in theories as General Relativity where GW are
characterized by spin-2 polarizations. Nevertheless, we include also possible
effects of parity violation [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18].
The geometrical dependence of the PTA response to kinematic anisotropies make
them a promising probe of the source of GW and of the physics of gravitation.

In fact, kinematic anisotropies are a guaranteed property of a SGWB of
primordial origin: their features are fully calculable, being determined by
the properties of the isotropic part of the SGWB. If the SGWB has a
cosmological origin related with early-universe physics, we can expect that
the relative motion among frames induces an effect analog to the large dipolar
anisotropy measured in the CMB [24, 25, 26, 27], whose amplitude is a factor
$1.2\times 10^{-3}$ times smaller than the isotropic background. For the case
of the CMB, the dipolar kinematic anisotropy is well larger in size than its
intrinsic anisotropies. The same can occur for a cosmological SGWB detectable
with PTA. For this reason it is worth characterizing kinematic effects, and
explore what information we can extract from their detection with PTA
experiments. We now show how the pulsar response to Doppler anisotropies
depends on the pulsar location in the sky; on the frequency dependence of the
isotropic part of the SGWB; and on the presence of parity violating effects in
the SGWB as considered in various early universe scenarios.

The SGWB intensity $I(f,\hat{n})$ and circular polarization $V(f,\hat{n})$
(see eq (2.10)) are scalars under boosts 444One can also study the response of
the GW experiment to kinematic anisotropies in other quantities, like the GW
energy density $\Omega_{\rm GW}$ [41].. Under a boost chararacterized by
velocity

$\vec{v}\,=\,\beta\hat{v}$ (3.1)

(with $\beta=v/c=v$ in our units with $c=1$) the GW frequency scales as [41]

$f\to{\cal D}^{-1}\,f\,,$ (3.2)

with

${\cal D}\,=\,\frac{\sqrt{1-\beta^{2}}}{1-\beta\,\hat{n}\cdot\hat{v}}\,.$
(3.3)

Hence, assuming that the GW intensity and circular polarization are isotropic
in the source rest frame, they develop kinematic anisotropies in a frame (like
ours) moving with velocity $\vec{v}$ with respect to the source.

The quantities entering in the integrands of eqs (2.15), (2.16) become

$\displaystyle\frac{I(f,\hat{n})}{\bar{I}(f)}\,=\,\frac{I({\cal
D}^{-1}\,f)}{\bar{I}(f)}$ $\displaystyle=$
$\displaystyle\left[1+\frac{\beta^{2}}{6}\left(\alpha_{I}+n_{I}^{2}+2n_{I}\right)\right]$
(3.4) $\displaystyle-\beta
n_{I}\,\hat{n}\cdot\hat{v}+\frac{\beta^{2}}{2}\left[(\hat{n}\cdot\hat{v})^{2}-\frac{1}{3}\right]\left(\alpha_{I}+n_{I}^{2}-n_{I}\right)\,,$
$\displaystyle\frac{V(f,\hat{n})}{\bar{V}(f)}\,=\,\frac{V({\cal
D}^{-1}\,f)}{\bar{V}(f)}$ $\displaystyle=$
$\displaystyle\left[1+\frac{\beta^{2}}{6}\left(\alpha_{V}+n_{V}^{2}+2n_{V}\right)\right]$
(3.5) $\displaystyle-\beta
n_{V}\,\hat{n}\cdot\hat{v}+\frac{\beta^{2}}{2}\left[(\hat{n}\cdot\hat{v})^{2}-\frac{1}{3}\right]\left(\alpha_{V}+n_{V}^{2}-n_{V}\right)\,,$

where, as in [41], we introduced parameters controlling the tilt of the
isotropic background:

$\displaystyle n_{I}(f)$ $\displaystyle=$
$\displaystyle\frac{d\,\ln\bar{I}(f)}{d\,\ln f}\hskip 22.76228pt,\hskip
14.22636pt\alpha_{I}(f)\,=\,\frac{d\,n_{I}(f)}{d\,\ln f}\,,$ (3.6)
$\displaystyle n_{V}(f)$ $\displaystyle=$
$\displaystyle\frac{d\,\ln\bar{V}(f)}{\,d\,\ln f}\hskip 14.22636pt,\hskip
14.22636pt\alpha_{V}(f)\,=\,\frac{d\,n_{V}(f)}{d\,\ln f}\,,$ (3.7)

and the isotropic bar quantities $\bar{I}(f)$, $\bar{V}(f)$ are defined after
eq (2.14). In writing equations (3.4) and (3.5) we expand the definition of
${\cal D}$ of eq (3.3) up to second order in the expansion parameter $\beta$,
and we assemble the results in a way that makes manifest how kinematic effects
give rise to dipolar and quadrupolar anisotropies 555They also give
contributions of order $\beta^{2}$ to the monopole, a small effect that we
neglect from now on., as controlled by the size of $\beta$. As suggested by
CMB results, we expect $\beta$ to be of order $10^{-3}$, making the detection
of kinematic anisotropies a demanding (but certainly interesting!) challenge
for PTA experiments.

Since $I$ and $V$ are scalars under boosts, kinematic anisotropies depend on
derivatives of these quantities, associated with slope parameters $n$ and
$\alpha$. Doppler effects are a probe of the frequency dependence of
correlators, a property that can be useful in a variety of situations (see
section 5).

From now on, in this section we consider two pulsars $a,b$ are located at
positions $\vec{x}_{a}\,=\,\tau_{a}\,\hat{x}_{a}$ and
$\vec{x}_{b}\,=\,\tau_{b}\,\hat{x}_{b}$ with respect to the Earth located at
the origin. The corresponding pulsar response functions $\Gamma_{ab}^{I,V}$
are analytically calculated plugging the expressions (3.4) and (3.5) into eqs
(2.15) and (2.16), and performing the angular integrals 666To carry on the
integrals we found convenient to make use of complex integration methods and
Cauchy theorem, as explained in detail in [92].. The results are easier to
handle introducing the combination

${y}_{ab}\,=\,\frac{1-\hat{x}_{a}\hat{x}_{b}}{2}\,=\,\frac{1-\cos{\zeta}}{2}\,,$
(3.8)

which depends on the relative angle
$\hat{x}_{a}\cdot\hat{x}_{b}\,=\,\cos{\zeta}$ of the pulsar positional vectors
with respect to the Earth (see eq (2.4)). We now analytically investigate how
the integrals depend on the pulsar positions with respect to the velocity
vector $\hat{v}$. We find exact results with a transparent geometrical
interpretation, which turns useful in developing strategies of detection in
section 5.

###  3.1 Pulsar response to GW intensity

The response function (2.15) to the GW intensity, expanded up to order
$\beta^{2}$, results

$\displaystyle\Gamma^{I}_{ab}$ $\displaystyle=$ $\displaystyle\Gamma_{ab}^{\rm
HD}+{\beta\,n_{I}}F_{ab}^{(1)}+{\beta^{2}}\left(\alpha_{I}+n_{I}^{2}-n_{I}\right)\,F^{(2)}_{ab}\,,$
(3.9)

with

$\displaystyle\Gamma_{ab}^{\rm HD}$ $\displaystyle=$
$\displaystyle\frac{1}{3}-\frac{y_{ab}}{6}+y_{ab}\ln y_{ab}$ (3.10)
$\displaystyle F^{(1)}_{ab}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{12}+\frac{y_{ab}}{2}+\frac{y_{ab}\ln
y_{ab}}{2(1-y_{ab})}\right)\,\left[\hat{v}\cdot\hat{x}_{a}+\hat{v}\cdot\hat{x}_{b}\right]\,,$
(3.11) $\displaystyle F^{(2)}_{ab}$ $\displaystyle=$
$\displaystyle\left(\frac{3-13y_{ab}}{20(y_{ab}-1)}+\frac{y_{ab}^{2}\ln
y_{ab}}{2(1-y_{ab})^{2}}\right)\,\left[(\hat{v}\cdot\hat{x}_{a})(\hat{v}\cdot\hat{x}_{b})\right]$
(3.12) $\displaystyle+$
$\displaystyle\left(\frac{1+2y_{ab}-4y_{ab}^{2}+y_{ab}^{3}+3y_{ab}\ln
y_{ab}}{12(1-y_{ab})^{2}}\right)\,\left[(\hat{v}\cdot\hat{x}_{a})^{2}+(\hat{v}\cdot\hat{x}_{b})^{2}\right]\,.$

The response function (3.9) includes a first part $\Gamma_{ab}^{\rm HD}$
(independent from $\beta$) corresponding to the classic Hellings Downs curve
[91] which we collect in eq (3.10). A second part (first determined in [46])
is weighted by the relative velocity $\beta$, and controls the dipolar
kinematic anisotropy. As anticipated above, it is proportional to the slope
$n_{I}(f)$ of the SGWB profile, as defined in eq (3.6), making Doppler effects
a probe of features of the SGWB frequency spectrum. Considering its
geometrical properties, we notice that eq (3.11) is symmetric under the
interchange of pulsar positions $a\leftrightarrow b$, and it vanishes if the
pulsar directions are orthogonal to the direction of the relative velocity
$\hat{v}$ among frames. In fact, differently from the Hellings Downs case, the
pulsar response to the kinematic dipole depends on the relative angle between
pulsars, but also on the angles made by the pulsar directions with the vector
controlling the velocity among frames [46, 47]. We find that only pulsars
whose direction vectors $\hat{x}$ have components along $\hat{v}$ respond to
kinematic anisotropies relative to the GW intensity $I$.

A third part of equation (3.9), proportional to $\beta^{2}$, controls the
kinematic quadrupole, and is suppressed by a factor $\beta^{2}$ with respect
to the isotropic background. The quadrupole can be enhanced if the frequency
dependence of the spectrum has features, which lead to a large parameter
$\alpha_{I}$ defined in eq (3.6) and entering in eq (3.9) (more on this
later). Also the quadrupolar contribution to the response to the GW intensity
vanishes when the pulsar directions are orthogonal to $\hat{v}$. However, it
depends on a different way on the angles among pulsars directions and
$\hat{v}$: compare eqs (3.11) with (3.12).

###  3.2 Pulsar response to GW circular polarization

The pulsar response function for a circularly polarized GW signal reads

$\displaystyle\Gamma^{V}_{ab}$ $\displaystyle=$
$\displaystyle{\beta\,n_{V}}\,G^{(1)}_{ab}+{{\beta^{2}}\left(\alpha_{V}+n_{V}^{2}-n_{V}\right)}G^{(2)}_{ab}\,,$
(3.13)

with

$\displaystyle G^{(1)}_{ab}$ $\displaystyle=$
$\displaystyle-\left(\frac{1}{3}+\frac{y_{ab}\ln
y_{ab}}{4(1-y_{ab})}\right)\,\left[\hat{v}\cdot(\hat{x}_{a}\times\hat{x}_{b})\right]$
$\displaystyle G^{(2)}_{ab}$ $\displaystyle=$
$\displaystyle\left(\frac{2y_{ab}^{2}-1-y_{ab}-3y_{ab}\ln
y_{ab}}{12(1-y_{ab})^{2}}\right)\,\left[\left(\hat{v}\cdot\hat{x}_{a}+\hat{v}\cdot\hat{x}_{b}\right)\left(\hat{v}\cdot(\hat{x}_{a}\times\hat{x}_{b})\right)\right]$
(3.14)

Figure 1:  We represent those contributions to PTA response functions to GW
intensity and circular polarization, which depend on the angle $\zeta$ among
pulsar directions. Namely, we plot the quantities within round parenthesis of
eqs (3.12) and (3.14). In black, the Hellings-Downs curve. Left panel: dashed
blue: dipole response to GW intensity; dot-dashed red: dipole response to GW
circular polarization. Right panel: dashed blue: quadrupole response to GW
intensity (the part proportional to $(\hat{v}\cdot
p_{\alpha})^{2}+(\hat{v}\cdot p_{\beta})^{2}$); dot-dashed red: quadrupole
response to GW circular polarization.

The response function (3.13) starts at order $\beta$: the PTA response to
circular polarization vanishes for an isotropic background, and we need to
include SGWB anisotropies for being sensitive to this quantity [77]. The
geometrical reason being an integration over all directions, which vanishes
for parity-odd isotropic signals associated with circular polarization (see
comment after eq (2.18)). This is similar in spirit to what happens for planar
interferometers as LISA, for which we need to probe anisotropies in order to
measure effects due to parity violation [73, 74, 75, 76].

Also in this case, kinematic anisotropies are proportional to parameters
controlling the frequency slope of the circular polarization function. The
expression (3.13) demonstrates that the PTA pair can detect circular
polarization only if $\hat{v}$ has components orthogonal to the plane formed
by the pulsar directions. This feature is opposite to what found above when
discussing pulsar response to GW intensity. This property – which we point out
for the first time – can then be useful for experimentally distinguish the two
contributions: see section 5.

It is not straightforward to represent in a plot the rich, multi-parameter
dependence of the response functions on the geometrical configuration of the
system. For definiteness, in Fig 1, left panel, we represent the pulsar
angular response to the kinematic dipole, both for intensity and circular
polarization, as a function of the angle between pulsar directions. In the
right panel, we represent the response functions to the kinematic quadrupole.
We plot only the quantities between round parenthesis in eqs (3.12) and
(3.14), without the geometrical factors depending on the relative velocity
$\hat{v}$ among frames. A more detailed analysis of the geometrical dependence
on the various quantities is postponed to section 5. For the moment, it is
sufficient to notice that the dependence on the pulsar separation of the
functions $F^{(i)}_{ab}$ and $G^{(i)}_{ab}$ is qualitatively different from
the Hellings-Downs curve, which we also represent for comparison.

###  3.3 Examples of SGWB frequency dependence

Besides the geometrical dependence of the results on the location of pulsars,
the PTA response to Doppler effects depends on the frequency dependence of the
isotropic part of the SGWB. In fact, the size and observational impact of
kinematic anisotropies can be enhanced if the SGWB is characterised by large
values for the slope parameters $n$ and $\alpha$ of eqs (3.6), at least for
certain ranges of frequencies. While current PTA data are compatible with a
power-law profile of the isotropic SGWB as function of frequency, in the
future more accurate data might favour other scenarios. For example, broken
power law profiles, well motivated by a variety of SGWB sources [93], have
also been considered in [1] for explaining the most recent data. The same is
true for more complex SGWB profiles motivated by primordial black hole
physics: see e.g. the first analysis carried out in [1, 94, 95, 96, 97] (and
also [98] for a recent review on inflationary primordial black hole models).

We briefly outline the consequences of these two scenarios for the physics of
kinematic anisotropies:

Figure 2:  Examples of intensity spectrum profiles, and combinations of slope
parameters characterizing kinematic anisotropies. In all cases,
$\beta=1.2\times 10^{-3}$. Left panel: broken power law profile of eq (3.15).
We choose $I_{0}=0.1$, $\sigma=100$, $\gamma=3$, and $\delta=8$. Black line:
the intensity. Dashed red line: the combination $\beta|n_{I}|$ characterizing
the amplitude of the dipolar anisotropy. Dot-dashed blue line: the combination
$\beta^{2}|\alpha_{I}+n_{I}^{2}-n_{I}|$, characterizing the amplitude of the
quadrupolar anisotropy. Notice that in proximity of the break of the power law
the amplitude of the quadrupolar anisotropy is comparable to the dipolar one.
Right panel: Ansatz (3.17) motivated by the physics of primordial black holes.
Color codes as above. In proximity of dips of intensity, the size of kinematic
anisotropies become large.

Broken power law: Consider the following broken power law Ansatz for the
intensity profile [99, 100]

$\displaystyle\bar{I}(f)\,=\,I_{0}\,\kappa^{\gamma}\,\left[1+\kappa^{\sigma}\right]^{-\frac{\gamma+\delta}{\sigma}}\,,$
(3.15)

where we denote $\kappa=f/f_{\star}$, and $I_{0}$, $\gamma$, $\delta$,
$\sigma$ are positive dimensionless parameters, while $f_{\star}$ a pivot
frequency within the PTA detection band. The associated tilt parameters result

$\displaystyle n_{I}$ $\displaystyle=$
$\displaystyle\frac{\gamma-\delta\,\kappa^{\sigma}}{1+\kappa^{\sigma}}\,,$
$\displaystyle\alpha_{I}$ $\displaystyle=$
$\displaystyle-\frac{\sigma(\gamma+\delta)\,\kappa^{\sigma}}{\left(1+\kappa^{\sigma}\right)^{2}}\,.$
(3.16)

The resulting slope $n_{I}$ of the intensity spectrum is equal to the
parameter $\gamma$ at small frequencies, and to $-\delta$ at larger
frequencies. For a large values of $\sigma$, at intermediate frequencies the
quantity $\alpha_{I}$ can increase by several orders of magnitude, enhancing
the quadrupolar kinematic anisotropies to a size comparable to the dipolar
ones (see [101] for related effects explored for the case of interferometers).
We represent in Fig 2, left panel, an example to visually demonstrate such
behaviour for the slope parameters.

Second-order GW from primordial black hole formation: Primordial black holes
can form in the early universe. Their formation requires to amplify the
spectrum of scalar fluctuations for a range of scales around a pivot frequency
$f_{\star}$, which we can consider in the PTA frequency ranges. At second
order in perturbations, these mechanisms induce a SGWB, whose properties
depend on the source scalar spectrum [102, 103, 104, 105]. For a narrow scalar
spectrum whose width is much smaller than $f_{\star}$, we expect that the GW
energy density $\Omega_{\rm GW}(f)$ increases as $(f/f_{\star})^{3}$ for small
frequencies $f\ll f_{\star}$ [106]. Then its profile has a dip (typically at
scales around $f/f_{\star}\simeq\sqrt{2/3}$), followed by a pronounced
resonance. The frequency profile then drops its amplitude for $f>2f_{\star}$
(see the discussion in [104, 105]). Given that intensity is connected to the
GW energy density through the relation $f^{3}\,\bar{I}(f)\propto\Omega_{\rm
GW}(f)$, we can consider the following simple Ansatz for the intensity
parameter (we call again $\kappa=f/f_{\star}$)

$I(\kappa)\,=\,\frac{(\kappa^{2}-4)^{2}\,(3\kappa^{2}-2)^{2}}{64}\hskip
22.76228pt,\hskip 22.76228pt0\leq\kappa\leq 2\,,$ (3.17)

as representative of the behaviour described above. In fact, the intensity
$I(\kappa)\propto\Omega_{\rm GW}(\kappa)/\kappa^{3}$ is constant at small
frequencies $\kappa\ll 1$, for then developing the features described above.
Hence, in this context,

$\displaystyle n_{I}$ $\displaystyle=$
$\displaystyle\frac{8\,\kappa^{2}\,(3\kappa^{2}-7)}{3\kappa^{4}-14\kappa^{2}+8}\,,$
(3.18) $\displaystyle\alpha_{I}$ $\displaystyle=$
$\displaystyle-\frac{16\,\kappa^{2}\,(21\kappa^{4}-48\kappa^{2}+56)}{\left(3\kappa^{4}-14\kappa^{2}+8\right)^{2}}\,.$
(3.19)

We represent in Fig 2, right panel, the resulting intensity and the slope
parameter combinations characterizing kinematic anisotropies. In proximity of
the dip in intensity which preceeds the resonance, the size of the slope
parameters become so large that dipolar and quadrupolar anisotropies become
comparable in size.

##  4 Kinematic anisotropies and modified gravity

Theories of gravity alternative to General Relativity allow for scalar and
vector metric components to participate to gravitational interactions, and to
contribute to the GW signal in terms of extra GW polarizations [107]. The PTA
responses to scalar and vector polarizations have been derived in [82, 83, 84,
85, 86, 87, 88] for the case of isotropic backgrounds (see also the general
treatment in [81]). Such extra polarizations induce correlations among pulsar
signals distinct from the Hellings Downs curve. These alternative predictions
are not favoured by recent detections [1], that can be used to set upper
bounds on scalar and vector contributions to the isotropic backgrounds.
Nevertheless, in view of future opportunities to further improve constraints
(or maybe even detect signals of modified gravity), it is interesting to
develop this topic, and analyse how scalar and vector GW polarizations
contribute to kinematic anisotropies.

Analogously to the spin-2 case, the PTA response to kinematic anisotropies
associated with extra GW polarisations are sensitive to the slope of scalar
and vector spectra, as well as to the presence of parity violating effects in
the vector sector. Doppler effects are unique probes of these features, which
can increase the opportunities of detection. Here we derive for the first time
analytical formulas describing the PTA response functions to kinematic
anisotropies, associated with scalar and vector polarizations.

The procedure to follow for determining the PTA response to kinematic
anisotropies is identical to what done in sections 2 and 3 for the tensor case
– the only difference being that we utilize the corresponding scalar and
vector GW polarization tensors, whose properties are described in Appendix A.
In particular, the scalar and vector polarization tensors are used to perform
angular integrals corresponding to eqs (2.15) and (2.16).

Scalar contributions to GW signal are characterized only by scalar intensity,
and there is no circular polarization. We can assume that the total isotropic
part for the intensity $\bar{I}(f)$ of the SGWB is made of a tensor and a
scalar part, $\bar{I}(f)\,=\,\bar{I}_{\rm tn}(f)+\bar{I}_{\rm sc}(f)$,
possibly hierarchical in size as $\bar{I}_{\rm sc}\ll\bar{I}_{\rm ten}$. Using
the same notation of section 3, and introducing the slope parameters for each
of the intensity contributions

$\displaystyle n_{tn}$ $\displaystyle=$ $\displaystyle\frac{f\,d\bar{I}_{\rm
tn}(f)}{\bar{I}(f)\,df}\hskip 19.91684pt,\hskip
19.91684ptn_{sc}\,=\,\frac{f\,d\bar{I}_{\rm sc}(f)}{\bar{I}(f)\,df}$ (4.1)
$\displaystyle\alpha_{tn}$ $\displaystyle=$ $\displaystyle\frac{f\,dn_{\rm
tn}(f)}{df}\hskip 19.91684pt,\hskip 19.91684pt\alpha_{sc}\,=\,\frac{f\,dn_{\rm
sc}(f)}{df}$ (4.2)

we find that the PTA response function to scalar part of the intensity is

$\Gamma_{ab}^{\rm sc}\,=\,\frac{1}{3}-\frac{y_{ab}}{6}+\beta\,\frac{n_{\rm
sc}}{12}\,\left[\hat{v}\cdot\hat{x}_{a}+\hat{v}\cdot\hat{x}_{b}\right]+\frac{\beta^{2}}{60}\left(\alpha_{\rm
sc}+n_{\rm sc}^{2}-n_{\rm
sc}\right)\,\,\left[(\hat{v}\cdot\hat{x}_{a})(\hat{v}\cdot\hat{x}_{b})\right]\,,$
(4.3)

where, beside the isotropic part, we consider the dipolar (proportional to
$\beta$) and quadrupolar (proportional to $\beta^{2}$) kinematic anisotropy
contributions. (Instead, the tensor part is identical to what discussed in
section 3.) Interestingly – and differently from the tensor case – the PTA
response to scalar kinematic anisotropies is independent from the angle
$\hat{x}_{a}\cdot\hat{x}_{b}$ among pulsars, while it depends on the angles
among the pulsar directions and the frame velocity vector $\hat{v}$. See Fig
3, left panel. Moreover, it is proportional to $n_{\rm sc}$, making it a
unique probe of the slope of the scalar contribution to the total intensity
(more on this in section 5).

Vector-tensor non-minimally coupled theories of gravity have also been
introduced in view of applications to dark energy and black hole physics (see
e.g. [108, 109, 110, 111]). The correlators among vector Fourier modes can be
decomposed into intensity and circular polarization, analogously to what done
in the tensor case in section 2. Using formulas (A.21) and (A.22), it is
straightforward to compute the PTA response to dipolar anisotropies of the
vector intensity and vector circular polarization, finding (the slope
parameters are defined analogously as above)

$\displaystyle\Gamma_{ab}^{\rm vector}$ $\displaystyle=$
$\displaystyle-\frac{7}{3}+\frac{8y_{ab}}{3}-\ln y_{ab}$ $\displaystyle+$
$\displaystyle{n_{\rm vec}\,\beta}\left\\{\left(y_{ab}-\frac{4}{3}-\frac{\ln
y_{ab}}{2(1-y_{ab})}\right)\left[\hat{v}\cdot\hat{x}_{a}+\hat{v}\cdot\hat{x}_{b}\right]+\left(-\frac{2}{3}-\frac{\ln
y_{ab}}{2(1-y_{ab})}\right)\left[\hat{v}\cdot(\hat{x}_{a}\times\hat{x}_{b})\right]\right\\}\,.$

Figure 3:  We represent part of the PTA response function to GW intensity and
circular polarization in modified gravity, including kinematic anisotropies,
as a function of the angle $\zeta$ among pulsar directions. We understand the
overall factors depending on the slopes $n$ and $\alpha$, as well as the
geometrical factors depending on the velocity among frames. Left panel: Black
line: PTA response to an isotropic scalar GW component. Dashed blue: dipole
response to scalar GW intensity. Dot-dashed red: quadrupole response to scalar
GW intensity. Right panel: Black line: PTA response to an isotropic vector GW
component. Dashed blue: dipole response to vector GW intensity. Dot-dashed
red: dipole response to vector GW circular polarization. In the right plot, we
plot the quantities within round parenthesis of eq (LABEL:GammaVe1).

The response to vector GW components does depend on the angular separation
among pulsar directions. See Fig 3, right panel. It is very large for small
angular separation among pulsars, where contributions of pulsar terms should
nevertheless be included, see e.g. [82, 49]. In the second line of eq
(LABEL:GammaVe1) we find in the first the pulsar dipolar response to vector
intensity, that depends on the projection of the pulsar directions on the
velocity vector $\hat{v}$. The second term controls the the pulsar dipolar
response to vector circular polarization, depending on the projection of
$\hat{v}$ on a direction perpendicular to the plane of the two pulsar
directions. These features are similar to what found for the spin-2 case, see
section 3.

##  5 Strategies of detection

So far, we learned several interesting features of kinematic anisotropies,
which make them potential probes of specific properties of the SGWB:

  * •

The geometrical configuration of the pulsar network enters in the PTA response
functions, which depend in a distinctive way on intensity and circular
polarization of the SGWB. While the response to GW intensity is maximal for
pulsars whose positional vectors are in the direction of the Doppler velocity
vector $\hat{v}$, the response to GW circular polarization is enhanced for
pulsars lying on a plane orthogonal to $\hat{v}$. This distinct behaviour in
the two cases can be used for selecting PTA sets aiming at distinguishing and
independently measure the GW intensity and circular polarization.

  * •

The overall size of kinematic anisotropies can be amplified in scenarios with
a rich frequency dependence of the isotropic part of the background, see e.g.
section 3.3. This properties applies to GW backgrounds associated with
standard spin-2 polarizations, but possibly also containing GW spin-0 and
spin-1 contributions. Hence, a detection of kinematic anisotropies can give us
a complementary probe of the slope of GW polarization components, which is
difficult or even impossible to extract from the study of the isotropic part
of the background only.

We now make concrete applications of our findings, in order to develop
specific strategies of detection of different physical effects. In a sentence,
the main feature we can exploit is the following:

  * $\triangleright$

The presence of kinematic anisotropies implies that signal correlations among
pulsars do not follow the standard Hellings Downs angular distribution, but
have deviations that depend on SGWB properties. Measurements of these
deviations, that are deterministic and analytically calculable, can be used to
extract information about the GW signal.

As explained in the Introduction, we are especially interested in using
kinematic anisotropies as a probe of: 1. parity violation, 2\. the frequency
slope of correlation functions, 3\. the presence of extra polarizations. We
discuss these three topics in what follows.

###  5.1 Kinematic anisotropies and SGWB circular polarization

The effects of of parity violation in the SGWB kinematic anisotropies can be
cleanly extracted from PTA data, once we make an appropriate choice of pulsars
to monitor. In fact, it is not even necessary to make combinations among
different signals, as proposed in the first work [77] discussing the topic
(see also [78]). Choosing carefully the pulsar system, we can reduce sources
of statistical errors by focussing only on pulsars located at convenient
positions to reveal the presence of parity violation. Eqs (3.13) and (3.14)
teach us that PTA are sensitive to circular polarization only through the
anisotropies of the SGWB (as first pointed out in [77]), and moreover the
corresponding signal is enhanced when maximising the quantity
$\left[\hat{v}\cdot(\hat{x}_{a}\times\hat{x}_{b})\right]$. (The same is true
for parity violating effects in possible spin-1 polarization contributions,
see section 4.)

In Figure 4, left panel, we represent
$\hat{v}\cdot(\hat{x}_{a}\times\hat{x}_{b})$ as a function of $\zeta$, for a
random set of pulsar pairs whose positional vectors lie on a plane orthogonal
with the velocity vector $\hat{v}$. The size of the quantity
$\hat{v}\cdot(\hat{x}_{a}\times\hat{x}_{b})$ is maximal for intermediate
values of $\zeta\simeq\pi/2$, while it reduces at the extremes $\zeta\simeq 0$
and $\zeta\simeq\pi$. In the right panel of the figure we represent the
response function $\Gamma^{V}_{ab}$ to circular polarization, which shows that
scattering effects are more pronounced for intermediate values of the angle
$\zeta$.

Figure 4:  We consider a random collection of pulsar pairs whose positional
vectors $\hat{x}_{a}$ and $\hat{x}_{b}$ lie on the a plane orthogonal to the
relative velocity $\hat{v}$. Our aim is to represent how geometrical factors
depending on $\hat{v}$ affect the response functions for circular
polarization, as a function of the angle $\zeta$ among pulsars. Left
combination $\hat{v}\cdot(\hat{x}_{a}\times\hat{x}_{b})$ for the random
realization, as a function of $\zeta$. Notice that the size of this quantity
decreases for $\zeta\to 0$ and $\zeta\to\pi$. Right: Black curve: Hellings
Downs distribution. Blue points: response function for our random realization
of pulsars, including the circular polarization kinematic dipole, with $\beta
n_{I}=1/10$. Notice that, for intermediate values of $\zeta$, the points are
scattered with respect to the black line.

Hence, to be maximally sensitive to circular polarization $V$, it is
convenient to focus on pulsars whose position vectors form a plane orthogonal
to the direction of the velocity vector $\hat{v}$, so that we completely
remove possible contaminations from kinematic effects due to GW intensity. The
formula for the optimal signal-to-noise ratio ${\rm SNR}$ for detecting the
presence of $V$ in the PTA time-delay correlators is derived in Appendix B. It
results

$\displaystyle{\rm SNR}_{V}$ $\displaystyle=$ $\displaystyle
2\pi\,T^{1/2}\,\left\\{\sum_{ab}\,\int
df\,\frac{\bar{V}^{2}(f)\,\left(\Gamma_{ab}^{V}(f)\right)^{2}}{S_{a}^{(n)}(f)\,S_{b}^{(n)}(f)}\right\\}^{1/2}\,,$
(5.1) $\displaystyle=$ $\displaystyle
2\pi\,T^{1/2}\,\beta\,\left\\{\sum_{ab}\,\int
df\,\frac{\bar{V}^{2}(f)\,n_{V}^{2}(f)}{S_{a}^{(n)}(f)\,S_{b}^{(n)}(f)}\left(\frac{1}{3}+\frac{y_{ab}\ln
y_{ab}}{4(1-y_{ab})}\right)^{2}\,\left[\hat{v}\cdot(\hat{x}_{a}\times\hat{x}_{b})\right]^{2}\right\\}^{1/2}$

where the sum is limited to pulsar pairs orthogonal to $\hat{v}$, as described
above, and $S^{(n)}$ is the noise spectral density function (see Appendix B
for more details). The general form for the circular polarization response
$\Gamma_{ab}^{V}$ is given in eq (3.13), and in the second line of eq (5.1) we
write its contributions up to the dipole. Besides being proportional to
$\beta$, it also depends on the slope of the quantity $\bar{V}(f)$, and might
be enhanced if $\bar{V}(f)$ is a steep function of frequency. In fact, the
study of kinematic anisotropies – given their unique dependence on the slope
of the spectrum, see comments between eqs (3.7) and (3.8) – might represent
the only way to probe the frequency-dependence of quantities associated with
circular polarization. In the most conservative cases, in case
$\bar{V}\simeq\bar{I}$ with no pronounced features, in order to detect parity
violating effects we would need a factor $1/\beta\simeq$ one thousand more in
sensitivity with respect to current experiments.

###  5.2 Kinematic anisotropies and SGWB intensity

Figure 5: We consider a random collection of pulsar pairs whose positional
vectors $\hat{x}_{a}$ and $\hat{x}_{b}$ lie on the same plane as the relative
velocity $\hat{v}$. Our aim is to represent how geometrical factors depending
on $\hat{v}$ affect the response functions, as a function of the angle $\zeta$
among pulsars. Left  combination
$\hat{v}\cdot\hat{x}_{a}+\hat{v}\cdot\hat{x}_{b}$ for the random realization,
as a function of $\zeta$. Notice that the size of this quantity decreases as
$\zeta\to\pi$. Right: Black curve: Hellings Downs distribution. Red points:
response function for our random realization of pulsars, including the
intensity kinematic dipole, with $\beta n_{I}=1/10$. Notice that, for small
values of $\zeta$, the points are scattered with respect to the black line.

Suppose we are interested in extracting from the kinematic anisotropy
contributions associated with the GW intensity $\bar{I}(f)$ as introduced in
section 3.1. The corresponding response function $\Gamma^{I}_{ab}$ of a pulsar
pair $(a,b)$ to intensity Doppler effects is given by eq (3.9), which we
rewrite here up to the dipole contributions:

$\displaystyle\Gamma^{I}_{ab}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{3}-\frac{y_{ab}}{6}+y_{ab}\ln
y_{ab}\right)+{\beta\,n_{I}}\left(\frac{1}{12}+\frac{y_{ab}}{2}+\frac{y_{ab}\ln
y_{ab}}{2(1-y_{ab})}\right)\,\left[\hat{v}\cdot\hat{x}_{a}+\hat{v}\cdot\hat{x}_{b}\right]\,,$
(5.2)

Recall the definition of the parameter $y_{ab}=(1-\cos{\zeta})/2$ is expressed
in terms of the angle $\hat{x}_{a}\cdot\hat{x}_{b}=\cos\zeta$ between the unit
vectors indicating the pulsar positions with respect to the Earth. The
function (5.2) depends on $\zeta$ through the first contribution between
parenthesis, which corresponds to the Hellings Downs curve $\Gamma_{ab}^{\rm
HD}$ as defined in eq (3.10). But eq (5.2) also depends on the angle that each
pulsar vector forms with the relative frame velocity $\hat{v}$. As explained
in section 3, only pulsars whose vector components lie along $\hat{v}$ are
sensitive to kinematic anisotropies relative to the GW intensity.

When plotting the quantity $\Gamma^{I}_{ab}$ as a function of $\zeta$ for each
pulsar pair in a given set, data points get scattered around the Hellings-
Downs curve of eq (3.10) (see [48, 47]). In fact, data are modulated by the
quantity $(\hat{v}\cdot\hat{x}_{a}+\hat{v}\cdot\hat{x}_{b})$ in eq (5.2); we
expect that point scattering around the Hellings-Downs line is maximal for
small values of $\zeta$, while it reduces for $\zeta\simeq\pi$, since in this
limit the coefficient of the kinematic dipole contribution vanishes:
$(\hat{v}\cdot\hat{x}_{a}+\hat{v}\cdot\hat{x}_{b})\simeq 0$. We represent in
Fig 5, left panel, the value of the quantity
$(\hat{v}\cdot\hat{x}_{a}+\hat{v}\cdot\hat{x}_{b})$ as a function of $\zeta$,
for a random set of pulsars whose positions are coplanar with the velocity
vector $\hat{v}$. As anticipated, the size of this quantity reduces as we
increase the angle $\zeta$. The right panel of the same figure shows the
resulting response function $\Gamma^{I}_{ab}$ as function of $\zeta$, which is
indeed scattered for small angular separations $\zeta$ with respect to the
Hellings Downs curve. Hence, the effects of kinematic anisotropies relative to
the GW intensity $I$ are maximal for pulsar pairs lying in the same plane as
$\hat{v}$, and with whose positional vectors form a small relative angle among
themselves. This represents an interesting difference with respect to the GW
circular polarization response that we discussed in section 5.1.

As manifest from Fig 5, the kinematic effects we are interested in are small.
On the other hand, by combining signals 777See the review [81] for a
discussion of possibilities to combine GW signals to extract specific physical
information from them., we can form null tests for Doppler effects associated
to the intensity of the SGWB. To find such combinations, we further exploit
the results of section 3. Suppose that GW are characterized by spin-2
polarization only, with no effects of parity violation. Then, consider two
pairs of pulsars. One pair, $ab$, lies on a plane parallel to $\hat{v}$: hence
it feels the effects of kinematic anisotropies, and the response function up
to the dipole is given by eq (5.2). The other pair, $cd$, lies on a plane
orthogonal to $\hat{v}$: hence it is blind to Doppler effects, and the pulsar
response function is determined only by the Hellings Downs curve of eq (3.10).

We then take the following combination of equal-time correlators for time
residuals (recall their definition in eq (2.14)):

$\displaystyle\frac{\left(\Gamma_{cd}^{\rm(HD)}\right)\langle
R_{a}(t)R_{b}(t)\rangle-\left(\Gamma_{ab}^{\rm(HD)}\right)\langle
R_{c}(t)R_{d}(t)\rangle}{\langle R_{c}(t)R_{d}(t)\rangle}$ $\displaystyle=$
$\displaystyle\beta\,\frac{\,F_{ab}^{(1)}\int{df\,n_{I}(f)\,\bar{I}(f)\,\sin^{2}{\left(\pi
ft\right)}}/{f^{2}}}{\int{df\,\bar{I}(f)\,\sin^{2}{\left(\pi
ft\right)}}/{f^{2}}}\,,$ (5.3)

where $\Gamma^{\rm HD}$ is the Hellings Downs function of eq (3.10), while
$F^{(1)}$ is given in eq (3.11). Hence, in the context we are considering, a
measurement of the combination (5.3) allows us to extract only the effects of
kinematic anisotropies, with no contaminations from the isotropic background.
Interestingly, given the dependence on the intensity slope $n_{I}$ in the
numerator of eq (5.3), the sensitivity to kinematic anisotropies get enhanced
for signals with features, see for example section 3.3. In this sense,
kinematic anisotropies allow us to probe features in the frequency dependence
of GW correlators.

Figure 6:  This plot complements Figs 5 and 4, by showing the contributions of
the kinematic quadrupole to the angular dependence of the PTA response
functions. Black lines: Hellings Downs curve. Left panel: The red points are a
random realisation of the quadrupole contribution to the intensity response
function, choosing the value $\beta^{2}(\alpha_{\rm I}+n_{\rm I}^{2}-n_{\rm
I})=1/50$. Right panel: The blue points are a random realization of the
quadrupole contribution to the circular polarization response function, with
$\beta^{2}(\alpha_{\rm V}+n_{\rm V}^{2}-n_{\rm V})=1/10$.

While so far we discussed the dipole only, similar considerations hold for the
quadrupole contributions, that as we have seen can be enhanced in scenarios
with features in the isotropic SGWB as function of frequency. See Fig 6 for
representative examples of kinematic quadrupole effects.

###  5.3 Kinematic anisotropies and scalar contributions to the GW signal

Models of modified gravity often predict the existence of light scalar fields
associated with gravitational interactions. They contribute to GW as
additional spin-0, scalar polarizations [107]. Scalars can contribute to the
total GW intensity. But if their amplitude $\bar{I}_{\rm sc}$ is well smaller
than the tensor intensity amplitude $\bar{I}_{\rm tn}$, it might be hard to
disentangle their contribution from analysing only the isotropic part of the
background.

Interestingly, if the isotropic scalar intensity $\bar{I}_{\rm sc}(f)$ has a
steep slope, or more generally it has pronounced features associated with its
frequency dependence, its consequences on kinematic anisotropies can allow us
to extract the presence of scalar polarization with PTA data.

In scenarios including contributions from the scalar polarization, we express
the total GW intensity (tensor plus scalar parts) as

$\bar{I}(f)\,=\,\bar{I}_{\rm tn}+\bar{I}_{\rm sc}(f)\,.$ (5.4)

Defining the intensity slope parameters as in eqs (4.1), (4.2), we find that
the PTA intensity response function to Doppler dipolar anisotropies in such
scalar-tensor theories can be decomposed as

$\displaystyle\Gamma^{I}_{ab}$ $\displaystyle=$ $\displaystyle\Gamma_{ab}^{\rm
ST,\,iso}+\frac{\beta}{12}\,\Gamma_{ab}^{\rm ST,\,dip}$ (5.5)

with

$\displaystyle\Gamma_{ab}^{\rm ST,\,iso}$ $\displaystyle=$
$\displaystyle\frac{\bar{I}_{\rm tn}(f)}{\bar{I}_{\rm tn}+\bar{I}_{\rm
sc}(f)}\left(\frac{1}{3}-\frac{y_{ab}}{6}+y_{ab}\ln
y_{ab}\right)+\frac{\bar{I}_{\rm sc}(f)}{\bar{I}_{\rm tn}+\bar{I}_{\rm
sc}(f)}\left(\frac{1}{3}-\frac{y_{ab}}{6}\right)$ (5.6)
$\displaystyle\Gamma_{ab}^{\rm ST,\,dip}$ $\displaystyle=$
$\displaystyle\left[n_{\rm sc}+{\,n_{\rm
tn}}\left({1}+{6\,y_{ab}}+\frac{6\,y_{ab}\ln
y_{ab}}{(1-y_{ab})}\right)\right]\,\left[\hat{v}\cdot\hat{x}_{a}+\hat{v}\cdot\hat{x}_{b}\right]\,.$
(5.7)

The quantity $\Gamma_{ab}^{\rm ST,\,iso}$ describes the pulsar response to the
isotropic part of the scalar-tensor intensity, while $\Gamma_{ab}^{\rm
ST,\,dip}$ the pulsar response to the dipolar kinematic anisotropy. Hence,
even if $\bar{I}_{\rm sc}/\bar{I}_{\rm tn}\ll 1$ – so that $\Gamma_{ab}^{\rm
ST,\,iso}$ is ‘almost blind’ to scalar contributions – a pronounced slope
dependence, leading to $n_{\rm sc}\geq n_{\rm tn}$ can make $\Gamma_{ab}^{\rm
ST,\,dip}$ sensitive to the scalar polarization. An appropriate combination of
signals associated to different pulsar pairs, following the method described
around eq (5.3), isolates the effects of scalar dipolar anisotropies in the
response functions. Consequently, the presence of scalar polarizations might
be detectable even if scalar polarizations contribute as small part to the GW
signal amplitude.

##  6 Outlook

The recent detection of a stochastic gravitational wave background from
several pulsar timing array collaborations opens a new window for
gravitational wave cosmology. If the stochastic gravitational wave background
has a cosmological origin, one of its guaranteed features are kinematic
anisotropies due to our motion with respect to the gravitational wave source.
In this work, we investigated the physical information that can be extracted
from a detection of kinematic anisotropies. We shown that the pulsar response
functions to Doppler effects depend on the stochastic background source and on
the underlying gravitational physics. Furthermore, we stressed for the first
time how the measured gravitational wave signal depends on the pulsar location
with respect to the relative velocity among frames. We presented our results
in a convenient way that emphasizes the geometrical dependence of the pulsar
response functions to different properties of the stochastic background.

Our findings can be useful in planning future measurements, or in elaborating
existing data, for using kinematic anisotropies to detect, or set more
stringent constraints, on parity violation in gravitational interactions, on
the presence of scalar/vector polarizations, or to develop independent probes
of the frequency profile of the stochastic background. Besides backgrounds
detectable at nano-Hertz frequencies with pulsar timing arrays, it would be
interesting to apply our findings also to astrometric measurements of
gravitational waves based on Gaia data, see e.g. the discussion in [112, 113,
114, 115, 116].

In our work, we assumed that the direction and amplitude of the velocity
vector among the cosmological source of gravitational waves and the solar
sytem baricenter coincides to what measured by the cosmic microwave background
dipole. On the other hand, our formulas can be used to directly test this
hypothesis, somehow analogously to the proposal [117] in the different context
of radio sources. In fact, the topic of radio galaxy and quasar measurements
of the cosmic dipole is a point of debate in recent literature [118, 119, 120,
121, 122, 123]. It would be interesting to use gravitational waves as an
independent probe of the direction of the kinematic dipole, and test
alternative scenarios based on large scale superhorizon isocurvature
fluctuations [124, 125], intrinsic anisotropies [126], or more general
deviations from the cosmological principle [127]. In this sense, this program
belongs to the ‘multimessenger cosmology’ framework advocated in [128, 129].

### Acknowledgments

It is a pleasure to thank Debika Chowdhury, Ameek Malhotra, Maria Mylova, and
Ivonne Zavala for useful input. GT is partially funded by the STFC grant
ST/T000813/1. For the purpose of open access, the author has applied a
Creative Commons Attribution licence to any Author Accepted Manuscript version
arising.

##  Appendix A Identities involving the polarization tensors

We derive useful identities involving the polarization tensors entering in the
Fourier transform of the GW signals, which we used in sections 2-4. We start
discussing the spin-2 case, to then consider respectively the spin-0 and
spin-1 cases.

Spin-2 polarization tensors: In the main text, section 3, we focussed on the
$(+,\times)$ basis for the polarization tensors. To develop our arguments, in
this Appendix it is convenient to introduce the $(R,L)$ circular basis:

${\bf e}_{ab}^{R/L}(\hat{n})\,=\,\frac{{\bf e}_{ab}^{+}(\hat{n})\pm\,i\,{\bf
e}_{ab}^{\times}(\hat{n})}{\sqrt{2}}\,.$ (A.1)

We assume that the polarization tensors ${\bf e}_{ab}^{+,\times}$ are real
quantities: hence $\left({\bf e}_{ab}^{R}\right)^{*}\,=\,{\bf e}_{ab}^{L}$.
Notice also the identity

$\displaystyle{\bf e}_{ab}^{R}(\hat{n})\,{\bf e}_{cd}^{L}(\hat{n})$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left({\bf e}_{ab}^{+}{\bf
e}_{cd}^{+}+{\bf e}_{ab}^{\times}{\bf
e}_{cd}^{\times}\right)+\frac{i}{2}\left({\bf e}_{ab}^{\times}{\bf
e}_{cd}^{+}-{\bf e}_{ab}^{+}{\bf e}_{cd}^{\times}\right)\,.$ (A.2)

We identify the circular polarization indexes with $\pm 1$:
$\lambda\,=\,R/L\,=\,\pm 1$. The polarization tensors in the circular basis
can be conveniently expressed as a product of polarization vectors:

${\bf e}_{ab}^{\lambda}(\hat{n})\,=\,{\bf e}_{a}^{\lambda}(\hat{n}){\bf
e}_{b}^{\lambda}(\hat{n})\,=\,\frac{\hat{p}_{a}+i\lambda\hat{q}_{a}}{\sqrt{2}}\,\frac{\hat{p}_{b}+i\lambda\hat{q}_{b}}{\sqrt{2}}\,.$
(A.3)

The real unit vectors $\hat{p}_{a}$, $\hat{q}_{a}$ are orthogonal among
themselves and to the GW direction $\hat{n}$. We can express $\hat{q}$ as

$\displaystyle\hat{q}_{a}$ $\displaystyle=$
$\displaystyle\epsilon_{abc}\hat{n}_{b}\hat{p}_{c}\,,$ (A.4)

with $\epsilon_{abc}$ the Levi-Civita tensor in three dimensions. (For what
comes next, we do not need to make any specific choice for the direction
$\hat{p}_{a}$.) It is easy to verify the following identity involving
symmetric tensors

$\hat{p}_{a}\hat{p}_{b}+\hat{q}_{a}\hat{q}_{b}\,=\,\delta_{ab}-\hat{n}_{a}\hat{n}_{b}\,.$
(A.5)

In fact, both the LHS and RHS are parallel to $\hat{p}_{a}$ and $\hat{q}_{a}$,
orthogonal to $\hat{n}_{a}$, and of trace 2. Moreover, the antisymmetric
combination $\hat{p}_{a}\hat{q}_{b}-\hat{p}_{b}\hat{q}_{a}$ is orthogonal to
$\hat{n}_{a}$, and satisfy the identity

$\hat{p}_{a}\hat{q}_{b}-\hat{p}_{b}\hat{q}_{a}\,=\,\epsilon_{abc}\,\hat{n}_{c}\,.$
(A.6)

We can convince ourselves of the validity of relation (A.6) by contracting
with $\hat{p}_{a}$, $\hat{q}_{b}$:

$\sum_{a}\hat{p}_{a}(\hat{p}_{a}\hat{q}_{b}-\hat{p}_{b}\hat{q}_{a})\,=\,\hat{q}_{b}\,=\,\hat{p}_{a}\epsilon_{abc}\,\hat{n}_{c}\,,$
(A.7)

and

$\sum_{a}\hat{q}_{a}(\hat{p}_{a}\hat{q}_{b}-\hat{p}_{b}\hat{q}_{a})\,=\,-\hat{p}_{b}\,=\,\hat{q}_{a}\epsilon_{abc}\,\hat{n}_{c}\,,$
(A.8)

compatibly with equation (A.6). See also [76].

Identities (A.5) and (A.6) ensure the relation (no summation over polarization
indexes)

$\displaystyle{\bf e}_{a}^{\lambda}(\hat{n})\left({\bf
e}_{b}^{\lambda}(\hat{n})\right)^{*}$ $\displaystyle=$
$\displaystyle\frac{p_{a}+i\lambda q_{a}}{\sqrt{2}}\,\frac{p_{b}-i\lambda
q_{b}}{\sqrt{2}}\,,$ (A.9) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(p_{a}p_{b}+q_{a}q_{b}\right)-\frac{i\lambda}{2}\left(p_{a}q_{b}-p_{b}q_{a}\right)\,,$
(A.10) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\delta_{ab}-n_{a}n_{b}\right)-\frac{i\lambda}{2}\epsilon_{abc}\,n_{c}\,.$
(A.11)

Hence:

$\displaystyle{\bf e}_{ab}^{\lambda}(\hat{n})({\bf
e}_{cd}^{\lambda}(\hat{n}))^{*}$ $\displaystyle=$ $\displaystyle{\bf
e}_{a}^{\lambda}(\hat{n})\left({\bf e}_{c}^{\lambda}(\hat{n})\right)^{*}{\bf
e}_{b}^{\lambda}(\hat{n})\left({\bf e}_{d}^{\lambda}(\hat{n})\right)^{*}$
(A.12) $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\delta_{ac}-n_{a}n_{c}-{i\lambda}\epsilon_{acf}\,n_{f}\right)\left(\delta_{bd}-n_{b}n_{d}-{i\lambda}\epsilon_{bdg}\,n_{g}\right)\,.$
(A.13)

Expanding the product, and passing from the circular $(R,L)$ polarization to
the original $(+,\times)$ polarization using eq (A.2), we find the identities:

$\displaystyle\sum_{\lambda\lambda^{\prime}}{\bf
e}_{ab}^{\lambda}(\hat{n}){\bf
e}_{cd}^{\lambda^{\prime}}(\hat{n})\delta_{\lambda\lambda^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\delta_{ac}-n_{a}n_{c})(\delta_{bd}-n_{b}n_{d})+\frac{1}{2}(\delta_{ad}-n_{a}n_{d})(\delta_{bc}-n_{b}n_{c})$
(A.14)
$\displaystyle-\frac{1}{2}(\delta_{ab}-n_{a}n_{b})(\delta_{cd}-n_{c}n_{d})\,,$

and

$\displaystyle\sum_{\lambda\lambda^{\prime}}{\bf
e}_{ab}^{\lambda}(\hat{n}){\bf
e}_{cd}^{\lambda^{\prime}}(\hat{n})\epsilon_{\lambda\lambda^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\delta_{ac}-n_{a}n_{c})\epsilon_{bdf}n_{f}+\frac{1}{2}(\delta_{bd}-n_{b}n_{d})\epsilon_{acf}n_{f}\,.$
(A.15)

We used these relations in the main text to obtain eqs (2.17) and (2.18),
controlling the PTA response respectively to GW intensity and circular
polarization.

Spin-0 polarization tensor: We can be brief here, since the spin-0 mode is
associated with a single polarization tensor

$\sigma_{ab}(\hat{n})\,=\,\hat{n}_{a}\hat{n}_{b}\,,$ (A.16)

which we used for deriving equation (4.3) in the main text.

Spin-1 polarization tensors: To describe spin-1 polarization tensors we can
use a $(+,\times)$ basis or a circular $(R,L)$ basis: they are related exactly
as in eq (A.1) for the spin-2 case. In the circular basis $(R,L)\,=\,\pm 1$,
the vector polarization tensors read

${\bf u}_{ab}^{\lambda}(\hat{n})\,=\,\frac{1}{\sqrt{2}}\left({\bf
e}_{a}^{\lambda}(\hat{n})\hat{n}_{b}+\hat{n}_{a}\,{\bf
e}_{b}^{\lambda}(\hat{n})\right)\,,$ (A.17)

with ${\bf e}_{a}^{\lambda}(\hat{n})$ the vectors appearing in eq (A.3). We
have the relation

$\displaystyle{\bf u}_{ab}^{\lambda}(\hat{n})\,\left({\bf
u}_{cd}^{\lambda}(\hat{n})\right)^{*}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left({\bf
e}_{a}^{\lambda}(\hat{n})\hat{n}_{b}+\hat{n}_{a}\,{\bf
e}_{b}^{\lambda}(\hat{n})\right)\left(\left({\bf
e}_{c}^{\lambda}(\hat{n})\right)^{*}\hat{n}_{d}+\hat{n}_{c}\,\left({\bf
e}_{d}^{\lambda}(\hat{n})\right)^{*}\right)\,,$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}{\bf e}_{a}^{\lambda}(\hat{n})\left({\bf
e}_{c}^{\lambda}(\hat{n})\right)^{*}\hat{n}_{b}\hat{n}_{d}+{\bf
e}_{b}^{\lambda}(\hat{n})\left({\bf
e}_{c}^{\lambda}(\hat{n})\right)^{*}\hat{n}_{a}\hat{n}_{d}$
$\displaystyle+{\bf e}_{a}^{\lambda}(\hat{n})\left({\bf
e}_{d}^{\lambda}(\hat{n})\right)^{*}\hat{n}_{b}\hat{n}_{c}+{\bf
e}_{b}^{\sigma}(\hat{n})\left({\bf
e}_{d}^{\lambda}(\hat{n})\right)^{*}\hat{n}_{a}\hat{n}_{c}\Big{]}\,,$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left[\delta_{ac}-\hat{n}_{a}\hat{n}_{c}-{i\lambda}\epsilon_{acf}\,n_{f}\right]\hat{n}_{b}\hat{n}_{d}$
$\displaystyle+$
$\displaystyle\frac{1}{4}\left[\delta_{bc}-\hat{n}_{b}\hat{n}_{c}-{i\lambda}\epsilon_{bcf}\,\hat{n}_{f}\right]\hat{n}_{a}\hat{n}_{d}\,,$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(\delta_{ac}\hat{n}_{b}\hat{n}_{d}+\delta_{bc}\hat{n}_{a}\hat{n}_{d}+\delta_{ad}\hat{n}_{b}\hat{n}_{c}+\delta_{bd}\hat{n}_{a}\hat{n}_{c}\right)-\hat{n}_{a}\hat{n}_{b}\hat{n}_{c}\hat{n}_{d}$
$\displaystyle-$
$\displaystyle\frac{i\lambda}{4}\left(\epsilon_{acf}\hat{n}_{b}\hat{n}_{d}+\epsilon_{bcf}\hat{n}_{a}\hat{n}_{d}+\epsilon_{adf}\hat{n}_{b}\hat{n}_{c}+\epsilon_{bdf}\hat{n}_{a}\hat{n}_{c}\right)\hat{n}_{f}\,.$
(A.20)

Proceeding as we did in deriving eqs (A.14) and (A.15) for the spin-2 case, we
can then obtain the following identities for vector polarizations in the
$(+,\times)$ basis:

$\displaystyle\sum_{\lambda\lambda^{\prime}}{\bf
u}_{ab}^{\lambda}(\hat{n}){\bf
u}_{cd}^{\lambda^{\prime}}(\hat{n})\delta_{\lambda\lambda^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\delta_{ac}\hat{n}_{b}\hat{n}_{d}+\delta_{bc}\hat{n}_{a}\hat{n}_{d}+\delta_{ad}\hat{n}_{b}\hat{n}_{c}+\delta_{bd}\hat{n}_{a}\hat{n}_{c}\right)-2\hat{n}_{a}\hat{n}_{b}\hat{n}_{c}\hat{n}_{d}\,,$
(A.21)

and

$\displaystyle\sum_{\lambda\lambda^{\prime}}{\bf
u}_{ab}^{\lambda}(\hat{n}){\bf
u}_{cd}^{\lambda^{\prime}}(\hat{n})\epsilon_{\lambda\lambda^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\epsilon_{acf}\hat{n}_{b}\hat{n}_{d}+\epsilon_{bcf}\hat{n}_{a}\hat{n}_{d}+\epsilon_{adf}\hat{n}_{b}\hat{n}_{c}+\epsilon_{bdf}\hat{n}_{a}\hat{n}_{c}\right)\hat{n}_{f}\,,$
(A.22)

that we used for deriving equation (LABEL:GammaVe1) in the main text.

##  Appendix B The optimal signal-to-noise ratio

In this Appendix we discuss the computation of the signal-to-noise ratio, in
terms of a match-filtering technique as discussed in the paper [46] and the
textbook [130]. We focus here on the case of spin-2 polarizations, but the
arguments are the same for spin-1 and spin-0. We assume that the noise
dominates over the signal. The latter can be extracted once we know its
properties.

The stationary two-point correlator for time-delays of pulsar measurements
summed over a PTA set of pulsar pairs $(a,b)$ is weighted by an appropriate
choice of filters

${\cal
Y}\,=\,\sum_{ab}\,\int_{-T/2}^{T/2}dt_{1}\int_{-T/2}^{T/2}dt_{2}\,z_{a}(t_{1})z_{b}(t_{2})\,Q_{ab}(t_{1}-t_{2})\,,$
(B.1)

where $T$ indicates the duration of the experiment. We pass to Fourier space,
and write the previous expression as

${\cal
Y}\,=\,\frac{1}{2}\sum_{ab}\,\int_{-\infty}^{\infty}df_{1}df_{2}\,\delta_{T}(f_{1}-f_{2})\,z_{a}^{*}(f_{1})z_{b}(f_{2})\,Q_{ab}(f_{2})+c.c.$
(B.2)

The ‘finite-time’ delta-function is

$\delta_{T}(f)\,=\,\frac{\sin{(\pi fT)}}{\pi f}$ (B.3)

We assume that $Q_{ab}(t)$ is real, and its Fourier transform obeys
$Q_{ab}^{*}(f)\,=\,Q_{ab}(-f)$. Decomposed in real and imaginary part, we can
express it as

$Q_{ab}(f)\,=\,Q_{ab}^{I}(f)+i\,Q_{ab}^{V}(f)\,,$ (B.4)

with $Q_{ab}^{I,V}$ two real functions of frequency. In defining the SNR, the
signal $S$ is the averaged value of ${\cal Y}$ when the GW signal is present;
the noise $N$ is root mean square of ${\cal Y}$ when the GW is absent. The aim
is to determine the optimal filter functions $Q^{I}$ and $Q^{V}$ for
extracting the GW signal from the noise.

We start expressing the signal $S$, as

$\displaystyle S\,=\,\langle{\cal
Y}\rangle\,=\,\frac{1}{2}\sum_{ab}\,\int_{-\infty}^{\infty}df_{1}df_{2}\,\delta_{T}(f_{1}-f_{2})\,\langle
z_{a}^{*}(f_{1})z_{b}(f_{2})\rangle\,Q_{ab}(f_{2})+c.c.$ (B.5)

Using formulas developed in section 2, we can write the two-point correlator
in Fourier space as

$\displaystyle\langle z_{a}^{*}(f_{1})z_{b}(f_{2})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
d^{2}\hat{n}\,D_{a}^{\lambda}D_{b}^{\lambda^{\prime}}\left(I(f_{1},\hat{n})\,\delta_{\lambda\lambda^{\prime}}-iV(f_{1},\hat{n})\,\epsilon_{\lambda\lambda^{\prime}}\right)\,\delta(f_{1}-f_{2})$
(B.6) $\displaystyle=$
$\displaystyle\pi\,\bar{I}(f_{1})\,\left(\Gamma_{ab}^{I}(f_{1})-i\,\Gamma_{ab}^{V}(f_{1})\right)\,.$
(B.7)

We plug this expression in eq (B.5), and use the decomposition (B.4), finding

$S\,=\,\pi\,T\,\sum_{ab}\,\int
df\,\bar{I}(f)\,\left[\Gamma_{ab}^{I}(f)Q_{ab}^{I}(f)+\Gamma_{ab}^{V}(f)Q_{ab}^{V}(f)\right]$
(B.8)

since $\delta_{T}(0)=T$. The computation of the noise $N$ is identical to the
textbook treatment of [130]. Characterizing the correlator of pulsar noise
$n_{a}$ in terms of the quantity $S_{a}^{(n)}$ as

$\langle
n_{a}^{*}(f_{1})n_{a}(f_{2})\rangle\,=\,\frac{1}{2}\delta(f_{1}-f_{2})\,S_{a}^{(n)}(f_{1})\,,$
(B.9)

the square of the noise is

$N^{2}\,=\,\frac{T}{4}\,\sum_{ab}\int
df\,Q_{ab}(f)Q_{ab}^{*}(f)\,S_{a}^{(n)}(f)\,S_{b}^{(n)}(f)\,.$ (B.10)

Hence, the signal-to-noise ratio results

$\displaystyle{\rm SNR}\,=\,T^{1/2}\,\frac{2\pi\,\sum_{ab}\,\int
df\,\bar{I}(f)\,\left[\Gamma_{ab}^{I}(f)Q_{ab}^{I}(f)+\Gamma_{ab}^{V}(f)Q_{ab}^{V}(f)\right]}{\left[\sum_{cd}\int
df\,\left[(Q_{cd}^{I}(f))^{2}+(Q_{cd}^{V}(f))^{2}\right]\,S_{c}^{(n)}(f)\,S_{d}^{(n)}(f)\right]^{1/2}}\,.$
(B.11)

We now determine the optimal choice of filter function that maximises the SNR.
Given a tensor with structure

$({\cal A}^{(1)},{\cal
A}^{(2)})\,=\,\left((A_{ab}^{(1)},A_{ab}^{(2)}),(A_{cd}^{(1)},A_{cd}^{(2)})\dots\right)$
(B.12)

where the dots run over the pulsar pairs, one considers the positive-definite
product

$\big{\langle}({\cal A}^{(1)},{\cal A}^{(2)}),({\cal B}^{(1)},{\cal
B}^{(2)})\big{\rangle}\,=\,\sum_{ab}\,\int
df\left(A_{ab}^{(1)}A_{ab}^{(2)}+B_{ab}^{(1)}B_{ab}^{(2)}\right)S_{a}^{(n)}(f)\,S_{b}^{(n)}(f)$
(B.13)

Using this tool, we can schematically express the SNR of eq (B.11) as

${\rm
SNR}\,=\,2\pi\,T^{1/2}\,\frac{\big{\langle}(Q^{(1)},Q^{(2)}),(\bar{I}\,\Gamma^{I}/S_{n}^{2},\bar{V}\,\Gamma^{V}/S_{n}^{2})\big{\rangle}}{\big{\langle}(Q^{(1)},Q^{(2)}),(Q^{(1)},Q^{(2)})\big{\rangle}\,.}$
(B.14)

We now have different options:

  * •

If we wish to maximise the SNR to the total signal (intensity and circular
polarization combined) we choose the filter
$(\bar{I}\,\Gamma^{I}/S_{n}^{2},\bar{V}\,\Gamma^{V}/S_{n}^{2})$ which
obviously maximises the scalar product of eq (B.14). Using this filter we
conclude that the optimal SNR is

${\rm SNR}_{I+V}\,=\,2\pi\,T^{1/2}\,\left\\{\sum_{ab}\,\int
df\,\frac{\bar{I}^{2}(f)\,\left(\Gamma_{ab}^{I}(f)\right)^{2}+\bar{V}^{2}(f)\,\left(\Gamma_{ab}^{V}(f)\right)^{2}}{S_{a}^{(n)}(f)\,S_{b}^{(n)}(f)}\right\\}^{1/2}\,.$
(B.15)

Its value depends on the GW intensity and circular polarization, and on the
geometric quantities constituting $\Gamma_{ab}^{I}$ and $\Gamma_{ab}^{V}$, as
discussed in the main text.

  * •

We can instead be interested to extract from the signal only components
associated for example to circular polarization, and determine the filter
better suited to this purpose. The filter is then
$(0,\bar{V}\,\Gamma^{V}/S_{n}^{2})$ (i.e. a filter whose Fourier transform has
only imaginary and no real component), and the associated SNR is

${\rm SNR}_{V}\,=\,2\pi\,T^{1/2}\,\left\\{\sum_{ab}\,\int
df\,\frac{\bar{V}^{2}(f)\,\left(\Gamma_{ab}^{V}(f)\right)^{2}}{S_{a}^{(n)}(f)\,S_{b}^{(n)}(f)}\right\\}^{1/2}\,.$
(B.16)

As discussed in the main text, the sensitivity to circular polarization is
then enhanced by selecting pulsars whose directions form a plane orthogonal to
the velocity vector $\hat{v}$ between SGWB source and detector frames.

  * •

Analogously, if we wish only to be sensitive to the GW intensity, we select a
filter with real part only, and the maximal SNR is

${\rm SNR}_{I}\,=\,2\pi\,T^{1/2}\,\left\\{\sum_{ab}\,\int
df\,\frac{\bar{I}^{2}(f)\,\left(\Gamma_{ab}^{I}(f)\right)^{2}}{S_{a}^{(n)}(f)\,S_{b}^{(n)}(f)}\right\\}^{1/2}\,.$
(B.17)

The sensitivity to intensity is enhanced by selecting pulsars whose directions
are parallel to the velocity vector $\hat{v}$ between SGWB source and detector
frames.

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