anisotropic components of the astrophysical SGWB (Cusin et al.2019b). Note that, to put this into practice, we need to understand better the time delays between formation and merger, which can change the host galaxy property if delays are long.
Finally, LISA will also allow one to study the astrophysical SGWB from other types of sources such as close white dwarf binaries (see e.g. Vecchio2002), which may also produce anisotropies in the galactic plane (Ungarelli and Vecchio 2001;
Kudoh and Taruya2005). We refer the reader to Amaro-Seoane et al. (2023) for an in-depth discussion on the astrophysical populations leading to this and the aforementioned astrophysical SGWBs.
5.3 Characteristics of the stochastic gravitational-wave background
decreases fork[kpwith a slope depending on the source characteristics; or it takes a shallower, but positive slope, in which case the peak occurs at a higher wave- number. This latter can correspond to an inverse lengthL1, though other time and length scales can also be relevant and show up in theXGWðkÞspectrum: for example, the characteristictime-scale over which the source is coherents\Dt(as opposed to the source durationDt), or differentlength-scales, relevant in the space distribution of the anisotropic stresses, other thanL.
Notable peaks in the spectrum are then expected for SGWB sources characterised by afinite, short durationDt.H1p . In particular, one example of sources giving rise to peaked spectra that are relevant for LISA (since they are typically peaking in the LISA band), are FOPTs related to the electroweak symmetry breaking, as described in Sect.6. On the other hand, for generation mechanisms that turn on at a given time H1p in the radiation era, but continue to source GWs throughout the universe evolutionDtH1p , the region ofk3increase is less and less relevant, being pushed towards the horizon today. One expects a wide frequency region over which the signal features a slower increase with k or, in some cases, is constant. The most noteworthy example for LISA is the SGWB produced by topological defects in the scaling regime, such as a NG cosmic strings network, which is exactly scale-invariant in the LISA band for wide regions of the model parameter space (see e.g. Figueroa et al.2013and the discussion in Sect. 7).
Another almost scale-invariant SGWB is the one generated by slow roll inflation.
This constitutes, however, an exception with respect to the cases described above.
The SGWB is in fact generated as the tensor perturbations reenter the horizon during the radiation and matter eras, and therefore the spectral shape does not depend on the causal evolution properties of some source anisotropic stresses, but on the amplification of vacuum tensor perturbations during inflation. In some scenarios in which the inflaton is coupled to an externalfield, the SGWB is indeed produced by the field anisotropic stresses, and one can obtain blue-tilted spectra whose tilt depends on the model. GWs actively generated at second order in perturbation theory from large scalar fluctuations could also have peculiar features depending on the inflaton dynamics. The scenarios pertaining to these categories that are relevant for LISA are presented in detail elsewhere in this paper (cf. Sects.8,10.4,10.3, and for a review see Caprini and Figueroa (2018) and references therein).
5.3.2 Anisotropies and propagation effects
Angular anisotropies in the energy density of the SGWB can be an efficient way to characterise its physical origin and properties. They provide a further tool to help in disentangling a SGWB of cosmological origin from an astrophysical one, besides the exploitation of their different frequency dependence. Angular anisotropies can be imprinted both at the epoch of the SGWB generation and at later times, during its propagation across cosmological perturbations. As such the anisotropies in the SGWB can provide a new way to characterise and distinguish various generation mechanisms of primordial SGWB and they allow one to probe the evolution of cosmological perturbations. Because the universe is transparent to GWs for sub-
Planckian energies, the case of a cosmological SGWB represents a privileged observable to probe the physics of the early universe, and its anisotropies can preserve the memory of the initial conditions of the universe right after inflation.
We are interested in anisotropies and inhomogeneities in the energy density of the SGWB, therefore we allow the monopole Eq. (30) to be dependent on space and direction of observation
XGW ¼ 1 4p
Z
d2n^xGWðg;~;x q;nÞ;^ ð34Þ
thus defining the energy density contrast as dGW dxGWðg;~;x q;nÞ^
xGWðg;qÞ ¼xGWðg;~;x q;nÞ ^ XGWðq;gÞ
XGWðq;gÞ ; ð35Þ where q¼2pf. Various approaches have been adopted to compute the angular anisotropies and their statistics (such as the angular power spectrum) both for the cosmological and astrophysical SGWB (Alba and Maldacena2016; Contaldi 2017;
Geller et al.2018; Bartolo et al. 2019a, 2020b; Cusin et al. 2017, 2018b, 2019a;
Pitrou et al.2020; Bertacca et al.2020; Cusin et al.2018a; Jenkins et al.2018,2019a;
Cusin et al.2019b,2020; Bartolo et al.2020a; Valbusa Dall’Armi et al.2021). A way to compute the SGWB anisotropies is to adopt a Boltzmann equation approach, similarly to CMB anisotropies (Contaldi2017; Bartolo et al. 2019a, 2020b; Cusin et al.2019a; Pitrou et al.2020). In such an approach one considers the generation of high-frequency GW modes and their propagation across a background of lower frequency (large-scale) cosmological perturbations (which can be either scalar or tensor in nature). As for CMB photons, therefore, the propagating GWs become the cosmological carrier of the underlying cosmic inhomogeneities. Such an approach allows one to put in evidence at least two distinguishing features for a cosmological SGWB (Bartolo et al. 2019a, 2020b): first the anisotropies imprinted at the pro- duction epoch can be characterised by a strong frequency dependent contribution;
secondly, if primordial non-Gaussianity are present in the background large-scale cosmological perturbations, then they will be left imprinted into the SGWB aniso- tropies. The bispectrum of the angular anisotropies ofdGWturns out therefore to be a new probe of primordial non-Gaussianity, potentially measurable at interferometers, beyond the CMB and large-scale structure measurements (Bartolo et al.
2019a,2020b). For these reasons, besides the information they provide for a SGWB from inflation, anisotropies can be a new probe for a whole series of phenomena.
They can be produced at the epoch of generation of GWs from a phase transition (Geller et al.2018; Kumar et al.2021), and they can characterise also the SGWB which is unavoidably produced by second-order curvature perturbations in PBH formation scenarios (Bartolo et al. 2020a). Specific imprints in the SGWB aniso- tropies can be also generated by decoupled relativistic particles in the early universe, thus reinforcing the SGWB as a new window into the particle physics content of the universe (Valbusa Dall’Armi et al.2021).
For a SGWB of astrophysical origin, the analytic derivation of energy density anisotropies can be found in Contaldi (2017), Cusin et al. (2017), Cusin et al.
(2018b), Cusin et al. (2019a), Pitrou et al. (2020), Bertacca et al. (2020). When adopting a Bolzmann-like description, one needs to add an emissivity term to the Vlasov equation for the graviton distribution function, accounting for the generation process at galactic scales (Cusin et al.2019a; Pitrou et al.2020). For extragalactic background components, the primary contribution to the energy density anisotropy comes from clustering (sources are embedded in the cosmic web), while a secondary source of anisotropy is due to line of sight effects (e.g. lensing, kinematic and volume distortion effects). Predictions for the energy density angular power spectrum have been presented in Cusin et al. (2018a), Jenkins et al. (2018), Jenkins et al. (2019a), Cusin et al. (2019b), Cusin et al. (2020), Bertacca et al. (2020) in the Hz band and in Cusin et al. (2020) in the mHz band. Anisotropies are typically suppressed by a factor 101–102 with respect to the monopole, the range of variability depending on the underlying astrophysical model for star formation and collapse, and the angular power spectrum scales as ‘1 on large scales. Different physical choices for the process of BH collapse and mass distribution lead to differences up to 50% on the angular power spectrum in the mHz band, non degenerate with a global scaling (Cusin et al. 2020). With LISA it may be possible to constrain the dipole and quadrupole components of the angular power spectrum, for sufficiently high SNR detection (i.e. sufficiently high monopole) (Alonso et al.2020a; Contaldi et al.2020).
As shown in Sect. 2.6, where we have described how to compute the angular cross-spectrum between GW and other cosmological probes, the study of the cross- correlation of the SGWB energy density fluctuation with the LSS (e.g. galaxy distribution) is an interesting subject to examine to distinguish the origin (cosmological versus astrophysical) of a given background component. Unlike a cosmological SGWB, the extragalactic astrophysical background is expected to be highly correlated with the large-scale structure (see e.g. Cusin et al.2017,2018b,a;
Jenkins et al.2019a; Jenkins and Sakellariadou2018; Jenkins et al.2018; Cusin et al.
2019b,2020; Jenkins and Sakellariadou2019; Jenkins et al.2019b; Bertacca et al.
2020; Pitrou et al. 2020; Mukherjee and Silk2020; Alonso et al.2020b; Adshead et al.2021; Ricciardone et al.2021; Braglia and Kuroyanagi2021; Valbusa Dall’Armi et al.2022). Ways to exploit this feature are discussed in Sect.3.
5.3.3 Polarisations
As any background of radiation, a SGWB is fully characterised in terms of Stokes parameters, intensity (proportional to the background energy density), andQ,U,V parameters describing polarisation. Classical diffusion of GW radiation from massive objects can generate a net polarisation out of an unpolarised flux, playing a role analogue to Thomson scattering for CMB photons (Cusin et al.2019a). The amount of polarisation that can be generated depends on the GW frequency, and it is more effective for large wavelength modes, for which wave effects are expected to be more important in an astrophysical context. An order of magnitude estimate of the effect gives that, in the mHz band, the net amount of polarisation generated by diffusion is suppressed by several orders of magnitude with respect to anisotropies in the intensity (Cusin et al.2019a). As polarisation cannot be effectively generated during
propagation and astrophysical background components are expected to be statisti- cally unpolarised at emission, the detection of a highly polarised background component is a smoking gun of its cosmological origin.
As we review in Sect.8.7.1, several inflationary mechanisms have been proposed that could produce a net circularly polarised SWGB, which is characterised by Stokes Vparameter. A net chiral polarisation can be measured with a network of ground- based (Seto and Taruya2007,2008; Crowder et al.2013; Smith and Caldwell2017) or space-based (Orlando et al.2021) interferometers.7
The measurement is more problematic in the case of a single planar instrument such as LISA. In this case, a left-handed GW with wave-vector~kproduces the same effects as a left-handed GW with wave-vector~kp, where~kphas been obtained from~k with a reflection on the plane of the detector. Therefore a difference between the two polarisations cannot be detected in the case of an isotropic SGWB.
A net polarisation can however be detected also by a planar instrument if the SGWB is not isotropic. As discussed in the previous subsection, It is natural to expect that a SGWB of cosmological origin has a dominant monopole component, with large-scale anisotropies of magnitude comparable to that of the CMB ones. This statement is, however, frame-dependent, and the most natural expectation is that the SGWB is isotropic in the CMB rest-frame. As seen in the CMB, the motion of the Solar System in this frame, with a velocityv’103, produces a dipole anisotropy, with an amplitude suppressed by a factorvwith respect to that of the monopole. Seto (2006), Domcke et al. (2020b) studied how the dipole signal might allow one to measure a net chirality with LISA. This can be done through the cross-correlation between the A and E channels, which vanishes both in the case of isotropic and of unpolarised SGWB. As estimated in Domcke et al. (2020b), the SNR associated with this measurement is
SNR’ v 103
P
k kXkGWh2 1:41011
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 3 years s
; ð36Þ
wherek¼ 1 refers to the right and left chirality, respectively, and whereTis the observation time.
5.3.4 Non-Gaussianity
There are two types of non-Gaussianity discussed in the context of GW observations.
One is the non-Gaussianity of inhomogeneities, which is defined in position or momentum space (see e.g. Sect.8.7.3). GWs generated at sub-horizon scales cannot produce correlation across the horizon due to causality, thus the SGWB is Gaussian.
Non-Gaussianity typically appears in GWs generated in the context of inflation, which could produce non-trivial spatial correlations stretched over the horizon (see Sects.8and10.2.3). See Bartolo et al. (2019a,2020b) for a detailed derivation of the non-Gaussianity expected in the SGWB, which is simply generated by the evolution
7 The detection of a circularly polarised SWGB at CMB scales was studied in Gluscevic and Kamionkowski (2010), Smith and Caldwell (2017), Gerbino et al. (2016), Thorne et al. (2018).
through the background large-scale underlying inhomogeneities, similarly to what happens for CMB photons. This is computed through the angular bispectra (i.e. the three point function) of the graviton energy density.
The other type of non-Gaussianity is the one in the time signal (sometimes referred to as a SGWB in the“popcorn”or“shot noise”regime), which could be a useful statistical measure for a SGWB formed by overlapped short-duration events, such as the astrophysical background. If GW events are not frequent enough to overlap in time, the observed strain has a non-Gaussian distribution. Among the cosmological sources, the SGWB from cosmic strings could show this non-Gaussian feature (Regimbau et al.2012) (see Method II in Sect.7.3). For non-Gaussianity of astrophysical sources, see Sect.12.2.
6 First-order phase transitions
Section coordinators: J. Kozaczuk, M. Lewicki. Contributors: M. Besancon, C. Caprini, D. Croon, D. Cutting, G. Dorsch, O. Gould, R. Jinno, T. Konstandin, J. Kozaczuk, M. Lewicki, E. Madge, G. Nardini, J.M. No, A. Roper Pol, P. Schwaller, G. Servant, P. Simakachorn.