Paper Title: A comparative study of radio signatures from winds and jets: Modelling synchrotron emission and polarization.



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The Aldous–Lyons Conjecture I: Subgroup Tests
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    August 5, 2024
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§ ABSTRACT 

Outflows driven by active galactic nuclei (AGN) are seen in numerous compact sources; however, it has remained unclear how to distinguish between the driving mechanisms, such as winds and jets. Therefore, our study aims to offer observational insights from simulations to aid in this distinction. Specifically, in this paper, we investigate the evolution of wide-angled, moderately relativistic, magnetized winds and analyze their non-thermal radio emission and polarization properties. We find that the evolution of winds varies depending on factors such as power, density, and opening angle, which in turn influence their observable characteristics. Additionally, different viewing angles can lead to varying observations. Furthermore, we note distinctions in the evolution of winds compared to jets, resulting in disparities in their observable features. Jets typically exhibit a thin spine and hotspot(s). Winds manifest broader spines or an “hourglass-shaped” bright emission in the cocoon, which are capped by bright arcs. Both display high polarization coinciding with the bright spine and hotspots/arcs, although these regions are relatively compact and localized in jets when compared to winds. We emphasize the importance of high resolution, as we demonstrate that emission features from both jets and winds can become indistinguishable at lower resolutions. The distribution of polarization is largely unaffected by resolution, though lower polarization becomes more noticeable when the resolution is decreased.




galaxies: active - ISM: jets and outflows - MHD - turbulence - methods: numerical








§ INTRODUCTION

Outflows driven by the Active Galactic Nuclei (AGN) have been observed through multi-wavelength studies in a plethora of sources <cit.>. These outflows can arise due to either collimated jets or wide-angled winds resulting from the activity of the central supermassive black hole <cit.>. Theoretical models and observations indicate that both jets and winds are launched by distinct mechanisms. The Blandford-Znajek mechanism is attributed to the generation of jets from the CSMBH <cit.>, rendering them collimated and highly magnetized. The formation of disc winds is explained by three popular models, which are radiation driving <cit.>, magnetic driving <cit.> and thermal driving <cit.>. These winds are wide-angled outflows, as presented in the models from <cit.> and <cit.>. 


The primary source of emission from both winds and jets is attributed to shocks that accelerate non-thermal electrons. These accelerated electrons generate incoherent synchrotron radiation, which is detected as radio emission <cit.>. 
A recent survey by <cit.> has shown that almost all high mass galaxies (M > 10^11 M_⊙) show radio emission beyond a threshold limit of 10^21W  Hz^-1. In radio-loud sources[“radio-loud” sources are characterized by high radio luminosity (e.g. P_1.4 GHz>>10^24 W Hz^-1) and high ratio-to-optical flux density ratio (>10), and vice-versa for radio-quiet sources <cit.>.], the jets are expected to dominate the radio emission, whereas the origin in radio-quiet AGN is widely debated <cit.>. In such sources, it can depend on various mechanisms, including synchrotron emission from electrons accelerated in low-power magnetized jets <cit.>, wide-angle winds <cit.>, and star formation <cit.>.


Polarimetry studies <cit.> have found distinctions in polarization features that enable differentiation between radio emissions arising from star formation and AGN-driven outflows. Several recent studies indicate that traditional definitions of radio-quiet may not always mean a lack of AGN winds or jets <cit.>; nonetheless, in several instances, the nature of the emission remains ambiguous. A potential solution lies in the radio emission and polarization study of resolved jets/winds, as it can aid in distinguishing between various radio emission mechanisms. The expectation stems from the understanding that jets and winds are launched through different mechanisms and are therefore expected to interact differently with their surroundings. Consequently, when these two outflows encounter a similar ambient medium, observable differences in their emission and polarization characteristics may arise, although this area remains largely unexplored. This serves as the motivation for our study (also discussed in <cit.>, hereafter referred to as Paper I) of jets and winds spanning a few kpc in scale, where we aim to analyze their observable properties.

In Paper I, we focused on magnetized jets interacting with turbulent magnetic fields. The effect of magnetized Shocked Ambient Medium (SAM) on the net polarization from the jet was also explored. In this subsequent paper, we now focus on the above features from winds launched in a magnetized ambient medium. In our studies, the jets and winds are launched in a hot halo at temperatures of 10^7K equipped with a turbulent magnetic field. Thus, our primary focus is on examining the evolution of jets/winds in an idealized scenario and understanding how this impacts their radio emission characteristics over time. This paper is organized as follows. The simulation setup for the study is briefly discussed in Sec. <ref>. Detailed results are presented in Sec. <ref>, where we specifically examine the dynamics, emission, and polarization characteristics in separate subsections. We discuss the distinctions in the emission and polarization features of jets from Paper I and winds from this study in Sec. <ref>. In Sec. <ref>, we discuss our findings and summarize our inferences regarding the jets and winds derived from our simulations.



§ SIMULATION SETUP



The setup for the ambient medium and the wind with mildly relativistic speeds in this study is kept similar to Paper I for the jets. Thus, initially, a hot halo is set up which is in the hydrostatic equilibrium in the presence of an external gravitational potential (ϕ(r)) as,

    d p_a/d r = - ρ_a (r)d ϕ (r)/dr;      p_a(r) = ρ_a (r)/μ m_a k_B T_h

from which we derive

    p_a(r) = (n_0 k_B T_h) exp[- ( μ m_a/k_B T_h) ϕ (r) ], T_h = 10^7  K

 where p_a and ρ_a = μ m_a n_h are respectively the pressure and density of the halo gas at spherical radius r, μ=0.6 is the mean molecular weight of the fully ionized gas, m_a is the atomic weight, and n_0 is the number density at r=0. The Boltzmann constant is shown using k_B, and T_h is the temperature of the halo. The total gravitational potential (ϕ(r)) is the sum of the baryonic and dark-matter potential, as described in Paper I in detail. Then, the turbulent magnetic fields are introduced into the ambient medium, and the winds are launched from the lower Z side of the simulation domain.


However, there are several differences, which are explained below. The simulation domain is a large box with dimensions of 8 kpc × 8 kpc × 4 kpc, corresponding to a grid of X × Y × Z, with a resolution of 800 × 800 × 400 cells. As the winds are expected to be broad, the wide box encompasses the cocoon and SAM for the winds by the time its head reaches the Z-altitude of 4 kpc. Similar to Paper I, we employ the RMHD module in pluto <cit.> along with a five-wave HLLD Riemann solver <cit.>. We use a piece-wise parabolic scheme for reconstruction <cit.> and a third-order Runge-Kutta method for time integration. In order to ensure numerical instability we use a more diffusive (HLL) Riemann solver and MIN-MOD limiter for shocked cells which are identified using a pressure condition with δ p/p_min > 6 in regions near the injection zone (|X, Y|≤ 0.5 and Z≤ 0.5 kpc). 

 
The turbulent magnetic field in the halo is incorporated by interpolating from a domain of size 4 kpc^3, as detailed in Paper I for compact jets. Given the larger simulation box here, the field is extended along the X and Y axes using periodic boundary conditions, as also done in Paper I for large-scale jet run. In this study, the maximum correlation length of the external magnetic field is 1000 pc. The investigation for varying the correlation lengths has already been performed in Paper I, and we anticipate similar outcomes in this regard. Consequently, our primary focus here is on examining the dynamics of winds with different powers launched in a similar turbulent magnetic field configuration. We use the constrained transport approach <cit.> to maintain the divergence-free nature of the magnetic field throughout the simulation. Additional specifics regarding the setup and numerical schemes can be found in Paper I, to which we direct the reader for further details.


The simulations performed in this study are listed in Table <ref>. The nomenclature of the simulations indicates the power of the injected wind and the correlation length of the ambient magnetic field, as explained in the footnote of Table <ref>. The half-cone opening angle for the wind (θ_W) is set at 70^∘ for a broader injection than the jets, which were launched with an angle of 7^∘. The radius of the injection zone is 0.2 kpc here; thus the diameter for the injected wind is resolved with 40 cells for a resolution of 10 pc in our runs. Simulations with high-density contrast (η_w) between the wind and the ambient medium (W43_L1000(D,HD)) are also performed to explore the effect of a dense wind on different physical measures. We also conduct runs for narrow winds, which are launched with a half-cone opening angle (θ_w) of 35^∘, and explore this for both light and dense wind cases (W43_L1000(N,HDN)). It can be seen that the winds exhibit mildly relativistic velocities in comparison to the jets in Paper I, which were launched with speeds reaching up to 0.99c. Outflows with velocities of around 0.25c, driven off by the accretion disc winds at distances of a few gravitational radii from the black hole, have been observed in the nuclear regions of several quasars <cit.>. These winds become mildly relativistic with distance and are likely to attain velocities similar to the values chosen in our study. For post-process analysis, we consider an electron population whose energy spectrum follows a power law with an index of 2.2, similar to the one assumed in Paper I. The synchrotron emissivities from radiation emitted by the acceleration of electrons in the presence of a magnetic field are estimated using the expressions given in Sec. 3 of Paper I. In the next section, we discuss the results of the dynamics and observable characteristics of winds from this study.













   











§ RESULTS
 



 §.§ Dynamics of wind


In this section, we discuss the structure of the wind, and how its characteristics vary in response to various factors, including wind power (P_w), density contrast between the injected wind and the ambient medium (η_w), and the wind's opening angle (θ_w). The general structure of light wind is illustrated through a schematic diagram and a 3D volume-rendered pressure map in Fig. <ref> and <ref>, respectively. Slices depicting density, pressure, and wind tracer[Tracers are passive scalar quantities between 0 and 1 in pluto, which gets advected with the fluid. The wind tracer (tr1) value at the injection zone is set to 1 at t=0, and zero elsewhere. See <http://plutocode.ph.unito.it/userguide.pdf> for further clarification.] in the Y-Z plane from different simulations are shown in Figs. <ref>, <ref>, and <ref>. These snapshots correspond to the times when the front head of the forward shock reaches close to the top of the simulation domain; indicating that the spatial scale of the winds is similar across all simulations.


    
  * Structure of the wind: The general structure of the wind is expected to align with the model presented in <cit.> and <cit.>. However, our study reveals deviations in the wind morphology, which primarily depend on the initial density ratio between the wind and the ambient medium (η_w). Specifically, dense winds (η_w ≳ 4 in this study) are stable and closely adhere to the wide-angle wind model, as depicted in above studies. Conversely, light winds (η_w ≲ 0.4) are more susceptible to external influences, resulting in instabilities and some deviations from this structure. Therefore, as the density of the wind decreases over time due to expansion, instabilities are likely to emerge. In this section, we focus primarily on the structure of the light wind, detailing the instabilities. A labeled schematic diagram for the structure of the light wind is shown in Fig. <ref>. Adjacent to it, the pressure map for W44_L1000 is presented to illustrate the various elements. The main features inside the wind cocoon include a Mach disk illustrating the wind's termination shock, and lateral streams emerging from the edges of the barrel (or conical-shaped region), which are shown using red arrows. The wind plasma is abruptly stopped at the termination shock of the Mach disk. However, the oblique shock near the outer edges of the barrel deflects the gas moving downstream. This leads to high-velocity lateral streams emerging around the barrel, which can also be seen from the 3D volume rendering of the pressure along with fluid velocity streamlines in Fig. <ref>. The shock reflections can be seen in the two streams in the pressure map in Fig. <ref>. Similar features can also be seen for other cases in Fig. <ref>. This wind structure bears similarity to the under-expanded jets <cit.> and has also been observed in the previous simulations of wide-angled winds/jets <cit.>.

    
  The backflows from the lateral streams are directed inwards i.e. in regions between the streams, as well as in the outward direction, as depicted using blue arrows in Fig. <ref>. Generally, the inward-directed backflows are weaker compared to the outward ones. As can be seen from the velocity streamlines shown in Fig. <ref>, these backflows lead to eddies/vortices in the cocoon. The eddies lead to velocity shears in the inner layers of the contact discontinuity, thereby facilitating some entrainment (or mixing) of the shocked ambient gas with the wind plasma. This causes wind tracer (tr1) values lower than 1 in these regions, which will be discussed in the next sections.
    
        
  * Dependence on the wind power: The logarithmic density, pressure, and wind tracer maps for simulations with different wind powers are shown in Fig. <ref>. The white and yellow contours in the right panels show the regions enclosed within the cocoon and forward shock, respectively. It can be seen that as the wind power decreases, the forward shock takes a more spherical shape. Conversely, high-power winds carry strong forward momentum, resulting in an elongated appearance along the Z direction. The SAM for W42 shows a double-layered structure where the density is very high in regions closer to the cocoon, and decreases outwards. The inner SAM consists of the old swept-up ambient medium and hence depicts high gas density. The outer SAM also indicates that the forward shock from W42 is not as powerful as the high-power winds by the time it reaches a similar altitude. This can affect the acceleration of electrons at the forward shock and in turn, the polarized emission, which is discussed later in Sec. <ref>.

   The backflows on the lateral sides of the cocoon generate eddies (as shown in Fig. <ref>), leading to a reduction in gas pressure in these regions. Simultaneously, a high-pressure region develops in the forward zone ahead of the Mach disk. The outward eddies act to confine the wind, resulting in collimation and instabilities in the fast streams from the edges of the barrel. Inward-directed backflows and high-pressure zones ahead of the Mach disk, as observed in W43 and W44, counteract this collimation. However, this is not the case for W42 due to weak inward-directed backflows. This suggests that low-power, light winds are likely to experience strong collimation from the surroundings, resulting in a considerably narrower cocoon, and conversely for the high-power ones. 

The wind tracer maps in the right panels indicate that the cocoons undergo some mixing with the ambient gas. The tracer values are 1 inside the conical zone and along the fast lateral streams. High tracer values (tr1>0.75) are also observed in the central and lateral regions of the cocoon, where the backflowing wind plasma is present. Some mixing can be seen on the lateral sides, and is significant near the cocoon head, resulting in lower wind tracer values (tr1≲0.25). Initially, the formation of fast streams and strong backflows in the lateral parts of the cocoon causes the ambient gas to fill most regions between the streams above the Mach disk. Inward-directed backflows cause shearing in the inner layers of SAM, leading to some entrainment of the external medium. This results in low wind tracer values near the top head of the cocoon. Stronger backflows in high-power winds cause more entrainment compared to low-power winds. The low-power wind W42 initially confines itself within a narrow cocoon, limiting the extent of its eddies and reducing mixing with the external medium.

    
    
  * Dependence on the density of the wind: Maps for different physical quantities for a dense and highly dense wind with a power of 10^43 are presented in Fig. <ref> (referred to as W43_L1000D and W43_L1000HD in Table <ref>, respectively). It is evident that, as the wind density increases, the Mach disk in the winds is able to advance more along the Z-direction. This is because the dense winds carry a high momentum flux and are, thus, weakly affected by interaction with the surroundings. Consequently, the lateral streams manifest at later times when the wind has progressed further than the case of the light wind in Fig. <ref>, resulting in shorter streams. This implies that with increasing density, the winds become more stable, closely resembling the wide-angled wind model from <cit.>.
    
   It can also be seen that the cocoon of dense winds lacks the characteristic “hump-like” feature at the head, as observed in the case of light winds in Fig. <ref>. While a small “hump” is discernible in the dense wind, it is notably absent in the highly-dense case. In light winds, the lateral streams are unstable due to confinement force from eddies in the lateral regions, and thus the backflowing plasma is directed both inwards and outwards.  The inward-directed backflows try to fill the volume ahead of the Mach disk, resulting in some plasma being pushed upward, creating a distinct “hump-like” structure within the cocoon. In contrast, when the wind is dense, and thus, relatively stable, the lateral streams emanating from the edges of the barrel are comparatively stable. As a result, strong backflows from high-velocity regions occur primarily along the lateral sides and are weaker inwards, leading to the absence of a prominent hump at the cocoon's head.

   The tracer maps displayed in the right panel indicate that some entrainment of the external gas (tr1≲0.25) is present near the cocoon head of the dense wind. This results from the formation of fast lateral streams and inward-directed backflows at later stages. However, in the case of the highly dense wind, the lateral streams are still in their initial phase, and thus no entrainment of the shocked ambient gas is seen here.
  
    
     
  * Dependence on the opening angle of the wind: Maps for a light and highly dense wind with a half-cone opening angle (θ_W) of 35^∘ and power 10^43 are shown in Fig. <ref>. It can be seen that a narrow light wind experiences strong collimation from its surroundings when compared to the wide-angled wind of similar power in Fig. <ref>. Consequently, the wind becomes more constrained, resulting in the formation of a slender, elongated high-pressure zone ahead of the Mach disk. This also leads to high-pressure knots in these regions and a high-pressure cocoon head. Similar high pressure can also be seen near the cocoon head of the dense narrow wind, resulting from collimation from the surroundings. The dense wind is stable than the light wind here, and so fast lateral streams have not yet developed. The narrow winds undergo weak mixing with the external medium, as can be seen from the tracer map in the right panel.
     
     


Similar to the jets in Paper I, the bow shock of the wind compresses the external magnetic fields. Fig. <ref> displays the probability density function (PDF) of the magnetic field within the cocoon and SAM across wind simulations with different powers. In all cases, the PDF for the SAM consistently lies on the left side of the cocoon, indicating comparatively weaker magnetic fields. However, it is noteworthy that the peaks in the PDF from the cocoon and SAM get close in magnitude as the wind power is decreased. This discrepancy arises due to the decay of the magnetic field within both the cocoon and the SAM as expansion occurs, as previously shown in Paper I. Consequently, the magnetic field within the cocoon of low-power winds, which are initiated with comparatively weaker magnetic fields (refer to Table <ref>), has significantly decayed to values comparable to those within the SAM. Conversely, the magnetic fields in the high-power winds, particularly for W44, are sustained at values higher than the SAM by the time the cocoon's head reaches the top of the simulation domain. The distinct magnetic field strengths in the cocoon and SAM lead to varying effects on the emission and polarization originating from the wind, as elaborated in the subsequent sections.



 §.§ Synchrotron emission and polarization from winds


In this section, we discuss the synchrotron emission and polarization from the wind simulations listed in Table <ref>. We present maps illustrating the logarithmic flux, polarization fractions (π_T and π_W), and the fractional change in polarization ((π_W - π_T)/π_W) for winds with different powers at θ_I=90^∘ (angle from the +Z-axis) and ϕ_I=0 (measured counter-clockwise from the X-Z plane) in Fig. <ref>. The polarization vectors are overlaid on the polarization map with black arrows. Similar to Paper I, π_T maps correspond to the total polarization obtained from the cocoon + SAM, and π_W shows the polarization from the cocoon only. The conditions for identifying the regions inside the forward shock and SAM are mentioned in Appendix <ref>. The fractional change in polarization due to the contribution of the SAM along the LOS is shown using Δπ = (π_W - π_T)/π_W, i.e., Δπ = 1 implies very strong depolarization due to the SAM, and 0 means no depolarization. Fig. <ref> showcase maps from dense (W43_L1000D) and highly dense (W43_L1000HD) winds. Additionally, in Fig. <ref>, maps for a narrow light and dense wind (W43_L1000N and W43_L1000HDN) are presented. In the paper, we assume ϵ_W and ϵ_SAM=0.1, implying that the electrons carry exactly 10% of the fluid energy density in the cocoon and SAM for almost all the cases. However, the acceleration efficiencies for non-thermal electrons in the cocoon and SAM can vary, which is explored in Fig. <ref>. Figs. <ref> and <ref> showcase maps depicting winds with different powers and the dense wind case, respectively, for an image plane at θ_I=60^∘. Below, we discuss the common characteristics of synchrotron emission and polarization, focusing on maps from the light winds at θ_I=90^∘ illustrated in Fig. <ref>. How different physical parameters (wind power (P_w), density contrast (η_w), opening angle (θ_w), viewing orientation of the observer (θ_I), etc.) can affect these features in the winds is discussed in the subsequent section (Sec. <ref>). The depolarization effect from the turbulent field in the SAM is discussed in Sec. <ref>.


















  §.§.§ General characteristics of emission morphology and polarization


Morphology of emission: 

As can be seen from Fig. <ref>, the general characteristics of wind emission at θ_I=90^∘ are diffuse spherical extended emission with a bright cylindrical structure that is capped by a horizontal bright band. However, the shapes of these substructures can vary depending on different parameters and also the observer's orientation, which we discuss in subsequent sections. The high emission in the central parts originates from the high-pressure zones situated between the lateral streams. Notably, nearly horizontal arcs of high emission are visible near the cocoon head (refer to H1, H2, and H3). Such a feature in the light winds arises due to elevated pressure and high magnetic fields, particularly the B_y component of the wind's magnetic field in the Y-Z planes[It is important to note that only the synchrotron emission resulting from the field components perpendicular to the line of sight contributes to the total emission at the designated image plane.]. 



Additionally, shearing of the fields (particularly B_y and B_z) can also cause bright filamentary structures (F1 and F2) in the regions between the lateral streams. The vertical filaments are primarily influenced by the poloidal component (i.e., B_z) of the magnetic field, as can be seen from the middle row in Fig. <ref>. The lateral regions, characterized by backflows and eddies (refer to Fig. <ref>), depict low pressure, consequently exhibiting weak emission.

Distribution of polarization: 
The polarization distribution (π_T) shows high polarization primarily from the forward shock, the central regions of the wind, and the horizontal arc, which coincides with a bright band in the synchrotron maps. High polarization at the forward shock arises from the compression of the external magnetic fields parallel to the shock. This outcome is similar to the findings observed for jets in Paper I. The polarization vectors align perpendicularly to the shock surface, indicating the compression of magnetic fields parallel to it. Within the SAM, there is a gradual decay of turbulent fields, resulting in low polarization levels due to the small-scale coherence of the fields. Regions above and below the high-polarized bright horizontal arc also display low-polarization values. This is due to turbulent fields in the SAM, and backflows in regions between lateral streams respectively in upper and lower zones. The poloidal fields between the lateral streams lead to low polarization in the cocoon, which is caused by the vector cancellation of orthogonally polarized components along LOS.


In π_W (also π_T) maps, there is a prevalence of high polarization, attributed to the injected toroidal magnetic field within the wind, as indicated by the vertical polarization vectors. However, certain regions near the cocoon head and lateral sides exhibit low polarization values, resulting from the generation of poloidal fields. The poloidal fields are introduced due to velocity shears caused by backflows in these regions (see Fig. <ref>). Additionally, the mixing with the surrounding SAM, facilitated by eddies in the backflows, introduces further poloidal fields, which are subsequently amplified by turbulent motions within the cocoon (see B_z map in Fig. <ref>). Such shearing of the fields near the contact surface leads to high polarization, particularly due to the small integration length in these regions. Consequently, the polarization vectors align perpendicularly to the projected surface of the cocoon in π_W maps. 



  §.§.§ Effect of different parameters on the observable features



    
  * Dependence on the wind power:
The emission maps in Fig. <ref> indicate that the bright region in the central parts gets wider with increasing wind power. This, as shown in Fig. <ref>, is attributed to the weak collimation of the lateral streams, resulting in wide high-pressure zones above the Mach disk for the high-power winds. As a result, the bright horizontal arc is more extended as the wind power is increased. These arcs near the cocoon's head display a double-layered appearance for light low-power winds here, which is clearly visible for W42 and W43 and is less prominent for W44. The upper arc is formed due to the shearing of the B_y component near the Y-Z midplane, resulting from inward-directed backflows. This is evident from the evolution of the B_y field displayed in the top row of Fig. <ref>, where fields are sheared to form an elongated, nearly horizontal structure. The lower arc is a consequence of the B_y component of the toroidal magnetic field of the wind, coinciding with high-pressure zones at the top edges of the lateral streams. We confirm that such features may not be as distinctly observable from other lines of sight (e.g., in the X-Z image plane), as it depends on the instantaneous flows in the wind cocoon. As indicated in Table <ref>, the same magnetization value of 0.1 across all winds results in relatively weak magnetic fields for low-power winds in comparison to high-power ones. Consequently, as the cocoon expands, the local magnetic field gradually decreases, reaching a level comparable to that in the SAM (Fig. <ref>). This results in nearly uniform emissions from the W42 wind. Conversely, high-power wind cocoons exhibit stronger magnetic fields than the SAM. Consequently, the emissions from the central parts of such cocoons are considerably higher compared to those emanating solely from the SAM.

From the polarization maps (π_T) in Fig. <ref>, it can be seen that the high polarization at the forward shock is particularly evident in the W43 and W44 cases, but not in W42 due to weak forward shock (see Fig. <ref>). The polarization map in W42 shows larger areas of low polarization when compared to cases with higher power. This occurs as the SAM in W42 is broader, which enhances the decay and disorder of the magnetic fields within it. Consequently, the bright arc near the cocoon head is notably visible in the W42 case.


 
  * Dependence on the density of the wind: 
 In order to compare how the wind density contrast can affect the observable features, we focus on the light, dense and highly dense wind cases with power 10^43. The maps from the light wind case are displayed in the middle column in Fig. <ref>. The emission and polarization maps for the dense and highly dense winds at θ_I=90^∘ image plane are illustrated in Fig. <ref>. It can be seen that the bright and highly polarized arc at the cocoon's head appears more extended for the dense winds when compared to the light wind case. This occurs due to more stability of the dense winds (as discussed in Sec. <ref>) when compared to the light winds. Also, the absence of a double-layered structure in the horizontal bright band, as observed in lighter wind cases, is evident here. This absence, as illustrated in the bottom row of Fig. <ref>, is attributed to the lack of enhancement in the B_y component within the Y-Z midplane, a phenomenon resulting from inward-directed backflows in lighter wind cases. Consequently, the B_y component of the toroidal field, coinciding with high-pressure zones at the top edges of the lateral streams (as depicted in the B_x map in the bottom row), contributes to the formation of the bright horizontal arc when integrated along the line of sight in this scenario. Some emission can be seen between the Mach disk and horizontal arc for dense wind, giving rise to a “mushroom-shaped” morphology. This, however, is not seen for the highly dense wind, as the whole regions above the Mach disk contribute to the wide bright band here. 

Also, the horizontal arcs are quite prominent in the dense wind cases when compared to the light wind case. It can be seen that the emission from the central parts of the light wind is almost uniform, making the arcs less prominent. Inward-directed backflows in the light winds introduce toroidal as well as poloidal components between the lateral streams (see the middle row in Fig. <ref>), leading to almost homogeneous emission from regions above the Mach disk, which however is not the case for the dense winds (bottom row in Fig. <ref>). Contrarily, backflows in the dense wind mainly occur in the lateral sides of the high-velocity streams, which also generate poloidal fields. This leads to low polarization values in these regions, as can be seen from the π_W maps.

 
  * Dependence on the opening angle of the wind:  
Fig. <ref> shows the emission and polarization maps for narrow light and highly dense winds, respectively, observed at an angle of θ_I=90^∘. The collimation of lateral streams in the light wind case leads to bright knots (labeled as N1). Both cases exhibit intense compact emissions labeled as H6 and N2, stemming from high-pressure zones near the cocoon head, which also coincide with high-polarization values. The highly dense wind case displays a horizontal bright and highly polarized arc from regions above the Mach disk (see H7 labeled region). However, in this instance, the width of the arc is relatively narrower, as the regions above are more constrained, resulting in a bright compact spot (N2) ahead. 
 

  * Different particle acceleration scenarios:
 In Fig. <ref>, the emission and polarization maps correspond to varying values of ϵ for the cocoon and SAM, i.e. implying different acceleration efficiency of the electrons in these regions. The top row shows the maps for higher ϵ_W (0.1) than ϵ_SAM (0.01); implying higher acceleration in the cocoon than SAM, and conversely in the bottom row. The electrons within the cocoon experience acceleration primarily at the Mach disk, while in the SAM, acceleration can be expected at the forward shock. When the efficiency of acceleration in the cocoon is notably high in the presence of magnetic fields <cit.>, it results in stronger radio emissions originating from the cocoon compared to that from the SAM (as shown in the top row of Fig. <ref>). Conversely, if the SAM exhibits substantially higher acceleration efficiency <cit.>, the observable radio emission characteristics from the wind cocoon become less discernible (see bottom row). 

The polarization maps (π_T) indicate that when the wind is strong in accelerating electrons in the cocoon, most of the regions from the wind display high polarization. Conversely, the region exhibiting low polarization from the wind becomes more pronounced in the bottom row. This observation suggests that for radiation-driven winds, where the magnetic fields inside the winds may not be very strong, it is improbable to obtain significant radio polarization from the wind. However, high polarization can be observed from the forward shock region in SAM only if the emission is strong enough to be detected.

Moreover, it should be noted that the acceleration efficiency of the electrons at the forward shock can also vary in time. For example, as can be seen from density and pressure maps in Fig. <ref>, the forward shock in W42 has considerably weakened compared to the other cases, developing a double-layered structure. This weakening diminishes the impact of the outer shock on the compression of external magnetic fields, consequently lowering the acceleration efficiency of electrons in this region. This leads to low polarization values at the forward shock for W42, as can be seen from Fig. <ref>. 


    
  
  * Dependence on the viewing angle: At an inclined view of θ_I=60^∘ in Fig. <ref> and <ref> the emission from the central parts of wind appears “hourglass-shaped”. The majority of this emission originates from regions situated above the Mach disk, with additional contributions coming from the areas surrounding the barrel zone, as illustrated at θ_I=90^∘ in Fig. <ref>. Consequently, in light winds (see e.g. W42), these structures are narrower due to the strong collimation and relatively broader for the high-power light winds. The injected toroidal magnetic field results in high polarization from these regions in all the cases. In the surrounding areas with weak flux, low polarization is attributed to vector cancellation arising from orthogonally polarized components, along with contributions from the SAM. It should be noted that an inclined LOS, the toroidal field components are not uniformly perpendicular or parallel to the line of sight. This also add on to the low-polarized regions seen around the central spines. The distinctiveness of low polarization in W42 is emphasized due to its broader SAM, leading to rapid decay and disorder of the turbulent fields.
  
The broad annular rings at the top of the “hourglass” structure (particularly visible on the approaching side for W43 and W44) represent high-pressure annular regions above the edges of the lateral streams coinciding with high toroidal field components. Due to weak inward-directed backflows in the dense wind when compared to the light winds, any emission from the regions between the arcs is considerably weaker. Nonetheless, the bright outer arcs display high polarization in all cases. These arcs are distinctly visible for W42, whereas they seem merged with the high-polarization regions from the SAM in others.


  


  §.§.§ Depolarization effect of the SAM




   
  * Dependence on the wind power: 
  The fourth row in Fig. <ref> depicts the fractional change in the polarization of the cocoon (Δπ) after the contribution from the SAM. More reddening for the W42 clearly indicates that the presence of a turbulent field in the SAM leads to more lowering in the polarization from the cocoon for the low-power winds. This is because the magnetic fields in the SAM are quite comparable to the cocoon for the low-power winds, as can be seen from the PDFs of the magnetic fields in Fig. <ref>. Contrarily, the high-power winds carry stronger magnetic fields than the SAM, and thus the depolarization effect is weaker. As found in Paper I, the magnetic fields in the cocoon and SAM decay with the expansion. When the fields in both regions become comparable, the depolarization from the SAM gets strengthened.
   

  * Dependence on ϵ_W:
It can be seen from Δπ maps in Fig. <ref>, the depolarization from the SAM becomes prominent for the case with ϵ_SAM=0.1 and ϵ_W=0.01. In the context of a radiation-driven wind, the non-thermal radio emission from the wind cocoon may not be very strong. Under such circumstances, the SAM's contribution takes precedence in the total non-thermal emission, as illustrated in the left panels of the figure. Consequently, the diminished polarization of the wind primarily results from the contribution of turbulent magnetic fields within the SAM.














 



§ COMPARING SYNCHROTRON AND POLARIZATION CHARACTERISTICS FROM JETS AND WINDS



The primary motivation of this section is to identify some noticeable observational features for the non-thermal synchrotron emission and polarization to distinguish between the jets and winds. Emission and polarization maps depicting jets and winds with varying powers and viewing angles are presented in Figs. <ref>-<ref> and Figs. <ref>-<ref>. To compare with observed results, these maps are convolved with a Gaussian filter with a different standard deviation of 100 pc and 300 pc and truncated at twice these scales[This implies FWHM of 240 and 720 pc, and truncated at scales of 400 and 1200 pc, respectively.]. The corresponding FWHM gives an angular resolution of 125 and 375 milliarcsec, respectively, for sources at a redshift (z) = 0.1 (assuming H_0= 67 kms^-1 Mpc^-1, and Ω_M=0.3). The latter values are comparable to the resolution scales reached with previous studies <cit.> and can be achieved in the local universe <cit.>. For a larger redshift of 2, they correspond to a resolution of around 27.4 and 82 mas, which is typically not achieved by the studies looking at this scale <cit.>; although can be possible using high-resolution data <cit.>. In these figures, we mainly focus on one order in magnitude for the flux dynamic range in the fiducial maps presented here and also show maps for the two-order-of-magnitude flux range for W42 and W44. The polarization vectors are depicted as black arrows. These snapshots correspond to times when the jet or wind head approaches near the top of the Z axis, and the forward shock is confined within the box. The correlation length of the external magnetic field is 1000 pc in all the simulations, except for J42, where, we ran the simulation with only a 500 pc coherence scale in Paper I.



 §.§ Characteristics in emission morphology

The winds and jets display some common traits as well as some differences in emission morphology, which are listed below:

    
  * In the emission morphologies, it is evident that bright features emerge from the central regions in both jets and winds, which are surrounded by weak extended emission (see Figs. <ref> and <ref>). A jet has a systematically narrow spine with defined hotspot, or hotspot complexes for kink-unstable jets (see h1, h2, and h3). Winds, on the other hand, display broader spines or “hourglass-shaped” bright emission. The bright annular rings/arcs observed at an inclined view of 60^∘ (R1, R2, and R3) manifest into bright horizontal arcs near the edges of the cocoon at 90^∘ (H2 and H5). Such rings/arcs are not seen for jets and are a particular feature in the winds.
    
    
  * Winds exhibit a diverse range of morphologies depending on their power and density. Light and low-power winds tend to become collimated, unlike high-power winds. As a result, the presence of a hotspot may not reliably distinguish jets from winds, as pronounced collimation in low-power winds can produce similar features (as observed for W42 in Fig. <ref>). We have confirmed that such strong collimation and hotspot-like regions do not form in highly dense (η_w=4) wide-angled winds with a power of 10^42. Additionally, the emission closely resembles that of the dense wind case illustrated in Fig. <ref>. This suggests that dense winds are more stable and thus, exhibit prominent bright annular rings (and bright distinguished horizontal arcs at θ_I=90^∘, as inferred from Fig. <ref>) compared to lighter winds. However, it is worth noting that the hotspot-like features can still form in dense narrow-angled winds, as depicted in Fig. <ref>. 
    


  
  * Effect of dynamic range: To illustrate the impact of dynamic range, we present the emission maps for W42 and W44 using a two-orders-of-magnitude flux range in Fig. <ref> (see top and bottom right plots). Notably, the extended emission is narrower for jets than for winds. It can be observed that the emission from regions near the forward shock in the SAM is lower compared to that from the central parts of the wind. Consequently, while the central bright regions are unlikely to be missed when observations are limited to one order of magnitude in dynamic range, outer regions, which are part of the SAM, can be overlooked.


  
  * Effect of resolution:
  The extent of bright emission from the central parts varies in both jets and winds with resolution. To facilitate comparison, we calculated the average width of the regions surrounding the central vertical axis, encompassing 80% of the maximum flux at the given Z' value, within a range of Z'=± 1 kpc in Fig. <ref>. At high resolutions, these widths measure 420 pc and 310 pc for J42 and J44, respectively, and approximately 1.2 kpc and 1.3 kpc for W42 and W44.  In low-resolution maps shown in Fig. <ref>, the widths of the bright spines have increased to quite comparable values of 920 pc and 1.2 kpc for J42 and W42, while for J44 and W44, these are 630 pc and 1.36 kpc, respectively. Furthermore, it is apparent that at low resolution, several features, such as the spine, hotspots, filaments, and arcs, have become less prominent. This leads to some resemblance in appearance between jets and some wind models. For example, in light winds, bright “hourglass-shaped” regions appear close to the spine of the jets. Interestingly, bright regions in W42 closely match the emission morphology from the jet in J44. Although a hotspot in W42 might resemble that of a jet, it is important to recognize the absence of a bright, thin spine typically associated with jets, which becomes evident at higher resolutions. Since the emission morphology is often used to distinguish between jets and winds, our study shows that some features can become blended when poor resolution is used. This underscores the importance of higher-resolution radio observations; however, noting that “only” utilizing the very highest resolution images may resolve away important diffuse structures that also contain important diagnostic information in radio morphology and polarization. 





 §.§ Distribution of polarization

Below, we list some common characteristics and differences in the polarization structure from the jets and winds.

    
  * Some notable features can be clearly seen from the maps depicted in Figs. <ref> and <ref>. High polarization is mainly detected along the central spine and hotspots/arcs, which are localized in jets and are more widespread in the winds. Additionally, high polarization is observed along the filament (F3) within the jet, although it is reduced due to smoothing.

    
  * In jets, particularly those with a higher power, the hotspots exhibit noticeable high-polarization values, typically found near the upper edges of the cocoon. These hotspot(s) exhibit compactness, resulting in polarization originating from them being concentrated at specific locations (e.g. refer to h1, h2 h3). However, in winds, high polarization mainly occurs from the central parts and top edges of the cocoon (refer to R1, R2, R3, H2, and H5 from emission maps). These arcs are surrounded by comparatively low-polarization values (e.g. see maps for W42 and W43 in Fig. <ref>). The lateral edges of the maps can also display high polarization values, as can be seen particularly in the middle row left panel in Fig. <ref>.
    
    
    
    
  



  * In Fig. <ref>, both winds and jets display low polarization values in their extended regions, which mainly arises because of the vector cancellation of orthogonally polarized components, and also some contributions from the SAM. It is clear that in the jets' cocoon, where stronger poloidal fields are generated due to pronounced backflows, experiences more internal depolarization compared to winds (e.g. see π_J maps in Fig. 9 of Paper I and π_W in Figs. <ref>-<ref>). The higher prevalence of low polarization in W42 is linked to its wider SAM and relatively narrower cocoon, as depicted in Fig. <ref>, resulting in more decay of turbulent fields.

 
  * Effect of dynamic range: The top and bottom right plots for total polarization (see Fig. <ref>) indicate that a two-order-of-magnitude flux range enables the observation of polarized emission from the outer parts of the SAM. This remains invisible with a lower dynamic range, as shown in the plots on the left side. These regions may exhibit either high or low polarization values in different wind power simulations. In the case of W42, the weak strength of the forward shock leads to low polarization, whereas relatively higher polarization is seen in W44. The lateral regions in W44 (which corresponds to SAM) also display high-polarization values, similar to those seen for the jets (see left panels), which were not observed at the lower dynamic range. At θ_I=90^∘, high values like these are not observed in 1-dex range maps (see Fig. <ref>). However, we confirm that they are seen, primarily near the bow shock's head, in 2-dex range maps.

   
  * Effect of resolution:
In Fig. <ref>, it is evident that decreasing the resolution has not altered the qualitative distribution of polarization characteristics. However, low-polarization values have become more prevalent. Polarization values near the spine and hotspots in jets appear lower compared to the high-resolution maps. In winds, the highly polarized arcs near the cocoon head seem to blend with regions from the SAM, and the resulting values are diminished after smoothing. This suggests that polarization can be an important factor in distinguishing between the jets and winds.





§ SUMMARY AND DISCUSSION



AGN-driven jets and winds are believed to be launched by distinct mechanisms <cit.>. Numerous simulations have been performed for both <cit.>, shedding light on various aspects relevant to real systems. It is found that the evolution and stability of jets are influenced by factors such as Mach number, Lorentz factor, and density contrast with respect to the ambient medium <cit.>. Conversely, the understanding of winds remains less clear, with their origin being a subject of ongoing debate. It is plausible that a single mechanism may not suffice to explain the observed ultrafast outflows (UFOs) in quasars, suggesting the contribution of multiple processes <cit.>. This underscores the necessity for incorporating more sophisticated physics and a deep understanding of physical conditions in AGNs to comprehensively account for the winds.


Observations have identified clear indications of AGN-driven winds, whether through direct or indirect means. For instance, an inverse relationship between the radio loudness and the column density of ionized wind is found in radio-loud AGN by <cit.>. This connection suggests a shared magnetic mechanism responsible for initiating both wind and jet formations <cit.>. Excess radio emissions within several quasars also appear to be predominantly influenced by the presence of either winds or small-scale jets <cit.>. 



Theoretical simulations can help establish a connection with observational findings, bridging the gap in our understanding of AGN-driven jets and winds. In our work (Paper I and this paper), we have attempted to gain insights into the evolution of jets and winds within a medium enriched with turbulent magnetic fields. We have examined the behavior of these at different energies, all of which are initiated with a toroidal magnetic field in our studies. Our analysis reveals that the evolution of jets and winds follows different behaviors, leading to observable distinctions in their features, which we summarize in the following section.



 §.§ Dynamical and observational perspectives on jets and winds


As indicated in Sec. <ref> of this study and Sec. 4.1 in Paper I, the evolution of jets and winds exhibits distinct evolution characteristics. Jets, characterized by high beaming and relativistic speeds, advance more rapidly in their surroundings compared to winds. In contrast, winds exhibit mild relativistic speeds with a wide-angle spread, providing a larger surface area for interaction with the surrounding medium. This wide-angle nature results in instabilities at the periphery of the barrel zone, particularly evident in light winds, leading to lateral streams from the sides of the barrel (see Fig. <ref>). These streams influence wind evolution by inducing backflows ahead of the Mach disk and contributing to mixing with the SAM. Dense winds, in contrast, demonstrate stability and propagate without being much affected by their surroundings. The emission and polarization features from the winds with varying parameters (such as power, density, and opening angle) are discussed in detail in Sec. <ref>.


We have analyzed polarization and emission from jets and winds with different models and compared them in Sec. <ref>. The insights from the observable distinguishing traits between jets and winds are outlined in Table <ref>. We find that several characteristic features, such as hotspots and spines, may not be good indicators for poorly resolved sources, highlighting the need for highly resolved observations. The polarization distribution appears to be a reliable indicator for distinguishing between jets and winds, as it is not significantly affected by resolution. Jets exhibit compact, bright emissions from narrow spines and hotspots, aligning with high polarization values. Winds display similar widespread features, with the polarized spine nearly hourglass-shaped and capped with polarized bright arcs. Additionally, our studies suggest that depolarization, due to poloidal fields generated in the cocoon, is more dominant in jets compared to winds. It should be noted that both jets and winds are initially launched with a toroidal magnetic field in our study; nevertheless, a poloidal component at the base region is expected to exist, giving rise to helical magnetic fields. Consequently, the amplification of the poloidal field is likely to be much more pronounced in jets compared to winds. Contrarily, a weak poloidal component is seen in winds, with a dominance of toroidal field elsewhere in the cocoon. In several radio sources, such transverse or toroidal components at the base of the jets (with poloidal fields) are associated with AGN wind <cit.>. The jets remain collimated over longer spatial scales, although the cocoons widen as jet power decreases. However, denser winds consistently remain broad. Lighter, low-power winds can become unstable at earlier stages, leading to collimation by the surroundings and resulting in narrower cocoons than those of their high-power counterparts.



 §.§ Limitations and future aspects

In our studies (Paper I and this work), we have focused on the evolution of magnetized compact collimated jets and wide-angled winds in a hot halo carrying turbulent magnetic fields. However, in realistic scenarios, such compact jets/winds are expected to interact with the multiphase ISM of the host galaxy. This interaction, in turn, may also influence their synchrotron and polarization characteristics, which we reserve for future exploration. In addition to investigating the morphology of emission and polarization, simulations can also provide valuable quantitative parameters, such as spectra originating from the acceleration of non-thermal electrons within the jets and winds. Particle acceleration processes are likely to differ between jets and winds. For example, significant acceleration in jets may occur along the spine and at the hotspot, while in winds, the Mach disk can accelerate particles. Furthermore, differences in the strength of forward shocks and SAM properties in both jets and winds may have varying effects on particle spectra originating from the SAM. This aspect is reserved for future research, wherein non-thermal electrons <cit.> will be introduced into the simulation domain and their evolution will be followed. Theoretical results can subsequently be juxtaposed with direct predictions from recent observational missions like LOw-Frequency ARray (LOFAR)[https://lofar-surveys.org/], eMERLIN[https://www.e-merlin.ac.uk/], Event Horizon Telescope (EHT)[https://eventhorizontelescope.org/], and Very Large Array (VLA)[http://www.vla.nrao.edu/], thereby facilitating a more clear understanding of jets and winds.




§ ACKNOWLEDGEMENTS

 We express our gratitude to the Accordo Quadro INAF-CINECA 2017 and `Pegasus' HPC facility at IUCAA, Pune, India, for high-performance computing resources. We also acknowledge the assistance of the INDO-ITALIAN mobility grant (INT/Italy/P-37/2022(ER)(G)). GB and PR acknowledge support by Next Generation EU through  PRIN MUR 2022 (grant n. 2022C9TNNX) and by the INAF Theory Grant “Multi-scale simulations of relativistic jets”. C.M.H acknowledges funding from a United Kingdom Research and Innovation grant (code: MR/V022830/1). MM expresses gratitude to Ankush Mandal for the valuable discussions.



§ DATA AVAILABILITY

The simulations generated for this study will be shared upon a reasonable request to the corresponding authors.




mnras
 



§ IDENTIFICATION OF FORWARD SHOCK AND SAM


The SAM region in different simulations is identified using the pressure (P), density (ρ), and jet-tracer (tr1) as:


    W42: (P>2 × P_0 or tr1>0 or ρ > 0.007 cm^-3) & tr1≤ 10^-3
    W43(N): (P>2 × P_0 or tr1>10^-20)  & tr1≤ 10^-3
    W43D: (P>1.6 × P_0 or tr1>10^-20)  & tr1≤ 10^-3
    W43HD: (P>1.5 × P_0 or tr1>10^-20)  & tr1≤ 10^-1
    W43HDN: (P>2 × P_0 or tr1>0)  & tr1≤ 10^-4
    W44: (P>4 × P_0 or tr1>10^-20) & tr1≤ 10^-3


Here, P_0 is the initial pressure in the simulation domain. The conditions in the brackets select the regions inside the forward shock. The SAM is then identified by the wind tracer limit at the end, which is also used to identify the regions inside the wind cocoon.





§ EVOLUTION OF MAGNETIC FIELDS IN THE LIGHT AND HIGHLY DENSE WINDS

Fig. <ref> illustrates the time evolution of the B_y component of the magnetic field for the light wind W43 in the top row. It can be seen that the inward-directed backflows cause shearing and amplification of B_y fields near the cocoon's head. The substructures of fields from opposite sides can combine, forming an elongated amplified magnetic component. Such phenomenon gives rise to the upper bright horizontal arcs in the light wind cases, depicted in Fig. <ref>.

 In the middle and bottom rows, we have shown the different magnetic field components in the Y-Z midplane for the light and highly dense winds, respectively. It is clear that the evolution of wind differs between these cases, as elaborated in Sec. <ref>, resulting in varying magnetic field structures. The B_x component represents the injected toroidal field and exhibits enhanced values in both cases. As shown in the top row, the inward-directed backflows enhance the B_y component of the field near the cocoon head in light wind. Moreover, the B_z component is also introduced above the Mach disk. On the contrary, a highly dense wind does not exhibit a significant enhancement in the magnetic field (B_y and B_z) above the Mach disk. The dominating toroidal component is notably higher in this case, particularly near the top edges of the lateral streams, which have partially emerged in this case.







§ JETS VS WINDS: SYNCHROTRON EMISSION AND POLARIZATION



The maps for the logarithmic synchrotron flux and polarization for the jets from Paper I and the winds from this study, observed at a viewing angle of θ_I=60^∘ are presented in this section. Specifically, the maps correspond to low resolution, where we employ a Gaussian filter with a FWHM of 720 pc for convolution, and are shown in Figs. <ref> and <ref>. Our focus primarily lies on displaying flux levels within a one-order dynamic range across all figures. However, for comparative purposes, we also include maps for W42 and W44 spanning a two-orders-of-flux range, which are shown in the top and bottom right plots. Regions below the lower limit of the flux color bar are omitted from both emission and polarization images.















 









 
  







 Paper Title: Axion-like Particle Effects on Photon Polarization in High-Energy Astrophysics.





§ INTRODUCTION


The Standard Model (SM) of particle physics provides a satisfactory theoretical baseline able to explain all known processes acting on elementary
particles. However, the SM cannot be considered as the ultimate theory of fundamental interactions since it does not really unify the electroweak and the strong interaction and does not contain gravity. Moreover, the SM describes neither dark matter nor dark energy, which together make up the majority of our Universe. Therefore, the SM represents a low-energy manifestation of a more fundamental theory describing and unifying all four fundamental interactions at the quantum level.

Several proposals have been presented with the aim of extending and completing the SM, such as
multidimensional Kaluza–Klein theories <cit.>, four-dimensional unsymmetrical models <cit.>, and superstring and superbrane theories <cit.>. A remarkable fact is that all these theories invariably predict the existence of axion-like particles (ALPs, for a review, see, e.g., <cit.>).

ALPs represent a generalization of the axion, the pseudo-Goldstone boson originating from the breakdown of the global Peccei–Quinn symmetry U(1)_ PQ, introduced to solve the strong CP problem (see, e.g., <cit.>). The axion is characterized by a strict relationship between its mass and its coupling to photons, and it interacts also with gluons and fermions so that the Peccei–Quinn mechanism can work. Instead, two aspects differentiate ALPs from axions: (i) the ALP mass m_a is unrelated to the two-photon-to-ALP coupling g_aγγ, and (ii) ALPs primarily interact with photons only, while other possible interactions can safely be neglected in the present context. ALPs are currently considered among the best candidates for being the long-sought dark matter <cit.>. Furthermore, ALPs produce two significant effects in the presence of external magnetic fields: (i) photon–ALP oscillations <cit.>, which are similar to the oscillations of massive neutrinos with different flavors with the difference that, in the ALP case, the external magnetic field is necessary for spin compensation between spin-0 ALPs and spin-1 photons, and (ii) the change of the polarization state of . Due to their faint interaction with photons, ALPs can hardly be detected over the limited distances of laboratory experiments. Instead, the astrophysical context allows us to overcome this restriction with photons that can originate at cosmological distances and can cross several magnetized media, where photon–ALP interaction can take place. Therefore, the astrophysical environment appears as the most promising way to study the ALP physics, especially in the high-energy (HE) and in the very-high-energy (VHE) band (see, e.g., <cit.> for a review). 

In particular, photon–ALP oscillations occur in different magnetized media, such as inside active galactic nuclei (AGN; see, e.g., <cit.> and Section <ref>). Specifically, photon–ALP oscillations explain the detection of photons with energies of up to ∼400   GeV originated from flat spectrum radio quasars (FSRQs, an AGN class), while conventional physics prevents any emission for energies above ∼20   GeV (see also Section <ref>): this fact represents the first hint at ALP existence <cit.>. Moreover, photon–ALP interaction solves the anomalous redshift dependence of the spectra of BL Lacs (another AGN class) <cit.>. In particular, within conventional physics, the values assumed by the BL Lac VHE-emitted spectral indexes decrease as the redshift grows, while no redshift dependence is expected, since cosmological effects are not present and selection biases are excluded, as discussed in <cit.>. The introduction of photon–ALP conversion inside the extragalactic space increases the Universe transparency <cit.>, partially avoiding the absorption of VHE photons due to their interaction with the extragalactic background light (EBL, see also Section <ref> ), and makes BL Lac VHE-emitted spectral indexes redshift independent. Since this is the only possibility according to the physical expectation, this result represents the second hint 
 at ALP existence <cit.>. Moreover, photon–ALP conversion inside the blazar jet, in the host galaxy, in the extragalactic space and in the Milky Way produces two features <cit.>: (i) spectral irregularities and (ii) photon excess in blazar spectra at VHE, which can be detected by current and planned observatories, such as ASTRI <cit.>, CTA <cit.>, GAMMA-400 <cit.>, HAWC <cit.>, HERD <cit.>, LHAASO <cit.>, and TAIGA-HiSCORE <cit.>. Furthermore, ALP-induced spectral irregularities produced inside the galaxy cluster turbulent magnetic field have been used to place constraints on the ALP parameter space <cit.>. In addition, photon–ALP interaction provokes modifications on stellar evolution <cit.>; it has been employed inside galaxy clusters as an explanation of the spectral distortions of the continuum thermal emission (T∼2  keV–8   keV) <cit.> and of the unexpected spectral line at 3.55   keV, viewed as dark matter decay into ALPs with their subsequent oscillations into photons <cit.>. ALPs have also been used in order to explain a blazar line-like feature <cit.>. The recent detection of the gamma-ray burst GRB 221009A by the LHAASO Collaboration <cit.> up to 18   TeV strongly challenges conventional physics, while the photon–ALP scenario naturally explains the VHE observation of this GRB, which represents the third hint and the firmest indication to date at ALP existence <cit.> (see also Section <ref> and <cit.>). Moreover, all three above-mentioned hints are derived with the same ALP parameters, which further strengthens the indication itself.

In this review, we concentrate on the other main effect of the photon–ALP interaction, i.e., the change of the polarization state of photons. This ALP effect has attracted less interest so far with respect to that producing spectral modifications. However, the existing and proposed polarimetric missions in both the X-ray band such as IXPE <cit.> (already operative), eXTP <cit.>, XL-Calibur <cit.>, NGXP <cit.>, XPP <cit.> and in the HE range like COSI <cit.> (approved to launch), e-ASTROGAM <cit.>, AMEGO <cit.> allow polarization studies to be carried out with unprecedented sensitivity. Therefore, these observatories represent a unique opportunity to perform additional studies on ALP physics. In particular, the ALP effects on the polarization of photons produced in GRBs have been studied in <cit.>. Attention to various ALP-induced polarization topics regarding different astrophysical sources, such as AGN, white dwarfs, and neutron stars, has been paid in <cit.>. Moreover, the photon–ALP interaction has been demonstrated as a unique way to measure the emitted photon polarization <cit.>. Quite recently, the impact of the photon–ALP interaction on the polarization of photons produced in the central region of typical galaxy clusters or generated at the jet base of generic blazars has been analyzed in <cit.>. Results obtained in <cit.> have been applied to concrete cases: (i) to Perseus and Coma concerning galaxy clusters, as shown in <cit.>, and (ii) to OJ 287, BL Lacertae, Markarian 501, and 1ES 0229+200 regarding blazars, as discussed in <cit.>, showing that, in all the cases, the initial photon polarization is strongly modified by the photon–ALP interaction with the production of features observable by present and planned missions <cit.>. The above-cited studies concerning blazars <cit.> and especially galaxy clusters <cit.> demonstrate that the measure of the polarization degree of photons produced by these astrophysical sources may provide us a new and an additional way to study ALPs and their effects. In fact, whenever the expectations of conventional physics about the photon polarization degree of astrophysical sources are in tension with possible future observations performed by missions in the X-ray band <cit.> and in the HE range <cit.>, this fact would represent a hint at ALP existence. This is especially true regarding galaxy clusters. As discussed below, photons diffusely produced in the central zone of galaxy clusters are expected to be unpolarized both in the X-ray and in the HE band within conventional physics, but the photon–ALP conversion makes them partially polarized, as shown in <cit.>. Therefore, a signal of partially polarized photons diffusely produced in a galaxy cluster would be a strong indication at ALP existence. Concerning blazars, we could achieve a similar conclusion if the detected polarization degree were extremely high, above the predictions of standard hadronic models (see below and <cit.>). In case no tension was found, the ALP parameter space (m_a, g_aγγ) could still be limited further. In this review, our aim is thus to describe the ALP-induced effects on photon polarization, which may be detected by missions in the X-ray band <cit.> and in the HE range <cit.> and which may provide us new ways to make progress in the ALP physics.

In particular, we introduce ALPs and the photon–ALP system in Section <ref>, and we illustrate polarization in Section <ref>. Section <ref> describes the astrophysical media crossed by the photon–ALP beam. In Section <ref>, the ALP-induced polarization effects are reported both in the X-ray and in the HE range for Perseus as an example for galaxy clusters and for OJ 287 as an example for blazars. In Section <ref>, the above-mentioned polarization effects are discussed along with their real detectability. Section <ref> is also devoted to describing possible future perspectives in the VHE range, where strong ALP signatures could be detected, along with a discussion about GRBs. Finally, we draw our conclusions in . Throughout this review, we employ the natural Lorentz–Heaviside (rationalized) units with ℏ = c = k_B = 1.



§ AXION-LIKE PARTICLES


ALPs are neutral, spin-zero pseudo-scalar bosons, whose primary interaction with photons (other interactions with fermions or gluons may exist but are subdominant and negligible in the present context) is described by the Lagrangian 

    ℒ_ ALP =  1/2 ∂^μ a  ∂_μ a - 1/2  m_a^2   a^2 -  1/4  g_aγγ  F_μνF̃^μν a  = 1/2 ∂^μ a  ∂_μ a - 1/2  m_a^2   a^2 + g_aγγ  E· B a,

where a denotes the ALP field, while E and B respectively represent the electric and magnetic components of the electromagnetic tensor F_μν, whose dual is F̃^μν.

Many bounds about the photon–ALP coupling g_a γγ and the ALP mass m_a have been derived in the literature <cit.>. The most reliable ALP limit is represented by g_a γγ < 0.66 × 10^- 10  GeV^- 1 for m_a < 0.02   eV at the 2 σ level from no detection of ALPs from the Sun <cit.>. Other bounds are more or less affected by the description of the astrophysical environment <cit.>.

Furthermore, the QED one-loop vacuum polarization effects, which become important in the presence of a strong external magnetic field, are accounted by the Heisenberg–Euler–Weisskopf (HEW) effective Lagrangian <cit.>

    ℒ_ HEW = 2 α^2/45 m_e^4 [ ( E^2 -  B^2 )^2 + 7 ( E· B)^2 ],

where α represents the fine-structure constant, while m_e is the electron mass.

We consider a photon–ALP beam of energy E propagating in the y-direction in a magnetized medium with an external magnetic field denoted by B and entering ), where E describes the photon electric field. Because of the off-diagonality of the mass matrix of the γ - a system, the propagation eigenstates differ from the interaction eigenstates, producing γ↔ a oscillations. This is similar to the oscillations of different flavor massive neutrinos, but in the case of the photon–ALP system, the external B field is essential to compensate for the spin mismatch between photons and ALPs. In particular, the functional expression of the photon–ALP coupling in Equation (<ref>) shows that a couples only with the component B_T of B transverse to the photon momentum k <cit.>. Furthermore, the short-wavelength approximation <cit.> holds in the present analysis since E ≫ m_a (we consider extremely light ALPs), so that the propagation of an unpolarized photon–ALP beam arising from ℒ_ ALP of Equation (<ref>) is described by the Von Neumann-like equation

    i d ρ (y)/d y = ρ (y)  ℳ^† ( E, y) - ℳ ( E, y)  ρ (y),

where ℳ (E,y) represents the photon–ALP mixing matrix, while ρ (y) is the polarization density matrix of the photon–ALP system. By defining ϕ, the angle that B_T forms with the z axis, ℳ reads as follows:

    ℳ (E,y) ≡(
    [        Δ_xx (E,y)        Δ_xz (E,y)  Δ_a γ(y)   sin ϕ;        Δ_zx (E,y)        Δ_zz (E,y)  Δ_a γ(y)   cos ϕ;  Δ_a γ(y)   sin ϕ Δ_ a γ(y)   cos ϕ         Δ_a a (E);                   ]),

with

    Δ_xx (E,y) ≡Δ_ (E,y)   cos^2  ϕ + Δ_∥ (E,y)   sin^2  ϕ,


    Δ_xz (E,y) = Δ_zx (E,y) ≡(Δ_∥ (E,y) - Δ_ (E,y) )  sin ϕ  cos ϕ,


    Δ_zz (E,y) ≡Δ_ (E,y)   sin^2  ϕ + Δ_∥ (E,y)   cos^2  ϕ,


    Δ_a γ(y) = 1/2g_aγγB_T(y),


    Δ_aa (E) = - m_a^2/2 E,

and

    Δ_ (E,y) = i/2  λ_γ (E,y) - ω^2_ pl(y)/2 E   + 2 α/45 π(B_T(y)/B_ cr)^2 E + ρ_ CMBE,


    Δ_∥ (E,y) = i/2  λ_γ (E,y) - ω^2_ pl(y)/2 E  + 7 α/90 π(B_T(y)/B_ cr)^2 E + ρ_ CMBE ,

[-15]where B_ cr≃ 4.41 × 10^13  G is the critical magnetic field and ρ_ CMB=0.522 × 10^-42.  describes the photon–ALP interaction, while Equation (<ref>) accounts for the ALP mass effect. The first term in Equations (<ref>) and (<ref>) describes photon absorption (e.g., due to the EBL <cit.>) with λ_γ being the γγ→ e^+e^- mean free path. In the second term of Equations (<ref>) and (<ref>), ω_ pl=(4 πα n_e / m_e)^1/2 represents the plasma frequency with n_e denoting the electron
number density. The third term in Equations (<ref>) and (<ref>) describes the QED one-loop vacuum polarization coming from ℒ_ HEW of Equation (<ref>), producing polarization variation and birefringence 
on the beam, and the fourth term expresses the photon dispersion on the CMB <cit.>. 

The solution of Equation (<ref>) can be expressed in terms of the transfer matrix 𝒰(E;y,y_0) of the photon–ALP system as

    ρ ( y ) = 𝒰(E; y, y_0 )  ρ_0  𝒰^†(E; y, y_0 ),

where ρ_0 is the density matrix at position y_0. The probability to find the photon–ALP beam in the final state ρ at position y starting in
the initial state ρ_0 at position y_0 reads as follows:

    P_ρ_0 →ρ (E,y) =  Tr[ρ 𝒰 (E; y, y_0)  ρ_0  𝒰^† (E; y, y_0) ],

with Tr ρ_0 =  Tr ρ =1 <cit.>.

We concentrate now on a photon–ALP system with the following characteristics: (i) fully polarized photons, (ii) no photon absorption (λ_γ→∞), (iii) homogeneous medium, and (iv) constant B field. Therefore, we can choose the z axis along the direction of B_T, resulting in ϕ=0 in Equation (<ref>). With these assumptions, the γ→ a conversion probability reads as follows:

    P_γ→ a (E, y) = (g_aγγB_T   l_ osc (E)/2π)^2  sin^2 (π (y-y_0)/l_ osc (E)),

with

    l_ osc (E) ≡2 π/[(Δ_zz (E) - Δ_aa (E) )^2 + 4  Δ_aγ^2 ]^1/2

being the photon–ALP beam oscillation length. The different energy dependence of the several Δ terms in Equation (<ref>) produces various regimes, which can be experienced by the photon–ALP beam. Hence, we can introduce the low-energy threshold E_L and the high-energy threshold E_H reading as follows:

    E_L ≡|m_a^2 - ω^2_ pl|/2 g_a γγ  B_T,

and 

    E_H ≡ g_a γγ  B_T [7 α/90 π(B_T/B_ cr)^2 + ρ_ CMB]^- 1,
 
respectively. In the energy interval E_L ≲ E ≲ E_H, the photon–ALP system lies in the strong-mixing regime, where the plasma, the ALP mass, the QED one-loop, and the photon dispersion on the CMB effects are all negligible with respect to the photon–ALP mixing term. In this case, P_γ→ a is maximal and energy independent and reduces to

    P_γ→ a (y) =  sin^2 ( g_aγγ B_T/2  (y-y_0) ).

Instead, both in the case E ≲ E_L, where the plasma contribution and/or the ALP mass term are prominent, and in the case E ≳ E_H, where the QED one-loop effect and/or the photon dispersion on the CMB dominate, the photon–ALP beam propagates in the weak-mixing regime, where P_γ→ a becomes energy dependent and progressively vanishes.

The usual situation of a generic photon initial polarization, structured magnetic fields, and non-homogeneous media represents a generalization of the above-considered case, which is reported here for the simplicity of the analytical expressions of the involved equations. However, in all our following calculations, we employ the appropriate photon polarization and the proper expressions of the magnetization, dispersion, and absorption properties of the crossed media.




§ POLARIZATION EFFECTS


The generalized polarization density matrix ρ associated with the photon–ALP beam can be expressed as 

    ρ (y) = ([ A_x (y); A_z (y);   a (y) ])
    ⊗([ A_x (y)   A_z (y)  a (y) ])^*,

where A_x (y) and A_z (y) represent the photon linear polarization amplitudes along the x and z axis, respectively, and a(y) is the ALP amplitude. Equation (<ref>) allows us to describe totally polarized (pure states), unpolarized, and partially polarized beams at once. In particular, pure photon states polarized in the x and z direction are expressed as

    ρ_x = (
    [ 1 0 0; 0 0 0; 0 0 0;   ]),         ρ_z = (
    [ 0 0 0; 0 1 0; 0 0 0;   ]),

respectively, and the ALP state as

    ρ_a = (
    [ 0 0 0; 0 0 0; 0 0 1;   ]),

whereas an unpolarized photon state is described by

    ρ_ unpol = 1/2(
    [ 1 0 0; 0 1 0; 0 0 0;   ]).

Partially 
 polarized photons are characterized by a polarization density matrix with an intermediate functional expression between Equations (<ref>) and (<ref>). 

The photonic part of the polarization density matrix of Equation (<ref>) can be expressed in terms of the Stokes parameters as <cit.>

    ρ_γ = 1/2(
    [  I+Q U-iV; U+iV  I-Q;      ]),

while the photon degree of linear polarization Π_L and the polarization angle χ are defined <cit.> as

    Π_L ≡(Q^2+U^2)^1/2/I = [ (ρ_11-ρ_22)^2+(ρ_12+ρ_21)^2]^1/2/ρ_11+ρ_22,

and

    χ≡1/2 arctan(U/Q)=1/2 arctan(ρ_12+ρ_21/ρ_11-ρ_22),

respectively, where ρ_ij with i,j=1,2 are the photon polarization density matrix elements. In the following Sections, we will show the behavior of the final Π_L, which turns out to be strongly modified by the photon–ALP interaction with respect to the initial photon degree of the linear polarization Π_L,0.

As shown by some theorems enunciated and demonstrated in <cit.>, in the absence of photon absorption, there exists a relationship between photon survival probability in the presence of photon–ALP interaction P_γ→γ and initial Π_L,0 (P_γ→ a=1-P_γ→γ due to the absence of photon absorption), and in particular, the following hold: 




  * In case of no photon absorption and a starting condition of only photons with an initial degree of linear polarization Π_L,0, the inequalities P_γ→ a≤ (1+Π_L,0)/2 and P_γ→γ≥ (1-Π_L,0)/2 hold. In case of initially unpolarized photons (Π_L,0=0), we observe P_γ→ a≤ 1/2 and P_γ→γ≥ 1/2.


  * In the previous conditions, Π_L,0 represents the measure of the overlap between the values assumed by P_γ→ a and P_γ→γ. In case of initially unpolarized photons (Π_L,0=0), P_γ→ a and P_γ→γ possess the common value of 1/2, at most.



We have employed the inequalities described in item 1 to verify the correctness of the behavior of P_γ→γ reported in several figures of this review (see the following sections).

As shown in <cit.>, the results presented in item 2 can be used to measure the initial Π_L,0 when only spectral data of an astrophysical source are known. The strategy works in the energy band in which the photon–ALP system lies in the weak mixing regime, where P_γ→ a and P_γ→γ are energy dependent and their assumed values satisfy the inequalities described in item 1. 

The procedure can be sketched as follows. We consider systems in an energy range where photon absorption is negligible. Starting from observational data of an astrophysical source with the observed spectrum Φ_ obs, we can infer P_γ→γ as 

    P_γ→γ=Φ_ obs/Φ_ em,

where Φ_ em represents the emitted spectrum, which can be either known or derivable from Φ_ obs (see <cit.> for more details). Furthermore, the absence of photon absorption by hypothesis ensures that P_γ→ a = 1 - P_γ→γ. Now, Π_L,0 can be obtained by measuring the interval of overlap between the values assumed by P_γ→γ and P_γ→ a.

In Figure <ref>, we show how the previous procedure works: in the left panels, an application in the optical-X-ray band is reported, while the right panels address a similar study but in the MeV range. We show how an observer would proceed to infer the initial Π_L,0 from spectral data. Note that cosmological effects are taken into account and E in Figure <ref> represents the photon energy, as observed from the Earth.



In the optical-X-ray band, we consider a high-frequency peaked blazar (HBL) placed inside a poor galaxy cluster at redshift z = 0.1 with an initial Π_L,0=0.3, while in the MeV range, we deal with an HBL but now placed inside a quite rich galaxy cluster at redshift z = 0.05 with Π_L,0=0.1. We assume the phenomenological emitted spectra derived in <cit.> and typical values concerning the magnetization properties of the astrophysical media crossed by the photon–ALP beam both in the optical-X-ray and in the MeV energy band.

In particular, in the optical-X-ray band, the photon–ALP interaction turns out to be negligible inside the poor galaxy cluster (see also Section <ref>), where the blazar is supposed to be located and inside the hosting elliptical galaxy, whose typical properties are described in Section <ref>. Instead, we have efficient photon–ALP conversion inside the jet of the blazar, whose characteristics are described in Section <ref>: For the magnetic field in the jet B^ jet, we assume a toroidal profile of Equation (<ref>) with magnetic field strength at the emission distance y_ em = 3 × 10^16  cm equal to B_0^ jet≡ B^ jet(y_ em)= 0.5   G, and for the electron number density n_e^ jet, we consider the profile expressed by Equation (<ref>), taking n_e,0^ jet≡ n_e^ jet(y_ em)=5 × 10^4   cm^-3. We hypothesize an efficient photon–ALP interaction inside the extragalactic space by assuming an extragalactic magnetic field
strength B_ ext = 1   nG and the B_ ext coherence properties described in Section <ref>. Lastly, the photon–ALP interaction turns out to be efficient also inside the Milky Way, where we adopt the Jansson and Farrar model <cit.> concerning the magnetic field B_ MW and that presented in <cit.> regarding the electron number density in the Galaxy n_ MW,   e (see also Section <ref>). For more details, we send the reader to <cit.>.



Instead, in the MeV range, we have an efficient photon–ALP conversion in the cluster since the blazar is supposed to be located inside a rich galaxy cluster, whose properties are reported in Section <ref>: we assume the cluster magnetic field B^ clu profile of Equation (<ref>), where n_e^ clu represents the cluster electron number density described by means of the β model expressed by Equation (<ref>). Concerning the parameters entering Equations (<ref>) and (<ref>), we assume k_L = 0.1   kpc^-1, k_H = 3   kpc^-1, q = -11/3, B^ clu_0 = 20   G, n_e,0^ clu = 0.1   cm^-3, η_ clu = 0.75, β_ clu = 2/3, a cluster core radius r_ core = 150   kpc, and a cluster radius of 1   Mpc (see Sections <ref> and <ref> for the definition of the parameters). In the present situation, we hypothesize a negligible photon–ALP conversion inside the extragalactic space by assuming B_ ext < 10^-15  G (see also Section <ref>). Concerning the photon–ALP beam propagation inside the blazar jet, host galaxy, and Milky Way, we consider the same models and parameter values assumed in the previous case (optical-X-ray band) apart from the jet magnetic field strength at the emission distance, which is now B_0^ jet≡ B^ jet(y_ em)= 0.1   G (see also Sections <ref>, <ref>, and <ref>). We send the reader to <cit.> for more details.



In both of the previous cases (optical-X-ray and MeV band), a typical realization of P_γ→γ is evaluated and reported in the top panels of Figure <ref>: we assume g_aγγ = 0.5 × 10^-11  GeV^-1 for both of the situations, but we take m_a = 5 × 10^-14  eV for the first blazar in the optical-X-ray band, while we deal with an alternative scenario by taking m_a = 2 × 10^-10  eV for the second blazar in the MeV range. The central panels of Figure <ref> show the observed binned spectra Φ_ obs (with a typical energy resolution <cit.>), simulated by multiplying the emitted spectra by P_γ→γ in the presence of photon–ALP interaction. In the bottom panels of Figure <ref>, we show P_γ→γ and P_γ→ a with P_γ→γ inferred from the simulated observed spectra Φ_ obs, reported in the central panels, by means of Equation (<ref>), where Φ_ em can be reconstructed by fitting the upper spectral bins in the central panels, since photon–ALP interaction produces an energy-dependent dimming in the observed spectra (see also <cit.>). We recall that P_γ→ a = 1 - P_γ→γ due to the absence of photon absorption. Finally, by employing the results presented in item 2, we can measure the initial Π_L,0, and we obtain Π_L,0 = 0.288 ± 0.016 in the X-ray band and Π_L,0 = 0.090 ± 0.018 in the MeV range. We observe that the inferred values are very close to the initially assumed ones, which demonstrates the power of the present method.

We want to stress that the above-described procedure is the only established possibility to measure the initial polarization of photons produced by astrophysical sources. Furthermore, this procedure can be employed in a wide energy range, where photon absorption is negligible, and only needs spectral observations of the source, so that flux-measuring observatories also become polarimeters.




§ PHOTON–ALP BEAM PROPAGATION IN DIFFERENT MEDIA


In the present section, we describe the main properties of the media crossed by the photon–ALP beam starting where photons are produced up to their arrival at the Earth: blazar, host galaxy, galaxy cluster, extragalactic space, and Milky Way. We concentrate mainly on the characteristics that are crucial for the photon–ALP system, such as the magnetization and absorption properties of the media, while we send the reader to the quoted papers for an exhaustive discussion.



 §.§ Blazars


Blazars are powerful astrophysical sources emitting in the whole electromagnetic spectrum, from radio waves up to the VHE band. They are a particular class of AGN that are basically extragalactic supermassive black holes (SMBHs) accreting matter from the surroundings. In these systems, two collimated relativistic jets can develop in opposite directions. In the accidental case where one of these jets turns out to be  in the Earth direction, the AGN is called blazar, while in other situations, the AGN shows a different phenomenology, as displayed in Figure <ref>, where the AGN structure is also shown. 



Blazars are divided into two sub-classes: flat spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs). FSRQs are powerful sources, where accretion from the disk surrounding the central SMBH is efficient. Furthermore, they show intense broad emission lines generated by clouds of gas in the so-called
broad-line region (BLR) encompassing the central SMBH and photoionized by the
ultraviolet photons produced by the disk. Moreover, FSRQs display absorption regions for VHE photons, which can interact with the optical/ultraviolet radiation emitted by the above-mentioned BLR and with the infrared radiation produced by the dusty torus (see Figure <ref>), preventing the observation of photons with energy above ∼20   GeV <cit.>. However, photons of up to ∼400   GeV have been observed from FSRQs <cit.>. Only ad hoc solutions have been proposed inside conventional physics, while photon–ALP oscillations provide a natural solution of the problem by partially preventing BLR absorption: this fact represents a hint at ALP existence <cit.>. Instead, BL Lacs are less powerful, and they show neither absorption regions nor significant emission lines due to the presence of a radiatively inefficient
accretion flow (RIAF) onto the central SMBH <cit.>.

In the following, we will concentrate on BL Lacs, where the emission region of photons is located at a distance of about y_ em= (10^16– 10^17)   cm from the central SMBH. Once produced at y_ em, photons propagate inside the BL Lac jet up to a distance y_ jet≃ 1   kpc, where they enter the host galaxy. During their path inside the jet, these photons can oscillate into ALPs inside the jet magnetic field B^ jet. The toroidal part of B^ jet, which is transverse to the jet axis <cit.>, turns out to be the dominant one at the distances under consideration, and its profile reads as follows:

    B^ jet ( y ) = B^ jet_0 (y_ em/y),

where B^ jet_0 denotes the magnetic field strength of the jet at y_ em. Another crucial information in order to evaluate photon–ALP conversion inside BL Lacs is the electron number density profile in the jet n_e^ jet, which, due to the conical shape of
the jet, is expected to be 

    n^ jet_e ( y ) = n^ jet_e,0(y_ em/y)^2,

where n^ jet_e,0 represents the electron number density at y_ em. Synchrotron Self Compton (SSC) diagnostics applied to blazar spectra suggests assuming n^ jet_e,0=5 × 10^4   cm^-3 <cit.>.

Two competing photon emission mechanisms have been proposed in order to explain blazar phenomenology: (i) the leptonic model (see, e.g., <cit.>) and (ii) the hadronic model (see, e.g., <cit.>). Both scenarios describe photon production in the optical up to the X-ray band as generated by electron synchrotron emission. At higher energies, leptonic models explain photon emission with inverse Compton scattering of synchrotron radiation or of external photons from the disk and/or clouds with the synchrotron producing relativistic electrons <cit.>. Instead, hadronic models attribute the high-energy photon generation to proton–synchrotron emission or photomeson production <cit.>. Hadronic models require higher values of B^ jet_0 and y_ em with respect to leptonic scenarios. Moreover, hadronic models predict a higher initial degree of linear polarization Π_L,0 with respect to leptonic scenarios, as shown in Figure <ref> for two particular BL Lacs.



The propagation of the photon–ALP beam is computed in the comoving frame of the jet so that a transformation E →γ E, with γ being the Lorentz factor, is required to pass to the fixed frames of the subsequently encountered regions. 

The values assumed by B^ jet_0, y_ em, γ and Π_L,0 affect the calculation of the photon–ALP beam propagation in the jet and will be discussed in Section <ref> for both leptonic and hadronic models. Then, the photon–ALP beam transfer matrix in the jet 𝒰_ jet can be evaluated by following the calculation reported in <cit.>.




 §.§ Host Galaxy


The effectiveness of the photon–ALP conversion inside the host galaxy depends on its nature and characteristics. Ideal galaxies in order to have an efficient photon–ALP conversion are spirals and even more starbursts. The reason lies in the high magnetic field strength inside the host B^ host, which can reach values of B^ host=𝒪(5-10)  μ G for spirals <cit.> and B^ host=𝒪(20-50)  μ G for starbursts <cit.>, and in the large coherence length L_ dom^ host of B^ host, which is around L_ dom^ host =𝒪(1-4)   kpc for both spiral and starburst galaxies outside the central region <cit.>. The magnetic field coherence can be modeled by employing a Kolmogorov-type turbulence power spectrum in a similar way as discussed in the following subsection concerning galaxy clusters (for more details about the parameter values, see also <cit.>). A possible example of photon–ALP conversion inside spirals or starbursts is represented by photons produced by GRBs, which are usually located in the central region of these galaxies <cit.>. 

In particular, the location of GRB 221009A—the brightest GRB to date—in the central region of a disk-like galaxy <cit.> allows for an efficient photon–ALP conversion inside the host, which is able to justify the detection by the LHAASO Collaboration of this GRB above 10   TeV <cit.>. The latter fact represents a strong hint at ALP existence, as demonstrated in <cit.> (see also Section <ref> and <cit.>).

Instead, blazars are often hosted inside elliptical galaxies, where B^ host is described by means of a domain-like model with the typical values B_ host≃ 5   G and L_ dom^ host≃ 150   pc <cit.>. In such a situation, the photon–ALP oscillation length turns out to be much larger than L_ dom^ host, resulting in an inefficient photon–ALP conversion, as shown in <cit.>. However, whatever the nature and the properties of the host galaxy are, we evaluate the associated photon–ALP beam transfer matrix 𝒰_ host.



 §.§ Galaxy Clusters


Galaxy clusters consist of 30 to more than 1000 galaxies with a total mass within the range (10^14–10^15)   M_⊙, making them the largest gravitationally bound structures in the Universe. Properties such as galactic content, density profile, shape, concentration, and symmetry allow galaxy clusters to be classified into three main classes: (i) regular clusters, (ii) intermediate clusters, and (iii) irregular clusters <cit.>. The approximate spherical symmetry and the satisfying knowledge of the density profile of regular clusters allow us to model their electron number density n_e^ clu and the linked magnetic field B^ clu carefully so that we can properly calculate the photon–ALP conversion. 

The strength of B^ clu is set to B^ clu = 𝒪 (1-10)   G by Faraday rotation measurements and synchrotron radio emission <cit.>. The morphology of B^ clu is turbulent and modeled with a Kolmogorov-type turbulence power spectrum M(k)∝ k^q, with index q=-11/3 and where k represents the wave number in the interval [k_L,k_H] with k_L = 2π/Λ_ max and k_H = 2π/Λ_ min being the minimal and maximal turbulence scales, respectively <cit.>. Consequently, the behavior of B^ clu can be modeled as <cit.>

    B^ clu(y)=ℬ( B_0^ clu,k,q,y ) ( n_e^ clu(y)/n_e,0^ clu)^η_ clu,

where ℬ represents the spectral function describing the Kolmogorov-type turbulence of the cluster magnetic field (for more details, see, e.g., <cit.>), η_ clu is a cluster parameter, and B_0^ clu and n_e,0^ clu are the magnetic field strength and the electron number density at the center of the cluster, respectively.

Several models are employed to describe the profile of n_e^ clu, such as the single β model typically used for non-cool-core (nCC) clusters, the double β model for cool-core (CC) clusters, or a modified version of them <cit.>. We will fix the behavior of n_e^ clu in Section <ref>, where we will concentrate on real situations.

Photons can be produced by some astrophysical source such as a blazar or a GRB and can pass through a galaxy cluster, or they can be diffusely generated in the central region of the cluster itself. In the latter situation, it is necessary to discuss their initial Π_L,0. In the X-ray band, photons are produced in the cluster central region via thermal Bremsstrahlung <cit.> with photons that turn out to be therefore unpolarized. In the HE range, several processes are proposed to explain possible photon emission, such as the cascade of VHE photons, inverse Compton scattering, and neutral pion decay (see, ). Even in the HE range, photons turn out to be effectively .

The transfer matrix 𝒰_ clu of the photon–ALP beam propagating from the central region of the cluster up to its virial radius can be evaluated by following the calculation reported in <cit.>. In the specific case in which photons are not diffusely produced in the central region of a rich cluster but are generated by another astrophysical source that does not lie inside a rich cluster—such a situation is frequent concerning blazars—the photon–ALP conversion in this region turns out to be inefficient.



 §.§ Extragalactic Space


The effectiveness of the photon–ALP conversion inside the extragalactic space is determined by the strength and morphology of the extragalactic magnetic field B_ ext. However, the properties of B_ ext are still poorly understood. While many configurations for B_ ext exist in the literature <cit.>, its limits are in the interval 10^- 7  nG≲ B_ ext≲ 1.7   nG on the scale of 𝒪 (1)   Mpc <cit.>.

The shape of B_ ext is modeled as a domain-like network, where B_ ext is homogeneous over a whole domain of size L_ dom^ ext, which is equal to its coherence length, and with B_ ext randomly and discontinuously changing its direction from one domain to the subsequent one, but maintaining approximately the same strength <cit.>. Since the exact orientation of B_ ext in each domain is unknown, the photon–ALP beam propagation becomes a stochastic process, and many realizations of it can be evaluated by varying the B_ ext orientations, in order to obtain the statistic properties of the system. However, only a single realization of the photon–ALP beam propagation process can be physically observed.

The domain-like model proposed for B_ ext is justified by several scenarios of outflows from primeval galaxies, further amplified by turbulence <cit.>. According to the latter scenarios, the usual values for B_ ext are B_ ext = 𝒪 (1)   nG and L_ dom^ ext = 𝒪 (1)   Mpc (for details, see <cit.>). In this fashion, we assume B_ ext = 1   nG and L_ dom^ ext in the interval (0.2–10)   Mpc with ⟨ L_ dom^ ext⟩ = 2   Mpc.

However, the commonly used discontinuous model for B_ ext produces unphysical results when the photon–ALP beam oscillation length l_ osc becomes smaller than L_ dom^ ext due to the photon dispersion on the CMB <cit.>, and consequently, the system turns out to be sensitive to the B_ ext substructure. In order to solve such a problem, we employ a more complex model developed in <cit.>, where the components of B_ ext change continuously, crossing from a domain to the subsequent one.

[-15]In the UV-X-ray and HE bands, photon absorption due to the interaction with EBL photons—we assume the conservative EBL model of Franceschini and Rodighiero <cit.>—is negligible, but it becomes substantial for E > 𝒪(100)   GeV <cit.>. Therefore, for , propagation in the extragalactic space produces additional photon–ALP oscillations when the photon–ALP conversion is efficient. Instead, for E > 𝒪(100)   GeV, where we have a large EBL absorption, the effect of the photon–ALP beam propagation in the extragalactic space is also to mitigate the EBL absorption of VHE photons. The reason lies in the fact that when photons become ALPs, they do not undergo EBL absorption so that the effective optical depth is reduced and more photons can reach the Earth.

In both of the previous situations, with large or negligible EBL absorption and in the case of both efficient or ineffective photon–ALP conversion, the transfer matrix 𝒰_ ext of the photon–ALP system in the extragalactic space can be calculated by following the analysis performed in <cit.>.



 §.§ Milky Way


Photon–ALP conversion in the Milky Way can be computed once the morphology and strength of the magnetic field B_ MW and the profile of the electron number density n_ MW,   e in the Galaxy are known.

Concerning B_ MW, we employ the model developed by Jansson and Farrar <cit.>, which includes a disk and a halo component, both parallel to the Galactic plane, and a poloidal `X-shaped' component at the galactic center. The most relevant component of B_ MW for the photon–ALP conversion turns out to be its regular part, to which we add for completeness the turbulent one developed in <cit.>. The alternative B_ MW model of Pshirkov et al. <cit.> does not substantially produce a different photon–ALP conversion in the Milky Way. Still, we prefer the model by Jansson and Farrar <cit.> since it determines the Galactic halo component with a higher level of accuracy. 

Regarding n_ MW,   e we employ the model presented in <cit.>, where the Galaxy is modeled with an extended thick disk standing for the warm interstellar medium, a thin disk accounting for the Galactic molecular ring, a Galactic Center disk, spiral arms and features for the Gum Nebula, Galactic Loop I, and the Local Bubble.

By following the procedure developed in <cit.>, we can evaluate the transfer matrix 𝒰_ MW of the photon–ALP system in the Milky Way.



 §.§ Overall Photon–ALP Beam Propagation


Once all the transfer matrices discussed in the previous subsections are known, we can combine them in order to calculate the total transfer matrix 𝒰 describing the photon–ALP beam propagation starting from the photon emission zone up to the Earth. When photons are diffusely generated in the central zone of a galaxy cluster, 𝒰 reads as follows:

    𝒰=𝒰_ MW 𝒰_ ext 𝒰_ clu,

while when photons are produced in the jet of a blazar, 𝒰 is expressed by

    𝒰=𝒰_ MW 𝒰_ ext 𝒰_ clu 𝒰_ host 𝒰_ jet.


In both of the previous cases, we can calculate the photon survival probability in the presence of photon–ALP interaction by specializing Equation (<ref>) as

    P_γ→γ  = ∑_i = x,z Tr[ρ_i  𝒰 ρ_ in 𝒰^†],

where ρ_x and ρ_z read from Equations (<ref>), and ρ_ in represents the beam initial polarization density matrix, which is specified in the following subsections for concrete cases, for which the initial photon degree of linear polarization Π_L,0 is known. Moreover, by recalling Equation (<ref>) with ρ_0 ≡ρ_ in, we obtain the final photon degree of the linear polarization Π_L and the polarization angle χ, thanks to Equations (<ref>) and (<ref>), respectively (see also <cit.>).




§ ALP EFFECTS ON PHOTON POLARIZATION


Hereafter, we show how photon–ALP interaction modifies the final degree of linear polarization Π_L and polarization angle χ for photons diffusely generated in the galaxy cluster central region or produced in the blazar jet. Here, we report two concrete examples: one for galaxy clusters, and specifically, we concentrate on Perseus, and one for BL Lacs and in particular, we focus on OJ 287. The latter two sources represent good observational targets to study ALP effects inside galaxy clusters <cit.> and blazars <cit.>, respectively. We send the interested reader to the above-quoted papers, where a deeper discussion is presented along with other concrete examples of both galaxy clusters and blazars <cit.>. We now describe the properties of Perseus and of OJ 287, which are crucial to evaluate the photon–ALP interaction effects on Π_L and χ.

Perseus is a deeply studied CC rich regular cluster located at redshift z = 0.01756 and represents the brightest galaxy cluster in the X-ray band. Its electron number density n_e^ clu can be expressed by

    n_e^ clu(y)=n_e,01^ clu( 1+y^2/r_ core,1^2)^-3/2β_ clu,1+ 
    
    n_e,02^ clu( 1+y^2/r_ core,2^2)^-3/2β_ clu,2,

with n_e,01^ clu=3.9 × 10^-2  cm^-3, r_ core,1=80   kpc, β_ clu,1=1.2, n_e,02^ clu=4.05 × 10^-3  cm^-3, r_ core,2=280   kpc, and β_ clu,2=0.58 <cit.>. The Perseus magnetic field B^ clu is described by Equation (<ref>) by taking B_0^ clu = 16   G, which represents an average estimate among those reported in the literature <cit.>. Due to the similarity of the stochastic properties of the Perseus B^ clu with those of the Coma cluster, we assume the following values regarding Perseus: q=-11/3, Λ_ min= 2   kpc and Λ_ max= 34   kpc <cit.>, and the average value η_ clu=0.5 entering Equation (<ref>). Finally, we take an initial degree of linear polarization Π_L,0=0 both in the UV-X-ray and in the HE band according to the discussion in Section <ref>. Then, the photon–ALP beam propagation can be evaluated as discussed in Section <ref>, and its transfer matrix reads from Equation (<ref>).

OJ 287 represents a typical low-frequency peaked blazar (LBL) located at redshift z=0.3056. Due to its high flux in both the X-ray and HE bands, OJ 287 turns out to be a favorable observational target for polarimetric studies in both energy ranges <cit.>. In the following, we consider the typical values regarding the LBL parameters: for the leptonic model, we take B^ jet_0 = 1   G, y_ em = 3 × 10^16  cm and γ = 10, while for the hadronic model, we assume B^ jet_0 = 20   G, y_ em = 10^17  cm and γ = 15 <cit.>. The profiles of B^ jet and n_e^ jet are represented by Equations (<ref>) and (<ref>), respectively. As discussed in Section <ref>, the initial Π_L,0 for OJ 287 is plotted in Figure <ref> for both the leptonic and the hadronic scenario. OJ 287 is hosted inside an elliptical galaxy with a resulting inefficient photon–ALP conversion in the host, as discussed in Section <ref>. Moreover, OJ 287 is not found within a rich galaxy cluster with a consequent negligible ALP effect also in this zone (see ). However, for completeness, we consider all the contributions for the evaluation of the total photon–ALP beam transfer matrix, which is expressed by Equation (<ref>). A remark regarding photons emitted by blazars should be kept in mind: due to the limited spatial resolution of real polarimeters, photons coming from different zones inside the transverse section of the blazar jet cannot be distinguished, and all photons are then collected together. Although the latter fact might in principle wash out the ALP-induced polarization features, the studies developed in <cit.> demonstrate that this is not the case in realistic situations. In order to be conservative, the magnetic field profile expressed by Equation (<ref>) and a propagation distance of 1   pc inside the jet are assumed. We send the reader to the papers quoted above <cit.> for a deeper discussion regarding the latter problem and the adopted strategy to take the polarimeter technological limitations into account. 

We can now show the final photon survival probability P_γ→γ, the corresponding photon degree of the linear polarization Π_L, and the polarization angle χ arising from photon–ALP oscillations inside the magnetized media analyzed in Section <ref> (blazar jet, host galaxy, galaxy cluster, extragalactic space, Milky Way) for the two above-discussed scenarios: (i) photons are diffusely generated in the central zone of a galaxy cluster—we take Perseus as an example, and (ii) photons are produced at the jet base of a BL Lac—we consider OJ 287 as an illustration. In order to assess the robustness of the results, several realizations of the photon–ALP beam propagation process are evaluated by considering different magnetic field configurations due to limited knowledge especially regarding B^ clu and B_ ext. The probability density function f_Π associated with the final Π_L can be computed by collecting all the values assumed by Π_L in the various realizations and gives us information about the most probable scenarios.

Regarding the ALP parameters, we assume g_aγγ=0.5 × 10^-11  GeV^-1 and the two ALP masses: (i) m_a ≲ 10^-14  eV and (ii) m_a = 10^-10  eV, which are within the CAST bound <cit.>. For m_a ≲ 10^-14  eV, the plasma term dominates over the ALP mass term for the chosen model parameters, while the opposite is true in the case m_a = 10^-10  eV. Results about P_γ→γ, Π_L and χ are obtained for photons produced at redshift z with emitted energy E and observed energy E_0=E/(1+z) in the two intervals: (i) UV-X-ray band and 

Note that, in the following figures, binned data about Π_L and χ are also shown in order to provide an idea of how a possible observation of ALP-induced features may look like. The binning procedure is based on the realistic energy resolution of current and/or planned polarimeters both in the X-ray band <cit.> and in the HE range <cit.> (see <cit.> for more details).



 §.§ UV-X-ray Band


In the UV-X-ray band, the photon–ALP beam propagates in the weak mixing regime due to the strength of the plasma and/or ALP mass term inside the mixing matrix of Equation (<ref>). As a result, P_γ→γ, Π_L and χ turn out to possess an energy-dependent behavior in the case of both diffuse photon production in the central region of the Perseus cluster and photon emission at the jet base of OJ 287.

In particular, in the case m_a = 10^-10  eV, the ALP mass term is so strong that it dominates over the photon–ALP mixing term. As a result, the photon–ALP conversion turns out to be inefficient in the UV-X-ray band for m_a = 10^-10  eV, apart from a moderate dimming of Π_L,0 for OJ 287 in the case of the hadronic emission model due to the high central value of B^ jet (see <cit.> for more details). This is the reason why we only consider the case m_a ≲ 10^-14  eV in this subsection. 

Regarding the Perseus cluster, we report P_γ→γ, Π_L and χ in the top subfigure of Figure <ref> and the corresponding f_Π associated with the final Π_L in the bottom subfigure (see <cit.> for more details). Similarly, concerning OJ 287, we show P_γ→γ, Π_L and χ in the top subfigure of Figure <ref> and the linked f_Π in the bottom subfigure, but we now consider the two possible emission mechanisms inside blazars (see Section <ref>): (i) the leptonic model in the left panels and (ii) the hadronic model in the right panels (see <cit.> for more details). In both cases, i.e., for both Perseus and OJ 287, we concentrate on two benchmark energies in the evaluation of f_Π: (i) E_0 = 1   keV and (ii) E_0 = 10   keV.




In the case considered here, i.e., g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV, the energy dependence of P_γ→γ, Π_L, and χ in the top subfigures of Figures <ref> and <ref> is caused by the strength of the plasma term, which is not negligible with respect to the photon–ALP mixing term, especially inside the cluster in the case of Perseus and in the BL Lac jet regarding OJ 287. As a result, the photon–ALP beam propagates in the weak mixing regime, which protracts for a few energy decades due to the large variation with respect to the distance of B^ clu and n_e^ clu regarding Perseus and of B^ jet and n_e^ jet concerning OJ 287 (see  and <cit.> for details). In addition, the previous figures demonstrate that the final Π_L turns out to be heavily modified with respect to the initial one Π_L,0 for both Perseus and OJ 287. The binned data in the top Subfigures of Figures <ref> and <ref> concerning Π_L and χ show that the ALP-induced features can be detected by IXPE <cit.>, eXTP <cit.>, XL-Calibur <cit.>, NGXP <cit.>, and XPP <cit.> for E_0 ≳ 2   keV regarding Perseus and for E_0 ≳ 0.5   keV concerning OJ 287. Regarding OJ 287, the top subfigure of Figure <ref> shows that the hadronic model produces a larger energy variability of Π_L and χ with respect to the leptonic scenario due to the higher value of B^ jet_0 and the wider extension of the jet region.



The behavior of f_Π reported in the bottom subfigures of Figures <ref> and <ref> shows a broadening of the final Π_L with respect to the initial Π_L,0 for both Perseus and OJ 287. Perseus appears as a strong target to study ALP effects on photon polarization, since the final value Π_L = 0 is never the most probable expected result, while conventional physics predicts Π_L = Π_L,0 = 0. Instead, for OJ 287, the most probable final value of Π_L is not very different from the initial Π_L,0 apart from the case of the leptonic model at E_0 = 10   keV, where the most probable result turns out to be Π_L ≳ 0.8, which is extremely difficult to explain within conventional physics. For more details, we send the reader to the discussion developed in <cit.> concerning Perseus and to that presented in <cit.> regarding OJ 287.
















 §.§ High-Energy Band


In the HE band, both the scenario with ALP parameters g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV and that with g_aγγ=0.5 × 10^-11  GeV^-1, m_a = 10^-10  eV produce sizable ALP effects on P_γ→γ, Π_L and χ in the case both of photon diffuse production in the central zone of the Perseus cluster and of photon emission at the jet base of OJ 287.

Concerning the Perseus cluster, we show the behavior of P_γ→γ, Π_L and χ in the top subfigure of Figure <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV and in the top subfigure of Figure
 <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a = 10^-10  eV. The associated f_Π are reported in the bottom subfigures of Figures <ref> and <ref>, respectively. Regarding OJ 287, we likewise report P_γ→γ, Π_L, and χ in the top subfigure of Figure <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV and in the top subfigure of Figure <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1,  m_a = 10^-10  eV. The corresponding f_Π are shown in the bottom subfigures of Figures <ref> and <ref>, respectively. For OJ 287, we assume (i) the leptonic emission model in the left panels of Figures <ref> and <ref> and (ii) the hadronic one in the right panels of Figures <ref> and <ref> (see Section <ref> and <cit.> for more details). Moreover, f_Π is evaluated for both Perseus and OJ 287 at the two benchmark energies: (i) E_0 = 300   keV and (ii) E_0 = 3   MeV.




We first consider the scenario g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV. In the present situation, the photon–ALP beam propagates in the strong mixing regime, since the photon–ALP mixing term dominates over all other terms inside the mixing matrix of Equation (<ref>) (see Section <ref>). Consequently, P_γ→γ, Π_L, and χ are energy independent in almost all the considered energy band in the case both of photon production in the central region of Perseus, as shown by the top subfigure of Figure <ref>, and of photon generation in the jet of OJ 287, as reported in the top subfigure of Figure <ref>. We can also infer that, in all the considered cases, the final Π_L turns out to be strongly modified with respect to the initial Π_L,0. The binned data show a lower uncertainty than in the UV-X-ray band due to fact that the photon–ALP system lies in the strong mixing regime, which makes both Perseus and OJ 287 good observational targets for missions such as COSI <cit.>, e-ASTROGAM <cit.>, and AMEGO <cit.>. Moreover, regarding OJ 287, we do not observe substantial differences between the leptonic and the hadronic emission model.




As far as the behavior of f_Π is concerned, for both Perseus and OJ 287, the bottom subfigures of Figures <ref> and <ref> respectively show that the most probable final value of Π_L is larger than the initial one Π_L,0 with a substantial broadening. For both Perseus and OJ 287, the most promising energy range to study ALP effects on photon polarization appears around E_0 = 3   MeV, where the most probable value for Π_L is Π_L ≳ 0.8, which is larger than that predicted by conventional physics in all considered cases.





We now move to the scenario g_aγγ=0.5 × 10^-11  GeV^-1, m_a = 10^-10  eV. In this situation, the photon–ALP system lies in the weak mixing regime, since the ALP mass term turns out to be non-negligible with respect to the photon–ALP mixing term inside the mixing matrix of Equation (<ref>) (see Section <ref>). As a result, P_γ→γ, Π_L, and χ become energy dependent, as shown in the top subfigure of Figure <ref> regarding Perseus and in the top subfigure of Figure <ref> concerning OJ 287. Furthermore, we observe that the final Π_L turns out to be altered compared with the initial Π_L,0 especially in the case of Perseus and in that of the leptonic model for OJ 287. As already observed in the X-ray band for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV, the wide energy range, where the photon–ALP system lies in the weak mixing regime, is due to the high variation of the magnetic field and electron number density profiles in the crossed media (see also Sections <ref> and <ref> and <cit.> for more details). The binned data in the top subfigures of Figures <ref> and <ref> show that observatories like COSI <cit.>, e-ASTROGAM <cit.>, and AMEGO <cit.> may detect ALP-induced features on Π_L for E_0 ≳ 3   MeV concerning Perseus, and similar features for E_0 ≳ 0.2   MeV regarding OJ 287 in the case both of the leptonic and of the hadronic model.










The behavior of f_Π reported in the bottom subfigure of Figure <ref> for Perseus and in the bottom subfigure of Figure <ref> for OJ 287 shows a broadening of the final Π_L with respect to the initial Π_L,0 in all the cases. Moreover, the best energy interval for studies of ALP-induced features on Π_L appears to be E_0 ≳ 3   MeV concerning the Perseus cluster, where the most probable value for the final Π_L is Π_L ≳ 0.2, while conventional physics predicts Π_L = Π_L,0 = 0. Even OJ 287 represents a good observational target, since the most probable values of the final Π_L are larger than the initial ones, but in this case, conventional physics could justify moderately high values of Π_L albeit with some difficulty.



§ DISCUSSION AND FUTURE PERSPECTIVES


In Section <ref>, the ALP impact on the polarization of photons coming from Perseus has been reported as an example of ALP effects on galaxy clusters in the X-ray and in the HE band (for details, see <cit.>). Similarly, OJ 287 has been presented as an illustration of the ALP-induced features on photon polarization in BL Lacs (see <cit.> for details). Present and planned observatories in the X-ray band <cit.> and in the HE range <cit.> possess the capability to detect possible ALP signatures as those analyzed in Section <ref>, as shown by the realistic binned data reported therein. An analysis about ALP effects on photon polarization in the VHE band has been performed in <cit.>: we will briefly discuss this topic in Section <ref> along with its future crucial importance for the possible identification of an ALP signal.

The ALP-induced features on the final photon degree of linear polarization Π_L discussed in Section <ref> mainly originate inside the galaxy cluster central region and/or in the blazar jet, while the effect of other crossed media is subdominant, as shown . Specifically, a negligible photon–ALP conversion inside the extragalactic space (when B_ ext < 10^-15  G) does not substantially change the results apart from a smaller broadening of Π_L shown in the behavior of its associated probability density function f_Π <cit.>.

In particular, photons diffusely emitted in the central region of galaxy clusters are expected to be observed as unpolarized within conventional physics both in the X-ray and in the HE range (i.e., with Π_L = 0, see Section <ref>). Instead, the photon–ALP interaction model consistently predicts Π_L > 0 for them in both energy ranges. Therefore, galaxy clusters appear among the best observational targets for studies concerning ALP-induced effects on photon polarization. Specifically, a possible future detection of photons diffusely emitted from galaxy clusters with Π_L > 0 would represent a hint at ALP existence in addition to the already-existing ones <cit.>, since conventional physics along with other scenarios like Lorentz invariance violation <cit.> is unable to explain Π_L > 0 (see <cit.> for details). BL Lacs are also promising targets especially in the cases where the photon–ALP interaction model predicts Π_L > 0.8 (see Section <ref>), since, in this case, even the hadronic emission model can hardly justify such high polarization degrees within conventional physics (for details, see <cit.>). However, in case of lower Π_L values detected from BL Lacs, the exclusion of an explanation within conventional physics and the demonstration of the need for ALPs may be less robust. Note that the magnetic field in the jet B^ jet is expected to be quite turbulent, which implies a lower initial photon degree of linear polarization Π_L,0 especially within the leptonic emission model, as shown by a recent theoretical study <cit.> and confirmed by IXPE results <cit.>. Consequently, the final Π_L predicted by the photon–ALP interaction model is reduced in the cases where the initial Π_L,0 remains the most probable final value, as shown by the behavior of the corresponding f_Π in Section <ref>.

Other possible targets to evaluate ALP effects on photon polarization are represented by (i) white dwarfs, where ALP bounds can be derived from polarization measurements <cit.>; (ii) neutron stars, where ALPs produce polarization signatures detectable in the X-ray band <cit.>; and (iii) GRBs, where the photon–ALP conversion over cosmological distances strongly modifies the initial photon polarization degree, as shown in the pioneering study performed in <cit.>. We will focus our attention in Section <ref> on GRBs due to their importance also for the Universe transparency problem regarding the recent detection by the LHAASO Collaboration of GRB 221009A up to 18   TeV <cit.>, which is difficult to justify within conventional physics, but is instead explained by the photon–ALP interaction (see Section <ref> and <cit.> for details).

[-25]As far as the ALP parameter space (m_a, g_aγγ) is concerned, we have analyzed two cases: (i) m_a ≲ 10^-14  eV , g_aγγ = 0.5 × 10^-11  GeV^-1 and (ii) m_a = 10^-10  eV, . Both the previous possibilities are within the CAST bound <cit.>. The former ALP scenario, i.e., the case with m_a ≲ 10^-14  eV, is disfavored by some recent bounds <cit.>, but this possibility cannot be excluded, as discussed in <cit.>. Instead, the latter choice, i.e., the case with m_a = 10^-10  eV, meets all existing bounds and is compatible with the ALP parameters employed in previous studies, where ALP hints have been found <cit.>. This is the reason why the case m_a = 10^-10  eV, g_aγγ = 0.5 × 10^-11  GeV^-1 appears to be more promising. As discussed in <cit.>, other possibilities, such as an ALP with mass m_a = 10^-11  eV and coupling to photons g_aγγ= 0.5 × 10^-11  GeV^-1, will produce an intermediate behavior between the two cases considered in Section <ref> and can meet all current ALP bounds. Note that, while an increment (decrement) of the strength of g_aγγ implies a corresponding increase (decrease) of the ALP effects on the final Π_L and χ at a given energy, a variation in the value of m_a produces the modification of the energy threshold, starting from which the ALP-induced polarization features become relevant. In particular, as shown in Section <ref>, for m_a ≲ 10^-14  eV down to extremely low ALP masses, we do not expect modifications in the ALP-induced effects on Π_L and χ, since the ALP mass term turns out to be subdominant with respect to the plasma term in Equation (<ref>). For 10^-14  eV≲ m_a ≲ 10^-10  eV, the ALP mass term starts to become important and progressively overcomes the strength of the plasma term with a resulting translation at higher energies of the threshold above which the ALP-induced effects become sizable. For the above-mentioned ALP mass m_a = 10^-11  eV, substantial effects on Π_L and χ are expected starting from E_0 ≳𝒪(2-5)   keV, while, as shown in Section <ref>, for m_a = 10^-10  eV, the ALP impact on Π_L and χ is relevant for E_0 ≳𝒪(0.2-0.5)   MeV. Lastly, for m_a progressively increasing above 10^-10  eV, the ALP-induced effects on Π_L and χ will correspondingly appear at larger and larger energies up to disappear in the HE range and to become sizable only in the VHE band and beyond.

As far as polarization measurements are concerned, unfortunately, we do not presently have many observational data regarding the astrophysical sources considered in this review, i.e., galaxy clusters, blazars, and GRBs. In particular, polarization data in both the X-ray and HE bands regarding diffuse emission from galaxy clusters are substantially missing. Since, as discussed above, these latter astrophysical [-15]sources are extremely important for analyzing ALP-induced effects on photon polarization and for studying ALP physics in general, a dedicated observational campaign by IXPE <cit.> could provide vital information in this regard. Concerning blazars, we now have recent polarimetric data obtained in the X-ray band by IXPE (see, e.g., <cit.>), which promises to provide us with further important results in the near future. Regarding GRBs, some polarization measurements have been obtained in the X-ray band <cit.>. For all the astrophysical sources discussed in this review, further polarimetric data are expected from new missions proposed both in the X-ray band <cit.> and in the HE range <cit.>.






 §.§ Very High Energy Band


In the VHE band, we consider the case of a BL Lac emitting within the leptonic model and placed inside a rich galaxy cluster, as studied in <cit.>; therefore, this represents an alternative possibility with respect to those contemplated in Sections <ref> and <ref>. In the present situation, photon–ALP interaction inside the galaxy cluster is relevant, while the conversion inside the host is subdominant (see Section <ref> for more details). The total transfer matrix of the photon–ALP beam is represented by Equation (<ref>). In the following, two possibilities are evaluated: (i) a BL Lac with average properties (see below) is located at a redshift z=0.03; (ii) the same BL Lac is placed at z=0.4. The BL Lac redshift represents a crucial parameter to evaluate the photon–ALP system behavior in the VHE band: since, for E_0 ≳ 100   GeV, the absorption of VHE photons due to their interaction with the EBL is not negligible, as it occurs instead for lower energies, as in Sections <ref> and <ref> (see also Section <ref>), a larger redshift means a higher photon absorption.

In the present scenario, the profiles of the magnetic field in the jet B^ jet and of the electron number density n_e^ jet are provided by Equations (<ref>) and (<ref>), respectively, by taking the following parameter values: B^ jet_0 = 0.5   G, y_ em = 3 × 10^16  cm, n_e,0^ jet=5 × 10^4   cm^-3, and γ = 15. Photons are assumed to be produced within the leptonic model as unpolarized, i.e., with the initial degree of the linear polarization Π_L,0=0 (see <cit.> for more details). Moreover, we consider a CC galaxy cluster, hosting the BL Lac, with the electron number density profile

    n_e^ clu(y)=n_e,0^ clu( 1+y^2/r_ core^2)^-3/2β_ clu,

with the typical parameter values n_e,0^ clu=5 × 10^-2  cm^-3, r_ core= 100   kpc, and β_ clu=2/3 <cit.>. The cluster magnetic field B^ clu reads from Equation (<ref>) by taking the following typical parameter values: B_0^ clu=15   G, η_ clu=0.75, q=-11/3, k_L=0.2   kpc^-1, and k_H=3   kpc^-1 <cit.>. Furthermore, we assume an extragalactic magnetic field strength B_ ext < 10^-15  G, resulting in an inefficient photon–ALP conversion inside the extragalactic space. Concerning the ALP parameters, we take g_aγγ=0.5 × 10^-11  GeV^-1 and m_a = 10^-10  eV. 

For a BL Lac with the above-discussed properties, we show P_γ→γ and Π_L in the top subfigure of Figure <ref> and the corresponding f_Π associated with the final Π_L in the bottom subfigure. The left panels of Figure <ref> report the results for the BL Lac placed at z=0.03, while the right panels show the same results but for the BL Lac located at z=0.4 (see <cit.> for more details). 




The top subfigure of Figure <ref> shows that P_γ→γ and Π_L are energy dependent, since the photon–ALP system lies in the weak mixing regime due to the strength of the QED one-loop term, which is not negligible at high energies especially inside the BL Lac jet. Specifically, the final Π_L turns out to be strongly modified with respect to the initial one Π_L,0 in both cases of the BL Lac placed at z=0.03 and at z=0.4. Now, a new aspect must be considered in the VHE band with respect to the cases at lower energies discussed in Sections <ref> and <ref>: the top subfigure of Figure <ref> shows that P_γ→γ considerably decreases due to the EBL absorption starting from E_0 ≳ 3   TeV at z=0.03 and from E_0 ≳ 200   GeV at z=0.4. Correspondingly, Π_L starts to grow up to its limit value Π_L=1, which is reached at E_0 ≳ 30   TeV for z=0.03 and at E_0 ≳ 5   TeV for z=0.4, where photons are fully polarized and P_γ→γ≲ 10^-2. The different behavior with respect to the redshift is due to the fact that the EBL absorption grows as both the energy and the redshift increase. As deeply discussed in <cit.>, we have therefore found a peculiar feature: in the presence of photon–ALP interaction, photons, observed at energies where EBL absorption is very strong, are fully polarized. The full explanation of this important feature regarding Π_L, and shown in the top subfigure of Figure <ref>, is provided in <cit.>: in short, after photon–ALP interaction inside the source, we have both photons and ALPs entering the extragalactic space. Then, when EBL absorption is extremely high, all remaining photons are lost during their propagation in the extragalactic space, and we therefore collect only those photons that are reconverted back from ALPs in the Milky Way magnetic field B_ MW. Still, only the coherent component of B_ MW is efficient in producing photon–ALP conversion so that photons reconverted back from ALPs inside the Milky Way turn out to be fully polarized.

The behavior of f_Π in the bottom subfigure of Figure <ref> confirms the robustness of the previous discussion. In particular, upper panels deal with the case of small EBL absorption, while the lower ones describe the situation illustrated above, in which Π_L tends to its limit value Π_L=1 with a concomitant high EBL absorption. Note that the behavior of f_Π in the left panels is similar to that in the right ones, but different energies are considered (see the legend inside the panels) due to the above-described behavior of the EBL absorption as a function of energy and redshift. Since conventional physics cannot explain the scenario Π_L=1, a possible detection of fully polarized VHE photons not only would be an additional hint at ALPs in addition to those discussed in <cit.>, but also would represent the proof of ALP existence. Therefore, this ALP-induced feature identified in <cit.> producing fully polarized photons at VHE possesses a high discovery power. However, this possibility nowadays represents only a hope for the future, as currently known techniques cannot measure photon polarization in the VHE band but are limited to tens of GeV at most <cit.>.



 §.§ Gamma-Ray Bursts


ALP effects on the polarization of photons produced by GRBs have been analyzed in a pioneering study <cit.>, where photon–ALP conversion is evaluated inside the source, in the extragalactic space, inside possibly crossed galaxy clusters, and in the Milky Way within the tens of KeV-few MeV band. In particular, the benchmark energy E_0 = 100   keV and the ALP parameters g_aγγ= 10^-11  GeV^-1, m_a = 10^-13  eV are taken into account.

GRBs are extremely intense explosions originating at cosmological distances and are characterized by a prompt and an afterglow phase. By considering the typical parameters of the prompt phase—which is analyzed in <cit.>—regarding the magnetic field strength and the electron number density, the photon–ALP conversion inside the source turns out to be negligible. Regarding the magnetic field structure in extragalactic space, galaxy clusters, and Milky Way, domain-like models with different parameter values inside the several crossed media are assumed in <cit.>. In particular, by calling B_ ext, B_ clu, and B_ MW the magnetic field strengths in extragalactic space, galaxy clusters, and Milky Way, respectively, and L_ dom^ ext, L_ dom^ clu, and L_ dom^ MW the corresponding magnetic field coherence lengths, the following values are assumed: B_ ext=𝒪(1)   nG, B_ clu=𝒪(1)  μ G, B_ MW=𝒪(1-5)   G and , L_ dom^ clu=𝒪(10)   kpc, L_ dom^ MW=𝒪(10)   kpc (see <cit.> for more details).

For the chosen parameter values concerning both the photon–ALP system and the crossed media, the photon–ALP beam propagates in the strong mixing regime in all the previous zones within the considered energy band. Consequently, both P_γ→γ and Π_L turn out to be energy independent with Π_L which is modified with respect to the initial Π_L,0. Therefore, the only informative quantity in the present situation is represented by f_Π, which is shown in Figure <ref> for photon–ALP conversion inside the extragalactic space. In particular, the behavior of f_Π for different initial values of Π_L,0 is shown in the top subfigure of Figure <ref> for a GRB placed at redshift z=0.3 and in the bottom subfigure for a GRB located at z=2. Note that, in Figure <ref>, the initial degree of linear polarization is denoted by Π_0 and the photon–ALP coupling by g_aγ, but these quantities possess the same meaning of Π_L,0 and g_aγγ, respectively, which are used in the whole review.



Figure <ref> shows that the initial Π_L,0 (we recall the different notation here with ) is strongly modified and broadened in a similar way as it occurs in the systems considered in Sections <ref> and <ref>. In particular, the bottom subfigure shows that this effect increases with the redshift. Moreover, as shown in <cit.>, the photon–ALP beam propagation inside galaxy clusters and in the Milky Way further increases the broadening effect of the initial Π_L,0 to a point that, at high redshift, the most probable value of the final Π_L is no more the initial one, apart from the case of the initial Π_L,0≡Π_0 = 1.

These findings demonstrate that GRBs are very important astrophysical sources where ALP-induced effects on photon polarization can be studied, 
in addition to galaxy clusters and BL Lacs (see Sections <ref> and <ref>). The recent detection by the LHAASO Collaboration of photons with energy of up to 18   TeV produced by GRB 221009A <cit.>—which is difficult to justify in the context of conventional physics—shows the need to extend the analysis of the ALP impact on GRB polarization even at higher energy bands in a similar way as discussed in Sections <ref> and <ref>. Moreover, the explanation of the GRB 221009A detection up to 18   TeV within the photon–ALP interaction model developed in <cit.> demonstrates the importance of the photon–ALP conversion also inside the galaxy hosting the GRB. Therefore, an analysis similar to that performed in <cit.>, by considering the photon–ALP beam propagation also in the host galaxy and updated models for the photon–ALP conversion inside extragalactic space, galaxy clusters, and Milky Way, appears very promising in order to discover further ALP-induced features on photon polarization and even possible new hints at ALP existence or to improve constraints on the ALP parameter space.



§ CONCLUSIONS


In this review, we have described the ALP-induced effects on photon polarization, which can provide a different and complementary way to study the ALP physics with respect to the analysis of the ALP consequences on astrophysical spectra. As mentioned above, there exist three hints at ALP existence linked to the ALP effects on astrophysical spectra, with the most recent and strongest indication coming from the detection by the LHAASO Collaboration of GRB 221009A up to 18   TeV <cit.>, which can hardly be justified within conventional physics but is naturally explained by photon–ALP interaction <cit.> (see also <cit.>). In the near future, we expect that new spectral data from observatories such as ASTRI <cit.>, CTA <cit.>, GAMMA-400 <cit.>, HAWC <cit.>, HERD <cit.>, LHAASO <cit.>, and TAIGA-HiSCORE <cit.> will be able to increase our knowledge of the ALP properties.

The study of the ALP effects on the polarization of photons generated by astrophysical sources such as galaxy clusters, blazars, GRBs, neutron stars, and white dwarfs can add further information by shedding light on the ALP parameter space with current spectral hints suggesting m_a = 𝒪(10^- 10)   eV and g_a γγ = 𝒪(10^- 11)   GeV^- 1. As discussed in this review, galaxy clusters represent very promising observational targets in this respect. In fact, according to conventional physics, photons are emitted unpolarized in the central region of galaxy clusters both in the X-ray and in the HE band, resulting in an expected zero polarization degree for the observed photons, while the photon–ALP interaction makes them partially polarized with ALP parameters in the same ballpark of those producing spectral effects. Therefore, a possible future signal of partially polarized photons diffusely produced inside galaxy clusters would be a further strong hint at ALP existence. A negative signal would instead be useful to constrain the ALP parameter space further. In any case, as discussed in this review, future observations of the photon polarization degree from other astrophysical sources like blazars and GRBs are also very promising. Missions such as IXPE <cit.> (already operative), eXTP <cit.>, XL-Calibur <cit.>, NGXP <cit.>, XPP <cit.> in the X-ray band and like COSI <cit.>, e-ASTROGAM <cit.>, and AMEGO <cit.> in the HE range may test our findings about ALP effects on photon polarization. 

Finally, an ALP direct detection could be achieved by laboratory experiments such as ALPS II <cit.>, IAXO <cit.> and STAX <cit.>, thanks to the techniques developed by Avignone and collaborators <cit.> and possibly with the ABRACADABRA experiment <cit.>.



 




























The work of the author is supported by a contribution through Grant No. ASI-INAF 2023-17-HH.0 and by the INAF Mini Grant `High-energy astrophysics and axion-like particles', PI: Giorgio Galanti.

Data used in this paper can be asked from the authors of the quoted papers. 










The author thanks Giacomo Bonnoli, Silvano Molendi, and Lara Nava for the discussions and all collaborators in this field, especially Enrico Costa, Alessandro De Angelis, Marco Roncadelli, and Fabrizio Tavecchio.

The author declares no conflict of interest. 



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 Paper Title: Separating deterministic and stochastic gravitational wave signals in realistic pulsar timing array datasets.

 

    
   




    Dipartimento di Fisica “G. Occhialini", Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy

    i.ferranti@campus.unimib.it
    INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
    INAF - Osservatorio Astronomico di Cagliari, via della Scienza 5, 09047 Selargius (CA), Italy
    INAF - Osservatorio Astronomico di Brera, via Brera 20, I-20121 Milano, Italy
    ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands
    
Not anymore
Ferranti et al





    











  Recent observations by pulsar timing arrays (PTAs) suggest the presence of gravitational wave (GW) signals, potentially originating from supermassive black hole binaries (SMBHBs). These binaries can generate two kinds of signals: a stochastic gravitational wave background (GWB) or deterministic continuous gravitational waves (CGW). Being able to correctly recognize and separate them is crucial for accurate signal recovery and astrophysical interpretation. This paper aims at investigating the interaction between stochastic GWB and deterministic CGW signals within current analysis pipelines. We focus on understanding potential misinterpretations and biases in the parameter estimation when these signals are analysed separately or together. To this end, we performed several realistic simulations based on the European PTA 24.8yr dataset. We first injected either a GWB or a CGW into five datasets (three GWB realisations and two CGW realisations) with identical noise and analysed each signal type independently, and then analysed data sets containing both a stochastic GWB and a single resolvable CGW. We compared parameter estimation using different search models, including Earth term (ET) only or combined Earth and pulsar term (ET+PT) CGW templates, and correlated or uncorrelated power law GWB templates. We show that, when searched independently, the GWB and CGW signals can be misinterpreted as each other, and only a combined search is able to recover the true signal present. For datasets containing both a GWB and a CGW, failure to account for the latter biases the recovery of the GWB, but when performing a combined search, both GWB and CGW parameters can be recovered without any strong bias. Care must be taken with the method used to perform combined searches on these multi-component datasets, as the CGW PT can be misinterpreted as a common uncorrelated red noise. However, this can be avoided by direct searches for a correlated GWB plus a CGW (ET+PT). Our study underscores the importance of combined searches to ensure unbiased recovery of GWB parameters in the presence of strong CGWs. This is crucial to interpret the signal recently found in PTA data, and is a first step towards a robust framework for disentangling stochastic and deterministic GW components in future, more sensitive datasets.
    
    
 


  

  

  

  

  





   Separating deterministic and stochastic gravitational wave signals in realistic pulsar timing array datasets
    Irene Ferranti1,2, Golam Shaifullah1,2,3, Aurelien Chalumeau1,2,5 & Alberto Sesana1,2,4
    August 5, 2024
============================================================================================================







§ INTRODUCTION


Pulsar Timing Array (PTA) collaborations search for gravitational waves (GWs) with frequencies in the nanoHertz (nHz) band (10^-9-10^-7Hz). The primary target of PTA experiments is a stochastic gravitational wave background (GWB) generated by the incoherent superposition of the GWs emitted by all the super-massive black hole binaries (SMBHBs) that populate our universe <cit.>. The signal manifests itself in the data as a red noise process common to all the pulsars in the array, with a specific angular correlation, first derived by <cit.>. Recently, the PTA community found evidence for a signal with both these characteristics: the collaborations reporting this finding were the European PTA collaboration along with the Indian PTA collaboration <cit.>, the North American Nanohertz Observatory for Gravitational Waves collaboration <cit.>, the Parkes PTA collaboration <cit.> and the Chinese PTA collaboration <cit.>. 

While the Hellings and Downs (HD hereinafter) correlation is distinctive of a GW origin of the signal, separating it from possible correlated noise of different nature <cit.>, its source cannot yet be confidently established. In fact, besides SMBHBs, a nHz cosmic GWB can also be produced by a number of physical processes occurring in the early universe, such as cosmic strings networks <cit.>, primordial curvature perturbations <cit.>, or QCD phase transitions in the early universe <cit.>, all of which can provide a viable explanation of the observed signal <cit.>.
Moreover, the HD correlation is not unique of stochastic GWBs; as shown in <cit.> and in <cit.>, the signal generated by a single SMBHB and observed by an array of pulsars results in an angular correlation with the HD expectation value, albeit with a somewhat different variance. Note that the quality of the current data does not allow a description of the signal variance and  therefore, the possibility that a particularly loud source – a massive and nearby SMBHB – can be mistaken by a GWB is real. By loud source, we mean a source that can be singled out from the noise and, if present, from the GWB by means of a deterministic, template based search. Since the SMBHBs producing such signals are far from mergers and persistent in the data, their signal is generally referred to as continuous gravitational wave <cit.>.

CGWs searches have been performed on the latest data release of EPTA <cit.> and on the NANOGrav 15yr dataset <cit.>. They both found evidence in favour of the presence of a CGW with a 2.5-3σ significance, compared to a model where only pulsar noise is present in the data. This value is comparable to the evidence in favour of the presence of a GWB, which has a 3-to-4σ significance). Some simulations carried out in <cit.> already showed how a CGW can be misinterpreted as a GWB and, vice versa, a GWB can be misinterpreted as a CGW. In both the EPTA and NANOGrav analyses, when a GWB is also searched for, evidence for a CGW significantly decreases, indicating that its presence is at the very least not necessary to explain the observations. However, the degeneracy between the pulsar response to the two signals does not allow to conclusively rule out the presence of a CGW. 

The presence of these resolvable sources in the PTA data on the one hand poses a number of interesting challenges, on the other hand opens the door to invaluable opportunities. In fact, while the analysis pipelines currently used to search for a GWB rely on the assumption that the background is a Gaussian, isotropic, and stationary process characterised by a power-law Fourier spectrum <cit.>, the presence of a CGW can result in an excess of power at specific frequencies, which produces some level of anisotropy and non Gaussianity <cit.>. Thus, the standard analysis may depend on inapplicable assumptions and yield biases on the estimated GWB parameters <cit.>. This is a recognised issue within the PTA community and analysis techniques are being developed to go beyond the power law, isotropic Gaussian assumption <cit.> and to pull out of that data GWBs and CGWs simultaneously <cit.>.

Within this context, it is of paramount importance to study the interplay of CGWs and GWB when both are present in the data and what challenges arise in the interpretation of the analysis results. From an exquisitely scientific perspective, conversely, the prospects of detecting CGWs is particularly appealing. In fact due to their deterministic nature, PTAs are able to estimate the source parameters, in particular the sky location via triangulation <cit.> and possibly the distance to the source <cit.>. The detection of one or more resolvable sources <cit.> would be very precious as it would allow to track the orbital evolution of the binary and thus obtain direct information on the Massive Black Holes (MBHs) coupling to the environment <cit.>. This information cannot be well constrained by a measure of the GWB power spectral density, because it is strongly degenerate with the MBH mass and eccentricity distribution. In addition, CGWs offer the exciting opportunity to search for electromagnetic counterparts of the GW signal, opening the prospects of low frequency GW multimessenger astronomy <cit.>. Although PTAs can realistically localise the CGW source only within several deg^2 <cit.> they must be originated by particularly massive and/or nearby SMBHBs, making the search of their galaxy host viable <cit.>. Identifying the host and detecting accretion activity onto the binary will shed light about the physics governing the formation and (thermo)dynamical evolution of SMBHBs and the accretion discs surrounding them. 

In this paper, we carry out the first detailed investigation of the interplay between stochastic GWBs and deterministic CGWs in realistic PTA datasets, assessing the performance of currently used analysis pipelines when both signals are present in the data. We do so by building state of the art mock datasets capturing the complexity of real data, and by injecting them with as realistic as possible GW signals, mimicking the one found in the recent EPTA analyses <cit.>. Realistic GWBs are injected as the incoherent sum of individual waves consistent with astrophysical observations <cit.>, and CGW are modelled according to the candidate identified in <cit.>. Datasets are then analysed with the standard models implemented in the  <cit.> analysis suite, which models the GWB as a common red process characterized by a single power law spectrum, while we also test CGW models including only the Earth term or the full Earth plus pulsar components. 

The paper is organized as follows. In Section <ref> we describe in detail the simulations: the properties of the adopted datasets, the injected signals and the model used for the recovery. In Section <ref> we present the main results of our analysis. Using the methods described in the previous section, we carried out several investigations. We first assess the importance of including the pulsar term in the CGW model (Section <ref>). We then study the degeneracy of signal recovery on datasets containing only a GWB or a CGW (Section <ref>), showing that datasets injected only with a CGW can produce evidence of a GWB (Section <ref>) and vice versa (Section <ref>), and that a joint search is able to pin down the correct nature of the signal (Section <ref>). The performance of a combined analysis of a GWB and a CGW on datasets containing both signals is then evaluated in Section <ref> and we discuss our main findings in Section <ref>.
 Throughout the paper we assume a Λ cold dark matter cosmology with parameters (H_0, Ω_m, Ω_Λ)=(70 km s^-1Mpc^-1, 0.3, 0.7). 
















  


§ METHODS

In this section, we describe both the simulations structure and the models and methods used in the analysis.



 §.§ Simulations




Our simulations are based on the second data release of EPTA <cit.>. In particular, we have reproduced the DR2full dataset: 25 pulsars with a maximum observation time of 24.8 yr. The main properties of the real time of arrivals (ToAs) of each pulsar are reproduced: the observation time, the number of ToAs, the timing model (TM), the multiband measures – the frequency coverage is identical to that of the EPTA DR2full pulsars – and the noise levels. The uneven sampling of the ToAs is also reproduced, but large gaps, which are common in the real data streams, are not present in the simulations (see Fig. <ref> for a comparison between real and simulated residuals).
All the properties of the arrival times were injected in the dataset using the  package <cit.>. The same package is used to inject the three components of the noise budget: white noise (WN), achromatic – independent of the observation radio frequency –  red noise (RN) and chromatic red noise (due to dispersion measure variations, and customarily referred to as DM). 

To model the white noise, the ToAs are sampled from a Gaussian distribution centred at the value predicted by the TM and with a width given by:

    σ_i = √(EFAC_i^2σ_ ToA^2 + EQUAD_i^2),

where σ_ ToA is the errorbar associated to each ToA according to the template-fitting errors obtained in EPTA DR2, the EFAC takes into account for the ToA measurement errors and the EQUAD accounts for any putative extra source of white noise. The two parameters EFAC and EQUAD are specific for each observing backend. In the simulations, each pulsar's backend, as well as the ToAs' uncertainties, are identical those of DR2full; therefore, each pulsar has ToAs taken from ∼12 different backends, which means 12 different couples of values for EFAC and EQUAD. The values used for EFAC and EQUAD are the maximum likelihood values obtained by the single pulsar noise analysis of EPTA DR2full <cit.>.

In addition to the white noise, achromatic and chromatic red noise contribute to the single-pulsar noise budget. These are time-correlated noise components modelled as a stationary Gaussian process with a power-law power spectral density (PSD) of the form

    P(f; A, γ) = A^2/12π^2(f/ yr^-1)^-γ yr^3.

For achromatic red noise, in each ToA, the delay induced in the residuals is proportional to the red noise PSD, 

    Δ_ RN∝√(P(f; A_ RN, γ_ RN)).


This noise component accounts for the long-term variability of the pulsar spin along with other sources, such as unmodelled objects orbiting the pulsar. For chromatic red noise, on the other hand, the induced delay also depends on the observing radio frequency because it is due to the interaction of the radio signal with the ionised interstellar medium (IISM), the interplanetary medium in the solar system, and the Earth's ionosphere. This effect is taken into account during the observations and inside the timing model which considers its value at a reference epoch together
with its first and second derivatives. However, the inhomogeneous and turbulent nature of the IISM also induces stochastic variations in the DM value, which are modelled as chromatic red noise. The residual DM delay is modelled as 

    Δ_ DM∝√(P(f; A_ DM, γ_ DM))×ν^-2

where ν is the observing radio frequency. The values of A_ RN, γ_ RN, A_ DM and γ_ DM used in the simulations are  reported in Table <ref>, together with the number of Fourier components used in the modelling of these processes. These values – amplitude, slope and number of Fourier components – correspond to maximum likelihood values obtained from the single pulsar noise analysis of EPTA DR2full. An example of simulated data compared to the real EPTADR2 dataset is shown in Fig. <ref>.






  §.§.§ Gravitational wave signal injection


Having specified the properties of the adopted PTA, we now turn to the description of the injected GW signals. The signal produced by a SMBHB is composed by two terms, customarily referred to as Earth term (ET) and pulsar term (PT), which are respectively associated to the GW strain hitting the Earth at the ToA of the considered pulse and the strain hitting the pulsar at the time of emission of the same pulse.

Throughout the paper we will always assume circular SMBHBs, evolving under the effect of GW backreaction only (i.e. we neglect any possible coupling with the local stellar or gaseous environment). Under these assumptions, the delays induced in the ToAs at an epoch t are fully defined by 8 parameters: the chirp mass ℳ[Note that in the quadrupolar approximation employed here for the injected waveform, the signal does not depend on the individual binary masses.], the observed frequency f_gw, the redshift z, the inclination angle , the sky location (θ, ϕ) – respectively polar and azimuthal angle – , the initial phase ϕ_0 and the polarization angle ψ. Following <cit.>, we model them as:


    s(t, Ω̂) = F^+(Ω̂)Δ s_+(t) +F^×(Ω̂)Δ s_×(t),

where 

    Δ s_A(t) = s_A(t_p) - s_A(t_e),

the functions F^A(Ω̂) – with A indexing the polarization and Ω̂ being the unit vector pointing from the GW source to the Solar System Barycenter ˆ
(SSB) – are the antenna pattern functions that depend on the relative position of the pulsar and the source <cit.>. s_A(t_p) and s_A(t_e) are the PT and ET respectively. They both take the form


    s_+(t) = ℳ^5/3/d_Lω(t)^1/3(-sin(2(Φ(t)-Φ_0))(1+cos^2)cos(2ψ) 
       - 2cos(2(Φ(t)-Φ_0))cos sin(2ψ))
       s_×(t) = ℳ^5/3/d_Lω(t)^1/3(-sin(2(Φ(t)-Φ_0))(1+cos^2)sin(2ψ) 
       +2cos(2(Φ(t)-Φ_0))cos cos(2ψ)),

where 

    Φ(t) = Φ_0 + 1/32ℳ^5/3(ω_0^-5/3-ω(t)^-5/3),

and 

    ω(t) = (ω_0^-8/3-256/5ℳ^5/3t)^-3/8.

The only difference between PT and ET is that while the latter is the delay induced at the time of observation on Earth t_e, the former has to be evaluated at the time of emission of the radio pulse at the pulsar t_p = t_e - L(1 + Ω̂·p̂), where L is the pulsar distance and p̂ is the pulsar position in the sky. We inject in our data two different GW components: a stochastic GWB and a resolvable CGW. Since the evolution of the signal over the observation time is small, the phase approximation is used both in the injection and in the recovery: ω_ Earth = ω_0 at each epoch t.

Stochastic GWB. The GWB is injected as a realistic superposition of GWs coming from a cosmic population of SMBHBs. The injection pipeline starts from a catalog of SMBHBs, computes the delays induced by each binary in each pulsar ToA according to Eq. (<ref>) and then sums all the delays in the time domain. The distribution of the sources' ℳ, f_gw, z and  determines the observed power spectrum generated by the SMBHBs population: the binned total characteristic strain, where the frequency bin width is Δ f = 1/T with T the time span of the pulsar array, can be evaluated as <cit.>

    h_c^2(f_i) = ∑_j∈Δ f_ih_j^2(f_r)f_r/Δ f_i,

where f_i is the central frequency of the bin Δ f_i, and the sum runs over all the binaries for which the observed frequency f=f_r/(1+z)∈Δ f_i, where f_r is the GW frequency in the binary rest frame. The strain of each GW signal is given by 

    h(f_r) = 2√(1/2(a()^2 + b()^2))(Gℳ)^5/3(π f_r)^2/3/c^4d(z),
 
where a() = 1+cos^2, b() = -2cos and d(z) is the comoving distance of the binary, which is computed as 

    d(z) = c/H_0∫_0^z1/√(Ω_m (1+z)^3 + Ω_Λ)dz.

For the population of circular GW driven SMBHBs, assumed here, the characteristic strain given by Eq. (<ref>) can be approximated as a power law. It is therefore customary to describe the signal as

    h_c(f) = A_ GW(f/ yr^-1)^α,

where α=(3-γ)/2=-2/3 being γ the PSD induced in the timing residual (cf. Eq. <ref>).

The power spectrum h_c^2(f_i) thus depends on the number of emitting sources per unit redshift, mass and frequency d^3N/(dzdℳdf_r). A list of ∼ 120k binaries is constructed by extracting ℳ, f_gw, z as Monte Carlo samples of a numerical distribution d^3N/(dzdℳdf_r), obtained from the observation-based models described in <cit.> and <cit.>. In those papers, the authors construct hundreds of thousands of SMBHB populations in agreement with current constraints on the galaxy merger rate, the relation between SMBHs and their hosts, the efficiency of SMBH coalescence and the accretion processes following galaxy mergers. From this set of models, we selected 100 populations producing a GWB amplitude A ∼ 2.4×10^-15, consistent with the signal observed by EPTA <cit.>. The other parameters of the binaries are then randomly sampled such that sources are isotropically distributed in the sky – cos θ∈Uniform(-1, 1) and ϕ∈Uniform(0, 2π) –, their orbital angular momenta are uniformly distributed on the unit sphere – cos ι∈Uniform(-1, 1) –, the phase and the polarization angle are random – ϕ_0 ∈Uniform(0, 2π), ψ∈Uniform(0, π). 


Resolvable CGW. In some of the dataset, an individually resolvable CGW is injected. The waveform is again injected in each pulsar in the time domain residual according to Eq. (<ref>). The only difference, compared to the SMBHBs contributing to the unresolved GWB is that the amplitude of the signal is large enough that the source can be identified above the pulsar noise and the unresolved GWB by a signal search employing a deterministic filter to match the signal (see next Section). We stress that we always inject both the ET and the PT for every SMBHB, regardless of whether it provides an unresolvable contribution to the GWB or it is a resolvable CGW.

Datasets. For the purpose of this investigation, seven different datasets were produced, with the exact same noise realization, but different gravitational signals:

    
  * dataset injGWB01, injGWB02, injGWB03. These three datasets contain a GWB-only. They are produced injecting three background selected from the 100 populations mentioned above. The three populations are chosen among those displaying a smooth behaviour in the lowest frequency bins, without particularly loud sources.

    
  * dataset injCGW5, injCGW20. These two datasets only contain a single loud CGW-only. The injection procedure is similar to the GWB datasets, but the population file used contains only one binary. The two injected sources have Earth term GW frequencies of 5nHz (injCGW5) and 20nHz (injCGW20). The parameters of the 5nHz source are chosen to approximately reproduce frequency and amplitude of the candidate CGW found in the EPTA DR2new, <cit.>, which in fact has a frequency of ∼5nHz and an amplitude h∼1.1× 10^-14. The source is then located in the Virgo cluster; this choice determines sky location and distance, and consequently the chirp mass. Inclination, initial phase and polarization angles are randomly extracted as in the GWB case. The source injected in injCGW20 has identical parameters except for the higher frequency (20nHz instead of 5nHz), thus resulting in a higher strain (h∼ 2.7× 10^-14). The delays induced in pulsar PSR J1022+1001 by these two CGWs are shown in Fig. <ref>.
    
    
  * dataset injGWB03+CGW5, injGWB03+CGW20. These two datasets are injected with a GWB and a CGW. The GWB is in both cases the one resulting from population 03, while the CGWs are either at 5nHz or 20nHz.










 §.§ Signal modelling and analysis methods

All the analyses are carried using the  <cit.> package. The starting point of the analysis is the likelihood function of the timing residuals δt

    P(δ𝐭|ξ, ζ) = exp[-1/2(δ𝐭 - 𝐌ϵ - 𝐬)^T𝐂^-1(δ𝐭 - 𝐌ϵ - 𝐬)]/√((2π)^ldet𝐂).

The term Mϵ is the timing model error expressed as a linear function of the offset from the nominal timing model parameters, ϵ. ξ are the timing model parameters and ζ are the stochastic components parameters. The likelihood function sampled to search for the gravitational signal is marginalized over the timing model parameters as described in <cit.>. 𝐂=𝐂(ζ) is the covariance matrix that contains all the contributions from stochastic processes, which are:

    
  * WN, modelled according to Eq. (<ref>). All the white noise parameters are fixed at the injected values in the GW searches;
    
  * RN, modelled as a Gaussian process with amplitude of the Fourier components given by Eq. (<ref>)  (<cit.>;
    
  * DM, modelled as a Gaussian process with amplitude of the Fourier components given by Eq. (<ref>) and amplitude depending on the observed radio frequency (see Eq. <ref>). As for RN, the DM is not present in all the pulsars and, if present, the number of Fourier bins used in the Gaussian process depends on the pulsar and it is equal to the number of components used in the injection. For a summary of the noise model characteristics, see Table <ref>.
    
  * common red noise (CRN), if searched for. A red noise process which is common to all the pulsars can be accounted for in the covariance matrix. If included in the analysis, it is also modelled as a Gaussian process. The amplitude of the components are modelled in two ways: as a power law function of the frequency, and therefore the parameters are the slope γ_ CRN and the amplitude A_ CRN; or as a binned spectrum, and thus the parameters are the amplitudes of the common process in each frequency bin f_i. In both cases, the number of components used in the Gaussian process is 30.
    

Although we inject the GWB as  a superposition of unresolved SMBHBs, we follow the standard PTA analysis practice and model it in our likelihood function as a CRN characterized by an amplitude and a spectral index, as in Eq. (<ref>), which can be directly translated in a characteristic strain of the form given by Eq. (<ref>). Note that, depending on the dataset, we use two different flavours of CRN. In most searches, the GWB is modelled as a common uncorrelated red noise (CURN). This means that inter-pulsar correlations induced by the GWB are neglected and the GWB posterior distribution is then reconstructed a posterirori with the reweighting technique <cit.>, when possible. The advantage of this model is that it is much faster since it keeps the correlation matrix in the likelihood function diagonal and it naturally provides the Bayes factor of the reconstructed model versus the  sampled model, besides the posterior distribution of both models. However, the reweighting technique is not always efficient and, as we will see later, using a CURN to model a GWB is not always appropriate; therefore, when needed, we use the HD-correlated CRN model which we referred to as GWB model. In this case, GWB induced inter-pulsar spatial correlation are also modelled, according to the HD curve <cit.>

    Γ_ab=3/21-cosθ_ab/2ln( 1-cosθ _ab/2)-1/41-cosθ_ab/2 +1/2

where a and b are indexes identifying two pulsars in the array and θ_ ab is their angular separation in the sky. Note that searching for an HD-correlated CRN is much more computationally expensive, since the correlation matrix is not diagonal anymore.

The last term in the likelihood function is s, which is the delay induced by an individually resolvable CGW, and is analytically described by Eqs.(<ref>)–(<ref>) which are implemented in the CGW model of . An approximation that can be adopted is to consider in the CGW model the ET only, neglecting the PT. This is computationally very convenient, since the former only depends on the 8 source parameters. Conversely, the latter also depends on the distance of each pulsar in the array. Since those  distances are generally known with large uncertainties (of the order of 20%), they must be considered as additional free parameters in the search. Therefore, the complete model has at least one more parameter per each pulsar with respect to the ET only model. Moreover, as prescribed in <cit.>, in order to ease the search of the best parameters, one extra parameter is added per each pulsar, which is an initial phase at the pulsar, Φ_p. We will assess the performance of an ET only search on realistic signals including ET and PT in Section <ref>.

We mainly adopt Bayesian techniques in our analysis, although we also exploit the frequentist optimal statistics, when useful. The priors used in the search are listed in table <ref> and are all uninformative, except for the pulsar distances' priors that exploit information obtained from independent electromagnetic observations. The prior distributions have a width which is taken as the measurement error listed in <cit.>; for the pulsars that are not included in this paper, the relative error is fixed at 20%. The mean of the prior is chosen such that the pulsar distance used for the injection is within 1σ of the prior distribution for each pulsar. Parameter estimation is then performed by sampling the posterior distribution with the Markov
Chain Monte Carlo (MCMC) sampler , <cit.>. For the 25 pulsar PTA we employ, the dimensionality of the posterior distribution is 62 (2 parameters – slope and amplitude – for each red noise and dispersion measure process included in the model) plus the GW model dimension (2 for the GWB power-law model, 8 or 58 for an ET only or a full ET+PT CGW model search respectively). Model selection is done by computing the Bayes factor, either with the reweighting technique, when it is efficient, or with the product space method implemented in  <cit.>.


































  


§ RESULTS


Using the methods described in the previous section, we carried out several investigations. In Section <ref>, we assess the importance of modelling a CGW including both the Earth and pulsar terms. In Section <ref>, we investigate the degeneracy between the analysis results obtained with a GWB-only dataset and with a CGW-only dataset. In Section <ref>, it is shown how CGW-only datasets can produce strong evidence in favour of an HD-correlated GWB. Viceversa, in Section <ref>, it is shown how a GWB-only dataset can be misinterpreted as a continuous wave. This degeneracy is solved in Section <ref>, by means of a combined analysis that searches simultaneously for a GWB and a CGW on both GWB-only and CGW-only datasets . Finally, in Section 
<ref>, we evaluate the performance of a combined analysis of a GWB and a CGW on datasets containing both signals. 






 §.§ On the importance of the pulsar term in CGW modelling





















As described in section <ref>, the signal imprinted by a CGW in the ToAs is the sum of two terms: the Earth term and the pulsar term. The ET is common to all pulsars and only depends on the 8 parameters describing the source. Conversely, the PT depends on the distance of each pulsar and the phase of the GW signal at the pulsar's location. 
This means that modelling of the full CGW ET+PT in a search performed on an array of 25 pulsars (like the one we are considering here) requires 50 extra free parameters, that must be added to the 8 source parameters and to the 62 parameters of the individual pulsars noise components. It is therefore interesting to compare searches of CGW modelled as ET only and ET+PT on realistic datasets, to see whether the computational efficiency of the ET only search bears any significant drawback.


The two templates have been tested on both CGW-only datasets (injCGW5 and injCGW20), and the main results are shown in Fig. <ref>. From the figure it is clear that the performance of the ET only template depends on the CGW frequency. In fact, when applied to the injCGW20 dataset (right panel of Fig. <ref>), the ET only model performs well on all parameters that it is sensitive to, i.e. all source parameters except the chirp mass and the luminosity distance. Conversely, the same model applied to the injCGW5 dataset results in some non-negligible biases in the recovery. As it can be seen from the left panel of Fig. <ref>, the injected values for the frequency and sky location are not compatible with the 90% C.I. of the recovered posterior distributions. 

The different performance of the ET only template in the two cases is due to the different degree of separability of the ET and PT terms for the two injected sources. For the injected system chirp mass, ℳ=1.16· 10^9M_⊙, the average frequency difference between ET and PT is Δ f∼ 
3.8×10^-6nHzyr^-1Δ t∼ 1.0× 10^-2 nHz∼ 0.02Δ f_bin for the 5nHz CGW and Δ f∼ 1.4× 10^-3nHzyr^-1Δ t∼ 4.0 nHz∼ 3.2Δ f_bin for the 20nHz CGW. Therefore, while in the 20nHz CGW, ET and PT falls at very distinct frequencies, they blend together in the same frequency bin for the 5nHz CGW. In the former case, the ET only template only picks the common power produced by the ET at its frequency, disregarding the PT term; while in the latter, the ET only template is trying to fit a superposition of ET and PT essentially overlapping at the same frequency. This concept is demonstrated in Fig. <ref>, where the reconstructed waveforms are compared to the injected ones. In the 20nHz case (injCGW20 dataset), the ET only template almost perfectly fits the ET of the full injected waveform, while in the 5nHz case (injCGW5), the ET only template absorbs power from both ET and PT and the reconstructed waveforms sits, for many pulsars, somewhat in between the ET and the full injected signal and is also way more uncertain as shown by the much larger credible region.





Conversely, the full ET+PT template is able to correctly reconstruct the signal and, because it is sensitive to the frequency evolution, it also allows to constrain both the source chirp mass and the luminosity distance. The performance is very different for the 5 nHz CGW and the 20 nHz CGW: for the former the chirp mass recovery is ℳ=0.52^+9.98_-0.48× 10^8M_⊙, while for the latter it is ℳ=1.01^+0.07_-0.06×10^9M_⊙, where we give the median and the 68% credible interval. The superior recovery of the chirp mass in the 20 nHz case is due to the much larger frequency evolution of the source between ET and PT, allowing to place a much tighter constrain on ḟ and thus on M.
Another consequence of the large frequency evolution of the 20 nHz CGW is that the model is sensitive to the pulsar distances. This is shown in Fig.<ref> where the reconstructed waveform (left) and the posterior distribution of the pulsar distance (right) are shown for PSR J1713+0747 for the injCGW20 (top row) and injCGW5 (bottom row) datasets. In the first case, the posterior distribution on the distance is narrower than the prior and centred at the injected value: this is allowed by a correct modelling of the frequency evolution, which is reflected in the excellent reconstruction of the waveform. Conversely, for the 5 nHz case, the posterior distribution of the pulsar distance is almost indistinguishable from the prior, but signal reconstruction still matches the injection, as the frequency evolution is too small for the search to be sensitive to it. We note that the GW signal reconstruction and distance determination is not always as good for the 20 nHz CGW. This is shown in the middle row of Fig.<ref> for pulsar PSR J1918-0642. In this case the posterior distribution resembles the prior, and while the reconstructed signal does not strongly disagree with the injection, the median fails to capture the signal shape accurately. In fact, only about 13 pulsars, whose evolution is accurately reconstructed, significantly contribute to the chirp mass recovery.



In addition to better and unbiased recovery of the CGW parameters, there is another rationale for preferring the complete CGW template over the ET-only template: in a joint CGW + GWB search, the contribution to the CGW signal due to the pulsar term, if not taken into account by the CGW model, is absorbed by the GWB model, since it is a common red noise process. It will therefore add extra spurious power to the GWB, biasing the recovery of its parameters. An example of this behaviour is shown in Fig. <ref>, where the GWB+CGW search is applied to the dataset injCGW5: when the CGW is searched with the full template, the posterior distribution of the GWB parameters is unconstrained, when the search is done with the ET-only template, the posterior distribution of the GWB parameters remains well constrained because it absorbs all the power coming from the unmodelled pulsar term.

We therefore conclude that while the ET-only template can be used to search for possible CGW candidates and provides a reasonable estimate of the source parameters <cit.>, the full ET+PT model must be preferred, especially in a joint search with a stochastic GWB. Despite the much lower computational efficiency, we will therefore always model our CGWs as ET+PT in the following.











 §.§ Degeneracy between GWB and CGW

In this section, searches for GWB, CGW and combined searches of the two signals together are applied to the 3 GWB realizations (injGWB01, injGWB02, injGWB03) and to the 2 CGW realizations (injCGW5, injCGW20). The aim is to study the degeneracy between the two signals and how it can be solved.





  §.§.§ CGW misinterpreted as a GWB


We start by performing a GWB analysis on the three injGWB and the two injCGW datasets. To this end we make use of both Bayesian and frequentist techniques following this procedure:

    
  * a power law-shaped and HD-correlated stochastic process is assumed for the GWB, then the slope γ_ GWB and the amplitude log_10A_ GWB of the powerlaw process are sampled, together with the noise parameters to obtain the posterior distribution;
    
  * the Bayes factor ℬ^ GWB_ CURN is evaluated to compare the evidence for an HD-correlated and an uncorrelated red noise process. The bayes factor is computed either with the reweigthing method or with the product space when reweighting is not efficient;
    
  * the averaged angular correlation between the pulsars is estimated with the optimal statistics method averaging over 10 angular bins, each one containing 30 pulsar pairs. A powerlaw-shaped background with variable γ is assumed;
    
  * the SNR of the HD cross-correlation marginalized over the noise parameters is obtained with the optimal statistics method, assuming a powerlaw-shaped background with variable γ.


1.4 





The results of the powerlaw spectrum parameter estimation are listed in the last two columns of Table <ref> for all five datasets and two examples of the recovered GWB parameters' posterior distributions are shown in Fig. <ref> for datasets injGWB02 and injCGW5. In general, the GWB search returns well constrained posteriors regardless of the actual content of the dataset, which in practice means that a CGW can easily be detected as (and mistaken for) a GWB. However, the signal recovered from the two CGW-only datasets (injCGW5, injCGW20) is higher and flatter than the one obtained from the background datasets. This can be clearly seen both from the A and γ values of Table <ref> and from the 2D contours in Fig. <ref>. This is expected since the extra power due to the two single sources is at frequencies which are higher than the loudest GWB frequency bins that dominate the powerlaw spectrum recovery in the GWB-only datasets.






The Bayes factors reported in Table <ref> show that an HD-correlated common process (i.e. a GWB) is generally strongly favoured regardless of the content of the data. In fact, dataset injCGW5 produces a Bayes factor ℬ^ GWB_ CURN of ∼ 10 and HD S/N of 3.0^+0.5_-0.5 (fixing γ to 3), while injCGW20 
gives ℬ^ GWB_ CURN of ∼ 60 and S/N = 3.2^+0.6_-0.6 (fixing γ to 2). Interestingly, injGWB03 does not favour an HD correlated process, displaying ℬ^ GWB_ CURN of ∼ 0.15. This is consistent with <cit.>, who found that, due to the stochastic nature of the signal, individual realizations of the same SMBHB population can result in Bayes factor that differ by several orders of magnitude.

Although the Bayes factors obtained on the  injCGW5 and injCGW20 are compatible with a GWB detection, the resulting HD S/N is significant only in a small range of values of γ that are consistent with the posterior distributions obtained by the Bayesian search, as shown in Fig. <ref>. By fixing γ to the nominal value of 13/3, the HD S/N drops to 1.3^0.4_-0.4 and to 1.1^1.0_-0.9 for the injCGW5 and injCGW20 datasets respectively. Conversely, the 
injGWB01 and injGWB02 datasets (also shown in Fig. <ref>) result in distributions of the HD S/N that are consistently high for all values of γ≥ 4.


Moreover, at higher values of γ, the CGW-only dataset has an increasing evidence for a dipolar-correlated signal, whose S/N surpasses that of HD. This behaviour is not observed in the GWB-only datasets, for which the dipole S/N always stays lower than the HD S/N. Finally, two examples of HD reconstruction are shown in Fig. <ref> for datasets injGWB02 and injCGW5. While the quadrupolar pattern can be seen in both datasets, the scatter around the expected relation appears to be much larger when a CGW only is present in the data, consistent with <cit.> and <cit.>. Moreover, the HD correlation for the injCGW5 dataset is reconstructed assuming γ=3, while it tends to disappear for higher values of γ.














  §.§.§ GWB misinterpreted as a CGW


We now search for a CGW in the same 5 datasets analysed in the previous section. In the search, we estimate the posterior distributions of all the noise parameters – except for the WN, which is fixed – and the CGW parameters – 8 source parameters plus 50 additional parameters for the pulsars' distances and phases. We then use hypermodeling to evaluate the Bayes factor between a model containing a CGW versus a model featuring intrinsic pulsar noise only. 

We can see from Table <ref> that also in this case, regardless on whether the dataset contains a CGW or a GWB, a CGW is recovered with high confidence, with a Bayes factor ℬ^ CGW_ PSRN>1000 in all cases. Table <ref> also shows that the CGW frequency and amplitude are fairly well constrained. The performance of the CGW search on the injCGW datasets has already been assessed in Section <ref>. As shown in Fig. <ref> when a full ET+PT template is used for the CGW, the model can correctly recover the injected signals and estimate their parameters, although with differences depending on the frequency of the source. Here it is particulary interesting to check the CGW recovery on the three injGWB datasets, which is shown in Fig. <ref>: all source parameters are fairly well constrained, including the sky location, which is different in each dataset. These parameters, however, do not correspond to any specific source present in the datasets, as can be seen in the three panels of Fig. <ref>. The recovered CGW frequency always corresponds to the frequency bin that is the most constrained by the HD-correlated free spectrum search; moreover, the amplitude recovered is always compatible with the GWB amplitude in that frequency bin. Thus, it can be concluded that the CGW parameters are constrained because the CGW model absorbs part of the HD-correlated common red noise produced by the incoherent superposition of all the SMBHBs composing the GWB. 

Finally, we note that the CGW amplitudes recovered from the three injGWB datasets are respectively 1.2^+1.5_-0.7× 10^-14, 1.4^+0.6_-0.5× 10^-14  and 1.8^+0.5_-0.4× 10^-14 (median and 68% credible interval). These values are all well compatible with the amplitude of the DR2new CGW candidate, which is 1.0^+0.5_-2.6× 10^-14 (median and 90% credible interval) <cit.>.














  §.§.§ Breaking the degeneracy: combined search for a GWB and a CGW




We now perform a joint CGW + GWB search on the five datasets considered thus far. We perform two flavours of combined searches: (i) common uncorrelated red noise plus CGW (CURN+CGW), and (ii) HD-correlated common red noise plus CGW (GWB+CGW). The CGW is always modelled as the sum of ET and PT, and together with the signal parameters, all the RN and DM parameters are also searched over.

The results of these two searches are reported in Table <ref> for all datasets, while posterior distributions of selected parameters of the model components are shown in Fig. <ref> for datasets injGWB02 and injCGW5. In figure, orange and light purple distributions are obtained respectively from the GWB+CGW and CURN+CGW searches, and results of GWB-only and CGW-only searches are also shown for comparison. To see how searching for both type of signals simultaneously allows to properly characterize the actual content of the data, let us discuss the injGWB and injCGW separately. 

In the injGWB cases, when a common red noise component is added to the model alongside the CGW, the evidence for the latter strongly decreases. This can be seen both from the drop of the Bayes factors in the first three lines of Table <ref>, and from the posterior distributions of the CGW parameters in the left panel of  Fig. <ref>. In Section <ref>, when searching for a CGW on a dataset containing only a GWB, we found  ℬ^ CGW_ PSRN>1000. This is because the GWB produces a common noise that the model prefers to assign to the CGW signal (common to all pulsars) rather than to the individual pulsars' RN components. 
When a CURN is included in the model, ℬ^ CURN+CGW_ CURN drops significantly but is still slightly larger than unity, suggesting that the presence of CGW component is still preferred. This can also be seen in the posterior distribution of the CGW amplitude shown in purple in the left panel of  Fig. <ref> that, while featuring a long tail that extends to the lower end of the prior range, still shows a significant peak around logA≈-14.3. This behaviour is shared by all three injGWB datasets and is due to the fact that the GWB manifests itself primarily as a common red noise process, with a weaker, emerging 'off diagonal' HD-correlated component. The CURN models it as a common red noise process, but is unable to account for the emerging HD correlation, which the model then assigns to the CGW.
Conversely, when a GWB is added to the model, we get ℬ^ GWB+CGW_ GWB<1 (first three lines of Table <ref>), strongly suggesting that the GWB component is now able to describe the full content of the data. This is also clear by looking at the CGW posterior parameters in   the left panel of  Fig. <ref>. The orange distributions show that both the frequency and amplitude posteriors are almost perfectly flat, with the latter displaying a cut-off, which we find to be at a similar value for all three realizations. From this we can extract  a 95% upper limit of the CGW amplitude which is ≈4.0×10^-15, ≈3.8×10^-15 and ≈4.6×10^-15 for datasets injGWB01, injGWB02 and injGWB03 respectively. Note that these upper limits are lower than the GWB level, since the total GW content of the dataset is correctly accounted for by the GWB component. Finally, the recovered source is not localized in the sky, with the posterior distributions of the sky location almost uniform across the sphere, consistent with the prior. 







Differently from the CGW parameters, the GWB parameters' posterior distributions are not significantly affected by the addition of the CGW component to the model: the GWB posterior distributions obtained from the GWB+CGW search and with the GWB-only search are well consistent with each other (see left panel of Fig. <ref> for the injGWB02 dataset and table <ref> for all the datasets) confirming again that the GWB component of the model is correctly accounting for all the HD-correlated power. Only a small variation can be noticed in the CURN+CGW recovery of the background parameters: the posterior distributions are slightly wider with a longer low amplitude tail. This is because the CURN cannot absorb any HD correlated power, which is (spuriously) assigned to the CGW component.  


 
Conversely, when the same analysis is applied to the injCGW datasets, the posterior distributions of the CGW parameters are always well constrained, both with the CURN+CGW and with the GWB+CGW search, as shown in the right panel of Fig. <ref>. Searching for CGW-only, CURN+CGW or GWB+CGW, in this case, gives consistent results for the CGW parameter, with no significant changes in the posterior distributions' shape and width (see right panel of Fig. <ref>). Moreover, GWB+CGW is able to recover the sky location with a precision which is comparable to the one obtained with the CGW-only search and the model comparison strongly favours the GWB+CGW model over the GWB-only model (see table <ref>). As expected, the GWB posterior distribution is instead completely unconstrained when the model applied in the analysis contains both a common red noise and a CGW: the power that generates the GWB signal recovered with the GWB-only search is now correctly completely absorbed by the CGW component (see Fig. <ref>).





  §.§.§ Noise-only datasets

As a further check, we tested the analysis pipelines on a dataset containing only the noise and no GW signal, neither deterministic nor stochastic. The noise realization is the one contained in all the datasets analysed. In this case, neither a GWB nor a CGW is detected. As for the GWB, we find a value of the Bayes factor ℬ^ HD_ CURN∼1.5 and the posterior distributions of A and γ are poorly constrained, with a 95% upper limit on the GWB amplitude of ∼ 4.47× 10^-15. As for the CGW, the Bayes factor ℬ^ CGW_ PSRN is ∼ 0.5 and the CGW parameters are totally unconstrained, with the amplitude showing a 95% upper limit of ∼4.5×10^-15. Although many more noise realizations should be tested to draw robust statistical conclusions, these tests show on fully realistic datasets that current PTA detection pipelines are unlikely to return a spurious detection in absence of an actual GW signal in the data.





 §.§ Joint GWB+CGW analysis on combined CGW+CGW datasets


In this section we present the results of simultaneous searches for a GWB and a CGW, applied to datasets injGWB03+CGW5 and injGWB03+CGW20. We remind the reader that both datasets contain the same noise and GWB realisation from dataset injGWB03. 

In each dataset a loud CGW is superimposed to the GWB: in one case the CGW is at 5nHz (injGWB03CGW5), in the second one it is at 20nHz (injGWB03CGW20). This is done in order to see how the joint analysis performs in presence of a slowly evolving source (the 5nHz CGW) and on a fast evolving one (the 20nHz CGW), and how the two of them interact with the GWB.









  §.§.§ Bias induced in the GWB recovery by the presence of a resolvable source in the SMBHBs population

As already discussed by <cit.> and  <cit.>, the presence of a particularly loud source among the systems contributing to the GWB, if not accounted for in the analysis, can significantly bias the recovery of the overall GWB signal. As a preliminary step in our investigation, we therefore perform a GWB only search on the injGWB03CGW5 and injGWB03CGW20 datasets. As expected, the analysis produces a biased recovery of the GWB parameters: the results are shown in the right panels of Fig. <ref> and <ref>: in both cases, the recovered background is flatter and higher than the injected signal due to the extra power introduced by the loud source. In the two following sections, we demonstrate how this bias can be solved by performing a joint analysis of the GWB and the CGW components.





  §.§.§ Results for the injGWB03+CGW5 dataset


















The results presented here are obtained by sampling the posterior distributions of the CURN+CGW model and then reweighting them to obtain the posteriors of the GWB+CGW model.
The results of the combined search applied to the dataset injGWB03+CGW5 provides good overall results, with the majority of the CGW parameters recovered within 2σ of the injected values. 

The posterior distributions of the CGW parameters are shown in the left panel of Fig. <ref>, while the recovery of the background parameters is shown in the right panel. The recovery of the CGW parameters is successful: the frequency and the sky localization of the source are well constrained – the width of the posterior distributions is ≲10% of the prior width for all three parameters – and compatible within 68% with the injected values: injected frequency is f = 5.02nHz, recovered frequency is f = 5.04^+0.04_-0.04nHz; injected sky location is (12^ h19^ min50^ s, 12.38^∘), recovered sky location is (12^ h 33^+20  min_-19, 10^+5_-5^∘). However, the 3D source location cannot be properly constrained, since the luminosity distance posterior distribution is  extremely wide due to its strong covariance with the chirp mass. Both posterior distributions are roughly bimodal, with a narrow peak close to the injected value – in particular for the chirp mass – and a long, bumpy tail extending to low values of luminosity distance and chirp mass. Note that the posterior of the luminosity distance spans the interval (0.02, 24)Mpc within the 90% credible region; for such a nearby putative source, ancillary information coming from galaxy catalogues can effectively enable the identification of the source host <cit.>. The inclination angle, initial phase and polarization angle are well constrained and compatible with the injected values.

Focusing now on the GWB parameters, the posterior distribution in the right panel of Fig. <ref> shows how accounting for the resolvable source in the model solves the bias in the GWB parameters' recovery. In fact, the posterior distribution obtained with the combined search is well consistent with the one obtained with the GWB-only search applied to the GWB-only dataset injGWB03: they peak at similar values and have comparable widths, showing that the parameters of the stochastic and of the deterministic components are not significantly correlated and the interplay between the two signals does not affect the search.





  §.§.§ Results for the injGWB03+CGW20 dataset













 





The joint search applied to the injGWB03CGW20 dataset unveils some new interesting behaviour of the recovery pipeline. In fact, in this case, when a joint CURN+CGW (ET+PT) search is performed and the GWB is obtained with the reweighting technique, the estimate of the GWB parameters is completely off, as demonstrated by the filled purple distributions shown in the right panel of 
Fig. <ref>. The result is indeed quite similar to a GWB search only (shown in red), meaning that a significant fraction of the CGW power is absorbed in the common background component and the CURN+CGW search followed by reweighting fails in this case, producing strong biases in the results.



The reason of this failure is the following: as shown in Section <ref>, due to the significant difference in the 20nHz CGW between the Earth and the pulsar term GW frequency, in order to correctly account for both terms with the CGW model, the majority of the 50 PT parameters (i.e. 25 pulsar distances and 25 individual PT GW phases) have to be constrained. This is a difficult task for the sampler, which in the CURN+CGW case converges to an easier, alternative solution: the pulsar terms are all absorbed by the common uncorrelated red noise component, which is possible because they are indeed a common sinusoidal process that is uncorrelated, since the pulsars are all at different sky locations and distances from Earth. This justifies the much higher and flatter recovered background shown in the right panel of Fig. <ref>: the recovered amplitude is 1.83^+0.28_-0.26×10^-14, while the injected value is 2.4×10^-15. 

This interpretation is confirmed by the recovered CGW parameters (not shown). Most of them are biased and in particular the chirp mass, which determines the frequency evolution of the signal, is much lower than the injected value: the obtained value is 6. 9^+18.0_-5.0×10^6 M_⊙, which corresponds to a 95% upper limit on the frequency derivative of 4.2×10^-6nHz·yr^-1, ∼300 times smaller than the ḟ of the injected SMBHB. This is consistent with the fact that the values of the CGW parameters are such that the recovered time domain signal matches the injected Earth term and completely neglects the pulsar term contribution. This is confirmed by the shape of the reconstructed CGW shown in  Fig.<ref>. In the top panel we can see that in the CURN+CGW search the recovered CGW waveform for pulsar PSR J1713+0747 reproduces exactly the injected ET contribution. In fact, being HD-correlated, in presence of a CGW component in the model, the Earth term is not absorbed by the CURN component. Indeed, the recovered background is slightly lower than the one obtained by a GWB-only search on the same dataset (shown by the red distributions in the right panel of Fig. <ref>), because in that case both the Earth and pulsar terms are absorbed by the background component. 

We stress that this bias indeed corresponds to the maximum of the posterior distribution. This can be explained by the fact that the CURN model does not exactly match the injected common red noise, which is HD correlated, and therefore it prefers to absorb the pulsar term, since it is much cheaper to describe it as a two-parameter CURN signal, rather than to determine the 2×25 parameters describing each individual sinusoidal pulsar term.

In this dataset, the injected signal components are  properly recovered only by a direct GWB+CGW (ET+PT) search, which is more computationally expensive due to the off-diagonal HD terms that must be computed when evaluating the correlation matrix in Eq. (<ref>). In fact, in a GWB+CGW search the pulsar term is not absorbed by the HD-correlated common red noise component and both signals are properly reconstructed. In the right panel of Fig.<ref> the GWB recovery obtained by this search is shown in filled purple: it agrees very well with that obtained with the GWB-only template applied to the GWB-only dataset (empty purple), although it is slightly larger, which can be explained by the fact that the GWB+CGW model contains 58 more parameters describing the single source. A tiny correlation with this high number of parameters can easily explain the less precise recovery of the background parameters.

The correct recovery of the GWB parameters is a direct consequence of the correct recovery of the CGW parameters. In the left panel of Fig. <ref>, the posterior distributions of the 8 source parameters are shown and in Fig. <ref> the median and 90% credible region of the recovered CGW signal is shown for pulsar PSR J1713+4707. From the recovery, it can be seen how the model is able to account for the frequency evolution of the signal: the frequency recovered is 19.99^+0.21_0.18nHz (injected is 20nHz) and the chirp mass 1.14^+0.11_0.10×10^9M_⊙ (injected is 1.16× 10^9M_⊙). This allows the luminosity distance to be much better constrained than in the low frequency case, with a median and 68% credible interval of 32.7^+15.1_-9.3Mpc. Since the injected value is 21.6Mpc, the value obtained is a mild overestimate, which is probably due to the anti correlation with the inclination angle and the initial phase, which are both slightly underestimated. The 2D sky location recovery is also improved with respect to the 5nHz case, resulting in a declination angle of 12^ h38^+17 min_-16 and a right ascension of 11.5^+3.0_-3.4^∘.

  


§ DISCUSSION AND CONCLUSIONS


In this paper, we have carried out the first comprehensive study of the interplay between stochastic GWBs and deterministic CGWs in realistic PTA datasets, assessing the performance of currently used analysis pipelines. To this end we constructed mock datasets featuring all the complexity of EPTA DR2full <cit.>; we simulated up to 24.8 years of unevenly sampled, multi frequency ToAs for the 25 pulsars of DR2full and injected WN, RN and DM consistent with the maximum likelihood values found in the DR2full single pulsar analysis <cit.>. The injected GW signals were also as realistic as possible, tailored to mimic the findings of the recent EPTA analyses <cit.>. A GWB with nominal amplitude of A_ GW=2.4× 10^-15 was injected as the incoherent sum of individual waves coming from observationally based cosmic populations of SMBHBs <cit.>, while the deterministic CGW was modelled according to the candidate signal identified in the analysis of EPTA DR2new <cit.>. In all cases, SMBHBs were assumed to be circular and both the ET and PT of the GW induced time delays were injected. These GW signals were searched with the standard models implemented in the  analysis suite. The GWB was therefore modelled as a common red process characterized by a single power law spectrum, while the model CGW matched the one used for injecting the signal (i.e., circular, GW driven SMBHB), either including only ET or ET+PT. With this set-up we carried a number of investigations, whose results are summarized in the following.

We first compared the performance of ET-only and ET+PT templates on two datasets containing only a CGW either at 5nHz (injCGW5) or at 20nHz (injCGW20). In both cases, the ET+PT template was able to fully recover the signal, providing an unbiased estimate of all its parameters, including chirp mass, distance and sky location. Things were not so straightforward when the ET only template was employed. In this case, the 5nHz CGW dataset resulted in a biased recovery of the frequency and the sky location, whereas in the 20nHz case, the recovery was unbiased. In fact, in the latter case Earth and pulsar term can be easily disentangled and the ET-only template correctly matches the ET component of the signal. Conversely, in the 5 nHz case, ET and PT fall in the same frequency bin and the ET template tries to fit a mixture of the two, leading to a bias in the signal reconstruction. In both cases, the ET-only model is not sensitive to the chirp mass and therefore to the luminosity distance, preventing 3D source localization.

We also stress that in both cases, modelling ET only leaves behind some unmodelled residual (e.g. the full PT in the 20 nHz case), which will likely be absorbed by the GWB template in a joint search, leading to a biased recovery of the background parameters. These are the reasons why the complete CGW template (ET+PT) must be preferred to the ET-only, even if this means having 2 times the number of pulsars extra parameters (50 in our case) and therefore much larger computational costs for the sampling.

We therefore adopted the full ET+PT template for the CGW and turned to the investigation of datasets in which only one of the two types of signal was injected (either a GWB or a CGW) and the analyses were performed separately with one model at the time (eiter a GWB or a CGW). We analysed 5 datasets: 3 realizations of a realistic GWB (dataset injGWB01,injGWB02, injGWB03) and 2 realizations of single resolvable sources (datasets injCGW5, injCGW20). All datasets contained the same noise realization. These searches showed that a GWB can easily be misinterpreted as a CGW and vice versa. In fact, all sources returned evidence in favour of either GW signal and well constrained recovery of the model parameters. However, when a combined search is performed – meaning that the model used for the recovery contains both an HD-correlated GWB and a CGW – this degeneracy is broken: when only a GWB is present in the data, the posterior distributions of the CGW parameters obtained with the combined search are unconstrained and the CGW amplitude shows an upper limit; on the contrary, when a CGW is the only signal injected, the combined search gives unconstrained posterior distributions for the GWB parameters. No CGW evidence is absorbed by the HD-correlated background if the single source is present in the data. Therefore, the combined search is able to correctly recover the signal that is actually present in the data. Focusing on GWB-only datasets, we highlight that the CGW evidence is totally absorbed when an HD-correlated GWB is present in the model, while it does not completely disappear if only a CURN component is modelled. In this case, even if poorly constrained, the CGW parameters posterior distributions still peak at specific values. Therefore, modelling the common red noise as HD-correlated is necessary to totally absorb the spurious CGW. These results are similar to the ones obtained with the real EPTA DR2new data <cit.>: CURN + CGW searches peaking at a CGW amplitude of ∼10^-14, and peak disappearing once the background is modelled as HD-correlated and the Bayes factor ℬ^ GWB+CGW_ GWB being of order unity. Our simulations show that this behaviour arises when a CGW is indeed not present in the data, strengthening 
  the conclusion reached in 
the EPTA DR2 analysis.

We then analysed datasets containing both a stochastic GWB and a CGW, either at 5 nHz (injGWB03+CGW5) or at 20 nHz (injGWB03+CGW20). The low frequency CGW is an almost monochromatic source, while the high frequency one is drifting and the signal frequency significantly changes between the  PT and the ET. Performing a combined search on these types of datasets is particularly important because a single resolvable source, if not properly taken into account in the search, will induce a bias in the GWB parameters recovery <cit.>.

The results of the combined searches indicate that the bias in the GWB recovery induced by the presence of the single source is resolved when the CGW is correctly accounted for by the deterministic CGW model, so long as it contains both ET and PT. This result is obtained with both datasets, injGWB03CGW5 and injGWB03CGW20. However, in the injGWB03CGW5 case, the same unbiased result can be obtained with two methods: by performing a CURN+CGW search and then reweighing to obtain the HD+CGW posterior distribution, and by directly sampling the HD+CGW posterior distribution. The first method is computationally advantageous, as the CURN model is considerably faster to compute than the HD one. In the injGWB03CGW20 case, however, the first method (reweighting) is unable to recover the unbiased posterior distribution, as the CGW template solely accounts for the ET contribution, while the PT is more easily absorbed by the CURN. The posterior distribution on the CURN parameters is therefore strongly biased with respect to the GWB injection. Performing a direct sampling of the HD+CGW model, on the contrary, gives an unbiased recovery of the GWB parameters: the posterior distributions of the CURN and HD parameters are consequently disjoint, which makes the reweighing technique extremely inefficient. Therefore, while computationally much more expensive, the only method that is able to converge to unbiased results is the direct HD+CGW search. As for the GWB, the CGW recovery is successful, with most of the parameters constrained and compatible with the injection.

Given the recent PTA results indicating the presence of a gravitational signal in the data, it is crucial to be able to distinguish between the two types of signal that can be produced by SMBHBs. This work represents a first thorough study of this problem. Despite the limited parameter exploration, which is constrained by the high computational cost of the analyses, the investigation decisively points towards the importance of performing combined searches. Indeed, in the cases analysed, combined searches were able to both identify which signal was present in the data, without misinterpretation, and to guarantee an unbiased recovery of the GWB parameters in the presence of a loud individual source in the SMBHB population. Looking ahead, it will be important to extend this analysis to richer and more sensitive datasets, investigating the possibility of disentangling multiple searches, also in the presence of eccentricity or environmental coupling. In fact, the high amplitude of the currently observed signal implies that the future SKA might be able to resolve several tens of individual SMBHBs at S/N>5 <cit.>. Being able to successfully identify and disentangle those signals will have invaluable payouts for the understanding of SMBHB evolution and dynamics in the context of multimessenger astrophysics. Given the complexity and high dimensionality of these sources, it is of paramount importance to design more efficient search techniques <cit.>.


       I.F. thanks Stanislav Babak for his hospitality and mentoring at the APC, where this investigation was initiated.
       We thank the B-Massive group at Milano-Bicocca University for useful discussions and comments. A.C., A.S., G.M.S. acknowledge the financial support provided under the European Union’s H2020 ERC Consolidator Grant “Binary Massive Black Hole Astrophysics” (B Massive, Grant Agreement: 818691). A.C. acknowledges financial support provided under the European Union's Horizon Europe ERC Starting Grant "A Gamma-ray Infrastructure to Advance Gravitational Wave Astrophysics" (GIGA).

  

aa







 Paper Title: A Sea of Black Holes: Characterizing the LISA Signature for Stellar-Origin Black Hole Binaries.






Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA
Department of Life and Physical Sciences, Fisk University
Institut für Theoretische Astrophysik, ZAH, Universität Heidelberg, Albert-Ueberle-Straße 2, D-69120,Heidelberg, Germany
Physics and Astronomy Department Galileo Galilei, University of Padova, Vicolo dell’Osservatorio 3, I–35122, Padova, Italy
INFN - Padova, Via Marzolo 8, I–35131 Padova, Italy
TAPIR, Mailcode 350-17, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA



Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA



Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA
Department of Life and Physical Sciences, Fisk University


Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA


Institut für Theoretische Astrophysik, ZAH, Universität Heidelberg, Albert-Ueberle-Straße 2, D-69120,Heidelberg, Germany
Physics and Astronomy Department Galileo Galilei, University of Padova, Vicolo dell’Osservatorio 3, I–35122, Padova, Italy
INFN - Padova, Via Marzolo 8, I–35131 Padova, Italy


TAPIR, Mailcode 350-17, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA


Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA



§ ABSTRACT 
Observations by the LIGO, Virgo and KAGRA (LVK) detectors have provided new insights in the demographics of stellar-origin black hole binaries (sBHB). A few years before gravitational-wave signals from sBHB mergers are recorded in the LVK detectors, their early coalescence will leave a unique signature in the ESA/NASA mission Laser Interferometer Space Antenna (LISA). Multiband observations of sBHB sources between LISA and LVK detectors opens an unprecedented opportunity to investigate the astrophysical environment and multi-messenger early-alerts. In this study, we report the sBHB sources that will be present in the LISA data derived directly from the hydrodynamic cosmological simulation Illustris. By surveying snapshots across cosmological volume, metallicity and look-back time, we find that about tens to thousand sBHB candidates will be present in the LISA data for various combinations of mission lifetime.
For estimates consistent with the LVK rates, we find that only 20 sBHBs across Illustris snapshots will be detected with significant confidence for a 10-year LISA mission, while a 4-year LISA mission would detect only 2 sBHBs.
Our work paves the way for creating LISA mock data and bench marking LISA detection pipelines directly using cosmological simulations.









§ INTRODUCTION
 
The Laser Interferometer Gravitational-Wave Observatory (LIGO) made the first direct detection of gravitational waves in 2015, revealing a merger of black holes with masses 36 and 29 M_⊙ <cit.>. These black holes, dubbed stellar origin binary black holes (sBHB) by the gravitational wave community, are much more massive than the roughly 10 M_⊙ black holes in accreting binaries that had been detected electromagnetically <cit.>.
This discovery expanded our understanding of black hole populations, providing insight into their formation and evolution <cit.>. However, the formation channel of these binaries is still under debate. The current favored mechanisms involve formation via: a) isolated binaries in the galactic field <cit.> ; b) dynamical interactions within dense stellar environments  <cit.>; and c) binaries within gaseous AGN accretion disks <cit.>.

sBHBs are so massive that they are a new source class for the Laser Interferometer Space Antenna (LISA), an ESA/NASA space-based gravitational wave observatory set to launch in 2035 <cit.>. LISA is sensitive from around 10^-4 to 0.1 Hz, a frequency range that captures the inspiral phase of the sBHB population. In the LISA band, low frequency sBHBs are relatively widely-separated, take millions of years to merge, and may generate an unresolved stochastic background <cit.>. The higher frequency sources are more tightly-bound, sweeping through the LISA band in the last phase of inspiral months to days before merging within ground-based gravitational wave detectors, making them a prime multiband source  <cit.>. 

Multiband observations alone enable a more accurate measurement of masses, spins, eccentricity, and distance <cit.>, but their unique power is in providing better spatial resolution and enough lead time before merger to conduct rigorous observational campaigns to identify electromagnetic counterparts <cit.>. Multiband data is vital to differentiate between sBHB formation channels as well as to constrain their final evolutionary sequence <cit.>.




To investigate the possibility that stellar origin binaries arise in an isolated binary stellar population, <cit.> used the cosmological hydrodynamic simulation suite Illustris <cit.> to predict the astrophysical and LIGO-observable rate of sBHB mergers, given a range of binary stellar population models. Illustris provided a self-consistent time-evolving map of star-formation rate, mass, and metallicity to be used as input to a binary population synthesis code, making it possible to paint a stellar population onto the simulation, let it evolve, and track the formation and ultimate merger of sBHBs.

Our work extends this study, identifying those sBHB mergers within the Illustris simulation volume that are observable with LISA. Formally, the time to merge for sBHBs in the LISA band can be from months to millions of years, but we narrowed our investigation to those sBHBs that transition from LISA to ground-based detectors within the lifetime of the LISA mission; these sources are the loudest in the LISA band and are therefore the most promising for multiband and multi-messenger observations <cit.>.
Figure <ref> is a subset of sBHB mergers within the <cit.> database that may be seen by LISA and merge within LIGO. These binaries are chosen from the snapshot at the highest redshift(z =   13.4), because it contains sBHBs that merge within a wide variety of lookback times, showing the robustness of <cit.>'s database for studying multi-messenger astronomy.

The paper proceeds as follows: Section <ref> describes our methods, including a brief review of the simulation, the stellar population input from  <cit.>. In addition, section <ref> discusses the signal-to-noise calculations for our binary database. In section <ref> we discuss the results, including the multiband rates (section <ref>), and number of detectable multiband sources (section <ref>). Finally, the conclusion and discussion are within section <ref> where we summarize our results, point out caveats, and identify areas for future work.





§ METHODS
 

Illustris is a set of cosmological hydrodynamic N-body simulations run using the AREPO moving mesh code <cit.>. To model galaxy formation and evolution in a cosmological context, the simulation self-consistently treats dark matter, supermassive black holes, gas, and stars, including prescriptions for gas cooling, star formation, metal enrichment, supernovae and associated feedback, as well as massive black hole formation and evolution (including gas accretion, mergers, and several modes of feedback). Illustris is a cosmological volume of 106.5^3 Mpc^3 initialized in a Wilkinson Microwave Anisotropy Probe (WMAP-9) cosmology with the following parameters <cit.>: Ω_M = 0.2726, Ω_Λ = 0.7274, Ω_b =0.0456, h = 0.704,  and H_0 = 70.4 kms^-1Mpc^-1, where Ω represents the mean density relative to the critical density of the Universe, and M, Λ, and b refer to matter, the cosmological constant, and baryons, respectively. h is the dimensionless Hubble parameter <cit.>, and H_0 the Hubble constant. The Illustris suite is composed of 6 simulations: Illustris-(1,2,3), which are the hydrodynamical simulations modeling galaxy formation and include all the star formation, gas and massive black hole physics described above, and Illustris-DM-(1,2,3), which are the dark matter-only variant. We use Illustris-1 for its higher mass resolution; the Illustris-1 simulation contains 1.6 × 10^12 particles over all snapshots, with a dark matter mass resolution of 6.3 × 10^6 M_⊙, a baryonic matter mass resolution of 1.3 × 10^6 M_⊙ and a softening length of ∼ 710 pc for stars and supermassive black holes (SMBHs) <cit.>.

Although the Illustris simulation suite is state-of-the-art, it cannot resolve individual stars and stellar remnants. Therefore, <cit.> maps a population of sBHB, each stellar particle in snapshots spanning redshifts 0–16 using an updated version of the public binary stellar population synthesis code, Binary Star Evolution (BSE) <cit.> called OBSE <cit.>. For more detail about the procedure to seed Illustris with sBHB, please see <cit.>; we briefly describe the process here. <cit.> constructed a library of 72 distinct binary stellar population models, each 'book' in the library consisting of millions of sBHBs representing a particular metallicity and a set of assumptions about the less well-constrained binary-evolution physics, such as mass loss in the common-envelope phase. 



 
Using this library, a sBHB distribution is assigned to Illustris-1 star particles. The metallicity of a star particle determines which stellar population 'book' to use, and the mass of the star particle determines how many sBHB to select. Each sBHB is characterized by its masses, separation, and eccentricity, which is used to calculate the binary merger timescale. By following only the sBHB that merge within a Hubble time, this earlier work resulted in a simulated census of the astrophysical merger rates for these sources, as well as estimates of the number of binaries and mass distribution of sBHB mergers that LIGO will see for various stellar population models, as shown in Figure 6 of <cit.>. 

Note that for our paper, we adopt models D and DK as our fiducials; these assume a delayed supernova model, a treatment of Hertzsprung gap (HG) donors, and natal kicks for compact binaries <cit.>. The difference between model D and DK is that D employs a method described in <cit.> to rescale natal kicks for black holes to take into account the fraction of the stellar envelope that can fall back onto the compact object during the explosion, f_b, which can range from 0 to 1 <cit.>, as shown by equation <ref>

    v_BH = v_NS(1-f_b),


where v_NS is the velocity distribution in <cit.>. Model DK applies the velocity distribution found by  <cit.> for natal kicks of neutron stars to the black holes.




The sBHB merger rate in model D is 181 Gpc^-3yr^-1 at a redshift of 0.2. At the time of publication, this rate was the most consistent with O1 LIGO observations. However, LIGO data from its observing run 3 (O3) places the sBHB merger rate between 17.9 and 44 Gpc^-3yr^-1 at a redshift of 0.2 <cit.>. This filters into our results as an overestimate of the LISA and LIGO multiband rate when we use model D. In contrast, model DK offers a much closer match to O3 rates with a sBHB merger rate of 29 Gpc^-3yr^-1 at redshift 0.2. This similarity is shown in Figure <ref> which plots the merger rates for model D, DK, and LIGO'S rate for O3 against redshift. The dashed line represents model D's rate and the lower line is model DK's rate. LIGO's rate is shown by the dotted gray line with the inner contours containing 50% of the data and the outer contours containing 90% of the data. Model DK's rate consistently falls within the 50% contours across the full redshift range and aligns more closely to LIGO's rate for z > 0.3. We note that the star formation rate history in the Illustris-1 simulation is  overall in good agreement with observations <cit.>, except for an overestimate of ∼40% at low redshift (z ≲0.2). However, this overestimate has a negligible impact on the merger rate density of sBHBs <cit.>, because only stars with nearly solar metallicity form at such a low redshift: the efficiency of BHB mergers drops by about three orders of magnitude if the metallicity Z is >0.2 Z_⊙ <cit.>.



 §.§ Calculating the Signal-to-Noise Ratio in the LISA Band
 

We began with 108 snapshots ranging from redshift 0.0072 to 13.6, each consisting of a database of the sBHB formed at that redshift. This database contains information on: 1. the progenitor stellar particle ID from Illustis; 2. metallicity; 3. black hole component masses; 4. formation redshift; 5. sBHB merger time.

Since each snapshot contains only those sBHB that formed within a particular snapshot time interval, the varying sBHB merger time implies that a sBHB at a previous snapshot may still exist at a given redshift, but will be missing in the snapshot; indeed, the sBHBs seeded in a given snapshot will merge in the future over a wide range of timescales. It is worth reiterating that the database only contains sBHBs that merge within a Hubble time. Consequently, sBHBs in the simulated population that do not merge by the present day are missing from the current database. We leave the analysis of the quasi-static sBHB population for a future paper. 







To identify sBHB mergers that will be observable by both LISA and LIGO, we first restrict the database to those sBHBs that merge within a horizon redshift of 0.5; this decision greatly reduced the number of sBHBs in the Illustris volume from 7.7 × 10^9 to 1.19 × 10^9. Our choice is conservative and encompasses the expected horizon distance for sBHBs (see Figure 2 of <cit.>). Figure <ref> displays the black hole mass distribution for those systems that merge within a Hubble time and merge below a redshift of 0.5 in model D. We also forced the m_1 to always be greater than m_2, which was not the case in our database. In these models, the most common sBHB merger arises between black holes with commensurate masses of roughly 15–25 M_⊙.
We caution that neither model D nor DK contain sBHBs with component masses m_ BH > 43 M_⊙. LIGO has certainly observed black holes more massive than the upper limit of these models <cit.>. It may well be that the high mass end of the observed BHB merger mass spectrum represents additional processes, such as hierarchical merging, that may be at play and are therefore not well-represented by BSE <cit.>. We leave a multiband treatment of the higher mass black holes observed by LIGO to a future paper.

Using the redshift-limited database, we extract the masses and lookback time for each black hole binary merger and determine the gravitational wave frequency, f_ obs, of these binaries at a time T_ obs before the merger, using:


    f_ obs =  c^3/8π G M_c( 5GM_c/c^3)^3/81/T_ obs^3/8,


where c is the speed of light, G is the gravitational constant, M_c is the chirp mass, and T_ obs is the time until the BHB merges in seconds <cit.>. We adopt 4 and 10 years for T_ obs, the projected duration of science operations for the LISA mission.  As outlined in <cit.>, this choice produces a louder SNR in the LISA band while enabling a relatively contemporaneous multiband source.
(<cit.> considered a T_ wait, the time between detecting an event with a space-based and a ground-based detector. In our work, T_ obs is the same as T_ wait.)
The masses in equation <ref> are in the detector frame, and are converted from the source frame found in the database to detector frame masses using: m_ detector = m_ source(1+z), where z is the sBHB merger redshift. 





§ RESULTS
 



 §.§ Multiband Rate for detectable sBHBs
  

With this information we can then calculate the signal-to-noise ratio (SNR). For this, we use the python ISA Sensitivity package given an observation duration and cosmology <cit.>. This code models LISA's sensitivity to laser shot noise, acceleration noise, and test-mass force noise over a mission duration, and calculates the characteristic strain, and SNR of a binary given the component masses, distance, and an optional sky location and inclination. Note that unless specified, the SNR is averaged over inclination angle, polarization, and sky position. Additionally, since the sBHBs did not include spin, the gravitational wave signal is approximated by a phenomenological waveform model for non-spinning binaries, referred to as PhenomA <cit.>.  In the database, the LISA SNR for a observation run of 10 years technically ranged from 0 to 1× 10^7, with the anomalously high SNR populated by a sBHB at a luminosity distance of less than a kpc. Figure <ref> maps the average SNR as a function of total mass and luminosity distance assuming a mission duration of 10 years. In this figure we see, as expected, that nearby high mass binaries yield the highest SNR. For example sBHB system with M_ tot = 80   M_⊙ will visible up to 200 Mpc in LISA with SNR ≳20. Such high-mass binaries will be visible in LISA out to ∼1 Gpc with an average SNR ∼6. 

The luminosity distance for high-mass sBHB in Figure <ref> at a SNR threshold of 8 is consistent with the multiband detection radius reported in <cit.>.

    


Now that we have the SNR for each sBHB in hand, we can determine the multiband event rates and the expected number of detections for LIGO and LISA as a function of SNR.

We calculate the expected astrophysical rate of a population above a certain SNR threshold using equation <ref> from <cit.>:

    dN/dzdt=d^2 n(z)/dzdV_cdz/dtdV_c/dz1/1+z,


where V_c is the co-moving volume, n(z) is the number of sBHBs at a redshift z, 1/1+z converts dz/dt to the observer frame, and d^2 n(z)/dzdV_c is the rate found in the simulation volume above a certain SNR threshold. This can be found directly from the simulation, expressed by:


    d^2 n(z)/dzdV_c = N(z)/Δ z V_sim,


where N(z) is the number of sBHBs merging within a redshift bin, Δ z is the width of the redshift bin and V_ sim is the simulation volume, which in our case is (106.5  Mpc)^3.

The multiband rate of sBHB mergers as a function of SNR is shown in Figure <ref> for 4-year (purple) and 10-year (teal) LISA mission for model D and model DK.


During a 4-year observational run, our results for model D indicate that LISA and LIGO will detect 27 sBHBs yr^-1 with SNR ≥ 4. However, this number drops to 2 yr^-1 when we increase the SNR threshold to 10 for model D. Model DK also shows the same pattern but is six times smaller than model D, i.e 5 yr^-1 for SNR ≥ 4 to 0.3 yr^-1 for an SNR ≥ 10. 

If we extend the observational run to 10 years, the detectable multiband rate increases to 132yr^-1 with an SNR threshold of 4 to 6 yr^-1 for SNR ≥ 10. For model DK we see only 22 yr^-1 for SNR ≥ 4 to 1 yr^-1 for SNR ≥ 10. 





 §.§ Expected Number of Multiband sBHBs
 

Converting the multiband rate from equation <ref> to the number of detectable sBHBs, we used:

    N = dN/dzdt T_ obs,

where dN/dzdt is the rate from equation <ref> and T_ obs is the LISA mission lifetime.
Figure <ref> displays the number of sBHB that can be detected by both LIGO and LISA given a 4- and 10-year LISA mission lifetime. We see once again that the detectable multiband BH rate decreases as the SNR thresholds increase for both a 4-year and 10-year mission. For a 4-year LISA run, model D (left panel) exhibits a detectable number of multiband sBHB mergers that ranges from 6–106 as the SNR threshold decreases from 10 to 4, while model DK (right panel) yields between 1–18. For a 10-year mission, the number of sBHB mergers in model D ranges from 59–1321, and from 10–220 for model DK.


From our results, we conclude that the multiband detection rates (per year) for a 10-year LISA mission increases by ∼4x for both the models across all SNR thresholds when compared to a 4-year LISA mission. This is consistent with an average increase in SNR of sBHB by around √(10/4) between the two LISA mission lifetimes, thus increasing the detection volume by a factor of ∼4. Similarly, the total number of multiband detections in a 10-year LISA mission is ∼10x higher at SNR threshold of 10 and ∼12x higher at SNR threshold of 4 when compared to 4-year LISA mission. The mild increase at low SNR thresholds compared to high SNR thresholds is due to an increase in the intrinsic sBHB merger rates with redshift in the local universe <cit.>.  







 §.§ Comparison to Previous Work



Previous work has been done to estimate sBHB multiband rates and expected numbers for LISA <cit.>.
<cit.> presented a semi-analytic data-driven approach using LIGO observations from O1, assuming two underlying black hole mass distributions and intrinsic merger rates: a Salpeter mass function with a merger rate of 100   Gpc^-3yr^-1, and log-uniform mass function with a rate of 35   Gpc^-3yr^-1. These higher merger rates inferred from O1 naturally lead to a larger number of predicted multiband events compared to our models.

<cit.> examined the multiband detection of GW150914-analogs assuming an O1 event rate as well, however this work was primarily focused comparing the effect of different noise curves for eLISA, a precursor to the mission architecture adopted by LISA. For a 10 year mission, they predicted that the number eLISA-LIGO multiband sBHB detections ranged from 8–400, depending on the mission configuration.

<cit.>  updated the data-driven approach of the Sesana work using LIGO/Virgo/KAGRA observations from O1 and O2; the Salpeter mass function yielded a merger rate of 57^+40_-25  Gpc^-3yr^-1, while the log-uniform yielded a rate of 19^+13_-8.2  Gpc^-3yr^-1. For a 10-year LISA mission, their finding of 4-22 multiband LISA+LVK detections above an SNR of 8 is most consistent with our model DK, while model D is roughly 6 times higher.

Additionally, <cit.> created binary population synthesis simulations with spinning black hole binaries for isolated systems that evolved through the common-envelope phase. They perform multiple models, each varying the natal kicks distributed using a Maxwellian distribution with a dispersion between 0 and 265 km/s, and noted that the expected multiband detection rates can vary by more than an order of magnitude depending on the natal kick choice, consistent with our findings.

<cit.> created a Monte Carlo-sampled database of mock sBHB based on GWTC-3 observations with a rate of 19^+8.4_-8.5 Gpc^-3yr^-1. For a 4-year LISA mission and an SNR threshold of 8 they predict 3-15 events, which falls within the range of both of our models.




§ CONCLUSION
 

In this paper, we aimed to predict the expected rate and signature of stellar origin binary black holes that will be detectable by both LISA and LVK, with the hope that the early warning of a merger will help constrain sky location and help facilitate an alert of an impending merger for electromagnetic and ground-based GW detectors. Combining two binary population synthesis models with a cosmological hydrodynamic simulation, we found the multiband rate and expected number of sBHB detectable as a function of SNR and LISA mission duration. Table <ref> summarizes our findings for four different mission durations. In this table, we define marginal detections as those with SNR ≥ 4, while confirmed sBHB detections must have SNR ≥8. Model DK produces the minimum value in each column while model D yields the maximum value. We note that model DK, the binary population synthesis model with larger supernova natal kicks, yields overall sBHB merger rates that are more consistent with LVK observations through O3. Although it may be tempting to conclude that small black hole natal kicks can be ruled out by LVK observations, the number of competing open questions in binary stellar evolution – from the strength of core overshooting, to the impact of unstable mass transfer, to the core-collapse explosion process itself – preclude the ability to do so. Advances in binary stellar population modeling will allow a better mapping of the unknown parameter space in binary stellar evolution and will therefore better enable comparisons to gravitational wave observations <cit.>.

Our work provides a novel approach for building LISA mock data catalogs of sBHB directly using cosmological simulations. Such synthetic catalogs provide useful benchmark for LISA data-analysis pipelines <cit.>. To maximize the LISA SNR of a given sBHB, we deliberately modeled those that would merge over a timescale that is roughly the LISA mission duration; there are certainly sBHBs that will merge in a ground-based gravitational wave band hundreds of years in the future, but these sources are much fainter in the LISA band when so far from merger, and are therefore not ideal multiband sources.
We also neglected sBHBs that did not merge by the end of the simulation; this population of monochromatic, or nearly monochromatic, sBHBs may well be buried in the confusion foreground of Galactic compact object binaries <cit.>. 

This study considers only the field sBHB formation channel, yet LVK observations may hint that multiple formation channels could be at play <cit.>. Cosmological simulations of the scale needed to make universal predictions are as yet unable to resolve AGN disks and dense stellar systems such as nuclear star clusters and globular clusters. We will explore building a framework to seed a simulation with sBHB from multiple formation channels in a future work.



§ ACKNOWLEDGMENTS

KHB acknowledges and appreciates support from NSF NRT-2125764. M.M. acknowledges financial support from the European Research Council for the ERC Consolidator grant DEMOBLACK, under contract No. 770017 (PI: Mapelli) and from the German Excellence Strategy via the Heidelberg Cluster of Excellence (EXC 2181 - 390900948) STRUCTURES. This work used the resources provided by the Vanderbilt Advanced Computing Center for Research and Education (ACCRE), a collaboratory operated by and for Vanderbilt faculty at Vanderbilt University.




aasjournal




 Paper Title: Searching for New Cataclysmic Variables in the Chandra Source Catalog.

 

   

   
   Kazan Federal University, Kremlevskaya Str.18, 420008, Kazan, Russia

     IlhIGaliullin@kpfu.ru
         Department of Astronomy, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA, 91125, USA
        Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195, USA
        Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
        Anton Pannekoek Institute for Astronomy, University of Amsterdam, 1090 GE Amsterdam, The Netherlands
        IPAC, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA
        Division of Physics, Mathematics, and Astronomy,
            California Institute of Technology, Pasadena, CA 91125, USA 
             
             
    Ilkham Galiullin, Antonio C. Rodriguez et al.
    
   



 
  
  
   
     Cataclysmic variables (CVs) are compact binary systems in which a white dwarf accretes matter from a Roche-lobe-filling companion star. In this study, we searched for new CVs in the Milky Way in the Chandra Source Catalog v2.0, cross-matched with Gaia Data Release 3 (DR3). 
  
      We identified new CV candidates by combining X-ray and optical data in a color-color diagram called the “X-ray Main Sequence".  We used two different cuts in this diagram to compile pure and optically variable samples of CV candidates. We undertook optical spectroscopic follow-up observations with the Keck and Palomar Observatories to confirm the nature of these sources.   
  
      We assembled a sample of 25,887 Galactic X-ray sources and found 14 new CV candidates. Seven objects show X-ray and/or optical variability. All sources show X-ray luminosity in the 10^29-10^32 erg s^-1 range, and their X-ray spectra can be approximated by a power-law model with photon indices in the Γ∼ 1-3 range or an optically thin thermal emission model in the kT ∼ 1-70 keV range. We spectroscopically confirmed four CVs, discovering two new polars, one low accretion rate polar and a WZ Sge-like low accretion rate CV. X-ray and optical properties of the other 9 objects suggest that they are also CVs (likely magnetic or dwarf novae), and one other object could be an eclipsing binary, but revealing their true nature requires further observations.  
  
    These results show that a joint X-ray and optical analysis can be a powerful tool for finding new CVs in large X-ray and optical catalogs. X-ray observations such as those by Chandra are particularly efficient at discovering magnetic and low accretion rate CVs, which could be missed by purely optical surveys. 

   

   Searching for New Cataclysmic Variables in the Chandra Source Catalog
    Ilkham Galiullin1,
          Antonio C. Rodriguez2,
          Kareem El-Badry2,
          Paula Szkody3,
          Abhijeet Anand4,
          Jan van Roestel5,
          Askar Sibgatullin1,
          Vladislav Dodon1,
          Nikita Tyrin1,
          Ilaria Caiazzo2,
          Matthew J. Graham2,
          Russ R. Laher6,
          Shrinivas R. Kulkarni2,
          Thomas A. Prince7,
          Reed Riddle2,
          Zachary P. Vanderbosch2
          Avery Wold 2
          
    Received XXX; accepted XXX
====================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================










§ INTRODUCTION


Cataclysmic variables (CVs) are compact semi-detached binaries in which a white dwarf (WD) accretes matter from a Roche-lobe filling companion star <cit.>. CVs have orbital periods, typically spanning 66 min – 10 hr, except for ultra-compact AM Canum Venaticorum (AM CVn) stars, which possess ≲ 1 hr periods <cit.>. In magnetic CVs,  the magnetic field of the WD (B≳ 1 MG) significantly influences the accretion onto the surface of the WD. Magnetic CVs are split into intermediate polars (IP) with truncated disks, and polars, where the strong field (B ∼ 10-250 MG) prevents disk formation. Studies of CVs augment our understanding of various astrophysical phenomena, including their role as potential progenitors for Type Ia supernovae <cit.>. They also present optimal non-relativistic laboratories to study accretion processes <cit.>. Furthermore, due to their short orbital periods, some CVs and many ultra-compact AM CVn systems are classified as prospective "verification binaries” for future planned space-based gravitational wave observatories such as LISA <cit.>.

From the theoretical standpoint, CVs are predicted to form as a result of Common Envelope Evolution (CEE) <cit.> that first brings the stars closer together. Angular momentum loss (AML) is the key factor in the long-timescale evolution of CVs. As AML occurs, the separation between the stars decreases, leading to a reduction in the orbital period. The evolutionary model considers two mechanisms for AML: Magnetic wind Braking (MB) <cit.> and Gravitational-wave Radiation (GR) <cit.>. Although the model initially aligned well with the observational data of that era, subsequent findings revealed numerous discrepancies between theory and observations <cit.>. Consequently, empirical revisions were proposed to rectify these inconsistencies <cit.>.

CVs exhibit a broad range of observational features, therefore detailed multiwavelength studies are necessary to draw accurate conclusions about their nature and population. X-ray emission analysis is a powerful tool for searching for and studying CVs. The first X-ray observations of CVs took place with HEAO-1 <cit.> and Einstein <cit.>. Early X-ray surveys led to the discovery of a unique, previously unidentified population of magnetic CVs <cit.>. The all-sky X-ray survey conducted by the eROSITA telescope aboard the Spektrum-Roentgen-Gamma (SRG) mission <cit.> is proving to be groundbreaking by cataloging millions of sources <cit.> and has already been successful in discovering a collection of new and rare CVs due to its ability to reach much fainter fluxes <cit.>.


Despite not being an all-sky X-ray survey, the Chandra observatory provides unique sub-arcsecond on-axis resolution which is comparable with optical surveys <cit.>. The Chandra Source Catalog v2.0 (hereafter, CSC2)  provides the uniformly calibrated, science-ready data products compiled from the archival Chandra data <cit.>. Such data collection enables the study of properties of X-ray sources with spectral and timing analyses and classifying new objects in the catalog. In addition to new X-ray surveys and catalogs, all-sky optical photometric surveys provide a wealth of new data. The Zwicky Transient Facility (ZTF) is the deepest Northern sky time-domain photometric survey, conducted at Palomar Observatory on the Samuel Oschin 48-inch telescope. The 47 deg^2 field-of-view camera images the sky roughly every two days down to ∼20.5 mag in g, r, and i filters <cit.>. 

In this paper, we aim to search for and study new CVs in the CSC2, cross-matched with the  third  Gaia Data Release (hereafter, Gaia DR3). In Section <ref>, we describe the construction steps of a sample of Galactic X-ray sources based on the CSC2–Gaia DR3 catalog. In Section <ref>, we describe the selection of new CV candidates in the CSC2–Gaia DR3 catalog using two different cuts in the X-ray and optical color-color diagram, known as the “X-ray Main Sequence". We compile pure and optically variable samples of CV candidates. In Section <ref>, we present the follow-up observations conducted with the 5m Hale Telescope at Palomar Observatory and the 10m Keck I Telescope on Mauna Kea. We also present detailed X-ray timing and spectroscopic analyses. In Section <ref>, we present the sample of 14 CV candidates found in the CSC2–Gaia DR3 catalog. We separately discuss the nature of four spectroscopically confirmed CVs and four variable CV candidates. Our results are summarized in Section <ref>.





§ SAMPLE OF GALACTIC X-RAY SOURCES



We began by constructing a sample of Galactic X-ray sources from the publicly available CSC2 <cit.>. We cross-matched CSC2 with the Gaia DR3  catalog <cit.>, keeping only objects with significant parallaxes and proper motions. We identified the known sources in our sample by cross-matching with the Simbad database and other publicly available CV catalogs. All sample creation steps are presented in detail below.






 §.§ Chandra Source Catalog

As a first step, we used the master source table from the CSC2 with a total of about ∼ 317,000 unique source detections. We set the following criteria[For detailed flag descriptions see <cit.> and CSC2 web-page: <https://cxc.cfa.harvard.edu/csc/columns/flags.html>] to select the point-like X-ray sources with measured parameters:


    
  * extent_flag=False; only point-like sources.
    
  * conf_flag=False; source regions do NOT overlap.
    
  * pileup_flag = False; ACIS pile-up fraction does NOT exceed ∼ 10% in all observations.
    
  * sat_src_flag=False; the source is NOT saturated in all observations.
    
  * streak_src_flag=False; the source is NOT located on an ACIS readout streak in all observations.
    
  * dither_warning_flag = False; highest statistically significant peak in the power spectrum of the source region count rate does NOT occur at the dither frequency of the observation.
    
  * man_add_flag=False and man_inc_flag=False; the source was NOT manually added nor included in the catalog via human review.
    
  * man_match_flag=False; source detections were NOT manually matched between overlapping stacked observations via human review.
    
  * man_pos_flag=False; best-fit source position was NOT manually modified via human review.
    
  * man_reg_flag=False; source region parameters were NOT manually modified via human review.
    


Using these criteria, we selected 271,438 unique objects from CSC2.




 §.§ Cross-match with Gaia


For the next step, we cross-matched 271,438 objects with the Gaia DR3 catalog <cit.> to select only Galactic sources. As a search radius for each object, we used the individual X-ray positional error (with a search radius equal to err_ellipse_r0, the semi-major axis of the 95% confidence ellipse). We only kept sources meeting the following criteria:

    
  * RUWE < 1.4 <cit.>; good fit to the astrometric observations.
    
  * Only ONE Gaia object must be within the search radius.
    
  * Parallax and proper motion are measured with a signal-to-noise ratio S/N ≥ 3.


These selection steps reduced the number of sources to 26,079. To remove any potential  high proper motion stars, we applied an additional filtering criterion:


    
  * pm×Δ t ≤ err_ellipse_r0;  The angular motion of the Gaia object (caused by a proper motion (pm)) during the time interval Δ t does not exceed the search radius (err_ellipse_r0).   The time interval Δ t is an epoch difference between Gaia DR3 (Epoch 2016) and CSC2 (Epoch 2000) catalogs.


This filter finally resulted in 25,887 objects in the CSC2–Gaia DR3 catalog.




 §.§ Identification of Known Sources



To identify known sources in the CSC2–Gaia DR3 catalog, we cross-matched our objects with the Simbad database <cit.> using Gaia coordinates and a 3 search radius. We grouped our objects based on the Simbad hierarchical structure[<http://simbad.cds.unistra.fr/guide/otypes.htx>], making seven general types: STARS (single and binaries), ACCRETING BINARIES (X-ray binaries, CVs, Nova and Symbiotic Stars), GALAXY, ISM, PROPERTIES, BLENDS (not well-defined objects) and UNKNOWN (object is not presented in Simbad database). In Appendix <ref>, we present a detailed description of our general object types compiled from the Simbad hierarchical structure. We found Simbad classifications for 12,181[While creating the catalog of known systems, we had multiple matches (more than one) within the search radius for 810 of the 12,181 objects, giving about 7% of possible spurious associations in the catalog. However, for most objects with multiple matches a Simbad database provided a similar main object type (as defined in Table <ref>). We adopted the Simbad object type from the nearest source for those objects.]  of the 25,887 objects. 32 of these 12,181 were known CVs or CV candidates. 

To check that we did not miss any known CVs, we independently cross-matched our 25,887 objects with the following catalogs using a 2 search radius: i) Ritter and Kolb catalog <cit.> (Final version; December 31, 2015); ii) Cataclysmic Variables Catalog (2006 edition; <cit.>); iii) The Open Cataclysmic Variable catalog <cit.> (only confirmed CVs are kept in the list). We combined the information from the Simbad database and these three CV catalogs to make the final list of known CVs. Out of 25,887 sources, we finally compiled a sample of 40 known CVs. This sample of 40 consists of the CVs from Simbad plus those from the three catalogs mentioned above.

We note that 52 objects (out of 25,887) were classified as GALAXY. These objects are primarily quasars and galaxies hosting active galactic nuclei that show significant parallax and proper motion. The high proper motions and parallaxes of these extragalactic objects could be due to variability-induced motion <cit.>. Detailed investigation of these objects is beyond the scope of this paper. 





§ CV SELECTION WITH THE X-RAY MAIN SEQUENCE







 
We combined X-ray and optical data to search for new CV candidates in the CSC2–Gaia DR3 catalog. We used the “X-ray Main Sequence", a phase space of  X-ray-to-optical flux ratio (F_X/F_opt) and a Gaia optical BP–RP color <cit.>, to select CV candidates and create a pure sample of these objects. The empirical cut in the “X-ray Main Sequence" is given by the following functional form,

    log(F_X/F_opt)_cut= A + B × (BP-RP),

where A= -3.5 and B=1 are linear parameters from <cit.>. We computed the F_X/F_opt ratio based on X-ray and Gaia optical fluxes. The X-ray fluxes were calculated in the 0.5–7 keV energy band from CSC2. The aperture-corrected (90% enclosed counts fraction) energy flux is calculated using the power-law model with a photon-index of Γ=2 and the individual Galactic hydrogen column density N_H(Gal)  in the direction of the source. Gaia G band magnitudes were converted to optical fluxes using the AB magnitude system and a central wavelength of 6,000 Å. Out of 25,887 sources in the CSC2–Gaia DR3 catalog, we selected only sources with measured X-ray fluxes and optical magnitudes:


  * flux_powlaw_aper90_b/error ≥ 2; Source X-ray flux in the 0.5–7 keV energy band is measured with signal-to-noise ratio more than S/N ≥ 2.

  * Gaia G band, BP and RP magnitudes are measured having signal-to-noise ratio more than S/N ≥ 2.

These selection steps finally resulted in 19,665 sources (with 36 known CVs).



 §.§ First Selection: A pure sample of CV candidates 



To compile a pure sample of CV candidates and avoid contamination from stellar objects, we searched for optimal values for linear parameters A and B in Eq. <ref>. We created two samples of known sources classified based on the Simbad hierarchical structure: one consisting of CVs (36 sources) and the other of stellar objects (STARS, 4781 sources). For the STARS sample, we included only objects with sub-types classified as "YSO", "Binaries", "Massive Stars", "Main-sequence Stars", or "Evolved Stars" (see Table <ref>). The left panel of Figure <ref> shows an F_X/F_opt and Gaia BP–RP color diagram for known CVs (red squares) and STARS (blue dots). It's evident that known CVs are primarily located in the upper left corner of the diagram and could be visually separated from STARS. We computed the purity and completeness using a sample of these known CVs and STARS (see Appendix <ref> for more details). Based on our completeness/purity analysis we selected A = -1.5, and B=0.7 that gives the highest purity (≈100 %) and ≈52% completeness.
  
We only searched for new CV candidates among objects without precise classification from the Simbad database. Out of 19,665 objects, we selected 13,956 sources classified as one of the following object types: i) UNKNOWN (9,868 sources); ii) PROPERTIES (355 sources); iii) STARS, sub-type "Star" (3,720 sources); iv) STARS, sub-type "WD" (10 sources); v) STARS, sub-type "BD" (3 sources). The right panel of Figure <ref> shows the F_X/F_opt and Gaia BP–RP color diagram for these objects. We found 11 new CV candidates (out of 13,956 objects) with our selection based on our most optimal values for  A = -1.5 and B=0.7 (see objects above the solid red line in the right panel of Figure <ref>).

We note that we independently found three objects (recently confirmed as CVs) as CV candidates using the F_X/F_opt and Gaia (BP–RP) color diagram. During the writing of this manuscript, we learned that these systems had been previously discovered. However, these objects have only been mentioned in one or two papers, so we included those objects in our analysis and discussed them individually in Appendix <ref>. 




 §.§ Second Selection: An optically variable sample of CV candidates with ZTF photometry



Many CVs appear as periodic sources in optical light curves. Furthermore, accretion can lead to outbursts, which can also be observed in their optical light curves, particularly in non-magnetic CVs. Therefore, even after compiling a sample of 11 CV candidates based purely on an F_X/F_opt and Gaia BP–RP color diagram, we also searched for possible variability in a larger sample of sources. We used the empirical cut on the F_X/F_opt and Gaia BP–RP color diagram from <cit.> (A = -3.5 and B=1) and searched for possible variability in 375 sources (see objects above the dashed black line in the right panel of Figure <ref>).

We searched for periodic and outbursting-type variability in sources based on ZTF optical photometry for both  g and r light curves, considering only data with a signal-to-noise ratio greater than 5, SNR_ZTF > 5. We retrieved ZTF light curves for our objects and used the barycentric corrected modified Julian date (MJD) in our analysis. Light curves have a photometric precision of ∼0.015 mag at the bright limit of 14 mag, decreasing to a precision of ∼0.1 mag at the faint limit of 20.5 mag. We searched for periodic variability using the Lomb-Scargle periodogram, as implemented in the  tool to search for periods ranging from 5 min to 10 days. Once the period search was performed, we excluded all sources with orbital periods that are multiples or fractions of the sidereal day. To first order, the periods are reported up to a precision of 30 sec, which is determined by the exposure time. To calculate the significance of a period, we first computed the median absolute deviation (MAD) of the periodogram. Then, we took the periodogram value at the highest peak and divided it by the MAD. This ratio served as our significance metric. We plotted the distribution of significance metrics and kept only those sources in the upper 80^th percentile of the distribution. To search for outbursting sources, we looked for ZTF light curves with more than 5 outburst points Δ mag≥ 1, with each point more than 3 standard deviations from the median.

In total, we had 178 (out of 375) sources with ZTF photometry. Using our selection criterion described above, we found five periodic sources (see Figures <ref>, <ref>, <ref> and <ref>). With ZTF photometry, only one outbursting source (2CXO J024131.0+593630) was identified, which we discuss in Appendix <ref>. Based on the analysis of optical ZTF light curves, we extended our final sample of CV candidates to include a total of 14 candidates[Two periodic sources were already presented in the sample of 11 new CV candidates (selected by F_X/F_opt and Gaia BP–RP color). Therefore, we added only three variable sources (out of five) in our final sample to include 14 CV candidates in total.].  

Out of 14 sources, 2 objects were marked as variables with a Gaia flag (phot_variable_flag=VARIABLE). These objects show outbursting-type variability. Figure <ref> shows optical light curves based on Gaia photometry (SNR_ Gaia > 3). Table <ref> (column 9) summarizes the variability properties of the sources in the final sample. We note that the number of spurious matches in our final sample is effectively zero due to the ∼1–2" localization of X-ray sources made possible by Chandra. Furthermore, our candidates that show optical variability are even less likely to be false matches, since optical variable sources, particularly accreting compact objects, are typically associated with X-ray emission (e.g., see Figure 2 of <cit.>). 














§ DATA AND ANALYSIS



































 §.§ Optical Spectra



We performed optical spectroscopic follow-up observations for four sources in our sample, focusing on those located in the Northern Hemisphere that were accessible to our telescopes and exhibited strong optical variability and/or a high F_X/F_opt ratio. We used both the Double Spectrograph <cit.> on the 5m Hale Telescope at Palomar Observatory and the Low-Resolution Imaging Spectrometer <cit.> at the 10m Keck I Telescope on Mauna Kea. During all DBSP observations, we used the 600/4000 grism on the blue side and the 316/7500 grating on the red side. A 1.5 slit was used, and the seeing on all occasions was 1.5 – 2.0. All P200/DBSP data were reduced with [<https://dbsp-drp.readthedocs.io/en/stable/index.html>], a Python-based pipeline built upon the existing  package <cit.> and optimized for reducing spectral data. During all LRIS observations, we used the 600/4000 grism on the blue side and the 600/7500 grating on the red side. We used a 1.0 slit, and the seeing on all occasions was around 0.7. LRIS data were reduced with , an IDL-based pipeline optimized for LRIS long slit spectroscopy <cit.>. All data were flat fielded and sky-subtracted using standard techniques, wavelength-calibrated with internal arc lamps, and flux-calibrated using a standard star. Table <ref> shows all data acquired for the four CV candidates. Single spectra were taken at a time for each object (no coadds), with typical exposure times of 600 seconds for Keck I/LRIS and 900 seconds for P200/DBSP.

Figure <ref> shows Chandra X-ray images and optical images from Pan-STARRS <cit.> of the same sky region for four spectroscopically confirmed CVs in our sample. Figures <ref> - <ref> show the optical spectra of these sources. We identified prominent emission lines and computed their equivalent widths (EWs) by fitting Gaussian profiles to them. Table <ref> shows the EWs of selected lines and the He II(4686) / Hβ ratio[2CXO J044048.3+292434 was observed twice, and both observations show equivalent width measurements which are in agreement to within 1σ. Therefore, in Table <ref>, we only report the values from the first exposure.]. 

We conducted an extensive cross-match analysis between our 14 objects and publicly available optical data from the Sloan Digital Sky Survey (SDSS) and the Dark Energy Spectroscopic Instrument (DESI) collaboration <cit.>. With 2 search radius, we found no matches between SDSS/APOGEE catalogs (APOGEE DR17 StarHorse VAC, <cit.>; APOGEE DR17 astroNN VAC, <cit.>; APOGEE DR17 DistMass VAC, <cit.>; APOGEE Net VAC <cit.>; MaStar Stellar Parameters VAC, <cit.>) and our CV sample. We cross-matched our objects with the early data release (EDR) of DESI collaboration <cit.>, which contains spectra of more than approximately 1 million objects <cit.>. We found no identification of our objects with this catalog.









 §.§ X-ray spectra and luminosities



 
        




For each of the 14 sources in our sample, we queried the archival Chandra data using the observation identifier (obsid) from source observation summary results in the CSCview application[<https://cxc.cfa.harvard.edu/csc2.0/gui/intro.html>]. For some objects, we additionally searched for new Chandra observations within a 10 search radius using the Chaser[<https://cda.harvard.edu/chaser/>] webtool and included them in our analysis. The Chandra archival observations were processed by following standard Chandra Interactive Analysis of Observations (CIAO) <cit.>  threads[<http://cxc.harvard.edu/ciao/index.html>] (CIAO version 4.16.0 and CALDB version 4.11.0). The chandra_repro tool was used to perform the initial calibration, reprocess the data, and create a level 2 event file. To extract the source and background spectra, we used the specextract tool in CIAO. A point-source aperture correction was applied for each source spectra by setting the parameter correctpsf=yes. We computed the radii of the Chandra point spread function (PSF) from the PSF map, which was constructed with mkpsfmap tool for 90% of encircled counts fraction (ECF) and the 2.3 keV energy (parameters ecf=0.9 and energy=2.3 respectively). Source and background regions were centered at source positions from CSC2. The circle with the radius equal to the Chandra PSF radius was used for the source regions, and annuli with the inner and outer radii equal to two and four times the Chandra PSF radius were used for background regions. 

We performed the spectral fit using  the  v12.13 package <cit.> in the 0.5–7 keV energy band. We binned the X-ray spectra to have at least three counts per bin and used C-statistics <cit.> to get best-fit results. We used the error command in XSPEC to compute 1σ confidence intervals for the parameters. For some sources in our sample, we estimated only the 1σ lower limit (or upper limit) of the parameters. Each spectrum was approximated using two spectral models: the power-law model ( in ) and the optically thin thermal plasma emission model ( in ). To take into account interstellar absorption, we used the Tuebingen-Boulder ISM absorption model ( in ) and the solar elemental abundances from <cit.>.

All sources in our sample are located at different distances, leading to various Galactic hydrogen column densities. We computed the distances of the sources from Gaia parallaxes, and their errors were calculated using the standard error propagation. Using the source distances, we adopted the E(B-V) values from the Bayestar dust map <cit.>. If data from the Bayestar dust map were not available, we utilized the Galactic plane reddening map <cit.>. We computed the hydrogen column density by assuming Cardelli's extinction law (R_V≈ 3.1) and the relation N_H (map)≈ R_V× E(B-V) × 2.21×10^21 cm^-2 <cit.>. For sources having more than 100 counts in their X-ray spectra, we computed hydrogen column density (N_H(fit)) and metal abundance directly from the fit. We found no significant deviation between the best-fit results of N_H(fit) and N_H(map) computed from the Bayestar dust map. For other sources, we approximated their spectra with a fixed N_H(map) and assuming solar metallicity. In the cases when N_H(map) was not available at the specified distance of the source or when the distance fell outside the interval classified as reliable, we fixed the hydrogen column density to the Galactic value <cit.>.


The best-fit results obtained from approximating the X-ray spectra with  and  models are presented in Appendix <ref> (see Table <ref> and Figure <ref>). For sources with more than one Chandra observation, we analyzed the X-ray spectra for each observation separately. However, we found no significant parameter variations between these separate observations subsequently performed a joint fit of the spectra in XSPEC by combining the parameters across multiple observations. We computed absorption corrected fluxes in the 0.5–7 keV energy band for the power-law model using the cflux command and converted them to luminosities. The approximate errors of the unabsorbed X-ray luminosities were computed from the uncertainties in the distances and fluxes using standard error propagation. Table <ref> (columns 10 and 11) shows the distances and unabsorbed X-ray luminosities in the 0.5–7 keV energy band.




 §.§ X-ray variability and timing analysis



We searched our sample of 14 CV candidates for X-ray variable sources using the CSC2 variability indices in the 0.5–7 keV energy band: var_intra_index_b ≥ 8 (variability within a single observation) and/or var_inter_index_b ≥ 8 (variability across multiple observations). We found only one source matching these criteria (2CXO J063805.1–801854). We also checked the significance of the difference between the maximum and minimum fluxes between observations using the ratio R=(F_max-F_min)/(σ_F_max^2 + σ_F_min^2)^1/2 and found only one source having R ≥ 3 (2CXO J123727.5+655211).

Out of the 14 CV candidates in our sample, we performed an X-ray timing analysis for three sources with more than 300 source counts in their spectra (see Table <ref>). If sources had more than one archival Chandra observation, we used the observation with the longest exposure time for the timing analysis. We extracted the background-subtracted X-ray light curves with the dmextract tool in CIAO, using the same source and background regions as those used for the X-ray spectral analysis. Barycentric corrections were applied for the event files by using axbary tool in CIAO. We used the XRONOS sub-package in FTOOLS[<http://heasarc.gsfc.nasa.gov/ftools>] for timing analysis <cit.>. First, we calculated the power spectrum density using a Fast Fourier Transform (powspec command in the XRONOS package) to search for periodic signal. The epoch-folding (EF) technique was then applied to constrain the period of the sources <cit.> using the efsearch command in the XRONOS package. We estimated the best-fit period from the EF method by fitting a Gaussian function to the χ^2 vs period diagram. The period error was estimated using the bootstrap technique similarly as described in <cit.>. To determine the significance of the detected period, we generated 1000 random Poisson, non-periodic light curves with mean source count rates and applied the powspec and efsearch tools on these randomly generated light curves. We computed the maximum value of the power spectrum density and the χ^2 (as defined in the EF method) for each generated light curve. A 3σ upper confidence level was calculated for the distributions of power spectrum densities and χ^2 values. A strong signal was considered present in the observed X-ray light curves if the results from the powspec and efsearch tools exceeded the 3σ confidence level determined from the simulations. We then used the efold tool to construct X-ray light-curves folded with the best-fit periods.

We found strong periodic X-ray signals only in one source out of three (see Table <ref>). Figure <ref> shows the X-ray pulse profile normalized by the mean count rate of that source. The pulse fraction (PF) was computed using maximum and minimum count rates, where PF=(F_max-F_min)/(F_max+F_min).  We discuss the nature of the X-ray periodic source in Section <ref>.




 §.§  Hertzsprung–Russell and X-ray luminosity – orbital period diagrams








To compute extinction-corrected absolute G band magnitudes for CV candidates, we directly adopted the interstellar extinction A_G  (at 6000 Å) from Bayestar or the Galactic plane reddening maps when available. Otherwise, we converted the hydrogen column densities presented in Table  <ref>  for the mekal model (see Section <ref>)  to interstellar extinction A_G  using the <cit.> extinction function. Figure <ref> shows the location of the CV candidates on the Hertzsprung–Russell (HR) diagram along with Gaia objects within 100 pc that have significantly measured parallaxes (> 3). 

We constructed an X-ray luminosity – orbital period diagram for six objects in our sample with periods computed from the optical and X-ray data (see Figure <ref>). To compare our objects with other known CVs, we cross-matched the CV catalog of Ritter and Kolb (Final Edition, frozen on 31 December 2015, <cit.>) with the the second ROSAT all-sky survey (2RXS) catalogue <cit.>, keeping only sources with a single match within 20. We then cross-matched with Gaia DR3 <cit.>, keeping only sources with significant parallaxes and proper motions (5σ), which yielded 259 sources. Next, we used the WebPimms[<https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl>] tool to convert ROSAT source count rates (0.1–2.4 keV) to unabsorbed X-ray luminosities in the 0.5–7 keV energy band, assuming a power-law model with a photon index of 1.7 and hydrogen column density of 3× 10^20 cm^-2. To show different CV types in Figure <ref>, we used the classification scheme of the Ritter and Kolb catalog, which categorizes them as into polar (AM), intermediate polar (DQ), or otherwise as non-magnetic. In Figure <ref>, we show different period minimum values for CVs from <cit.> (≈82 min) and <cit.> (≈76 min) as well as CV orbital period gap values from <cit.> (≈ 2.5-3.2 hr) and <cit.> (≈ 2.2-3.2 hr).







§ RESULTS AND DISCUSSION



Table <ref> shows the parameters of the 14 CV candidates found in the CSC2–Gaia DR3 catalog using  the F_X/F_opt and the Gaia BP–RP color diagram and the period search with ZTF optical photometry (see Section <ref>).  Figure <ref> shows that most of our CV candidates are located between the main sequence and the WD regions, which is consistent with the distribution of known CVs in the Gaia HR diagram <cit.>. The X-ray luminosities of all our objects vary in the ≈ 10^29 - 10^32 erg s^-1 range, which matches with the range of X-ray luminosities of some known CVs (see Figure <ref>). X-ray spectra of our CV candidates can be approximated by a power-law model with photon indices in the Γ≈ 1-3 range (or kT ∼ 1-70 keV for the optically thin thermal emission model). Such X-ray spectra are seen in some magnetic and non-magnetic CVs <cit.>. 

Our variability search indicates that seven sources out of these 14 CV candidates are variable in nature. The X-ray luminosity – orbital period diagram shows that most of our variable sources have X-ray luminosities and periods typically seen for magnetic and non-magnetic CVs (see Figure <ref>). Only two sources (2CXO J182117.2–131405 and  2CXO J185603.2+021259) are located in a distinct region of the phase space compared to the main sample of CVs, and exhibit low X-ray luminosity. These two objects might be a low-accretion rate  CV and an eclipsing binary[Such a binary would be an “active binary", like RS CVn systems, which have orbital periods from as low as 5–6 hours to ∼10 days. In these systems, X-rays originate from the hot corona of one or both stellar components <cit.>.]. We spectroscopically confirm four objects in our sample as new CVs. Their optical spectra show prominent emission lines or cyclotron humps, features typically observed in CVs (see Figures <ref> - <ref>). Below, we provide a detailed discussion of these CVs and other variable objects in our sample.


We note that some objects in our sample were previously suggested to be CV or quiescent low-mass X-ray binary (qLMXB) candidates using a multiwavelength machine-learning method, however, the results have low confidence <cit.>. Objects having Gaia optical light curves were also classified as possible CV candidates using a machine learning variability classifier in Gaia DR3 <cit.>. In Appendix <ref>, we discuss three recently discovered CVs, which we independently selected using the methods described in this paper. These examples demonstrate that the F_X/F_opt and Gaia BP–RP color diagram, as suggested by <cit.> can be a powerful method for searching and independently selecting accreting binary candidates. Without optical follow-up, it is challenging to distinguish between CVs and qLMXBs, as they display similar F_X/F_opt ratios and have overlapping X-ray luminosity ranges. Further spectroscopic follow-up to place constraints on the mass of the accreting object is required for the rest of the objects in our sample to clarify their nature.  






 §.§ Spectroscopically Confirmed CVs



Here, we discuss the nature of four objects from our final sample of CV candidates, for which we obtained optical spectra using Keck I/LRIS or Palomar/DBSP.




  §.§.§ 2CXO J044048.3+292434, a new polar



We present the optical spectrum of J044048.3+292434 (hereafter J04404) in the right panel of Figure <ref>. It shows Hydrogen Balmer and Helium emission lines along with a high-excitation He II 4686Å emission line, yielding a high line ratio  of He II / Hβ≈ 0.64, which is typically seen in polars (see Table <ref>).

J04404 shows a flux ratio of about F_X/F_opt≈ 0.6, and an X-ray luminosity L_X ≈ 7.1× 10^30 erg s^-1 (see Table <ref>). The X-ray spectrum can be approximated by a power-law model with an unusually shallow photon index Γ≈ 0.5 (or kT  55.5 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>), a characteristic observed in some magnetic CVs <cit.>. J04404 is located close to the period minimum of CVs in the X-ray luminosity–period diagram (see Figure <ref>). Its optical and X-ray properties indicate that  J04404 is a polar near the CV period minimum.

In the left panels of Figure <ref>, we present the long-term ZTF light curve of  J04404 and the phase-folded light curve at the best period (1.30 h = 78 min). It is evident that J04404 underwent a state change, commonly observed in polars, around MJD = 59000, where the average optical r filter magnitude increased from r∼21 to r∼19 (see top left panel of Figure <ref>). 
 
In some cases, polars can show two peaks in their optical light curve within a single orbital period due to cyclotron beaming[A two-component cyclotron model has been used to model the optical and near-infrared light curves of polars, such as in <cit.>. A two-component cyclotron model leads to two peaks in a single orbital period, and can be due to any of the following: two magnetic poles of different field strength, optical depth, and/or viewing angles.]. The phase-folded optical light curve of J04404 shows ∼ 2 mag peak-to-peak modulation, which is characteristic of cyclotron beaming following a state change in polars (see the bottom left panel of Figure <ref>). In order to determine if the orbital period of J04404 is 1.30 hr or 2.60 hr, we utilized the fact that the Hydrogen Balmer and He II emission lines in polars typically vary by ≈1000 km/s within a single orbital period <cit.>. In the case of J04404, two spectra taken 1.4 hr apart do not show a significant radial velocity shift. In the bottom right panel of Figure <ref>, we show both spectra of J04404, taken 1.4 hr apart. We also show a ≈1000 km/s shift, which would be expected if 1.4 hr were approximately half of the orbital period. However, we note that if the system were viewed nearly perpendicular to the accretion stream, such a radial velocity shift would not be observed. Time-resolved spectroscopy for 2.6 hr is necessary to confirm the orbital period of this system, leading us to currently label the orbital period as either 1.3 or 2.6 hr.





  §.§.§ 2CXO J044147.9-015145, a low accretion rate non-magnetic CV



In Figure <ref> (left panel), we present the long-term ZTF optical light curve of 2CXO J044147.9-015145 (hereafter J04414). A preliminary Lomb-Scargle period analysis reveals no periodicity in the ZTF  data, indicating that further data are required to determine the possible orbital period of this system. The optical spectrum of J04414 is shown in Figure  <ref> (right panel). Hydrogen and Helium emission lines are present, with no detectable He II 4686Å emission line, giving a 3σ upper limit for the line ratio He II / Hβ < 0.05 (see Table <ref>). J04414 shows a flux ratio of F_X/F_opt≈ 0.3 and an X-ray luminosity L_X ≈ 3.1× 10^29  erg s^-1. The X-ray spectrum can be approximated by a power-law model with a photon index Γ≈ 1.8 (or kT ≈ 4.5 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). The optical and X-ray properties suggest that J04414 is a low accretion rate non-magnetic CV (WZ Sge type). This classification is based on the apparent WD Balmer absorption lines and weakly double-peaked central emission lines seen in the center of its  optical spectrum. There is no sign of the donor signature in the optical spectrum. The Gaia HR diagram (see Figure <ref>) also supports this classification, as J04414 is located in the WD region, where WZ Sge type systems are concentrated  <cit.>.






  §.§.§ 2CXO J123727.5+655211, a new polar



In Figure <ref> (top panels), we present the long-term ZTF light curve and the phase-folded light curve at the best period for 2CXO J123727.5+655211 (hereafter J12372). The ZTF light curve reveals a significant period at 2.12 hr. The color dependence of cyclotron beaming is evident, with the ZTF i filter showing the highest modulation (peak-to-peak ∼ 2 mag) and the ZTF g filter showing the lowest (peak-to-peak ∼ 1 mag) (see the top left panel of Figure <ref>). 
 
We present the optical spectrum of J12372 in Figure <ref> (bottom panel). The spectrum shows Hydrogen, Helium, and high excitation He II 4686Å emission lines, yielding a line ratio of He II / Hβ≈ 0.37 (see Table <ref>).

J12372 shows a flux ratio F_X/F_opt≈ 1.1 and an X-ray luminosity L_X ≈ 4.4× 10^31 erg s^-1 (see Table <ref>). The X-ray spectrum can be approximated by a power-law model with a photon index Γ≈ 1.4 (or kT ≈  38.3 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). X-ray timing analysis revealed no significant period in the longest exposure Chandra observation (see Table <ref>). J12372 shows significant X-ray flux variability (R≈ 6, see Section <ref>), with a factor of ≈ 2 between the Chandra observations, but no statistically significant variability between the spectral parameters was found. J12372 lies close to the CV period gap in the X-ray luminosity–period diagram (see Figure <ref>). Together with the optical light curves and spectrum, this indicates that J12372 is a polar.








  §.§.§ 2CXO J182117.2-131405, low accretion rate polar




Figure <ref> (top panels) shows the optical long-term ZTF light curve of 2CXO J182117.2–131405 (hereafter J18211) and the phase folded light curve. We determined  its best period to be 2.01 hr using the Lomb-Scargle period analysis. The phase-folded optical light curve shows cyclotron beaming  near phase 0.15 and ≈ 1 mag peak-to-peak modulation (see top right panel of Figure <ref>). This modulation could be either due to cyclotron beaming or an eclipse of the WD by the donor star. High-speed photometry is needed to further investigate these scenarios.

J18211 has a low number of X-ray counts, and its spectrum can be approximately fitted by a power-law model, resulting in a large uncertainty for a photon index  Γ∼ 1.7 (or kT  1.6 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). J18211 shows a flux ratio F_X/F_opt≈ 0.15 and a lower X-ray luminosity (L_X ≈ 6× 10^28 erg s^-1) than typically seen in CVs (see Table <ref>). J18211 is located near the edge of the CV period gap in the X-ray luminosity–period diagram (see Figure <ref>). In this part of the diagram, the canonical picture of CV evolution predicts that detached systems begin to undergo accretion again. Hence, J18211 could be a low accretion rate polar (previously referred to as a pre-polar).

We present the optical spectrum of J18211 in Figure <ref> (bottom panel). The spectrum is dominated by broad humps corresponding to cyclotron harmonics with no significant Hβ and He II 4686Å emission lines detected. Using the equation for the electron gyration frequency around a magnetic field, we can define the wavelength of cyclotron harmonics as the following:


    λ_n = 10710/n(100  MG/B)sinθ


With the spacing of two harmonics, it is possible to constrain the magnetic field strength, as the spacing must always be the ratio of integers. We must assume a cyclotron beaming angle, which should be close to 90^∘ since the amplitude of the cyclotron harmonics is so large (An angle θ > 60^∘ is a good approximation for cyclotron beaming factors greater than 2 <cit.>). Detailed cyclotron modeling is beyond the scope of this paper. In this work, we fixed the magnetic field at different values and approximately matched the location of the cyclotron humps in the spectrum to different cyclotron harmonics (see Figure <ref>). This combination of the cyclotron wavelengths observed in the spectrum of J18211 and the large beaming amplitude constrains the magnetic field to be B≈54-63 MG. This magnetic field strength is typical of low accretion rate polars <cit.>.






 §.§ Variable CV candidates



Here, we discuss four CV candidates classified based on their X-ray and optical variability, but for which optical spectra are not yet available.






  §.§.§  2CXO J063805.1–801854: A magnetic CV?


2CXO J063805.1–801854 (hereafter J06380) shows F_X/F_opt≈ 0.8 and X-ray luminosity L_X ≈ 2.4× 10^31 erg s^-1 (see Table <ref>). Its spectrum can be approximated by a power-law model with an unusually shallow photon index Γ≈ 0.2 (or kT  75 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). Such a shallow photon index is observed in the X-ray spectra of some magnetic CVs, such as polars or IPs <cit.>. X-ray timing analysis revealed a ≈ 13,200 sec periodicity (PF=61%), which is consistent with the period previously found by <cit.> in the same Chandra data (see Table <ref> and Figure <ref>). There is no available optical ZTF photometry for J06380. The Gaia optical light curve shows an outbursting-type variability with ≈ 3 mag outburst lasting around ≈ 100 days and a possible recurrence time of approximately ≈ 500 days (see Figure <ref>). Figure <ref> shows that J06380 lies within main sample of known CVs. The X-ray periodicity (in the range of CV orbital periods), along with a high X-ray luminosity and hard spectrum, suggests that J06380 could be a magnetic CV, likely an IP.




  §.§.§   2CXO J165219.0–441401: A possible dwarf nova?


2CXO J165219.0–441401 (hereafter J16521) shows F_X/F_opt≈ 1.2 and a high X-ray luminosity L_X ≈ 3× 10^32 erg s^-1 (see Table <ref>). The X-ray spectrum can be approximated by a power-law model with a photon index Γ≈ 1.5 (or kT ≈ 5.8 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). There is no ZTF photometry for J16521 and the Gaia optical light curve shows a ≈ 3 mag outburst with a possible ≈ 500 days recurrent time (see Figure <ref>). This suggests that J16521 could be a dwarf nova.




  §.§.§ 2CXO J173118.6+224248: A possible non-magnetic CV?


2CXO J173118.6+224248 (hereafter J17311) shows F_X/F_opt≈ 0.3 and an X-ray luminosity L_X ≈ 2.4× 10^30 erg s^-1 (see Table <ref>). The optical ZTF light curves show ≈0.5 mag variability with a period of about ≈1.8 hr (see Figure <ref>). However, the actual period may be twice this value, possibly due to ellipsoidal modulation. The X-ray spectrum can be approximated by a power-law model with a photon index Γ≈ 2.2 (or kT ≈ 3.0 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>), a characteristic observed in non-magnetic CVs. J17311 is located close to the main sequence of known CVs in the X-ray luminosity–orbital period diagram (see Figure <ref>). The X-ray properties and optical variability suggest that J17311 is a non-magnetic CV.





  §.§.§ 2CXO J185603.2+021259: A possible eclipsing binary?


2CXO J185603.2+021259 (hereafter J18560) shows F_X/F_opt≈ 0.05 and a lower X-ray luminosity (L_X ≈ 2.5× 10^29 erg s^-1), than observed in CVs (see Table <ref>). The ZTF optical light curve shows deep eclipses of approximately ≈ 1 mag in the r filter (or approximately ≈ 1.5 mag in the g filter) with a period of about 5.5 hr (see Figure <ref>).  Out of the eclipse sinusoidal variability with ≈ 0.1 mag is seen, possibly caused by the contribution from the donor. The X-ray spectrum can be approximated by a power-law model having a photon index Γ≈ 2.8 (or kT ≈ 1 keV for an optically thin thermal plasma model) (see Table <ref> and Figure <ref>). Figure <ref> shows that J18560 is located below the main CV sample, similar to J18211. These observed properties suggest that J18560 could be an eclipsing binary.






§ CONCLUSIONS



We searched for new Galactic CV candidates among 25,887 sources in the CSC2, cross-matched these with Gaia DR3, and studied their optical and X-ray properties. We summarize our findings below:




  – We used a modern concept suggested by <cit.> to classify new CVs among stellar objects based on a combination of F_X/F_opt and Gaia BP–RP color (so called the “X-ray Main Sequence") (see Figure <ref>). We further defined a new threshold cut to compile a pure sample of CV candidates and minimize contamination from stellar objects (see Eq.<ref> and Section <ref>).




  –  We excluded known Galactic stellar objects by cross-matching the CSC2 – Gaia DR3 catalog with the Simbad database and other publicly available CV catalogs (see Section <ref>). Out of 13,956 unclassified Galactic objects, we identified 14 new CV candidates using two different cuts on the “X-ray Main Sequence"  (see Table <ref> and Section <ref>).




  – All objects show X-ray luminosities in the range L_X ≈ 10^29-10^32 erg s^-1. Their X-ray spectra can be approximated by a power-law model with photon indices in the Γ∼ 1-3 range or by an optically thin thermal emission model in the kT ∼ 1-70 keV range (see Figure <ref> and Table <ref>). These X-ray properties are mostly typical of CVs (both magnetic and non-magnetic), but some objects might also be qLMXBs.





  – We performed optical follow-up observations with Keck I/LRIS and P200/DBSP and spectroscopically confirmed four objects as new CVs. Their optical spectra show prominent emission lines or cyclotron humps, typically seen in CVs (see Tables <ref> and <ref>). We determined orbital periods for three of these objects using ZTF photometry. Among these, we discovered two polars, one low accretion rate polar and one non-magnetic CV (WZ Sge type) (see Section <ref>). 




  – The X-ray and optical properties of four variable CV candidates suggest that  three are potential magnetic systems or dwarf novae, while one may be an eclipsing binary (see Section <ref>). Further spectroscopic follow-up is required to confirm the nature of these objects.





Our analysis shows that a multiwavelength approach is a powerful tool for discovering new CVs and studying their properties. We illustrated that the “X-ray Main Sequence" presented in <cit.> allows us to compile a sample of CV candidates for a detailed study. This approach can also be applied to other X-ray catalogs or upcoming all-sky SRG/eROSITA data to search for accreting binary systems. Further investigation of selection effects is necessary to use the “X-ray Main Sequence" to compile volume-limited samples of CVs and study their population.




This research has made use of data obtained from the Chandra Data Archive and software provided by the Chandra X-ray Center (CXC) in the application packages CIAO. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.

Based on observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grants No. AST-1440341 and AST-2034437 and a collaboration including current partners Caltech, IPAC, the Weizmann Institute of Science, the Oskar Klein Center at Stockholm University, the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National Laboratories, IN2P3, University of Warwick, Ruhr University Bochum, Northwestern University and former partners the University of Washington, Los Alamos National Laboratories, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW. The ZTF forced-photometry service was funded under the Heising-Simons Foundation grant No.12540303 (PI: Graham).

We are grateful to the staffs of the Palomar and Keck Observatories for their work in helping us carry out our observations. 

This work has made use of data from the European Space Agency (ESA) mission Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.


This research made use of matplotlib, a Python library for publication quality graphics <cit.>; NumPy <cit.>; Astroquery <cit.>; Astropy, a community-developed core Python package for Astronomy <cit.>; and the VizieR catalogue access tool, CDS, Strasbourg, France. 

IG, AS, VD, NT acknowledge support from Kazan Federal University. ACR acknowledges support from the National Science Foundation via an NSF Graduate Research Fellowship. The authors thank the anonymous referee for useful and inspiring suggestions which helped to improve the manuscript.






 



§ GENERAL OBJECT TYPES WITH A SIMBAD AND F_X/F_OPT-(BP-RP) SELECTION CUT 









Table <ref> shows our general object types based on the Simbad hierarchical structure. The columns are: (1) our main object type; (2) object sub-type; (3) title of Simbad hierarchical structure used to define the main object type and its sub-type; (4) number of sources per each object sub-type, including known sources and their candidates. No source in our sample had an object type from the sixth Simbad's hierarchy ("Gravitation"); therefore, it's not shown in Table <ref>. 

We investigated purity and completeness to construct a pure sample of new CV candidates and avoid contamination from stellar objects. Based on the Simbad hierarchical structure, we created two samples of known CVs (36 sources) and STARS (4781 sources) (see Section <ref>).

<cit.> computed purity and completeness for the red and blue galaxies classified by optical color. We applied a similar approach to define the purity and completeness as:





  * True positive (TP) = known CVs classified as CVs based on a given threshold (i.e., known CVs with an observed X-ray-to-optical ratio exceeding the threshold cut, (F_X/F_opt)_obs≥ (F_X/F_opt)_cut).


  * False positive (FP) = known STARS classified as CVs based on a given threshold (i.e., known STARS with an observed X-ray-to-optical ratio exceeding the threshold cut, (F_X/F_opt)_obs≥ (F_X/F_opt)_cut).


  * False negative (FN) = known CVs classified as STARS (i.e., known CVs with an observed X-ray-to-optical ratio below the threshold cut, (F_X/F_opt)_obs < (F_X/F_opt)_cut).



Purity (P) is computed as the fraction of true identification to all identifications, 

    P = TP / (TP + FP),

and the completeness (C) is calculated as the fraction of true detections to all that should have been classified, 

    C = TP / (TP + FN).

To find the optimal combination of linear parameters (A, B) for Eq.<ref>, we created a grid of parameters with the 0.1 steps in the range 0  A 3 and -8  B 3 and computed the purity and completeness. 

Figure <ref> shows the purity and completeness as a function of linear parameters (A, B). For illustrative purposes, we show the purity and completeness only for three B parameters. The B=0 case corresponds to the canonical selection cut based only on the X-ray-to-optical flux ratio without additional information from Gaia colors. For example, the cut F_X/F_opt = 0.1 (where A = -1 and B=0) gives ≈27% purity and ≈48% completeness. The drop of the purity above the A  -0.2 is caused by a low number of sources and is not statistically significant. The <cit.> empirical cut (A = -3.5 and B=1) shows ≈25% purity and ≈78% completeness. In our study, we chose to use A = -1.5 and B=0.7, which gives ≈100 % purity and ≈52% completeness.

We note that our threshold is defined based only on a small sample of confirmed CVs (36 objects). To study the CV population and create a volume-limited sample, the F_X/F_opt and Gaia BP–RP color diagram should be further investigated for selection effects with a large sample of confirmed CVs and stellar objects. A detailed investigation is beyond the scope of this paper.








§ RECENTLY DISCOVERED CVS

















We selected three objects from the CSC2–Gaia DR3 catalog using the X-ray main sequence as CV candidates (see Figure <ref> and Table <ref>).  In the course of this study, two of these objects (2CXO J024131.0+593630 and 2CXO J143435.3+334049) were recently discovered as CVs with SDSS spectra and ZTF light curves <cit.>. Another object, 2CXO J174133.7–284034[We included this system in our analysis because it appears as an X-ray source ("X") in Simbad, not explicitly as a CV. ], was confirmed as a CV via optical spectroscopy by <cit.>. However, we included this object in our analyses to search for possible X-ray variability through timing and spectral analyses. Here, we individually discuss the X-ray and optical properties of these objects.




 §.§ 2CXO J024131.0+593630, a non-magnetic CV




In the left panel of Figure <ref>, we present the long-term optical ZTF light curve of 2CXO J024131.0+593630 (hereafter J02413), which shows two significant (≈ 2 mag in r filter and ≈ 3 mag in g filter) short outbursts. A preliminary Lomb-Scargle period analysis reveals no periodicity in the ZTF and Gaia data, indicating that further observations are needed to determine the possible orbital period of this system.

We observed J02413 with P200/DBSP on 6 December 2023 for 900s. The optical spectrum of J02413 is presented in the right panel of Figure <ref>. Single-peaked Hydrogen Balmer emission lines are clearly visible, and a weak He II 4686Å emission line is also present, giving a low line ratio He II / Hβ≈0.12  (see Table <ref>). However, as noted in Table <ref>, the detection of He II 4686Å does not pass a visual inspection, so the value of the ratio should be interpreted as an upper limit. We also note that the detection (or non-detection) of He II 4686Å does not affect our final interpretation of this object. 

J02413 shows a high flux ratio F_X/F_opt≈ 1.7 and an X-ray luminosity L_X ≈ 1.5× 10^31 erg s^-1 (see Table <ref>). The X-ray spectrum of J02413 can be approximated by a power-law model with Γ≈ 1.8 (or kT ≈ 5.5 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). We found no significant period for J02413 from the X-ray timing analysis of the longest exposure Chandra observation (ObsID = 7152). J02413 also does not show X-ray flux variability and or changes in spectral parameters between observations. Our analysis, using optical and X-ray properties, suggests that J02413 is a CV, likely non-magnetic. This agrees with <cit.> conclusion, where J02413 is identified as a new non-magnetic CV (U Gem type). The spectrum of this object is not shown in <cit.>, making our spectrum a useful addition to the literature.




 §.§ 2CXO J143435.3+334049, low-accretion rate CV



In Figure <ref> (left panel), we present the long-term ZTF light curve of 2CXO J143435.3+334049 (hereafter J14343). A preliminary Lomb-Scargle period analysis reveals no periodicity in the ZTF  data, indicating that further observations are needed to determine the possible orbital period of this system.

We observed J14343 with P200/DBSP on 21 January 2023 for 900s. The optical spectrum of J14343 is presented in Figure <ref> (right panel). The WD Balmer absorption lines are seen with double-peaked emission lines at the center, though the peaks appear to be asymmetric in height and width for both the Hα and Hβ lines. We do not detect the high-excitation He II 4686Å emission line providing an upper limit for a line ratio  He II / Hβ < 0.15 (see Table <ref>). There is no sign of the donor signature in the optical spectrum. The spectrum of this object in Figure <ref> is different from the one shown in <cit.>, with the Balmer emission lines here being noticeably asymmetric. This could warrant future study of this system to further understand the nature of this source.

J14343 shows a flux ratio F_X/F_opt≈ 0.3  and a lower X-ray luminosity (L_X ≈ 4.8× 10^29 erg s^-1) than typically seen in CVs (see Table <ref>). We have not detected statistically significant flux and spectral parameter variability between Chandra observations. The X-ray spectrum can be approximated by a power-law model with a photon index Γ≈ 1.7 (or kT  5.2 keV for optically thin thermal emission model) (see Table <ref> and Figure <ref>). X-ray and optical properties suggest that J14343 is a low-accretion rate CV (WZ Sge type), similar to what <cit.> concluded in their analysis. 




 §.§ 2CXO J174133.7–284034, a non-magnetic CV


2CXO J174133.7–284034 (hereafter J17413 shows a flux ratio F_X/F_opt≈ 1.0 and an X-ray luminosity L_X ≈ 4.4× 10^31 erg s^-1 (see Table <ref>). The X-ray spectrum of J17413 can be approximated by a power-law model with Γ≈ 1.8 (or kT ≈ 5.6 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). We have not detected statistically significant flux and spectral parameter variability between Chandra observations. We found no significant period for J17413 with the X-ray timing analysis for the longest exposure Chandra observation (ObsID = 9565). <cit.> observed optical variability (≈ 1 mag) with no periodicity. <cit.> observed prominent emission lines and confirmed via optical spectroscopy that J17413 is a CV. Our X-ray analysis supports  <cit.> and <cit.>  conclusions and also suggests that J17413 is likely a non-magnetic system.





 




§ X-RAY SPECTRA FIGURES



Figure <ref> shows Chandra X-ray spectra of all objects from Tables <ref> and <ref>. Table <ref> shows the results of the approximation of X-ray spectra with two spectral models: power-law ( in  with photon index parameter, Γ) and  optically thin thermal plasma emission model ( in  with temperature kT, and metal abundance [Z] parameters).





























aa







 Paper Title: Spinning waveforms in cubic effective field theories of gravity.
































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        ^aCentre for Theoretical Physics, Department of Physics and Astronomy, 
        
        Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom^bNiels Bohr International Academy,
        Niels Bohr Institute, University of Copenhagen,
        
        Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmarka.brandhuber@qmul.ac.ukgraham.brown@qmul.ac.ukgang.chen@nbi.ku.dkg.travaglini@qmul.ac.ukp.vivesmatasan@qmul.ac.uk
        
        	QMUL-PH-24-11
        
        
        We derive analytic all-order-in-spin expressions  for  the leading-order time-domain waveforms  generated in the scattering of two Kerr black holes with arbitrary masses and spin vectors in the presence of all independent cubic deformations of Einstein-Hilbert gravity. These are  the  two parity-even interactions I_1 and G_3, and  the parity-odd ones 
        Ĩ_1 and G̃_3. 
        Our results are obtained using three independent methods: a particularly efficient  direct integration and tensor reduction approach; integration by parts combined with  the method of differential equations; and finally a residue computation. 
        For the case of the G_3 and G̃_3 deformations we can express the spinning waveform  in terms of the scalar waveform  with appropriately shifted impact parameters, which are reminiscent of Newman-Janis shifts. For I_1 and Ĩ_1 similar shifts occur, but  are   accompanied by additional contributions that cannot be captured by simply shifting the scalar I_1 and Ĩ_1 waveforms. 
        We also show the absence of leading-order corrections to gravitational memory.  
         Our analytic results are  notably compact,  and we compare the effectiveness of the three methods used to obtain them.  We also briefly comment on the magnitude of the corrections to observables due to cubic deformations.  
        
        
        
        
        
        Spinning  waveforms in cubic effective field theories of gravity 
        
            and Pablo Vives Matasan^a
            Submitted August 5, 2024
        ==================================================================
        
        
        
        § INTRODUCTION
        
        
        
        The high accuracy achieved in current observations of gravitational waves by the LIGO-Virgo-KAGRA collaboration, along with the potential of further  improvements  in upcoming experiments such as LISA, strongly motivates the pursuit of 
        increasing precision in theoretical calculations.
        One is then faced with the need to perform higher-loop calculations in Newton's constant 
        G, incorporating  spin effects, and also considering    potential modifications to the Einstein-Hilbert theory arising from as yet undiscovered higher-dimensional interactions. 
        
        
        
        Quadratic corrections in the curvatures are known to leave scattering amplitudes invariant <cit.>, hence the first  deformations to be studied appear at dimension six, that is they are cubic in the curvatures (see <cit.> for work on dimension-eight operators). 
        Specifically, we consider the deformations 
        
            I_1    R^αβ_μνR^μν_ρσR^ρσ_αβ ,      
            I_2 R^μνα_βR^βγ_νσR^σ_μγα .
        
        However,  instead of    I_2,  we  prefer working with  the combination 
        
            G_3  I_1 - 2 I_2  ,
        
        which is a topological term in six dimensions and has vanishing four-dimensional graviton amplitudes. 
        We will also study the effect of  the parity-odd couplings Ĩ_1 and G̃_̃3̃, which are obtained from the parity-even ones by replacing one of the Riemann curvatures R^μναβ by the dual R̃^μναβ=  
        (1/2)  ϵ^μνρσR_ρσ^αβ.
        Summarising, the effective action we will work with is 
        
            S = ∫d^4x  √(-g) (    -2/κ^2R      +  β_1 I_1 + β_2 G_3 + β̃_1 Ĩ_1 + β̃_2 G̃_3)  .
        
        Particular choices of the coefficients β_1, β_2, β̃_1, β̃_2  correspond to specific theories
        [For instance β_1 = -2/κ^2 α^'  2/48,  β_2 = -2/κ^2 α^'  2/24, β̃_1=β̃_2=0 reproduces the low-energy effective action of bosonic string theory.], but we will be agnostic and treat each deformation independently.  
        
        Cubic  deformations  have been studied  
        both using  general relativity approaches <cit.>, as well as 
        amplitude methods applied to the case of spinless binaries <cit.>, or systems where only one black hole is spinning and has a  much larger mass than the other black hole  <cit.>.
        Related work on other modified theories of gravity include <cit.>. We also note that in <cit.> a bound of ℓ_ EFT≤ 38.2 km was determined for the fundamental length scale of cubic theories, specifically for the  case where β_1 =β̃_1, β_2 =β̃_2=0. Furthermore,   <cit.> investigated the potential  detectability of cubic deformations of gravity in a gravitational-wave event from the merger of two stellar-mass black holes. 
        
        In a typical encounter, we expect both celestial bodies to have non-zero spin vectors, along with arbitrary masses, and this is the scenario we considered in <cit.>. 
        In that paper we focused   on  elastic processes, related to the two-to-two scattering amplitude of Kerr black holes,  and   using the KMOC formalism <cit.> we computed the leading-order  impulse (or momentum kick) and spin kick in a generic   hyperbolic encounter. These observables  vanish at tree level, and hence a one-loop calculation was required. 
        For the sake of gravitational-wave observations, the most relevant quantity to compute is the waveform, and this is what we  focus on in this paper. 
        
        
        Waveforms for hyperbolic spinning encounters in general relativity were derived to leading order using a worldline quantum field theory approach in <cit.> for the spinless case, reproducing the classic results of Kovacs and Thorne <cit.>. This was later extended to include spinning bodies in   <cit.>, with results valid  up to quadratic order in the spin. Results for the scattering waveform of spinless bodies at next-to-leading order were later derived using amplitude approaches in <cit.>,  while for spinning objects, 
        leading-order expressions 
        valid to high order in the spins were obtained in <cit.>, also using amplitude methods,   which could readily be extended to any spin order once the relevant Compton amplitudes are available. 
        
        
        In order to derive the waveforms  we will follow the adaptation of the KMOC approach presented in <cit.>. In that paper it was shown that, to leading order in the coupling, gravitational  waveforms can be  expressed as a Fourier transform to impact parameter space of the five-point amplitude describing the scattering of two spinning celestial objects with the emission of a graviton. In fact, only the physical singularities of this amplitude contribute to the waveform; as we approach the poles, the five-point amplitude factorises onto a four-point Compton amplitude and a three-point amplitude of two spinning objects with one graviton. 
        Compact expressions for  the Compton amplitude in the presence of cubic deformations were derived  in <cit.>, and we will use them to derive all-order in spin expressions for the tree-level waveforms of two Kerr black holes in a cubic background.
        
        
        After constructing the integrand from factorisation, we are left with the task of taking its Fourier transform. We will do this using three different approaches, which will give us the possibility of comparing their own merits. The first approach  is that of direct integration, building on  the seminal work of <cit.>.
        In this method, it is possible to write down ansätze for certain tensor (i.e. not necessarily scalar) integrals which are invariant under rescaling of the integration variable. Such integrals often contain poles on the integration contour and, for the rescaling argument to be valid, they  must  be regularised with a principal value prescription. This is possible  since  additional contributions arising from the   i ε prescriptions 
         are free of physical poles and hence cancel, as we have checked explicitly.  
         Higher-rank tensor integrals can then be produced efficiently with a generating function technique which is applicable thanks to the presence of a Fourier transform to impact parameter space. Effectively, integrals with  (loop-momentum dependent) numerators can be obtained by differentiating a master integral with a shifted impact parameter with respect to certain scalar auxiliary variables. This approach avoids  differentiation with respect to   four-momenta as well as lengthy Lorentz contractions, thereby leading to very compact results. 
        
         A second independent approach is based on integration by parts (IBP) reductions and the method of differential equations <cit.>. The waveforms corresponding to I_1 and its parity-odd version Ĩ_1
        contain spurious poles of the form (sinh x)/x, which we avoid by rewriting this function in an integral representation that is well suited for IBP reductions.  As a consequence, the Fourier transforms in the I_1 or Ĩ_1 cases are reduced to a single  simple master integral. We also note that this approach is systematic and  can be extended to one-loop waveforms with or without spin.  Finally, a third approach we  pursued employs Cauchy's residue theorem. This is straightforward to implement and highly  efficient, though it generally produces less compact  analytic results  compared to  the   other two methods, with  direct integration   yielding  the most compact results. 
        The three approaches give results in 
        perfect agreement. 
        
        Remarkably, we are able to write the spinning waveforms in terms of  Fourier transforms to impact parameter space where most (and, for the G_3 and G̃_3 cases, all)
        of the spin dependence is encoded in spin-dependent shifts of the impact parameter. These  shifts are reminiscent of the Newman-Janis shift,  which made an appearance  in amplitude contexts in <cit.>,  
        and more recently in the computation of spinning waveforms in <cit.>. 
        For the G_3 and G̃_3 waveforms, we are able to write the spinning waveforms entirely in terms of non-spinning ones (with shifted impact parameter). 
        
        The case of waveforms in the presence of parity-odd deformations is very easy to discuss and in fact does not require new calculations: parity-odd waveforms can be obtained from parity-even ones by simply swapping the “plus” and “cross” polarisations.  
        Investigating parity-odd deformations of gravity  is not merely an academic exercise, given the early 
         indications of parity violation in  the Cosmic Microwave Background and  the  
        large-scale structure of galaxies <cit.>.  
        
        A feature of all the leading waveforms in the presence of cubic deformations is that they do not modify the gravitational memory, which is easily proven using the connnection between the memory and soft theorems <cit.>. 
        
        
        Our  waveform results can be found in our   
        https://github.com/QMULAmplitudes/Cubic-Corrections-to-Spinning-Observables-from-Amplitudes Cubic Corrections to Spinning Observables from Amplitudes GitHub repository.  
        
        The rest of the paper is organised as follows. 
        In Section <ref> we summarise the expressions for the classical Compton amplitudes derived in <cit.>. 
        In Section <ref> we present the relevant formulae to derive the waveforms from  the factorisation diagrams of the five-point amplitudes. We also construct these factorisations using the classical Compton amplitudes and the classical three-point amplitudes of two spinning particles and one graviton. 
        In Section <ref> we present the direct integration method of the waveform, inspired by the work of <cit.>. The method is illustrated in great detail in the Appendices, while in this section  we use the results of the integrals to derive the  final expressions for the  waveforms with  G_3 and I_1 deformations. 
        In Section <ref> we present our second alternative derivation of the waveforms using a  systematic method proposed in <cit.> for the tensor integral generating functions <cit.>.    
        A third derivation of the waveforms is then briefly shown  in 
        Section <ref> using residues, much in line with our previous work <cit.>. Section <ref> illustrates the waveforms for various values of the relative velocities of the black holes and their spins, for the G_3 and I_1 deformations. These waveforms  clearly  show the absence of a  contribution to the gravitational memory, which we   demonstrate  in 
        Section <ref>. 
        Finally, in 
        Section <ref>  
        we conclude by comparing the size of cubic corrections to that of Post-Minkowskian (PM) corrections. 
        
        
        Two Appendices deal with the evaluation of  the  integrals needed to compute our waveforms. Specifically, in 
        Appendix <ref> we evaluate the necessary master integrals using a combination of explicit evaluations and educated guesses; and in 
        Appendix <ref> we present the generating function technique described earlier, which allows to compute with great economy all the necessary higher-tensor integrals. 
        
        
        
        
        § SUMMARY OF CLASSICAL SPINNING COMPTON AMPLITUDES WITH CUBIC DEFORMATIONS
        
        
        
        In this section we briefly review the Compton amplitudes in the presence of parity-even and parity-odd cubic deformations derived in <cit.>. 
        We denote by k_i and ε_k_i, i=1,2, the momenta and polarisations of the emitted gravitons, while m and a are the mass and ring radius of the black hole, with p being the classical momentum.  
        
        
        
         §.§ Parity-even deformations
        
        For the parity-even cubic deformations, the classical Compton amplitudes 
        in the gauge k_1 ε_k_2 = k_2 ε_k_1 =0 
        are found to be
        [From now on,   we will omit the coefficients of the cubic interactions β_1, β_2, β̃_1, β̃_̃2̃, which  can  be reintroduced at the end of the calculations. Furthermore, we  note that in our normalisations Newton's  constant is defined as   G κ^2 /(32π).]<cit.>
            M_I_1(p, k_1, k_2)    = i ( κ/2)^4   24 (k_1  k_2)^2 (ε_k_1ε_k_2) 
                {cosh ( a q ) [ 2 (pε_k_1) (pε_k_2) - m^2 (ε_k_1ε_k_2)  ]
                -i sinh (a q)/a  q[ 
            (pε_k_1 ) ϵ( ε_k_2 p a q) +(pε_k_2 ) ϵ( ε_k_1 p a q) 
            ] }  ,
        
            and
        
            M_I_2(p, k_1, k_2)    =
                i   ( κ/2)^4 6   (k_1  k_2)  (ε_k_1ε_k_2)^2  
                {cosh ( a q ) [ ( p (k_1 - k_2))^2 + m^2 (k_1  k_2)   ]
                -2i  sinh (a q)/a  q p (k_1 - k_2) ϵ( k_1 k_2  p a)  
            ] }  ,
        
            while 
            
            M_I_1(p, k_1^±±, k_2^∓∓)= M_I_2(p, k_1^±±, k_2^∓∓)=0 
                     ,
        
                where 
                q=k_1 + k_2.
                
        The amplitudes  (<ref>) and (<ref>) can also be rewritten in spinor-helicity variables:
        [Our spinor conventions are the same as in  <cit.>.]
            M_I_1(p, k_1^++, k_2^++)    = 
                i( κ/2)^4 3! [1  2]^4/q^2{ - 4 cosh(a q)   (p k_1)(p k_2) 
                +1/2p(k_1-k_2) sinh (a q)/a  q( [1|p|2⟩ [2|a|1⟩ - [2|p|1⟩ [1|a|2⟩) 
             }  ,
        
            and 
            
            M_I_2(p, k_1^++, k_2^++)    = i/2( κ/2)^4   3! [1  2]^4/q^2{cosh (a q) [ ( p (k_1 - k_2))^2 + m^2 (k_1  k_2)   ]
                        
               +1/2p(k_1-k_2) sinh (a q)/a  q( [1|p|2⟩ [2|a|1⟩ - [2|p|1⟩ [1|a|2⟩)  } .
        
        In the following we will focus on the G_3 deformations instead of I_2. The corresponding classical Compton amplitude in spinor-helicity variables is  
        
            M_G_3(p, k_1^++, k_2^++) = -3   i m^2  ( κ/2)^4 cosh (a q) [1  2]^4   
                .
        
        Note that this amplitude has no term proportional to (sinh a q) /(a q).
        
        
        
         §.§ Parity-odd deformations
        
        
        Finally, as discussed in <cit.>
        the amplitudes with parity-odd deformations can be obtained easily from the parity-even  ones:  for the case of positive (negative)  helicity, they are equal to those for the corresponding parity-even case multiplied by a factor of  +i (-i), and we recall that the only non-vanishing amplitudes are those where the two gravitons have the same helicity.  
        
        
        
        
        § CONSTRUCTING THE TIME-DOMAIN WAVEFORMS
        
        
         §.§ General results
        
        
        The derivation of the time-domain waveforms using the KMOC formalism was presented  in <cit.>.  
        We will work in the  far-field limit, that is at large observer distance r|x⃗|  and time t with fixed retarded time u t-r.  
        The quantity of interest  is then 
        
            ⟨ h_+± i h_×⟩   ⟨ h_μν^ out⟩^μν_(±±)1/4π r (h_+^∞± i h_×^∞)   ,
        
        for which one finds that
        [See for example Section 5 of <cit.> for a detailed derivation.]
            h^∞_+± i  h^∞_×  =    
              
              κ∫_-∞^+∞dω/2π e^-i ω u 
             [ 	 
            		 
               θ(ω) .W(b, k^±)  |_k= ω (1, 𝐱̂)
             +   
             θ(-ω) . W^∗(b,k^∓) |_k= -ω (1, 𝐱̂)]
            		,
        
        where  W=W(b⃗, k; h)  is the spectral waveform for the emission of a graviton of momentum k and helicity h, 
        which satisfies,  at tree level, 
            
            .    W^∗ (b, k^h)|_k=-ω(1, 𝐱̂) = 
            .    W(b, k^-h)|_k=ω(1, 𝐱̂)  .
        
        Using this result, one can easily combine  the two terms in (<ref>) into 
        
            h^∞_+± i  h^∞_× = 
            κ∫_-∞^+∞dω/2πe^- i ω u. 
            W(b, k^±)|_k=ω(1, 𝐱̂)  .
        
        At leading order (tree level) this can be neatly obtained from the  five-point classical bremsstrahlung amplitude ℳ_5(q_1, q_2, a_1, a_2; h) describing the scattering of two objects with ring radii a_1 and a_2 with the emission of a graviton with momentum k= q_1 + q_2 and helicity h: 
        
            W(b, k^h )   -i ∫dμ^(4)  e^i(q_1 b_1 + i q_2  b_2)  ℳ_5(q_1, q_2, a_1, a_2; h)   ,
        
        where  
        
            dμ^(D)d^Dq_1/(2π)^D-1d^Dq_2/(2π)^D-1  (2π)^D  δ^(D) (q_1 + q_2 - k) δ(2  _1 q_1 ) δ(2  _2 q_2 )  ,
        
        and   q_1,2= p_1,2- p_1,2^' are the momentum transfers. The barred variables 
        <cit.>  are defined as 
        
            p_1    = p̅_1 + q_1/2  ,      p_1^' = p̅_1 - q_1/2  , 
            
            		p_2    = p̅_2 +  q_2/2  ,      p_2^' = p̅_2 - q_2/2  ,
        
        and satisfy  
            p̅_1 q_1 =p̅_2 q_2 =0   .
        
        However, at leading order we can actually drop the distinction between p̅_i and p_i, hence we will henceforth drop bars. Furthermore, as in <cit.>,
        in order to simplify the notation  we will  work with the quantity 
        
            h^∞(u)    κ∫_-∞^+∞dω/2πe^- i ω u. 
            W(b, k)|_k=ω(1, 𝐱̂)
               = -iκ∫_-∞^+∞dω/2πe^- i ω u∫d^4q_1/(2π)^2δ(2  p_1 q_1 ) δ(2  p_2 (k-q_1) )  e^i(q_1 b_1 + q_2  b_2) ℳ_5   ,
        
        where we omitted the helicity dependence, 
        performed the q_2 integration using momentum conservation from (<ref>)
        and defined
        
            kωk̂ =ω(1, 𝐱̂)  .
        
        This is the main formula we will use. We also note that b b_1 - b_2 is the asymptotic impact parameter, which  satisfies 
        
            b v_1 = b v_2 = 0  .
        
        
         §.§ Kinematics of the five-point scattering
        
        
        The time-domain waveform (<ref>) requires knowledge of the classical part of the five-point bremsstrahlung amplitude, that is the amplitude of two spinning particles with  one emitted graviton. 
        The kinematics of the process is shown below: 
        
            [ [scale=12,baseline=([yshift=-1mm]centro.base)]
            			00 at (0+,0+) (centro) ;
            			 at (-3pt+,-3pt+) (uno) p_1;
            			 at (-3pt+,3pt+) (due) p_2;
            			 at (3pt+,3pt+) (tre) p_2^';
            			 at (3pt+,-3pt+) (quattro) p_1^';
            			 at (5.55pt+,) (cinque)
            			
            			k= q_1 + q_2;
            
            			 [thick,double,orange] (uno) – (centro);
            			 [thick,double,amber] (due) – (centro);
            			 [thick,double,amber] (tre) – (centro);
            			 [thick,double,orange] (quattro) – (centro);
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            			 [shading=radial] (centro) circle (1.5pt);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ]
        
        The  two Kerr black holes  have  masses m_1 and m_2  with 
        
            p_1^2 = (p_1^')^2 = m_1^2  ,       p_2^2 = (p_2^')^2 = m_2^2  .
        
        It is also convenient to  introduce four-velocities v_1 and v_2 as
        
            p_1  m_1 v_1  ,       p_2  m_2 v_2  ,
        
        so that  v_1^2 = v_2^2 = 1. The two black holes have ring radii a_1 and a_2, where for a single black hole we recall that the ring radius a^μ is related to the spin vector as a^μ S^μ/ m <cit.>, where 
        
            S^μ = 1/2 mϵ^μναβ p_ν S_αβ ,
        
        with S^μν being the spin tensor. 
        The ring radius satisfies the covariant spin supplementary condition    p a(p) =0. 
         Finally, we also define the Lorentz factor
        
            σ v_1  v_2  ≥ 1 .
        
        Note that σ=1/√(1-ẋ⃗̇^2), with  ẋ⃗̇ being the relative velocity of one of the two black holes  in the rest frame of the other. For instance, in  the rest frame of particle 1 we can write  v_1^μ=(1,0,0,0) and   v_2^μ=σ(1, ẋ⃗̇). 
        
        
        
         §.§ Waveform integrands
        
        
        An important simplification  in the calculation of the waveforms  consists in the fact that only the residues on the physical factorisation channels  are needed in order to obtain  the waveform. These can  be computed from the two factorisation diagrams 
        
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                         [right=0.6cm of cutLt] (cutL);
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            			;
        
        which  correspond to the two possible factorisations as q_1^2→ 0 or q_2^2→ 0, with q_1 + q_2 = k. 
        In each diagram, the key ingredients are the Compton amplitudes for two spinning objects, which were computed for the parity-even and parity-odd cubic deformations  in <cit.> and are quoted in Section <ref>. Using these, we now construct the integrands for the  parity-even and parity-odd cubic deformations of gravity. 
        
        For simplicity, from now on we focus on the case of an emitted graviton with positive helicity. The two factorisation diagrams are then given by 
        
            _I, q_1^2    = i/q_1^2 M_3(-p_1, q_1^–) M_I(-p_2, - q_1^++, k^++)    , 
            _I, q_2^2    = i/q_2^2 M_3(-p_2, q_2^–) M_I(-p_1, - q_2^++, k^++)   ,
        
        with  
        
            _5, I  = _I, q_1^2+ _I, q_1^2 +  contact  terms  .
        
        Here  M_I denotes  the Compton amplitude in the presence of a deformation I,  with I∈ (I_1, G_3, Ĩ_1, G̃_3), as given in Section <ref>. 
        M_3 is  
        a classical three-point amplitude with one graviton of momentum q and two massive 
        spinning particles with momenta p and -(p+q) and ring radius a. For the two possible helicities of the emitted gravitons one has <cit.>
            M_3 (p, q^++) = -i ( κ/2) e^a q(
                ⟨ξ | p | q]/⟨ξ q⟩)^2  ,      
                M_3 (p, q^–) = -i ( κ/2) e^-a q(
                ⟨ q  | p | ξ ]/[ q ξ] )^2  ,
        
        where ξ and ξ̃ are reference spinors. 
        Finally, we observe that the second factorisation diagram in (<ref>) can be obtained from the first as 
        
            _I, q_2^2 = ._I, q_1^2|_1↔2  ,
        
        where 1↔2 means swapping  the masses, spin vectors and momenta, in addition to also exchanging q_1 and q_2.
        
        
        With these ingredients it is very easy to construct the integrand of the waveforms. For the case of the G_3 deformation,  and choosing as reference spinor of the three-point amplitude the spinor of the external graviton, we get 
        
            _G_3, q_1^2 = - i (κ/2)^53  m_1^2 m_2^2/q_1^2  [k| v_1 q_1 | k]^2  e^- a_1  q_1cosh( a_2 (q_1 - k) )  .
        
        For I_1, the result is
        
            _I_1, q_1^2   =  -i (κ/2)^53  m_1^2 m_2^2/q_1^2[k| v_1 q_1 | k]^2/ q_1 ke^- a_1  q_1×
               [ - 4 cosh [ a_2 (k-q_1)] (-v_2 q_1) (v_2  k) - 2 i  v_2  (k+q_1) sinh a_2 (k-q_1)/a_2 (k-q_1)ϵ ( k v_2 q_1 a_2)]  .
        
        Note that because of the first  δ-function in (<ref>)
        we can simplify the above expression using v_1 q_1=0 into 
        
            _I_1, q_1^2   = -  i (κ/2)^512    m_1^2 m_2^2/q_1^2  (q_1 k)  
              [k| v_1 q_1 | k]^2(v_2  k )  e^- a_1  q_1×
               [   cosh [ a_2 (k-q_1)]  (v_2  k) -  i sinh a_2 (k-q_1)/a_2 (k-q_1)ϵ ( k v_2 q_1 a_2)]  .
        
        
        § TIME-DOMAIN WAVEFORMS FROM DIRECT INTEGRATION 
        
        
        There are several approaches to explicitly perform the integrations in (<ref>). 
        In this section  we follow  the direct integration approach of <cit.>, while in Section <ref> we present  an alternative derivation  based on Cauchy's  residue theorem. 
        
        
        The first step in the direct integration method consists in 
        performing the ω integration
        in (<ref>) using
        the delta function δ (2 p_2  (k-q_1)). This localises the graviton energy ω to the value ω^∗ given by  
        
            ω^∗ = q_1 v_2/k̂ v_2  ,
        
        where we recall from (<ref>) that 
           kωk̂.
        One then quickly arrives at <cit.>
            h^∞(u) = - i κ/2 m_1 m_2 (k̂ v_2)∫d^4 q_1/(2π)^3δ(2 q_1  v_1) e^i q_1b̃._5|_k = ω^∗k̂  ,
        
        with the following definitions: 
        
            b̃   b̃_1 - b̃_2  ,  
                
            b̃_i     b_i + u_i v_i  ,      
               u_i  u - k̂ b_i/k̂ v_i  ,      i=1,2.
        
        We note that the modified (shifted) impact parameter b̃ has the useful  property that 
        
            k̂b̃=0  ,
        
        since k̂b̃_i =u. We are then left to perform integrations of the form (<ref>). For clarity we have derived and collected all required  master integrals in Appendix <ref> 
        and <ref>, where a detailed description of the direct integration method is included. 
        
        
        
        
        
        
        
         
        
        
        
         §.§ The G_3 waveform
        
        We now discuss the G_3 waveform. 
        The relevant integral is (<ref>), where we can replace the five-point amplitude with the contributions of the two poles,  _G_3, q_1^2, _G_3, q_2^2 and discarding contact terms (see (<ref>)).   
        To proceed, we first rewrite  
        
            |k] √(ω) |k̂]  ,
        
        and recall that ω is localised at the q_1-dependent value ω^∗ given in (<ref>), and where | k̂] is ω-independent. 
        We also rewrite the cosh term in M_G_3(-p_2, - q_1^++, k^++) in terms of exponentials, with  the net effect of shifting the impact parameters. Doing so, we obtain for  
        the contribution to the time-domain waveform  arising from the first diagram in (<ref>), 
        
            h^∞, (1)_G_3 (u) = - i κ/2 m_1 m_2 (k̂ v_2)∫d^4 q_1/(2π)^3δ(2 q_1  v_1) e^i q_1b̃._G_3, q_1^2|_k = ω^∗k̂
               =- 3/2(κ/2)^6
                m_1 m_2/(k̂ v_2)^3∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^2/q_1^2[ e^i q_1  [  b̃ +  i (ã_1 - ã_2) ] + e^i q_1  [  b̃ + i (ã_1 + ã_2)]] 
                = - 3/2(κ/2)^6
                m_1 m_2
                /(k̂ v_2)^3 X_μ X_ν v_2 α v_2β[ 
                .^μναβ|_b̃→b̃ + i(ã_1 - ã_2) + .^μναβ|_b̃→b̃+ i(ã_1 + ã_2)] 
                = - 3/2(κ/2)^6
                m_1 m_2
                /(k̂ v_2)^3[ 
                .^(1)|_b̃→b̃ + i(ã_1 - ã_2) + .^(1)|_b̃→b̃+ i(ã_1 + ã_2)]  ,
        
        where we used the expression for _G_3, q_1^2 derived earlier in (<ref>). We also  defined
        
            X_μ = [k̂| v_1 σ_μ |k̂]   ,
        
        and
        
            ã_i   a_i - v_i a_ik̂/v_ik̂ ,
        
        which allows us, using (<ref>), to rewrite
        [In writing (<ref>) we have also used that q_1 v_1=0 and replaced a_1 q_1 with ã_1 q_1. This is useful in order to maintain the symmetry between the two factorisation diagrams in (<ref>) so that the second can easily be obtained from the first upon performing suitable replacements, see (<ref>). 
        ]
            a_2  (k-q_1) = -ã_2  q_1  .
        
        The definitions of the tensor integrals ^μ_1 …μ_n can be found in (<ref>). 
         
        The integrals in the second line of (<ref>) can be computed elegantly using the method of generating functions introduced in Appendix <ref>. In particular  (<ref>) and (<ref>) are relevant for this case, with the result
        
            ^(1)   =∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^2/q_1^2 e^i q_1 b̃
               = -1/8 π(∂^4/∂^2 t_1 ∂^2 t_21/|b̃_(1)+t_1 v_2(1) + t_2 X_(1)|)_t_1=t_2=0 .
        
        Here the subscript (1)  indicates the projection imposed by the delta function and implemented by the projector 
        
            P_1^μν=η^μν-v_1^μ v_1^ν ,
        
        which allows us to define
        
            V^μ_(1) = P_1^μν V_ν ,
        
        and
        
            |V_(1)| = √(-V_(1) V_(1))
            = √(-V  P_1  V) .
        
        We also note that X^μ_(1) = X^μ.
        
        With these definitions and performing the derivatives in (<ref>) we find
        
            ^(1)     = -3/8 π|b̃_(1)|^5[ 
            2 (X  v_2 (1))^2
                
            + 5/|b̃_(1)|^2(v_2(1)^2 (X b̃_(1))^2 
            + 4 (X  v_2 (1)) 
            (X b̃_(1))(b̃_(1) v_2(1))
            +7 (b̃_(1) v_2)^2/|b̃_(1)|^2 (X b̃_(1))^2)
            ]  ,
        
        which can be simplified further into
        [We have also rewritten b̃_(1) v_2(1) as b̃_(1) v_2, since for any two vectors m^μ and n^μ we have m_(1) n_(1) =m n_(1) = m_(1) n, with m_(1)^μ P_1^μν m_ν, with the projector P_1 defined in (<ref>).]
            ^(1) = -3/8 π[ 
            2 (X  v_2)^2/|b̃_(1)|^5  
            + 5 (X b̃)/|b̃_(1)|^7(v_2(1)^2 (X b̃) 
            + 4 
            (b̃_(1) v_2) (X  v_2) 
            +7 (b̃_(1) v_2)^2/|b̃_(1)|^2 (X b̃))
            ]  .
        
        Alternatively, we can evaluate the third line of (<ref>) by twice differentiating (<ref>), which gives  
        
            ^μναβ(b̃)     = - 3/8 π|b̃_(1)|^5[ 
                ( P_1^μν P_1^αβ + P_1^μα P_1^νβ + P_1^να P_1^μβ) 
                
               + 5/|b̃_(1)|^2  (
                P_1^μνb̃_(1)^αb̃_(1)^β + P_1^μαb̃_(1)^νb̃_(1)^β + P_1^ναb̃_(1)^μb̃_(1)^β
                +
                P_1^αβb̃_(1)^μb̃_(1)^ν + P_1^μβb̃_(1)^αb̃_(1)^ν + P_1^νβb̃_(1)^αb̃_(1)^μ)
                
                + 35/|b̃_(1)|^4   b̃_(1)^μb̃_(1)^νb̃_(1)^αb̃_(1)^β]   ,
        
        where the projector P_1^μν is defined in (<ref>), 
        
        b̃_(1)^μ in (<ref>) and 
        |b̃_(1)| 
        in (<ref>). 
        
        Note that all the terms proportional to P_1^μν in 
        (<ref>) do not contribute to  the contractions in (<ref>).
        Pleasingly, we find complete agreement of the contractions of this tensor integral with the result obtained from the generating function,  
        
            X_μ X_ν v_2 α v_2β ^μναβ(b̃) = ^(1) .
        
        We then have 
        
            h^∞, (1)_G_3 (u)  
                    = - 3/2(κ/2)^6
                m_1 m_2
                /(k̂ v_2)^3[ 
                .^(1)|_b̃→b̃ + i(ã_1 - ã_2) + .^(1)|_b̃→b̃+ i(ã_1 + ã_2)] 
                  .
        
        We  note that effectively we have rewritten the spinning waveform in terms of the scalar waveform, but with shifted impact parameters: 
        
            h_G_3^∞, (1) (u) =1/2[ . h_G_3^∞, (1) (u) |_a_i=0,  b̃→b̃+ i (ã_1 + ã_2)  +  . h_G_3^∞, (1) (u) |_a_i=0,  b̃→b̃+ i (ã_1 - ã_2)]   .
         
        We call the two contributions of these two terms as arising from two “sectors”, with shifted impact parameter
        
            (1)               (2)        
            b̂ = b̃ + i (ã_1+ ã_2)        b̂ = b̃ + i(ã_1- ã_2)   ,
        
        where we recall that ã_i was defined in (<ref>). 
        The result for I_1 will not have such a simple form, but it can be expressed in terms of the same two sectors. 
        
        Finally, the  complete waveform, including contributions from both factorisation channels, is 
        
            h^∞_G_3 (u) = h^∞, (1)_G_3 (u) + h^∞, (2)_G_3 (u)  ,
        
        where
        
            h^∞, (2)_G_3 (u) = .h^∞, (1)_G_3 (u)|_(m_1, a_1,  v_1, b_1)↔ (m_2, a_2,  v_2, b_2)  .
        
        We also observe  that under this exchange the variable b̃ defined in (<ref>) changes sign, b̃↔  - b̃. 
        
        
        
        
        
        
        
        
        
        
        
         §.§ The I_1 waveform
        
        
        In the presence of the I_1 deformation, the time-domain waveform  is 
        
            h^∞, (1)_I_1 (u) = - i κ/2 m_1 m_2 (k̂ v_2)∫d^4 q_1/(2π)^3δ(2 q_1  v_1) e^i q_1b̃._I_1, q_1^2|_k = ω^∗k̂  ,
        
        where we have, from (<ref>),
        
            ._I_1, q_1^2|_k = ω^∗k̂   =   -  i (κ/2)^512    m_1^2 m_2^2  (ω^∗)^3/q_1^2  (q_1k̂)  
              (X  q_1)^2(v_2 k̂ )  e^- a_1  q_1×
               [   cosh [ a_2 (ω^∗k̂-q_1)]  (v_2 k̂) -  i sinh a_2 (ω^∗k̂-q_1)/a_2 (ω^∗k̂-q_1)ϵ ( k̂ v_2 q_1 a_2)]  ,
        
        with ω^∗  given in (<ref>). 
        Therefore we obtain  
        
            h^∞, (1)_I_1 (u)
                = - (κ/2)^6 
            6m_1 m_2/(k̂ v_2)^3∫d^4q_1/(2π)^3δ(2 q_1 v_1) (X  q_1)^2  (q_1 v_2)^3/q_1^2  (q_1k̂)×
               [ ( v_2 k̂ + i ϵ(k̂ v_2 q_1 a_2)/q_1 ã_2)  e^i q_1  [  b̃ +  i (ã_1 + ã_2) ] + ( v_2 k̂  - i ϵ(k̂ v_2 q_1 a_2)/q_1 ã_2)  e^i q_1  [  b̃ +  i (ã_1 -  ã_2) ]]
                  ,
        
        where the shifted spin vectors  ã_i were defined in (<ref>). 
        
        We now  derive using the  method of generating functions presented in Appendix <ref>
        the two master integrals  needed to evaluate the waveform integral  in  (<ref>), corresponding to the first factorisation diagram. 
        Before doing so, we  comment that 
        its integrand   contains the spurious pole 1/ (q_1k̂), which originates from the propagator i/(q^2 + i ε) =  i / ( -2 q_1k̂ + i ε) in the Compton amplitude (<ref>) onto which the five-point amplitude factorises. We have checked explicitly that this pole is cancelled by a corresponding spurious pole in the second factorisation diagram, and have double-checked this conclusion by a detailed comparison with the full five-point amplitude obtained from a diagrammatic computation. In practice it is convenient to perform the integrations of the two factorisation diagrams separately, which requires a choice of regularisation of the pole. The sum of the two diagrams should be independent of the  regularisation, since the pole is spurious; 
        we have confirmed this expectation by performing the integration with either the Feynman i ε prescription or the Principal Value (PV)  prescription,  finding complete agreement.
        [In fact, using the PV prescription the spurious contributions vanish separately in each diagram; this is related to the tracelessness of the integral 𝒦^μν
        introduced in (<ref>), see end of Appendix <ref> for a thorough discussion of this point.
        ]
        In the following we will choose the PV prescription, and for completeness    we  give in Appendix <ref> the master integral 𝒥^μ in (<ref>) with both prescriptions (while for 𝒦^μν we only present the result in the PV prescription).
        
        The first type of integral we need is
        
            ^(1)   ∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^3/q_1^2 (q_1 k̂) e^i q_1 b̃ , 
               = [  ∂^4/∂^2 t_1 ∂^2 t_2(
             𝒥 v_2 )_b̃→b̃+t_1 v_2 + t_2 X]_t_1=t_2=0
                = 
            [  ∂^4/∂^2 t_1 ∂^2 t_2(v_2  K_1 k̂/8 π |b̃_(1)|  [(v_1 k̂)^2 |b̃_(1)|^2-(k̂b̃_(1))^2 ])_b̃→b̃+t_1 v_2 + t_2 X]_t_1=t_2=0  ,
        
        where we have used (<ref>) and 
        K_1^μν = |b̃_(1)|^2 P_1^μν + b̃_(1)^μb̃_(1)^ν from (<ref>).
        
        The second type is
        
            ℰ^(1)   ∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^3 ϵ(k̂ v_2 q_1 a_2)/q_1^2 (q_1 k̂)(q_1 ã_2) 
            e^i q_1 b̃
               = -[  ∂^4/∂^2 t_1 ∂^2 t_2(Y 𝒦 v_2 )_b̃→b̃+t_1 v_2 + t_2 X]_t_1=t_2=0 ,
        
        with 
        
            Y 𝒦 v_2 = 
              - ( ã_2  K_1k̂) (Y  K_1 v_2) - (ã_2  K_1 Y) (k̂ K_1 v_2)-
            (ã_2  K_1 v_2) (k̂ K_1 Y)/8 π  |b̃_(1)| ( ã_2  K_1ã_2 ) ( k̂ K_1k̂) ,
        
        where we have used
        Y^μϵ(k̂ v_2μ a_2), as well as (<ref>) and (<ref>).
        
        Putting everything together, we find that (<ref>) evaluates to
        
            h^∞, (1)_I_1 (u)
                = - (κ/2)^6 
            6m_1 m_2/(k̂ v_2)^3×
               [ ( (k̂ v_2) ^(1)+ i ℰ^(1))_b̃→b̃ +  i (ã_1 + ã_2) + 
            ( (k̂ v_2) ^(1)- i ℰ^(1))_b̃→b̃ +  i (ã_1 - ã_2)]
                  .
        
        As for the G_3 case, the complete I_1 waveform is 
        
            h^∞_I_1 (u) = h^∞, (1)_I_1 (u) + h^∞, (2)_I_1 (u)  ,
        
        where
        
            h^∞, (2)_I_1 (u) = .h^∞, (1)_I_1 (u)|_(m_1, a_1,  v_1, b_1)↔ (m_2, a_2,  v_2, b_2)  .
        
        Finally, we  note the appearance in (<ref>) of shifted impact parameters b̃ +  i (ã_1 ±ã_2),  similarly to (<ref>) in the G_3 case. However there is an important difference with that case: while  the G_3 waveform can be obtained from the scalar  integrated waveform by performing shifts in the impact parameter as dictated by the sectors, in the I_1 case these shifts occur at the level of the  integrand. For the cosh part of the I_1 integrand they translate to shifts that can be performed on the integrated result; however the part of the I_1 proportional to the sinh function vanishes in the scalar case, and thus provides a new contribution that cannot be obtained from the spinless waveform.  
        
        
        
        
        
        
         §.§ Waveforms for parity-odd deformations
        
        
        To derive the waveforms for parity-odd deformations we only need to recall from Section <ref> that the Compton amplitudes for parity-odd deformations are obtained from the parity-even ones by multiplying them by a factor of ± i depending on whether the two gravitons have positive or negative helicity. The same property is inherited by the cut five-point amplitudes which we use to construct the waveforms, that is 
        
            -i _5^ P.O. (k^+) = i (-i _5^ P.E.(k^+))  ,     (-i _5^ P.O. (k^-))^∗ = i (-i _5^ P.E.(k^-))^∗  ,
        
        from which it follows that 
        
            W^ P.O. (k^+) = i W^ P.E.(k^+)  ,     (W^ P.O. (k^-))^∗ = i (W^ P.E.(k^-))^∗  ,
        
        where P.O. and P.E. stand for parity odd and even, respectively.
        Combining (<ref>) with (<ref>) 
        we see that 
        
            .    (W^ P.O. (k^-))^∗|_k = - ω (1,𝐱̂)  = .W^ P.O. (k^+)|_k =  ω (1,𝐱̂)  = 
            .i  W^ P.E. (k^+)|_k = ω (1,𝐱̂)   .
        
        The two terms in  
        (<ref>), can then be combined as in the parity-even case, 
        and hence we conclude that 
        
            (h^∞,  P.O._+ + i    h^∞,  P.O._×)    = 
                κ∫_-∞^+∞dω/2πe^- i ω u. 
            W^ P.O.(b, k)|_k=ω(1, 𝐱̂)
               =
                i   (h^∞, P.E._+ + i    h^∞,  P.E._×)  .
        
        This implies that  
        
            h^∞,  P.O._+ = -    h^∞,  P.E._×  ,       
                h^∞,  P.O._× = h^∞,  P.E._+  ,
        
        that is,  the “plus” and “cross” polarisations are then swapped  in the way prescribed by (<ref>).
        
        
        
        § TIME-DOMAIN WAVEFORMS FROM  TENSOR INTEGRAL GENERATING FUNCTIONS, RELOADED
        
        Alternatively, in the spirit of modern multiloop amplitude calculations, we can compute the integrals by a systematic method proposed in <cit.>.
        For the tree-level waveform, we have two D-dimensional master integrals,    
        
            _1[𝐲]∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1) q_1^2  e^i q_1 b̂ ,      _2[𝐲]∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1) q_1^2 q_1k̂  e^i q_1 b̂ ,
        
        where 𝐲 (y_1, y_2, y_3,ŷ_4,ŷ_5,ŷ_6,ŷ_7), with 
        
            y_1   σ=v_1 v_2,     y_2    v_1k̂ ,    y_3    v_2k̂ , 
            ŷ_4   b̂k̂√(-b̂b̂)=0,    ŷ_5   b̂ v_1 √(-b̂b̂),    ŷ_6   b̂ v_2√(-b̂b̂),    ŷ_7b̂b̂ .
        
        One can rewrite the two master integrals as 
        
            _j[𝐲]∫_-∞^∞dt   e^it _j[𝐲, t]   :=∫_-∞^∞dt   e^it∫d^D-4q_1/(2π)^3δ(2 q_1 v_1) q_1^2 (q_1 k)^j-1δ(q_1b̂-t)
               .
        
        The differential equation in t for _j[𝐲, t] is very simple:
        
            ∂_t  _1[𝐲, t]= D-4/t_1[𝐲, t],       ∂_t  _2[𝐲, t]= D-5/t_2[𝐲, t] .
        
        It is easy to see that  the t dependence is factorised from the other variables, and we can solve for it directly as 
        
            _1[𝐲, t]= t^D-4 D-4_1[𝐲] ,       _2[𝐲, t]= t^D-5_2[𝐲]  .
        
        Then the two integrals in (<ref>) are rewritten as  
        
            _1[𝐲]   =(∫_-∞^∞dt   e^it t^-4 + D D-4)_1[𝐲]=-π _1[𝐲]  ,
            _2[𝐲]   =(∫_-∞^∞dt   e^it t^-5 + D)_2[𝐲]=iπ _2[𝐲]  ,
        
        where we have already used D = 4 - 2ϵ and taken the → 0 limit.
        The differential equations in four  spacetime dimensions  of the  𝐲-dependent part _1,2[𝐲] are 
        
            ∂_ŷ_5_1[𝐲]= -ŷ_5/ŷ_5^2+1_1[𝐲],       ∂_y_2_2[𝐲]= -1  y_2_2[𝐲].
        
        The solutions  are 
        
            _1[𝐲]=-π/(2π)^3√(-b̂b̂)√(ŷ_5^2+1),      _2[𝐲]=c_2  y_2 ,
        
        where c_2=0 from the boundary conditions. 
        Hence we can omit _2[𝐲] in the calculation and we are thus left with a single master integral. 
        
        All other integrals can be reduced to this master integral. For example,  consider the case of the integral
        
            𝒱[𝐲]   ∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1)q_1 v_2 q_1^2 q_1k̂  e^i q_1 b̂  .
        
        We can rewrite it as 
        
            𝒱[𝐲]=∫_-∞^∞dt  e^it 𝒱[𝐲, t]   =∫_-∞^∞dt   e^it∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1)q_1 v_2 q_1^2 (q_1 k)δ(q_1b̂-t)  .
        
        By  IBP reduction, the q_1-integral is reduced to the single master integral  _1[𝐲], and finally we get 
        
        
        
        
        
            𝒱[𝐲]
            	   =(-π)y_1 (y_2+ŷ_4 ŷ_5)-y_3 (ŷ_5^2+1)-ŷ_4 ŷ_6+y_2 ŷ_5 ŷ_6/y_2^2+2 ŷ_4 ŷ_5 y_2-ŷ_4^2_1[𝐲]  .
        
        We have checked that this formula is  consistent with (<ref>).   
        
        
        
        
        
        
        
        
        
         The integral in (<ref>) contains a spurious pole in the entire function (sinh x)/x (see <cit.> for general discussions). However we can eliminate it  by introducing the integral representation   <cit.>
            sinh x  x= ∫_0^1dz  cosh(z  x)  .
        
        Then there are four sectors in the waveform, with exponential factors e^i q b̂. The particular  form of the new variable b̂ depends on the  sector as follows:
        [We note that in the previous approach we employed to compute the I_1 waveform there were  two sectors, corresponding to the cases   (1) and  (2) below. We now have two additional sectors because 
         the integral representation    (<ref>) introduces a   dependence on the integration variable z.]
            (1)              (2)              (3)              (4)       
            b̂= b̃+i(ã_1+ ã_2)        b̂=b̃+i(ã_1- ã_2)       b̂= b̃+i(ã_1+ z ã_2)       b̂= b̃+i (ã_1- z ã_2)  .
        
        Then the waveform can be represented as a combination of tensor integrals:
            
            h^∞, (1)_G_3 (u)
                = - (κ/2)^6
            6m_1 m_2/(k̂ v_2)^3(1 4X^μ_1X^μ_2v_2^μ_3v_2^μ_4∑_i=1^2 ^(i)_μ_1μ_2μ_3 μ_4(y))   ,
               h^∞, (1)_I_1 (u)
                = - (κ/2)^6
            6m_1 m_2/(k̂ v_2)^3(v_2k̂ X^μ_1X^μ_2v_2^μ_3v_2^μ_4∑_i=1^2 '^(i)_μ_1μ_2μ_3 μ_4(y)
                                                  +X^μ_1X^μ_2v_2^μ_3v_2^μ_4Y^μ_5∑_i=3^4 '^(i)_μ_1μ_2μ_3 μ_4μ_5(y)) 
              ,
        
        where Y^μϵ(k̂ v_2μ a_2),  and  y (y_1, y_2, y_3, y_4, …, y_18)  denotes all the independent scalar products among external kinematic vectors v_1, v_2, k̂, ã_1, ã_2, b̃, with y_1, y_2, y_3 defined in (<ref>) and 
        
            y_4   b̃k̂√(-b̃b̃)=0,      y_5   b̃ v_1 √(-b̃b̃),     y_6   b̃ v_2√(-b̃b̃),      y_7   b̃b̃ ,  
            
                  y_8   ã_1k̂√(-b̃b̃)=0,     y_9   ã_1 v_1√(-b̃b̃),     y_10   ã_1 v_2√(-b̃b̃),      y_11   ã_1 ã_1 -b̃b̃    
             
                   y_12   ã_2k̂√(-b̃b̃)=0,      y_13   ã_2 v_1√(-b̃b̃),      y_14   ã_2 v_2√(-b̃b̃),     y_15   ã_2 ã_2 -b̃b̃    
             
                   y_16   ã_1 b̃ -b̃b̃,    y_17   ã_2 b̃ -b̃b̃,    y_18   ã_1 ã_2 -b̃b̃  .
        
        The tensor integrals are generated as  
        
            ^(i)_μ_1…μ_r(y)   = ∂_b̃^μ_1⋯∂_b̃^μ_r _1^(i)[𝐲]= ∂_b̃^μ_1⋯∂_b̃^μ_r∫_0^1dz  _1^(i)[𝐲]  ,  
            '^(i)_μ_1…μ_r(y)   = ∂_b̃^μ_1⋯∂_b̃^μ_r 𝒱^(i)[𝐲]=∂_b̃^μ_1⋯∂_b̃^μ_r∫_0^1dz 𝒱^(i)[𝐲]  .
        
        We note that one cannot use _2 to generate the tensor integral as its value is trivial. Furthermore,  it is not possible to   set y_4=0 in 𝒱^(i) and ^(i)_1 directly,  as ∂_b̃^μy_4 is nonvanishing.  The superscripts in 𝒱^(i) and ^(i)_1 denote the tensor generating function with b̂ in different sectors. Then we only need to perform the z integration for the simple cases of _1^(3,4)[𝐲] and 𝒱^(3,4)[𝐲]. By using IBP and partial fractioning, the z integral can  then  be reduced  to a basis of  three integrals:
        
            ∫_0^1 dz/𝒴,       ∫_0^1 dz/b̂b̂ 𝒴,       ∫_0^1  dz  z/b̂b̂ 𝒴  ,
        
        where  𝒴^2  -b̂b̂ ŷ_5^2-b̂b̂, which 
         are straightforward to evaluate directly.
        
        
        
        
        
        
        
        
        
        
        
        
        
        
            
            
            
            
            
        
        
        
        
        
        § TIME-DOMAIN WAVEFORMS FROM RESIDUES 
        
        A final approach, followed in <cit.>, 
        makes use of Cauchy's residue theorem. We start by rewriting the original expression for the waveform in (<ref>) as follows,
        
            h_I^∞(u) 
               = -iκ∫_-∞^+∞dω/2πe^- i ω (u-k̂_2)∫d^4q_1/(2π)^2δ(2  p_1 q_1 ) δ(2  p_2 (k-q_1) )  e^iq_1 ℳ_I   ,
        
        where k=ωk̂. Further, we  split up the waveform into contributions coming from the two cuts (<ref>), and rescale the momentum transfers as q_i=ωq̂_i,  
        
            h_I^∞, (i) (u) 
            = -iκ∫d^4q̂_1/(2π)^2∫_-∞^+∞dω/2π    ω^2 e^- i ω (u-k̂_2 -q̂_1)δ(2  p_1q̂_1 ) δ(2  p_2 (k̂-q̂_1) ) 
               ℳ_I,q_i^2(ωk̂,ωq̂_1,ωq̂_2) .
        
        The basic idea now is to rewrite the integral over q_1 as a contour integral encircling the physical poles. To do this, it is worth introducing a parameterisation which is best suited to the various sectors of the problem. 
        Explicitly, we can factor out the exponential dependence on the spin for each cut, in a similar manner to (<ref>),
        
            _I,q_1^2=e^-_1  q_1∑_l=±e^l _2 q_2_I,q_1^(l) ,    _I,q_2^2=e^-_2  q_2∑_l=±e^l _1 q_1_I,q_2^(l) .
        
        Focusing on the q_1 cut, this allows us to rewrite the waveform as 
        
            h_I^∞, (1) (u) 
               = -iκ∫d^4q̂_1/(2π)^2∫_-∞^+∞dω/2πω^2  
                δ(2  p_1q̂_1 ) δ(2  p_2 (k̂-q̂_1) ) 
                 ∑_l=±e^- i ω (u-k̂_2 -q̂_1-i_1q̂_1+ i l_2(k̂-q̂_1))ℳ_I,q_1^2^(l)(ωk̂,ωq̂_1,ωq̂_2)
               = -iκ∫d^4q̂_1/(2π)^2∫_-∞^+∞dω/2πω^4  
                δ(2  p_1q̂_1 ) δ(2  p_2 (k̂-q̂_1) ) 
                 ∑_l=±e^- i ω (_l-_l q_1)ℳ_I,q_1^2^(l)(k̂,q̂_1,q̂_2)  ,
        
        where we have extracted the ω dependence of ℳ_I,q_i^2^(l), which is simply a factor of ω^2, and we have defined
        
            _l u-k̂_2+ i   l _2k̂  ,   _l^μ^μ + i _1^μ +i   l _2^μ .
        
        To evaluate (<ref>), we parameterise the q_1 integral in each sector using <cit.>
            z_1 v_1 q_1,    z_2    v_2 q_1,     z__l_l q_1,    z_o_l o_l q_1 ,
        
            with 
            
            o_l^μϵ(v_1 v_2 _lμ ) .
        
        The Jacobian for this transformation is simply 
        
            | ∂ q_1^μ/∂ z_j |= 1/|o_l o_l| .
        
        With this, the q_1 cut contribution to the waveform becomes
        
            h_I^∞, (1) (u)   = -iκ/(4π)^2 m_1 m_2(∂/∂ u)^4 ∑_l=±∫d^4 z_j/|o_l o_l|δ(z_1) δ(z_2-v_2k̂)δ(z__l-_l) _I,q_1^2^(l)
               =-iκ/(4π)^2 m_1 m_2(∂/∂ u)^4 ∑_l=±∫_-∞^∞d z_o_l/|o_l o_l|_I,q_1^2^(l)|_z_1→0, z_2→ v_2k̂, z__l→_l .
        
        The final integral in z_o_l can be computed using Cauchy's residue theorem. There are (at most) three poles present in _I,q_1^2^(l):
        
            Physical Pole:    1/q_1^2∼1/(z_o_l-A)(z_o_l-A^⋆) ,  
               Spurious Poles:  1/q_1 k∼1/(z_o_l-B) ,  1/(k-q_1) a_2∼1/(z_o_l-C) ,
        
        where A,B,C are functions of the external kinematics. The residues in 1/q_1 k and 1/(k-q_1) a_2 must cancel upon summing over the sectors l and cuts since the poles are spurious.[ More precisely the residue in  1/(k-q_1) a_2 must cancel when summing over the sectors l in each cut. The residue in 1/q_1 k will only cancel after summing both cut contributions.] Thus, in practice, we only need to compute the residue on the physical pole 1/q_1^2. This way of computing the integrals also makes it clear why terms 
        with only spurious poles do not contribute to the waveform: they have vanishing residues. 
        
        One may check that the integral (<ref>) has no pole at infinity (after applying the ∂/∂ u derivatives) and that the integrand falls off sufficiently fast at infinity to close the contour above or below the axis. This contrasts with the case of  Einstein-Hilbert gravity <cit.> where the pole at infinity exists and must be evaluated using a principal value prescription. In fact, we could have predicted this from the start. In  <cit.>, it was shown that the pole at infinity is given by the leading soft theorem of the five-point amplitude. Our five-point amplitude is built from higher-derivative corrections to general relativity and cannot change this universal leading soft theorem. Indeed, we can check that our amplitude vanishes in the soft limit ω→ 0, which manifests in the time-domain waveform as vanishing linear memory, as discussed in Section <ref>. 
        
        To compute the contribution of the q_2 cut, we follow an identical procedure to the above but using the sectors for _q_2^2 in (<ref>). Upon summing both contributions, we obtain the full waveform for I_1 and G_3:
        
            h_I^∞ (u)= h_I^∞, (1) (u)+h_I^∞, (2) (u) .
        
        The expressions obtained using this method are lengthier than those found in Section <ref>. Reassuringly, they are in complete agreement.
        
        
        
        
        § SHOWCASE OF WAVEFORMS
        
        
         
        In this section we show several  plots of the waveforms for the  G_3 and I_1 deformations, with and without spin. We recall that the waveforms for the parity-odd interactions G̃_3 and Ĩ_1 are related to the parity-even ones by (<ref>), that is a simple swap of their real and imaginary parts; thus  separate plots for the case of parity-odd deformations are not needed. 
        We will   parameterise our external kinematics as follows:
        [Our waveforms are valid for arbitrary spin configurations; however in the figures shown in this section we will focus on the case where the spins of the two bodies are aligned.]
            v_1^μ=(σ,0,0,√(σ^2-1)) ,      v_2^μ   =(1,0,0,0) ,
               k̂^μ=(1,sinθcosϕ ,sinθsinϕ ,cosθ ) ,     b^μ   =b(0, 1, 0, 0) , 
               a_1^μ=a_1(0,0,1,0) ,     a_2^μ   =a_2(0,0,1,0) , 
               
               [k|_α̇=√(2)[         sinθ2; -cosθ2  e^-iϕ                             ]  .
        
        For the graphs below, we will choose θ=ϕ=π/3 and fix the spins to be either aligned or anti-aligned with the orbital angular momentum. We will also set b = 1 which means in practice 
        that the spins a_i and the retarded time u are measured in units of b. Finally we will ignore a prefactor so that the waveforms are plotted in units of κ^6 m_1 m_2 β_i.
        
        
        
        
        
         §.§ Spinless case
        
        
        We begin by showing the G_3 and I_1 waveforms for the scalar case, for various values of σ=v_1 v_2. 
        
        
        
        
         §.§ Spinning case
        
        We now show a few  G_3 and I_1 waveforms for  spinning objects. In Figures <ref> and <ref> we consider the cases where only the spin of one of the two bodies is nonvanishing, say a_1≠ 0 and a_2=0, and  do so for increasing values of a_1.
        In Figures <ref> and <ref> we show the G_3 and I_1  waveforms when both objects are spinning,  with their spins aligned, for various choices of the ratio a_2/a_1.
        Note that the  gravitational memory is absent  also in the spinning case. 
        We also recall that in setting b=1 we are measuring the spins a_i in units of b. Thus, the values we are plotting here, e.g. a_i=± 0.5, are rather large. Finally,  the waveforms also have poles at b=± a_1± a_2,
        [These poles arise  from treating  the  spin dependence   exactly and are  also present  in the scattering  angle <cit.>.] although  these points are not problematic since we assume |a_i|≤  G m_i< b.
        
        
        
        
        § (NO) GRAVITATIONAL MEMORY
        
        
        
        As it is well known, gravitational memory is related to soft limits of the five-point amplitude <cit.>, and a discussion of the memory in the General Relativity waveforms to all orders in spin was presented in <cit.>. We now wish to study if   cubic deformations alter in any way the gravitational memory. 
        Defining the memory  as 
        
            Δ (h^∞_+± i  h^∞_×) . (h^∞_+± i  h^∞_×)|_u→ + ∞ - . (h^∞_+± i  h^∞_×)|_u→ -  ∞  ,
        
        one finds that (see e.g. <cit.>)
        
            Δ (h^∞_+± i  h^∞_×)    = 
              -i  κ S_ W^ HEFT(k̂, -i ∂/∂ b; ±) δ_ HEFT  ,
        
        where
        
            δ_ HEFT∫dμ^(D) e^iq_ b ( -i _4^ HEFT)(q)  ,
        
        is the Fourier transform to impact parameter space of the four-point classical amplitude _4^ HEFT<cit.>, which now has to be evaluated in the presence of cubic deformations; while 
        and S_ W^ HEFT is the classical (or HEFT) Weinberg soft factor   <cit.>
            S_ W^ HEFT   = 
               - κ/2 1/ωε_μν^(h)(k) [
            		p_1^μ q^ν +p_1^ν q^μ/p_1k̂ - p_1^μ p_1^νqk̂/(p_1k̂)^2  -   1↔ 2]
                 .
        
         However, at leading order (tree level) in the presence of cubic deformations the classical four-point amplitude vanishes,  _4^ HEFT, (0)=0; indeed,  this quantity  starts receiving corrections at one loop, or 2PM, which were computed  in <cit.>. We conclude that the leading-order gravitational memory in the presence of cubic deformations vanishes. This is also confirmed by the explicit plots shown in 
         the 
         
         figures  in Section <ref>. 
        
        
        § DISCUSSION
        
        
        
        
         
        
        In this paper we have considered  parity-even and -odd cubic deformations to general relativity, and computed analytically the corresponding corrections to the waveforms to leading order in the deformations and in Newton's constant. We have done so following three independent approaches, yielding  results in complete agreement. The direct integration method provides the most compact expressions, though the IBP reduction and residue methods are possibly more easily extendable beyond tree level. The residue method is especially   rewarding in that it easily shows  the absence of contributions from spurious poles. 
        
        To conclude, we  briefly muse on the size  of the corrections due to the cubic deformations studied here and in <cit.>. Specifically, we   focus  on an observable quantity such as the impulse (or momentum kick) .Δℙ|_R^3, and   ask when the corrections arising from cubic deformations become comparable to those  in general relativity at a certain PM order. 
        
        Our cubic vertices have the form (schematically) ∫d^4x βκ^3 (∂^2h)^3. We  redefine the coupling constant in such a way that the three-point vertex scales as κ, thus introducing β = β̂ / κ^2; in terms of β̂ the vertex has the form κβ̂∫d^4x  (∂^2h)^3, and we  note that the dimension of β̂ is ( length)^4, so that  we will set β̂ = ℓ_ EFT^4. 
        We found in <cit.> that, 
        to leading order in the deformation and in G, the corrections to the impulse arising from cubic deformations have the form (see (6.61)–(6.64) of that paper)
        
            .Δℙ|_R^3∼βκ^6 m^3/b^6∼β̂/κ^2 G^3 m^3/b^6∼ m ×(ℓ_ EFT/b)^4 (Gm/b)^2   .
        
        On the other hand, to nPM order in general relativity the impulse  receives corrections    of the type 
        
            .Δℙ|_ EH,  n∼ m ×(Gm/b)^n
                  ,
        
        so that 
        
            .Δℙ|_R^3/.Δℙ|_ EH,  n∼(ℓ_ EFTb)^4/(Gmb)^n-2  .
        
        For hyperbolic encounters of interest we can choose for illustration  G m/b ∼  1/5, which ensures that  perturbation theory is still valid while also giving rise to a sizeable effect. By further requiring that, for observability,  .Δℙ|_R^3/ .Δℙ|_ EH,  n∼ 1, we arrive at  the estimate 
        
            ℓ_ EFT∼ b   (1/5)^n-2/4  .
        
        As an   example, we can ask when   the corrections  from cubic deformations  become comparable to those arising at 5PM (or four loops)  in general relativity.
        [Parts of  the 5PM corrections to the impulse and the scattering angle for non-spinning objects were recently computed in <cit.>.]
        From  (<ref>), and further choosing b∼ 100   km we arrive at the estimate  ℓ_ EFT∼ 30   km, which is within the regime of validity of the effective field theory,
        and also happens to be in agreement with the estimate   provided 
        in <cit.>. We also note that in <cit.>, bounds on the fundamental length scale  ℓ_R^4 in a quartic effective field theory considered earlier in <cit.> were established, resulting in ℓ_R^4≲ 150  km.  
        
        
        
        Similarly, we find that the waveform 
        h^∞ in general relativity at order n in the PM expansion scales as
        
            h^∞|_ EH,  n∼ b ×( G m/b)^n  ,
        
        where now  n=2 corresponds to tree level and n=3  to one loop. 
        For the leading cubic correction computed in this paper we found (see for example (<ref>) and (<ref>))
        
            h^∞|_R^3∼ b ×( G m/b)^2 ( ℓ_ EFT/b)^4  ,
        
        leading to similar conclusions as above for the impulse (<ref>). Furthermore, if we assume that ℓ_ EFT∼ G m we can say that the effects of cubic interaction are in practice four orders higher in the PM expansion. This may be challenging given the currently available state of the art results in the PM expansion, but results in the PN framework might already give access to higher orders in G.
        
        A final comment on the spin radii a_i. In the expressions for the impulse and waveform,  they appear through the shifted impact parameters (schematically) b̃ = b + a (omitting Lorentz indices and numerical factors). Recalling that for a physical Kerr black hole
        a/(G m) ≤ 1,  we can express this as
        
            b̃ = b ( 1 + a/G mG m/b)  .
        
        This implies that effectively every factor of a increases the  PM order by one,  since GM/b is the effective loop counting parameter. For example,  the linear in spin part of our results  should be accompanied  by the one-loop correction of the scalar waveform in the cubic theory.
        We hope to return to these questions in the  near future. 
        
        
        
        
        
        
        
        
        
        
        
        § ACKNOWLEDGEMENTS
        
        We would like to thank  
        Joshua Gowdy and Paolo Pichini 
        for stimulating conversations.
        GT thanks the Physics Department at the University of Rome “Tor Vergata” for their warm hospitality.   
        This work was supported by the Science and Technology Facilities Council (STFC) Consolidated Grants   ST/T000686/1 and  ST/X00063X/1 “Amplitudes, Strings  & Duality”.
        The work of GRB and PVM is supported by STFC quota studentships.   GC has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 847523 “INTERACTIONS”. 
        GT is also supported by a Leverhulme research fellowship RF-2023-279\ 9.
        No new data were generated or analysed during this study.
        
        
          
        
        
        § ALL TENSOR INTEGRALS WE NEED
        
        
        We summarise here relevant integrals needed to compute the time-domain waveforms in our paper. We focus here on integrals involving the 1/q_1^2 propagator, while integrals involving the 1/q_2^2 propagator can be  obtained by appropriate relabelling of the results quoted below. In the following we define q^μ q_1^μ and introduce the projector
        
            P_1^μνη^μν - v_1^μ v_1^ν .
        
        Henceforth, we denote projected four-vectors  and their modulus as
        
            V^μ_(1) P_1^μν V_ν  ,
        
            and 
            
            |V_(1)| √(-V_(1) V_(1)) = √(-V  P_1  V).
        
        The first and simplest integral we encounter is
        
            ℐ    = ∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃/q^2
                = ∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃_(1)/q^2
                = -1/8 π |b̃_(1)| ,
        
        where in the second line we exploited the fact that due to the δ-function we can insert for free the projector P_1 between q and b̃ in the exponent. We actually need tensor versions of this integral, 
        
            ℐ^μ_1 μ_2 …μ_n = 
                ℐ[q^μ_1 q^μ_2⋯ q^μ_n] = 
                ∫d^4q/(2 π)^3δ(2 q v_1 )   e^i q b̃ q^μ_1 q^μ_2⋯ q^μ_n/q^2 ,
        
        which can be simply obtained by acting  n times on ℐ  with the derivative operator
        
            -i ∂/∂b̃_(1) μ_i ,
        
        and using the identity 
        <cit.>
            ∂b̃_(1)^μ/∂b̃_(1)ν = P_1^μν .
        
        Two examples are 
        
            ^μ = -i ∂ℐ/∂b̃_(1)μ = i/8 πb̃_(1)^μ/|b̃_(1)|^3  ,
        
            and 
            
            ^μν = - ∂^2 ℐ/∂b̃_(1)μ∂b̃_(1)ν = |b̃_(1)|^2 P_1^μν+3 b̃^μ_(1)b̃^ν_(1)/8 π |b̃_(1)|^5 .
        
        In a similar fashion, we can obtain the other two more complicated tensor integral families. 
        The first one is 
        
            𝒥^μ∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃ q^μ/q^2  k̂ q ,
        
        which, as already mentioned at the end of Section <ref>, requires a regulator because of the 1/k̂ q pole.
        This is a spurious pole that cancels in the sum of the two diagrams contributing to  the waveform and, of course, the final result is independent of the regulator. For  illustration we will present here  the evaluation of 𝒥^μ using both the principal value (PV) and iε prescriptions, while in the main text we  only make use the PV prescription.
        
        We begin discussing  the PV prescription.
        We  denote the PV regulated integral by 𝒥^μ,  which can  be expressed in terms of the scalar integral 
        
            𝒥∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃/q^2  k̂ q ,
        
        as
        
            𝒥^μ = -i ∂𝒥/∂b̃_(1)μ .
        
        Following <cit.>,  we now notice that  𝒥 is invariant under rescaling of b̃. Indeed a rescaling b̃→α b̃ can be undone by the rescaling q → q/α which leaves the integrand unchanged. Hence, we infer that 
        
            b̃_(1)μ𝒥^μ = -i  b̃_(1)μ∂𝒥/∂b̃_(1)μ = 0  ,
        
        and we further notice that v_1 μ𝒥^μ = 0 since v_1  q = 0 due to the δ-function in the integral. Hence 𝒥^μ must live in a two-dimensional subspace orthogonal to v_1^μ and b̃_(1)^μ. In order to implement this,  it is convenient  to introduce the symmetric tensor <cit.>
            K_1^μν = |b̃_(1)|^2 P_1^μν + b̃_(1)^μb̃_(1)^ν ,
        
        obeying K_1^μν v_1ν = K_1^μνb̃_(1)ν = 0. This allows us to write an ansatz for 𝒥^μ as
        
            𝒥^μ = c   K_1^μνk̂_ν .
        
        We can now solve for c by observing that 
        
            𝒥^μk̂_μ = ℐ = -1/8 π |b̃_(1)| = c  k̂ K_1 k̂  ,
        
            from which it follows that 
            
            c = -1/8 π |b̃_(1)| k̂ K_1 k̂ .
        
        Therefore, we find that 
        
            𝒥^μ = K_1^μνk̂_ν/8 π |b̃_(1)|  [(v_1 k̂)^2 |b̃_(1)|^2-(k̂b̃_(1))^2 ] ,
        
        and from this result we can obtain 𝒥^μμ_1 …μ_n with any number of extra q^μ_i insertions by taking derivatives with respect to b̃_(1) μ_i as explained above.
        
        In order to obtain tensor integrals,  the form above is preferred as it makes the b̃_(1)-dependence fully transparent. On the other hand, one can obtain a more compact expression  in terms of the shifted, unprojected impact parameter b̃ as follows. 
        First, it is important to note that b̃k̂=0 as assumed throughout the paper. Note that any shift of b̃ by ã_i
        preserves this property
        since ã_i k̂=0.
        Under this assumption we can then write 
        
            b̃^μ = b̃_(1)^μ - k̂b̃_(1)/k̂ v_1 v_1^μ  ,
        
        from which it follows that 
        
            |b̃  |^2 -b̃b̃ =  |b̃_(1)|^2 - 
            (k̂b̃_(1))^2/(k̂ v_1)^2 .
        
        We  can then  rewrite  𝒥^μ in the more compact form (if b̃k̂=0)
        
            𝒥^μ =  K_1^μνk̂_ν/8 π  (v_1 k̂)^2 |b̃_(1)|   |b̃  |^2 .
        
        Next, we examine  the same integral,  this time  regulated with an iε prescription:
        
            𝒥_ε^μ   ∫d^4q/(2 π)^3δ(2 q  v_1) q^μ e^i q ·b̃/q^2  (k̂ q - i ε)
                =   i ∫_-∞^0 dτ∫d^4q/(2 π)^3δ(2 q  v_1) q^μ e^i q · (b̃ + τk̂)  + τε/q^2
                = -1/8 π∫_-∞^0 dτb̃^μ_(1) + τk̂^μ_(1)/[ -(b̃+τk̂)  P_1  (b̃+τk̂)]^3/2 ,
        
        where in the second line we have used the Schwinger trick and in the last line we have used (<ref>). Performing the τ integration gives
        
            𝒥_ε^μ =
            |b̃_(1)| k̂^μ_(1) - v_1 k̂ b̃^μ_(1)/8 π (v_1 k̂) |b̃_(1)| (|b̃_(1)| v_1 k̂ + b̃_(1)k̂ ) ,
        
        and similarly, replacing i ε→ -i ε we find
        
            𝒥_-ε^μ =
            |b̃_(1)| k̂^μ_(1) + v_1 k̂ b̃^μ_(1)/8 π (v_1 k̂) |b̃_(1)| (|b̃_(1)| v_1 k̂ - b̃_(1)k̂ ) .
        
        It can then be checked that these integrals are related to the PV integral (<ref>) as
        
            𝒥^μ = 1/2( 𝒥^μ_ε + 𝒥^μ_-ε)  .
        
        Finally, we tackle (with PV regularisation) the most complicated integral,
        
            𝒦^μν = ∫d^4q/(2 π)^3δ(2 v_1  q)
            e^i q b̃ q^μ q^ν/q^2  k̂ q  a_2  q_2 
            = ∫d^4q/(2 π)^3δ(2 v_1  q)
            e^i q b̃ q^μ q^ν/q^2  k̂ q  (- ã_2  q) ,
        
        where 
        we have introduced the shifted spin radii 
        
            ã_i^μ = a_i^μ - a_i k̂/v_i k̂ v_i^μ  ,      ã_i k̂ = 0   ,
        
        for i=1,2.
        Also note that q_2^μ = k^μ - q_1^μ = ω^* k̂^μ - q^μ = v_2  q/v_2 k̂k̂^μ - q^μ, and therefore we can rewrite a_2  q_2 = - ã_2  q.
        𝒦^μν has the following important properties:
        
            𝒦^μν v_1ν = 𝒦^μνb̃_ν
                = 𝒦^μνb̃_(1)ν =𝒦^μνη_μν = 0  .
        
        The first is due to the δ-function, the second and third property are consequence of the rescaling invariance of 𝒦^μ, and the fourth property (tracelessness) can be seen as follows: taking the trace and
        choosing to work in the rest frame v_1=(1,0,0,0), we find
        
            𝒦^μνη_μν    = ∫d^4q/(2 π)^3 δ(2 q v_1 )
            e^i q b̃/k̂ q   (- ã_2  q)
                = -∫_-∞^0dt_1 dt_2 ∫d^3q⃗/2 (2 π)^3  e^-i q⃗ (b⃗̃⃗+t_1 k⃗̂⃗ + t_2 ã⃗_2)
                = -1/2∫_-∞^0 dt_1 dt_2  δ^(3)(b⃗̃⃗+t_1 k⃗̂⃗ + t_2 ã⃗_2)  ,
        
        which vanishes for generic choices of the vectors.
        
        Due to the properties mentioned above, the integral (<ref>) must be a symmetric tensor living in the two-dimensional subspace orthogonal to v_1 and b̃, 
        
            𝒦^μν = c_1 K_1^μν + 2 c_2 (ã_2  K_1)^(μ (k̂ K_1)^ν) ,
        
        where 
            K_1^μν |b̃_(1)|^2 P_1^μν + b̃_(1)^μb̃_(1)^ν  .
        
            Notice that no other structures are allowed in this ansatz, since the result must be symmetric under swapping k̂↔ -ã_2 and be rescaled by 1/(α_1 α_2) if we perform the replacement  {k̂, ã_2}→{α_1 k̂, α_2 ã_2}. 
        Requiring also tracelessness, we instantaneously find
        c_1 = -c_2 ( ã_2  K_1k̂).
        On the other hand
        
            𝒦^μνk̂_μã_2ν = - ℐ = 1/8 π |b̃_(1)| = c_2 ( ã_2  K_1ã_2 )( k̂ K_1k̂) = - c_2 ( ã_2  K_1ã_2 ) (v_1 k̂)^2 |b̃|^2  ,
        
        and hence we find
        
            𝒦^μν = -( ã_2  K_1k̂) K_1^μν -2 (ã_2  K_1)^(μ (k̂ K_1)^ν)/8 π  |b̃_(1)| ( ã_2  K_1ã_2 ) ( k̂ K_1k̂) .
        
        If we further assume b̃k̂=0, which is
        true for all possible shifted versions of the impact parameter b̃± i ã_1 ± i ã_2,
        we find
        
            𝒦^μν = ( ã_2  K_1k̂) K_1^μν -2 (ã_2  K_1)^(μ (k̂ K_1)^ν)/8 π (v_1 k̂)^2 |b̃|^2 |b̃_(1)| ( ã_2  K_1ã_2 )  .
        
        Finally, we comment on potential terms in the waveform integrand that do not have physical poles 1/q_i^2 with i=1,2, but instead have a pole of the form 1/(w  q) (or a product of such factors). In such cases the same manipulation as in (<ref>) guarantees that such terms make vanishing contributions to the waveform. In  other words,  the tracelessness of integrals like 𝒦^μν in (<ref>) guarantees that terms with only spurious poles do not contribute to the waveform. This fact was also noticed in the Einstein-Hilbert case in <cit.>. For this reason, it is not necessary to use the full five-point amplitude,  but the sum of the two factorisation channels (<ref>) suffices. Similarly, all polynomial terms that only give rise to (derivatives of) δ-functions can be ignored.
        
        
        
        
        
        § TENSOR INTEGRALS FROM GENERATING FUNCTIONS
        
        We now describe an alternative (and faster) way to compute tensor integrals. All of the integrals we are interested in  have the form
        
            _m_1… m_n^ [] (b̃)     ∫d^4q/(2 π)^3 δ(2 q v_1 )  
            e^i q b̃ (q_1 c_1)^m_1⋯  (q_1 c_n)^m_n/ ,
        
        for some denominator , and where c_i, i=1, … , n are four-vectors. These can be obtained very easily from the integral 
        
            ^[] (b̃; t_1, …, t_n)    ∫d^4q/(2 π)^3 δ(2 q v_1 ) e^i q ( b̃ + ∑_i=1^n t_i c_i)/
                = ^[] (b̃ + ∑_i=1^n t_i c_i; 0, …, 0)
               ,
        
        as
        
            _m_1… m_n^[]   = .(-i)^∑_i=1^n m_i∂^∑_i=1^n m_i/∂^m_1t_1 ⋯∂^m_nt_n^[] (b̃; t_1, …, t_n)|_t_1 = t_2 = ⋯ = t_n = 0  .
        
        For instance, for the case of the G_3 waveform we need an integral of the form 
        
            ∫d^4q/(2 π)^3 δ(2 q v_1 )  
            e^i q b̃ (q  v_2)^2(q  X)^2/q^2  ,
        
        where X^μ[k̂| v_1 
        σ^μ | k̂]. This can be evaluated as 
        
            . ∂^4/∂^2t_1  ∂^2t_2∫d^4q/(2 π)^3 δ(2 q v_1 ) e^i q ( b̃ + t_1v_2 + t_2 X)/q^2|_t_1 = t_2  
            = 0
                
            = -1/8 π(∂^4/∂^2t_1  ∂^2t_21/
            |b̃_(1) + t_1 v_2(1) + t_2 X_(1)|)_t_1 = t_2 = 0 ,
        
        where we used (<ref>), and as usual we adopt the notation defined in (<ref>) for projected vectors (but note that X_(1) = X). 
        
          
        Summarising, in this approach we can easily relate tensor integrals to simpler ones (not necessarily scalar integrals, as seen in the previous section), without the need to perform Lorentz contractions and with the advantage of having to perform differentiations with respect to scalar parameters rather than vectors. 
        
        JHEP


