The gravitational wave GW interferometers LISA and ET are expected to be functional in the next decades possibly around the same time. They will operate over different frequency ranges with similar integrated sensitivities to the amplitude of a stochastic GW background SGWB. We investigate the synergies between these two detectors in terms of a multiband detection of a cosmological SGWB characterised by a large amplitude and a broad frequency spectrum. By investigating various examples of SGWBs such as those arising from cosmological phase transition cosmic string primordial inflation we show that LISA and ET operating together will have the opportunity to assess more effectively the characteristics of the GW spectrum produced by the same cosmological source but at separate frequency scales. Moreover the two experiments in tandem can be sensitive to features of early universe cosmic expansion before bigbang nucleosynthesis BBN which affects the SGWB frequency profile and which would not be possible to detect otherwise since two different frequency ranges correspond to two different preBBN or postinflationary epochs. Besides considering the GW spectrum we additionally undertake a preliminary study of the sensitivity of LISA and ET to soft limits of higher order tensor correlation functions. Given that these experiments operate at different frequency bands their synergy constitutes an ideal direct probe of squeezed limits of higher order GW correlators which can not be measured operating with a single instrument only.
['Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory']
Exploring cosmological gravitational wave backgrounds through the synergy of LISA and ET
{'Exploring cosmological gravitational wave backgrounds through the synergy of LISA and ET': "Alisha Marriott-Best a 1 , Debika Chowdhury b 2 , Anish Ghoshal c 3 , Gianmassimo Tasinato a,d 4  \na  \nPhysics Department, Swansea University, SA2 8PP, United Kingdom b Indian Institute of Astrophysics, II Block, Koramangala, Bengaluru 560034, India c Institute of Theoretical Physics, Faculty of Physics, ul. Pasteura 5, 02-093 Warsaw, Poland d Dipartimento di Fisica e Astronomia, Universit'a di Bologna, INFN, Sezione di Bologna, I.S. FLAG, viale B. Pichat 6/2, 40127 Bologna, Italy", 'Abstract': 'The gravitational wave (GW) interferometers LISA and ET are expected to be functional in the next decade(s), possibly around the same time. They will operate over different frequency ranges, with similar integrated sensitivities to the amplitude of a stochastic GW background (SGWB). We investigate the synergies between these two detectors, in terms of a multi-band detection of a cosmological SGWB characterised by a large amplitude, and a broad frequency spectrum. By investigating various examples of SGWBs, such as those arising from cosmological phase transition, cosmic string, primordial inflation, we show that LISA and ET operating together will have the opportunity to assess more effectively the characteristics of the GW spectrum produced by the same cosmological source, but at separate frequency scales. Moreover, the two experiments in tandem can be sensitive to features of early universe cosmic expansion before big-bang nucleosynthesis (BBN), which affects the SGWB frequency profile, and which would not be possible to detect otherwise, since two different frequency ranges correspond to two different pre-BBN (or post-inflationary) epochs. Besides considering the GW spectrum, we additionally undertake a preliminary study of the sensitivity of LISA and ET to soft limits of higher order tensor correlation functions. Given that these experiments operate at different frequency bands, their synergy constitutes an ideal direct probe of squeezed limits of higher order GW correlators, which can not be measured operating with a single instrument only.', '1 Introduction': 'The detection of gravitational waves (GWs) from astrophysical sources by the LIGO-Virgo collaboration in 2015 [1] opened up a new window into GW astronomy. For cosmology, upcoming upgrades of LIGO-Virgo [2] and proposed future detectors such as LISA [3], BBO-DECIGO [4], the Einstein Telescope (ET) [5, 6], and Cosmic Explorer (CE) [7] will also open up a new possible observational window into the early universe. Unlike photons, the gravitons (primordial GWs) that were produced in the early Universe can propagate freely throughout cosmic history and therefore would constitute ideal messengers of the history of the Universe [8-10].  \nIn fact, the recent hints of detection of a stochastic gravitational wave background (SGWB) in the nanoHertz regime by Pulsar Timing Array collaborations [11-14] initiated the era of experimental characterization of the SGWB. Still, much has to be done to distinguish between different sources of SGWB, astrophysical or cosmological (see e.g. [15,16] for recent topical reviews on SGWB sources and detection techniques).  \n3  \nanish.ghoshal.at.fuw.edu.pl  \nThe next generation of gravitational wave (GW) detectors promises to improve upon the current experimental sensitivity to SGWB in frequency ranges between milli-Hz and deca-Hz, much higher than the nano-Hz regime probed by pulsar timing arrays. These higher frequency regimes are more suitable for detecting GWs produced by various early universe cosmic sources, such as those arising from phase transitions, cosmic strings, cosmological inflation etc. Early universe scenarios can lead to a SGWB with intriguing properties such as a rich frequency profile, chirality, non-Gaussianity, all of which are important to accurately characterise for future targets (see e.g. [17] for a comprehensive recent review). It is essential to develop tools to detect and better characterise the SGWB in a frequency range which can be tested with high sensitivity by future experiments, say between 10 -5 ≤ f/ Hz ≤ 10 2 . In this work, we explore such a possibility of studying the early-universe sources of SGWB spanning this frequency range by exploiting synergies between the Laser Interferometer Space Antenna (LISA) [17-19] and the Einstein Telescope (ET) [5,6,20,21] experiments, for a multi-band detection of the SGWB. Both experiments are planned to take data in the next decade, and will have similar sensitivities to the amplitude of SGWB. Hence it is interesting to inquire what we can gain from detecting a SGWB with both the experimental facilities.  \nWhile LISA will have its maximal sensitivity for frequencies f in the milli-Hertz (mHz) regime, ET will be more sensitive to signals in the deca-Hertz range. If the SGWB has a broad enough frequency profile and a sufficiently large amplitude, it will be advantageous to have both the experiments detecting its features in different frequency ranges. We can then measure the properties of the GW source more accurately, and study aspects of early-universe cosmology which cannot be probed by each single experiment.  \nIn the context of beyond the Standard Model (BSM) of particle physics there are several concrete predictions of SGWBs over multi-band frequency ranges as we will discuss below. Firstly, the very well understood temperature anisotropies in the Cosmic Microwave Background Radiation (CMBR) superimposed on the perfectly smooth background implies that the universe at the very beginning has undergone an accelerated expansion, a phenomenon also known as the cosmic inflation [22-25]. However, the history of the primordial universe post inflation (plausibly after temperature T ≤ 10 14 GeV ) and before the beginning of Big Bang Nucleosynthesis (BBN), that is the at temperature above the SM plasma temperature of T ≳ 1 MeV, remains unconstrained by any observational data at the moment. The standard assumption that the pre-BBN universe is filled with radiation and becomes radiation-dominated after the end of the inflationary phase, is often challenged by open problems in the Standard Model (SM) of particle physics, e.g., the microscopic origin of dark matter (DM), the explanation for the observed matter-antimatter asymmetry, the flavor puzzle, or the ultraviolet SM Higgs field dynamics, the Strong CP problem (see e.g. Ref. [26,27] for a review). Introducing new BSM physics which resolves these puzzles of modern particle physics and cosmology often is associated with new energy scales (other than the electroweak or Planck scales) and on many occasions new degrees of freedom (like new particles) or interactions which sometimes generate deviations from the standard radiation domination era before the onset of BBN.  \nWe start our work with section 2 explaining why and how a detection of the SGWB in synergy between LISA and ET can improve the signal-to-noise ratio (SNR) on the measurements of parameters characterizing a SGWB with a broad frequency spectrum. We then move on to section 3 to discuss and analyse several early universe scenarios that are able to produce GWs spanning over a broad frequency range. By means of a Fisher analysis, we quantitatively demonstrate how a detection of GW with the two experiments together can help us to measure specific model parameters. Section 4 discusses the notion of integrated sensitivity curves, which offer a simple visual aid to demonstrate the advantages of the synergy between the two experiments with regard to detecting SGWB with certain frequency shapes. Cosmological SGWB can be characterised by non-Gaussian features, which motivate the study of n -point correlation functions going beyond the GW power spectrum and energy density. Given that LISA and ET operate in different frequency ranges which  \ncorresponding to different energy scales of tensor Fourier modes, in section 5 we address the problem of the detectability of soft limits of n -point correlation functions, discussing the response function of the LISA-ET system to such observables. We conclude in section 6. A technical appendix complements our arguments. We work with natural units c = ℏ = 1. We fix the h in the Hubble parameter as h = 0 . 67.', '2 Synergies between LISA and Einstein Telescope': 'The aim of this section is to start discussing in practical terms the possibility of making a synergetic detection of a SGWB with the Laser Interferometer Space Antenna (LISA) and the Einstein Telescope (ET) instruments. In the next section we will describe theoretical motivations to do so.', 'Gravitational waves and their detection': "Gravitational waves (GWs) are associated with spin-2 fluctuations h ij of the Minkowski metric:  \nds 2 = -dt 2 +( δ ij + h ij ( t, ⃗x )) dx i dx j . (2.1)  \nWe decompose h ij into Fourier modes as  \nh ij ( t, ⃗x ) = ∑ λ ∫ + ∞ -∞ df ∫ d 2 ˆ ne -2 πif ˆ n⃗x e 2 πift e λ ij (ˆ n ) h λ ( f, ˆ n ) , (2.2)  \nimposing the condition  \nh λ ( -f, ˆ n ) = h ∗ λ ( f, ˆ n ) , (2.3)  \nwhich ensures that h ij ( t, ⃗x ) is a real function. The quantities f , ˆ n , and λ denote respectively the GW frequency, direction, and polarization ( λ = + , × ). In the previous expressions, we have decomposed the GW momentum as ⃗ k = 2 πf ˆ n , with f being the GW frequency, and ˆ n being its direction. We assume that the polarization tensors e λ ij are real quantities. We adopt a (+ , × ) basis, and use the normalization: ∑ ij e λ ij e λ ' ij = 2 δ λλ ' .  \nWe assume in this section that the SGWB is isotropic, stationary, and Gaussian. The GW energy density is expressed in terms of the function Ω GW ( f ), defined starting from the two-point function for GW Fourier modes (see e.g. [28]). It is defined as  \nΩ GW ( f ) = ( 4 π 2 3 H 2 0 ) f 3 I ( f ) , (2.4)  \nwhere the GW intensity I ( f ) is given by  \n〈( h λ ( f, ˆ n ) ) ∗ h λ ' ( f ' , ˆ n ' ) 〉 = δ λλ ' 2 δ (ˆ n -ˆ n ' ) 4 π δ ( f -f ' ) I ( f ) . (2.5)  \nWe shall now discuss how the interferometers LISA and ET respond to the presence of GWs. Their behaviour depends on the so-called response functions, and on the sources of noise which affect a possible GW detection. For the case of LISA, this topic is explained in a clear and pedagogical way in [29], which we briefly review here (see also [30] for a more systematic discussion). We extend their analysis to include ET 5 and the synergy between the two detectors. We mainly use the notation and conventions of [29], adapting them to the present context.  \nWe assume that both the LISA and the ET instruments have shapes corresponding to equilateral triangles 6 . The GW is detected as an effect of the time difference between signals measured at different vertices of  \nthe triangular interferometer. We shall denote the three vertices of a triangle (it can be the LISA or the ET instrument) with the combination of letters ( abc ). Let us consider the vertex a as reference. The instrument measures the phase difference Φ  \nΦ a bc = ∆ φ a bc + n a bc (2.6)  \nof the GW signals travelling along the arms ( ab ) and ( ac ), plus the contribution n of noise. In what follows, we will neglect the time dependence of the positions of the detectors.  \nThe interferometer response and the GW signal contribution can then be expanded in Fourier modes as  \n∆ φ a bc ( t ) = ∫ + ∞ -∞ df e 2 πift ∆˜ φ a bc ( f ) , (2.7)  \nwhere the signal Fourier mode ∆ ˜ φ is given by the combination of the spin-2 mode h λ ( f, ˆ n ) and the interferometer response F λ a bc , as contained in the following definition  \n∆˜ φ a bc ( f ) = ∑ λ ∫ d 2 nh λ ( f, ˆ n ) F λ a bc ( f, ˆ n ) . (2.8)  \nThe quantity F λ a bc is expressed as  \nF λ a bc ( f, ˆ n ) = e -2 πi f ˆ n · ⃗x a 2 e λ ij (ˆ n ) [ F ij ( ˆ ℓ ab · ˆ n, f ) -F ij ( ˆ ℓ ac · ˆ n, f ) ] , (2.9)  \nwith the unit vector ˆ ℓ corresponding to the direction of the detector arm. The geometry of the detector enters into the functions F ij . Their expressions depend on the type of interferometer one considers - space-based (LISA) or ground-based (ET). For the case of LISA, they read  \nF ij LISA ( ˆ ℓ · ˆ n, f ) = ˆ ℓ i ˆ ℓ j 2 e -if (3+ ˆ ℓ · ˆ n ) / (2 f ⋆ ) sinc ( f 2 f ⋆ (1 -ˆ ℓ · ˆ n ) ) + ˆ ℓ i ˆ ℓ j 2 e -if (1+ ˆ ℓ · ˆ n ) / (2 f ⋆ ) sinc ( f 2 f ⋆ (1 + ˆ ℓ · ˆ n ) ) , (2.10)  \nwhere the pivot scale f ⋆ = 1 / (2 πL ) - with L being the length of the interferometer arms - is of the order of the milli-Hz frequencies probed by LISA. In contrast, the expression for F is much simpler for a ground-based detector such as ET, and corresponds to the following low-frequency limit of the previous equation:  \nF ij ET ( ˆ ℓ · ˆ n, f ) = ˆ ℓ i ˆ ℓ j . (2.11)  \nStarting from the above formulae, we can measure the phase difference of signals travelling between the arms ( ab ) and ( ac ), and correlate signals measured at different vertices by computing their two-point functions. They depend on the GW intensity I ( f ) (see eq (2.5)), weighted by the instrument response to the GW signal, and on possible noise sources. The signal two-point function in Fourier space reads  \n⟨ Φ a bc ( f )Φ x yz ( f ' ) ⟩ = δ ( f -f ' ) 2 [ R a bc , x yz ( f ) I ( f ) + N a bc , x yz ( f ) ] , (2.12)  \nwith N a bc , x yz being the correlated noise. The signal response functions are  \nR a bc , x yz ( f ) = ∫ d 2 ˆ n 4 π [ F + a bc ( f, ˆ n ) F + x yz ( -f, ˆ n ) + F × a bc ( f, ˆ n ) F × x yz ( -f, ˆ n ) ] . (2.13)  \nAt a given frequency f , we assume that there is neither noise correlation nor contaminations between the two detectors LISA and ET - correlated noise is present only between arms of the same interferometer - hence the functions N a bc , x yz are zero for correlations among vertices of two different experiments. In this section, moreover, we do not take into account correlations of the signal intensity I ( f ) among arms of the two different interferometers. In fact, the latter have best sensitivities in different frequency ranges, hence we expect that at a given frequency f the signal intensity can be at best probed by one individual experiment only. Namely, we consider only signal correlations to exist between the arms of each interferometer. Under these hypotheses, the phase covariance of the correlated signals among the different vertices of the two equilateral triangles (LISA and ET) results in a block-diagonal 6 × 6 matrix:  \n         C 1 C 2 C 2 0 0 0 C 2 C 1 C 2 0 0 0 C 2 C 2 C 1 0 0 0 0 0 0 C 3 C 4 C 4 0 0 0 C 4 C 3 C 4 0 0 0 C 4 C 4 C 3          (2.14)  \nThe upper block corresponds to the LISA and the lower block to the ET equilateral triangle. The quantities  \nC i = S i + N i ,  \nare combinations of (possible) GW signal ( S i ) and instrumental noise ( N i ) at each detector. This matrix can easily be diagonalized, leading to the definition of six orthogonal channels. In analogy with the names traditionally assigned to the LISA channels, these are called ( A ℓ , E ℓ , T ℓ , A e , E e , T e ) . They are given by  \nC A ℓ = C E ℓ = C 1 -C 2 , (2.15)  \nC T ℓ = C 1 +2 C 2 , (2.16)  \nC A e = C E e = C 3 -C 4 , (2.17)  \nC T e = C 3 +2 C 4 . (2.18)  \nStarting from these considerations, we can obtain the response functions for the diagonal channels, in our approximation of static set-up 7 . In the case of LISA, the response functions depend on the frequency; in the small-frequency limit they can be expressed as  \nR A ℓ = R E ℓ = 9 20 -169 1120 ( f f ⋆ ) 2 + O ( f/f ⋆ ) 4 , (2.20)  \nR T ℓ = 1 4032 ( f f ⋆ ) 6 + O ( f/f ⋆ ) 8 . (2.21)  \n⃗x A = { 0 , 0 , 0 } , ⃗x B = L { 1 / 2 , √ 3 / 2 , 0 } , ⃗x C = L {-1 / 2 , √ 3 / 2 , 0 } . (2.19)  \nFor simplicity we choose the same arm directions for the triangle forming ET, which of course has much shorter arm lengths. As mentioned above, in this work we do not consider effects of the relative motion between LISA and ET detectors.  \n<!-- image -->  \nFigure 1: The numerical LISA response functions for A,E ( left panel ) and T ( right panel ) orthogonal channels, (dashed lines) as well as the corresponding analytical fits of eqs (2.22) , (2.23) (continuous lines).  \n<!-- image -->  \nThe complete frequency dependence of the response functions R A ℓ , E ℓ and R T ℓ for LISA can be easily obtained numerically, as explained in [29] (see Fig 1). Suitable analytical approximations for these two quantities are  \nR fit A ℓ , E ℓ ( f ) = 9 20 ( 1 + ( f 1 . 25 f ⋆ ) 3 ) -2 / 3 , (2.22)  \nR fit T ℓ ( f ) = 1 10 ( f 2 . 8 f ⋆ ) 6 ( 1 + ( f 2 . 8 f ⋆ ) 6 ) -4 / 3 , (2.23)  \nwhich are also represented in Fig 1. The T -channels T ℓ and T e are either weakly sensitive or not sensitive at all to the GW signal (and the sensitivity in any case vanishes in the small-frequency limit). For the groundbased interferometer ET, the response functions are independent from frequency, and are proportional to the zero-frequency limit of eqs (2.20), (2.21): see e.g. [31] for details.", 'The optimal signal-to-noise ratio': "After the characterization of the geometrical properties of the interferometer system as described above, we investigate the optimal signal-to-noise ratio (SNR) for detecting a SGWB with the two instruments LISA and ET working together. We assume that the two detectors take data approximately for the same amount of time T (although not necessarily simultaneously, exploiting the stationarity of the SGWB). The optimal signal-to-noise ratio (SNR) for measuring a SGWB with LISA and ET in synergy is built using techniques based on Wiener filtering, combining information obtained from the independent channels A ℓ,e and E ℓ,e . We follow [29] (see also [30]), extending it to the general case where we work with two instruments together. Working in the weak-signal limit, denoting with S i the signal on each independent channel, N i the noise, and Q i the filter, the signal-to-noise ratio (SNR) in Fourier space reads  \nSNR = √ T 2 ∑ i ∫ ∞ -∞ df S i ( f ) Q i ( f ) √ ∑ i ∫ ∞ -∞ df N 2 i ( f ) Q 2 i ( f ) , (2.24)  \nwhere the sums are over the four channels A ℓ,e and E ℓ,e which are most sensitive to the signal. As mentioned above, the quantity T indicates the duration of the measurements, which we consider to be comparable in  \nthe two experiments. We now wish to determine the optimal filter which maximizes eq (2.24). We define a positive definite inner product  \n( P i , Q i ) = ∑ i ∫ ∞ -∞ df P i ( f ) Q i ( f ) N 2 i ( f ) , (2.25)  \nwhich acts on the four vectors ( P i ), where i = A ℓ,e , E ℓ,e . The SNR can then be expressed as  \nSNR = √ T 2 ( S i /N 2 i , Q i ) √ ( Q i , Q i ) , (2.26)  \nand the filter that maximizes the previous expression, up to an overall factor, is Q i = S i /N 2 i . The optimal SNR is (we now integrate over positive frequencies only)  \nSNR =   T ∑ i = A ℓ,e ,E ℓ,e ∫ + ∞ 0 df S 2 i ( f ) N 2 i ( f )   1 / 2 . (2.27)  \nWe can now decompose the integrand in the previous formula as  \n∑ i = A ℓ,e ,E ℓ,e S 2 i ( f ) N 2 i ( f ) = [ ( R A ℓ ( f ) N A ℓ ( f ) ) 2 + ( R E ℓ ( f ) N E ℓ ( f ) ) 2 ] I 2 ( f ) + [ ( R A e ( f ) N A e ( f ) ) 2 + ( R E e ( f ) N E e ( f ) ) 2 ] I 2 ( f ) . (2.28)  \nIt is convenient to assemble the above result as (recall that Ω GW is defined in eq (2.4))  \n∑ i = A ℓ,e ,E ℓ,e S 2 i ( f ) N 2 i ( f ) = Ω 2 GW ( f ) Σ 2 LISA ( f ) + Ω 2 GW ( f ) Σ 2 ET ( f ) , (2.29)  \nwith  \nΣ LISA ( f ) = ( 4 π 2 3 H 2 0 ) f 3 [ ( R A ℓ ( f ) N A ℓ ( f ) ) 2 + ( R E ℓ ( f ) N E ℓ ( f ) ) 2 ] -1 / 2 , (2.30)  \nand analogously for ET.  \nThe result depends on the instrument response R to the signal, and on the noise curve N for each independent channel. The functions Σ LISA ( f ) and Σ ET ( f ) are called nominal sensitivity curves - see [33] for a general discussion - and we represent them in Fig 2 for the two experiments under consideration. Besides the numerically evaluated sensitivity curves, we also represent analytical approximations for the curves in Fig 2. The analytical fit we use for LISA is obtained from [34], while the one for ET is a new result of the present work, and we discuss it in Appendix A. For each experiment, the function Σ encapsulates a weighted combination of response functions and noise on each channel, and is useful for visually understanding the sensitivity of the instruments. For an extended discussion on sensitivity curves, see section 4, where we will also discuss the more refined notion of integrated sensitivity curves in this context. To summarize, the square of the total SNR is the sum of the squares of the individual SNRs:  \nSNR tot = √ T ∫ ∞ 0 df [ Ω 2 GW ( f ) Σ 2 LISA ( f ) + Ω 2 GW ( f ) Σ 2 ET ( f ) ] = √ SNR 2 LISA +SNR 2 ET , (2.31)  \nFigure 2: Nominal sensitivity curves for LISA and ET. For the latter, we represent the so-called ET-D curve. The approximate analytical fit for ET is discussed in Appendix A. Recall we take h = 0 . 67 .  \n<!-- image -->  \na formula which will be used in what follows. This expression demonstrates that, by working in synergy, the two detectors can reach higher values of SNR than each experiment operating individually.  \nInterestingly, the very same results can be obtained in terms of a likelihood function associated with a measurement of Ω GW carried out by the two experiments together. This method is also useful for applications to Fisher matrix forecasts. We assume the following structure for the Gaussian likelihood:  \nln L = const -1 2 ∫ ∞ 0 df df ' ( ˆ Ω GW ( f ) -Ω th GW ( f ) ) C -1 ( f, f ' ) ( ˆ Ω GW ( f ' ) -Ω th GW ( f ' ) ) , (2.32)  \nwhere ˆ Ω GW is the measured value, and Ω th GW is the theoretical prediction from various sources for the quantity Ω GW we wish to test. The inverse of the covariance matrix corresponding to the GW measurement by the two experiments together is  \nC -1 ( f, f ' ) = T δ ( f -f ' ) ( 1 Σ 2 LISA ( f ) + 1 Σ 2 ET ( f ) ) . (2.33)  \nConsidering ˆ Ω GW ( f ) to be the quantity to measure, we can compute the following quantity corresponding to a continuous version of the Fisher matrix:  \nF ( f, f ' ) = -δ 2 ln L δ Ω th GW ( f ) δ Ω th GW ( f ' ) (2.34)  \n= C -1 ( f, f ' ) . (2.35)  \nThe optimal SNR can then be computed in terms of a convolution integral  \nSNR 2 opt = ∫ + ∞ 0 dfdf ' Ω th GW ( f )Ω th GW ( f ' ) F ( f, f ' ) . (2.36)  \nSubstituting the inverse covariance function (2.33), this result coincides with eq (2.31).  \nThe concept of Fisher matrices, of course, can be used more directly to make forecasts on the prospective error bars associated with measured quantities, see e.g. [35,36]. Suppose we are interested in measuring the components of a parameter vector Θ i , with i being an index running over the number of model parameters we are interested in. The corresponding Fisher matrix is  \nF ij = -δ 2 ln L δ Θ i δ Θ j , (2.37)  \nwhere we consider the quantity given in eq (2.32) as likelihood function. Then the errors on the measurements of Θ i are at least  \n∆Θ i = √ ( F ) -1 ii . (2.38)  \nThis is a standard formula that we utilise in later sections. After discussing how to use the two instruments together to measure SGWB signals, in the next section we will provide motivations and examples of broad multiband SGWB sources, which can benefit from a joint detection by LISA and ET.", '3 Examples of SGWB with broad frequency profiles': "In this section we discuss examples of cosmological SGWB sources, which are able to produce a broad GW signal with a sizeable amplitude spanning several decades in frequency. We consider, in succession, GW sources from first order cosmological phase transitions, cosmic strings, and primordial inflation. We are interested in SGWB spectra enhanced within the broad frequency band  \nB tot = 10 -5 ≤ f/ Hz ≤ 445 . (3.1)  \nThe lower part of the interval (3.1) corresponds to the region of maximal sensitivity of LISA (the milli-Hz), while the upper part corresponds to the region (the deca-Hz) where ET is more sensitive to a GW signal (see both panels of Fig 2). We intend to demonstrate that important physical information about the GW source and the universe's evolution history can be extracted by measuring in synergy the SGWB within the broad frequency interval (3.1).  \nWe are not interested though in measuring the finer details of the frequency dependence of the GW spectrum (for methods to do so, see e.g. [34,37]). Instead, we wish to characterise the overall frequency profile of the spectrum, including the properties of a SGWB which extends all the way between the lower and the upper regions of the frequency band in eq (3.1). A GW spectrum is particularly interesting for us if it has a structure evolving in frequency throughout the entire interval (3.1). In such a case, the synergy between the two experiments LISA and ET can be especially useful for better characterizing the signal and extracting its physical properties, compared to measurements made with a single instrument (LISA or ET). We explore this topic quantitatively by means of Fisher forecasts on the detectability of the properties of the SGWB shape.  \nA useful SGWB template to keep in mind for the GW energy density of eq (2.4), with the properties we need, is the so-called broken power law (BPL) function which well describes, at least up to first approximation, GW spectra produced by several early universe phenomena [38]. This template applies well to GW  \nFigure 3: Left panel: Examples of SGWB from phase transitions (PT), see section 3.1. Middle panel: Examples of SGWB from cosmic strings (CS), see section 3.2. Right panel: Examples of SGWB from cosmic inflation, see section 3.3. Besides LISA and ET nominal curves, in blue and green lines are represented the benchmark scenarios discussed in each of the aforementioned sections.  \n<!-- image -->  \nspectra from phase transitions and cosmic strings (see sections 3.1 and 3.2). We adopt the frequency shape parametrization of [39]:  \nΩ GW ( f ) = Ω ⋆ ( f f ⋆ ) n 1 [ 1 2 + 1 2 ( f f ⋆ ) σ ] n 2 -n 1 σ . (3.2)  \nThe quantities f ⋆ and Ω ⋆ in eq (3.2) control the position of the break and the amplitude of the spectrum around the break. The quantities n 1 , 2 are related to the spectral indexes before and after the break, while σ controls the smoothness of the break - the smaller σ is, the smoother is the transition. If the break occurs somewhere the middle of band (3.1), it will be interesting to detect it with the two experiments in synergy for the possibility of measuring both the indexes n 1 and n 2 .  \nIf n 1 and n 2 have opposite sign, the break position f break and the corresponding value of Ω break GW are given by  \nf break = ( -n 1 /n 2 ) 1 /σ f ⋆ , (3.3)  \nΩ break GW = Ω ⋆ [ ( -n 2 /n 1 ) n 1 / ( n 1 -n 2 ) 2 + ( -n 1 /n 2 ) n 2 / ( n 2 -n 1 ) 2 ] ( n 2 -n 1 ) /σ . (3.4)  \nCurrently, we have an indirect bound on the amplitude of a cosmological SGWB signal in the frequency band (3.1), because its amplitude should not exceed the big-bang nucleosynthesis (BBN) bound Ω GW ≤ 1 . 7 × 10 -6 [40]. Moreover, at the frequency scales of ground-based interferometers - around deci-Hertz - the Ligo-Virgo-Kagra collaboration currently sets the upper bound Ω GW ≤ 6 × 10 -8 at the reference ground-based frequency of 25 Hz [41], for a flat GW spectrum. In this work, we consider the BBN bound as reference for the maximal amplitude of the SGWB even when studying GW sources active after BBN.  \nAfter these preliminary considerations, we can start looking at concrete early universe sources of GW. We do not plan to be exhaustive, but to discuss selected examples of sources which lead to a broad GW spectrum, and whose detection would benefit from synergies between LISA and ET. We focus on the theoretical aspects  \nof the discussion, and also present Fisher estimates on the capability of the two instruments together to better detect properties of the SGWB profile. Although the template (3.2) is simple and general enough to accommodate several early universe sources - as discussed in sections 3.1 and 3.2 - for selected cases in the context of inflation we go beyond the profile of eq (3.2), and we consider a different broad ansatz for Ω GW -the so-called log-normal profile - to better describe the frequency dependence of the SGWB (see section 3.3).", '3.1 Cosmological Phase transitions': 'Several well-motivated models of particle physics predict the existence of scalar sectors beyond the Standard Model, whose potentials are characterised by local minima. The energy release of strong first order phase transitions (PTs) between different vacua produces a stochastic background of GW. The detection of such a background would provide invaluable information on physics beyond the Standard Model, and on the early cosmic evolution of our universe. We refer to [17,39,42,43] for complete discussions and reviews on PT and their consequences for different aspects of GW production. There are essentially three mechanisms 8 for GW production: collisions among bubbles of different vacua [45-47], sound waves in the primordial plasma [48,49], and turbulent motion [50-52]. The resulting shape in frequency of the SGWB spectrum has a characteristic peak structure associated with the duration and properties of the PT responsible for the GW emission. The SGWB from strong first order PT increases from small towards large frequencies, reaches a maximum associated with the Hubble size during the PT in early universe, and then decreases in amplitude.  \nAccording to the recent discussion in [39], in the first case of bubble collisions we can expect a SGWB with a broken power-law profile as in eq (3.2), while the other two cases are better described by a double broken power law template. At first approximation - since as mentioned earlier we are not interested in fine details of the SGWB frequency dependence, but only on its overall structure within the broad interval (3.1) - we do not take into account the differences between the latter and the former template. We consider the BPL profile in eq (3.2) as describing reasonably well the overall frequency dependence of a SGWB from first order PT, which can in principle span over the frequency band (3.1). In this context, the parameters in the template (3.2) depend on the GW production mechanisms, as well as on the particle physics models sourcing the PT in the first place (see the recent work [39] for a more detailed analysis).', 'Position and height of the peak': 'For the case of bubble collisions, many studies over the years have clarified the role of bubble dynamics and surrounding relativistic fluid shells for GW production (see e.g. [43] which contains a complete review). The dynamics of fluid shells might be important, requiring to go beyond the so-called thin-shell approximation (see [53, 54] for latest developments). In the limit of strong phase transitions, the inverse duration of the transition, denoted as β/H ⋆ (normalized against the Hubble parameter at the transition epoch), and the temperature T ∗ at transition are related to the BPL amplitude Ω ⋆ and break position f ⋆ by the formulae [39]  \nΩ ⋆ ≃ 2 H 2 ∗ 10 6 β 2 ; f ∗ Hz ≃ 1 10 8 √ Ω ⋆ ( T ∗ 100 GeV ) . (3.5)  \nWe refer to [39] for details. Hence, by tuning appropriately the transition temperature and its duration, the position of the break (see eq (3.3)) might be placed freely within the band (3.1).  \nA precise measurement of the value of f break informs us of when the PT occurs during cosmological history, and the time-scale of its duration. As an explicit example, let us assume n 1 = -n 2 ≥ 0 (so that f break = f ⋆ )  \nand a high SGWB amplitude Ω ⋆ = 10 -6 . The transition temperature corresponds to the electroweak value T ∗ = 10 2 GeV - for a break in the LISA band at f ⋆ = 10 -5 Hz, the lower extremum of the interval (3.1). On the other hand, we find an intermediate scale of T ∗ = 10 9 GeV for a break within the ET band at f ⋆ = 5 × 10 2 Hz, the upper extremum of the interval (3.1) (see the recent discussion [55]). Such intermediate-case PTs are very well motivated from scenarios of BSM involving axion physics with classic Peccei-Quinn symmetry breaking scales around 10 9 -10 11 GeV [56-59]. Interestingly, for this first order PT, the energy scale of new physics the ground-based detectors are sensitive to, roughly coincides with the lowest possible energy scale at which the Peccei-Quinn (PQ) symmetry U (1) PQ has to be broken in QCD axion models which also address the strong CP problem of the SM [60-63]. The involved axion scalar field is a viable cold dark matter (DM) candidate [64-66], and even more generally, axion-like particles (ALPs) are well-motivated, since they are naturally present as pseudo Nambu-Goldstone bosons in many BSM extensions with a spontaneously broken global U (1) symmetry, e.g. in several string theory avatars [67-69]. Yet another motivation for intermediate scale PT comes from neutrino mass generation also known as seesaw mechanism connected to the scale of baryogenesis via leptogenesis, see e.g. [70-74]. In fact, while standard thermal leptogenesis is a simple and elegant mechanism, it requires a small window of right-handed neutrino masses in the high energy regime 10 9 -10 11 GeV. Hence, GW detectors could probe these energy ranges which can not be probed by accelerator experiments (see the analysis [75] in terms of existing GW data).  \nMoreover, it is also possible to push T ⋆ to high values considering non-minimal Higgs scenarios, or scalar setups belonging to dark sectors beyond the Standard Model (see e.g. [76], and also see [77] for an early, complete analysis of the possibility of tuning the scale of the transition to intermediate values and its consequences for interferometer detections and physics beyond the standard model). The possibility of detecting a break in the spectrum somewhere within the entire range (3.1) would be an important opportunity to study the physics of PTs and probe high energy physics beyond the electroweak scale.', 'The spectral indexes, and our benchmark scenarios': 'Table 1: Benchmark values for the scenarios corresponding to cosmological phase transitions.  \nInterestingly, the values of the tilts n 1 , 2 and of the smoothing quantity σ depend more specifically on the PT scenario and GW source under consideration. Measuring both indexes n 1 and n 2 accurately is then essential to reconstruct the details of the physics leading to the PT: such a measurement can be achieved by the synergy of LISA and ET, as we are going to demonstrate. We analyze two benchmark scenarios, PT1 and PT2, as summarized in Table 1. For uncorrelated primordial sources, one finds a slope n 1 = 3 in the deep infrared (see e.g. [78] for a detailed analysis). But more generally, the slope depends sensitively on the GW production. We first consider a benchmark scenario PT1, with spectral index n 1 = 3 in the infrared; in the UV, we consider n 2 = -1 as predicted in scenarios where GWs are produced by sound waves of the bubble surrounding plasma, or by effects of turbulent behaviour in the fluid. For the second benchmark scenario, PT2, we consider the model of [79] in the context of highly relativistic fluid shell dynamics, which finds n 1 = -n 2 ≃ 2 . 4 and σ ≃ 1 . 2. These two scenarios are presented in the left panel of Fig 3.  \nFigure 4: Fisher forecasts for the phase transition benchmark scenarios PT1 and PT2, summarized in Table 1.  \n<!-- image -->', 'What can we learn about PT by LISA and ET in synergy?': 'In both of the aforementioned benchmark models, the position of the break of the broken power-law spectrum is located somewhere in the middle between LISA and ET bands, see left panel of Fig 3. The plots suggests that a measurement in synergy between LISA and ET would allow us to get important information on the position of the break and the value of the spectral tilts, and thereby on particle physics models leading to a first order PT. Notice that, importantly, the signal profile lies well below the nominal sensitivity curve of both experiments. Nevertheless, it can be detected by integrating over frequencies: recall the expression for the SNR (eq (2.27)). Such an integration allows one to acquire sufficiently high values of SNR even if the signal lies well below the nominal sensitivity curves of the experiment. In fact, this property suggests the definition of broken power law sensitivity curve, as depicted with light green colour in Fig 3, left and middle panel: the GW signals lie well above such a curve. We will reconsider this topic in section 4, in the context of frequency-integrated sensitivity curves.  \nIn fact, we can carry out a Fisher analysis using a likelihood whose structure is given in eq (2.32), and assuming the BPL ansatz (3.2) with the aforementioned two sets of benchmark values for the parameters, summarised in Table 1. The benchmark values of Ω ⋆ are selected in a way such as to show how the two instruments together can achieve good accuracy in the measurements of the template parameters. The results are shown in Fig 4. From now on, we will present Fisher plots obtained using the GetDist package [80].  \nFor both the scenarios PT1 and PT2 as shown in Fig 4, each of the two experiments - LISA and ET - can measure with good accuracy only one of the two spectral indexes n 1 or n 2 . The two experiments in synergy, though, can measure both these quantities well, with an accuracy of at least 10 percent. Apart from the spectral indexes, the parameter σ controlling the degree of smoothness of the transition can also be measured accurately by the synergy of the two experiments. This implies that, by working with LISA and ET together, we can obtain much richer information on the physics of PT occurring at high temperature scales. Additionally, for both scenarios, the ET experiment by itself cannot accurately measure the amplitude of the SGWB. Only in synergy with LISA can it do so, measuring with a 10 percent accuracy all the parameters characterizing our benchmark models. Moreover, the synergy of the two experiments can help in alleviating degeneracies in the parameter measurements - see for example the measurement of σ and Ω ⋆ by ET only in the second row of Fig 4.  \nIn conclusion, this analysis demonstrates quantitatively, by means of the Fisher plots of Fig 4, the advantages of measuring the profile of SGWB over a broad interval, for reconstructing the physics of the PT and the details of the frequency profile around the peak.', '3.2 Cosmic strings': 'Another opportunity for determining the pre-BBN cosmic history of the universe is associated with the detection of GW sourced by a network of cosmic strings. Cosmic strings are basically one-dimensional objects produced by the spontaneous breaking of a U (1) symmetry in the early universe [81, 82], or sometimes considered as fundamental objects, for instance, in superstring theory [83-87]. The essential feature in the GW emmittted by the cosmic strings is that they are sources of very long-standing over the entire history of the evolution of the universe [88-93]. Let us try to understand why: after the formation of the network of cosmic strings, it assumes a constant fraction of the total energy budget of the universe, this is very popularly known as the scaling regime [94-99]. Consequently as long as the strings exist in the universe it will keep on emitting GW during and this happens through most of the universe history. Since frequency of cosmic sources of GW represents time in the early universe (higher frequency means earlier time) this generates a GW spectrum spanning many orders of magnitude in frequencies. Therefore a possible measurement of the  \nGW spectrum from high to lower frequencies will determine the universe expansion rate from early to later times by investigating the features on the cosmic string GW spectrum [100-105]. A detailed study of the impact of various pre-BBN cosmological epochs on the GW spectrum emitted from local and global cosmic strings was carried out in Refs. [105] which clearly predicts the multi-band frequency spectrum of the GW detectors. In fact, the SGWB from cosmic strings can be conveniently studied in synergy between LISA and ET. (See e.g. [106,107] and references therein for a recent assessment in the context of LISA physics.)  \nThe process of string loop formation, evolution, and decay into GWs is quite complex. It is usually studied numerically, although accurate semi-analytical fits for the frequency shape of the SGWB can be determined (see e.g. the recent account [107]). The SGWB characteristics depend on the string tension ( Gµ ), normalized against the gravitational constant G , and on the loop size α , normalized against its time of formation. Typically, the SGWB frequency profile initially increases during the first phase of the decay of string loops into GWs, up to a maximum at the frequency [38]  \nf max ≃ 3 × 10 -8 ( Gµ 10 -11 ) -1 Hz . (3.6)  \nThe value of the quantity Gµ is quite model dependent, but f max usually occurs at frequencies well below the band (3.1) we are interested in. For example, for a specific model of loop distribution, the Ligo-VirgoKagra collaboration sets a bound Gµ ≤ 4 × 10 -15 [108]. Then, at larger frequencies, the SGWB becomes nearly constant, or slightly decaying with an approximately constant slope. We refer the reader to [38] for a more complete discussion and references therein. As a consequence, if future measurements favour a SGWB entering from the left side of the band (3.1) with a negative slope, they will provide circumstantial evidence for a cosmic string origin of the signal. Going beyond the discussions of local cosmic strings as above, there are several additional instances of CS sources as: meta-stable cosmic strings [109], global cosmic strings [110] or cosmic superstrings [111], current-carrying [112] and superconducting strings [113]. Various other topological defects like monopoles and textures can interact with cosmic strings [114]: a separate dedicated analysis would be required since for each of them carries features of top of the standard flat spectrum, which can be tested and searched for in the broad band interval of eq (3.1).  \nMotivated by the previous considerations we shall now discuss two benchmark models. In our first cosmic string benchmark scenario - CS1 - we assume a constant power-law profile in the frequency band 3.1, with a spectral index n 1 = -0 . 1. If the amplitude of Ω GW , proportional to ( Gµ ) 2 /H 2 0 , is sufficiently large, then such a power law profile can be probed with both the LISA and ET instruments. We present this case in the middle panel of Fig 3. Note that the constant slope lies well below the nominal sensitivity curves. Nevertheless, as mentioned above, it can be detected with sufficient SNR by integrating over frequencies. For this reason we represent in the same plot in light green the corresponding broken-power-law sensisitivity curve: more on this in the next section.  \nTable 2: Benchmark values for each cosmic string scenario.  \nLet us also consider another interesting possibility offered by the synergy of LISA with ET: an accurate test of the early expansion of our universe. By measuring the frequency profile of the spectrum we can probe  \n(or constrain) early epochs of non-standard cosmic expansion, preceding big bang nucleosynthesis. Early matter domination eras, kination domination, or the early presence of extra degrees of freedom beyond the Standard Model can modify the string network evolution, and the corresponding dynamics of GW production (see e.g. [115] and references therein for a comprehensive review, and [105] for a dedicated analysis). Nonstandard early cosmological epochs lead to sudden changes, as breaks and features in slope at frequency 9 [38]  \nf break ≃ ( 9 × 10 -3 Hz ) ( T rd GeV )( 10 -12 αGµ ) 1 / 2 . (3.7)  \nBy making appropriate choices of the string network properties, the break position can occur within the interval (3.1). Right after the break, the slope of the spectrum changes to a slope depending on ω - the equation of state during the non-standard cosmological expansion. For ω ≥ 1 / 4, the tilt n 2 is given by: n 2 = -2(3 ω -1) / (3 ω +1) [107]. Hence, knowledge of the spectral tilts n 1 , 2 and of the break position f break offers us crucial information not only on the cosmic string properties, but also on the evolution of the universe prior to BBN. In the middle panel of Fig 3, we show an explicit example of this phenomenon for the benchmark scenario dubbed CS2 in Table 2, where we have chosen ω = 5 / 9.  \nOther than continuous symmetries which when broken leads to GW (as discussed above), domain walls (DWs) [116] are topological defects, are formed when a discrete symmetry in some BSM scenario is broken after inflation. As well studied, during the scaling regime when the DW network evolves and expands along with its surroundings, the energy density stored is ρ DW = c σH [82,117], where σ is the surface tension of the wall and c = O (1) is a scaling parameter. DWs keep on emitting GWs until they annihilate at a temperature given by T = T ann [118-121]. The peak frequency of the resulting GW spectrum from DW annihilation tells us about the horizon size at the time of DW annihilation, f peak = f H ( T ann ). Another important feature is that the frequencies f ≫ f peak the amplitude of the GW spectrum scales as f -1 . Studying closely the approximation for the GW spectrum at the formation time T = T ann are shown in Ref. [117, 122] from which it can be understood the GW spectrum depends on on α ∗ ≡ ρ DW ( T ann ) /ρ r ( T ann ) which is the energy density in the domain walls relative to the radiation energy density ρ r of the universe at the time of DW annihilation. The microscopic physics parameters of the DW model are the relative energy density in DWs, α ∗ , and the temperature at which they annihilate, T ann . Depending upon if the DWs annihilate completely into dark radiation or into visible sector radiation (SM radiation), the energy density can be constrained as the equivalent number of neutrino species [122] ∆ N eff which is constrained by BBN (∆ N eff < 0 . 33) [123] and CMB (∆ N eff < 0 . 3) [124,125].  \nWe point out that the symmetry breaking scale probed by GW in this context is different than in PT scenarios, due to the different microscopic physics involved, see comparative analysis in Ref. [126].  \nFigure 5: Fisher forecasts for the cosmic string benchmark scenarios CS1 and CS2, as summarized in Table 2.  \n<!-- image -->', 'What can we learn about CS from synergies between LISA and ET?': 'The two benchmark scenarios CS1 and CS2 - a single power-law and a broken power-law - are summarized in Table 2. Again, the value of Ω ⋆ in the two cases (and f ⋆ in CS2) are selected with the aim of demonstrating the advantages offered by the synergy of the two instruments. The corresponding Fisher analysis is collected in the plots in Fig 5. Similar to the case of phase transitions, this plot demonstrates the advantages of synergetic measurements with the two experiments for accurately measuring the parameters of each benchmark scenario. Only the two instruments together can measure with a 10 percent accuracy the entire set of parameters. Moreover, a detection in synergy can reduce apparent degeneracies characterizing the detection with single instruments for the case CS1 (see Fig 5, first row).', '3.3 Cosmological inflation': 'Cosmological inflation is a well studied early universe phenomenon capable of producing a stochastic background of gravitational waves (see e.g. [127] for a textbook account). While most minimal models of inflation predict a SGWB amplitude too small to be detected by LISA, there are several well-motivated scenarios which can raise the amplitude of the spectrum to an observable level in the band (3.1). These scenarios are based on multiple field dynamics involving vector and axion fields [128, 129], spontaneous breaking of space-time symmetries [130-132], or secondary effects associated with primordial black hole production (PBH) [133,134] (for reviews see e.g. [135,136]). Different inflationary sources can provide distinct frequency profiles for Ω GW , which can be distinguished when measured by GW experiments. In general, the frequency profile of a SGWB produced by inflation is much richer in features than SGWB produced by other phenomena. It may include a log-normal profile, multiple peaks, or shapes characterised by oscillatory features (see e.g. [137] for examples, and a classification of SGWB templates suitable for describing GW from different inflationary scenarios). Moreover, quite interestingly, the SGWB characteristics depend on the very early cosmological history preceeding BBN, which gets imprinted in the SGWB frequency spectrum. Hence, before focusing on developing forecasts to detect a specific template of inflationary SGWB by means of synergies of LISA and ET, we theoretically further motivate how the frequency profiles of SGWB from inflation can be teach us about early universe evolution prior to BBN.  \nInflationary first-order tensor perturbations: The primordial spectral index n T , defined in terms of log derivative of the power spectrum along the momentum scale, is very crucial in characterizing the inflationary primordial tensor power spectrum. As well known the standard single-field slow-roll inflation predicts a redtilted spectrum, with n T satisfying the slow roll consistency relation n T ≈ -r/ 8 [138], there exists several other scenarios, including blue-tilted spectra ( n T > 0), well-motivated in various UV-complete cosmological models [139-146]. Since this GW spectrum exists all through-out the cosmic history (produced during inflation) the spectrum is a perfect target for multi-band frequency ranges as we allude to in our analysis.  \nJust like the long standing GW spcectrum from cosmic strings, GW from inflation are also ideal targets for probing the period of pre-BBN history, or more simply put, the background equation of state of the universe in the post inflationary era. Let us discuss some examples where the the background equation of state deviates from the standard prediction for radiation domination (1 / 3): for instance in models of quintessential inflationary theories [147-150], or in generic non-oscillatory inflation models [151]. Here the scalar field like the inflaton or some spectator field stays to roll for a long duration of time even much after inflation stops. As a consequently of this the primordial Universe experiences a phase which is known as kination [152,153], during which the kinetic energy fraction of this scalar field is the dominant component of energy budget in the Universe. But this does not happen for too long time as this decreases very fast with expansion rate as ρ ϕ ∝ a -6 before the onset of the traditional radiation domination phase in the universe.  \nThe corresponding background equation of state or also known as the barotropic parameter of the universe during kination is given as w = 1, stiffer than the barotropic parameter during RD ( w = 1 / 3) or matter domination (MD) ( w = 0). It is during this period in the early universe when the tensor perturbation modes if they re-enter the horizon they get boosted in the amplitude instead of being the standard flat spectrum, as is the case when the tensor modes re-entering the horizon during radiation domination. This holds to be true whenever the barotropic parameter if stiffer than radiation-domination ( w > 1 / 3), the exact slope of the GW spectrum depending upon its value. Particularly for frequencies which correspond to the period of kination or such stiff era when the barotropic parameter of the Universe lies in the range 1 / 3 < w < 1 [154-158] or hyper-kination, or via parametric resonance Refs. [159, 160]. Particularly, in [161], it was investigated and found that in order to get a detectable signal in LISA - the stiff period in the post-inflationary epochs must be in the range 0 . 46 ≲ w ≲ 0 . 56 with a high inflationary scale H inf ∼ 10 13 GeV and the reheating temperature in the range 1 MeV ≲ T reh ≲ 150 MeV assuming no blue-tilting ( n T ∼ 0). A realization of this possibility in UV-complete inflationary models was actively studied in details in Ref. [162, 163] along with other interesting predictions. However, intermediate cosmological eras can imprint signatures on such GW spectrum [114, 153, 155, 157, 164, 165]. In this paper we will consider the possibility the in the presence of both LISA and ET operating together how better probes of such thermal histories be realised. Other wellstudied examples of such scenarios are that of a long-lived heavy scalar field generating an early matter era [103, 115, 166-171], a very fast rolling scalar field generating a kination era [147, 153, 156, 165, 172-176], or a supercooled phase transition [70,177-186] or when an extended particle physics sector undergoes decays and scatterings, or when an extended distribution of Primordial Blackholes (PBHs) evaporates in the early universe, as studied in detail in Refs. [187-191].  \nScalar-induced GW: Yet another important and well studied source of cosmological GWs or the so-called scalar induced SGWB (second-order tensor perturbation) [133,134,192-195] particular with boosted interests in very recent times [196-198,198-208,208,209,209-211,211,212,212,213,213-220] due to the fact that this comes hand in hand with the production of the PBH scenarios from primordial inhomogenities.  \nJust like the cosmic strings or the first-order inflationary GW, the induced SGWB is also a very wellrecognised tool to test the thermal history of the universe but however with totally different shapes and features determined by different aspects of microphysics involved as we will describe. Ref. [215] extended the investigations of the induced SGWB for radiation and early matter dominated universes to more arbitrary barotropic parameters w > 0 and predicts multi-band GW frequency spectrum including motivations for Primordial Blackhole domination and its evaporation [190, 191]. Particularly in Ref. [215] it is shown that for an adiabatic perfect fluid the shape of the peak of the spectrum depends on the value of w . In similar lines, Ref. [221] shows the impact in the GW spectrum due to the change in the effective degrees of freedom in thermal history, like those occurring during the QCD and electroweak phase transitions in early universe. In a more general set up, Ref. [222] shows that the infrared side of the GW spectrum has a universal slope given a certain w .  \nThe main interest is the broad frequency profile of these scalar induced GW which is characterised by the epochs at which the high density scalar fluctuations re-enter the horizon (and may collapse to form PBH). In this manner scalar-induced GW probe the thermal history of the universe associated at different (scalar fluctuation or inhomogenity scales or various parts of the scalar potential and the rolling of the scalar field) scales corresponding to different frequencies, see Ref. [223]. We remark the difference between this probe of cosmic history with that of inflationary first-order and cosmic strings lies in the shape of the resultant GW spectrum as well as the microphysics involved.  \nParticle production during and after inflation: Axion or more general pseudoscalar inflation models with  \nparticle production [224-237] are characterised by a pseudoscalar inflaton χ which respects an approximate shift symmetry [231] and a Chern-Simons coupling of the form χF ˜ F to a U (1) gauge field. F denotes the field strength of the gauge field and ˜ F is its dual. This Cherns-Simon like coupling leads to a tachyonic production of a transverse mode of the gauge fields leading to huge particle production of the gauge fields and boosted primordial GW spectrum [128, 144, 238-252]. It is a well known prediction of primordial first-order GW across multi-band frequencies deciphering the particle production. Going beyond the axion inflation, even in models in which an axion or an axion-like particle is not the inflaton, a SGWB can be produced [253-259]. After the axion begins to roll, it induces a tachyonic instability for one of the dark photon helicities, causing vacuum fluctuations to grow exponentially. This effect generates time-dependent anisotropic stress in the energy-momentum tensor, which ultimately sources the tensor perturbations. The GW formation ends when the tachyonic band closes at temperature [253]  \nT ∗ ≈ 1 . 2 √ m a M P g 1 / 4 ∗ ( αθ ) 2 / 3 , (3.8)  \nwhere α is the coupling with the dark photon, θ is the initial misalignment angle, and m a is the mass of the axion. For well motivated and suitable values of axion mass and axion coupling and the initial angle one easily gets a broad multi-band frequency profile corresponding to this tachyonic phase in cosmic history, in the range between LISA and ET as is evident from eqn (3.8).  \nAfter theory motivations aimed at underlying how the detection of GW from inflation can help in characterizing the thermal history of the universe, we focus on forecasting the detectability of a specific SGWB template, with a section, we focus log-normal profile  \nΩ GW = Ω ⋆ exp [ -ln 2 ( f/f ⋆ ) 2 ρ 2 ] , (3.9)  \ncharacterised by three free parameters: Ω ⋆ , f ⋆ , ρ . These parameters control the amplitude, position, and sharpness of the peak, respectively. Such an SGWB profile can be produced by axion or axion spectator models of inflation [260-262] mentioned above: hence it is theoretically well motivated. Explicit particle physics models leading to eq (3.9) are based on the dynamics of the aforementioned models based spectator axion field χ , which rolls during a fraction ∆ N of e-folds of inflation. In this epoch, the axion excites vector gauge fields, through the coupling χF ˜ F . The dynamics and energy density of the latter produces a sizeable SGWB with the lognormal profile (3.9). The SGWB peak f ⋆ occurs at scales corresponding to modes leaving the horizon during the epoch of fastest roll of χ . The height of the peak is sensitive to the quantity ˙ χ , while its width depends on ∆ N . Hence, each of the parameters characterizing the ansatz (3.9) has a clear physical interpretation in terms of quantities characterizing well-motivated underlying scenarios.  \nFigure 6: Fisher forecasts for the cosmic inflation benchmark scenarios Inf1 and Inf2, as summarized in Table 3.  \n<!-- image -->  \nTable 3: Benchmark values for cosmological inflation scenarios described by Ansatz (3.9) .  \nIn Table 3 we collect two representative scenarios (Inf1 and 2) and the associated benchmark values for the parameters corresponding to the log-normal ansatz (3.9). We present their profiles in the right panel of Fig 3, where we also depict in blue the corresponding log-normal sensitivity curve (to be discussed in the next section). The parameters are selected such that their profiles peak in the middle between the LISA and ET bands, with different amplitudes and different peak sharpness. In Fig 6 we plot the corresponding Fisher forecasts. LISA alone would be able to measure the parameters with good accuracy in scenario Inf1. On the other hand, scenario Inf2 would benefit much from the synergy between the two experiments.  \nIn conclusion, the synergy between LISA and ET can help in distinguishing and characterizing early universe sources of SGWB. In the next two sections, we develop and expand upon this topic, considering further concepts and observables to exploit the potential of making detections with the two experiments together.', '4 The notion of integrated sensitivity curves': 'Is there a simple, intuitive way to know whether a given SGWB profile can be detected by GW experiments? The answer is affirmative, thanks to the notion of a sensitivity curve. In this section we discuss various versions of sensitivity curves for LISA, ET, and the two experiments operating together.  \nThe concept of nominal sensitivity curve offers a visual tool to intuitively understand whether a certain GW source with its frequency profile can be detected by a GW experiment. If a given GW signal has a sufficiently large amplitude to cross the sensitivity curve, it will automatically be detected by the particular experiment with a signal-to-noise ratio of greater than unity. The frequency profile of the nominal sensitivity curve depends on the noise sources affecting a given experiment, and on the response of the latter to a GW input. We already discussed and represented the nominal sensitivity curves in section 2 (see Fig 2).  \nBy inspecting Fig 2, we can see that the frequency regions of maximal sensitivity for LISA and ET are different (we call them B LISA and B ET ), and span the ranges  \nB LISA ≃ 10 -5 ≤ f/ Hz ≤ 10 -1 , (4.1)  \nB ET ≃ 10 0 ≤ f/ Hz ≤ 445 . (4.2)  \nAt face value, the detectors do not cover well (due to poor sensitivity) the intermediate region in between say 8 × 10 -2 ≤ f/ Hz ≤ 2. Also, the minimal nominal sensitivity of LISA to Ω GW is around one order of magnitude larger than ET. We have cut off the upper limit of the frequency range for ET at 445 Hz, because beyond this frequency the sensitivity of the instrument decreases beyond the previously mentioned BBN limit of Ω GW ≤ 1 . 7 × 10 -6 , and therefore those frequencies are not of interest to us in this work.  \nHowever, in representing the nominal sensitivity curves as discussed above, we do not make use of the crucial fact that the SGWB signal is extended over decades of frequency ranges. If such a broad frequency profile exists we may integrate over the entire frequency range to allow us obtain more crucial information on the signal by having more large SNR. A broad frequency profile suggests that we can integrate over  \nFigure 7: Green lines: power-law sensitivity (PLS) curves for LISA ( left panel ), ET ( central panel ) and the two experiments combined ( right panel ). For the nominal sensitivity curves in purple we use the analytical fits of Appendix A.  \n<!-- image -->  \nthe frequency range, allowing us to obtain further information on the signal by collecting more SNR (see formula (2.31)). We already came across this feature while discussing Fisher forecasts in section 3. For this reason, [263] introduced the notion of power law sensitivity curve 10 as a useful visual device to understand whether a given signal can be detected by a GW experiment. Following [263], we start by assuming that a given signal is described by a power law profile  \nΩ GW = Ω ⋆ ( f/f ⋆ ) n T , (4.3)  \nover the frequency band we are interested in, with f ⋆ a given reference frequency. The spectral tilt n T is not exactly known though: we assume that it can vary over an interval between two fiducial values.  \nFor each value of the tilt, we can determine the minimal value of the signal amplitude Ω ⋆ the ensures that the corresponding SNR overcomes a certain threshold. We shall consider  \nSNR = 5 , T = 3years . (4.4)  \nThen, we determine the envelope of the resulting curves associated with the various spectral tilts, and draw for each frequency the maximal signal amplitude after scanning over all the tilts.  \nThe result is the so-called power law sensitivity curve (PLS) which we represent in Fig 7. We allow n T to vary between -9 / 2 < n T < 9 / 2, with these numbers chosen for better visualization purposes. The left and central panels show the standard PLS curves associated with LISA and ET, built following the aforementioned algorithm. Every power law spectrum passing above these curves can be detected by LISA (left panel) or ET (central panel) with SNR > 5. In drawing these curves, we have assumed that the signal is a power law over the entire range of frequencies (3.1), but we measure it with one experiment only. Notice that, differently  \nFigure 8: Representation of the broken power-law (BPLS) sensitivity curves, as discussed in the main text.  \n<!-- image -->  \nfrom the nominal curves, the integration over frequencies associated with the notion of PLS curves leads to a sensitivity for ET comparable with the one of LISA.  \nIn plotting the combined PLS in the right panel of Fig 7, we take advantage of the fact that both LISA and ET can measure the same signal independently, hence they can provide a multiband detection of a given SGWB. Again, we assume that the SGWB signal is a power law for the entire frequency range (3.1). We form the total SNR tot using eq (2.31), and use this quantity to draw the combined PLS in the right panel of Fig 7.  \nThe plots in Fig 7 demonstrate that the PLS curves gain orders of magnitude in sensitivity with respect to the nominal curves. If a power law SGWB signal passes above the PLS (but below the nominal sensitivity curve) it can nevertheless be detected by the experiment. Notice that when LISA and ET operate together, the PLS has a low amplitude also within the frequency interval between the sensitivity bands of the individual instruments, where the system would seem to have low sensitivity (see comment after eq (4.2)). In fact, if a power law profile crosses in the middle of each of two experiment bands, e.g. Ω GW ∼ 10 -10 at a frequency f ∼ 10 -2 (above the PLS of the plot of Fig 7, right panel), it certainly crosses the sensitivity curve of one or the other experiment, as it grows towards larger or smaller frequencies through the band (3.1). This feature, due to our hypothesis that the signal is a power-law in the entire band (3.1), explains the much improved PLS sensitivity in the intermediate regions in the right panel of Fig 7, when compared to the PLS sensitivity of the individual experiments.', 'Broken power-law and log-normal sensitivity curves': 'After reviewing the concept of power law sensitivity curves, we shall now discuss the other families of integrated sensitivity curves. As discussed in section 3, we are interested in SGWB profiles described by a broken powerlaw, with the frequency dependence given by the function (3.2), or log-normal profiles associated with ansatz (3.9). For this reason, we go beyond the concept of the PLS curve [263] analysed above, and we discuss the concept of BPLS sensitivity curve as introduced in [265]. (See also [264] for previous related arguments.) We can ask what is the sensitivity of GW experiments towards detecting a SGWB with a particular SNR  \nFigure 9: Log-normal integrated sensitivity curves, built following the method discussed in the main text.  \n<!-- image -->  \ndetection threshold, assuming a broken power-law profile within the range (3.1).  \nIn this case, there are several parameters we can vary: the spectral tilts n 1 , 2 of the growing and decreasing part of the spectrum, the position f ⋆ of the break, and the parameter σ controlling the smoothness of the transition. We independently vary over the spectral tilts n 1 , 2 in the interval -9 / 2 ≤ n 1 , 2 ≤ 9 / 2, over 1 < σ < 10 . 2, as well as over the values of f ⋆ in the ranges of sensitivity of the system. We determine the minimal amplitude in eq (3.2) to ensure we reach an SNR=5 for each set of values of the parameters we examine, and we draw the envelope of the corresponding curves. In the left and central panel of Fig 8 we focus on the individual experiments LISA and ET, varying f ⋆ respectively within the B LISA and B ET bands of eq (4.1). On the right panel we consider the two experiments together, and vary the break position f ⋆ over the entire range (3.1).  \nWhile for the LISA-only and ET-only cases the BPLS curves result in similar shapes and amplitude as the PLS curves in the previous subsection (compare the left and central panels of Fig 7 and Fig 8), the BPLS curve for the two experiments together is very different compared to the PLS curve for the two experiments (compare the right panels of Fig 7 and Fig 8). The reason for this difference is that the break of the BPL might occur in between the LISA and ET bands, where the sensitivity is reduced, and BPL spectra with large spectral tilts (in absolute value) might enter only partially within the sensitivity region of an experiment. Nevertheless, the plot suggests that we can accurately detect BPL SGWB profiles with a break in the middle of LISA and ET bands, and with a relatively small amplitude at the break position. This is a property that we have already explored in the previous section with a Fisher analysis of selected benchmark models for phase transition and cosmic string scenarios.  \nThe method outlined above can also be applied to other SGWB profiles, such as the log-normal described by ansatz (3.9). In fact, we can build integrated sensitivity curves varying over the parameters characterizing eq (3.9). We do so in Fig 9, varying over 0 . 4 < ρ < 1, while f ⋆ varies over the sensitivity bands of the experiments as described in the BPLS case. We notice that in this case the sensitivity of the combined LISA+ET system is not as good as the BPLS curves of Fig 8 in the intermediate band between LISA and  \nET maximal sensitivities. This could be because a lognormal spectrum with a sharp peak in the intermediate band may not enter the sensitivity regions of either of the two experiments at all. Therefore, detecting such a spectrum by combining SNR values from the two instruments would also be difficult unless it had an extremely high amplitude.  \nAs we have learned, the concept of integrated sensitivity curves offers an immediate tool to understand (or guess) the results of more sophisticated analyses based on Fisher forecasts, regarding the combined sensitivity of LISA and ET to SGWB signals. It allows us to visually understand, in a semi-quantitative manner, why the two experiments operating together are more powerful for detecting and characterizing a SGWB. Besides the Ω GW profile, we shall now proceed to discussing another observable which can benefit from joint GW detection: the non-Gaussianity of the SGWB.', '5 The case of a non-Gaussian signal': 'In this section, we wish to discuss at a preliminary level yet another observable which can benefit from synergies between LISA and ET: the non-Gaussian features of the SGWB. Non-Gaussianities arise whenever non-linearities are important in the formation and characterization of the SGWB. It is a well-studied theme in the context of the cosmic microwave background. Yet, its physics needs to be developed further for SGWB sources.  \nWe focus here on non-Gaussianities in the GW statistics produced during inflation (but see [266] for related studies in the context of PT). See e.g. [267], section 5 for a review. It is well-known that intrinsic non-Gaussianities of the SGWB are difficult to directly measure with interferometers [202, 268]. In fact, Shapiro time-delay effects ruin phase correlations that are essential for characterizing most of the non-Gaussian shapes which are possible to directly detect through n -point function measurements. Exceptions are shapes corresponding to soft limits of correlation functions, such as squeezed [269, 270] or collapsed limits [271] of three or higher point functions. In this case, momenta characterizing the Fourier modes entering correlation functions get aligned, and avoid the previously mentioned dephasing time-delay effects (see [270, 271] for extended discussions of aspects of the physics involved) 11 .  \nIntuitively, the synergy between the LISA and ET detectors - which operate at well-separated frequency scales - represents an invaluable opportunity to probe soft limits of GW higher point functions. Soft limits contain a large wealth of physics information (see e.g. [276]), which would be interesting to acquire. Here we take a small step in this direction, and investigate the response of the LISA-ET system to the collapsed limit of the four-point function (the system has vanishing response to the squeezed limit of 3-point function [277]). We assume that the GW four-point correlator is described by the following ansatz:  \n⟨ h λ 1 ( f 1 , ˆ n 1 ) h λ 2 ( f 2 , ˆ n 2 ) h λ 3 ( f 3 , ˆ n 3 ) h λ 4 ( f 4 , ˆ n 4 ) ⟩ f 1 ≪ f 3 = δ λ 1 λ 2 δ λ 3 λ 4 2 δ ( f 1 -f 2 ) δ ( f 3 -f 4 ) δ (3) (ˆ n 1 + ˆ n 2 ) δ (3) (ˆ n 3 + ˆ n 4 ) δ (3) (ˆ n 1 -ˆ n 3 ) S ( f 1 , f 3 ) . (5.1)  \nThe above four-point correlator in Fourier space describes a closed quadrilateral with momenta aligned and two-by-two equal in magnitude, enhanced in a soft counter-collinear limit with a frequency f 1 much smaller  \nthan f 3 . Since f LISA ≪ f ET , such a soft regime can be probed by our set-up. We shall not discuss theoretical motivations and model building perspectives leading to the ansatz (5.1). This will be covered elsewhere, including further analysis of its consequences for GW experiments. Instead, we shall enquire how the LISAET system in synergy responds to the collapsed correlator described by eq (5.1). We wish to measure the four-point amplitude S ( f 1 , f 3 ) in synergy between LISA and ET. The response of the system can be obtained by a generalization of the analysis reviewed in section 2.  \nThe four-point function corresponding to the measured signal - the generalization of eq (2.12) to higher point correlations - is  \n⟨ Φ a 1 ,b 1 c 1 ( f 1 )Φ a 2 ,b 2 c 2 ( f 2 )Φ a 3 ,b 3 c 3 ( f 3 )Φ a 4 ,b 4 c 4 ( f 4 ) ⟩ = δ ( f 1 -f 2 ) δ ( f 3 -f 4 ) 2 [ R (4) ( f 1 , f 3 ) S ( f 1 , f 3 ) + N ] , (5.2)  \nwith R (4) ( f i ) being the four-point response function, and N the noise (to avoid cumbersome expressions, we drop indexes labelling the interferometer channels). In writing the previous formula, we make use of our assumption (5.1) for the GW response function. Geometrically, the above quantity correlates measurements made at two arms of LISA and two arms of ET, in the limit f 1 ≪ f 3 . R (4) ( f ) is a generalization of the two-point response function of eq (2.13), formally given by the integral  \nR (4) ( f 1 , f 3 ) = ∫ d 2 ˆ n 4 π [ F + a 1 b 1 c 1 ( f 1 , ˆ n ) F + a 2 b 2 c 2 ( -f 1 , ˆ n ) + F × a 1 b 1 c 1 ( f 1 , ˆ n ) F × a 2 b 2 c 2 ( -f 1 , ˆ n ) ] LISA × [ F + a 3 b 3 c 3 ( f 3 , ˆ n ) F + a 4 b 4 c 4 ( -f 3 , ˆ n ) + F × a 3 b 3 c 3 ( f 3 , ˆ n ) F × a 2 b 4 c 4 ( -f 3 , ˆ n ) ] ET . (5.3)  \nAs for the case of two-point response functions, orthogonal channels can be found and orthogonal response functions can be numerically evaluated. The complication is that the response function we are dealing with is a four-dimensional tensor. It has four indexes corresponding to each interferometer channel we correlate in the soft limit - two for LISA and two for ET. The result depends on the relative positions among all arms. As a very first step to address the subject, here we shall fix the ET arms along the directions ( ab ) and ( ac ) of eq (2.19), and we do not attempt to diagonalise the channels in the ET sector. We instead diagonalise the two remaining indexes corresponding to the LISA channels.  \nWe again call the orthogonal LISA channels as A ℓ , E ℓ , T ℓ . The computation of such a four-point response function is similar to the diagonalization discussed in section 2, although the results have different amplitude and frequency dependence with respect to the two-point ones. The reason is that we have to deal with extra angular integrations when computing the integrals in eq (5.3). For definiteness, we fix the positions of the vertexes of the interferometer as in Footnote 7. It would be interesting and important to extend our analysis to more general arm orientations.  \nIn the small-frequency limit, the four-point response for the orthogonal channels A ℓ = E ℓ and T ℓ channels result  \nR (4) A ℓ = R E ℓ = 27 140 + O ( f/f ⋆ ) 2 , (5.4)  \nR (4) T ℓ = -17 295680 ( f f ⋆ ) 4 + O ( f/f ⋆ ) 6 . (5.5)  \nNotice that their amplitudes are different from the two-point cases of eqs (2.20) and (2.21), and the R T ℓ now starts at small frequencies with a ( f/f ⋆ ) 4 contribution, instead of a ( f/f ⋆ ) 6 as in eq (2.21). The complete  \n<!-- image -->  \nFigure 10: The response function for the collapsed four-point function, as measurable by the LISA and ET system, for the A , E ( left panel ), T ( right panel ) channels in the LISA sector. Dashed lines: numerical results. Continuous lines: the analytical approximations of eqs (5.6) , (5.7) .  \n<!-- image -->  \nfrequency dependence of the response functions can be easily obtained numerically: see Fig 10. Suitable analytical approximations for these two quantities are give by  \nR (4) A ℓ , E ℓ ( f ) = 27 140 ( 1 + ( f 1 . 1 f ⋆ ) 3 ) -2 / 3 , (5.6)  \nR (4) T ℓ ( f ) = ( f 3 . 3 f ⋆ ) 4 ( 0 . 00315 ( f f ⋆ ) -0 . 0104 ) ( 1 + ( f 3 . 3 f ⋆ ) 7 . 2 ) -1 . (5.7)  \nThese formulae provide the starting point for probing soft limits of GW correlation functions by considering synergies among detectors operating at different frequencies. This is a topic with several theoretical and phenomenological ramifications that we plan to develop elsewhere.', '6 Outlook': "In the next decades the GW interferometers LISA and ET will hopefully be working around the same time. They will operate over different frequency ranges, but will have similar integrated sensitivities to the amplitude of the SGWB. It is important to embark on the quest to investigate what new physics we may learn from synergies between these two detectors. We take a first small step towards that direction in this paper. Particularly we have focused on cosmological sources of GWs, leading to a SGWB characterised by a large amplitude and a broad frequency spectrum spanning several decades in frequency. Operating at different frequency scales, LISA and ET together will have the opportunity to detect distinct features of GWs produced by the same cosmological source. We quantitatively demonstrated this possibility by discussing various earlyuniverse examples motivated by phase transitions, cosmic strings, and inflation, showing that the synergy of the two detectors can improve our measurements of the parameters characterising a cosmological GW source. Moreover, the two experiments operating in tandem can be sensitive to features of early universe cosmic expansion before big-bang nucleosynthesis, which affect the SGWB frequency profile. This probe of early universe of the pre-BBN epoch is challenging if not impossible to test otherwise. Besides considering the GW spectrum, we additionally made a preliminary study of the sensitivity of LISA and ET to soft limits of higher  \norder GW correlation functions. Given that these experiments operate over different frequency bands, their synergy constitutes an ideal direct probe of squeezed limits of non-Gaussian GW correlators, and of its rich physical content.  \nWe leave the important discussion of astrophysical SGWB and/or astrophysical noise sources to a future study. It is well known that astrophysical sources of SGWB can also lead to a broad spectrum of GWs, typically characterised by a broken power law profile. Its shape is controlled by the type of sources of GWs (see e.g. [278] for a review). In order to extract the signal and distinguish between a cosmological SGWB from the one generated by the astrophysical foregrounds, it is necessary to subtract the astrophysical signals expected with sensitivities of BBO and ET or CE windows of frequency ranges [279, 280]. As well known a binary white dwarf galactic and extra-galactic astrophysical foreground also present in LISA is the dominant component as shown in Refs. [33,281,282]. This issue is quite well studied for the case of galactic white dwarfs in the LISA band (see e.g. [283] for a recent analysis). Therefore it should be possible to be subtracted [284-286] in order to disentangle our alluded cosmological effects and signal. In the entire analysis in our present work we assume that such subtractions will be possible. If LISA and ET operate in synergy, and if the cosmological sources lead to a sufficiently broad GW spectrum, it would be possible to obtain extra information about the signal at ET frequencies in order to 'dig out' the properties of GWs in the LISA band through matched filtering techniques.  \nOther important simplifications we made are related to the fact that we neglected the relative motion between the two detectors, and we made simplifying assumptions about the directions of the interferometer arms. Also, we considered the noise models to be fixed, and we did not marginalize over the noise parameters. All these hypotheses will need to be extended in a more complete analysis. We leave all these interesting questions to future studies.  \nUshering in the era of gravitational wave astronomy with the planned network of GW detectors worldwide aspires to and perhaps will be able to achieve measurement precisions that are orders of magnitude better with respect to the present day detectors. This new era of GW detectors, particularly with LISA and ET will make the dream of testing fundamental BSM microphysics, e.g. scales of new physics symmetry breaking, scale of primordial cosmic inflation, probing pre-BBN cosmic epochs, a reality forthcoming in a not-so-distant future.", 'Acknowledgments': 'The work of AMB and GT is partially funded by STFC grant ST/X000648/1. DC would like to thank the Indian Institute of Astrophysics and the Department of Science and Technology, Government of India, for support through the INSPIRE Faculty Fellowship grant no. DST/INSPIRE/04/2023/000110. DC would also like to thank the Indian Institute of Science for support through the C. V. Raman postdoctoral fellowship. DC acknowledges the use of high-performance computational facilities at the Supercomputer Education and Research Centre (SERC) of the Indian Institute of Science and the NOVA HPC cluster at the Indian Institute of Astrophysics. For the purpose of open access, the authors have applied a Creative Commons Attribution licence to any Author Accepted Manuscript version arising. Research Data Access Statement: No new data were generated for this manuscript.', 'A Analytical fits to nominal curves': 'For LISA we use the analytical fits to the nominal sensitivity curve of [34], using h = 0 . 67 (see Fig 2 right panel). In this appendix, we report the expressions for the fit to the nominal ET-D 12 sensitivity curve for the Einstein telescope, which we have presented in the left panel of Fig 2. The fit is given by  \nΩ GW ( f ) = 0 . 88 × ( t 1 + t 2 ) × t 3 t 4 t 5 t 6 , (A.1)  \nwith,  \nt 1 = [ 9 x -30 +5 . 5 × 10 -6 x -4 . 5 +28 × 10 -13 x 3 . 2 ] × ( 1 2 -1 2 tanh (0 . 06 ( x -42)) ) , t 2 = [ 1 × 10 -13 x 1 . 9 +20 × 10 -13 x 2 . 8 ] × 1 2 tanh (0 . 06 ( x -42)) , t 3 = 1 -0 . 475 exp ( -( x -25) 2 50 ) , t 4 = 1 -5 × 10 -4 exp ( -( x -20) 2 100 ) , t 5 = 1 -0 . 2 exp    -( ( x -47) 2 ) 0 . 85 100    , ,  \nt 6 = 1 -0 . 12 exp    -( ( x -50) 2 ) 0 . 7 100    -0 . 2 exp ( -( x -45) 2 250 ) +0 . 15 exp ( -( x -85) 2 400 )  \nwhere x = f/ 1 Hz.  \nWe determined this fit by trial and error. Note that the nominal sensitivity curve provided for the Einstein Telescope, which we are fitting with the above function, is for a pair of interferometers with an opening angle of 90 · [6]. 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