Listening to the sound of dark sector interactions with gravitational wave standard sirens
Weiqiang Yang, Sunny Vagnozzi, Eleonora Di Valentino, Rafael C. Nunes, Supriya Pan, David F. Mota
LListening to the sound of dark sector interactions with gravitational wave standardsirens
Weiqiang Yang, ∗ Sunny Vagnozzi,
2, 3, 4, † Eleonora Di Valentino, ‡ Rafael C. Nunes, § Supriya Pan, ¶ and David F. Mota ∗∗ Department of Physics, Liaoning Normal University, Dalian, 116029, P. R. China The Oskar Klein Centre for Cosmoparticle Physics, Stockholm University,Roslagstullsbacken 21A, SE-106 91 Stockholm, Sweden The Nordic Institute for Theoretical Physics (NORDITA),Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden Kavli Institute for Cosmology (KICC) and Institute of Astronomy,University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais,Avenida dos Astronautas 1758, São José dos Campos, 12227-010, SP, Brazil Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway
We consider two stable Interacting Dark Matter – Dark Energy models and confront them againstcurrent Cosmic Microwave Background data from the
Planck satellite. We then generate luminos-ity distance measurements from O (10 ) mock Gravitational Wave events matching the expectedsensitivity of the proposed Einstein Telescope. We use these to forecast how the addition of Gravi-tational Wave standard sirens data can improve current limits on the Dark Matter – Dark Energycoupling strength ( ξ ). We find that the addition of Gravitational Waves data can reduce the currentuncertainty by a factor of . Moreover, if the underlying cosmological model truly features DarkMatter – Dark Energy interactions with a value of ξ within the currently allowed σ upper limit, theaddition of Gravitational Wave data would help disentangle such an interaction from the standardcase of no interaction at a significance of more than σ . PACS numbers: 98.80.-k, 98.80.Cq, 95.35.+d, 95.36.+x, 98.80.Es.
I. INTRODUCTION
Dark matter (DM) and dark energy (DE), despite theirunknown nature, are two key ingredients of the standardcosmological model. Within the concordance Λ CDM cos-mological model, widely supported by a number of inde-pendent observations (for instance [1, 2, 3, 4, 5]), therole of DM is played by a cold pressureless fluid, whereasthe role of DE is played by a cosmological constant. Inthe standard picture, DM and DE evolve separately, eachobeying a separate continuity equation, and do not inter-act if not gravitationally. However, from a microphysicalperspective, most field theoretical descriptions of DM andDE lead to interactions between the two. In fact, even ifabsent at tree level, a DM-DE interaction will inevitablybe generated at loop level if not explicitly forbidden bya fundamental symmetry [6]. There exist a plethora ofmodels featuring DM-DE interactions, usually referred to ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] See e.g. [7, 8, 9, 10, 11] for examples of explicit particle realiza-tions of DM-DE interactions. Moreover, a number of modified as interacting dark energy (IDE) models, most of whichphenomenological in nature. For an incomplete selec-tion of works examining IDE models from the model-building, theoretical, and observational perspectives, seefor instance [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75,76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91,92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105,106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116]. Fora recent comprehensive review on IDE models, see [117].In 2015, the detection of the gravitational wave (GW)event GW150914 [118] by the LIGO collaboration offi-cially inaugurated the era of GW astronomy, which hasopened an unprecedented window onto tests of funda-mental physics [119, 120, 121]. In particular, the de-tection of the GW event GW170817 [122] and its elec-tromagnetic (EM) counterpart GRB170817A [123] hasmarked the dawn of the multi-messenger astronomy era,and has already been successfully utilized to place tight gravity models can be recasted as interacting DM-DE modelswhen expressed in the Einstein frame. See e.g. [12, 13, 14] forthe case of f ( R ) gravity [15, 16, 17], e.g. [18] for the case ofmimetic gravity and variants thereof [19, 20, 21, 22, 23, 24], ande.g. [25, 26, 27, 28, 29, 30, 31] for discussions in the context ofother theories. a r X i v : . [ a s t r o - ph . C O ] J u l constraints on various aspects of fundamental physics, seee.g. [124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134,135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146](see also the reviews [147, 148]).One particularly intriguing use of GWs is the possibil-ity of exploiting them as standard sirens (SS) [149, 150](see e.g. Chapter 1, Sec. 13 of [147] for a review), apossibility first proposed in [151] to measure the Hubbleconstant H . In fact, an accurately measured GW signalallows one to reconstruct the luminosity distance to thesource d L , and hence can be used as a distance indica-tor. This possibility was in fact first exploited in [152]to provide the first ever SS measurement of H , and forthe first time this new measurement was combined withthe Planck
Cosmic Microwave Background (CMB) datain [153]. The SS measurement of d L is best performedif the observation of an EM counterpart allows one todetermine the redshift z of the source [151], but is inprinciple possible through a more complicated statisti-cal approach even if direct redshift information is lack-ing [151, 154, 155] (although such procedure is not freefrom complications, see for instance [156]).The possibility of using SS to constrain the late-timedynamics of the Universe, including the Hubble con-stant H , the matter density parameter Ω m , and thedark energy equation of state w x , has been contem-plated in a number of works: for an incomplete list, seee.g. [157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167].Moreover, the first work combining the GWs probe andparticle collider constraints was put forward in [168]. Inparticular, the work of [169] first explored the possibilityof reconstructing interactions between DM and DE usingfuture data from LISA.In this work, it is our goal to revisit the possibilityof using GW SS measurements to study interactions be-tween DM and DE. Basing ourselves on the formalismof [166] (see also [170, 171, 172] for later work), we explorehow the use of future GW data can improve constraintson DM-DE interactions, within the context of two stableinteracting DE models proposed by the present authorsin [173].This paper is then organized as follows. In Sec. II weprovide a very brief overview of IDE models, focusing inparticular on the two IDE models presented in [173] andwhich we will consider in this work. In Sec. III we de-scribe the data we will make use of in this work: we con-sider both current data in the form of CMB anisotropyand Baryon Acoustic Oscillation (BAO) distance mea-surements, as well as mock luminosity distance measure-ments from future GW standard sirens data, and brieflydescribe our methodology for generating the mock data.In Sec. IV we present our results forecasting the abil-ity of future GW data to improve current constraints onthe IDE models under consideration. We conclude withclosing remarks in Sec. V. II. STABLE INTERACTING DARK ENERGYREVISITED
In the following, we provide the basic equations de-scribing the evolution of the Universe in presence of DM-DE interactions. As usual, we work within the frameworkof a homogeneous and isotropic Universe described bya spatially flat Friedmann-Lemaître-Robertson-Walker(FLRW) line element. We take the gravitational sectorof the Universe to be described by Einstein’s theory ofGeneral Relativity, whereas we consider a matter sectorminimally coupled to gravity. The energy budget of theUniverse is provided by five species: baryons, photons,neutrinos, dark matter (DM) and dark energy (DE). In particular, we allow DM and DE to interact, with thespecific form of the interaction to be described shortly.In the absence of interactions, the energy-momentumtensors of the DM and DE components are separatelyconserved, i.e. ∇ µ T µνi = 0 , where i stands for DM orDE. In the presence of non-gravitational DM-DE inter-actions, a convenient phenomenological parametrizationof the effect of these interactions is obtained by modifyingthe conservation equations for the stress-energy tensorsas follows [42, 48, 55, 178, 179]: ∇ µ T µνi = Q νi , (cid:88) i Q µi = 0 , (1)where again i stands for either DM ( i = c ) or DE ( i = x ).The four-vector Q µi specifies the coupling between thedark sectors, and characterizes the type of interaction.We take Q µi to assume the following form: Q µi = ( Q i + δQ i ) u µ + a − (0 , ∂ µ f i ) , (2)where u µ is the velocity four-vector, Q i is the backgroundenergy transfer, and f i is the momentum transfer poten-tial. From now on, we take Q i ≡ Q . In addition, weconsider the simplest possibility wherein the momentumtransfer potential is zero in the rest frame of DM, seee.g. [42, 48, 55, 178, 179] for more details.All that is left, therefore, is to specify the functionalform of Q . In what follows, we will consider a particularclass of IDE models, whose appeal is their being free fromearly-time linear perturbation instabilities. These modelswere proposed by us in [173], and we refer the readerthere for more details. We consider the same models,keeping the same labels IDErc1 and
IDErc2 respectively,whose coupling functions Q take the form [173]: Q = 3(1 + w x ) Hξρ c ( IDErc1 ) , (3) Q = 3(1 + w x ) Hξ ( ρ c + ρ x ) ( IDErc2 ) . (4) We fix the total neutrino mass to M ν = 0 .
06 eV , as done in the
Planck baseline analyses. Given the currently very tight upperlimits on M ν , of order . [5, 174, 175, 176, 177], we do notexpect the introduction of massive neutrinos to the picture willchange our conclusions substantially. In Eqs. (3,4), w x denotes the DE equation of state(EoS), H is the Hubble expansion rate, whereas ρ c and ρ x are the DM and DE energy densities respectively. Fi-nally, the parameter ξ controls the strength of the DM-DE coupling, with ξ = 0 representing the standard caseof no interaction. The introduction of the factor (1 + w x ) in Eqs. (3,4), absent in earlier IDE works, allows for sta-ble early-time perturbations independently of the valueof w x (whereas earlier models usually featured stableperturbations only for either quintessence-like or phan-tom DE EoS, limiting the possibilities in terms of DEfluid allowed to interact), see [173, 180, 181] for furtherdiscussions. In particular, as shown in [173], both the IDErc1 and
IDErc2 models are stable for ξ > . Ouraim is to explore how the addition of future luminositydistance measurements from GW standard sirens can im-prove constraints from current cosmological data on theDM-DE coupling strength ξ . III. OBSERVATIONAL DATA ANDMETHODOLOGY
In the following, we describe in more detail the ob-servational datasets (both current and future) we in-clude in this work. In terms of current observationaldatasets, we make use of measurements of Cosmic Mi-crowave Background (CMB) temperature and polariza-tion anisotropies and their cross-correlations from the
Planck
Planck likelihood [186]. We refer to this dataset as “CMB” (notethat this dataset is usually referred to as
PlanckTT-TEEE + lowTEB in the literature).To forecast the constraining ability of future GW stan-dard sirens (SS) luminosity distance measurements, wegenerate mock data matching the expected sensitivityof the Einstein Telescope. The Einstein Telescope is aproposed third-generation ground-based GW detector,whose main objectives will be to test General Relativityin the strong field regime and advance precision GW as-tronomy [187]. Although full instrumental details are stillunder study, the Einstein Telescope will likely be locatedunderground and will feature three O (10 km) long armsarranged following an equilateral triangle. Each arm willbe composed of two interferometers, optimized for oper-ating at frequencies of O (1 − and O (0 . − respectively [187]. After 10 years of operation, the Ein-stein Telescope is expected to detect O (10 ) GW SSevents.Our goal is then to generate a luminosity distancecatalogue matching the expected sensitivity of the Ein-stein Telescope after 10 years of operation. We gener-ate triples ( z i , d L ( z i ) , σ i ) , with z i the redshift ofthe GW source, d L the measured luminosity distance,and σ i the uncertainty on the latter. There are threeaspects to take into consideration when generating thismock data: the fiducial cosmological model enters both in z i (or more precisely into the redshift distribution of ex-pected sources) and d L , the expected type of GW sourcesenter in z i , and finally the instrumental specifications en-ter in σ i .We now very briefly summarize the procedure adoptedfor generating the mock GW data and further details onthe generation of the mock GW standard sirens datasetare presented in Appendix A. In addition, we encouragethe reader to consult [166] for further technical details onthe procedure, which is the same as that adopted here.The first step is to specify the expected GW sources. Weconsider a combination of black hole-neutron star andbinary neutron star mergers. The ratio of number ofevents for the former versus the latter is taken to be . ,with mass distributions specified in [166]. Following [166,188, 189, 190, 191], we then model the merger rate of thesources R ( z ) , and from their merger rate we are able todetermine their redshift distribution P ( z ) (see [166] fordetailed formulas). Once we have P ( z ) , we sample values of redshifts from this distribution: these will bethe redshifts of our mock GW events z i .Note that going from merger rate to redshift distribu-tion requires a choice of fiducial cosmological model, asthe expression for P ( z ) contains both the comoving dis-tance and expansion rate at redshift z , χ ( z ) and H ( z ) respectively (see [166]). Since our goal is to explore howGW data can improve our constraints on IDE models,we generate two different mock GW datasets choosingas fiducial models first the IDErc1 and then the
IDErc2 models. We adopt fiducial values for the cosmological pa-rameters given by the best-fit values of the same parame-ters obtained analysing the models with the CMB datasetpreviously described. Using the same choice of fiducialmodel(s) and parameters, and the values of redshifts wesampled from P ( z ) , we then compute the luminosity dis-tances at the respective redshifts, d L ( z i ) , through: d L ( z i ) = (1 + z i ) (cid:90) z i dz (cid:48) H ( z (cid:48) ) . (5)Having obtained z i and d L ( z i ) , all that is left to com-plete our mock GW luminosity distances catalogue is σ i .We determine these error bars following the Fisher matrixapproach outlined in [166]. In summary, this is achievedby modelling the observed GW strain as linear combina-tion of the two GW polarizations weighted by the twobeam pattern functions. Instrumental specifications en-ter in determining exact functional form of the beam pat-tern functions. We follow [164, 190, 192] in determiningthe functional form of the beam pattern functions forthe Einstein Telescope. We then compute the Fouriertransform of the observed GW strain, h ( f ) . In particu-lar, the phase and amplitude of the Fourier transform (infrequency domain) are estimated for the post-Newtonianwaveform TaylorF2 at 3.5 post-Newtonian order (3.5 PNin the standard notation). When generating the data,during the integration of the signal-to-noise ratio we as-sume a minimum frequency f min = 1 Hz and a maximumfrequency f max = 2 f isco , where f isco is the orbital fre-quency of the last stable orbit associated with each sim-ulated event.The crucial point to note is that amplitude of theGW signal is inversely proportional to the distance lu-minosity distance d L ( z ) (see appendix A). Thus, assum-ing that the errors on d L ( z ) are uncorrelated with errorson the remaining waveform parameters (which is trueif the distance to source does not correlate with otherparameters), it is possible to show that the instrumen-tal uncertainty on the luminosity distance is given by σ ins d L ( z i ) ∝ d L ( z i ) / SNR i , where SNR is the signal-to-noiseratio associated to the i th event (see Appendix A forfurther details on the estimation of SNR and the instru-mental uncertainty).In addition to instrumental uncertainties, GWs are alsolensed along their journey from the source to us. Thisresults in an additional lensing uncertainty which follow-ing [190] we take to be . z i d L ( z i ) . Adding in quadra-ture the instrumental and lensing errors we obtain thetotal error σ i . Finally, we randomly displace the previ-ously determined luminosity distances d L ( z i ) by quan-tities ∆ i ∼ N (0 , σ i ) , i.e. the displacements are drawnfrom normal distributions with mean and standard de-viation σ i . For further details, we invite the reader toconsult [166]. The resulting mock dataset is referred toas “GW”.This concludes the generation of the GW SS luminos-ity distance catalogue ( z i , d L ( z i ) , σ i ) . In Fig. 1 we showthe mock GW data generated assuming as fiducial mod-els the IDErc1 (left) and
IDErc2 (right) models, withthe black solid curves representing the theoretical pre-diction for d L ( z ) . We model the GW likelihood as aproduct of Gaussians in d L ( z i ) (with standard deviation σ i ), one Gaussian for each GW SS distance measurementin our mock catalogue. Therefore, we have modelled the1000 events as being independent, and have neglected thecross-covariance between these measurements. More ac-curate analyses once real data is available should take thiscross-covariance into account, as well as possible mod-elling systematics. For the time being, we have followed amore simple treatment and neglected these effects, whichwe will return to in future work.We analyse two dataset combinations: CMB andCMB+GW. As previously stated, we generate the mockGW data using the best-fit values of the cosmologicalparameters from the CMB-only analysis. The only ex-ception is the DM-DE coupling ξ (whose best-fit valuewould nominally be ξ = 0 ). We take the fiducial valueof ξ to be ξ = 0 . and ξ = 0 . when generating theGW data for the IDErc1 and
IDErc2 models respectively.Both these values are within the confidence level(C.L.) upper limits obtained analysing the CMB dataset.The reason is that we want to check whether a non-zerocoupling currently allowed by CMB data at
C.L. ispotentially discernible from zero coupling once GW SSdata is added. In other words, we want to understandwhether the addition of GW SS data can improve theuncertainty on ξ to the point that we can detect non- zero ξ , should the true model chosen by Nature reallyfeature dark sector interactions with a strength allowedby current data.Finally, at a later stage we consider Baryon AcousticOscillations (BAO) distance measurements from the 6dFGalaxy Survey [193], the Main Galaxy Sample of DataRelease 7 of the Sloan Digital Sky Survey [194], and theCMASS and LOWZ samples of Data Release 12 of theBaryon Oscillation Spectroscopic Survey [195]. We referto this dataset as “BAO”. Since BAO data is usually usedin combination with CMB data to break parameter de-generacies, our goal is to check whether the CMB+GWcombination will yield parameter constraints comparableto, or better than, the CMB+BAO combination. We fur-ther discuss our use of BAO data in Appendix B, wherewe also provide a more detailed description of the BAOmeasurements.In principle, we could also have chosen to includetype-Ia Supernovae (SNeIa) luminosity distance measure-ments. One could in fact expect that the most importantgain in combining GW and SNeIa distance measurementswould be the fact that the two suffer from completely dif-ferent systematic uncertainties. However, given that inour use of GW data we have not attempted to modelsystematic uncertainties, it is unlikely that such a gainwould be appreciable in our analysis. For this reason,and in the interest of conciseness, we have chosen not toinclude SNeIa measurements.We work within the framework of a 8-dimensional Λ CDM+ w x + ξ cosmological model, described by theusual 6 Λ CDM parameters (the baryon and CDM physi-cal densities Ω b h and Ω c h , the angular size of the soundhorizon at recombination θ s , the amplitude and tilt ofthe primordial power spectrum of scalar fluctuations A s and n s , and the optical depth to reionization τ ), the DEEoS w x , and the DM-DE coupling ξ . We impose flatpriors on all these parameters unless otherwise stated,with prior ranges shown in Tab. I. Note that ξ > isrequired for the perturbations of both the IDErc1 and
IDErc2 models to be stable [173]. We sample the pos-terior distribution of the parameter space using MarkovChain Monte Carlo (MCMC) methods, and generatingMCMC chains through the publicly available MCMCsampler
CosmoMC [196]. The convergence of the generatedchains is monitored through the Gelman-Rubin statistic R − [197]. IV. RESULTS
We now examine the improvements in the constraintson the IDE parameters (especially the DM-DE coupling ξ ) within the IDErc1 and
IDErc2 models, brought uponby the addition of GW SS data. As described in Sec. III,initially we constrain the parameters of the two IDE mod-els using CMB data alone. Using the obtained best-fitvalues for all cosmological parameters (except the DM-DE coupling ξ , see above), we generate two mock GW d L ( G p c ) IDErc with CMB d L ( G p c ) IDErc with CMB
FIG. 1. Mock d L ( z ) measurements resulting from 1000 simulated GW events assuming a fiducial IDErc1 model (left panel)and
IDErc2 model (right panel). The fiducial cosmological parameters are the best-fit values obtained when constraining thesemodels against CMB data alone, except for ξ which is fixed to ξ = 0 . and ξ = 0 . for the IDErc1 and
IDErc2 modelsrespectively.
55 60 65 70 75 H . . . . ξ . . . . . Ω m . . . . σ w x H ξ Ω m σ IDErc1 wth CMBIDErc1 wth CMB+GW
FIG. 2. 1D marginalized and 2D joint posterior distributions for selected parameters (including a selection of derived parame-ters) of the
IDErc1 model whose determination is particularly improved by the inclusion of the GW dataset: w x , ξ , Ω m (thetotal matter density parameter today), σ , and H (in km s − Mpc − ). Contours are obtained using only CMB data (grey)and CMB+GW data (red). SS catalogues (one for each IDE model). We finally com-bine CMB and GW data and examine the improvementon the uncertainties of the cosmological parameters withrespect to the case where only CMB data is used, andwith respect to the case where the CMB+BAO dataset combination is considered.
Parameter Prior range Ω b h [0 . , . c h [0 . , . τ [0 . , . n s [0 . , . A s ] [2 . , θ s [0 . , w x [ − , ξ [0 , TABLE I. Prior ranges imposed on the 8 cosmological pa-rameters of the interacting DE-DM models considered in thiswork.
A. Results for model
IDErc1
In Tab. II, we report observational constraints on theparameters of the
IDErc1 model from the CMB andCMB+GW dataset combinations. In Fig. 2 we insteaddisplay a triangular plot showing the 1D marginalizedand 2D joint posterior distributions for selected parame-ters (including a selection of derived parameters) whosedetermination is particularly improved by the inclusionof the GW dataset: w x , ξ , Ω m (the total matter den-sity parameter today), σ , and H . Fig. 2 also showsthe correlations/degeneracies between these parametersand how some of these degeneracies are broken by theaddition of GW data.The introduction of GW data clearly leads to a sub-stantial improvement in the determination of certain pa-rameters, such as Ω m and H , whose error bars havebeen reduced by about a factor of . In fact, while theCMB alone determines H and Ω m with an accuracy of and respectively, we expect an improvement re-ducing these uncertainties down to on H and on Ω m with the addition of GW SS data. This improvementis not surprising, as within a flat Universe and within theredshift range probed by GW SS data, the backgroundexpansion and hence the luminosity distance-redshift re-lation is mostly governed by Ω m and H . The same istrue for ξ , as a small amount of energy transfer betweenDM and DE is sufficient to alter the background expan-sion by an appreciable amount. From Tab. II, we see thatit should be possible to detect a non-zero ξ = 0 . at asignificance of about σ . From Fig. 2, the improvementin the determination of ξ after marginalizing over theother cosmological parameters is appreciable. This im-proved determination is particularly helped by the factthat the ξ - Ω m and ξ - H degeneracies are almost brokenby the addition of GW data. Perhaps surprisingly, thereis only a slight improvement in the determination of theDE EoS w x .From Fig. 2, it is clear that the power of GW SS datagoes beyond solely an improvement in the determinationof background quantities. In fact, we see that the inclu-sion of GW SS data has halved the uncertainty on σ ,which probes the growth of structure. At a first glance Parameters CMB CMB+GW Ω c h . +0 . . − . − . . +0 . . − . − . Ω b h . +0 . . − . − . . +0 . . − . − . w x − . +0 . . − . − . − . +0 . . − . − . ξ . +0 . . − . − . . +0 . . − . − . Ω m . +0 . . − . − . . +0 . . − . − . σ . +0 . . − . − . . +0 . . − . − . H . +3 . . − . − . . +0 . . − . − . TABLE II. Observational constraints on selected cosmologicalparameters within the
IDErc1 model. Constraints on H arereported in units of km s − Mpc − and Ω m = Ω b + Ω c is thetotal matter density today. this might seem surprising, since σ cannot be “directly”probed by background measurements. In fact, the im-provement in the determination of σ is mostly “indirect”:a better determination of background quantities whichare strongly degenerate with σ will naturally lead to animproved determination of the latter. From Fig. 2, wesee that the improvement in the determination of Ω m isparticularly helpful in this sense, since the introductionof GW SS data has almost completely broken the σ - Ω m degeneracy.In Fig. 3 we can compare the constraints obtained com-bining the CMB first with BAO measurements, and thenwith GW SS data. We can see that the CMB+GWcombination will perform as well as the CMB+BAOcase for most of the parameters, and will improve byabout a factor of the determination of H and Ω m .For example, for the CMB+BAO case we have σ H ∼ . − Mpc − and σ Ω m ∼ . , while we fore-cast σ H ∼ . − Mpc − and σ Ω m ∼ . forCMB+GW.However, it is worth commenting on a subtle pointconcerning the CMB+GW versus CMB+BAO compari-son. This comparison is in reality not totally fair withregards to BAO measurements. It is in fact a compari-son between measurements at completely different timescales: future GW data against current BAO data. Theestimated O (10 ) GW SS events will only be availableat the end of the Einstein Telescope science run, whichwill likely conclude between 2040 and 2060, a couple ofdecades from now. On the other hand, already withinthe next decade we will have sub-percent BAO measure-ments from a host of state-of-the-art large-scale structuresurveys such as DESI [182], Euclid [183], and LSST [184].These measurements will considerably improve limits oncosmological parameters, including the DM-DE coupling ξ . One can safely extrapolate that such limits would sur-pass those of the CMB+GW combination in constrain-ing power. Moving beyond these experiments, we do notyet know what the future of large-scale structure sur-veys has in store (especially not in 2060), but it is safeto say that the limits on dark sector interactions fromBAO measurements on that time scale should consider-ably surpass those from the Einstein Telescope (unless
66 68 70 72 74 H . . . . ξ . . . . Ω m . . . . σ w x H ξ Ω m σ IDErc1: CMB+BAOIDErc1: CMB+GW
FIG. 3. 1D marginalized and 2D joint posterior distributions comparing the CMB+BAO case (red contours) with CMB+GW(green contours), for selected parameters (including a selection of derived parameters), assuming the
IDErc1 model. the estimated O (10 ) GW SS events were somehow toopessimistic and we should serendipitously detect a sig-nificantly larger number of GW SS events). In any case,while we can certainly say that the addition of GW SSdata to current CMB data will lead to important im-provements in our constraints on dark sector interactions,one should keep in mind that the comparison with BAOdata we have performed here is carried out between mea-surements living on completely different time scales.
B. Results for model
IDErc2
Our findings for the
IDErc2 model are summarized inTab. III and Fig. 4, which are completely analogous totheir
IDErc1 counterparts. The results we find are ex-tremely similar to those we found for the
IDErc1 model.In particular, we find that the introduction of GW datasubstantially reduces the error bars on quantities gov-erning the background evolution such as Ω m and H ,whose uncertainties are reduced by factors larger than .In particular, while H is determined with an accuracyof and Ω m of by the CMB, we predict an im-provement up to . on H and on Ω m with theinclusion of the GW SS data.Partially by breaking the ξ - Ω m and ξ - H degenera-cies, and also by better constraining the background evo- lution which is sensitive to ξ , GW data allow for a sub-stantially more precise determination of ξ . In fact, wefind that the σ uncertainty on ξ has roughly halved af-ter using GW SS data. In particular, we have found thatit should be possible to detect ξ = 0 . at > σ , i.e. at relatively high significance. As for the IDErc1 model,we find that there is only a marginal improvement in thedetermination of w x , even if slightly larger in this case.On the other hand, we again find that the introduction ofGW SS data halves the uncertainty on σ . Again, this isan “indirect” effect mostly brought upon by the fact thatthe σ - Ω m degeneracy is almost broken by the additionof GW data.Finally, comparing the constraints obtained combiningthe CMB and BAO measurements versus CMB and GWSS data, we find results completely analogous to thoseobtained for the IDErc1 model (which for concisenesswe don’t show here). That is, CMB+GW will performbetter than CMB+BAO, improving also in this case con-siderably the determination of H and Ω m . However,we wish to remind the reader once more that the GWand BAO data we have considered live at completely dif-ferent time scales (the final data from the Einstein Tele-scope is projected to be available between 2040 and 2060,whereas we have only considered current and not fore-casted BAO data), and thus the comparison is perhapsnot completely fair towards BAO measurements. Fore-
55 60 65 70 75 H . . ξ . . . . Ω m . . . . σ w x H ξ Ω m σ IDErc2 wth CMBIDErc2 wth CMB+GW
FIG. 4. As in Fig. 2, for the
IDErc2 model.
Parameters CMB CMB+GW Ω c h . +0 . . − . − . . +0 . . − . − . Ω b h . +0 . . − . − . . +0 . . − . − . w x − . +0 . . − . − . − . +0 . . − . − . ξ . +0 . . − . − . . +0 . . − . − . Ω m . +0 . . − . − . . +0 . . − . − . σ . +0 . . − . − . . +0 . . − . − . H . +3 . . − . − . . +0 . . − . − . TABLE III. As in Tab. II, for the
IDErc2 model. casts from next-generation BAO measurements (or evenfrom BAO measurements available in 2060, although wecurrently have no idea what the projected sensitivity forBAO measurements will be at that point) would certainlylead to tighter constraints than those we have forecastedfor the CMB+GW combination.
V. SUMMARY AND CONCLUSIONS
Interacting dark energy models, wherein dark matterand dark energy interact through couplings other thangravitational, have received renewed interest recently.While originally motivated by the possibility of thereinaddressing the coincidence problem, recently several of these models have been shown to potentially be able toaddress a number of discrepancies between high- andlow-redshift determinations of cosmological parameters( e.g. the H tension and the σ tension). Current ob-servational datasets place relatively tight limits on thestrength of the DM-DE coupling ξ (with C.L. up-per limits of order ξ (cid:46) . depending on the IDE modelunder consideration), but are far from ruling out the pos-sibility of DM-DE interactions.With this in mind, our goal in this paper has been thatof investigating how future distance measurements fromGW standard sirens might improve constraints on IDEmodels. We considered two IDE models featuring sta-ble early-time linear perturbations, originally proposedin [173]. After generating O (10 ) mock GW standardsirens luminosity distance measurements matching theexpected sensitivity of a 10-year run of the Einstein Tele-scope, a third-generation ground-based GW detector, wehave studied how the addition of these GW distance mea-surements can improve current determinations of cosmo-logical parameters within IDE models based on CMBdata from Planck . Our results show that the introductionof GW data is extremely helpful in pinning down back-ground quantities such as H and Ω m . We find thatGW standard sirens distance measurements can reducethe uncertainty on the DM-DE coupling ξ by up to afactor of or more. We show that DM-DE interactionswith strength within the current σ upper limit shouldbe detectable at potentially more than σ with futureGW data from the Einstein Telescope.In conclusion, in this work we have demonstrated thatfuture GW standard sirens distance measurements fromthe Einstein Telescope are expected to provide a power-ful window onto the physics of dark sector interactions.Therefore, it might in principle just be a matter of timebefore we might be able to convincingly detect dark sec-tor interactions, which the current discrepancies betweenhigh- and low-redshift cosmological probes might be hint-ing to. The methodology we have adopted in this paper isuseful for analysing several cosmological models beyondinteracting dark energy. It might be promising to adopta similar methodology to investigate modified theoriesof gravity, although in such case one must take into ac-count the fact that the waveform and hence the observedGW strain will be modified. It would also be interest-ing to investigate the sensitivity of future GW detectorsother than the Einstein Telescope, in combination withfuture ground-based CMB surveys such as Simons Ob-servatory [198] and CMB-S4 [199] as well as BAO mea-surements from future large-scale structure surveys suchas DESI [182], Euclid [183], LSST [184]. A more accu-rate modelling might potentially improve the impact ofGW data [163] (or, conversely, a smaller number of GWevents would be required to reach the sensitivity we haveforecasted in this work). We plan to develop these andrelated issues in future work.
ACKNOWLEDGMENTS
The authors wish to thank the referee for her/his veryuseful comments, which have helped us to improve thequality of the discussion. W.Y. acknowledges financialsupport from the National Natural Science Foundationof China under Grants No. 11705079 and No. 11647153.S.V. acknowledges support by the Vetenskapsrådet(Swedish Research Council) through contract No. 638-2013-8993 and the Oskar Klein Centre for CosmoparticlePhysics, and from the Isaac Newton Trust and the KavliFoundation through a Newton-Kavli fellowship, andthanks the University of Michigan, where part of thiswork was conducted, for hospitality. E.D.V. acknowl-edges support from the European Research Council inthe form of a Consolidator Grant with number 681431.S.P. acknowledges partial support from the FacultyResearch and Professional Development Fund (FRPDF)Scheme of Presidency University, Kolkata, India. D.F.M.thanks the Research Council of Norway for their support.
APPENDIX A: FURTHER DETAILS ON THEGENERATION OF THE MOCK GW STANDARDSIRENS DATASET
In this Appendix, we provide further technical detailson the well-known methodology used to generate themock GW standard sirens dataset, which we briefly de-scribed in Sec. III. We also encourage the reader to con-sult [166]. This method was used to obtain the uncertain-ties on the luminosity distance measurements associatedto the binary black hole-neutron star and binary neutronstar merger events.During the coalescence phase, the GW waveform isvery well described by the stationary phase approxima-tion, wherein it takes the form: h ( f ) = Af − / e i Φ( f ) , (6)where the amplitude A is given by: A = 1 d L (cid:113) F (1 + cos ( ι )) + 4 F × cos ( ι )) × (cid:112) π/ π − / M / c . (7)In the above, M c , ι , F + , and F x are respectively the red-shifted chirp mass, angle of inclination of the binary or-bital angular momentum with the line-of-sight, and twoantenna pattern functions associated with the EinsteinTelescope (see Eq. (18) in [192] for the exact functionalform of the antenna pattern functions). Finally, d L ( z ) is the luminosity distance to the redshift of the merger,and is our physical observable of interest here. In Eq. (6),the function Φ( f ) is the inspiral phase of the binary sys-tem, computed perturbatively within the so-called post-Newtonian formalism. Here, we follow the standard as-sumption of assuming a correction up to 3.5 PN ordergiven by the TaylorF2 waveform (for more details see theexpansion of the coefficients in [200]).Once the waveform is well defined, the other rele-vant quantity for generating the mock catalogue is thesignal-to-noise ratio (SNR) associated with each simu-lated event. The SNR is given by: SNR ≡ Re (cid:90) f max f min df | h ( f ) | S n , (8)where S n ( f ) is the spectral noise density of the EinsteinTelescope detector (see Eq. (19) in [192]). The uppercutoff frequency f max is determined by the last stableorbit (isco), which marks the end of the inspiral regimeand the onset of the final merger. We assume f max =2 f isco Hz, where f isco = 1 / / πM c . The lower cutofffrequency f min is instead determined by the sensitivity ofthe Einstein Telescope, so we set f min = 1 Hz .We are now ready to estimate the instrumental erroron d L ( z ) , which is given by σ ins d L (cid:39) d L / SNR . The factorof 2 has been introduced to take into account the effect ofthe inclination angle for which the GW amplitude is max-imum. On the other hand, GWs are lensed in the sameway photons are, resulting a weak lensing effect error0which we model as σ lens d L = 0 . zd L ( z ) [192]. In doing so,we have not considered possible errors induced from thepeculiar velocity due to the clustering of galaxies. Sinceour simulated events are at relatively high z , this is a safeassumption, as significant corrections due to peculiar ve-locities of galaxies are significant only for z (cid:28) . In fact,at high z the dominant source of uncertainty is the onedue to weak lensing. Finally, the total uncertainty σ d L on the luminosity distance measurements associated toeach event are obtained by combining the instrumentaland weak lensing uncertainties in quadrature, as follows: σ d L = (cid:113)(cid:0) σ ins d L (cid:1) + (cid:0) σ lens d L (cid:1) = (cid:115)(cid:18) d L ( z )SNR (cid:19) + (0 . zd L ( z )) . (9)To conclude, as discussed in Sec. III, when generatingthe luminosity distance measurements themselves (beforeeven generating their uncertainties), one has to modelthe distribution of GW events. The redshift distributionof the sources, taking into account evolution and stellarsynthesis, is well described by: P ( z ) ∝ πχ ( z ) R ( z ) H ( z )(1 + z ) , (10)where χ is the comoving distance and R ( z ) describes thetime evolution of the burst rate and is given by R ( z ) =1 + 2 z for z < , R ( z ) = 3 / − z ) for ≤ z ≤ , and R ( z ) = 0 for z > .Finally, the input masses of the neutron stars and blackholes are randomly sampled from uniform distributionswithin [1 −
2] M (cid:12) and [3 −
10] M (cid:12) respectively. Whengenerating our mock GW events, we have only consideredmergers with
SNR > . APPENDIX B: A COMMENT ON BAO DATA
In this Appendix, we provide further details on theBAO measurements we used throughout the work. Weuse both isotropic (for [193] and [194]) and anisotropic(for [195]) measurements of the BAO scale. In general,BAO data measure the ratio between a distance scale D and the length of a standard ruler r s . What exactly thedistance scale is depends on whether the measurement isisotropic or anisotropic, and we will return to this pointlater. As for the standard ruler, we make the assumptionthat r s coincides with the sound horizon at baryon drag,usually denoted by r d ( z d ) . This assumption is strictlyspeaking only valid for Λ CDM [201, 202, 203, 204] andmight be broken when considering more exotic models.However, this should not be an issue for the IDE mod-els under consideration, given that the deviation from Λ CDM therein is small (especially given the size of theallowed values of ξ ). Therefore, in our work we havemade the assumption that the standard ruler for BAOmeasurements is given by the sound horizon at baryondrag.We now comment more on the distance scale appearingin BAO measurements. An isotropic BAO measurementwill constrain the ratio D V ( z eff ) /r s between the dilationscale D V (at the effective redshift of the survey z eff ) andthe length of the standard ruler r s . The dilation scale asa function of redshift is given by: D V ( z ) = (cid:20) (1 + z ) D A ( z ) czH ( z ) (cid:21) , (11)where D A ( z ) and H ( z ) denote the angular diameter dis-tance and Hubble rate at redshift z respectively. On theother hand, anisotropic BAO measurements can separatecorrelations along and across the line-of-sight. In thiscase, correlations along the line-of-sight will constrain theproduct H ( z ) r s , whereas correlations across the line-of-sight will constrain the ratio D A ( z ) /r s .One final point worth mentioning is that BAO mea-surements are usually obtained by analysing the 2-pointcorrelation function (or power spectrum, or both) of alarge-scale structure tracer sample (for example lumi-nous red galaxies). Computing the correlation func-tion or the power spectrum requires a choice of fidu-cial cosmology, within which one can compute fiducialvalues for all the scales of interest: D fid V ( z ) , D fid A ( z ) , H fid ( z ) , and r fid s ( z ) . Technically speaking, BAO mea-surements then actually constrain quantities such as [ D V ( z eff ) /r s ] / [ D V ( z eff ) /r s ] fid (where of course the quan-tity [ D V ( z eff ) /r s ] fid is a constant once the fiducial cos-mology is specified), and correspondingly for anisotropicmeasurements. [1] A. G. Riess et al. [Supernova Search Team], Observa-tional evidence from supernovae for an accelerating uni-verse and a cosmological constant,
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