Using a multi-messenger and multi-wavelength observational strategy to probe the nature of dark energy through direct measurements of cosmic expansion history
Jing-Zhao Qi, Shang-Jie Jin, Xi-Long Fan, Jing-Fei Zhang, Xin Zhang
UUsing a multi-messenger and multi-wavelength observational strategy to probe thenature of dark energy through direct measurements of cosmic expansion history
Jing-Zhao Qi, Shang-Jie Jin, Xi-Long Fan, ∗ Jing-Fei Zhang, and Xin Zhang † Department of Physics, College of Sciences, & MOE Key Laboratory of Data Analytics and Optimization for Smart Industry,Northeastern University, Shenyang 110819, China School of Physics and Technology, Wuhan University, Wuhan 430072, China
In the forthcoming decades, the redshift drift observations in optical and radio bands will provideaccurate measurements on H ( z ) covering the redshift ranges of 2 < z < < z <
1. Inaddition, gravitational wave (GW) standard siren observations could make measurements on thedipole anisotropy of luminosity distance, which will also provide the H ( z ) measurements in theredshift range of 0 < z <
3. In this work, we propose a multi-messenger and multi-wavelengthobservational strategy to measure H ( z ) based on the three next-generation projects, E-ELT, SKA,and DECIGO, and we wish to see whether the future H ( z ) measurements could provide tightconstraints on dark-energy parameters. It is found that E-ELT, SKA1, and DECIGO are highlycomplementary in constraining dark energy models using the H ( z ) data. We find that E-ELT,SKA1, and DECIGO can tightly constrain Ω m , w (or w ), and H , respectively, and thus thecombination of them could effectively break the cosmological parameter degeneracies. The jointE-ELT+SKA1+DECIGO data give σ ( w ) ≈ .
02 in the w CDM model and σ ( w ) ≈ .
03 in the CPLmodel, which are better than the results of
Planck w a in the CPL model. I. INTRODUCTION
One of the most important missions in modern cos-mology is to understand the fundamental nature of darkenergy. To achieve this goal, the basic premise is to pre-cisely measure the equation of state (EoS) of dark energy[characterized by w ( z ) with z being redshift] through cos-mological observations. However, the most difficult pointfor studying dark energy lies in the fact that the EoS ofdark energy w ( z ) is not a direct observable, which canonly be inferred from the various actual cosmological ob-servations, such as the measurements for the expansionhistory of the universe [characterized by the Hubble pa-rameter H ( z )]. More unfortunately, the Hubble param-eter H ( z ) at different z is actually also very difficult tobe measured, and instead usually the measurements ofthe distance–redshift relation are used to infer the EoSof dark energy.However, in the application of distance–redshift rela-tion measurements, there is a severe disadvantage in in-ferring the EoS of dark energy, i.e., lots of information islost because of the two integrals relating cosmic distanceto the EoS of dark energy. In addition, actually, it isalso rather difficult to measure the absolute cosmologicaldistances, no matter for the standard candle method [us-ing type Ia supernovae (SNe Ia) to measure luminositydistance] and the standard ruler method [using baryonacoustic oscillation (BAO) to measure angular diameterdistance].The successful detections of gravitational waves (GWs)bring some new lights on cosmological research. Through ∗ Electronic address: [email protected] † Electronic address: [email protected] the analysis for the GW’s waveform, absolute luminositydistance can be measured, which is extremely importantfor cosmology and referred to as standard siren . Thecampaign in observing the event of binary neutron star(BNS) merger (known as GW170817 [1] as a GW event)by detecting GWs and electromagnetic waves (EMWs) invarious bands opened a new era of multi-messenger as-tronomy [2]. The observations of GWs and their EMcounterparts enable us to establish an absolute lumi-nosity distance–redshift relation, which can play an ex-tremely important role in breaking the cosmological pa-rameter degeneracies generated by the traditional EMWcosmological probes [3–11]. Nevertheless, even thoughthe GW standard sirens are used, the information lossstill exists in inferring the EoS of dark energy due tothe two integrals in the expression of distance relating to w ( z ). Thus, the best way of constraining w ( z ) is to di-rectly measuring H ( z ), in stead of the distance–redshiftrelation. Actually, the ways of directly measuring H ( z )by using GW observations have been proposed.For instance, Seto et al. [12] proposed that using theDECihertz Interferometer Gravitational wave Observa-tory (DECIGO) to observe BNSs during about a 10-yearobservation could measure the phase and frequency evo-lutions of GW signal which are proportional to H ( z ).However, this approach strongly depends on the knowl-edge of waveform and is limited by the estimation erroron the individual source mass. Another approach of di-rectly measuring H ( z ) through GW observation is to usedipole components of luminosity distance arising fromthe matter inhomogeneities of large-scale structure andthe local motion of observer [13]. This idea was proposedby Bonvin et al. [14, 15] for the SN observation, and fur-ther developed by Nishizawa et al. [16] for the GW ob-servation. Compared with the SN observation, the GWobservation can have a larger number of samples, smaller a r X i v : . [ a s t r o - ph . C O ] F e b systematic errors, and higher redshifts ( z ∼
3) in the fu-ture. Therefore, in the GW multi-messenger astronomyera, we can use this method to measure H ( z ), and toexplore the nature of dark energy.On the other hand, actually using EMWs can also di-rectly measure H ( z ). In addition to the traditional wayssuch as the differential age method [17] and radial BAOmethod [18], in the future the most promising way is touse the redshift drift, also known as the Sandage-Loeb(SL) test, which was originally proposed by Sandage [19]and further improved by Loeb [20]. By the upcoming ex-periments such as the European Extremely Large Tele-scope (E-ELT) and the Square Kilometre Array (SKA),the accurate measurements of the redshift drift will beachieved through two different means. In the opticalband, E-ELT with high-resolution optical spectrographenables the measurement of redshift drift in the redshiftrange of 2 < z < α absorp-tion lines of distance quasars. By observing the neutralhydrogen 21-cm emission signals of galaxies in the radioband instead of the Lyman- α absorption lines, the SKA1-mid can measure the redshift drift in the redshift rangeof 0 < z < H ( z ), which is rather importantfor the studies on dark energy. We wish to investigatewhat the synergy between GW and EMW observationswill bring to cosmology in the next decades. For thispurpose, we put forward a multi-messenger and multi-wavelength observational strategy to provide accuratemeasurements on H ( z ), and further use these H ( z ) datato constrain dark energy models. We show that the com-bination of E-ELT (EMW, optical band), SKA (EMW,radio band), and DECIGO (GW multi-messenger, stan-dard sirens) can offer an excellent scheme for studyingdark energy because of their complementarity. II. METHODA. Redshift drift from optical and radioobservations
In an observing time interval (∆ t ), the shift in thespectroscopic velocity of a source (∆ v ) can be expressedas [19, 21]∆ v = c ∆ z z = cH ∆ t (cid:18) − E ( z )1 + z (cid:19) , (1) where E ( z ) ≡ H ( z ) /H is the dimensionless Hubble ex-pansion rate, c is the speed of light, and H = H ( z = 0)is the Hubble constant.By observing the Lyman- α absorption lines of distantquasar, E-ELT with high-resolution optical spectrographcould measure the velocity shift in a redshift range 2 4, and q = − . z > S/N is the signal-to-noise ratio of the Lyman- α spectrum, and N QSO is the number of observed quasarsat the effective redshift z QSO . In this work, we generate30 mock data in five redshift bins (the redshift intervalis ∆ z = 0 . 5) at the E-ELT observational redshift range,with the assumptions of time span ∆ t = 30 yr, S/N =3000, and N QSO = 6 at each effective redshift. In thissimulation, the fiducial cosmology we adopt is the Λ colddark matter (ΛCDM) model with Ω m = 0 . 315 and H =67 . − Mpc − from the Planck α absorption lines, it can measure the redshift drift inthe redshift range of 0 < z < v in redshift binscentered on z i = [0 . , . , . 3] with uncertainties σ ∆ v , re-spectively, of 3%, 5%, and 10%. B. Dipole anisotropy of luminosity distance fromGW multi-messenger observation In a homogeneous and isotropic universe, in principle,the observations of luminosity distances to astronomicalobjects over the sky should not be directional. How-ever, in fact, due to the matter inhomogeneities of thelarge-scale structure and the local motion of observer [13],there are tiny anisotropies leading to the appearance ofcorrection term for the luminosity distance d L , given by[14–16] d L ( z, n ) = d (0) L ( z ) + d (1) L ( z ) , cos θ (3) d (0) L ( z ) ≡ π (cid:90) d n d L ( z, n ) , (4) d (1) L ( z ) ≡ π (cid:90) d n ( n · e ) d L ( z, n ) , (5)where cos θ = n · e , n is the angular position of theluminous object, and e is the unit vector directed to-ward dipole coming from the peculiar velocity of the ob-server. In a homogeneous and isotropic universe, d (0) L ( z )can be reduced to the traditional meaning of luminositydistance, d (0) L ( z ) = (1 + z ) (cid:90) z dz (cid:48) H ( z (cid:48) ) . (6)The dipole d (1) L ( z ) can be expressed as d (1) L ( z ) = | v | (1 + z ) H ( z ) , (7)with the direction of the dipole specifically chosen as e = v / | v | , and the value of | v | is estimated as369 . ± . − in the CMB frame [26]. Thus, adirect measurement of H ( z ) can be given by the dipoleanisotropy of luminosity distance. The error of H ( z ) isrelated to the error of the luminosity distance, and canbe estimated as [14–16]∆ H ( z ) H ( z ) = ∆ d (1) L ( z ) d (1) L ( z ) = √ (cid:34) d (1) L ( z ) d (0) L ( z ) (cid:35) − (cid:34) ∆ d (0) L ( z ) d (0) L ( z ) (cid:35) . (8)For detailed description of the above formulae deriva-tion, we refer the reader to Refs. [14–16]. In this paper,to get ∆ d (0) L /d (0) L , we consider the dipole anisotropy ofluminosity distance from the GW standard sirens pro-duced by the BNS and observed by the DECIGO de-tector. The systematic error of the averaged luminositydistance is [16] (cid:34) ∆ d (0) L ( z ) d (0) L ( z ) (cid:35) = σ ( z ) + σ ( z ) + σ ( z ) , (9)where σ inst is the uncertainty associated with the DE-CIGO described in detail in Sec. IIIA of Ref. [16]. σ lens is the lensing error adopted as σ lens ( z ) = 0 . (cid:20) − (1 + z ) − . . (cid:21) . . (10)The peculiar-velocity error σ pv is a kind of Doppler effectcaused by the motions of galaxies, and it is estimatedas [27] σ pv ( z ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (1 + z ) H ( z ) d (0) L ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ v , gal , (11)where σ v , gal = 300 km s − represents the 1-dimensionalvelocity dispersion of the galaxy [28].If we have ∆ N ( z ) independent binary NS-NS systemsin the vicinity of the redshift z , the mean error of H ( z ) re-duces to ∆ H ( z ) / √ ∆ N , which is dependent on the num-ber distribution of NS binary system at different redshift Ω m H [ k m s − M p c − ] ( ( / 7 6 . $ ' ( &