A crucial test of the phantom closed cosmological model
aa r X i v : . [ a s t r o - ph . C O ] S e p MNRAS , 1–5 (2020) Preprint 23 September 2020 Compiled using MNRAS L A TEX style file v3.0
A crucial test of the phantom closed cosmological model
S. I. Shirokov ⋆ and Yu. V. Baryshev, SPb Branch of Special Astrophysical Observatory of Russian Academy of Sciences, 65 Pulkovskoye Shosse, St. Petersburg, 196140, Russia Saint Petersburg State University, 7/9 Universitetskaya Nab., St. Petersburg, 199034, Russia
Accepted 2020 September 18. Received 2020 September 18; in original form 2020 July 14
ABSTRACT
We suggest a crucial direct-observational test for measuring distinction between thestandard Λ CDM model and recently proposed phantom dark energy positive curva-ture cosmological model. The test is based on the fundamental distance–flux–redshiftrelation for general Friedmann models. It does not depend on the CMBR data, onthe large-scale structure growth models, and also on the value of the Hubble constant H . Our crucial test can be performed by future gamma-ray burst observations withTHESEUS space mission and by using gravitational-wave standard siren observationswith modern advanced LIGO–Virgo and also forthcoming LISA detectors. Key words: cosmological parameters – gamma-ray bursts – gravitational waves.
In the last few years there has been growing evidence for anumber of “tensions” between the derived parameters of theearly Universe and measured parameters of the late localUniverse. Both the cosmic microwave background radiation(CMBR) data and the local Universe observations have re-vealed underlying discrepancies that cannot be ignored.It comes from comparison of the measured Hubble Con-stant H for the early and late Universe, so-called “the H tension” Riess et al. (2020); Verde et al. (2019); Lin et al.(2019), and also, from uncertainty in curvature density pa-rameter – “the curvature tension” Handley (2019).Recent analysis of the combined observational data,using a high-redshift Hubble diagram for Type Ia super-novae, quasars, and gamma-ray bursts, also discovered atension with the flat Λ CDM model (Lusso et al. 2019;Risaliti & Lusso 2019; Demianski et al. 2019, 2017a,b).Further evidence on modern “crisis” in cosmology wasfound from recent combined analysis of the Planck CMBpower spectra anisotropy and large-scale structure (LSS)data (Di Valentino et al. 2020b,a). Their results cast doubton the standard values of basic Λ CDM parameters for in-flation, non-baryonic dark matter, and dark energy. Insteadof the flat universe with cosmological constant Λ , they sug-gest to consider the phantom dark energy closed (PhDEC)cosmological model.The conclusion of the recent works Di Valentino et al.(2020b,a); Handley (2019); Riess et al. (2020); Verde et al.(2019); Lin et al. (2019) is that either Λ CDM needs to bereplaced by a drastically different model, or else there is sig- ⋆ E-mail: [email protected] nificant but still undetected systematics. The new theoreti-cal suggestions call for new observations and stimulate theinvestigation of alternative theoretical models and solutions.Here we suggest a crucial cosmological test for mea-suring distinction between the standard Λ CDM and pro-posed PhDEC cosmological models. It does not use theCMBR data, the LSS growth, and the value of theHubble constant. General analysis of CMBR independenttests for alternative cosmological models was consideredby Baryshev & Teerikorpi (2012); Shirokov et al. (2020). Inthis letter, we apply this approach to the phantom dark en-ergy positive curvature cosmological model, which now isconsidered as possible alternative to the standard Λ CDMmodel.
Recent discussion on the Λ CDM “tensions” points to con-sideration of a wider class of cosmological models, which arebased on reanalysis of the CMBR data and N-body sim-ulations of the LSS growth (Riess et al. 2020; Verde et al.2019; Lin et al. 2019; Di Valentino et al. 2020b,a; Handley2019). Also, problems in the galaxy formation theory raisethe issue on necessity for consideration a more general initialconditions in the cosmological N-body simulations Peebles(2020); Benhaiem et al. (2019).The PhDEC model was suggested in Di Valentino et al.(2020b,a) as a “new standard” Universe model. They per-formed reanalysis of the Planck-2018 temperature and polar- © S. I. Shirokov et al. ization CMBR angular power spectra, the cosmic shear mea-surements, and the baryon acoustic oscillation data. Theyalso emphasized existing uncertainties in determination ofthe Hubble constant value H (Riess et al. 2020; Verde et al.2019; Lin et al. 2019).In view of the tension between different approaches toCMBR and LSS data analyses, here we suggest a robustobservational test of the reality of the phantom dark en-ergy closed (positive curvature) cosmological model, whichis independent from CMBR data, LSS data and the deter-mination of the local value of the Hubble constant.As the independent test of the PhDEC model, weconsider the high-redshift Hubble diagram, which is basedon the directly observed flux–distance–redshift relationfor a sample of standard candles (Amati et al. 2018;Demianski et al. 2019). Such test belongs to the class ofclassical crucial cosmological tests, which related to thebasis of the cosmological models (Baryshev & Teerikorpi2012). It allows us to perform robust testing Friedmann–Lemaitre–Robertson–Walker (FLRW) primary quantitiessuch as curvature constant, density and equation of state(EoS) for dark energy and dark matter (Shirokov et al.2020; Demianski et al. 2019).This crucial test can be performed by future gamma-ray burst (GRB) observations with THESEUS space mis-sion (Amati et al. 2018), and by using gravitational-wavestandard siren observations with LIGO–Virgo and LISAadvanced detectors (Schutz 1986; Holz & Hughes 2005;Abbott et al. 2017). Friedmann–Lemaitre–Robertson–Walker expanding spacecosmological model is a direct consequence of the general rel-ativity and cosmological principle of the homogeneous andisotropic distribution of matter.The general form of the Friedmann equation is given bythe formula H − π G ρ = − kc S , or − Ω = − Ω k , (1)where k = (− , , + ) is the curvature constant for open,flat, or closed space, respectively, S is the scale factor, H = ( d S / d t )/ S is the Hubble parameter, ρ = Σ i ρ i is the to-tal density of all cosmological homogeneous non-interactingfluids ρ i .The total matter density parameter is defined as Ω = Σ i Ω i , where Ω i = ρ i / ρ c with the critical density ρ c = H / π G . The curvature density parameter is defined as Ω k = kc / S H = ( Ω − ) . (2)We use definition of Ω k which has the same sign as the curva-ture constant k , i.e. positive curvature space ( k = + ) corre-sponds to the positive curvature density parameter Ω k > . The w CDM model is defined as the FLRW model thatcontains two cosmological non-interacting fluids, having EoS Note that in number of papers it is also used another definition Ω ′ k = − Ω k , so positive curvature space ( k = + ) corresponds tothe negative curvature density parameter Ω ′ k < . of the cold matter p = , and the quintessence (dark energy) p = w ρ c (with w < ). Thus, the normalised Hubble param-eter h ( z ) = H ( z )/ H is given by Eq. 1 in the form h ( z ) = q Ω m ( + z ) + Ω DE ( + z ) − Ω k ( + z ) , (3)where Ω i is the corresponding density parameter at presentepoch, and w is the dark energy EoS parameter.For w = − one have p = − ρ c and constant cosmologicalvacuum density Ω Λ = Ω DE = const. In this case, w CDMmodel is called the Λ CDM model. For curvature constant k = and ( Ω Λ = . , Ω m = . ) one have the standard Λ CDM model.If dark energy parameter w < , the model is calledquintessence w CDM. For w < − , one have so-called “phan-tom” w CDM model. In the case of curvature constant k = + ,the model is called phantom dark energy closed FLRWmodel. Our test of the PhDEC model is based on the measure-ment of the luminosity distance d L in the Friedmann models,which is given by the following expression d L ( z ) = cH ( + z ) q | Ω k | I k (cid:18)q | Ω k | ∫ z d z ′ h ( z ′ ) (cid:19) , (4)where k = (− , , + ) is the curvature constant, Ω k is thecurvature density parameter, I k ( x ) = sinh ( x ) for Ω k < , and I k ( x ) = x for Ω k = , and I k ( x ) = sin ( x ) for Ω k > .Let us consider the relative luminosity distance D rel , i.e.the ratio of the luminosity distance of a considered cosmolog-ical model to the luminosity distance of the fixed standardflat Λ CDM model ( Ω k = , Ω Λ = . , Ω m = . , w = − ). So,we get the equation D rel ( z ) = d L ( z ) d Λ CDM ( z ) = F ( z ; Ω k , Ω DE , w ) , (5)which depends only on three parameters ( Ω k , Ω DE , and w )of the considered cosmological model. Thus, the relative lu-minosity distance D rel does not depend on the Hubble con-stant H . The matter density parameter Ω m does not enterto the Eq. (5) and is determined by the Friedmann relationEq. (1) as Ω m = Ω k − Ω DE + . (6)The relative luminosity distance modulus is given byrelation D rel ( z ) , i.e. it is equal to the difference betweena considered model of luminosity distance modulus and thestandard Λ CDM luminosity distance modulus ∆ µ L ( z ; Ω k , Ω DE , w ) = F ( z ; Ω k , Ω DE , w ) . (7)Eq. (7) depends on the three parameters of a stud-ied FLRW model Ω k , Ω DE , and w (for fixed standard Λ CDM model) and it can be used as a test of the phan-tom dark energy closed cosmological model independentlyfrom the Di Valentino et al. (2020b,a) analysis.
To estimate the predicted relative luminosity distance mod-ulus ∆ µ L ( z ) for a number of phantom dark energy FLRW MNRAS , 1–5 (2020) crucial test of the PhDEC cosmological model models having closed, flat, and open geometry we con-sider following parameters: Ω k = (− . , . , + . ) , Ω DE = ( . , . , . , . ) , and w = (− , − ) .Fig. 1 shows predicted quantitative difference betweenstandard Λ CDM and phantom w CDM models for the case Ω DE = ( . , . ) , w = (− , − ) in redshift logarithmic scale.On the right panel, the available median data on high-redshift long GRBs from the Amati et al. (2019) 193-GRBsample are also shown.Table 1 contains values of the relative luminosity dis-tance modulus ∆ µ L ( z ) for phantom dark energy EoS with w = − . The high-redshift relative luminosity distance modulus ∆ µ L ( z ) (Eq. 7 and Fig. 1) can be used as a crucial test ofthe phantom dark energy closed universe model (PhDEC)that recently was suggested as a possible alternative to thestandard flat Λ CDM model (Di Valentino et al. 2020b,a).An advantage of ∆ µ L ( z ) test is that it does not de-pend on the CMBR anisotropy analysis and on theory ofthe density fluctuation growth. Our suggested test is basedon directly observed flux–distance–redshift relation for gen-eral FLRW model. The ∆ µ L ( z ) -test depends only on threeparameters ( Ω k , Ω DE , and w ) of the tested FLRW mod-els, which determine the specific shape of the correspondingcurves.Our calculations of the ∆ µ L ( z ) for the number of FLRW w CDM models give quantitative estimation of needed mea-surement accuracy for detecting the distinction between thestandard flat Λ CDM and PhDEC models. In our test onemust use the whole shape of the function (Eq. 7) withinstudied redshift interval ( . < z < ). According to ourcalculation presented at Fig. 1, the accuracy of distancemodulus measurements ∆ µ ∼ . at z ∼ is enough fordistinction between considered models. Future observationsof high-redshift GRBs and gravitational waves (GW) willbring such possibility for redshift interval . < z < .For small redshifts ( z < . ) the Hubble Diagram is usedas a measure of the local Hubble constant value H becauseof the linear relation z = ( H / c ) r is fulfilled for all models.So according to Eq. (7) the ∆ µ L ( z ) -test does not depend onthe value of the H .To performing the ∆ µ L ( z ) -test, one need observations ofthe objects having standard luminosity in the high-redshiftinterval . < z < . Possible candidates for such objects arethe long GRBs. Fig. 1 (right) shows how the existing sam-ple of 193 long GRBs (Amati et al. 2019) with calibratedluminosity by the Amati relation fits the models (with errorbars for median intervals of ∆ log z = . ). High luminosityquasars also can be used as a standard candles for the Hub-ble diagram analysis (Lusso et al. 2019; Risaliti & Lusso2019). Future THESEUS GRB observations will reduce theerror bars and can distinguish between standard Λ CDM andPhDEC.Perspective candidates for ∆ µ L ( z ) -test are the gravita-tional wave standard sirens, which are the GW analog ofthe optical “standard candles” (Schutz 1986; Holz & Hughes2005; Abbott et al. 2017). We should emphasized that im-portant obstacle for measuring distances to the standard luminosity objects is the necessity of accounting the grav-itational lensing bias (Broadhurst et al. 2020). However,forthcoming surveys of the weak and strong gravitationallensing, like EUCLID, will help to overcome this problem(Cervantes-Cota et al. 2020). ACKNOWLEDGEMENTS
We thank referee for useful comments and suggestions. Thework was performed as part of the government contract ofthe SAO RAS approved by the Ministry of Science andHigher Education of the Russian Federation.
DATA AVAILABILITY
The data underlying this article will be shared on reasonablerequest to the corresponding author.
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Left:
The relative distance modulus ∆ µ L ( z ) verse redshift is shown. Predicted quantitative difference between standard Λ CDMand phantom w CDM models for the case Ω DE = ( . , . ) , w = (− , − ) and k = (− , , + ) . Right:
The one with medians of the 193-GRBsample (Amati et al. 2019) for ∆ log z = . . Phantom dark energy EoS parameter w = − Ω k − . . + . − . . + . − . . + . − . . + . Ω DE Ω m z = . -0.02 -0.02 -0.03 0.04 0.04 0.03 0.11 0.10 0.09 0.18 0.17 0.160.20 -0.05 -0.06 -0.07 0.06 0.05 0.03 0.19 0.17 0.16 0.35 0.33 0.310.30 -0.08 -0.10 -0.12 0.07 0.05 0.03 0.25 0.23 0.20 0.50 0.46 0.420.50 -0.14 -0.17 -0.20 0.06 0.02 -0.02 0.34 0.28 0.23 0.78 0.67 0.590.60 -0.17 -0.21 -0.24 0.05 0.01 -0.04 0.36 0.29 0.23 0.89 0.75 0.640.80 -0.22 -0.27 -0.31 0.03 -0.03 -0.09 0.40 0.30 0.21 1.10 0.87 0.701.00 -0.26 -0.32 -0.37 0.01 -0.07 -0.14 0.42 0.29 0.19 1.27 0.95 0.722.00 -0.37 -0.47 -0.55 -0.05 -0.18 -0.29 0.46 0.24 0.07 1.86 1.08 0.683.00 -0.41 -0.53 -0.64 -0.07 -0.23 -0.38 0.49 0.21 -0.01 2.25 1.12 0.604.00 -0.43 -0.57 -0.69 -0.07 -0.26 -0.43 0.52 0.19 -0.06 2.57 1.13 0.545.00 -0.44 -0.59 -0.73 -0.07 -0.28 -0.47 0.54 0.18 -0.10 2.84 1.13 0.496.00 -0.45 -0.61 -0.76 -0.07 -0.30 -0.49 0.56 0.17 -0.13 3.08 1.14 0.458.00 -0.45 -0.63 -0.79 -0.06 -0.32 -0.53 0.59 0.16 -0.17 3.48 1.14 0.4010.00 -0.46 -0.65 -0.82 -0.06 -0.33 -0.56 0.62 0.15 -0.20 3.82 1.15 0.35 Table 1.
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