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Prospects of High Redshift Constraints on Dark Energy Models with the Ep,I – Eiso Correlation in Long Gamma Ray Bursts

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Prospects of High Redshift Constraints on Dark Energy Models with the Ep,I – Eiso Correlation in Long Gamma Ray Bursts MNRAS 000, 000{000 (2021) Preprint June 16, 2021 Compiled using MNRAS LATEX style file v3.0 Prospects of high redshift constraints on dark energy models with the Ep;i { Eiso correlation in long Gamma Ray Bursts M.Demianski,1;2 E.Piedipalumbo,3;4? D.Sawant5;6 and L.Amati 7 1Institute for Theoretical Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland 2Department of Astronomy, Williams College, Williamstown, MA 01267, USA 3Dipartimento di Fisica, Universit`adegli Studi di Napoli Federico II, Compl. Univ. Monte S. Angelo, 80126 Naples, Italy 4I.N.F.N., Sez. di Napoli, Compl. Univ. Monte S. Angelo, Edificio 6, via Cinthia, 80126 - Napoli, Italy 5Dipartimento di Fisica e Scienze della Terra, Universit`adegli Studi di Ferrara 6 IIT, Mombay, India 7INAF-IASF, Sezione di Bologna, via Gobetti 101, 40129 Bologna, Italy Accepted XXX. Received YYY; in original form ZZZ ABSTRACT So far large and different data sets revealed the accelerated expansion rate of the Universe, which is usually explained in terms of dark energy. The nature of dark energy is not yet known, and several models have been introduced: a non zero cosmological constant, a potential energy of some scalar field, effects related to the non homogeneous distribution of matter, or effects due to alternative theories of gravity. Recently, a tension with the flat ΛCDM model has been discovered using a high-redshift Hubble diagram of supernovae, quasars, and gamma-ray bursts. Here we use the Union2 type Ia supernovae (SNIa) and Gamma Ray Bursts (GRB) Hubble diagram, and a set of direct measurements of the Hubble parameter to explore different dark energy models. We use the Chevallier-Polarski- Linder (CPL) parametrization of the dark energy equation of state (EOS), a minimally coupled quintessence scalar field, and, finally, we consider models with dark energy at early times (EDE). We perform a statistical analysis based on the Markov chain Monte Carlo (MCMC) method, and explore the probability distributions of the cosmological parameters for each of the competing models. We apply the Akaike Information Criterion (AIC) to compare these models: our analysis indicates that an evolving dark energy, described by a scalar field with exponential potential seems to be favoured by observational data. Key words: cosmology: cosmological parameters { cosmology: dark energy { cosmology: cosmological observations{ stars: gamma-ray burst: general 1 INTRODUCTION discovered scalar field (Sahni et al. 2003; Alam et al. 2003), to effects connected with inhomogeneous distribution of mat- Starting at the end of the 1990s, observations of high-redshift ter and averaging procedures (Clarckson & Maartens 2010), supernovae of type Ia (SNIa) revealed the accelerated expan- or modifications of the Einstein General Theory of Relativity sion of the Universe (Perlmutter et al. 1998, 1999; Riess et al. (De Felice & Tsujikawa 2010; Capozzilello & De Laurentis 1998, 2007; Astier et al. 2006; Amanullah et al. 2010). This 2011; Clifton et al. 2012; Capozzilello & D'Agostino & Lu- unexpected result has been confirmed by statistical analy- arXiv:1911.08228v2 [astro-ph.CO] 15 Jun 2021 ongo 2019). In the last cases, in general, the effective EOS sis of observations of small-scale temperature anisotropies of dark energy is not constant, but depends on redshift z. of the Cosmic Microwave Background Radiation (CMBR) Therefore populating the Hubble diagram up to high red- (WMAP 2013; Planck Collaboration 2016). The observed ac- shifts remains a primary task to test the consistency of the celerated expansion is usually related to a non zero cosmo- ΛCDM model (see for instance Lusso et al. 2019; Risaliti & logical constant or to existence of so called dark energy, a Lusso 2019; Lusso 2020; Lusso et al. 2020b, for discussions cosmic medium with positive energy density but sufficiently about the ΛCDM tension ) and, therefore, to shine new light large negative pressure, which now provides about 70% of the on the nature of dark energy. So far dark energy models are matter energy in the Universe. The nature of dark energy is, poorly tested in the redshift interval between the farthest however, not known. The models of dark energy proposed observed Type Ia supernovae and that of the Cosmic Mi- so far range from a non-zero cosmological constant (Peebles crowave Background. In our high redshift investigation we 1984; Carroll 2001), to a potential energy of some not yet consider the Union2 SNIa data set, and the long gamma-ray burst (GRB) Hubble diagram, constructed by calibrating the correlation between the peak photon energy, Ep;i, and the ? E-mail: [email protected] isotropic equivalent radiated energy, Eiso (Demianski et al. © 2021 The Authors 2 M. Demianski et al. 2017a,b). Here we consider an extended Ep;i- Eiso correla- 2.1 Parametrization of the dark energy EOS tion, to take into account possible redshift evolution effects, Within the Friedman equations dark energy appears through modeled through power law terms. It turns out that at least its effective energy density, ρ , and pressure, p : a part of this evolution can be caused by gravitational lensing de de effects (see, for instance, Shirokov et al. 2020). We consider 2 8πG also a sample of 28 direct measurements of the Hubble pa- H = (ρm + ρde) ; (1) rameter, compiled by (Farooq & Ratra 2013). These measure- 3 ments are performed through the differential age technique, first suggested by (Jimenez & Loeb 2002), which uses red a¨ 4πG = − (ρm + ρde + 3pde) ; (2) passively evolving galaxies as cosmic chronometers. Here we a 3 probe the dynamical evolution of dark energy, by considering where a is the scale factor, H =a=a _ the Hubble parameter, some proposed so far competing models of dark energy: and ρm is the dark matter energy density. Here and through- out this paper the dot denotes the derivative with respect to i) an empirically parametrized EOS of dark energy, usually the cosmic time, and we have assumed a spatially flat Uni- using two or more free parameters. Among all the proposed verse in agreement with what is inferred from the CMBR parametrization forms of the dark energy EOS, we consider anisotropy spectrum (Planck Collaboration 2016). The con- the CPL (Chevallier & Polarski 2001; Linder 2003), which is tinuity equation for any cosmological fluid is : now widely used , ρ_i pi ii) a quintessence dark energy: a model where a self inter- = −3H 1 + = −3H [1 + wi(t)] ; (3) acting scalar field plays the role of dark energy and drives the ρi ρi acceleration (Peebles & Ratra 1988a; Ratra & Peebles 1988b; where the energy density ρi, the pressure pi, and the EOS pi Tsujikawa 2013), of each component is defined by wi = . For ordinary non- iii) early dark energy: models where a non negligible fraction ρi relativistic matter w = 0, and the cosmological constant can of dark energy exists already at early stages of evolution of be treated as a medium with w = −1. Let us recall that the Universe (Khoraminazad et al. 2020). −3 3 ρm = Da , where the parameter D ≡ ρm0a0 is determined Our statistical analysis is based on the Monte Carlo Markov by the current values of ρm and a. If we explicitly allow the Chain (MCMC) simulations to simultaneously compute the possibility that the dark energy evolves, the importance of its full probability density functions (PDFs) of all the param- equation of state is significant and it determines the Hubble eters of interest. The structure of the paper is as follows. function H(z), and any derivation of it as needed to obtain In Sect. 2 we describe the different models of dark energy the observable quantities. Actually it turns out that: p 3 considered in our analysis. In Sect. 3 we describe the obser- H(z; θ) = H0 (1 − Ωm)g(z; θ) + Ωm(z + 1) ; vational data sets that are used in our analysis. In Sect. 4 we (4) describe some details of our statistical analysis and present results. In Sect. 5 we present constrains on dark energy mod- where els that could be derived from future GRB Hubble diagram ρde(z; θ) 3 R z w(x,θ)+1 dx g(z; θ) = = e 0 x+1 ; samples. General discussion of our results and conclusions are ρde(0) presented in Sect. 6. w(z; θ) is any dynamical form of the dark energy EOS, and θ = (θ1; θ1::; θn) are the EOS parameters. It is worth noting that in Eq. (4) we are neglecting the 4 radiation term Ωr(1+z) , but in this way we introduce errors much smaller than the observational uncertainties of all our datasets. Of course in the case of the early dark energy, when 2 DIFFERENT MODELS OF DARK ENERGY we have to investigate a dark energy component just in early times, we must include the radiation term. Although seemingly consistent with the current standard Using the Hubble function we define the luminosity dis- model where the cosmic acceleration is due to the Einstein's tance dL as cosmological constant, Λ, the precision of current data is not c Z z dy sufficient to rule out an evolving dark energy term. If then the dL(z; θ) = (1 + z) (5) H H(y; θ) cosmological constant is not responsible for the observed ac- 0 0 c Z z dy celerated expansion of the Universe, we are considering some = (1 + z) : p 3 of the proposed models of a dynamical field that is generat- H0 0 (1 − Ωm)g(y; θ) + Ωm(y + 1) ing an effective negative pressure. Moreover this could also (6) indicate that the cosmological Copernican principle cannot Using the luminosity distance, we can evaluate the distance be applied at certain scales, and that radial inhomogeneities modulus, from its standard definition (in Mpc): could mimic the accelerated expansion.
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