MNRAS 000, 000–000 (2021) Preprint June 16, 2021 Compiled using MNRAS LATEX style file v3.0

Prospects of high constraints on dark 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 , University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland 2Department of , 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 , which is usually explained in terms of dark energy. The of dark energy is not yet known, and several models have been introduced: a non zero , a of some scalar field, effects related to the non homogeneous distribution of , or effects due to alternative theories of . Recently, a tension with the flat ΛCDM model has been discovered using a high-redshift Hubble diagram of supernovae, , 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 (EOS), a minimally coupled 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: : cosmological parameters – cosmology: dark energy – cosmology: cosmological observations– : 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 , 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 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; 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 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 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 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 as cosmic chronometers. Here we a 3 probe the dynamical evolution of dark energy, by considering where a is the , H =a/a ˙ the Hubble parameter, some proposed so far competing models of dark energy: and ρm is the 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 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 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 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 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. Within the Friedman- Lemaitre-Robertson-Walker (FLRW) paradigm, all possibil- µ(z) = 25 + 5 log dL(z, θ). (7) ities can be characterized, as far as the background dynam- In this we consider the so-called CPL parametrization ics is concerned, by the dark energy EOS, w(z). The main of the dark energy EOS (Chevallier & Polarski 2001; Linder task of is to search for evidence for 2003) given by w(z) 6= −1. This is usually done in terms of an appropriate −1 parameterization of the EOS. w(z) = w0 + w1z(1 + z) , (8)

MNRAS 000, 000–000 (2021) Dark energy models with the Ep,i – Eiso correlation 3 where w0 and w1 are real numbers that represent the EOS of the Friedman equations are known (Rubano & Scudellaro present value and its overall time evolution, respectively. 2002; Demianski et al. 2005, 2011; Piedipalumbo et al. 2012). Assuming that ϕ is minimally coupled to gravity, the cosmo- logical equations are written as 2.2 A scalar field quintessence model  2   a˙ 8πG −3 1 2 One of the more interesting physical realizations of the dark = Da + ϕ˙ + V (ϕ) , (10) a 3 2 energy is the so called quintessence, i.e. a self-interacting scalar field minimally coupled with gravity (Peebles & Ratra a¨  a˙ 2 8πG  1  1988a; Ratra & Peebles 1988b; Tsujikawa 2013). Such field 2 + = − ϕ˙ 2 − V (ϕ) , (11) induces the repulsive gravitational force dynamically, driv- a a 3 2 ing the accelerated expansion of the Universe. Moreover, it also influences the growth of structures, arisen from gravita-  a˙  ϕ¨ + 3 ϕ˙ + V 0(ϕ) = 0 , (12) tional instability. Quintessence could cluster gravitationally a on very large scales (' 100 Mpc), and leaves an imprint on anisotropy of the microwave background radiation (Caldwell and 0 denotes partial derivative with respect to ϕ. Here we & Steinhardt 1998) , and, at smaller scales, its fluctuations consider the potential analyzed in (Demianski et al. 2011) are damped and do not modify the evolution equation for and (Piedipalumbo et al. 2012), ( r ) the perturbations in the dark matter (Ma et al. 1999). Since 3 redshift space distortions in the clustering of galaxies provide V (ϕ) ∝ exp − ϕ , (13) 2 constrains on the growth rate of matter perturbations, δm, which depend on the scalar field dynamics, and on the scalar for which the general exact solution exists: actually it turns field equation of state, it is possible to test quintessence mod- out that els from this kind of data (see for instance Alcaniz & Lima 2 3 t 2 2001; Copeland, Sami, & Tsujikawa 2006; Bueno Sanchez a (t) = [(3H0 − 2)t + 4 − 3H0], (14) 2 & Perivolaropoulos 2010, and references therein). Indeed in 2 2(3H − 2)t2 + 4 − 3H  (Demianski et al. 2005) we showed that some scalar mod- 0 0 H(t) = 2 , (15) els with exponential potential, including the one used in the 3t ((3H0 − 2)t + 4 − 3H0) 2  present analysis, are fully compatible with the power spec- (4 − 3H0) (3H0 − 2)t + 4 − 3H0 ΩM = 2 2 , (16) trum of the CMBR anisotropy, and the parameters of large [2(3H0 − 2)t + 4 − 3H0] scale structure determined by the 2-degree Field Red- 2 2  (3H0 − 2)t 4(3H0 − 2)t + 3(4 − 3H0) shift Survey (2dFGRS). It is worth noting that, in absence Ω = (17) ϕ [2(3H − 2)t2 + 4 − 3H ]2 of the matter term, it is possible to deeply connect extended 0 0 r theories of gravity, as f(R) theories and scalar tensor the- 2  6.48  ϕ(t) = − log 2 , (18) ories in the so called Jordan frame, to minimally coupled 3 (3H0 − 2) t − 3H0 + 4 scalar field with appropriate self interaction potential. Actu- where H is a constant. In order to determine the integration ally, to any f(R) theory in the so called Jordan frame, in the 0 constants we set the present time t = 1, so we are using Einstein frame corresponds a minimally coupled scalar field, 0 the as a unit of time, so a = a(1) = 1, throughout the conformal transformation 0 which is a standard choice, and finally H0 = H(1). Because of our choice of time unit H0 does not have the same value as 2 gµν → g˜µν = Ω gµν , the standard Hubble constant H0. In this model all the basic cosmological parameters can be written in terms of H0 only, p 0 where Ω ≡ exp (ϕ) = f (R), with a potential given by so we find that: (Maeda 1989): 2(4 − 3 H ) Ω ≡ Ω (t = 1) = 0 , (19) M0 M 9 H2 f(R) − Rf 0(R) 0 V (ϕ ˜) = . (9) 2 [f 0(R)]2 (3 H0 − 2) (3 H0 + 4) Ωϕ0 ≡ Ωϕ(t = 1) = 2 . (20) Here tilted quantities refer to the Einstein frame and 0 denotes 9 H0 partial derivative with respect to R. It is worth highlight- The scalar field EOS evolves with time and the parameter ing, however, that this connection only holds in the absence w is given by of a matter term. Indeed, even if it is possible to derive a conformal transformation in presence of a matter term, the 1 3(3 H0 − 4) wϕ = − + 2 , (21) self-interaction potential is not expressed as in Eq. (9), and 2 6(4 − 3 H0) + 8 (3 H0 − 2) t matter becomes coupled to the scalar field. Therefore the so that today we have physical equivalence of these two conformally equivalent the- ories remains an open question in theoretical cosmology (see 8 − 3H0 wϕ0 = − . (22) for instance Ma et al. 1999; Capozziello et al. 2006; Demianski 4 + 3H0 et al. 2008). Many quintessence models have been proposed, As we see in Fig. (1 ), the scalar field EOS in the past is equal considering different kinds of potentials driving the dynam- to wϕ = −1 so that the dark energy behaves as a subdom- ics of the scalar field. Here we take into account the specific inant cosmological constant, but only recently has started class of exponential–type potential; in particular we consider dominating the expansion of the universe, as illustrated in an exponential potential for which general exact solutions Fig. (2). It turns out that it undergoes a transition from a

MNRAS 000, 000–000 (2021) 4 M. Demianski et al.

-0.5 the recent cosmological past, in order to trigger a dark en- ergy dominant era. A large class of models of this type has -0.6 been proposed (Karwal & Kamionkowski 2016; Niedermann & Sloth 2020). Since the main parameter of an early dark en- -0.7 ergy model is Ωe, it parametrizes the evolution of dark energy. In different parameterizations Ω have been estimated from w e φ several observations, as nucleosynthesis, , -0.8 or the peak separation in the power spectrum of the cosmic microwave radiation anisotropy(Das et al. 2011; Doran, Lil- -0.9 ley, & Wetterich 2002; Doran & Robbers 2006; Di Valentino 2021). Following (Doran & Robbers 2006; Pettorino et al. -1.0 2013) we use parametrized representation of the dark energy -1 0 1 2 3 density fraction, Ωde, which depends on the present matter Log(1+z) fraction, Ωm, the early dark energy density fraction, Ωe , and the present dark energy equation of state w0: Figure 1. Redshift dependence of the equation of state parameter 3w0  wϕ corresponding to the best fit value of H0, as illustrated in Ωe (z + 1) − 1 − Ωm + 1 Ωde(z, Ωm, Ωe, w0) = −3w + the statistical analysis section: we see that wϕ smoothly transits Ωm(z + 1) 0 − Ωm + 1 between two asymptotic constant values. The vertical lines indicate +Ω 1 − (z + 1)3w0  . (23) respectively the SNIa (blue line) and the GRB range of redshift e (green line). It turns out that in these models the Hubble function takes the form: 2 H (z, Ωm, Ωe, w0, Ωγ ,Neff ) = Ωde(z, Ωm, Ωe, w0) + 10 -0.6 4 !   3 -0.7 3 4 7 4 8 +Ωm(z + 1) + Ωγ (z + 1) Neff + 1 (24). wφ -0.8 8 11

6 -0.9 Here Neff is defined so that the total relativistic energy den- -1.0 0.0 0.5 1.0 1.5 2.0 sity (including and any other ) is Log(ρ) 4 Log(1+z) given in terms of the photon density ργ at T  1 MeV by 2 the relation: 4 7  4  3 0 ρ = N ρ . (25) eff 8 11 γ -2 In this equation Neff = 3 for three standard neutrinos that were thermalized in the early Universe and decoupled -1 0 1 2 3 Log(1+z) well before electron-positron annihilation. Moreover Ωγ = −2 −5 ωγ h , and we set ωγ = 2.47 10 . In Figs. (3) and (4) Figure 2. The scalar field density energy ρ (red line) is compared H(z, Λ) − H(z, θ) ϕ we plot relative residual curves of , and with the matter density (blue line): we see the transition from a H(z, Λ) subdominant phase, during the matter-dominated era, to a domi- µ(z, Λ) − µ(z, θ) , for all the competing models described nant phase, at the present accelerated expansion. It turns out that µ(z, Λ) the range of of this transition corresponds to the passage above, where H(z, Λ) is the Hubble function in the standard of the equation of state from wϕ = −1 to its present value. flat ΛCDM model, and H(z, θ) is the Hubble function in each of the other models parametrized by θ. We plot also curves of µ(z,Λ)−µ(z,θ) for the CPL model and scalar field models: subdominant phase, during the matter-dominated era, to a µ(z,Λ) it turns out that the range of redshifts larger than z = 1 is dominant phase, associated to the present accelerated expan- very important to break degeneracies among the flat standard sion. It is clear that extending the Hubble diagram beyond ΛCDM and other models considered in this paper. The values the SNIa range of redshift allows us to fully investigate the of the parameters have been chosen in order to highlight the transition of the EOS from w = −1 to its present value. φ differences among models.

2.3 Early dark energy 3 OBSERVATIONAL DATA In this section we consider some proposed cosmological mod- els that allow a non negligible amount of dark energy at early In our analysis we use the SNIa and GRB Hubble diagrams, times (Doran & Robbers 2006): they are often connected with and a list of 28 direct H(z) measurements, compiled by (Fa- the existence of scaling or attractor-like solutions, in which rooq & Ratra 2013). the dark energy density follows the density of the dominant component of matter-energy in the Universe. These models 3.1 Supernovae naturally predict a non-vanishing dark energy fraction of the total energy at early times, Ωe, which should be substantially SNIa observations gave the first strong indication that now smaller than its present value. Therefore these models need the expansion rate of the Universe is accelerating. First re- an exit mechanism, allowing the scaling solutions to end in sults of the SNIa teams were published in (Riess et al. 1998)

MNRAS 000, 000–000 (2021) Dark energy models with the Ep,i – Eiso correlation 5

CPL CPL

0.10 0.20 5

0.08 0.15 4 0.06 0.10 3 δμ ( z )% δ H ( z )% δ H ( z )% 0.04 0.05 2 0.02 0.00 1 0 2 4 6 8 0.00 0 z 0 1 2 3 4 5 6 0 1 2 3 4 5 6 z z scalar field

EDE 0.1 8 0.0

6 -0.1 δμ ( z )% -0.2 4

δ H ( z )% -0.3 0 2 4 6 8 2 z EDE 0 0 1 2 3 4 5 6 z 0.00 -0.05 H(z, Λ) − H(z, θ) Figure 3. We show relative residual curves of -0.10 H(z, Λ) for all considered competing models described above. For the flat -0.15 -0.20 ΛCDM model we take Ωm = 0.3, and H0 = 70. For other models δμ ( z )% the values of the parameters have been chosen in order to highlight -0.25 the differences between models. In particular, for the CPL model -0.30 Ωm = 0.3, h = 0.7, w0 = −1.2, w1 = 0.2 (red line) and Ωm = -0.35 0 2 4 6 8 0.3 ,h = 0.7, w0 = −1.2, w1 = −0.2 (blue line); for the EDE −5 model Ωm = 0.3, h = 0.7, w0 = −1, Ωe = 0.08, ωγ = 2.47 10 z (red line), Ωm = 0.27, h = 0.7, w0 = −1, Ωe = 0.03, ωγ = µ(z, Λ) − µ(z, θ) 2.47 10−5(blue line). For the scalar field model H = 0.94 (red Figure 4. We show relative residual curves of for 0 µ(z, Λ) line) and H0 = 0.98. all our considered models described above. For the flat ΛCDM model we take Ωm = 0.3, and H0 = 70. For other models the values of parameters have been chosen as in Fig.(3). For the EDE and (Perlmutter et al. 1999). Here we consider the recently model we take Ωm = 0.3, h = 0.7, w0 = −1, Ωe = 0.08, ωγ = −5 updated Supernovae Cosmology Project Union 2.1 compila- 2.47 10 (red line), and Ωm = 0.27, h = 0.7, w0 = −1, Ωe = 0.03, −5 tion (Suzuki et al. (The Cosmology Project) 2012), ωγ = 2.47 10 (blue line). which is an update of the original Union compilation. The Union2.1 dataset is one of the largest compilations of super- novae of type Ia, originally consisting of 833 SNIe drawn from (Nesseris & Perivolaropoulos 2005). For example, in the case different samples, and reduced to 580 after several selection of a flat CPL model {θp} = {Ωm, w0, w1}. criteria. It spans the redshift range 0.015 6 z 6 1.4. To com- pute the χ2 function related to the distance modulus we can easily relate the m(z) to the so called 3.2 Gamma-ray bursts Hubble free luminosity distance DL(z) = H0dL(z) through Gamma-ray bursts are visible up to high redshifts thanks the relation: to the enormous energy that they release, and thus may be good candidates for our high-redshift cosmological investiga- mth(z) = M¯ + 5 log DL(z) . (26) 10 tion (see, for instance, Lin et al. 2015, 2016a,b; Dainotti et al. Here M¯ is the zero point offset, it depends on the absolute 2016; Amati et al. 2016a; Izzo et al. 2015; Wei & Wu 2018; Si magnitude M and on the present value of the Hubble param- et al. 2018; Fana Dirirsa et al. 2019; Khadka & Ratra 2020; eter H0: Zhao et al. 2020), or (Zhao et al. 2020; Cao et al. 2021; Muc- cino et al. 2021b; Khadka & Ratra 2020). However, GRBs M¯ = M − 5 log h + 42.38 , (27) 10 may be everything but standard candles since their peak lu- where M is the . The cosmological model minosity spans a wide range, even if there have been many parameters can be determined by minimizing the quantity efforts to make them distance indicators using some empirical correlations of distance-dependent quantities and rest-frame N 2 2 X (µobs(zi) − µth(zi, {θp})) observables (Amati et al. 2008). χ ({θ }) = . (28) SNIa p σ2 Actually GRBs show non thermal spectra which can be i=1 µ i empirically modeled with the Band function (Band 1993), 2 2 2 Here σµ i =σ ¯µ i + σint, where σint is a fit parameter; in our which is a smoothly broken power law with parameters α, statistical analysis we will marginalize over σint. The theo- the low-energy spectral index, γ , the high energy spectral retical distance modulus is defined as index and the roll-over energy E0. Their spectra show a peak corresponding to a value of the photon energy E = E (2+α); µ (z , {θ }) = 5 log D (z , {θ }) + ν , (29) p 0 th i p 10 L i p 0 indeed it turns out that for GRBs with measured spectrum where ν0 = 42.38 − 5 log10 h, and {θp} denotes the set and redshift it is possible to evaluate the intrinsic peak en- of parameters that appear in different dark energy models ergy, Ep,i = Ep(1 + z) and the isotropic equivalent radiated

MNRAS 000, 000–000 (2021) 6 M. Demianski et al.

Dependence on redshift bins 1. We concentrated on observations for which the exposure

Ep,i−Eiso Correlation total GRBs normalization slope scatter time was at least 2/3rd of the whole event duration.

z < 0.5 13 1.97±0.07 0.60±0.06 0.201±0.052 2. Given the broad energy band and good calibration, 0.5 < z < 1 38 2.05±0.05 0.51±0.07 0.238±0.032 Konus-WIND and Fermi/GBM were chosen whenever avail- 1 < z < 1.5 38 2.02±0.06 0.50±0.05 0.172±0.025 1.5 < z < 2 28 1.95±0.16 0.56±0.11 0.230±0.041 able. For Konus-WIND, the measurements were taken from 2 < z < 2.5 28 2.13±0.11 0.44±0.07 0.174±0.031 the official catalog (Ulanov et al. 2005) and from GCN 2.5 < z < 3 16 1.78±0.11 0.68±0.08 0.101±0.031 3 < z < 3.5 13 1.99±0.13 0.55±0.10 0.135±0.048 archives (http : //gcn.gsfc.nasa.gov/gcn3−circulars.html). 3.5 < z < 4 7 2.16±0.15 0.44±0.10 0.069±0.008 4 < z 12 2.06±0.13 0.51±0.11 0.116±0.058 In the case of Fermi/GBM, the observations were derived from (Gruber 2012) and from several other papers, as, for Table 1. Dependence of the Ep,i−Eiso correlation on different red- instance, (Ghirlanda et al. 2004). The observations from shift bins. SUZAKU were not considered as the uncertainties in the cal- ibration are higher and also because it works in a narrow energy band. energy, defined as: 3. The SWIFT BAT observations were chosen when no other preferred missions (Konus-WIND, Fermi/GBM) were Z 104/(1+z) 2 −1 able to provide information. They were considered only for Eiso = 4πdL(z, θ) (1 + z) EN(E)dE . (30) 1/(1+z) GRBs with the value of Ep,obs that was within the energy band of the instrument. For Swift GRBs, the Ep,i value de- Here N(E) is the Band function: rived from BAT spectral analysis alone were conservatively taken from the results reported by the BAT team (Sakamoto  E αB  E   A exp − , (αB ) E0 > E,  100keV E0 et al. 2008). The GCN circulars were also used when needed.  αB −βB N(E) = (αB −βB )E E βB A 100keV exp (αB − βB ) 100keV ,  When more than one mission provides good observations  (α − β ) E E. B B 0 6 based on these criteria, the values and uncertainties of all Even if Ep,i and Eiso span several orders of magnitude, it those observations (hence more than one set for some finely turned out that they are strongly correlated, according to the observed GRBs) are taken into account. When the observa- relation (Amati et al. 2002). tions were to be included in the data sample, it has been checked that the uncertainty on any value is not below 10% in order to account for the instrumental capabilities, etc. So,  E   E (1 + z)  log iso = b + a log p,i , (31) when the error was lower, it has been assumed to be 10%. 1 erg 300 keV When available, the Band model (Band 1993) was considered where a and b are constants. This correlation, as other corre- since the cut-off power law tends to overestimate the value of lations, is characterized by an extra-Poissonian scatter, σint, Ep,i. GRBs have been observed by different detectors, that distributed around the best fit law. It is clear from Eqs. (30 are characterized by different thresholds and spectroscopic and 31) that it is possible to use GRBs to investigate cos- sensitivity, therefore they can spread relevant selection biases mological models, if we can calibrate the Ep,i - Eiso correla- in the observed correlation. This is ongoing debated topic: in tion in a model independent way, overcoming the so called the past, there were claims that a large fraction (70−90%) of circularity-problem, which affects the estimation of the lu- BATSE GRBs without redshift is inconsistent with the corre- minosity distance from all the GBR correlations. Actually lation for any redshift (Band & Preece 2005; Nakar & Piran it turns out that GRBs can be used as cosmological tools 2005). However other authors ((Ghirlanda et al. 2008; Nava through the Ep,i – Eiso correlation; however the computa- et al. 2011)) showed that, in fact, most BATSE GRBs with tion of Eiso is based on a fiducial cosmological model. Differ- unknown redshift were consistent with the Ep,i – Eiso correla- ent and alternative techniques have been recently developed tion. We also note that inconsistency of a high percentage of in literature (see for instance Montiel, Cabrera, & Hidalgo GRBs of unknown redshift would imply that most GRBs with 2021; Amati et al. 2019; Luongo & Muccino 2021; Izzo et al. known redshift should also be inconsistent with the Ep,i – Eiso 2015; Wang, Dai, & Liang 2015; Liang et al. 2008; Kodama relation, and this fact was never observed. Moreover, (Amati et al. 2008; Wei 2010; Lin et al. 2015)): here we standard- et al. 2009) showed that the normalization of the correlation ize our GRB dataset updating a method previously adopted, varies only marginally for GRBs observed by different instru- as discussed in a later section. The Ep,i – Eiso data sample ments with different sensitivities and energy bands, while in used in this analysis was build up by Amati and Sawant, other papers as, for instance, (Ghirlanda et al. 2010) and they collected the spectral information of GRBs with mea- (Amati et al. 2016a) it is shown that the parameters of the sured redshift from February 1997 to October 2015 (Amati correlations are independent of redshift. If the whole GRB et al. 2016a). This database includes redshift z, both energy sample is divided into redshift bins, as shown in Table1 and indices α and γ, the peak energy Ep,i computed from the Fig. (1), it turned out that the possible evolutionary effects break energy E0, t90, exposure time, the fluence and the value are within the intrinsic scatter and, therefore, do not affect of peak flux. The redshift distribution covers a broad range the correlation. Similar results were obtained in (Dainotti & 0.033 6 z < 9.0, thus extending far beyond that of type Ia SN Amati 2018; Demianski et al. 2017a), and (Amati & Della z 6 1.7. For the oldest GRBs (BeppoSAX, BATSE, HETE-2) Valle 2013; Demianski & Piedipalumbo 2011) and other GRBs up to mid 2008, the data was adapted from Furthermore, the Swift satellite, thanks to its capability of (Amati et al. 2008). As already discussed in (Demianski et al. providing quick and accurate localization of GRBs, thus re- 2017a), the criteria behind selecting the measurements from ducing the selection effects in the observational chain leading a particular mission are based on the following conditions: to the estimate of GRB redshift, has further confirmed the

MNRAS 000, 000–000 (2021) Dark energy models with the Ep,i – Eiso correlation 7

Figure 6. Dependence of the Ep,i−Eiso correlation on different en- ergy ranges (based on various satellite missions).

Figure 5. Dependence of time integrated bolometric Ep,i−Eiso cor- relation on differeninst redshift b.

Dependence on energy bins

Ep,i−Eiso Correlation total GRBs normalization slope scatter

BeppoSAX (2–700 keV) 11 2.11±0.08 0.48±0.08 0.138±0.057 HETE–2 (2–500 keV) 18 1.88±0.06 0.51±0.06 0.133±0.041 Konus–WIND (20 keV–10 MeV) 72 2.03±0.06 0.54±0.04 0.168±0.019 SWIFT (15–150 keV) 32 1.96±0.03 0.55±0.04 0.113±0.031 FERMI (10–30 MeV) 51 2.11±0.06 0.45±0.05 0.236±0.026

Table 2. Dependence of the E −E correlation on different en- p,i iso Figure 7. Monochromatic Ep,i−Eiso correlation from the Swift mis- ergy bins (based on various satellite missions). sion

reliability of the Ep,i – Eiso correlation (Amati et al. 2009; 3.2.1 Calibrating the Ep,i−Eiso correlation and fitting its Ghirlanda et al. 2010; Amati & Dichiara 2013; Martone et parameters al. 2017; Amati et al. 2019). If one divides the GRB sam- Here we update a local regression technique, inspired by ple on the basis of different high energy satellite missions, the standardization of the SNIa with Cepheid variables, it turns out that the correlation always remains within the and based on Union 2.1 SNIa sample, which we adopted in same E −E fit parameters range (slope ∼ 0.5), as it can p,i iso previous works (Demianski et al. 2017a),(Demianski et al. be seen from Table2, and Fig.(6). 2017b),(Demianski & Piedipalumbo 2011). We actually con- Moreover, based on time-resolved analysis of BATSE, Bep- sider a sort of extended Ep,i−Eiso correlation, introducing poSAX and Fermi GRBs, it was found that the E – E cor- p,i iso in the calibration terms representing the z-evolution, which relation also holds within each single GRB with normalization kiso are assumed to be power-law functions: giso(z) = (1 + z) and slope consistent with those obtained with time-averaged 0 0 kp Eiso spectra and energetics/luminosity (Basak & Rao 2013; Fron- and gp(z) = (1 + z) , so that Eiso = and Ep,i = giso(z) tera et al. 2013; Lu et al. 2012), confirming the physical ori- Ep,i gin of the correlation, and providing clues to its explanation. are the de-evolved quantities (see also Shirokov et al. gp(z) Therefore, it turns out that, at the present stage, the fit val- 2020).Therefore we consider a correlation with three param- ues of the Ep,i – Eiso correlation parameters are marginally eters a, b, and kiso − akp: affected by selection and/or evolutionary effects, which are  E   E  less than the intrinsic dispersion ((Amati & Della Valle 2013; log iso = b + a log p,i + Amati et al. 2016a; Dainotti & Amati 2018; Demianski et al. 1 erg 300 keV 2017a; Sawant & Amati 2018)). + (kiso − akp) log (1 + z) . (32)

MNRAS 000, 000–000 (2021) 8 M. Demianski et al.

�� ���

�� ���

�� ��� / ��� ) -��� � ��� ) / ��� ��� �� ���� -��� ��� �� ��� � ��� ( � ( ��� ��� -��� ��� -��� �� -��� -��� ���� ���� ���� ���� ���� �� -��� -��� -��� -��� ��� ��� ��� ��� ��� ��� ��� ��� ���(� ���� /��� ���) ���(�+�)

Figure 9. Redshift distribution of log Ep,i/(300keV ) for the sub- Figure 8. De-evolved (red points) and evolved/original Ep,i−Eiso correlation (blue points); there is no noticable evolution. sample used for our calibration.

We can simplify the redshift dependence term in Eq. (32), �� introducing a single average coefficient γ:  E   E  log iso = b + a log p,i + 1 erg 300 keV �� +γ log (1 + z) . (33)

Calibrating this 3D relation means determining the coeffi- ��

cients a, b, and γ and the intrinsic scatter σint. It is worth / ��� )

noting thatlow values of γ would indicate negligible evolu- ��� tionary effects. In order to calibrate our de-evolved relation �� we consider a 3D Reichart likelihood: � ��� ( 53.5 53.0 P 2 2 2 2 3D 1 log (σint + σyi + a σxi ) 52.5 LReichart(a, γ, b, σint) = + �� 2 log (1 + a2) 52.0 2 51.5 1 X (yi − axi − γzi − b) + 2 2 2 2 . (34) 0.20 0.25 0.30 0.35 0.40 2 σint + σxi + a σxi �� We maximized our likelihood with respect to a and γ since b ��� ��� ��� ��� ��� can be evaluated analytically by solving the equation ���(�+�)

∂ 3D Figure 10. Redshift distribution of log Eiso/(erg) for the subsample L (a, kiso, α, b, σint) = 0 , (35) ∂b Reichart used for our calibration. we obtain    −1 X yi − axi − γzi X 1 b = . 2 2 2 2 2 2 2 2 filters on E and E , reducing the sample to 60 objects, as σint + σyi + a σxi σint + σyi + a σxi p,i iso (36) shown in Figs. (9 and 10), where the Ep,i and Eiso are homo- geneously distributed in the sample. Note that the measured We also used the MCMC method to maximize the likeli- values of Ep,i and Eiso are not systematically larger at lower hood and ran five parallel chains and the Gelman-Rubin con- redshift than at higher redshifts. +0.04 vergence test. We obtain a = 1.92 ± 0.09 , b = 52.7−0.03 , From the Reichart likelihood analysis we find that a = +0.04 σint = 0.35 , γ = −0.07 ± 0.14, thus confirming that the +0.14 +0.04 −0.05 2.0−0.13, b = 52.5−0.03 ,σint = 0.38 ± 0.06 , γ = 0.02 ± 0.09. evolutionary effects, not included in the intrinsic dispersion, The fit values of the correlation parameters are fully compat- can be, at this stage, neglected, as shown in Fig. (8). ible (at 3 σ) with the results obtained from the full dataset. Moreover it is worthwhile to note that we have been forced to limit our analysis to the redshfit range 0.015 z 1.414, 3.2.2 Further investigations on the calibration of the E – 6 6 p,i where we apply our calibration technique. In order to extend E correlation iso our analysis to a wide range of redshifts we build up a further In this section we discuss some aspects related to the calibra- calibration procedure which updates the procedure described tion of the Ep,i – Eiso relation, and its impact on reliability in (Demianski & Piedipalumbo 2011; Demianski et al. 2017b), of the GRBs as distance indicators. Actually, we apply some based on an approximate function for the luminosity distance.

MNRAS 000, 000–000 (2021) Dark energy models with the Ep,i – Eiso correlation 9

250 46 44 200 42

40 150 μ ( z ) H ( z ) 38

36 100 34

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 z z

Figure 11. Comparison between the observational data and theo- retical predictions for the distance modulus (left panel), and the Hubble function, H(z), (right panel).

app This function dL (z) has the form c z(z + 1)2 dapp(z) = , (37) L p 3 2 2 Figure 12. Distance modulus µ(z) for the calibrated GRB Hubble H0 d3z + (d2z + d1z + 1) diagram obtained by fitting the Ep,i – Eiso relation. where d1, d2 and d3 are constants. To estimate these parame- ters we have simultaneously fitted the SNIa samples and the H(z) measurements. Actually it turns out that we can de- particular, recalling that our correlation relation can be writ- termine an approximate function for the H(z), from dapp(z), ten as a linear relation, as in Eqs. (32, 33), the error on the L h i Eiso according to the relation: distance dependent quantity y = log 1 erg was estimated as 1 d  dapp(z)  = L . (38) q Happ(z) dz 1 + z 2 2 2 2 2 2 σ(y) = a σx + σax + σb + σint, (40) We find that d1 = 1.18±0.03, d2 = 0.30±0.04, d3 = 0.49±0.1. app h Ep,i i In Fig. (11), we plot the data with the best fit dL and where x = log 300 keV , σb is properly evaluated through the app H (z) respectively. Eq. (36), which implicitly defines b as a function of a and σint, app Therefore we can use the approximate function dL (z) and is then added in quadrature to the uncertainties of the to calibrate the Ep,i – Eiso correlation, following the pro- other terms entering Eq.(39) to obtain the total uncertainty. cedure described above. Using all the data, we find that It turns out that a = 1.84 ± 0.08, b = 52.7 ± 0.05 ,σ = 0.37 ± 0.04 , int  5    E  γ = 0.03 ± 0.1. As the last check we applied the same fil- 5 log d (z) = b + a log p,i + L 2 300 keV ters on Ep,i and Eiso as before, selecting a sub-sample of 128 + (γ + 1) log (1 + z) − log (4πS ) + µ } , (41) GRBs, and calibrated the Ep,i – Eiso correlation. We find bolo 0 that a = 1.94 ± 0.09, b = 52.4 ± 0.04 , σint = 0.38 ± 0.05 , where µ0 is a normalization parameter. Actually the distance γ = 0.02 ± 0.15. Also in this case it turns out that the cal- modulus of GRBs are not absolute, thus this cross-calibration ibration parameters a, γ and σint are fully consistent with parameter is needed to match the GRB Hubble diagram and our SNIa-calibration technique, and that all the possible sys- the one of supernovae. If µ0 cannot be determined, we can tematics and evolutionary effects are within the intrinsic dis- only use the shape of the Hubble Diagram to constrain the persion σint, as already discussed in literature ( see, for in- cosmological parameters such as Ωm and ΩΛ, with no infor- stance, (Amati et al. 2008; Amati & Della Valle 2013; Amati mation on H0, which is degenerate with the µ0 parameter. & Dichiara 2013)). However it is worth noting that when fu- It turns out that µ0 depends on the fiducial cosmological ture GRB missions will substantially increase the number of model and its parameters; however in all models considered GRBs usable to construct the Ep,i – Eiso correlation up to in our analysis µ0 ' 0.4. In Fig. (12) we plot the GRB Hub- redshift z ' 10 , they may shed new light on the properties ble diagram, and in Fig. (13) we show that this diagram of this important correlation. matches the Union 2.1 Hubble diagram, then µ0 has been evaluated for the CPL best fit model, discussed above. In order to make this comparison clearer, we use the auxiliary 3.2.3 Bilding up the Hubble diagram z variable y = , which map the z-range [0, ∞) into the After fitting the correlation and estimating its parameters, 1 + z we used them to construct the GRB Hubble diagram. We y-interval [0, 1]. It turns out that the GRBs are the natural recall that the luminosity distance of a GRB with redshift z continuation of SNIa in the Hubble diagram. The listed data is are available on request to the authors.  1/2 Eiso(1 + z) dL(z) = . (39) 4πSbolo 3.3 Direct H(z) measurements

The uncertainty of dL(z) was estimated through the propaga- The direct measurements of Hubble parameters are comple- tion of the measurement errors of the pertinent quantities. In mentary probes to constrain the cosmological parameters and

MNRAS 000, 000–000 (2021) 10 M. Demianski et al.

Parameters Priors % Ωm (0, 1) w0 (−3, 3) !% w1 (−3, 3)

) Ωe (0, 0.1)  ( H0 (0.5, 1.5) μ ! Table 3. Priors for parameters estimate in the MCMC numerical $% analysis.

$ likelihood function L(p): exp (−χ2 /2)   ! " # L(p) ∝ SNIa/GRB NSNIa/GRB 1/2 & (2π) 2 |CSNIa/GRB | exp (−χ2 /2) × H . (43) Figure 13. GRB (blue points) and SNIa (red points) superposed N /2 1/2 (2π) H |CH | Hubble diagrams after µ0 has been evaluated for the best fit CPL z Here model (see the section below). Here y = . 1 + z N 2 X  th  −1  th  χ (p) = xi − xi (p) Cij xj − xj (p) , (44) i,j=1 p is the set of parameters, N is the number of data points, th investigate the dark energy (Jimenez & Loeb 2002). The Hub- xi is the i − th measurement; xi (p) indicate the theo- a˙ ble parameter, defined as H(z) = , where a is the scale fac- retical predictions for these measurements that depend on a the parameters p. C is the covariance matrix (specifically, tor, depends on the differential variation of the cosmic time ij C indicates the SNIa/GRBs/H covariance ma- with redshift as SNIa/GRB/H trix). Eq. (43) includes σint term to allow for intrinsic scatter 1  dt −1 in the data sets. H(z) = − . (42) In our analysis we include only flat priors on the typical 1 + z dz parameters of the considered cosmological models (see Table 3, with the exception of the Hubble constant, h, for which we  dt  The can be measured using the so-called cosmic consider a gaussian prior accounting for the local determi- dz chronometers. dz is obtained from spectroscopic surveys with nation of the Hubble constant by the SHOES collaboration high resolution, and the differential evolution of the age of the (hS , σS ) = (0.742, 0.036) Riess et al.(2009). We actually con- Universe dt in the redshift interval dz can be measured pro- sider the term : vided that optimal probes of the aging of the Universe, that  2 2  is, the cosmic chronometers, are identified. The most reliable exp −(hS − h) /4σS L(h) = . (45) cosmic chronometers observable at high redshift are old early- p 2 4πσS type galaxies that evolve passively on a timescale much longer We sample the space of parameters by running five paral- than their age difference, which formed the vast majority of lel chains and use the Gelman- Rubin diagnostic approach to their stars rapidly and early and have not experienced subse- test the convergence. As a test probe, it uses the reduction quent major episodes of formation or merging. Moreover, factor R, which is the square root of the ratio of the variance the Hubble parameter can also be obtained from the BAO between-chains and the variance within-chain. A large R indi- measurements: by observing the typical acoustic scale in the cates that the between-chains variance is substantially greater line-of-sight direction, it is possible to extract the expansion than the within-chain variance, so that a longer simulation rate of the Universe at a certain redshift. We used a list of 28 is needed. We require that R converges to 1 for each param- direct H(z) measurements in the redshift range z ∼ 0.07−2.3 eter. We set R − 1 of order 0.05, which is more restrictive compiled by (Farooq & Ratra 2013). than the often used and recommended value R − 1 < 0.1 for standard cosmological investigations. We discarded the first 30% of the point iterations at the beginning of any MCMC run, and thinned the chains that were run many times. We finally extracted the constrains on cosmographic parameters by coadding the thinned chains. The histograms of the param- 4 STATISTICAL ANALYSIS eters from the merged chains were then used to infer median To test the dark energy models described above, we use a values and confidence ranges. In Tables (4), (5), and (6) we Bayesian approach based on the MCMC method. In order to present the results of our analysis. In Fig. (16) we plot a 2D set the starting points for our chains, we first performed a confidence region for the CPL model: it is worth noting that, preliminary and standard fitting procedure to maximize the using only the GRB Hubble diagram and the H(z) sample,

MNRAS 000, 000–000 (2021) Dark energy models with the Ep,i – Eiso correlation 11 1.2

1.0

0.8

1 0.6 w

0.4

0.2 Figure 14. Redshift dependence of the EDE EOS corresponding to the best fit values of the parameters. 0.0 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4

w0

Figure 16. 2D confidence regions in the w0-w1 plane for the CPL model, obtained from the GRB Hubble diagram and the H(z) sam- ple. It is worth noting that the ΛCDM model of dark energy is disfavoured at more than 3σ.

0.0

) -0.2 z ( q

Figure 15. Redshift dependence of the Ω parameters for the EDE model corresponding to the best fit values of the parameters: -0.4 Ωm(z) is shown in blue, and ΩDE (z) in red. the ΛCDM model of dark energy is disfavoured at more than -0.6 3σ, as indicated also by the Hubble diagram of quasars at 0.0 0.2 0.4 0.6 0.8 1.0 high redshifts (Risaliti & Lusso 2019; Lusso & Risaliti 2017). In Figs. (14) and (15) we plot the redshift behaviour of the z Ω parameter and the effective early dark energy (EDE) EOS, corresponding to the best fit values of the parameters, ob- Figure 17. Redshift dependence of the q(z) tained in our statistical analysis. for the CPL model, corresponding to the best values of the relative It is well known that the likelihood-statistics alone does parameters obtained combining the GRB Hubble diagram and the H(z) measurements. not provide an effective way to compare different cosmolog- ical models. In this section we compare the different models presented in the previous sections and check if we can discrim- inate between them. We use the Akaike Information Criterion (AIC) (Akaike 1974), (Liddle 2007), and its indicator while ∆ 10 indicates a strong evidence. ∆ 2 is 2n (n + 1) AIC > AIC 6 AIC = −2 ln L + 2n + p p , (46) an indication that the two models are consistent. In our case max p N − n − 1 tot p we have found that the model with the lower AIC is the expo- where Ntot is the total number of data and np the number of nential scalar field: it turns out that ∆AIC ' 5 if we consider free parameters (of the cosmological model) . It turns out that the CPL model and ∆AIC = 9 for the early dark energy. the lower is the value of AIC the better is the fit to the data. Moreover, it turns out that also without the SNIa data, com- To compare different cosmological models we introduce the bining the GRB Hubble diagram with the H(z) compilation, difference ∆AIC = AICmodel −AICmin. This difference corre- it is possible to set the transition region from the decelerated sponds to different cases: 4 < ∆AIC < 7 indicates a positive to the accelerated expansion in all the tested cosmological evidence against the model with higher value of AICmodel, models, as indicated in Figs. (17), (20), and (21).

MNRAS 000, 000–000 (2021) 12 M. Demianski et al.

CPL Dark Energy

Id hxi x˜ 68% CL 95% CL hxi x˜ 68% CL 95% CL

SNIa /GRBs/H(z) GRBs/H(z)

Ωm 0.29 0.29 (0.28, 0.30) (0.27, 0.31) 0. 17 0.18 (0.16, 0.195) (0.15, 0.26)

w0 -1.03 -1.02 (-1.1, -0.96) (-1.14, -0.88) -0.84 -0.858 (-0.94, -0.74) (-1.07, -0.68)

w1 0.03 0.03 (-0.15,0.24) (-0.32, 0.38) 0.8 0.86 (0.63, 0.95) (0.39, 0.99)

h 0.69 0.69 (0.68, 0.70) (0.67, 0.71) 0.67 0.67 (0.65, 0.7) (0.62, 0.74)

Table 4. Constrains on the CPL parameters from different data: combined SNIa and GRB Hubble diagrams, and H(z) data sets (Left Panel); and GRB Hubble diagram and H(z) data sets (Right Panel ). Columns show the mean hxi and medianx ˜ values and the 68% and 95% confidence limits.

Scalar Field Quintessence

Id hxi x˜ 68% CL 95% CL hxi x˜ 68% CL 95% CL

SNIa /GRBs/H(z) GRBs/H(z)

H0 0.98 0.98 (0.97, 1.0) (0.95, 1.03) 0. 97 0.98 (0.96, 1.05) (0.95, 1.1)

h 0.69 0.69 (0.68, 0.70) (0.67, 0.71) 0.67 0.67 (0.65, 0.7) (0.62, 0.74)

Table 5. Constrains on the scalar field parameters from different data: combined SNIa and GRB Hubble diagrams, and H(z) data sets (Left Panel); and GRB Hubble diagram and H(z) data sets (Right Panel ). Columns show the mean hxi and medianx ˜ values and the 68% and 95% confidence limits. It is worth noting that Ωm = 0.24 ± 0.02 in the case of the SNIa /GRBs/H(z) samples, and Ωm = 0.25 ± 0.03 for the GRBs/H(z) samples.

Early Dark Energy

Id hxi x˜ 68% CL 95% CL hxi x˜ 68% CL 95% CL

SNIa /GRBs/H(z) GRBs/H(z)

Ωm 0.28 0.28 (0.26, 0.30) (0.25, 0.32) 0. 17 0.18 (0.16, 0.195) (0.15, 0.26)

w0 -0.62 -0.6 (-0.7, -0.53) (-0.95, -0.5) -0.84 -0.858 (-0.94, -0.74) (-1.07, -0.68)

Ωe 0.003 0.003 (0.001,0.07) (0.005, 0.09) 0.008 0.009 (0.0006, 0.0095) (0.0004, 0.0015)

h 0.71 0.71 (0.7, 0.72) (0.69, 0.73) 0.67 0.67 (0.65, 0.7) (0.62, 0.74)

Table 6. Constrains on the Early Dark Energy parameters from different data: combined SNIa and GRB Hubble diagrams, and H(z) data sets (Left Panel); and GRB Hubble diagram and H(z) data sets (Right Panel ). Columns show the mean hxi and medianx ˜ values and the 68% and 95% confidence limits.

5 PROSPECTS WITH THESEUS will substantially increase the number of GRBs that could be used to construct the Ep,i – Eiso correlation up to redshift z ' 10 and will allow a better calibration of this correlation. So far we have shown that the E – E correlation has sig- p,i iso Here we consider a simulated sample of 772 objects to con- nificant implications for the use of GRBs in cosmology and strain our models, their redshift distribution is shown in Fig. therefore GRBs are powerful cosmological probe, complemen- (18). tary to other probes. GRB missions, like, the proposed THE- SEUS observatory (Amati et al. 2018; Cordier et al. 2018), These simulated data sets have been obtained by imple-

MNRAS 000, 000–000 (2021) Dark energy models with the Ep,i – Eiso correlation 13

Figure 18. Simulated data for a THESEUS like mission: we plot Figure 19. Simulated data for a THESEUS like mission: we plot the GRB redshift distribution of the sample of 772 GRBs used in the GRB redshift distribution of the sample of 1500 GRBs used in our analysis. our analysis. menting the Monte Carlo approach and taking into account 0.2 the slope, normalization, dispersion of the observed Ep,i and Eiso correlation, the distribution of the uncertainties in the 0.0 measured values of Ep,i and Eiso, and finally the observed red- shift distribution of GRBs. In this simulations we took into account the sensitivity limits and spectroscopy thresholds and -0.2

sensitivity of the THESEUS monitors (SXI and XGIS). This ) z mock sample is based on parameters of the observed sample ( -0.4 and corresponds to the actual data sets and to the data sets q expected to be available within 3-4 years from THESEUS. 0.6 The cosmological parameters assumed for the simulations are - the median, or average, values found in the reported analy- sis on real data and, indeed, Tables (7), and (8) show that -0.8 the analysis on the simulated data recover very well these assumed cosmological parameters. It turns out that with our -1.0 mock sample of GRBs we are able to constrain much better 0.0 0.2 0.4 0.6 0.8 1.0 the cosmological parameters. Actually, in Figs. (22), (23) and (24), we show the 2D confidence regions in the w0-w1 plane z for the CPL model, obtained from the simulated GRB Hub- Figure 20. Redshift dependence of the deceleration parameter q(z) ble diagram and the H(z) sample, compared with the same for the scalar field model, corresponding to the best values of the confidence region obtained from the real datasets, it turns out relative parameters obtained combining the GRB Hubble diagram that the evolving dark energy, described by the exponential and the H(z) measurements. scalar field potential is the favoured model. Moreover, in the case of the CPL model, we tested the efficiency of our probes looking at the Figure of Merit (FoM), that is the inverse of thus confirming that with future data we will able to bet- area of the w0 −w1 contour: for this purpose we simulate new ter constrain the dark energy EOS as described by the CPL mock samples of the CPL cosmological model corresponding parametrization. to the best fit values in Table (7), with the probability density function for the distribution in redshift corresponding to the histograms in Figs. (18) and (19), and containing respectively 772 and 1500 objects. To each simulated GRB we estimate 6 DISCUSSION AND CONCLUSIONS the error on the distance modulus as: The Ep,i – Eiso correlation has significant implications for the s  2 use of GRBs in cosmology to test different models of dark en- 2 z 2 σµ(z) = σsys + σm . (47) ergy beyond the standard ΛCDM . Here we considered an ex- zmax tended Ep,i – Eiso correlation, which takes into account possi- Here zmax is the maximum redshift of the sample, and ble redshift evolution effects (related, for instance to the grav- σsys the intrinsic scatter. In our case (zmax, σsys, σm) = itational lensing along the GRB line of sight), parametrized (8.5, 0.3, 0.05). We get F oM = 1.9 when using 772 GRBs, as power law terms. Using recently updated data set of 193 and it increases up to F oM = 4.8 if 1500 GRBs are used, high-redshift GRBs, we applied a local regression technique

MNRAS 000, 000–000 (2021) 14 M. Demianski et al.

CPL Dark Energy

Id hxi x˜ 68% CL 95% CL

Simulated GRBs/H(z)

Ωm 0.18 0.19 (0.16, 0.21) (0.15, 0.27)

w0 -0.86 -0.86 (-0.94, -0.79) (-1.07, -0.72)

w1 0.82 0.82 (0.71,0.93) (0.62, 0.98)

h 0.67 0.67 (0.65, 0.69) (0.61, 0.74)

Table 7. Constrains on the CPL parameters from our simulated GRB Hubble diagram and H(z) data.

Scalar Field Quintessence

Id hxi x˜ 68% CL 95% CL

Simulated GRBs/H(z)

H0 0.98 0.981 (0.96, 1.00) (0.94, 1.02)

h 0.68 0.68 (0.68, 0.69) (0.65, 0.70)

Table 8. Constrains on the scalar field parameters from our simulated GRB Hubble diagram and H(z) data. It turns out that Ωm = 0.26 ± 0.02

to calibrate the Ep,i – Eiso relation. The values of the cali- Hubble diagram, which we adopted as a tool to constrain bration parameters are statistically fully consistent with the different cosmological models: to investigate the prospective results of our previous work (Demianski et al. 2011, 2017a), for high redshift constrains on dark energy models with the and confirm that for this correlation the evolution effects re- Ep,i – Eiso correlation is the prime objective of our analy- main within the intrinsic scatter of the correlation around sis. We considered the CPL parametrization of the EOS, an the best fit values, σint. This result has been confirmed also exponential potential of a self interacting scalar field, and, with different calibration techniques, based on an approxi- finally a model with dark energy at early times. We com- mation function able to reproduce the luminosity distance in pare these different models, by using the Akaike Information different cosmological models. Moreover in order to further Criterion (AIC) and its indicator. In our case we have found investigate this question here we follow a different approach: that the model with the lower AIC is the exponential scalar in order to remove possible systematics, we apply some filters field: ∆AIC ' 5 if we consider the CPL model and ∆AIC = 9 on the Ep,i – Eiso correlation, selecting a subsample where for the early dark energy. Therefore this model is slightly they are homogeneously distributed in the redshift space and preferred by the present data. Moreover, even if the cosmo- we showed that none of the measured values of Ep,i – Eiso logical constrains from the currently available GRB Hubble are systematically larger at lower redshifts than at higher red- diagram are not so restrictive, future GRB missions, like the shifts. Again we fit the calibration within this smaller sample, proposed THESEUS observatory will increase the number and the results are fully consistent with the results obtained of GRB usable to construct the Ep,i – Eiso correlation up on the whole dataset. Therefore these results justify exten- to redshift z ' 10. We actually considered a mock sample, sion of this calibration to the whole sample, at least on the consisting of 772 objects, obtained taking into account the basis of the current stage of knowledge. The fitted calibration slope, normalization, dispersion of the observed correlation, parameters have been used to construct a high redshift GRB the distribution of the uncertainties in the measured values

MNRAS 000, 000–000 (2021) Dark energy models with the Ep,i – Eiso correlation 15 0.2 1.5

0.0

-0.2 1.0 �σ ) z (

q �σ �σ -0.4 1 w 0.5 -0.6

0.0 0.2 0.4 0.6 0.8 1.0 z 0.0

Figure 21. Redshift dependence of the deceleration parameter q(z) for the EDE model, corresponding to the best values of the relative parameters obtained combining the GRB Hubble diagram and the -1.2 -1.0 -0.8 -0.6 H(z) measurements. w0

1.0 Figure 23. 2D confidence regions in the w0-w1 plane for the CPL model, obtained from the simulated GRBs HD and the H(z) mea- surements (bottom panel) and the full real datasets (upper panel).

0.5 1.2

1.0 �σ 1 0.0 �σ �σ w 0.8 �σ�σ

1 0.6 -0.5 w �σ 0.4

-1.0 0.2 -1.4 -1.2 -1.0 -0.8 -0.6

w0 0.0 -1.5 -1.0 -0.5 0.0 Figure 22. 2D confidence regions in the w0-w1 plane for the CPL model, obtained from the full real datasets. w0

Figure 24. 2D confidence regions in the w0-w1 plane for the CPL of Ep,i and Eiso, and finally the observed redshift distribu- model, obtained from the GRBs HD and the H(z) measurements tion of GRBs. In these simulations we took into account the . sensitivity limits and spectroscopy thresholds and sensitivity of the THESEUS monitors. The mock sample corresponds to the data sets expected to be available within few years objects respectively, and we get F oM = 1.9 when using 772 from THESEUS. It turns out that in this case we are able to GRBs, while it increases by a factor 2.5 if 1500 GRBs are constrain much better the cosmological parameters, and the used. This confirms, indeed, that with future data we will exponential scalar field potential is confirmed as the favourite be able to better constrain the evolution of the dark energy model. We finally tested the efficiency of our probes looking EOS, in a complementary way to type Ia SN, and in synergy at the Figure of Merit (FoM). In the case of the CPL model: with other independent high redshift cosmological probes, as, we simulated new mock samples consisting of 772 and 1500 for instance quasars (Lusso 2020; Lusso et al. 2020b).

MNRAS 000, 000–000 (2021) 16 M. Demianski et al.

Acknowledgments Corasaniti, P.S., and Copeland, E. J., 2006, Phys. Rev. D, 67, 063521 MD is grateful to the INFN for financial support through the Cordier, B., G¨otz, D., Motch, C., SVOM Collaboration, 2018, Fondi FAI GrIV. EP acknowledges the support of INFN Sez. Mem. SAIt, 89, 26 di Napoli (Iniziativa Specifica QGSKY ). LA acknowledges Dainotti M. G., Postnikov S., Hernandez X., Ostrowski M., 2016, support by the Italian Ministry for Education, University and ApJL, 825, L20. doi:10.3847/2041-8205/825/2/L20 Research through PRIN MIUR 2009 project on ”Gamma ray Dainotti, M.G., and Amati,L., 2018, Publications of the Astronom- bursts: from progenitors to the physics of the prompt emission ical Society of the Pacific, 987, 051001 process.” (Prot. 2009 ERC3HT). Das S., Sherwin B. D., Aguirre P., Appel J. W., Bond J. R., Car- valho C. S., Devlin M. J., et al., 2011, PhRvL, 107, 021301. doi:10.1103/PhysRevLett.107.021301 De Felice, A., Tsujikawa, S., 2010, Living Rev. Rel., 13, 3 DATA AVAILABILITY Demianski M., Piedipalumbo E., Rubano C., Tortora C., 2005, Astronomy&, 431, 27. doi:10.1051/0004- The data underlying this article will be shared on reasonable 6361:20041508 request to the corresponding author. Demianski M., Piedipalumbo E., Rubano C., Scudellaro P., 2008, Astronomy&Astrophysics, 481, 279. doi:10.1051/0004- 6361:20067045 Demianski, M., E. Piedipalumbo, D. Sawant, L. Amati, 2017a, References Astronomy & Astrophysics, 598, A112 Akaike, H., 1974, IEEE Transactions on Automatic Control 19, Demianski, M., E. Piedipalumbo, D. Sawant, L. Amati, 2017b, 716 Astronomy & Astrophysics, 598, A113 Alam, U., Sahni, V., Saini, T.D., Starobinsky, A.A., 2003, MNRAS, Demianski, M., Piedipalumbo, E., 2011, MNRAS, 415, 3580 344, 1057 Demianski, M., Piedipalumbo, E., Rubano, C., 2011, MNRAS, 411, Alcaniz J. S., Lima J. A. S., 2001, ApJL, 550, L133. 1213 doi:10.1086/319642 Di Valentino E., 2021, MNRAS, 502, 2065. Amanullah, R., Lidman, C., Rubin, D., Aldering, G., Astier, P., doi:10.1093/mnras/stab187 Barbary, K., Burns, M. S., Conley, A., et al.,2010, ApJ, 71, Doran M., Lilley M., Wetterich C., 2002, PhLB, 528, 175. 712 doi:10.1016/S0370-2693(02)01218-2 Amati, L., Frontera, F., Tavani, M., in’t Zand, J.J.J, Antonelli, Doran, M., Robbers, G. , 2006, JCAP, 06, 026 A., Costa, E., Feroci, M., Guidorzi, C., Heise, J., Masetti, N., Fana Dirirsa F., Razzaque S., Piron F., Arimoto M., Axelsson M., Montanari, E., Nicastro, L., Palazzi, E., et al., 2002, Astronomy Kocevski D., Longo F., et al., 2019, ApJ, 887, 13 &Astrphysics, 390, 81 Farooq, O., Ratra, B., 2013, ApJ, 766, L7 Amati, L., Guidorzi, C., Frontera, F., Della Valle, M., Finelli, F., Frontera, F., Amati, L., in’t Zand, J.J.M., et al., 2013, ApJ, 754, Landi, R., Montanari, E., 2008, MNRAS, 391, 577 138 Amati, L., Frontera, F., Guidorzi, C.: 2009, A&A 508, 173 Ghirlanda, G., Ghisellini, G., Firmani, C., & Lazzati, D., 2004, Amati, L., Della Valle, M., 2013, IJMP D, 22, 1330028 ApJ, 616, 331. Amati, L., Dichiara, S., 2013, Acta Polytechnica 53, 687 Ghirlanda, G., Nava, L., Ghisellini, G., Firmani, C., Cabrera, J.I., Amati, L., Sawant, D., Della Valle, M., 2016a, Astronomical & 2008, MNRAS, 387, 319 Astrophysical Transactions, 29, 193 Ghirlanda, G., Nava, L., Ghisellini, 2010, Astron- Amati, L., O’Brien, P., Goetz, P., Bozzo, E., Tenzer, C. , et al., omy&Astrophysics 511, 43 2018, Advances in Space Research, in press. Gruber, D. for the Fermi/GBM collaboration, Proceedings of Amati L., D’Agostino R., Luongo O., Muccino M., Tantalo M., Scince (GRB 2012), 7 2019, MNRAS, 486, L46 Izzo, L., Muccino, M., Zaninoni, E., Amati, L., Della Valle, M., Astier, P., Guy, J., Regnault, N., Pain, R., Aubourg, E., et al., 2015, Astronomy & Astrophysics, 582, A115 2006, Astronomy & Astrophysics, 447, 31 Liang N., Xiao W. K., Liu Y., Zhang S. N., 2008, ApJ, 685, 354. Band, D., 1993, ApJ, 413, 281 doi:10.1086/590903 Band, D., Preece R., 2005 ApJ, 627, 319 Jimenez, R., and Loeb, A., 2002, ApJ, 573, 37. Basak, R., & Rao, A. R. 2013, ApJ, 775, 31 Karwal, T., Kamionkowski, M., 2016, Phys. Rev. D, 94, 103523. Bueno Sanchez J. C., Perivolaropoulos L., 2010, PhRvD, 81, Khadka N., Ratra, B., 2020, MNRAS.tmp, 103505. doi:10.1103/PhysRevD.81.103505 doi:10.1093/mnras/staa2779 Caldwell R.R., Dave R., Steinhardt P.J., 1998, Phys. Rev. Lett. Khoraminazad, H, Viel, M., Baccigalupi, C., Archidiacono, M., 80, 1582 2020, JCAP 7, 39 Cao, S, Ryan, J., Khadka, N., Ratra, B. 2021, MNRAS, 501, 1520 Kodama Y., Yonetoku D., Murakami T., Tanabe S., Tsutsui R., Capozziello S., Cardone V. F., Piedipalumbo E., Rubano C., 2006, Nakamura T., 2008, MNRAS, 391, L1. doi:10.1111/j.1745- CQGra, 23, 1205. doi:10.1088/0264-9381/23/4/009 3933.2008.00508.x Capozziello, S., De Laurentis, M., 2011, Phys. Rep., 509, 167 Liddle, A.R., MNRAS, 377, L74 (2007) Capozziello, S., D’Agostino, R., Luongo, O., 2019, Int. J. Mod. Lin, H.-N., Li, X., Liu, Y., Wang, S., Chang, Z., 2015, MNRAS, Phys. D, 28, 1930016 453, 128 Carroll, S., M., ”The Cosmological Constant”, 2001, Living Rev. Lin, H.-N., Li, X., Chang, Z., 2016a, MNRAS, 459, 2501 Relativity 4, 1 Lin, H.-N., Li, X., Chang, Z., 2016b, MNRAS 455, 2131 Clarkson, C., Maartens, R., 2010, Classical and Quantum Gravity, Linder, E.V. 2003, Phys. Rev. Lett., 90, 091301 27, 124008 Lu, R., Wei, J., Liang, E., et al., 2012, ApJ, 756, 112 Clifton,T., Ferreira, P.G., Padilla, A., Skordis, C., 2012, Phys. Luongo O., Muccino M., 2021, MNRAS.tmp. Rep., 513, 1 doi:10.1093/mnras/stab795 Chevallier, M., Polarski, D., 2001, Int. J. Mod. Phys. D, 10, 213 Lusso, E., Risaliti, G., 2017, Astronomy & Astrophysics, 602, A69 Copeland E. J., Sami M., Tsujikawa S., 2006, IJMPD, 15, 1753. Lusso, E., Piedipalumbo, E., Risaliti, G., Paolillo, M., Bisogni, S., doi:10.1142/S021827180600942X Nardini, E., Amati, L., 2019, Astronomy & Astrophysics, 628,

MNRAS 000, 000–000 (2021) Dark energy models with the Ep,i – Eiso correlation 17

L4 Wei H., 2010, JCAP, 2010, 020. doi:10.1088/1475- Lusso, E., Frontiers in Astronomy and Space Sciences, 2020, 7, 8 7516/2010/08/020 Lusso E., Risaliti G., Nardini, E., et al. Quasars as standard can- Wei, J.-J., Wu, X.-F., 2017, IJMPD, 26, 1730002 dles III. Validation of a new sample for cosmological studies, Weinberg, S., 1989, Rev. Mod. Phys., 61, 1 arXiv:2008.08586 WMAP Collaboration, 2013, ApJS, 208, 19H Ma, C., Caldwell, R.R., Bode, P., Wang, L., 1999, ApJ, 521, L1 Zhao W., Zhang J.-C., Zhang Q.-X., Liang J.-T., Luan X.-H., Zhou Maeda, K., 1989, Phys. Rev. D39, 3159 Q.-Q., Yi S.-X., et al., 2020, arXiv, arXiv:2009.03550 Martone, R., Izzo, L., Della Valle, M., Amati, L., Longo, G., Gotz, Zhao W., Zhang J.-C., Zhang Q.-X., Liang J.-T., Luan X.-H., Zhou D., 2017, Astronomy & Astrophysics, 608, A52 Q.-Q., Yi S.-X., et al., 2020, ApJ, 900, 112 Montiel A., Cabrera J. I., Hidalgo J. C., 2021, MNRAS, 501, 3515. doi:10.1093/mnras/staa3926 Muccino M., Izzo L., Luongo O., Boshkayev K., Amati L., Della Valle M., Pisani G. B., et al., 2021, ApJ, 908, 181. doi:10.3847/1538-4357/abd254 Nakar, E., Piran T., 2005, MNRAS, 360, L73 Nava, L., Ghirlanda, G., Ghisellini, V., Celotti, A., 2011, A&A 530 21 Nesseris, S., and Perivolaropoulos, L., 2005, Phys. Rev. D, 72, 123519 Niedermann, F., Sloth, M.S., 2020, Phys. Rev. D, 102, 063527. Peebles, P. J. E., 1984, ApJ, 284, 439 Peebles, P. J. E., Ratra, B., ApJ, 1988a, 325,17 Perlmutter, S., Aldering, G., Della Valle, M., Deustua, S., Ellis, R. S., et al., 1998, Nature, 391,51 Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R. A., Nugent, P., et al., 1999, ApJ, 517, 565 Pettorino, V., Amendola, L., Wetterich, C., 2013, Phys. Rev. D, 87, 083009 Piedipalumbo, E., Scudellaro, P., Esposito, G., Rubano, C., 2012, and Gravitation, 44, 2611 Planck Collaboration, 2016, Astronomy & Astrophysics, 594, A13 Ratra, B., Peebles, P. J. E., 1988b, Phys. Rev. D, 37, 3406 Riess, A.G., Filippenko, A.V., Challis, P., Clocchiatti, A., Diercks, A., et al., 1998, AJ, 116,1009 Riess, A.G., Strolger, L.G., Casertano, S., Ferguson, H.C., Mobasher, B. et al., 2007, ApJ, 659, 98 Riess A. G., Macri L., Casertano S., Sosey M., Lampeitl H., Fer- guson H. C., Filippenko A. V., et al., 2009, ApJ, 699, 539. doi:10.1088/0004-637X/699/1/539 Risaliti, G., Lusso, E., Cosmological constraints from the Hubble diagram of quasars at high redshifts, 2019, Nature Astronomy, Advanced Online Publication, 10.1038/s41550-018-0657-z Rubano, C., Scudellaro, P., 2002, Gen.Rel.Grav., 34, 307 Sahni, V., Saini, T.D., Starobinsky, A.A., Alam, U., 2003, JETP Lett., 77, 201 Sakamoto T., Barthelmy S. D., Barbier L., Cummings J. R., Fen- imore E. E., Gehrels N., Hullinger D., et al., 2008, ApJS, 175, 179. doi:10.1086/523646 Sawant, D., & Amati, L., 2018, Perspective GRB cosmology with time resolved Ep,i(t)-Liso(t) correlation, The Fourteenth Mar- cel Grossmann Meeting On Recent Developments in Theoret- ical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories, held 12-18 July 2015 in Rome, Italy. Edited by Massimo Bianchi, Robert T Jansen and Remo Ruffini. Published by World Scientific Publishing Co. Pte. Ltd., 2018, pp. 2913-2917 DOI: 10.1142/97898132266090365 Shirokov S. I., Sokolov I. V., Lovyagin N. Y., Amati L., Baryshev Y. V., Sokolov V. V., Gorokhov V. L., 2020, MNRAS, 496, 1530. doi:10.1093/mnras/staa1548 Si, S.-K., Qi, Y.-Q., Xue, F.-X., Liu, Y.-J. et al., 2018, ApJ, 863, 50 Suzuki et al. (The Supernova Cosmology Project), 2012, ApJ, 746, 85 Tsujikawa, S., 2013, Class. Quan. Grav., 30, 241003 Ulanov, M. V., Golenetskii, S. V., Frederiks, D. D., et al. 2005, NCCGSPC, 28, 351 Wang F. Y., Dai Z. G., Liang E. W., 2015, NewAR, 67, 1. doi:10.1016/j.newar.2015.03.001

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