The Djorgovski-Gurzadyan dark energy integral equation and the Hubble diagram
aa r X i v : . [ a s t r o - ph . C O ] O c t Astronomy & Astrophysicsmanuscript no. 39246 © ESO 2020October 12, 2020 L etter to the E ditor The Djorgovski-Gurzadyan dark energy integral equation and theHubble diagram
H.G. Khachatryan , , A. Stepanian Center for Cosmology and Astrophysics, Alikhanian National Laboratory, Yerevan, Armenia Yerevan State University, Yerevan, ArmeniaReceived: 24 August 2020; Accepted: 15 September 2020
ABSTRACT
We consider the observational aspects of the value of dark energy density from quantum vacuum fluctuations based initially on theGurzadyan-Xue model. We reduce the Djorgovski-Gurzadyan integral equation to a di ff erential equation for the co-moving horizonand then, by means of the obtained explicit form for the luminosity distance, we construct the Hubble diagram for two classes ofobservational samples. For supernova and gamma-ray burst data we show that this approach provides viable predictions for distancesup to z ≃
9, quantitatively at least as good as those provided by the lambda cold dark matter ( Λ CDM) model. The Hubble parameterdependence H ( z ) of the two models also reveals mutual crossing at z = . Key words. cosmological parameters - cosmology: observations - dark energy - cosmology: theory
1. Introduction
In his pioneering work Zeldovich (1967) assigned the cosmolog-ical constant Λ , if non-zero, as a contribution of quantum vac-uum fluctuations with the ǫ = − p equation of state (EOS; energy ǫ and pressure p ). After the discovery of the accelerated expan-sion of the Universe a number of models and approaches wereproposed (see Copeland et al. 2006), typically involving eithertheoretical or phenomenological input parameters.Among the proposed approaches, Gurzadyan-Xue (GX)models do not have any free or empirical parameters, andthey predict a value of Λ fitting its observed value. This classof models (see Khachatryan et al. 2007; Khachatryan 2007;Vereshchagin 2006,a; Vereshchagin & Yegorian 2006b, 2008)also covers those parameters with a variation of fundamentalconstants, and has been tested regarding the Hubble diagram us-ing the supernova (SN) and gamma-ray burst (GRB) observa-tional data Mosquera Cuesta et al. (2008).Djorgovski and Gurzadyan (DG) (Djorgovski & Gurzadyan(2007)) did an in-depth refinement while comparing these mod-els to the observational data; in other words, they reconsideredthe concept of one of the key parameters in the GX formula,the maximum distance L max assigned to the quantum fluctua-tions. By setting this distance to co-moving horizon, they gota parameter-free integral equation, which they solved iteratively.In this paper we switch the DG equation to a di ff erential one,avoiding the arising of an integration constant, and obtain a valueof dark energy density again without any other assumptions ex-cept on the empirical values of the matter density and the cur-vature of the Universe; we also derive the Hubble diagram foravailable SN and GRB data samples.The paper is organized as follows. In the first part we derivethe main formulae and get a numerical solution for parameter y ( z ). Then we construct Hubble diagrams for SN and GRB sam- Send o ff print requests to : H.G. Khachatryan, e-mail: [email protected] ples and compare Hubble parameter graphs for the lambda colddark matter ( Λ CDM) model and the GX-DG model. While theobservational data seem to support the EOS parameter w ( z ) ≈ − w ( z ) and see that it slightly varies withredshift. We also consider some further cosmological insights onthe considered dark energy scalings.
2. Dark energy versus luminosity distance
In their original work Gurzadyan & Xue (2002, 2003) proposedconsidering, in the dark energy only, the contribution of l = quanti-tative discrepancies of the theoretical density with the empiricaldark energy value. Thus, they derived a simple equation for darkenergy density, Ω Λ = π D H L max ! , (1)where D H = c / H is the Hubble horizon scale, i.e., the distancethat light would travel during Hubble time 1 / H .The key issue with Eq.(1) is what to choose for the maxi-mum distance L max associated with the fluctuations. In some pa-pers (Vereshchagin (2006,a); Vereshchagin & Yegorian (2006b,2008); Khachatryan et al. (2007); Khachatryan (2007)) the scalefactor is used as a maximum distance and that assumption gave Ω Λ ≈ .
29 for dark energy density, while the observed value is ≈ . L max , namely using the Article number, page 1 of 4 & Aproofs: manuscript no. 39246 distance to the horizon measured in co-moving units D ∞ ( z ) = D H Z ∞ z h (´ z ) d ´ z (2) h ( z ) = X i Ω (0) i (1 + z ) + ω i ) , where D ∞ and h ( z ) = H ( z ) / H are the distance to the horizonand a dimensionless Hubble parameter, respectively. As a resultthey obtained only a 20% di ff erence between the theoretical andobserved values of the dark energy density. This integral equa-tion can be rewritten in the form of di ff erential equation for y ( z )parameter d y dz = − q Ω m (1 + z ) + Ω r (1 + z ) + π y , (3) y ′ (0) = − ,y ( z ) = D ∞ ( z ) D H . This equation assumes that in the early Universe z → ∞ theHubble parameter h ( z ) → ∞ .The initial condition for y ′ follows from the Friedmann equa-tions and can be rewritten as P Ω (0) i =
1. Here we consider onlyflat k = Ω k = k .Eq.(4) cannot be solved analytically and the numerical solu-tion is depicted in Fig.(1).To construct the Hubble diagram the luminosity distance andthe distance moduli are needed. They can be expressed by y ( z )in a simple form d L ( z ) = D H (1 + z )( y (0) − y ( z )) , (4) d M ( z ) = ( d L ( z )10 pc ) Fig. 1.
Numerical solution of Eq.(4) for redshift z up to 10. Model pa-rameters are set to Ω m = . , Ω r = . · − .
3. Hubble diagram
In this section we construct the Hubble diagram for the Union2.1sample of SN Ia (Amanullah et al. (2010); Suzuki et al. (2012)).We used a standard parameter set for the numerical solution of y ( z ) (see Fig.(1)). It shows that the standard model with a cos-mological constant predicts values that are a bit higher for dis-tance moduli of SNs (Fig.(2)). The χ values for Λ CDM modelare somewhat smaller than for the DG-GX model, χ = . χ = .
29, respectively.
Fig. 2.
Hubble diagram for 580 SNs of the Union2.1 SN Ia sam-ple (Amanullah et al. (2010); Suzuki et al. (2012)). The red solid linerepresents the GX-DG cosmological model and the green solid linethe Λ CDM model. The Hubble constant value is taken as H = km s − M pc − . We also fit the models with GRB data from Demianski et al.(2017), which were calibrated independently using Amati rela-tion and can be used with any dark energy model. The Amatirelation (Amati (2006)) refers to an empirical relation betweenisotropic equivalent radiated energy E iso and the observed pho-ton energy of the peak spectral flux E p , i :log E iso er g ! = b + a log " E p , i KeV . (5)The Amati relation is known to be not applicable for anyGRBs with measured parameter pairs E iso , E p , i (see Qin & Chen(2013); Lin et al. (2015)); moreover, the regression parameters a , b slightly di ff er when obtained by di ff erent authors. Althoughthe Amati relation was nevertheless concluded to be the prefer-able to other empirical relations for GRBs while constructing theHubble diagram Wang et al. (2020), more detailed work is in-deed needed in the future in the comparative analysis of variousempirical relations and their possibly combined applications.For GRBs we see that the GX-DG model fits better: for the Λ CDM model we have χ = .
41 and for the GX-DG model χ = .
21. This can be seen even in Fig.(3), where the GRBdistance moduli values are smaller and therefore the GX-DGmodel fits better. To trace the statistical properties of the GRBdata we fitted and then calculated χ , for example for the GRBdata from Liu & Wei (2015) as well; we obtained lower valuesfor χ (0 . , . Λ CDM models, re-spectively.Then we plot in Fig.(4) the Hubble parameter dependence onthe redshift for di ff erent values H = . , . km s − M pc − for the Λ CDM and the GX-DG model; we chose the Hubbleparameter because the
Planck satellite data on the Cosmic Mi-crowave Background (CMB)
Planck Collaboration (2018) pro-vide the Hubble constant value H = . ± . km s − M pc − ,while the Milky Way Cepheid survey (Riess et al. (2018, 2019))suggests H = . ± . km s − M pc − . We note that thegraph with the GX-DG model for the Hubble constant value H = . km s − M pc − crosses at around z = . Λ CDM model with Hubble constant value H = . km s − M pc − ; i.e., at redshifts lower than z = . Article number, page 2 of 4.G. Khachatryan, A. Stepanian: The Djorgovski-Gurzadyan dark energy integral equation
Fig. 3.
Hubble diagram for 580 SNs of the Union2.1 SN Ia sample and162 calibrated GRBs from Demianski et al. (2017). The red line repre-sents the GX-DG cosmological model and the green line the Λ CDM model. the Hubble parameter for GX-DG model approximate Λ CDMmodel’s graph with H = . km s − M pc − and at higher red-shifts tending to that of H = . km s − M pc − . Fig. 4.
Hubble parameter H ( z ) dependance on the redshift z . The redlines represent GX-DG cosmological model and the green lines referto the Λ CDM model (dashed H = . km s − M pc − , dotted H = . km s − M pc − . The interpretation of this result, and whether it is related tothe Hubble parameter issues in Verde et al. (2019), is not clear.We can predict, to be confirmed or ruled out at future observa-tions, that at z = . w , or some other related parameter undergoes a changein properties.
4. Equation of state parameter
In the standard Λ CDM model dark energy is adopted as the cos-mological constant. Therefore, it is important to have an equa-tion for the EOS parameter w ( z ) dependence on redshift for theGX-DG model; for Λ CDM w ( z ) ≈ −
1. From Eq.(4) we can eas-ily derive a formula for the EOS parameter w ( z ) = + z )3 h ( z ) y ( z ) − . (6)In Fig.(5) we can see the dependance of the EOS parameteron the redshift. It is negative in a wide range of redshifts, but ataround z ≈ Fig. 5. E ff ective EOS parameter w ( z ) as a function of redshift z .
5. Information evolution
In this section we link the GX-DG model to the “in-formation” description of the cosmological evolution(Gurzadyan & Stepanian (2019)). Thus, the vacuum matterdensity is ρ ∝ ( L pl ) − ( L max ) − , (7)where L pl is the Planck length and L max is the upper bound of thevacuum modes, which is considered the cosmological horizon inco-moving coordinates (Djorgovski & Gurzadyan 2007).Within the framework of the information evolution (IE), itis assumed that the evolution of the Universe can be describedaccording to the increase of a discretized quantity, the informa-tion. This concept is obtained based on the notion of BekensteinBound (Bekenstein (1981)), which introduces an upper bound ofinformation I BB ≤ π RE ~ c ln 2 , (8)where R and E are the characteristic radius and energy of a sys-tem under the consideration. Thus, the evolution of the Universe( T ) is discretized as follows: T = { , , ... j , ..., π I } , I = c Λ G ~ . (9)Accordingly, at each stage ( j ) of IE, the Universe is denoted bythe characteristic mass M j and radius R j I BB ( M j , R j ) = I BB ( π c j ~ G H j , cH j ) j = , ..., π c ~ G Λ , (10)where cH j is the Hubble horizon at that stage. On the one hand, I BB is inversely proportional to the total density of the Universeand Eq.(9) can be written as T = ( ρ = c ~ G , ...ρ j = j ρ ..., ρ π I = π I ρ = Λ c π G ) . (11)On the other hand, the evolution of the Universe can be describedbased on the area of the cosmological horizon at each stage A j ,i.e., T = ( A = l p , ... A j = jA , ..., A π I = (3 π I ) A = π Λ ) , (12) Article number, page 3 of 4 & Aproofs: manuscript no. 39246 so that the information content of Universe increases until thearea of the Hubble horizon reaches π Λ , the de Sitter cosmologi-cal horizon. Consequently, at this stage the density becomes Λ c π G .Thus, in the context of IEInformation ∝ Area ∝ (density) − . (13)Then, the “maximum number of relevant radial modes” in theGX model is defined as ρ ∝ N max ( N max + , N max = L max L pl . (14)Comparing these relations with the notion of area in IE leads usto the following degrees of freedom I j = A j A pl = π ( cH j ) G ~ c . (15)Thus, by comparing Eq.(7) with the notions of “area of cos-mological horizon” and “matter density” we can conclude thatthe results of GX-DG and IE models, although assuming entirelydi ff erent approaches, are similar. It should be noted that this sim-ilarity exists only if ρ in Eq.(7) corresponds to the total densityof the Universe (according to Eq.(11)).Obviously, the two approaches deal with certain globalscaling parameters and they should be reduced to standardFriedmann-Lemaitre-Robertson-Walker universe with the con-tinuous geometrical structure.
6. Conclusions
We reduced the Djorgovski & Gurzadyan (2007) integral equa-tion to a di ff erential equation to describe dark energy with ob-servable parameters. Solving that di ff erential equation we ob-tained the exact value of dark energy density. This allowed us toconstruct the Hubble diagram for SN and GRB samples and toshow that the model estimates the observed distances up to z ≃ Λ CDM model with a cosmolog-ical constant. The dark energy EOS parameter predicts a slightvariation with redshift which is still not observable, however.The behavior of the GX-DG model and comparison withobservational data shows that if there are no unknown ex-perimental systematics (Niedermann & Sloth (2008); Efstathiou(2014, 2020)), the GX-DG model can be adopted along withthe standard Λ CDM model. The revealed feature of the dia-grams, namely, the crossing of the curves of both models at z = . ff erent behaviors of parameters at redshifts z < . z > . ff erential equation for dark energy densityEq.(4) has one integration invariant that can be related to the in-variants of the general equations of GX models (Khachatryan(2007)).
7. Acknowledgments
We are thankful to the referee for very useful comments and sug-gestions that helped to improve the paper.
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