Quasar X-ray and UV flux, baryon acoustic oscillation, and Hubble parameter measurement constraints on cosmological model parameters
MMNRAS , 1–8 (2019) Preprint 21 April 2020 Compiled using MNRAS L A TEX style file v3.0
Quasar X-ray and UV flux, baryon acoustic oscillation, andHubble parameter measurement constraints oncosmological model parameters
Narayan Khadka, (cid:63) and Bharat Ratra, † Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66502, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We use the 2015 Risaliti and Lusso compilation of 808 X-ray and UV flux measure-ments of quasars (QSOs) in the redshift range . ≤ z ≤ . , alone and in conjunc-tion with baryon acoustic oscillation (BAO) and Hubble parameter [ H ( z ) ] measure-ments, to constrain cosmological parameters in six cosmological models. The QSOdata constraints are significantly weaker than, but consistent with, those from the H ( z ) + BAO data. A joint analysis of the QSO + H ( z ) + BAO data is consistent withthe current standard model, spatially-flat Λ CDM, but mildly favors closed spatialhypersurfaces and dynamical dark energy.
Key words: (cosmology:) cosmological parameters – (cosmology:) observations – (cosmology:) dark energy
Type Ia supernova (SNIa) apparent magnitude measure-ments provided the first convincing evidence for acceleratedcosmological expansion (see Scolnic et al. (2018) for a re-cent discussion). Supporting evidence soon came from othercosmological probes, the most significant being cosmic mi-crowave background (CMB) anisotropy data (Plank Collab-oration 2018), baryon acoustic oscillation (BAO) distancemeasurements (Alam et al. 2017), and Hubble parameter[ H ( z ) ] observations (Moresco et al. 2016; Farooq et al. 2017).If general relativity is an accurate model of gravitation, hy-pothetical dark energy is responsible for the observed accel-eration of the cosmological expansion. There are many dif-ferent dark energy models. In this paper we consider threeof them and also consider flat and non-flat spatial hypersur-faces in each case, for a total of six cosmological models.The simplest observationally-consistent dark energymodel is the flat Λ CDM model, the current standard model(Peebles 1984). In this model the accelerated expansion ispowered by the spatially homogenous cosmological constant( Λ ) energy density which is constant in time. This model isconsistent with most observations when about of thecurrent cosmological energy budget is contributed by darkenergy, with about 25 % coming from the cold dark matter(CDM), and the remaining 5 % due to baryons. The standard (cid:63) E-mail: [email protected] † E-mail: [email protected] model assumes flat spatial hypersurfaces. Current observa-tions allow a little spatial curvature, so we can generalisethe standard model to the non-flat Λ CDM model which al-lows for non-zero spatial curvature energy density.While the Λ CDM model is consistent with many obser-vations, its assumption of a time-independent and spatially-homogeneous dark energy density is difficult to theoreticallymotivate. Also, observations do not require that the dark en-ergy density be time independent, and models in which thedark energy density decreases with time have been studied.Here we consider two dynamical dark energy models, theXCDM parametrization in which an X -fluid is the dynami-cal dark energy and the φ CDM model in which a scalar field φ is the dynamical dark energy. We also study spatially flatand non-flat versions of both the XCDM parametrizationand the φ CDM model.The main goal of our paper is to use the Risaliti & Lusso(2015) quasar (QSO) X-ray and UV flux measurements toconstrain cosmological parameters. Risaliti & Lusso (2015) For discussion of observational constraints on spatial curvature,see Farooq et al. (2015), Chen et al. (2016), Yu & Wang (2016),Wei & Wu (2017), Rana et al. (2017), Ooba et al. (2018a,b,c),DES Collaboration (2018a), Witzemann et al. (2018), Yu et al.(2018), Park & Ratra (2018a,b,c,d, 2019), Mitra et al. (2018),Penton et al. (2018), Xu et al. (2019), Zheng et al. (2019), Ruanet al. (2019), Giamb´o et al. (2019), Coley (2019), Eingorn et al.(2019), Jesus et al. (2019), Handley et al. (2019), and referencestherein. © a r X i v : . [ a s t r o - ph . C O ] A p r N. Khadka, B. Ratra consider cosmological parameter constraints in the non-flat Λ CDM model; here we also consider cosmological param-eter constraints in five other cosmological models. In ad-dition, we examine the effect of different Hubble constantpriors on the cosmological parameter constraints. By study-ing constraints in a number of models, we are able to drawsomewhat model-independent conclusions about the QSOdata constraints. We find that the Risaliti & Lusso (2015)QSO data by themselves do not provide very restrictive con-straints on cosmological parameters. However, the QSO con-straints are largely consistent with those that follow from the H ( z ) + BAO data, and when jointly analyzed the QSO dataslightly tighten and shift the H ( z ) + BAO data constraints.The QSO + H ( z ) + BAO data are consistent withthe standard flat Λ CDM cosmological model although theymildly favor closed spatial hypersurfaces over flat ones anddynamical dark energy over a cosmological constant.While current QSO data by themselves do not pro-vide restrictive cosmological parameter constraints, the newRisaliti & Lusso (2019) compilation of 1598 QSO measure-ments will provide tighter constraints, that should be im-proved upon by near-future QSO data. Currently, CMBanisotropy, BAO, SNIa, and H ( z ) data provide the mostrestrictive constraints on cosmological parameters. To testconsistency, and to help tighten cosmological parameter con-straints, it is essential that additional cosmological probes,such as the QSO data studied here, be developed.This paper is organized as follows. In Sec. 2 we describethe models that we use. In Sec. 3 we discuss the data that weuse to constrain cosmological parameters in these models.In Sec. 4 we describe the methodology adopted for theseanalyses. In Sec. 5 we present our results and conclude inSec. 6. In this paper we constrain cosmological parameters of thespatially-flat and non-flat versions of three different darkenergy cosmological models, for a total of six cosmologicalmodels. For the dark energy we consider a cosmological con-stant Λ in the Λ cold dark matter ( Λ CDM) model, as wellas a decreasing dark energy density modeled as an X -fluidin the XCDM parametrization, or as a scalar field φ in the φ CDM model.In the Λ CDM model the Hubble parameter, as a func-tion of redshift z , is H ( z ) = H (cid:113) Ω m ( + z ) + Ω k ( + z ) + Ω Λ , (1)and Ω m + Ω k + Ω Λ = . (2)Here H is the Hubble constant, Ω m and Ω k are the currentvalues of the non-relativistic matter and the spatial curva-ture energy density parameters and Ω Λ is the dark energydensity parameter. For the spatially-flat Λ CDM model thefree parameters are chosen to be Ω m and H . For the spa-tially non-flat Λ CDM model the free parameters are chosento be Ω m , Ω Λ , and H .In the XCDM parametrization, the dark energy densityis dynamical and decreases with time. The equation of statefor the dark energy fluid is P X = ω X ρ X . Here P X is the pressure of the X -fluid, ρ X is the energy density of that fluid,and ω X is the equation of state parameter whose value isnegative ( ω X < − / ). In this model the Hubble parameteris H ( z ) = H (cid:113) Ω m ( + z ) + Ω k ( + z ) + Ω X ( + z ) ( + ω X ) , (3)and Ω m + Ω k + Ω X = . (4)Here Ω X is the current value of the X -fluid energy densityparameter. For the spatially-flat case the free parameters are Ω m , ω X , and H . For the non-flat case the free parametersare Ω m , Ω k , ω X , and H . In the ω X = − limit the XCDMmodel is the Λ CDM model.In the φ CDM model dark energy is modeled as a scalarfield φ with potential energy density V ( φ ) (Peebles & Ratra1988; Ratra & Peebles 1988; Pavlov et al. 2013). In thismodel the dark energy density is dynamical and decreaseswith time. A widely used V ( φ ) is of the inverse power lawform, V ( φ ) = κ m p φ − α , (5)where m p is the Planck mass, α is a positive parameter, and κ = (cid:18) α + α + (cid:19) (cid:20) α ( α + ) (cid:21) α / . (6)In this model the equations of motion are (cid:220) φ + (cid:219) aa (cid:219) φ − ακ m p φ − α − = , (7)and, (cid:18) (cid:219) aa (cid:19) = π G (cid:0) ρ m + ρ φ (cid:1) − ka , (8)where a is the scale factor, overdots denote derivativeswith respect to time, k is positive, zero, and negative forclosed, flat, and open spatial hypersurfaces, ρ m is the non-relativistic matter density, and the scalar field energy densityis ρ φ = m p π [ (cid:219) φ + κ m p φ − α ] . (9)So, the Hubble parameter in this model is H ( z ) = H (cid:113) Ω m ( + z ) + Ω k ( + z ) + Ω φ ( z , α ) , (10)where the scalar field energy density parameter Ω φ ( z , α ) = π G ρ φ H , (11) Discussion of constraints on the φ CDM model may be tracedthrough Chen & Ratra (2004), Samushia et al. (2007), Yasharet al. (2009), Samushia & Ratra (2010), Samushia et al. (2010),Chen & Ratra (2011b), Campanelli et al. (2012), Farooq & Ratra(2013), Farooq et al. (2013), Avsajanishvili et al. (2015), S`ola etal. (2017), S`ola Peracaula et al. (2018, 2019), Zhai et al. (2017),Sangwan et al. (2018), Singh et al. (2019), Mitra (2019), andreferences therein. MNRAS000
Type Ia supernova (SNIa) apparent magnitude measure-ments provided the first convincing evidence for acceleratedcosmological expansion (see Scolnic et al. (2018) for a re-cent discussion). Supporting evidence soon came from othercosmological probes, the most significant being cosmic mi-crowave background (CMB) anisotropy data (Plank Collab-oration 2018), baryon acoustic oscillation (BAO) distancemeasurements (Alam et al. 2017), and Hubble parameter[ H ( z ) ] observations (Moresco et al. 2016; Farooq et al. 2017).If general relativity is an accurate model of gravitation, hy-pothetical dark energy is responsible for the observed accel-eration of the cosmological expansion. There are many dif-ferent dark energy models. In this paper we consider threeof them and also consider flat and non-flat spatial hypersur-faces in each case, for a total of six cosmological models.The simplest observationally-consistent dark energymodel is the flat Λ CDM model, the current standard model(Peebles 1984). In this model the accelerated expansion ispowered by the spatially homogenous cosmological constant( Λ ) energy density which is constant in time. This model isconsistent with most observations when about of thecurrent cosmological energy budget is contributed by darkenergy, with about 25 % coming from the cold dark matter(CDM), and the remaining 5 % due to baryons. The standard (cid:63) E-mail: [email protected] † E-mail: [email protected] model assumes flat spatial hypersurfaces. Current observa-tions allow a little spatial curvature, so we can generalisethe standard model to the non-flat Λ CDM model which al-lows for non-zero spatial curvature energy density.While the Λ CDM model is consistent with many obser-vations, its assumption of a time-independent and spatially-homogeneous dark energy density is difficult to theoreticallymotivate. Also, observations do not require that the dark en-ergy density be time independent, and models in which thedark energy density decreases with time have been studied.Here we consider two dynamical dark energy models, theXCDM parametrization in which an X -fluid is the dynami-cal dark energy and the φ CDM model in which a scalar field φ is the dynamical dark energy. We also study spatially flatand non-flat versions of both the XCDM parametrizationand the φ CDM model.The main goal of our paper is to use the Risaliti & Lusso(2015) quasar (QSO) X-ray and UV flux measurements toconstrain cosmological parameters. Risaliti & Lusso (2015) For discussion of observational constraints on spatial curvature,see Farooq et al. (2015), Chen et al. (2016), Yu & Wang (2016),Wei & Wu (2017), Rana et al. (2017), Ooba et al. (2018a,b,c),DES Collaboration (2018a), Witzemann et al. (2018), Yu et al.(2018), Park & Ratra (2018a,b,c,d, 2019), Mitra et al. (2018),Penton et al. (2018), Xu et al. (2019), Zheng et al. (2019), Ruanet al. (2019), Giamb´o et al. (2019), Coley (2019), Eingorn et al.(2019), Jesus et al. (2019), Handley et al. (2019), and referencestherein. © a r X i v : . [ a s t r o - ph . C O ] A p r N. Khadka, B. Ratra consider cosmological parameter constraints in the non-flat Λ CDM model; here we also consider cosmological param-eter constraints in five other cosmological models. In ad-dition, we examine the effect of different Hubble constantpriors on the cosmological parameter constraints. By study-ing constraints in a number of models, we are able to drawsomewhat model-independent conclusions about the QSOdata constraints. We find that the Risaliti & Lusso (2015)QSO data by themselves do not provide very restrictive con-straints on cosmological parameters. However, the QSO con-straints are largely consistent with those that follow from the H ( z ) + BAO data, and when jointly analyzed the QSO dataslightly tighten and shift the H ( z ) + BAO data constraints.The QSO + H ( z ) + BAO data are consistent withthe standard flat Λ CDM cosmological model although theymildly favor closed spatial hypersurfaces over flat ones anddynamical dark energy over a cosmological constant.While current QSO data by themselves do not pro-vide restrictive cosmological parameter constraints, the newRisaliti & Lusso (2019) compilation of 1598 QSO measure-ments will provide tighter constraints, that should be im-proved upon by near-future QSO data. Currently, CMBanisotropy, BAO, SNIa, and H ( z ) data provide the mostrestrictive constraints on cosmological parameters. To testconsistency, and to help tighten cosmological parameter con-straints, it is essential that additional cosmological probes,such as the QSO data studied here, be developed.This paper is organized as follows. In Sec. 2 we describethe models that we use. In Sec. 3 we discuss the data that weuse to constrain cosmological parameters in these models.In Sec. 4 we describe the methodology adopted for theseanalyses. In Sec. 5 we present our results and conclude inSec. 6. In this paper we constrain cosmological parameters of thespatially-flat and non-flat versions of three different darkenergy cosmological models, for a total of six cosmologicalmodels. For the dark energy we consider a cosmological con-stant Λ in the Λ cold dark matter ( Λ CDM) model, as wellas a decreasing dark energy density modeled as an X -fluidin the XCDM parametrization, or as a scalar field φ in the φ CDM model.In the Λ CDM model the Hubble parameter, as a func-tion of redshift z , is H ( z ) = H (cid:113) Ω m ( + z ) + Ω k ( + z ) + Ω Λ , (1)and Ω m + Ω k + Ω Λ = . (2)Here H is the Hubble constant, Ω m and Ω k are the currentvalues of the non-relativistic matter and the spatial curva-ture energy density parameters and Ω Λ is the dark energydensity parameter. For the spatially-flat Λ CDM model thefree parameters are chosen to be Ω m and H . For the spa-tially non-flat Λ CDM model the free parameters are chosento be Ω m , Ω Λ , and H .In the XCDM parametrization, the dark energy densityis dynamical and decreases with time. The equation of statefor the dark energy fluid is P X = ω X ρ X . Here P X is the pressure of the X -fluid, ρ X is the energy density of that fluid,and ω X is the equation of state parameter whose value isnegative ( ω X < − / ). In this model the Hubble parameteris H ( z ) = H (cid:113) Ω m ( + z ) + Ω k ( + z ) + Ω X ( + z ) ( + ω X ) , (3)and Ω m + Ω k + Ω X = . (4)Here Ω X is the current value of the X -fluid energy densityparameter. For the spatially-flat case the free parameters are Ω m , ω X , and H . For the non-flat case the free parametersare Ω m , Ω k , ω X , and H . In the ω X = − limit the XCDMmodel is the Λ CDM model.In the φ CDM model dark energy is modeled as a scalarfield φ with potential energy density V ( φ ) (Peebles & Ratra1988; Ratra & Peebles 1988; Pavlov et al. 2013). In thismodel the dark energy density is dynamical and decreaseswith time. A widely used V ( φ ) is of the inverse power lawform, V ( φ ) = κ m p φ − α , (5)where m p is the Planck mass, α is a positive parameter, and κ = (cid:18) α + α + (cid:19) (cid:20) α ( α + ) (cid:21) α / . (6)In this model the equations of motion are (cid:220) φ + (cid:219) aa (cid:219) φ − ακ m p φ − α − = , (7)and, (cid:18) (cid:219) aa (cid:19) = π G (cid:0) ρ m + ρ φ (cid:1) − ka , (8)where a is the scale factor, overdots denote derivativeswith respect to time, k is positive, zero, and negative forclosed, flat, and open spatial hypersurfaces, ρ m is the non-relativistic matter density, and the scalar field energy densityis ρ φ = m p π [ (cid:219) φ + κ m p φ − α ] . (9)So, the Hubble parameter in this model is H ( z ) = H (cid:113) Ω m ( + z ) + Ω k ( + z ) + Ω φ ( z , α ) , (10)where the scalar field energy density parameter Ω φ ( z , α ) = π G ρ φ H , (11) Discussion of constraints on the φ CDM model may be tracedthrough Chen & Ratra (2004), Samushia et al. (2007), Yasharet al. (2009), Samushia & Ratra (2010), Samushia et al. (2010),Chen & Ratra (2011b), Campanelli et al. (2012), Farooq & Ratra(2013), Farooq et al. (2013), Avsajanishvili et al. (2015), S`ola etal. (2017), S`ola Peracaula et al. (2018, 2019), Zhai et al. (2017),Sangwan et al. (2018), Singh et al. (2019), Mitra (2019), andreferences therein. MNRAS000 , 1–8 (2019) onstraints from QSO, H ( z ) , and BAO data where G is the gravitational constant, and Ω φ ( z , α ) has tobe numerically computed. For the spatially non-flat φ CDMmodel the free parameters are Ω m , Ω k , α , and H . For thespatially-flat φ CDM model the free parameters are Ω m , α ,and H . In the limit α → , the φ CDM model reduces to the Λ CDM model.
We use three different data sets to constrain cosmologicalparameters. The main purpose of our paper is to use the808 QSO X-ray and UV flux measurements of Risaliti & Lusso (2015), extending over a redshift range of . ≤ z ≤ . , to determine cosmological parameter constraints, andto compare these QSO cosmological parameter constraintsto those determined from more widely used BAO distancemeasurements and H ( z ) observations. The BAO and H ( z ) data we use are listed in Tables 1 and 2 of Ryan et al. (2018)and consist of 11 BAO measurements over the redshift range . ≤ z ≤ . and 31 H ( z ) measurements over the redshiftrange . ≤ z ≤ . . As described in Risaliti & Lusso (2015), the method of anal-ysis depends on the non-linear relation between the X-rayand UV luminosities of quasars. This relation is log ( L X ) = β + γ log ( L UV ) , (12)where log = log and L X and L UV are the QSO X-ray andUV luminosities. β and γ are free parameters to be deter-mined by using the data.Expressing the luminosity in terms of the flux, we obtain log ( F X ) = β + ( γ − ) log ( π ) + γ log ( F UV ) + ( γ − ) log ( D L ) , (13)where F X and F UV are the X-ray and UV fluxes respectively.Here D L is the luminosity distance, which is a function ofredshift and cosmological parameters, which will allow us toconstrain the cosmological model parameters. The luminos-ity distance D L ( z , p ) is given by H (cid:112) | Ω k | D L ( z , p )( + z ) = sinh [ g ( z )] if Ω k > , g ( z ) if Ω k = , sin [ g ( z )] if Ω k < , (14)where p is the set of cosmological model parameters, g ( z ) = H (cid:112) | Ω k | ∫ z dz (cid:48) H ( z (cid:48) ) , (15)and H ( z ) , which is a function of cosmological model param-eters, is given in Sec. 2 for the six cosmological models westudy in this paper.We determine the best-fit values and uncertainty of the Also see Risaliti & Lusso (2016), Lusso & Risaliti (2017), andBisogni et al. (2017). For a newer compilation of QSO data see ? .For cosmological parameter constraints derived from QSO data,also see L´opez-Corredoria et al. (2016), Lusso et al. (2019), Melia(2019), and Lazkoz et al. (2019). parameters for a given model by maximizing the likelihoodfunction. For QSO data, we have the observed X-ray fluxand we can predict the X-ray flux at given redshift as afunction of cosmological parameters by using eqs. (13) and(14). So, the likelihood function ( LF ) for QSO data is ln ( LF ) = − (cid:213) i = (cid:34) [ log ( F obs X , i ) − log ( F th X , i )] s i + ln ( π s i ) (cid:35) , (16)where ln = log e and s i = σ i + δ , where σ i and δ are the mea-surement error on F obs X , i and the global intrinsic dispersion re-spectively. We treat δ as a free parameter to be determinedby the data, along with the other two free parameters, β and γ , which characterise the L X - L UV relation in eq. (12). Ineq. (16) F th X , i is the corresponding model prediction definedthrough eq. (13), and is a function of F UV and D L ( z i , p ) .Our determination of the BAO and H ( z ) data con-straints follows Ryan et al. (2019). The likelihood functionfor the uncorrelated BAO and H ( z ) data is ln ( LF ) = − N (cid:213) i = [ A obs ( z i ) − A th ( z i , p )] σ i , (17)where A obs ( z i ) and σ i are the measured quantity and errorbar at redshift z i and A th ( z i , p ) is the corresponding model-predicted value. The measurements in the first six lines ofTable 1 of Ryan et al. (2019) are correlated and the likeli-hood function for those data points is ln ( LF ) = − [ A obs ( z i )− A th ( z i , p )] T C − [ A obs ( z i )− A th ( z i , p )] , (18)where C − is the inverse of the covariance matrix C (Ryanet al. 2019) = .
707 23 .
729 325 .
332 8 . .
386 3 . .
729 5 . . . . . .
332 11 . .
777 29 . .
271 14 . . . . . . . .
386 6 . .
271 16 . .
12 40 . . . . . . . . (19)For all parameters except for H , we assume top hat pri-ors, non-zero over ≤ Ω m ≤ , ≤ Ω Λ ≤ . , − . ≤ k ≤ . , − ≤ ω X ≤ , ≤ α ≤ ( ≤ α ≤ . for QSO only), − ≤ ln δ ≤ , ≤ β ≤ , and − ≤ γ ≤ . Here k = − Ω k a where a is the current value of the scale factor. Forthe Hubble constant we use two different Gaussian priors, H = ± . km s − Mpc − corresponding to the results of amedian statistics analysis of a large compilation of H mea-surements (Chen & Ratra 2011a), and H = . ± . kms − Mpc − from a recent local expansion rate measurement(Riess et al. 2016). This is consistent with earlier median statistics analyses (Gottet al. 2001; Chen et al. 2003), as well as with many other recentmeasurement of H (L’Huillier & Shafieloo 2017; Chen et al. 2017;Wang et al. 2017; Lin & Ishak 2017; DES Collaboration 2018b;Yu et al. 2018; G´omez-Valent & Amendola 2018; Haridasu et al.2018; Zhang 2018; Zhang & Huang 2018; Dom´ınguez et al. 2019). Other local expansion rate observations find slightly lower H values and have somewhat larger error bars (Rigault et al. 2015;Zhang et al. 2017; Dhawan et al. 2017; Fern´andez Arenas et al.2018; Freedman et al. 2019), but see Yuan et al. (2019).MNRAS , 1–8 (2019) N. Khadka, B. Ratra
The likelihood analysis is performed using the Markovchain Monte Carlo (MCMC) method as implemented in theemcee package (Foreman-Mackey et al. 2013) in Python 3.7.By using the maximum likelihood value LF max we computethe minimum χ value − ( LF max ) . In addition to χ we also use the Akaike Information Criterion AIC = χ + d (20)and the Bayes Information Criterion BIC = χ + d ln N (21)(Ryan et al. 2018), where d is the number of free parame-ters, and N is the number of data points. The AIC and
BIC penalize models with a larger number of free parameters. H ( z ) + BAO constraints Results for the H ( z ) + BAO data set are listed in Tables 1–3.The unmarginalized best-fit parameters are in Tables 1 and2. The marginalized one-dimensional best-fit parameter val-ues with σ error bars are given in Table 3. These results areconsistent with those of Ryan et al. (2019). The slight dif-ferences between the two sets of results are the consequenceof the different analysis techniques used and the Gaussianpriors used for the Hubble constant. Our computations aredone using the MCMC method while Ryan et al. (2019) useda grid-based χ technique.The one-dimensional likelihoods and two-dimensional σ , σ , and σ confidence contours for all parameters deter-mined by using H ( z ) + BAO data are shown in red in Figs.1–12. Some of the plots for the φ CDM model differ slightlyfrom the corresponding plots of Ryan et al. (2019) becauseof the difference we discussed above. The H ( z ) + BAO datareduced χ values are ∼ H ( z ) data. Use of the QSO data to constrain cosmological parametersis based on the assumed validity of the L X - L UV relationin eq. (12). This assumption is tested by Risaliti & Lusso(2015). By fitting this relation in cosmological models wehave found, in agreement with Risaliti & Lusso (2015), thatthe slope γ ∼ . ± . , the intercept β is between 8 and 9,and the global intrinsic dispersion δ = . ± . . The globalintrinsic dispersion is large and so cosmological parameterdetermination done using these data is not as precise as thatdone by using, for example, the SNIa data. But the mainadvantage of using the quasar sample is that it covers a verylarge redshift range and eventually with more and betterquality data it should result in tight constraints.The QSO data determined cosmological parameter re-sults are given in Tables 1, 2, and 4. The unmarginalizedbest-fit parameters are given in the Tables 1 and 2 for the H = ± . − Mpc − and . ± .
74 km s − Mpc − priors respectively. The one-dimensional likelihoods and thetwo-dimensional confidence contours are shown in grey inthe left panels of Figs. 1–12. The cosmological parameterconstraints are insensitive to the H prior used. Risaliti & Lusso (2015) have determined cosmological parameters forthe non-flat Λ CDM model. Our QSO data constraints inthe Ω Λ – Ω m sub-panels of our Figs. 3 and 4 agree wellwith the corresponding constraints in Fig. 6 of Risaliti & Lusso (2015). For the non-flat Λ CDM model we find Ω m = . + . − . and Ω Λ = . + . − . , also in good agreement withthe corresponding Risaliti & Lusso (2015) values of Ω m = . + . − . and Ω Λ = . + . − . .The cosmological parameters obtained by using theseQSO data have relatively high uncertainty for all modelsbut they are mostly consistent with the results obtained byusing the BAO + H ( z ) data set, which are shown in red inFigs. 1–12. The QSO data reduced χ values are also small ∼ χ values do not change significantly from modelto model. H ( z ) + BAO constraints From Figs. 1–12 we see that the constraints from H ( z ) +BAO data and those from QSO data alone are largely mutu-ally consistent, except for the H = . ± .
74 km s − Mpc − prior case in the flat Λ CDM and flat φ CDM models, see thebottom left sub-panels in the left panels of Figs. 2 and 10.The H ( z ) + BAO data constrain cosmological parametersquite tightly while the QSO data result in very loose con-straints on these parameters. Although the QSO data aloneare not able to provide restrictive constraints, they can helptighten constraints when used in combination with H ( z ) +BAO data.Given that the QSO and the H ( z ) + BAO constraintsare mostly consistent, it is reasonable to determine jointQSO + H ( z ) + BAO constraints. These results are given inTables 1, 2, and 5. The QSO + H ( z ) + BAO one-dimensionallikelihoods and two-dimensional confidence contours for allthe cosmological parameters are shown in blue in the rightpanels of Figs. 1–12. These figures also show the H ( z ) +BAO data constraint contours in red. Adding the QSO datato the H ( z ) + BAO data and deriving joint constraints oncosmological parameters, results in bigger effects for the caseof the H = . ± .
74 km s − Mpc − prior (Figs. 2, 4, 6, 8,10, & φ CDM model for both priors (Figs.9 & Ω m = . ± . to . ± . ( Ω m = . ± . to . ± . ) for flat (non-flat) models and the H = ± . − Mpc − prior and to lie in the range Ω m = . ± . to . ± . ( Ω m = . ± . to . ± . ) for flat (non-flat) models and the H = . ± .
74 km s − Mpc − prior.In some cases these results differ slightly from the H ( z ) +BAO data results of Table 3. These results are consistentwith those derived using other data.The Hubble constant is found to lie in the range H = . + . − . to . + . − . ( H = . + . − . to . + . − . ) km s − Mpc − for flat (non-flat) models and the H = ± . − Mpc − prior and to lie in the range H = . ± . to . + . − . ( H = . + . − . to . ± . ) km s − Mpc − forflat (non-flat) models and the H = . ± .
74 km s − Mpc − prior. As expected, for the H = . ± .
74 km s − Mpc − prior case, the measured value of H is pulled lower than the MNRAS000
74 km s − Mpc − prior case, the measured value of H is pulled lower than the MNRAS000 , 1–8 (2019) onstraints from QSO, H ( z ) , and BAO data Table 1.
Unmarginalized best-fit parameters of all models for the H = ± . km s − Mpc − prior.Model Data set Ω m Ω Λ Ω k ω X α H a δ β γ χ AIC BIC
Flat Λ CDM H ( z ) + BAO 0.29 0.71 - - - 67.56 - - - 32.47 36.47 39.95QSO 0.20 0.80 - - - 68.00 0.32 8.29 0.59 468.94 478.94 502.41QSO + H ( z ) + BAO 0.30 0.70 - - - 67.97 0.32 8.53 0.58 497.01 507.01 530.74Non-flat Λ CDM H ( z ) + BAO 0.30 0.70 . - - 68.23 - - - 27.05 33.05 38.26QSO 0.12 1.13 − . - - 68.00 0.32 8.57 0.58 466.13 478.13 506.30QSO + H ( z ) + BAO 0.30 0.70 . - - 68.33 0.32 8.52 0.58 496.52 508.52 536.99Flat XCDM H ( z ) + BAO 0.30 0.70 - − . - 67.24 - - - 27.29 33.29 38.50QSO 0.21 0.79 - − . - 68.00 0.32 8.41 0.59 468.35 480.35 508.52QSO + H ( z ) + BAO 0.30 0.70 - − . - 67.62 0.32 8.53 0.58 496.90 508.90 537.37Non-flat XCDM H ( z ) + BAO 0.32 - − . − . - 67.42 - - - 24.91 32.91 39.86QSO 0.021 - − . − . - 68.00 0.32 8.65 0.58 463.10 477.10 509.96QSO + H ( z ) + BAO 0.32 - − . − . - 67.76 0.32 8.76 0.57 494.65 508.65 541.87Flat φ CDM H ( z ) + BAO 0.32 - - - 0.10 67.23 - - - 27.42 33.42 38.63QSO 0.2 - - - 0.07 68.00 0.32 8.31 0.59 469.04 481.04 509.21QSO + H ( z ) + BAO 0.30 - - - 0.03 66.69 0.32 8.86 0.57 497.03 509.03 537.50Non-flat φ CDM H ( z ) + BAO 0.33 - − . - 1.20 65.86 - - - 25.04 33.04 39.99QSO 0.20 - − . - 0.30 68.00 0.33 8.20 0.59 471.06 485.06 517.92QSO + H ( z ) + BAO 0.29 - − . - 0.47 69.57 0.31 9.01 0.57 494.73 508.73 541.95 a km s − Mpc − . Table 2.
Unmarginalized best-fit parameters of all models for the H = . ± . km s − Mpc − prior.Model Data set Ω m Ω Λ Ω k ω X α H a δ β γ χ AIC BIC
Flat Λ CDM H ( z ) + BAO 0.30 0.70 - - - 69.11 - - - 33.76 38.76 41.24QSO 0.20 0.80 - - - 73.24 0.32 8.26 0.59 468.94 478.94 502.41QSO + H ( z ) + BAO 0.30 0.70 - - - 69.09 0.32 8.53 0.58 503.30 513.30 536.76Non-flat Λ CDM H ( z ) + BAO 0.30 0.78 − . - - 71.56 - - - 28.80 34.80 40.01QSO 0.12 1.13 − . - - 73.24 0.32 8.55 0.58 466.13 478.13 506.30QSO + H ( z ) + BAO 0.30 0.78 − . - - 71.66 0.32 8.61 0.58 497.85 509.85 538.32Flat XCDM H ( z ) + BAO 0.29 0.71 - − . - 71.27 - - - 30.68 36.68 41.89QSO 0.21 0.79 - − . - 73.24 0.32 8.39 0.59 468.35 480.35 508.52QSO + H ( z ) + BAO 0.29 0.71 - − . - 71.37 0.32 8.51 0.58 499.84 511.84 540.31Non-flat XCDM H ( z ) + BAO 0.32 - − . − . - 71.22 - - - 28.17 36.17 43.12QSO 0.021 - − . − . - 68.00 0.32 8.65 0.58 463.10 477.10 509.96QSO + H ( z ) + BAO 0.31 - − . − . - 72.14 0.32 8.72 0.58 498.07 512.07 545.29Flat φ CDM H ( z ) + BAO 0.33 - - - 0.09 69.31 - - - 33.36 39.36 44.57QSO 0.2 - - - 0.13 73.24 0.32 8.32 0.59 469.04 481.04 509.21QSO + H ( z ) + BAO 0.30 - - - 0.05 70.20 0.33 8.98 0.57 506.97 518.97 547.44Non-flat φ CDM H ( z ) + BAO 0.32 - − . - 1.14 69.23 - - - 27.62 35.62 42.57QSO 0.20 - − . - 0.30 71.00 0.33 8.20 0.59 473.45 487.45 520.31QSO + H ( z ) + BAO 0.32 - − . - 1.17 73.51 0.32 9.99 0.53 497.58 511.58 544.80 a km s − Mpc − . prior value because the H ( z ) and BAO data favor a lower H . For the non-flat Λ CDM model the curvature energydensity parameter is measured to be Ω k = . ± . and − . ± . for the H = ± . − Mpc − and . ± .
74 km s − Mpc − priors respectively. The curvatureenergy density parameter is found to be Ω k = − . + . − . and − . + . − . for the non-flat XCDM and non-flat φ CDMmodels for the H = ± . − Mpc − prior and Ω k = − . + . − . and − . + . − . for the non-flat XCDM and non-flat φ CDM models for the H = . ± .
74 km s − Mpc − prior. It is interesting that in all cases closed spatial hy-persurfaces are favored, albeit just barely in the non-flat Λ CDM model with the H = ± . − Mpc − prior,and only at 1 σ for most other cases, but at more than 2 σ for the non-flat φ CDM model and the H = . ± . − Mpc − prior. This preference for closed spatial hyper-surfaces is largely driven by the H ( z ) + BAO data (Park & MNRAS , 1–8 (2019)
N. Khadka, B. Ratra
Table 3.
Marginalized one-dimensional best-fit parameters with 1 σ confidence intervals for all models using BAO and H ( z ) data. H a prior Model Ω m Ω Λ Ω k ω X α H a H = ± . Flat Λ CDM . + . − . - - - - . + . − . Non-flat Λ CDM . + . − . . + . − . . + . − . - - . + . − . Flat XCDM . + . − . - - − . + . − . - . + . − . Non-flat XCDM . + . − . - − . + . − . − . + . − . - . + . − . Flat φ CDM . + . − . - - - . + . − . . + . − . Non-flat φ CDM . + . − . - − . + . − . - . + . − . . + . − . H = . ± . Flat Λ CDM . + . − . - - - - . + . − . Non-flat Λ CDM . + . − . . + . − . − . + . − . - - . + . − . Flat XCDM . + . − . - - − . + . − . - . + . − . Non-flat XCDM . + . − . - − . + . − . − . + . − . - . + . − . Flat φ CDM . + . − . - - - . + . − . . + . − . Non-flat φ CDM . + . − . - − . + . − . - . + . − . . + . − . a km s − Mpc − . Table 4.
Marginalized one-dimensional best-fit parameters with 1 σ confidence intervals for all models using QSO data. H a prior Model Ω m Ω Λ Ω k ω X α H a δ β γ H = ± . Flat Λ CDM . + . − . - - - - + . − . . + . − . . + . − . . + . − . Non-flat Λ CDM . + . − . . + . − . − . + . − . - - + . − . . + . − . . + . − . . + . − . Flat XCDM . + . − . - - − . + . − . - + . − . . + . − . . + . − . . + . − . Non-flat XCDM . + . − . - . + . − . − . + . − . - + . − . . + . − . . + . − . . + . − . Flat φ CDM . + . − . - - - . + . − . + . − . . + . − . . + . − . . + . − . Non-flat φ CDM . + . − . - − . + . − . - . + . − . + . − . . + . − . . + . − . . + . − . H = ± . Flat Λ CDM . + . − . - - - - . + . − . . + . − . . + . − . . + . − . Non-flat Λ CDM . + . − . . + . − . − . + . − . - - . + . − . . + . − . . + . − . . + . − . Flat XCDM . + . − . - - − . + . − . - . + . − . . + . − . . + . − . . + . − . Non-flat XCDM . + . − . - . + . − . − . + . − . - . + . − . . + . − . . + . − . . + . − . Flat φ CDM . + . − . - - - . + . − . . + . − . . + . − . . + . − . . + . − . Non-flat φ CDM . + . − . - − . + . − . - . + . − . . + . − . . + . − . . + . − . . + . − . a km s − Mpc − . Table 5.
Marginalized one-dimensional best-fit parameters with 1 σ confidence intervals for all models using QSO+ H ( z ) +BAO data. H a prior Model Ω m Ω Λ Ω k ω X α H a δ β γ H = ± . Flat Λ CDM . + . − . . + . − . - - - . + . − . . + . − . . + . − . . + . − . Non-flat Λ CDM . + . − . . + . − . . + . − . - - . + . − . . + . − . . + . − . . + . − . Flat XCDM . + . − . - - − . + . − . - . + . − . . + . − . . + . − . . + . − . Non-flat XCDM . + . − . - − . + . − . − . + . − . - . + . − . . + . − . . + . − . . + . − . Flat φ CDM . + . − . - - - . + . − . . + . − . . + . − . . + . − . . + . − . Non-flat φ CDM . + . − . - − . + . − . - . + . − . . + . − . . + . − . . + . − . . + . − . H = . ± . Flat Λ CDM . + . − . . + . − . - - - . + . − . . + . − . . + . − . . + . − . Non-flat Λ CDM . + . − . . + . − . − . + . − . - - . + . − . . + . − . . + . − . . + . − . Flat XCDM . + . − . - - − . + . − . - . + . − . . + . − . . + . − . . + . − . Non-flat XCDM . + . − . - − . + . − . − . + . − . - . + . − . . + . − . . + . − . . + . − . Flat φ CDM . + . − . - - - . + . − . . + . − . . + . − . . + . − . . + . − . Non-flat φ CDM . + . − . - − . + . − . - . + . − . . + . − . . + . − . . + . − . . + . − . a km s − Mpc − . Ratra 2018d; Ryan et al. 2019). Mildly closed spatial hy-persurfaces are also consistent with CMB anisotropy mea-surements (Ooba et al. 2018a,b,c; Park & Ratra 2018a,b,c,2019).The cosmological constant density parameter for the flat (non-flat) Λ CDM model is determined to be Ω Λ = . ± . ( . ± . ) and . ± . ( . ± . ) for the H = ± . − Mpc − and . ± .
74 km s − Mpc − priorsrespectively.The parameters that govern dark energy dynamics move MNRAS000
74 km s − Mpc − priorsrespectively.The parameters that govern dark energy dynamics move MNRAS000 , 1–8 (2019) onstraints from QSO, H ( z ) , and BAO data closer to those of a time-independent Λ when we jointlyanalyze the QSO data with the H ( z ) + BAO data, com-pared to the corresponding QSO data alone case. From theanalyses of the QSO + H ( z ) + BAO data the equation ofstate parameter for the flat (non-flat) XCDM parametriza-tion is determined to be ω X = − . ± . ( − . + . − . ) and − . ± . ( − . + . − . ) for the H = ± . − Mpc − and . ± .
74 km s − Mpc − priors respectively. The pa-rameter α in the flat (non-flat) φ CDM model is determinedto be α = . + . − . ( . + . − . ) and . + . − . ( . + . − . ) forthe H = ± . − Mpc − and . ± .
74 km s − Mpc − priors respectively. Of these 8 cases, 6 favor dark energy dy-namics over a time-independent cosmological constant en-ergy density; for the ± . − Mpc − prior dynamicaldark energy is favored at 1.3 σ to 1.7 σ in the flat φ CDM andnon-flat XCDM and φ CDM models, while for the . ± . − Mpc − prior, dark energy dynamics is favored at 1.5 σ to 2.6 σ in the flat and non-flat φ CDM models and the flatXCDM cases. Other data also favor mild dark energy dy-namics (Ooba et al. 2018d; Park & Ratra 2018b,c).
We have used the Risaliti & Lusso (2015) compilation of 808X-ray and UV QSO flux measurements to constrain cosmo-logical parameters in six cosmological models. These QSOdata constraints are much less restrictive than, but mostlyconsistent with those obtained from the joint analyses of 31Hubble parameter and 11 BAO distance measurements.We find that joint analyses of the QSO and H ( z ) + BAOdata tightens (and in some models, alters) constraints oncosmological parameters derived using just the H ( z ) + BAOdata. In general, the tightening effect is more significant inmodels with a larger number of free parameters. The jointQSO + H ( z ) + BAO data constraints are consistent with thecurrent standard flat Λ CDM model, although they weaklyfavor closed over flat spatial hypersurfaces and dynamicaldark energy over a cosmological constant.While cosmological parameter constraints from theQSO data we have used here are not that restrictive, thenew Risaliti & Lusso (2019) QSO data compilation (of 1598measurements over the redshift range . ≤ z ≤ . ) will re-sult in more restrictive cosmological parameter constraintsthat near-future QSO data should improve upon. ACKNOWLEDGEMENTS
We thank Elisabeta Lusso for her generous help, and wethank Lado Samushia, Chan-Gyung Park, and Joe Ryan foruseful discussions. We are grateful to the Beocat ResearchCluster at Kansas State University team, especially DaveTurner and Adam Tygart. This research was supported inpart by DOE grant DE-SC0019038.
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Figure 1.
Flat Λ CDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panel shows1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots for onlycosmological parameters Ω m and H , without the QSO-only constraints. These plots are for the H = ± . − Mpc − prior. H . . . . m H . . . . . . QSOQSO + H ( z ) + BAOH ( z ) + BAO . . . . H . . m H QSO + H ( z ) + BAOH ( z ) + BAO
Figure 2.
Flat Λ CDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panel shows1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots for onlycosmological parameters Ω m and H , without the QSO-only constraints. These plots are for the H = . ± .
74 km s − Mpc − prior.MNRAS , 1–8 (2019) N. Khadka, B. Ratra
60 65 70 75 H . . . . m H . . . .
30 0 .
33 0 . . . .
64 7 8 9 10 11
QSOQSO + H ( z ) + BAOH ( z ) + BAO H . . . m H . . . . QSO + H ( z ) + BAOH ( z ) + BAO
Figure 3.
Non-flat Λ CDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panelshows 1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots forcosmological parameters Ω m , Ω Λ , and H , without the QSO-only constraints. These plots are for the H = ± . − Mpc − prior.The black dotted straight lines corresponds to the flat Λ CDM model.
69 72 75 78 H . . . . m H . . . .
30 0 .
33 0 . . . .
64 7 8 9 10 11
QSOQSO + H ( z ) + BAOH ( z ) + BAO . . . . H .
30 0 . m H .
64 0 .
72 0 .
80 0 . QSO + H ( z ) + BAOH ( z ) + BAO
Figure 4.
Non-flat Λ CDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panelshows 1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots foronly cosmological parameters Ω m , Ω Λ , and H , without the QSO-only constraints. These plots are for the H = . ± .
74 km s − Mpc − prior. The black dotted straight lines corresponds to the flat Λ CDM model. MNRAS000
74 km s − Mpc − prior. The black dotted straight lines corresponds to the flat Λ CDM model. MNRAS000 , 1–8 (2019) onstraints from QSO, H ( z ) , and BAO data
60 65 70 75 H X . . . . m H X .
30 0 .
33 0 . . . . .
63 7 8 9 10
QSOQSO + H ( z ) + BAOH ( z ) + BAO H X . . . m H . . X QSO + H ( z ) + BAOH ( z ) + BAO
Figure 5.
Flat XCDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panel shows1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots for onlycosmological parameters Ω m , ω X , and H , without the QSO-only constraints. These plots are for the H = ± . − Mpc − prior.The green dotted straight lines represent ω x = − .
66 69 72 75 78 H X . . . . m H X .
30 0 .
33 0 . . . . .
63 7 8 9 10
QSOQSO + H ( z ) + BAOH ( z ) + BAO . . . . H X . . . m H . . X QSO + H ( z ) + BAOH ( z ) + BAO
Figure 6.
Flat XCDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panel shows1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots for onlycosmological parameters Ω m , ω X , and H , without the QSO-only constraints. These plots are for the H = . ± .
74 km s − Mpc − prior. The green dotted straight lines represent ω x = − .MNRAS , 1–8 (2019) N. Khadka, B. Ratra
60 65 70 75 H k X . . . . m H k X .
30 0 .
33 0 . . . .
64 7 8 9 1011
QSOQSO + H ( z ) + BAOH ( z ) + BAO H k X . . . . m H . . . k . . . . X QSO + H ( z ) + BAOH ( z ) + BAO
Figure 7.
Non-flat XCDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panelshows 1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots foronly cosmological parameters Ω m , Ω k , ω X , and H , without the QSO-only constraints. These plots are for the H = ± . − Mpc − prior. The black dashed straight lines and the green dotted straight lines are Ω k = 0 and ω x = − lines. H k X . . . . m H k X .
30 0 .
33 0 . . . .
64 7 8 9 10
QSOQSO + H ( z ) + BAOH ( z ) + BAO .
57 0 .
07 2 .
57 5 . H k X . . . . m H . . . k . . . . X QSO + H ( z ) + BAOH ( z ) + BAO
Figure 8.
Non-flat XCDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left pnnelshows 1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots for onlycosmological parameters Ω m , Ω k , ω X , and H , without the QSO-only constraints. These plots are for the H = . ± .
74 km s − Mpc − prior. The black dashed straight lines and the green dotted straight lines are Ω k = 0 and ω x = − lines. MNRAS000
74 km s − Mpc − prior. The black dashed straight lines and the green dotted straight lines are Ω k = 0 and ω x = − lines. MNRAS000 , 1–8 (2019) onstraints from QSO, H ( z ) , and BAO data
60 65 70 75 H . . . . m H . . . .
00 0 .
30 0 .
33 0 . . . .
64 7 8 9 10
QSOQSO + H ( z ) + BAOH ( z ) + BAO . . . . H . . m H . . . . QSO + H ( z ) + BAOH ( z ) + BAO
Figure 9.
Flat φ CDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panel shows1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots for onlycosmological parameters Ω m , α , and H , without the QSO-only constraints. These plots are for the H = ± . − Mpc − prior.
68 72 76 H . . . . m H . . . .
00 0 .
30 0 .
33 0 . . . .
64 7 8 9 10
QSOQSO + H ( z ) + BAOH ( z ) + BAO H . . . . . m H . . . QSO + H ( z ) + BAOH ( z ) + BAO
Figure 10.
Flat φ CDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panel shows1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots for onlycosmological parameters Ω m , α , and H , without the QSO-only constraints. These plots are for the H = . ± .
74 km s − Mpc − prior.MNRAS , 1–8 (2019) N. Khadka, B. Ratra
60 65 70 75 H k . . . . m H . . . . k . . . . .
30 0 .
33 0 . . . .
64 7 8 9 10
QSOQSO + H ( z ) + BAOH ( z ) + BAO H k . . m H . . k . . . . QSO + H ( z ) + BAOH ( z ) + BAO
Figure 11.
Non-flat φ CDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panelshows 1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots foronly cosmological parameters Ω m , Ω k , α , and H , without the QSO-only constraints. These plots are for the H = ± . − Mpc − prior. The black dashed straight lines are Ω k = 0 lines. H k . . . . m H . . . . k . . . . .
30 0 .
33 0 . . . .
64 7 8 9 10
QSOQSO + H ( z ) + BAOH ( z ) + BAO . . . . H k . . m H . . . . k . . . . QSO + H ( z ) + BAOH ( z ) + BAO
Figure 12.
Non-Flat φ CDM model constraints from QSO (grey), H ( z ) + BAO (red), and QSO + H ( z ) + BAO (blue) data. Left panelshows 1, 2, and 3 σ confidence contours and one-dimensional likelihoods for all free parameters. Right panel shows magnified plots for onlycosmological parameters Ω m , Ω k , α , and H , without the QSO-only constraints.These plots are the for H = . ± .
74 km s − Mpc − prior. The black dashed straight lines are Ω k = 0 lines. MNRAS000