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This content was downloaded from IP address 186.217.236.55 on 29/05/2019 at 18:08 JCAP05(2018)073 ). z ( q 063 and . 0 hysics ± P 870 . le , which determines t z ic 094, 0 t . 0 b ± ar 806 . ) data complement each other z ( H strop A , a second degree parametrization for the z https://doi.org/10.1088/1475-7516/2018/05/073 and S.H. Pereira c,d,e , for each approach, as 0 t z osmology and and osmology ) up to third degree on z ) and a linear parametrization for the deceleration parameter ( C z ( c.l., respectively. Then, such approaches provide cosmological-model C D H σ R.F.L. Holanda 1712.01075 a,b redshift surveys, supernova type Ia - standard candles This paper aims to put constraints on the transition redshift 058 at 1 . rnal of rnal 0 ± ou An IOP and SISSA journal An IOP and 2018 IOP Publishing Ltd and Sissa Medialab 973 Universidade Estadual Paulista (Unesp),R. Campus Geraldo Experimental Alckmin de 519, Itapeva, Itapeva,Universidade SP, Estadual 18409-010 Paulista Brazil (Unesp),Departamento Faculdade de de Engenharia F´ısicae Qu´ımica,Av. de Dr.Guaratinguet´a,SP, Guaratinguet´a, Ariberto 12516-410 Brazil Pereira da CunhaDepartamento 333, de Experimental, F´ısicaTe´oricae Universidade Federal do Rio GrandeAv. do Sen. Norte, Salgado Filho 3000,Departamento Natal, de RN, F´ısica,Universidade Federal 59078-970 deAv. Brazil Sergipe, Marechal Rondon S/N, 49100-000 S˜aoCristov˜ao,SE, Departamento Brazil de F´ısica,Universidade Federal deR. Campina Apr´ıgioVeloso 882, Grande, Campina Grande, PB, 58429-900E-mail: Brazil , [email protected] , [email protected] [email protected] . b c e d a c J.F. Jesus, Model independent constraints on transition redshift J For each case,on we the show parameter that spacecombining type and the tighter Ia type constrainspossible Ia supernovae for to supernovae and the constrain observations transition the and0 values redshift Hubble of are parameter obtained. measurementsindependent By estimates it for is this parameter. Keywords: ArXiv ePrint: Received December 5, 2017 Revised March 26, 2018 Accepted May 17, 2018 Published May 30, 2018 Abstract. the onset of cosmicto acceleration, perform in our cosmological-model analyses,comoving distance we independent consider frameworks. aHubble In flat parameter order universe and assume a parametrization for the JCAP05(2018)073 6 ) theories that T ( , the deceleration is the scale factor f a 1. The deceleration a/a − = ˙ ) and = H R j ( f ), where 3 ˙ a a ( / 3 a a ... − = j – 1 – )4 z ( C D and the jerk parameter 2 ˙ )4 a 2 while the accelerating ΛCDM model has )6 z / ), measurements [17–19]. The simplest theoretical model supporting such ( z ¨ a/ z ( ( a q H = 1 − H q ) data6 = z from comoving distance, from from q ( t t t z z z H However, some works have tried to explore the history of the universe without to appeal 2.1 2.3 2.2 3.1 3.2 JLA SNe Ia compilation7 to any specific cosmologicalcosmokinetic model. models [42–47], Suchnomenclature approaches and comes are we from sometimes will the called refer(or fact cosmography its to that kinematics) or the them is complete simply described study just as of by kinematic the the expansion models. Hubble of expansion the This rate Universe generalize the general relativitymodels [32–34], [35–39], models string based [ 40]particular on and model, extra Kaluza-Klein the cosmological theories dimensions, parameters], [41 of as can among observational be brane data. others. determined world by Having using statistical adopted analysis a parameter in the Friedmann-Roberson-Walker (FRW) metric.is The homogeneous only and assumptionuniverse isotropic. is has that In space-time parameter such allows parametrization, to a study simple the dark transition matter from dominated a decelerated phase to an accelerated one, 1 Introduction The idea of aservations late [1–8] time and accelerating confirmed universe by isBackground other (CMB) indicated independent radiation by [9–11], observations type Baryonic such Ia Acousticble as Oscillations supernovae parameter, Cosmic (BAO) (SNe Microwave [12–16] Ia) and ob- Hub- accelerating phase is based onter a component cosmological constant [22–24], Λmodel term the [20, have so-called21] been plus ΛCDM constrained aadded. model. more Cold and Dark Beyond Mat- more The a accuratelyrecently cosmological [11, constant in parameters18, Λ order of25] based ason to such model, new a explain observations several dark the are other energyThe accelerated models fluid nature expansion. [26, have of27] been such Theeven endowed exotic also to with most fluid suggested mass a popular is negative dimensioncorrectly unknown, ones pressure one sometimes describe are filling fermionic attributed an based the fields. to acceleratedries whole a expansion [28], There universe. single of modifications are scalar the of also field Universe, Newtonian modified or such theory gravity as: (MOND) theories [29–31], that massive gravity theo- Contents 11 Introduction 2 Basic equations3 3 Samples 4 Analyses and results7 5 Conclusion 16 JCAP05(2018)073 0 1 is ). ≤ . 28 11 q 21 . . z . 0 0 ) = ( 0 z +0 − dz q 17 ± ( . ± 1 4 60 H . . ) and a − 98 using z . 71 ( . 0 0 = 0 = 0 H t − t ≈ z z . In seventies, . For the same t 0 = by using galaxy z found by Cunha as a new cosmic H 110 070 0 . . t 36, 35 25 q . t 0 . . z 1 0 z +0 − +1 − 85 and 609 12 and . ≤ − . . 0 0 0 0 q − q = 0 ). By using the results from ± approach, the quantity t = ≤ z dz 0 52 0 . q . 0 1 − − would be the main role of cosmology 0 = q 045 and . 0 q 0 is of fundamental importance in modern values in these cosmological-model inde- , the values for ). One of the first analyses and constraints and t t ± z z z z 0 ( 1 from 182 SNe Ia of Riess et al. [4], q q H ), Xu, Li and Lu [51] used 307 SNe Ia together 715 . z + 09 05 . . 0 – 2 – 0 0 cosmic chronometer q − +0 − (1 + = 43 . ) = 0 z/ q z 1 ( q q = 0 + t 0 ) in a given redshift interval ( z q dt ). By combining luminosity distances from SNe Ia [56] and 7, . z from SNLS data set [3] and 0 ( ) = q z ( 68 21 . . . In the so called q 0 ∼ − +3 − 0 using a sample of 307 SNe Ia from Union compilation [6]. Also, for q 61 ) data and found . dz/dt z the universe undergoes a transition to accelerated phase [18, 19, 44, 48– 27 09 )] ( . . z t 0 = 0 H z +0 − t z 49 . (1 + / 16, [1 . = 0 0 − t For a linear parametrization, In order to study the deceleration parameter in a cosmological-model independent frame- Concerning the deceleration parameter, several studies have attempted to estimate at As one may see, the determination of the deceleration parameter is a relevant subject in In the present work we study the transition redshift by means of a third order z = ) measurements, it is possible to determine z ≤ − ( 0 ˙ and Lima [48] were q on cosmological parameters ofby employing kinematic Bayesian models marginal was likelihoodmented analysis. done the analysis by Since by then, Elgarøy including several and new authors46] Multam¨aki[ data have imple- sets and also different parametrizations for and deceleration-acceleration transition redshift from Davis et al.and data set [5].the linear Guimar˜aes,Cunha parametrization, anda Lima Rani [49] joint et found al. analysisparametrization [50] of of found type age ofwith BAO galaxies, and strong gravitational lensing and SNe Ia data. For a work, it is necessarytages to and use disadvantages. some parametrization Onecontent for advantage of is it. the that This itthe universe. methodology is cause has independent One of both of disadvantage advan- theparameter the of may accelerated matter depend this expansion. on and the formulation energy Furthermore, assumed is form the of that value of it the does present not deceleration explain cosmology. As amodels. new cosmic parameter [54], it should be used to test several cosmological which redshift while the jerkwithout parameter restricting allows to to a specific study model. departures from the cosmic concordance50, model, 54, 55 ]. A model independent determination of the present deceleration parameter modern cosmology, as well as the determination of the Hubble parameter clusters of ellipticalobservations. morphology Such based parametrization on has their also Sunyaev-Zeldovich been effect studied (SZE) in [46, and48]. X-ray Sandage [53] foretold that the determination of parametrization, Holanda, Alcaniz and Carvalho [52] found for the forthcomingdiscriminator decades. has been The advocated bycosmological inclusion some parameters authors of in [54]. a thea/a model An transition alternative independent method redshift fashion to is access by the means of the study of obtained from spectroscopic surveysage and evolution the of only the quantity toBaryon universe be ( Oscillation measured Spectroscopic is Surveya the (BOSS) cosmological-model differential [14–16], independent Moresco determination et of al. the55] [ transition have redshift obtained as pendent frameworks. Inthe such observational approach data we and, obtain consequently, tighter an constraints interesting on complementarity the between parameter spaces. parametrization of the comoving distance, a second order parametrization of (see also [17–19]). linear parametrization of H JCAP05(2018)073 (2.1) (2.4) (2.6) (2.7) (2.8) (2.5) (2.2) (2.3) through a t z ) as given by: z ( q and the dimensionless Hubble ). Section3 presents the data , z ) = 0, leading to: 0 t ( 1 z q . ( . − 1 , q 1 c/H , . ) , ) , 1 z − ) − z ) ( 0 ≡ − dz 0 0 t ( dH 0 t z z C  ) and z z ( C z H ( ) = 1 z dz 1 = dz D d z ( d z z − | H ) ) − | E ( H ) the Hubble parameter in km/s/Mpc. For ,  z z  H z z C z 1 + ) ) dz L 0 H 0 ( d z z d Z , as the third cosmological number. To begin Z ( ( t = dD H c z – 3 – E ≡  H dz 2 dz ) = L ln ln ¨ a z ) = d ) = (1 + ) = (1 + ), at least around a redshift interval involving the d aH ( D ) = z  z z z (  C z , ( ( ( − ( , can be defined as C C L L H t D = d = d H d E z d t D t z ) = z ≡ z ( C q ) and D z ( q (in Mpc), is given by: . Thus, we have: ) L z 0 from luminosity distance, ( d H t H z ≡ ) z ( E is the comoving distance: C being the speed of light in km/s and d c Therefore, from a formal point of view, we may access the value of The paper is organized as follows. In2, section we present the basic equations concerning Let us assume a flat Friedmann-Robertson-Walker cosmology. In such a framework, the parameter, mathematical convenience, we choose tothe work with dimensionless dimensionless distances, quantities. Then, we define set used and the analyses are presented in section4.2 Conclusions are left Basic to equations section5. Let us discussvision (from by a including more the observational transition viewpoint) redshift, the possibility to enlarge Sandage’s from which the transition redshift, with from which follows From (2.2) we also have: parametrization of both transition redshift.which As is a directly related third toIn method the which luminosity follows we distance, we in can present order the also to three study parametrize different the methods the transition considered redshift. co-moving here. distance, with, consider the general expression for the deceleration parameter the obtainment of and luminosity distance, where JCAP05(2018)073 via ). If 2 (2.9) z (2.16) (2.15) (2.10) (2.11) (2.12) (2.13) (2.14) d ) by a ( z ( H C and D t z . , .  in terms of 3 3 . d z 3 . 1 t 9 = 1 d 2 − z d ) 1 2  − t 1 2 d h z 2 3 z . d t t and z ⇒ z + 4 . ) 2 t 2 ) t . From a statistical point of view, 2 2 t . + 9 z. z t d t d z z z d 3 1 z (2 + 3 z 3 3 t z ) we need an expression for h d + d z 3 z + 4 2 + 4 3 d ( 1 d 2 2 1 h 1 + 2 6 d + 3 d H + (2 + 3 (2 + 3 1 t 3 t z 2 − − = z d 2 z z 3 2 2 t 2 9 d 2 ) = 1 + z d z 1 + 2 d z 1 + 2 1 4 2 = ( ) – 4 – − z d − + z − E  p ( 1 + 2 z ) z + C = 2 z ± z ) = ( d = 3 t ) D  E d ) = z 0 ) C dz , z ( ) over the redshift, the linear expansion, gives no transi- 3 z z 2 H ( ln z D H d ( − 1 + 2 d E (  2 3 H  d d 2 = ) = by considering the comoving distance, we can write − t z z ) = (1 + t ( ) given by (2.10), we find z z ) over the redshift. = z ( E z ( t L ( 1 + z E , maybe it is better to write the coefficient from eq. (2.2) by means of D H t ) z t z z ( are free parameters. Naturally, from eqs. (2.7) and (2.9), one obtains H 3 d and from comoving distance, from 2 t t d z z The simplest expansion of ) measurements, respectively, in order to determine z ( Therefore, the transition redshift is undefined in this case. From (2.8), we have third degree polynomial such as: where In order to put limits on 2.1 Solving eq.) (2.8 with where we may see there are twoeq. possible) (2.8 solutions as to Then Finally, from eqs. (2.5), (2.9) and (2.12) the dimensionless luminosity distance is Equations) (2.14 and ( 2.13)H are to be compared with luminosity distances2.2 from SNe Ia and In order to assess tion. To realize this, let us take aiming to constrain one wants to avoid dynamicaluses assumptions, an one must expansion to of resort to kinematical methods which JCAP05(2018)073 ) z ( = 0 H 2 (2.17) (2.18) (2.19) (2.22) (2.23) (2.24) (2.20) (2.21) h , 2 t z 2 such as h 2 . h + 0 0 4 t z < > − 1 h 2 1 , from which follows the ∆ ∆ = 0 ∆ h 2 , h 2 , ≡ 0 i z coefficients from a fit to . 2 1 and this value is not possible ∆ 2 1 and h ) = 1 + 2 − z − h t , , h t 2 0 z √ + ] . . , z 2 h 2 0 < 1 1 :

) dz h z 1 t h h t  + 1 2 2 z z h and z z 1 = 0 h h h z − − 2 2 + 2 1 1 h h 1 1 − arctan 2 1 h h 1 − +2 h 1 + (1 + − 1 1 h z )( h h 1 + 1 + − t  0 − + z 1 Z t r r ∆ ∆+ [1 h – 5 – ∆ z 2 √ √ + ) expansion, namely, the quadratic expansion: = ) = 1 + 2 ± − z h z z 2 h ) ( (1 + √ 1 1 + ( 0 h 0   2 z − − E H 1 1 ). ( ⇒ + 2 dz  h h : z t E = = = 2 t − 2 t = 1 + z z t t z z ∆+ ∆ ) 1 2 z z 0 2 t z 0 √ √ h h Z z ( h )(1 + 2  H arctan

z H h , = + h ( t ln C C z ∆ +2 z + 2 z 1 2 2 − ∆ D D 1 1 ) from (2.23) just in terms of h 1 √ h √ z h = (          C L 1 + = D D = C t z D 1 + In order to constrain the model with SNe Ia data, we obtain the luminosity distance However, in order to obtain the likelihood for the transition redshift, we must Equation (2.21) already is an interesting result, and shows the reliability of the quadratic Let us now try the next simplest from eqs. (2.5), (2.6) and). (2.17 We have reparametrize the eq. (2.17)that to from show eq. its (2.21) explicit we dependency may on eliminate this the parameter parameter. Notice also In this case, inserting (2.10) into (2.16), we are left with: from which follows an equation for whose solution is: into) (2.19 does not furnish any information about the transition redshift. which gives three possible solutions, according to the sign of ∆ thus we may write luminosity distance model as a kinematic assessment of transition redshift. It is easy to see that taking We may exclude the(negative negative root, scale which factor). would give Thus, if one obtains the data, one may obtain a model independent estimate of transition redshift from JCAP05(2018)073 , (i) dt 1 1 − (2.28) (2.26) (2.27) (2.29) (2.25) a ) mea- t t z ( − e H ∞ x R ) data are z ≡ ( ) , H ))] a, x z ). From (2.1) one may z ( q , which can be constrained (1 + 1 1 q , , q t 0 . z q and 1  0 , q t − 1 z q − dz 1 − ) q 0 0 = 0 q z z Γ( ( 0 z, 1+ q q 1 ) − q z ) 1 + 1 + 1 + 0 , q z q 0 0 (1 + q – 6 – Z z and 1  ), as q − ) = e z z 1 ( ( q q q 1 0 ) = q q [Γ( z ) as a function of just 1 ( ) = exp )()2.6 is given by z − q ( z z ) data, as can be seen on table 1 of [59] and figure 1b here. E − ( ( L by means of a parametrization of z = 0 ( q C 1 E t t D ) data in comparison to Farooq and Ratra [18] compilation, q z z D H 1 z q ( e H ) parametrization that allows for a transition, one may find ) and dependence in z z ) = ( ( z z q ( C C D ) D z ( q ) is the incomplete gamma function defined by [57] as Γ( ) data 0. z from a, x ) ( z t ( H z a > From (2.26) (or from (2.27) and (2.8)) it is easy to find: At the present time, the most important methods for obtaining The data we work here are a combination of the compilations from Sharov and E See [54] for a review. 1 where Γ( with which is the simplest while the comoving distance from which follows from observational data. 3 Samples 3.1 Hubble parameter data in termsical of tests redshift because yields it onebackground is of cosmological inferred the models. from most astrophysical straightforward cosmolog- observations alone, not depending on any Vorontsova [58] andVorontsova Moresco [58] et added 6 al. [55] as described on Jesus et al. [59]. Sharov and find If we assume a linear through “cosmic chronometers”,measurements for of peaks example, of acoustic thefunction oscillations of of differential baryons luminous (BAO) age red and (iii) galaxies of through (LRG). galaxies correlation (DAG), (ii) which had 28 measurements.surements Moresco in et comparison al. [55], to onet Sharov their and al. turn, Vorontsova [59] [58]. have have added By arrived 7 combining at new both 41 datasets, Jesus 2.3 Now let us see how to assess JCAP05(2018)073 ) ), z ( z ( H (4.1) H ), z ( C 4) and three . D 0 ≤ , for z 1 q ≤ or (b) 2 05 . h , 2 , 2 )] ) (4.2) d describes the supernova color ,j C ) data for each model. b) 41 ) data for each model. were estimated from eq. (4.2), with z C ( z × mod ( µ H β , θ H t + , z 1 0 obs X , i ,H i 2 × H z σ 10) may be given by ( α / H − – 7 – − B (pc) ) data, the likelihood distribution function is given ,i L M z d ( ( ( obs − H 10 H [ ∗ B m =1 41 i X ) model (table1) and the error bars comes from the diagonal of the 4). See figure 1a and more details in next section. = . z = 5 log corresponds to the observed peak magnitude in the rest-frame = ( 1 µ C µ ∗ B 2 H D χ m 1), all three seasons from the SDSS-II (0 . 0 < z < (a) 2 . z < , where 2 H 2 is the specific parameter for each model, namely χ ,j − describes the time stretching of the light-curve, e . a) SNe Ia distance moduli from JLA. The data for 1 mod ∝ X θ H As explained on [56], we may assume that supernovae with identical color, shape and L ) parametrizations, respectively. z ( 4 Analyses and results In our analyses, weproportional have to used the flat likelihoods. priors For overby the parameters, so the posteriors are always data compilation. The lines represent the best fit from SNe+ 3.2 JLA SNeThe Ia JLA compilation compilationcollaborations. [56] consists Actually,associated of this distances 740 compilation for SNe produced thecosmological SDSS-II Ia recalibrated constraints, and limited SNe from SNLS by samplesthe Ia the systematic in measurement light-curves uncertainty SDSS-II order uncertainties, and in to [60] as,light improve the for and the curves instance, accuracy band-to-band SNLS have of and highAdaptive [61] quality Light-curve survey-to-survey Templates) and relative methodredshift were flux samples [56, obtained calibration. ( 62–64]. byyears using The from The an SNLS data improved (0 set SALT2 (Spectral includes several low- Figure 1 SNe Ia parameters fromcovariance the matrix. The lines represent the best fit from SNe+ galactic environment have, on average,case, the the same distance intrinsic modulus luminosity for all redshifts. In this where q at maximum brightness, where JCAP05(2018)073 , , - t 2 n d z, z (4.3) ( ) over 2 . Also, z µ t ( z H ). For z is fixed. We ( , by [68]. 0 H )] (4.4) H ,j = 70 km/s/Mpc. may depend on the mod 0 , θ B H t COSMOMC M . z, z ( , where

µ as described on [56], 2 M 2 SN χ − ˆ software by [66]. This MCMC µ 10 ) while for JLA ) − z e 10 ( SN ∝ H θ < . ( emcee ˆ µ to sample from our likelihood in ) yields slightly better constraints. For [ SN z 1 ( L stellar − constraint comes just from H M C t , due to the recent tension from different limits z 3 T emcee if otherwise d 0 )] H – 8 – ,j M mod by the position of walkers of the other group. is the dimension of parameter space. The main idea comes just from from SNe Ia are improved and one may see how SNe 0 , θ n t + ∆ t z H 1 1 B B language with the only z, z is the covariance matrix of 10 pc) computed for a fiducial value M M ( / ( µ ) version. The results of our statistical analyses can be seen on C as one of our fiducial parameter. The other parameters from ,j − = t ), z ) Python B mod M ) yields strong constraints over all parameters, especially ) as SN M z , θ ∆ ( θ t Python , are nuisance parameters. According to], [61 ( 1 B H ˆ µ B stellar z, z through Monte Carlo Markov Chain (MCMC) analysis. A simple and ( . M 0 2 M L = [ in its / d 2 H ( 3 χ , α, β, M 2 SN 10 − and χ e β ) parametrization. Almost all the = ( ) complement each other in order to constrain the transition redshift. In figure6, z , ∝ z ( 2 tuning parameters, where is part of the great MCMC sampler and CMB power spectrum solver ( α getdist / C SN L H θ D ) = 5 log For SNe Ia from JLA, we have the likelihood We used the freely available software In order to obtain the constraints over the free parameters, we have sampled the like- In figures2–4, we have the combined results for each parametrization. As one may In figures5–7, we show explicitly the independent constraints from JLA and ,j ) and JLA yields similar constraints. For + 1) See [67] for agetdist comparison among various MCMC sampling techniques. z for band, 2 3 n ( ( mod t of the Goodman-Weare affine-invariantposition sampler (parameter is vector in thesition parameter so of space) called of the “stretch a otherit move”, walker walkers. suitable (chain) where is Foreman-Mackey for the et determined parallelization,walker al. by by in the modified splitting one po- this group the method, is walkers in determined in order two to groups, make then the position of a method has advantage overonly the on simple one Metropolis-Hastings scale (MH)ers, method, parameter while since of MH it the methodn depends proposed is distribution based and on the also parameter on covariance the matrix, number that of is, walk- it depends on powerful MCMC method is thewhich so-called was Affine implemented Invariant in MCMC Ensemble Sampler by [65], B host stellar mass ( lihood where θ dimensional parameter space.plot all We have the used constraintssoftware on flat each priors model over in the the parameters. same figure, In we have order used to the freely available see, we have always chosen figures2–8 and on table1. each model iscombination later of obtained JLA+ as derived parameters. As one may see in figures2–4, the we find negligibleemphasize difference that in the constraints the over choose not SNe to Ia include other parameters constraints over for each model. We have also to over the Hubble constant.constraints At over the end of this section, we compare our results with different the cosmological parameters. Asz one may see in figure5, SNe Ia sets weaker constraints over figures6 and7, the constraints over H Ia and JCAP05(2018)073 13 013 045 046 . 16 12 . . . 046 047 . . 12 11 . . 0 0 . . 0 0 0 0 3 2 0 +0 − . . is better +0 − ± 3 +0 − ± ± ± 2 +3 − h 023 6 023 . . ) 058 . 065 . 0 1 . z 080 062 0 0064 . . ( 0 0 — — — — . q 0 0 ± ± 0 ± ± ± 6 ± ± . ± constraint. As one 033 071 . 68 973 . 434 . . 0 094 446 0 . . 0 19 0 1408 H 3 0 − . − − is better from JLA. The 0 0 q . The other parametrizations t z 013 047 045 16 13 . . . 045 046 . . 13 12 . . for each parametrization. As one 0 2 051 052 . . ) parametrizations. The parameters 0 0 . . 0 . 0 0 z t +0 − 0 3 +0 − ( +0 − z ± +0 − ± ± q ± 023 ) 023 . 6 063 . z 026 . . 0 . 082 065 ( 0 0066 . 1 . 0 — — — — 0 . ) and 0 0 ± H 0 ± z ± ± ± ( ± ± 8 ± is still allowed to vary. All constraints are . H 039 071 . t 870 . 192 68 . ), z . 101 522 0 . . z 0 – 9 – 19 0 ( 1412 3 0 − 1 for their model independent approach. Only our . − C . 0 0 is better from JLA. The constraint over D 1 ± from Riess et al. (2011) [69], Moresco et al. [55] found h ). 4 0 . z ( ) for 013 046 090 089 H 16 . . . z 033 031 . 0085 0090 . 19 18 . . . . ( 0 are consistent through the three different parametrizations, 0 H . . 0 0 0 = 0 0 . 0 0 +0 − 0 3 H +0 − t +0 − +0 − ± ± ± z H ± ) 044 016 . z 5 094 . . . ( 0 023 0044 080 0 . . 0065 . 1 0 — — — — C . 0 0 0 ± 0 ± D ± ± ± ± ± ) data, plus 1 ± . z 073 ( 253 . 806 c.l. correspond to the minimal 68.3% and 95.4% confidence intervals. . 69 . is better from 069 105 ) in their parametrization, however, they did not compare with SNe Ia 0 c.l. . H . 0 19 0299 σ z 1 0 . 1412 3 ) model yields the strongest constraints over − ( . q σ − 0 z − 0 for ΛCDM and ( H q . So we decided to deal only with the JLA full data. 1 06 t and 2 . . 0 z ). In figure7, one may see that the constraint over σ − +0 z ). from . Constraints from JLA+ ( z 64 ( t . H ) parametrization is compatible with their model-independent result. Their ΛCDM q z Another interesting result that can be seen in table1 is the For all parametrizations, the best constraints over the transition redshift comes from Figure8 summarizes our combined constraints over 1 Table shows the full numerical results from our statistical analysis. As one may see the By using 30 1 B z ) data, as first indicated by]. [54 Moresco et al. [55] also found stringent constraints 0 M ( 2 2 z 1 t 1 3 0 = 0 ( C h h q q H z d d α β M ∆ Parameter t D result is compatiblewith with all our parametrizations, although it is marginally compatible from constraint over may see, the compatible at 1 may see, the constraints over H over constraints. SNe Ia constraints vary little forfrom each the parametrization. (faster) In JLA fact, binnedconstraints, we data we also have have [56], found found however, constraints that whenespecially the comparing full with JLA the yields (slower) full stronger JLA constraints over the parameters, z are important for us to realize how much one may see that the constraint over without bold faces wereand the treated 1 as derived parameters. The central values correspond to the mean Table 1 JCAP05(2018)073 . 3 z 3 d + 2 z 2 d + z ) = z ( C D ) for z ( H – 10 – . Combined constraints from JLA and Figure 2 JCAP05(2018)073 ). 2 z 2 h + z 1 h (1 + 0 H ) = z ( H ) for z ( H – 11 – . Combined constraints from JLA and Figure 3 JCAP05(2018)073 . z 1 q + 0 q ) = z ( q ) for z ( H – 12 – . Combined constraints from JLA and Figure 4 JCAP05(2018)073 . 3 z 3 d + 2 z 2 d + z ) = z ( C D ) for z ( H – 13 – . Constraints from JLA and Figure 5 JCAP05(2018)073 ). 2 z 2 h + z 1 h (1 + 0 H ) = z ( H ) for z ( H – 14 – . Constraints from JLA and Figure 6 JCAP05(2018)073 . z 1 q + 0 q ) = z ( q ) for z ( H – 15 – . Constraints from JLA and Figure 7 JCAP05(2018)073 , ) = z z ( 0 H H value. t z ) parametrization z ( H is strongly dependent on the , the Hubble parameter 5 km/s/Mpc. The constraints ) data combined. Blue solid line C t . z z 1 were obtained, independent of a ( D t H ± z 1 . = 69 0 H ), – 16 – ) parametrization. z z ( ( with the Planck’s result, while it is incompatible at q C σ D ) as third, second and first degree polynomials on z . The determination of ( t q z lower value. It is interesting to note, from our table1 that, 74 km/s/Mpc, the Planck collaboration analysis yields . σ 1 ± 24 . values estimated from Cepheids [70] and from CMB [ 71]. While Riess = 73 ) parametrization, orange long-dashed line corresponds to z 0 0 ( H C H D 62 km/s/Mpc, a 3.4 . . Likelihoods for transition redshift from JLA and are quite stringent today from many observations, [70 71]. However, there is some 0 0 ± In the present work, we wrote the comoving distance H with the Riess’ result. 93 . σ with a little smaller uncertainty for et al. advocate and green short-dashed line corresponds to over tension among 66 although we are working withredshifts, model our independent result parametrizations is and inall data better our at agreement results intermediate with are the compatible high within redshift 1 result from Planck. In fact, cosmological model adopted,such thus parameter the in search aserve for model as methods independent a that way test are for allow of several the fundamental cosmological determination importance, models. and of since it the would deceleration parameter 3 5 Conclusion The accelerated expansionservations. of Several Universe models isdecelerated proposed phase confirmed in to by literature thetransition current different satisfactorily occurs accelerated explain from sets phase. one the of phase Ais transition to cosmological more called another, from significant and ob- the question the transition parameter is that when redshift, measures the this transition corresponds to respectively (see equations (2.9), (2.17)Only and a (2.26)), and flat obtained, universespecific for was each cosmological assumed case, model. the and the As estimates observational for data, we have used Supernovae type Ia and Figure 8 JCAP05(2018)073 , ] and 063 Λ 12 . . : Ω 0 1 Ω ] 1 ]. , ± , ≥ M ] z (1998) Ω 870 < z < (2008) 749 SPIRE . 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