Falsifying Λ CDM: Model-independent tests of the concordance model with eBOSS DR14Q and Pantheon
FFalsifying Λ CDM: Model-independent tests of the concordance model with eBOSSDR14Q and Pantheon
Arman Shafieloo,
1, 2, ∗ Benjamin L’Huillier, † and Alexei A. Starobinsky
3, 4, ‡ Korea Astronomy and Space Science Institute, Yuseong-gu, Daedeok-daero 776, Daejeon 34055, Korea University of Science and Technology, Yuseong-gu 217 Gajeong-ro, Daejeon 34113, Korea L. D. Landau Institute for Theoretical Physics RAS, Moscow 119334, Russia National Research University Higher School of Economics, Moscow 101000, Russia
We combine model-independent reconstructions of the expansion history from the latest Pan-theon supernovae distance modulus compilation and measurements from baryon acoustic oscillationto test some important aspects of the concordance model of cosmology namely the FLRW metricand flatness of spatial curvature. We then use the reconstructed expansion histories to fit growthmeasurement from redshift-space distortion and obtain constraints on (Ω m , γ, σ ) in a model inde-pendent manner. Our results show consistency with a spatially flat FLRW Universe with generalrelativity to govern the perturbation in the structure formation and the cosmological constant asdark energy. However, we can also see some hints of tension among different observations withinthe context of the concordance model related to high redshift observations ( z >
1) of the expansionhistory. This supports earlier findings of [1, 2] and highlights the importance of precise measurementof expansion history and growth of structure at high redshifts.
I. INTRODUCTION
The concordance model of cosmology is based on Ein-stein’s general theory of relativity (GR), which enabled usto build a theory of the Universe that is testable and canbe falsified. The concordance flat ΛCDM model, whichis based on GR and the assumptions of isotropy and ho-mogeneity of the Universe, has been very successful atexplaining various astronomical observations from a veryearly epoch (at least, from the Big-Bang nucleosynthe-sis time). This predictive model explains the dynamicsof the Universe with only 6 free parameters. Ω b andΩ dm (baryonic and dark matter densities) are the mat-ter parameters. Assuming a flat universe and cosmolog-ical constant being responsible for late time accelerationof the Universe, we can derive Ω Λ = 1 − (Ω b + Ω dm ). τ representing the epoch of reionization, H the Hub-ble parameter, n s the spectral index of the primordialspectrum and A s the overall amplitude of the primordialspectrum are the other 4 parameters of this model. Outof these parameters, the first four dictate the dynamicof the Universe and the other two represent the initialcondition through the primordial fluctuations given by P R ( k ) = A s (cid:16) kk ∗ (cid:17) n s − , where k ∗ is the pivot point. Hav-ing the form of the primordial fluctuations and the ex-pansion history of the Universe one can determine thegrowth of structure for this model on linear scales fol-lowing the linearised perturbation equation and also run N -body simulations to study the small scales and non-linear regime. Despite the simplicity of the model, mostastronomical observations are in great agreement withthe concordance model and so far there has not been any ∗ shafi[email protected] † [email protected] ‡ [email protected] strong observational evidence against it [e.g., 3–5]. In thispaper we test some important aspects of the concordancemodel of cosmology in light of the most recent cosmo-logical observations in a model-independent manner. Atthe background level, we derive the H r d parameter, testdark energy as the cosmological constant Λ, the FLRWmetric and the flatness of the Universe. At the pertur-bation level, we then use model independent reconstruc-tion of the expansion history from supernovae data tofit growth of structure data and put model independentconstraints on some key cosmological parameters, namelyΩ m , γ , and σ . In § II we describe the background expan-sion and our tests on Λ dark energy, FLRW metric andflatness of the spatial curvature. Analysis on the growthof structure and testing general theory of relativity arepresented in § III, and our conclusions are drawn in § IV.
II. BACKGROUND EXPANSION: TESTING Λ ,THE FLRW METRIC, AND THE CURVATURE At the background level, it is possible to test dark en-ergy as Λ, the FLRW metric, and the curvature of theUniverse. In a FLRW universe with a dark energy com-ponent of equation of state w ( z ), the luminosity distancecan be written for any curvature Ω k d L ( z ) = cH (1 + z ) D ( z ) , (1)where D ( z ) = 1 √− Ω k sin (cid:18)(cid:112) − Ω k (cid:90) z d xh ( x ) (cid:19) (2) a r X i v : . [ a s t r o - ph . C O ] O c t is the dimensionless comoving distance, and h ( z ) = (cid:18) H ( z ) H (cid:19) = Ω m (1 + z ) + Ω k (1 + z ) + (1 − Ω m − Ω k ) exp (cid:18) (cid:90) z w ( x )1 + x d x (cid:19) (3)is the expansion history. Having different observables ofthe cosmic distances and expansion history one can thenintroduce novel approaches to examine the FLRW metric,flatness of the Universe and Λ dark energy in a model-dependent [e.g., 6] or independent way [2, 7–12]. Notethat one can also test the metric and the curvature us-ing gravitational lensing [e.g. 13, 14] or cosmic parallaxes[15]. A. Model-independent reconstruction of theexpansion history from the Pantheon compilation
In order to reconstruct the D ( z ), D (cid:48) ( z ) and h ( z ) at anygiven redshift, we apply the iterative smoothing method[10, 16–18] to the the latest compilation of supernovaedistance modulus [Pantheon, 5]. Pantheon is the latestcompilation of 1048 SNIa, extending previous compila-tions with confirmed SNIa from the Pan-STARRS1 sur-vey.The method of smoothing is a fully model indepen-dent approach to reconstruct the D ( z ) relation directlyfrom the supernova data, without assuming any partic-ular model or a parametric form. The only parameterused in the smoothing method is the smoothing width∆, which is constrained only by the quality and quan-tity of the data. The smoothing method is an iterativeprocedure with each iteration providing a better fit tothe data. It has been discussed and shown that the fi-nal reconstructed results are independent of the assumedinitial guess [16–18]. In our analysis we start the smooth-ing procedure from various arbitrary choices of the ini-tial guess models and while their final results converge tothe same reconstruction, we select within the process, anon-exhaustive samples of the reconstructions that havea χ better than the best fit ΛCDM model. In [18] themethod of smoothing was modified to incorporate thedata uncertainties and hence making the approach error-sensitive. However, the formalism in [18] could take into account only the diagonal terms of the error matrix.While the quality of the data is improving continuouslyand non-diagonal terms of the covariance matrices canplay an important role in likelihood estimations, in thiswork we modify the smoothing method further by in-corporating the whole covariance matrix of the data into the smoothing procedure. While this improvementmight look like a minor modification, it is in fact a veryimportant step to make this model independent recon-struction approach complete and comprehensive to dealwith highly correlated data. In order to take into account the non-diagonal termsof the covariance matrix, we modified the method in thefollowing way. Starting with some initial guess ˆ µ , weiteratively calculate the reconstructed ˆ µ n +1 at iteration n + 1: ˆ µ n +1 ( z ) = ˆ µ n ( z ) + δµ T n · C − · W ( z ) T · C − · W ( z ) , (4)where the weight W and residual δµ n are defined as W i ( z ) = exp − ln (cid:16) z z i (cid:17) (5) δµ n | i = µ i − ˆ µ n ( z i ) , (6) T = (1 , . . . , , (7)and C SN is the covariance matrix of the data (in our case,Pantheon data). In case of uncorrelated data ( C ij = δ ij σ i )), we recover the formula introduced in [18] usedrecently in [10].The χ of the reconstruction ˆ µ n ( z ) is then defined as χ n = δµ T n · C − · δµ n , (8)and in this work we only consider reconstructions with χ < χ .The result of the smoothing procedure is thus H ˆ d L n ( z ) = 10 (ˆ µ n − / . Under the assumption of a flatUniverse, we can obtain h n ( z ) = 1 / (d D n ( z ) / d z ).We should clarify here that our selected reconstruc-tions of the expansion history from the iterative smooth-ing method are not posterior samples within a Bayesianframework. We in fact obtain a non-exhaustive sampleof plausible expansion histories, directly reconstructed bysupernova data and with no model assumption, which allgive a better χ to the Pantheon data than the best-fit ΛCDM model. This enables us to explore regionsof the physical space of the expansion history beyondthe flexibility of the concordance model (or other para-metric functional forms) that can fit the data reasonablywell. Note that the formalism given in this paper for themethod of smoothing is self-contained and has all neededinformation. Equations (4) to (8) contain the full formal-ism of the iterative smoothing method including the fullcovariance matrix of the data, which is now simply writ-ten in a matricial way (which is more compact). However,for more details and better understanding of the methodone can follow the given references. B. BAO measurements of cosmic distances andexpansion history
The radial mode of the BAO measures H ( z ) r d , whilethe transverse modes provide d A ( z ) /r d , where r d = c √ (cid:90) / (1+ z drag )0 d aa H ( a ) (cid:113) b r a (9)is the sound horizon at the drag epoch z drag . Wecombined the Baryon Oscillation Spectroscopic Survey(BOSS) DR12 consensus values [4] and the extended-BOSS (eBOSS) DR14Q measurements [19]. We notethat both BOSS DR12 and eBOSS DR14Q pro-vide H ( z ) r d /r d,fid and d A ( z ) r d,fid /r d with r d,fid =147 .
78 Mpc. We also include the Dark Energy SurveyDR1 (DES DR1) measurement of d A /r d at z = 0 .
81 [20].We use these BAO data along with our reconstructionsof the expansion history from supernova data as two in-dependent sets of observations to test some key aspectsof the concordance model.
C. Testing Λ Dark Energy
The solid black lines in Fig. 1 show the different recon-structed D ( z ), h ( z ) = 1 / D (cid:48) ( z ) and Om ( z ) from Pantheonsupernovae compilation where Om ( z ) is defined as [8]: Om ( z ) = h − z ) − H ( z ) r d and d A ( z ) /r d , to have a good sense of comparison within thecontext of the concordance model, we normalize themby H r d from Planck 2015 (TTTEEE+LowP+Lensing)best fit ΛCDM model, and show on the top panel D ( z ) = (1 + z ) H r d d A ( z ) / ( cr d ), in the middle panel h ( z ) = H ( z ) r d /H r d and the corresponding Om ( z ) onthe bottom panel. The magenta solid line shows the cor-responding D ( z ), h ( z ) and Om ( z ) for the best-fit Planck2015 Flat-ΛCDM model.While the reconstructed expansion history h ( z ) fromSNIa are fully consistent with the BAO data points atlow redshifts ( z ≤ . z ≥ .
5) where the reconstructed ex-pansion histories from the BAO data suggest lower h ( z )with respect to the best fit ΛCDM model from Planck.While the errorbars are still quite large, the BAO dataseem to follow the same trend in suggesting lower valuesof h ( z ) (with respect to the best fit ΛCDM model fromPlanck) at high redshifts. For illustration purpose we alsoshow the measurement of h ( z = 2 .
33) from the Lyman- α forest [21] which seems to agree with other BAO datapoints suggesting lower h ( z ) with respect to Planck bestfit ΛCDM model, although we did not include this datapoint in our analysis since the supernovae data do not FIG. 1: BAO data points normalized by H r d from [3]best fit ΛCDM model. The solid lines are thereconstructed expansion histories from the Pantheondata which are fully model independent, and the purpleline is the prediction from [3] for the best-fitconcordance ΛCDM model. They are color-coded bytheir ∆ χ with respect to the best-fit ΛCDM model,with earlier iterations having less negative δχ (yellow),and later iterations more negative ∆ χ (dark blue).reach such a high redshift. This data point is consistentwith the previous result from SDSS III [22]. This ten-sion is also visible clearly looking at the Om diagnosticin bottom plot of Fig. 1, which is also consistent with thefinding of [1]. If dark energy is a cosmological constant(and if the Universe is flat), the Om diagnostic should beconstant in redshift. Therefore, having different valuesfrom different observations suggests some tension amongthe data within the framework of the concordance model.Meanwhile, the comoving distances D ( z ) from BAOand SNIa are fully consistent together and with the best-fit Planck cosmology. Combining these results of the co-moving distances and expansion histories may show someinconsistency with flatness as we will see later in thiswork. D. Estimating H r d Ref. [10] estimated H r d in a model-independent wayby combining BAO measurements and reconstructions ofthe expansion history from supernovae. H r d is an im-portant parameter combining physics of the early (soundhorizon at the drag epoch) and late Universe (expansionrate). For each reconstruction n , we can calculate H r d in two different ways H r d | d A ,n = c z D n ( z ) r d d A ( z ) (11a) H r d | H,n = H ( z ) r d h n ( z ) , (11b)and their associated errors σ H r d | d A ,n = c z D n ( z ) σ d A /r d ( z )( d A ( z ) /r d ) (12a) σ H r d | H,n = σ Hr d ( z ) h n ( z ) , (12b)where, assuming a flat-FLRW universe, h ( z ) = 1 / D (cid:48) ( z ).Fig. 2 shows our estimation of H r d at the differ-ent BAO data points for the two estimations. In greenis shown the ΛCDM value from Planck 2015 [3]. Wecan then define two error-bars. The first one is theerror due to the supernova. At fixed redshift, we de-fine (cid:104) H r d (cid:105) X as the median over all reconstructions formethod X ∈ { d A , H } . We can then define the upperand lower limit as the minimal and maximal values of H r d | X,n . This error-bar is shown as a dashed line inFig. 2. The second error is due to the uncertainty onthe BAO (equations (12a) and (12b)), and is the uncer-tainty of the central value for a given reconstruction n .For each reconstruction n and method X , we have anerror σ H r d | X,n . They are of the same order for eachreconstruction, so we define the final BAO error as themaximum value over all reconstructions. This error-baris shown as a solid error-bar in Fig. 2.For the first method (in orange), the measurementsof H r d from combination of supernova and SDSS BAOdata are fully consistent with Planck. The DES datapoint, also using the transverse BAO mode, is an inde-pendent confirmation at intermediate redshift. However,for the second method, while at low redshift, the mea-surements are consistent with Planck, the eBOSS datapoints are systematically lower than the Planck best-fitat z ≥ . D ( z ) which do not use derivative.The second method however, uses the line-of-sightmode of the BAO, together with h ( z ) from supernovaedata which is a derivative. Since the Pantheon data be-come scarce at z ≥
1, the estimation of h ( z ) becomes lessprecise at this range having large error-bars. Combina-tion of these two results to large uncertainties for H r d from the second method. On the other hand, it can beseen from Fig. 1 that while h ( z ) from SNIa are higherthan the best-fit Planck ΛCDM model, h ( z ) from the z H r d ( k m / s ) Alam et al. (2016)Zhao et al. (2018)DES Collaboration (2017)Planck 2015 H ( z ) r d / h ( z ) c /(1 + z ) ( z ) r d / d A ( z ) Error BAOError SN
FIG. 2: Model-independent measurement of H r d estimated at the different BAO data points. The dottederror-bars show the range of possible central values fromdifferent reconstructions (SN error), while the soliderror bars show the uncertainty on the central value(BAO error).BAO (scaled with best fit Planck ΛCDM model) are actu-ally lower. This explains the lower values of H ( z ) r d /h ( z )at the eBOSS redshifts with respect to the other mea-surements.We can then estimate, for each reconstruction n andmethod X ∈ { d A , H } , the weighted average (cid:104) H r d (cid:105) X,n = T · C − n · H r d | X,n T · C − n · , (13)where H r d | X,n is a vector constituted of estimationsof H r d at different redshifts for iteration n , and C n isthe associated covariance matrix (due to the correlationin the BAO data). We report our results in Table I.The Planck 2015 value of H r d for the ΛCDM model is(9944 . ± .
4) km s − Mpc − . We should note an im-portant interpretation of this result. While all our re-constructions of the expansion history from supernovaedata have better χ with respect to the best fit ΛCDMmodel, our large uncertainties on H r d indicates thattight constraints on this quantity from model dependentapproaches (such as assuming ΛCDM model) have limi-tations in expressing the reality of the universe and esti-mating its key parameters. E. Test of the FLRW metric and the curvature [10] reformulated the O k diagnostic [7] by introducingthe Θ diagnostic so that it now only depends on the BAOTABLE I: Weighted average of H r d from the H and d A methods. Method H r d Error SN Error BAO (cid:104) H r d (cid:105) d A +33 . − . ± . − ) (cid:104) H r d (cid:105) H +875 . − . ± . − ) Flat FLRWError BAOError SN Alam et al. (2016)Zhao et al. (2018) z k FIG. 3: FLRW and curvature test: Θ( z ) (top) and O k ( z ) (bottom). The dotted error-bars show the rangeof possible central values from different reconstructions(SN error), while the solid error bars show theuncertainty on the central value (BAO error). For aflat-FLRW Universe, Θ( z ) ≡ O k ( z ) ≡ Ω k .and supernovae observables: O k ( z ) = Θ ( z ) − D ( z ) (14a)Θ( z ) = h ( z ) D (cid:48) ( z ) = 1 + zc H ( z ) r d d A ( z ) r d D (cid:48) ( z ) D ( z ) . (14b)For a FLRW Universe, O k ( z ) ≡ Ω k , and in case of flat-ness, O k ( z ) ≡ z ) ≡
1. We can then calculate foreach reconstruction n the associated O k,n ( z ) and Θ n ( z ).We calculated the median of O k and Θ over all recon-structions, and defined the SN error as the minimal andmaximal values, and the BAO error as the maximal er-ror over all reconstructions. Fig. 3 shows Θ( z ) (top) and O k ( z ) (bottom). Both are consistent with a flat FLRWmetric up to z (cid:39) . z ≥ .
5, which results in into poor constraintson h ( z ), the BAO seem to show some internal tensions.While d A ( z ) /r d are consistent with the Planck best-fit, H ( z ) r d are lower than expected. However, the Θ and O k statistics assume a FLRW metric, where d A and H are related to each other. Thus, discrepancy between d A and H combined with the higher h values at high-redshift( z ≥
1) yields lower values for Θ and O k . We should alsonote that in the case of supernova data, the Malmquistbias (if not treated carefully) can pull down the D ( z ) rela-tion at high redshifts. This might explain the large swingupward of h ( z ) and Om ( z ) (with respect to the best-fitΛCDM case) that we can see in Fig. 1, and consequentlythe apparent deviation from flatness observed in Θ and O k . While it is certainly important to study further thiseffect in the case of the Pantheon data, it is beyond thescope of this paper. III. GROWTH OF STRUCTURE VERSUSEXPANSION: TESTING GR
At the perturbation level, the cosmological growth ofstructure can also serve as a test of gravity [11, 23–34]. Inthe linear regime, the growth of structure in GR follows¨ δ + 2 H ˙ δ − πG ¯ ρδ = 0 , (15)where δ = ρ/ ¯ ρ − ρ . The growth rate f ( a ) = dln δ dln a (16)can be approximated for a wide range of cosmologies by[35–37] f ( z ) = Ω γ m ( z ) , (17)where Ω m ( z ) = Ω m (1 + z ) h ( z ) . (18)In general relativity (GR), γ (cid:39) . f σ is thus a power-ful probe of gravity. Observationally, redshift-space dis-tortion enables to measure the combination [e.g. 24] f σ ( z ) (cid:39) σ Ω γ m ( z ) exp (cid:18) − (cid:90) z Ω γ m ( x ) d x x (cid:19) , (19)where σ = σ ( z = 0) is the rms fluctuation in 8 h − Mpcspheres. Following this formalism, having model inde-pendent reconstructions of the expansion history and f σ ( z ) data, one can obtain constraints on Ω m , γ , and σ [38]. Note that we consider the estimated f σ datafrom BAO surveys as an independent and uncorrelatedmeasurements with respect to the supernova data thatwe used to reconstruct the expansion history.Note, however, that one should keep in mind thatEq. (17) is an approximate fit only. In particular, γ maynot be exactly constant for quintessence—dark energymodelled by a scalar field with some potential minimallycoupled to gravity [39]. Still both for ΛCDM and forquintessence-CDM this fit is good since d γ d z is small as faras Ω m is not too small, see also [40]. For modified gravitytheories like f ( R ) gravity, the situation can be different[41, 42]. A. Cosmological constraints on Ω m , γ, σ Following [38], we combined the Pantheon compilationwith the latest measurements of f σ : 2dFGRS [24], Wig-gleZ [43], 6dFGRS [44], VIPERS [45], the SDSS Maingalaxy sample [46], 2MTF [47], BOSS DR12 [48], Fast-Sound [49], and eBOSS DR14Q [19]. In this section, weassume a flat Universe, therefore h ( z ) = 1 D (cid:48) ( z ) . (20)It is worth noting that these measurements, comingfrom different surveys, were obtained assuming differentfiducial cosmologies. Therefore, we correct for the fidu-cial cosmology [32, 50]. The growth χ for the n th recon-struction h n ( z ) and parameters p = (Ω m , γ, σ ) is thusgiven by χ n,fσ ( p ) = δf σ n · C − fσ · δf σ n , (21)where C fσ is the growth covariance matrix, and the i thcomponent of the residual vector δf σ n is δf σ n | i = h n ( z i ) D n ( z i )(1 + z i ) H fid ( z i ) d A,fid ( z i ) (cid:100) f σ n ( z i | p , h n ) − f σ | i . (22)The total χ for reconstruction n and parameter p is then χ n, tot ( p ) = χ n,fσ ( p ) + χ n, SN , (23)where (cid:100) f σ ( z i | p , h n ) is the model corresponding to theexpansion history h n and parameters p and fid standsfor the fiducial cosmology used by the survey to estimatethe data point.The red contours in the ( σ , Ω m ) plane in Fig. 4 showthe 1 σ and 2 σ regions of the parameter space in the flatΛCDM case, that is, flat-ΛCDM expansion history and γ = 0 .
55. The blue contours show the allowed parameterspace in the model-independent case. Namely, for anypoint in the blue contours, one can find at least one re-construction h ( z ) which, combined to the corresponding(Ω m , γ, σ ), gives a better fit to the data than the best-fit ΛCDM. In the ( σ , Ω m ) plane, the model-independentcase is fully consistent with the ΛCDM case. Moreover,the flexibility of the model-independent approach allowsa larger area of the parameter space to be consistent tothe data. For instance, for larger values of σ and lowervalues of Ω m , one can find reconstructed expansion histo-ries that give a better total fit to the data (SNIa+growth)with respect to the best fit ΛCDM model. For the model-independent case, γ is fully consistent with 0.55, as ex-pected from GR. Moreover, lower value of γ , combinedwith lower value of Ω m and larger σ , can also providegood fit to the data.We then fix γ = 0 .
55, as we did for the ΛCDM case,and show in Fig. 4 the corresponding confidence contours m FIG. 4: Model independent cosmological constraints on(Ω m , γ, σ ) from growth and expansion data. The redcontours are the 1 σ and 2 σ confidence levels for theΛCDM case. The blue contours are associated to thecombination of the parameters and reconstructions ofthe expansion history that yield a better χ withrespect to the best-fit ΛCDM model. The dark-blueregion satisfy positive dark energy density condition asexpressed in equation (24). The green contours showthe model-independent case where we fixed γ = 0 . γ to vary, the region with low Ω m andhigh σ is now forbidden.Finally, following [38], we focus on combinations of h ( z ) and Ω m that respect the positive dark energy con-ditionΩ de ( z ) = h ( z ) − Ω m (1 + z ) ≥ ∀ z. (24)We show this region in dark-blue (free γ ) and dark-green (fixed γ ) in Fig. 4. Imposing equation (24)effectively forbids large values of Ω m , and dramati-cally reduces the allowed parameter space of the model-independent case. The allowed region of the parameterspace is then fully consistent with the model-dependentcase, as in [38]. This is a strong support from the data forcombination of ΛCDM and GR. Comparing our resultshere using most recent supernovae (Pantheon compila-tion) and BAO data (from eBOSS DR14) with what wasreported in [38] we can notice substantial improvementon the constraints on these three key cosmological pa-rameters. Based on our analysis we can now put strongmodel-independent upper bound limits on Ω m < .
42 and γ < .
58 and a lower bound limit on σ > .
70. Theseare in fact model independent constraints on these keycosmological parameters.
IV. SUMMARY AND CONCLUSIONS
We used the Pantheon supernovae compilation to re-construct the expansion history in a model-independentway, using an improved version of the iterative smooth-ing method [10, 16–18], which we modified to take intoaccount the non-diagonal terms of the full covariancematrix. We then combined the reconstructed expansionhistories to measurements of H ( z ) r d and d A ( z ) /r d fromBOSS DR12 and eBOSS DR14Q to model-independentlymeasure H r d and test the FLRW metric. Our measure-ments of H r d are consistent with the Planck 2015 val-ues, while the metric test is consistent with a Flat-FLRWmetric. However, for the eBOSS DR14Q data points,while d A ( z ) /r d is consistent with the prediction from thePlanck best-fit ΛCDM cosmology, the H ( z ) r d measure-ments are slightly but systematically lower. This yieldssome hints for a departure from flat-FLRW (Fig. 3) andsupports previous findings of [1] & [2].We then fit the growth data from redshift space dis-tortion, mainly from SDSS survey using the model-independent reconstructions of the expansion history,and put model-independent constraints on Ω m < . γ < .
58 and σ > .
70. Our measurements are fullyconsistent with the ΛCDM model with GR ( γ ≈ . ACKNOWLEDGEMENTS
We thank Gongbo Zhao and Yuting Wang for usefuldiscussions, and Dan Scolnic for providing the Pantheondata. This work benefited from the SupercomputingCenter/Korea Institute of Science and Technology Infor-mation with supercomputing resources including techni-cal support (KSC-2016-C2-0035 and KSC-2017-C2-0021)and the high performance computing clusters Polaris andSeondeok at the Korea Astronomy and Space ScienceInstitute. A.S. would like to acknowledge the supportof the National Research Foundation of Korea (NRF-2016R1C1B2016478). A. A. S. was partly supported bythe program P-28 “Cosmos” of the Russian Academy ofSciences (the project number 0033-2018-0013 of the Fed-eral Agency of Scientific Organizations of Russia). [1] V. Sahni, A. Shafieloo, and A. A. Starobinsky, ApJ ,L40 (2014), 1406.2209.[2] G.-B. Zhao, M. Raveri, L. Pogosian, Y. Wang, R. G.Crittenden, W. J. Handley, W. J. Percival, F. Beutler,J. Brinkmann, C.-H. Chuang, et al., Nature Astronomy , 627 (2017), 1701.08165.[3] Planck Collaboration XIII, A&A , A13 (2016),1502.01589.[4] S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J. A.Blazek, A. S. Bolton, J. R. Brownstein, A. Burden, C.-H.Chuang, et al., MNRAS , 2617 (2017), 1607.03155.[5] D. M. Scolnic, D. O. Jones, A. Rest, Y. C. Pan,R. Chornock, R. J. Foley, M. E. Huber, R. Kessler,G. Narayan, A. G. Riess, et al., ApJ , 101 (2018),1710.00845.[6] O. Farooq and B. Ratra, Physics Letters B , 1 (2013),1212.4264.[7] C. Clarkson, B. Bassett, and T. H.-C. Lu, Physical Re-view Letters , 011301 (2008), 0712.3457.[8] V. Sahni, A. Shafieloo, and A. A. Starobinsky,Phys. Rev. D , 103502 (2008), 0807.3548.[9] D. Sapone, E. Majerotto, and S. Nesseris, Phys. Rev. D , 023012 (2014), 1402.2236.[10] B. L’Huillier and A. Shafieloo, J. Cosmology Astropart.Phys. , 015 (2017), 1606.06832.[11] V. Marra and D. Sapone, Phys. Rev. D , 083510(2018), 1712.09676.[12] F. Montanari and S. R¨as¨anen, J. Cosmology Astropart.Phys. , 032 (2017), 1709.06022.[13] S. R¨as¨anen, K. Bolejko, and A. Finoguenov, Physical Re-view Letters , 101301 (2015), 1412.4976. [14] M. Denissenya, E. V. Linder, and A. Shafieloo, J. Cos-mology Astropart. Phys. , 041 (2018), 1802.04816.[15] S. R¨as¨anen, J. Cosmology Astropart. Phys. , 035 (2014),1312.5738.[16] A. Shafieloo, U. Alam, V. Sahni, and A. A. Starobinsky,MNRAS , 1081 (2006), astro-ph/0505329.[17] A. Shafieloo, MNRAS , 1573 (2007), astro-ph/0703034.[18] A. Shafieloo and C. Clarkson, Phys. Rev. D , 083537(2010), 0911.4858.[19] G.-B. Zhao, Y. Wang, S. Saito, H. Gil-Mar´ın, W. J. Perci-val, D. Wang, C.-H. Chuang, R. Ruggeri, E.-M. Mueller,F. Zhu, et al., ArXiv e-prints (2018), 1801.03043.[20] The Dark Energy Survey Collaboration, T. M. C. Ab-bott, F. B. Abdalla, A. Alarcon, S. Allam, F. Andrade-Oliveira, J. Annis, S. Avila, M. Banerji, N. Banik, et al.,ArXiv e-prints (2017), 1712.06209.[21] J. E. Bautista, N. G. Busca, J. Guy, J. Rich,M. Blomqvist, H. du Mas des Bourboux, M. M. Pieri,A. Font-Ribera, S. Bailey, T. Delubac, et al., A&A ,A12 (2017), 1702.00176.[22] T. Delubac, J. E. Bautista, N. G. Busca, J. Rich,D. Kirkby, S. Bailey, A. Font-Ribera, A. Slosar, K.-G. Lee, M. M. Pieri, et al., A&A , A59 (2015),1404.1801.[23] S. Nesseris and L. Perivolaropoulos, Phys. Rev. D ,023504 (2008), 0710.1092.[24] Y.-S. Song and W. J. Percival, J. Cosmology Astropart.Phys. , 004 (2009), 0807.0810.[25] V. Acquaviva and E. Gawiser, Phys. Rev. D , 082001(2010), 1008.3392. [26] S. Basilakos, International Journal of Modern Physics D , 1250064 (2012), 1202.1637.[27] A. Shafieloo, A. G. Kim, and E. V. Linder, Phys. Rev. D , 023520 (2013), 1211.6128.[28] A. Pavlov, O. Farooq, and B. Ratra, Phys. Rev. D ,023006 (2014), 1312.5285.[29] A. G´omez-Valent, J. Sol`a, and S. Basilakos, J. CosmologyAstropart. Phys. , 004 (2015), 1409.7048.[30] E. J. Ruiz and D. Huterer, Phys. Rev. D , 063009(2015), 1410.5832.[31] E.-M. Mueller, W. Percival, E. Linder, S. Alam, G.-B.Zhao, A. G. S´anchez, F. Beutler, and J. Brinkmann, MN-RAS , 2122 (2018), 1612.00812.[32] S. Nesseris, G. Pantazis, and L. Perivolaropoulos,Phys. Rev. D , 023542 (2017), 1703.10538.[33] J. Sol`a, A. G´omez-Valent, and J. de Cruz P´erez, ModernPhysics Letters A , 1750054-144 (2017), 1610.08965.[34] L. Kazantzidis and L. Perivolaropoulos, ArXiv e-prints(2018), 1803.01337.[35] O. Lahav, P. B. Lilje, J. R. Primack, and M. J. Rees,MNRAS , 128 (1991).[36] L. Wang and P. J. Steinhardt, ApJ , 483 (1998),astro-ph/9804015.[37] E. V. Linder, Phys. Rev. D , 043529 (2005), astro-ph/0507263.[38] B. L’Huillier, A. Shafieloo, and H. Kim, MNRAS ,3263 (2018), 1712.04865.[39] D. Polarski, A. A. Starobinsky, and H. Giacomini, J. Cos-mology Astropart. Phys. , 037 (2016), 1610.00363.[40] D. Polarski and R. Gannouji, Physics Letters B , 439(2008), 0710.1510.[41] R. Gannouji, B. Moraes, and D. Polarski, J. CosmologyAstropart. Phys. , 034 (2009), 0809.3374.[42] H. Motohashi, A. A. Starobinsky, and J. Yokoyama,Progress of Theoretical Physics , 887 (2010),1002.1141.[43] C. Blake, S. Brough, M. Colless, C. Contreras, W. Couch,S. Croom, T. Davis, M. J. Drinkwater, K. Forster, D. Gilbank, et al., MNRAS , 2876 (2011), 1104.2948.[44] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, G. B. Poole, L. Campbell, Q. Parker, W. Saun-ders, and F. Watson, MNRAS , 3430 (2012),1204.4725.[45] S. de la Torre and J. A. Peacock, MNRAS , 743(2013), 1212.3615.[46] C. Howlett, A. J. Ross, L. Samushia, W. J. Percival, andM. Manera, MNRAS , 848 (2015), 1409.3238.[47] C. Howlett, L. Staveley-Smith, P. J. Elahi, T. Hong, T. H.Jarrett, D. H. Jones, B. S. Koribalski, L. M. Macri, K. L.Masters, and C. M. Springob, MNRAS , 3135 (2017),1706.05130.[48] H. Gil-Mar´ın, W. J. Percival, L. Verde, J. R. Brown-stein, C.-H. Chuang, F.-S. Kitaura, S. A. Rodr´ıguez-Torres, and M. D. Olmstead, MNRAS , 1757 (2017),1606.00439.[49] T. Okumura, C. Hikage, T. Totani, M. Tonegawa,H. Okada, K. Glazebrook, C. Blake, P. G. Ferreira,S. More, A. Taruya, et al., PASJ , 38 (2016),1511.08083.[50] S. Alam, S. Ho, and A. Silvestri, MNRAS456