Observational constraints on the non-flat $\Lambda CDM$ model and a null test using the transition redshift

A natural extension of the standard cosmological model are models that include curvature as a free parameter. In this work we study in detail the observational constraints on the non-flat $\Lambda CDM$ model using the two main geometric tests: SNIa and Hubble parameter measurements. In general we show that the observational constraints on the parameters of the $\Lambda CDM$ model strongly depend on the curvature parameter. In particular, we study the constraints on the transition redshift ($z_{t}$) of a universe dominated by matter for a universe dominated by the cosmological constant. Using this observable we construct a new null test defining $\zeta = z_{t, flat}-z_{t, non flat}$. This test depends only on the data of the Hubble parameter, the Hubble constant and the matter density parameter. However, it does not depend on derivative of an observable as generally many tests in the literature. To reconstruct this test, we use the Gaussian process method. When we use the best-fit parameters values of $PLANCK/2018$, we find no evidence of a disagreement between the data and the standard model (flat $\Lambda CDM$), but if we use the value $H_{0}$ from $RIESS/2018$ we found a disagreement with respect at the standard model. However, it is important to note that the Hubble parameter data has large errors for a solid statistical analysis.


II. THE ΛCDM MODEL IN THE BACKGROUND
Considering the cosmological principle, the FLRW metric can be written as [24,25,26,27], ds 2 = −dt 2 + a(t) 2 dr 2 1 − kr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 , where a(t) is the scale factor and k is the spatial curvature which can be k = +1 for a closed universe, k = 0 for a at Universe and k = −1 for an open universe. In addition, if we consider Einstein's equations and a tensor energy-momentum of perfect uid, then we can derive the fundamental equation of cosmology [28], This equation can be rewritten using the redshift as: where we use the de nitions: and Ω k0 = −k a 2 H 2 . Additionally, we have the restriction: To calculate the transition redshift, z t , of a decelerated to accelerated universe we use the de nition of the deceleration parameter, Thus, using the de nition of H(z) and the condition for the transition redshift q(z t ) = 0. We can determine that [29], where we can observe that the transition redshift for a non-at universe ΛCDM is a analytical function of the parameters of relative densities [29]. Using equations (5) and (7) we can explicitly rewrite the hubble parameter using the variables, (Ω k0 e z t )as, This expression of the Hubble parameter is important because the statistics of the χ 2 constructed will explicitly depend on these parameters avoiding a extra propagation of errors. The importance of the curvature parameter in determining the transition redshift can best be observed by means of Figure 1. where we show the function z t (Ω m , Ω k0 ). This function is a well-behaved three-dimensional surface, except for extreme value of the curvature and very low values for the matter. But these regions are excluded by observational data. In the gure the intersection of the planes corresponds to the case where the curvature is zero. A quick inspection allows us to observe that if we consider curvature, then there are di erent ways to accommodate the measures on the surface of z t . In particular if we consider that the observations various determine z t preferably in the range (0.5 − 1.00), then the inclusion of the curvature allows that the value of z t can easily be accommodated outside of the line for a at universe. In this work we study the constraints associated with this degeneracy employing SNIa and Hubble parameter measurements.

III. NULL TEST FOR ΛCDM
We can de ne a new null test to distinguish between at and non-at ΛCDM models using the redshift transition To do this we put in evidence the z t of the expression for the Hubble parameter given by the equation (10) obtaining, In an analogous way we can determine for at transition redshift, so the null test can be formulated as: where we can see that if ζ = 0, then the at model is preferred, otherwise the model with curvature is preferred. In an explicit way we can write this null test as, Interestingly, our test includes the reconstructed data of the Hubble parameter and does not include derivatives of data such as other tests, for example [30]. Data derivatives in general are di cult to obtain and spread the error remarkably. However, our test explicitly includes the Hubble constant, H 0 , and the matter density parameter today Ω m0 .
To reconstruct the observable, H(z), we use only data from the Hubble parameter and as a statistical method we use the non-parametric method of Gaussian processes. This method is suitable for this case, since it does not assume a speci c model to reconstruct the function H(z). Once the H(z) function is obtained, the null test can be reconstructed. As a rst approximation to determine the errors of the ζ function we can use the theory of error propagation, To perform the reconstruction of Gaussian processes we use the popular public package GaP P , which has been applied to a large number of cosmological studies. For package details you can see the references [31]. For a recent use of this package to see the reference [32].

IV. OBSERVATIONAL CONSTRAINTS
For determine observational constraints on the non-at ΛCDM model it is essential to de ne the comoving distance as, To determine the luminosity distance including curvature is necessary to distinguish three cases, for this we de ne the transversal comoving distance, r t , 1 using the above de nitions we can determine the luminosity distance as,

A. Supernovae Ia
In this study we use the data from Supernovas Ia called "Pantheon" sample [33] which is the largest combined sample of SNIa and consists of 1048 data with redshifts in the range 0.01 < z < 2.3. It is a collection of SNe Ia discovered by the Pan-STARRS1 (PS1) Medium Deep Survey and SNe Ia from Low-z, SDSS, SNLS and HST surveys. This supernova Ia compilation uses The SALT 2 program to transform light curves into distances using a modi ed version of the Tripp formula [34], where µ is the distance modulus, ∆ M is a distance correction based on the host-galaxy mass of the SNIa and ∆ B is the distance correction based on predicted bias from simulations. Also α is the coe cient of the relation between luminosity and stretch, β is the coe cient of the relation between luminosity and color and M is the absolute B-band magnitude of a ducial SNIa with x 1 = 0 and c = 0. Also c is the color and x 1 is the light-curve shape parameter and m B is the log of the overall ux normalization. An uncertainty matrix C is de ned such that, where ∆ µ = µ obs − µ model and µ model is a vector of distance modulus from a given cosmological model and µ obs is a vector of observational distance modulus. The µ = m − M , where M is the absolute magnitude and m is the apparent magnitude, which is is given by where D L = H0 c d L andM = M + 5 log 1 0( c/H0 1M pc ) is an nuisance parameter, which depends on the Hubble constant H 0 and the absolute magnitude M . To minimize with respect to the nuisance parameter we follow a process similar at the references [33,35]. Therefore the χ 2M ,marg is, where, where ∆ m = m obs − m model and I is the identity matrix.

B. Hubble Parameter Measurements
There are two e cient and widely used forms to obtain Hubble parameters measurements: • Cosmic Chronometers(CC): This method is based on the expression of the di erential age of the universe as a function of redshift, This method was proposed by directly measuring the amount dz/dt and, consequently, the Hubble parameter. The most used data to measure this amount have been passively evolving galaxies with high-resolution spectroscopic data along with synthetic catalogs to limit the age of the oldest stars in the galaxy. A complete description of this methodology can be reviewed for the SDSS in the reference [37].
• The Radial BAO Size Method:This method is based on measurements of the scale of BAO. This method is more accurate with respect to CC. This accuracy is understandable because BAO mainly depends on a spatial measurement compared to the rst method where a time measure is required which increases systematic errors. However, this method of BAO requires assuming a prior in the radius of the sound horizon,r(z), so that This method depends on the ducial model, usually the model associated with CM B, is the ΛCDM model.
In the literature there are di erent compilations of samples of the Hubble parameters data, we use the sample presented by [38] what includes data of CC and BAO. These data cover a range 0.2 < z < 2.35 in redshift. We can construct the statistics χ 2 H as, where H obs,i are the observational data and H model are the theoretical values determined by equation (3) and the σ obs,i are the errors of the observational data.

C. Combining Data
We combine the data by adding the χ 2 of each dataset, so we get We can construct the probability contours through the marginalization process, thus, for example, for the case of (Ω m0 , Ω Λ0 ) we integrate on the likelihood with respect For other sets of parameters we proceed analogously.

V. RESULTS AND DISCUSS
We investigate the observational constraints on the di erent combinations of cosmological parameters. In Figure 2, we show the observational constraints using SNIa and the Hubble parameter data. In all cases we use a marginalization over the Hubble parameter in the range 50 < H 0 < 80. The gure on the right side show the constraints due to the sum of the data. It is worth mentioning the lower right gure, which shows the dependence between the curvature parameter and the transition redshift. But it is also evident that the at model is within 1σ. The gure 3, on the left side, we show the strong degenescescence between the matter density parameter and the parameter of curvature for SNIa data, in the middle gure we show the con dence contours for the matter versus the transition redshift. This result updates the calculation shown in the reference [39]. The discontinuous curve represents the at universe.
In Figure 4 we show the dependence of the parameters with respect to the Hubble constant, H 0 . For all cases we use a marginalization interval on the matter density parameter of 0.07 < Ω m0 < 0.6. On the right side we show the constraints of the sum SN Ia + H. We can see that the variation range of, z t , corresponds to 0.4 < z t < 1.00 and of the curvature parameter for −0.2 < Ω k0 < 0.2. In Figure 5, we show the PDFs for all the parameters studied. The best-tting values are shown in Table I with 1σ. In Figure 6, we show the three-dimensional constraints for the parameter space (Ω k0 , z t , H 0 ). In this gure we shown that the geometric tests can restrict the parameters values on a closed surface for the non-at ΛCDM model, but the at model ΛCDM cannot be ruled out.
In gure 7. we shown the result of the null test using the Gaussian process method. In the gure on the left side we use the best-tting values of the P LAN CK/2018 [6], H 0 = 67.4 ± 0.5 and Ω m0 = 0.315 ± 0.007. We can see that the at ΛCDM

VI. CONCLUSION
In the present work we have investigated the observational constraints on the non-at ΛCDM model using as observational data the SNIa and Hubble data. We have emphasized the study on the transition redshift, which can be used to construct a null test. This test is sensitive to the values of H(z), H 0 and Ω m0 , but does not include derivatives of cosmological observables, which prevents excessive propagation of errors.
In general, we have shown a strong dependence between the constraints of the curvature parameter and all other parameters. Our results are quite general, because we do not use xed values of the parameters, but we have used a marginalization process integrating in a large interval to allow considerable changes of the parameters. In all case the at ΛCDM model cannot be excluded. A subsequent study may include other observables such as BAO, QSO and structure growth data to constraints the transition redshift with curvature, i.e., the (z t , Ω k0 ) plane.
On the other hand, the null test is quite sensitive to the values of the parameters, for the best-tting of Planck/2018 the at ΛCDM model is preferred, but if use the Hubble constant value local of Riess et al. 2018 our null test excludes the at model with 2σ. If we consider values less for the matter content, for example, Ω m0 = 0.28, we can alleviate the rejection of the at model at 1σ. To reconstruct the null test we only use data from the Hubble parameter measurements and the non-parametric method of Gaussian processes. We do not include SN Ia, since it would imply reconstruct derivatives and would spread the error excessively. Our null test may be interesting to study other cosmological models, such as a quintessence model or interaction between dark energy and dark matter models.  In all case we marginalized 50 < H0 < 80). The blue line represents the universe with at curvature. In the gure on the left we show constraints on the Hubble constant using data from the Hubble parameter and on the left side we add data from Supernovas Ia. In all cases we marginalize the matter density parameter in the interval 0.07 < Ωm0 < 0.6.