Observational constraints on Myrzakulov gravity
Fotios K. Anagnostopoulos, Spyros Basilakos, Emmanuel N. Saridakis
OObservational constraints on Myrzakulov gravity
Fotios K. Anagnostopoulos a Spyros Basilakos b , c Emmanuel N. Saridakis c , d , e a Department of Physics, National & Kapodistrian University of Athens, Zografou Campus GR 157 73,Athens, Greece b Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efesiou 4, 11527,Athens, Greece c National Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece d CAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, University ofScience and Technology of China, Hefei, Anhui 230026, P.R. China e School of Astronomy, School of Physical Sciences, University of Science and Technology of China, Hefei230026, P.R. China
E-mail: [email protected] , [email protected] A BSTRACT : We use data from Supernovae (SNIa) Scolnic sample, from Baryonic Acoustic Os-cillations (BAO), and from cosmic chronometers measurements of the Hubble parameter (CC),alongside arguments from Big Bang Nucleosynthesis (BBN), in order to extract constraints onMyrzakulov F ( R , T ) gravity. This is a connection-based theory belonging to the Riemann-Cartansubclass, that uses a specific but non-special connection, which then leads to extra degrees of free-dom. Our analysis shows that both considered models lead to ∼ σ compatibility in all cases. Forthe involved dimensionless parameter we find that it is constrained to an interval around zero, how-ever the corresponding contours are slightly shifted towards positive values. Furthermore, applyingthe AIC, BIC and the combined DIC criteria, we deduce that both models present a very efficientfitting behavior, and are statistically equivalent with Λ CDM cosmology, despite the fact that Model2 does not contain the latter as a limit. Finally, we use the obtained parameter region and we re-construct the corresponding dark-energy equation-of-state parameter as a function of redshift. Aswe show, Model 1 is very close to Λ CDM scenario, while Model 2 resembles it at low redshifts,however at earlier times deviations are allowed.K
EYWORDS : Modified gravity, Observational Constraints, Torsional Gravity a r X i v : . [ g r- q c ] D ec ontents According to the concordance cosmological model the universe experienced two epochs of accel-erated expansion, one at early and one at late times. Although the latter can be explained by thepresence of a cosmological constant, the related theoretical problem, the possibility of a dynamicalbehavior, and especially the inability of the cosmological constant to describe the early acceleratedphase, led to the incorporation of some form of modification. As a first possibility one can main-tain general relativity as the underlying theory and modify the matter content of the universe byintroducing extra fields, such as the inflaton at early times [1, 2] and/or the dark energy sector atlate times [3, 4]. As a second possibility one modifies the gravitational sector itself, constructinga theory that possesses general relativity as a particular limit but which in general exhibits extradegrees of freedom [5, 6].There are many ways to construct gravitational modifications, each one modifying a particularfeature of general relativity. Modifying the dimensionality gives rise to the braneworld theories[7], modifying the Einstein-Hilbert Lagrangian, gives rise to F ( R ) gravity [8, 9], F ( G ) gravity[10, 11], Lovelock theories [12, 13], etc, while adding a scalar field coupled with curvature invarious ways gives rise to Horndeski/Galileon theories [14–16]. Additionally, starting from the– 1 –quivalent, teleparallel, formulation of gravity [17, 18] one can construct modifications using tor-sional invariants, such as in F ( T ) gravity [19, 20], in F ( T , T G ) gravity [21], or in scalar-torsiontheories [22, 23]. Moreover, one can construct the general class of metric-affine theories [24–26],which incorporates a general linear connection structure, or proceed to the introduction of non-linear connections such as in in Finsler and Finsler-like theories [27–32].Inspired by these, one could start from such affinely connected metric theories, and in particu-lar from their Riemann-Cartan subclass [33], and construct a theory using a specific but non-specialconnection, which would lead to non-zero torsion and non-zero curvature at the same time, andthus offering the extra degrees of freedom typically needed in any gravitational modification [34].Myrzakulov gravity can thus lead to a good phenomenology, being able to describe the universeevolution at early and late times [35–39].One basic question in modified gravities is the determination of the arbitrary function thatenters in the theory. Although some general features can be deduced through theoretical consider-ations, such as the absence of ghosts and instabilities, or the existence of Noether symmetries, themost powerful tool is the use of observational data [40–60]. Hence, in this work we are interestedin using expansion data such as Supernovae type Ia data (SNIa), Baryonic Acoustic Oscillations(BAO), and Hubble Cosmic Chronometers (CC) observations, in order to impose constraints onMyrzakulov gravity. The plan of the work is the following. In Section 2 we present Myrzakulovgravity and its cosmological applications. In Section 3 we describe the various datasets and theinvolved statistical methods. Then in Section 4 we preform our analysis and we present the re-sults, namely the constraints on the various parameters. Finally, in Section 5 we summarize andconclude. In this section we present a brief review of Myrzakulov gravity, or F ( R , T ) gravity [34, 35], ex-tracting additionally the relevant cosmological equations. The central idea of this modified gravity is the modification of the underlying connection. Inparticular, it is known that imposing a general connection ω abc one defines the curvature and thetorsion tensor respectively as [21] R ab µν = ω ab ν,µ − ω ab µ,ν + ω ac µ ω cb ν − ω ac ν ω cb µ , (2.1) T a µν = e a ν,µ − e a µ,ν + ω ab µ e b ν − ω ab ν e b µ , (2.2)where e µ a ∂ µ is the tetrad field related to the metric through g µν = η ab e a µ e b ν , where η ab = diag(1 , − , − , − Γ abc is the only connection that givesvanishing torsion, and from now on we use the label “LC” to denote the curvature (Riemann) tensorcorresponding to Γ abc , namely R ( LC ) ab µν = Γ ab ν,µ − Γ ab µ,ν + Γ ac µ Γ cb ν − Γ ac ν Γ cb µ . On the other handone can use the Weitzenb¨ock connection W λµν = e λ a e a µ,ν which is curvatureless, leading only to– 2 –orsion as T ( W ) λµν = W λνµ − W λµν (we use the label “W” to denote quantities corresponding to W λµν .From the above it is implied that the Ricci scalar corresponding to the Levi-Civita connection is R ( LC ) = η ab e µ a e ν b (cid:104) Γ λµν,λ − Γ λµλ,ν + Γ ρµν Γ λλρ − Γ ρµλ Γ λνρ (cid:105) , (2.3)while the torsion scalar corresponding to the Weitzenb¨ock connection is T ( W ) = (cid:16) W µλν − W µνλ (cid:17) (cid:16) W µλν − W µνλ (cid:17) + (cid:16) W µλν − W µνλ (cid:17) (cid:16) W λµν − W λνµ (cid:17) − (cid:16) W µνν − W νµν (cid:17) (cid:16) W λµλ − W λλµ (cid:17) . (2.4)As it is known, the former is used in the Lagrangian of General Relativity and in all curvature-basedmodified gravities, e.g. in F ( R ) gravity [8], while the latter is used in the Lagrangian of teleparallelequivalent of general relativity and in all torsion-modified gravities, e.g. in F ( T ) gravity [6].In Myrzakulov gravity one uses a non-special connection which has non-zero curvature andtorsion simultaneously [35]. Hence, the resulting theory will in general possess extra degrees offreedom, even if the imposed Lagrangian is simple, which is not the case of general relativity or ofteleparallel equivalent of general relativity that both have two degrees of freedom corresponding tothe massless graviton. The action of the theory is S = (cid:90) d xe (cid:34) F ( R , T )2 κ + L m (cid:35) , (2.5)where e = det( e a µ ) = √− g , κ = π G is the gravitational constant, and where we have introducedthe matter Lagrangian L m too for completeness. Note that in the arbitrary function F ( R , T ) the R and T are the curvature and torsion scalars corresponding to the non-special connection used,which read as [21] T = T µνλ T µνλ + T µνλ T λνµ − T νµν T λλµ , (2.6) R = R ( LC ) + T − T νµν ; µ , (2.7)where ; marks the covariant differentiation with respect to the Levi-Civita connection. Therefore, T depends on the tetrad field, its first derivative and the connection, while R depends on the tetradand its first derivative, and on the connection and its first derivative, with an additional dependenceon the second tetrad derivative due to the last term of (2.7). Thus, using (2.4),(2.6),(2.7) we canfinally write T = T ( W ) + v , (2.8) R = R ( LC ) + u , (2.9)where v is a scalar depending on the tetrad, its first derivative and the connection, while u is a scalardepending on the tetrad, its first and second derivatives, and the connection and its first derivative.The quantities u and v quantify the effect of the specific but non-special imposed connection.In the case where this connection becomes the Levi-Civita one, then u = v = − T ( W ) , and theabove theory becomes the usual F ( R ) gravity, which in turn coincides with general relativity under F ( R ) = R . On the other hand, in the case where the connection is the Weitzenb¨ock one, then wehave that v = u = − R ( LC ) and hence the theory coincides with F ( T ) gravity, which in turnbecomes the teleparallel equivalent of general relativity for F ( T ) = T .– 3 – .2 Cosmology Let us now apply the above into a cosmological framework and extract the corresponding equationsthat determine the universe evolution. As it was shown in [35], in order to avoid complicationsrelated to the additional variation in terms of the connection, it proves convenient to apply a mini-super-space procedure. Hence, we apply the homogeneous and isotropic flat Friedmann-Robertson-Walker (FRW) geometry ds = dt − a ( t ) δ i j dx i dx j , (2.10)which corresponds to the tetrad e a µ = d iag [1 , a ( t ) , a ( t ) , a ( t )], where a ( t ) is the scale factor. In thiscase one can easily find that R ( LC ) = (cid:16) ¨ aa + ˙ a a (cid:17) and T ( W ) = − (cid:16) ˙ a a (cid:17) . Furthermore, we use thestandard replacement L m = − ρ m ( a ) [61–63]. Lastly, following the discussion on the dependence of u and v above, we consistently impose that u = u ( a , ˙ a , ¨ a ) and v = v ( a , ˙ a ).In this work we are interested in exploring the cosmological behavior that arise purely fromthe non-special connection of Myrzakulov gravity. Hence, we focus on the simplest case where theinvolved arbitrary function is trivial, namely F ( R , T ) = R + λ T with λ a dimensionless parameter(we omit the coupling coefficient of R since it can be absorbed into κ ). Note that we do notconsider an explicit cosmological constant term in the Lagrangian. Inserting the above mini-super-space expressions into the action (2.5), for this Lagrangian choice we acquire S = (cid:82) Ldt , where L = κ [ λ + a ˙ a − a κ [ u ( a , ˙ a , ¨ a ) + λ v ( a , ˙ a )] + a ρ m ( a ) . (2.11)Extracting the equations of motion for a , alongside the Hamiltonian constraint H = ˙ a (cid:104) ∂ L ∂ ˙ a − ∂∂ t ∂ L ∂ ¨ a (cid:105) + ¨ a (cid:16) ∂ L ∂ ¨ a (cid:17) − L =
0, we finally acquire the Friedmann equations3 H = κ ( ρ m + ρ de ) (2.12)2 ˙ H + H = − κ ( p m + p de ) , (2.13)where ρ de = κ (cid:34) Ha u ˙ a + v ˙ a λ ) −
12 ( u + λ v ) + au ¨ a (cid:16) ˙ H − H (cid:17) − λ H (cid:35) (2.14) p de = − κ (cid:34) Ha u ˙ a + v ˙ a λ ) −
12 ( u + λ v ) − a u a + λ v a − ˙ u ˙ a − λ ˙ v ˙ a ) − a (cid:16) ˙ H + H (cid:17) u ¨ a − Ha ˙ u ¨ a − a u ¨ a − λ (2 ˙ H + H ) (cid:21) , (2.15)with H = ˙ aa the Hubble parameter, p m the matter pressure, and with the subscripts a , ˙ a , ¨ a denotingpartial derivatives with respect to this argument. Hence, in the theory at hand, we obtain an effectivedark energy sector which arises from the non-special connection. Additionally, given the matterconservation equation ˙ ρ m + H ( ρ m + p m ) = ρ de + H ( ρ de + p de ) = , (2.16)which implies that the effective dark energy sector is conserved.– 4 –he above Friedmann equations can efficiently describe the late-time acceleration. A firstobservation is that in the case where λ =
0, namely in the case where the Lagrangian is just the cur-vature (nevertheless the non-special connection leads to non-zero torsion too), and for the choice u = c ˙ a − c , with c , c constants, then we have ρ de = − p de = c κ ≡ Λ . Hence, the scenario at handincludes Λ CDM cosmology as a sub-case, although we have not considered an explicit cosmologi-cal constant, since the cosmological constant arises effectively from the connection structure of thetheory. Thus, we expect that a realistic model would be a deviation from the above scenario.Finally, it proves convenient to introduce the decceleration parameter as q = − − ˙ HH , (2.17)which quantifies the cosmic acceleration. Defining additionally the density parameters Ω m = κ ρ m / (3 H ) and Ω de = κ ρ de / (3 H ), as well as the equation-of-state parameters w m ≡ p m /ρ m and w de ≡ p de /ρ de , we can extract the useful expression2 q − = Ω m w m + Ω de w de . (2.18)Hence, in the standard case of dust matter, namely for w m ≈
0, we obtain w de = q − − Ω m ) . (2.19)This expression allows one to find the evolution of the dark energy equation-of-state parameter,knowing the solution of the Friedmann equations, or knowing the observable values of H ( z ) (where z is the redshift defined through 1 + z = a / a setting the current value of the scale factor to a = Choosing u = c ˙ a − c and v = c ˙ a − c , with c , c constants, we obtain3 H = κ ( ρ m + ρ de ) (2.20)2 ˙ H + H = − κ ( p m + p de ) , (2.21)with ρ de = κ (cid:104) c − λ H (cid:105) (2.22) p de = − κ (cid:104) c − λ (2 ˙ H + H ) (cid:105) , (2.23)where c ≡ c + c . Hence, in this scenario the geometrical sector constitutes an effective darkenergy sector with the above energy density and pressure, and an equation-of-state parameter ofthe form w de = − + λ ˙ Hc − λ H . (2.24)Interestingly enough, we can see that w de can be both larger or smaller than -1, and thus the effectivedark energy can be quintessence or phantom like.– 5 –his model has two parameters, namely c , λ , but one of them can be eliminated using thepresent value of the matter density parameter Ω m (from now on the subscript “0” denotes thecurrent value of a quantity), since (2.20) at present gives:1 = Ω m + c H − λ. (2.25)Additionally, the deceleration parameter (2.18), using (2.22),(2.23), becomes q ( z ) = − + Ω m (1 + z ) Ω m (1 + z ) + + λ − Ω m , (2.26)and thus its value at present is q = − + Ω m + λ ) . (2.27)Comparing with the corresponding value of Λ CDM scenario, namely q Λ = − + Ω m /
2, we verifythat for the special case of λ = As a second example let us consider a more general model with u = c aa ln ˙ a and v = s ( a )˙ a , with s ( a ) an arbitrary function. In this case (2.14),(2.15) give3 H = κ ( ρ m + ρ de ) (2.28)2 ˙ H + H = − κ ( p m + p de ) , (2.29)with ρ de = κ (cid:20) c H − λ H (cid:21) (2.30) p de = − κ (cid:34) c H + c HH − λ (2 ˙ H + H ) (cid:35) , (2.31)while w de = − + λ ˙ H − c HHc H − λ H . (2.32)Similarly to the previous example, for this case too w de can be quintessence-like or phantom-like.This model has two parameters, namely c , λ , but one of them can be eliminated using Ω m ,since (2.20) at present time leads to: 1 = Ω m + c H − λ. (2.33)The deceleration parameter (2.18) becomes q ( z ) = − + Ω m (1 + z ) − Ω m + λ ) + (1 + λ ) − (cid:2) (1 − Ω m + λ ) + (1 + λ )(1 + z ) Ω m (cid:3) / (2.34)and its current value is q = − + Ω m − Ω m + λ ) + (1 + λ ) − (cid:2) (1 − Ω m + λ ) + (1 + λ ) Ω m (cid:3) / . (2.35)– 6 – Data and Methodology
In this section we describe the various datasets that are going to be used in our analysis, and alsothe involved statistical methods. In particular, we will use data from direct measurements of theHubble parameter, from Supernovae Type Ia (SNIa), and from Baryonic Acoustic Oscillations.Finally, we present various information criteria that offer information on the quality of the fit.
From the latest H ( z ) data set compilation available in Ref. [64] we use only data obtained fromcosmic chronometers (CC). By using the differential age of passive evolving galaxies one can mea-sure the Hubble rate directly (see e.g. Ref. [65] and references therein). These galaxies are massivegalaxies that evolve “slowly” at certain intervals of the cosmic time, i.e with small fraction of “new”stars. A striking advantage of the differential age of passive evolving galaxies is that the resultingmeasurement of the Hubble rate comes without any assumptions for the underlying cosmology,with the exception of imposed spatial flatness. Our study incorporates N =
31 measurements ofthe Hubble expansion in the redshift range 0 . (cid:46) z (cid:46) . χ H function reads χ H (cid:0) φ ν (cid:1) = N (cid:88) i = (cid:104) H obsi − H th ( z i ; φ ν ) (cid:105) σ i , (3.1)where H obsi is the observed Hubble rate at redshift z i and σ i the corresponding uncertainty, while φ ν is the statistical vector that contains the free parameters of the examined model. The most common class of cosmological probes is the so-called “standard” candles. The latterare luminous extra-galactic astrophysical objects with observable features that are independent ofthe cosmic time. The most known standard candles and probably the most thoroughly studiedare Supernovae Type Ia (SNIa). In our analysis we use the most recent SNIa dataset available,i.e the binned Pantheon sample of Scolnic et. al. [66]. The full dataset is approximated veryefficiently with the binned N =
40 data points belonging to the redshift interval 0 . (cid:46) z (cid:46) . χ is χ S NIa (cid:16) φ ν + (cid:17) = µ SNIa C − , cov µ T SNIa , (3.2)where µ SNIa = { µ − µ th ( z , φ ν ) , ... , µ N − µ th ( z N , φ ν ) } . The distance modulus reads as µ i = µ B , i − M ,with µ B , i the apparent maximum magnitude for redshift z i . Here, M is a hyper-parameter [66] thatquantifies uncertainties of various origins, such as astrophysical ones, data-reduction pipeline, etc,and it is employed instead of the usage of α, β free parameters in the context of BBC method. Theobserved distance modulus is compared with the theoretical one, i.e µ th = (cid:32) d L ( z ; φ ν )Mpc (cid:33) + , (3.3)– 7 –ith d L ( z ; φ ν ) = c (1 + z ) (cid:90) z dxH ( x , φ ν ) (3.4)the luminosity distance for flat FRW geometry. It must be noted that M and the normalized Hubbleconstant h are degenerate in light of Pantheon dataset in an intrinsic way, as it is usual in standardcandles. Therefore, one should jointly employ other data-sets in order to obtain meaningful infor-mation regarding the present value H . Baryonic Acoustic Oscillations refer to the imprint left by relativistic sound waves in the earlyuniverse, providing an observable to the late-time large scale structure. The main idea is to measurethe aforementioned scale at different times (i.e redshifts), and thus obtain D A ( z ) and H ( z ). Theacoustic length scale corresponds to the co-moving distance that the sound waves could travel untilthe recombination z ∗ [67], namely r d = (cid:90) ∞ z ∗ c s ( z ) H ( z ) . (3.5)For the concordance model, the sound speed, c s , is given from an analytical expression. How-ever, for the models considered here there is not such an expression, therefore the scale r d willbe addressed as a free parameter. Furthermore, distances of different objects along the line ofsight correspond to different redshifts and thus depend on the combination H ( z ) r d , while distancestransverse to the line of sight are related with the combination D A ( z ) / r d .Employing large samples of tracers, (i.e galaxies), one can detect by statistical means the BAOpeak, (for details see [68] and references therein). In order to achieve this it is required to impose anunderlying cosmology, and hence the method is not model independent. However, the differencesthat may infiltrate at the final data products are much less than the statistical errors, and in mostcases the data points are calibrated with the quantity r d , f id / r d . In this work we employ the BAOsdata-set used by [69], that consists of N =
11 data points in the redshift range 0 . (cid:46) z (cid:46) . χ function reads as χ BAO (cid:16) φ ν + (cid:17) = s C − cov s T + N (cid:88) i = (cid:104) T ( z i ; φ ν ) − T obsi (cid:105) σ i , (3.6)with C − cov the inverse of the covariance matrix of the first 6 measurements available at [69]. Thevector s has as elements the s i , given as s i = d m − d obsi r d / r d , f id for odd i and s i = H ( z i ; φ ν ) − H obsi r d , f id / r d for even i. In all cases, r d , f id = .
78. Furthermore, for i ∈ { , } , T ( z i ; φ ν ) = D v ( z i ; φ ν ), T obsi = D obsv , i r d / r d , f id , with r f id , = . M pc and r f id , = . M pc respectively.For i = T ( z i ; φ ν ) = cH ( z i ; φ ν ) − . D m ( z i ; φ ν ) . / r d and for i = T ( z i ; φ ν ) = cH ( z i ; φ ν ) − r d .Finally, in the expressions above the following quantities have been used D M ( z i ; φ ν ) = D L ( z i ; φ ν )1 + z , (3.7a) D A ( z i ; φ ν ) = D L ( z i ; φ ν )(1 + z ) , (3.7b) D V ( z ; φ ν ) = (cid:34) cD A ( z ; φ ν ) z (1 + z ) H ( z ; φ ν ) (cid:35) / . (3.7c)– 8 – .1.4 Big Bang Nucleosynthesis Any cosmological scenario arising from modified gravity should preserve the standard thermalhistory of the universe. Hence, a basic and rough condition is applicable in the form of an extraprior. Specifically, we require that the following inequality holds [70–72]( H i ( z BBN ; φ ν ) − H Λ ( z BBN ; Ω m )) H Λ ( z BBN ; Ω m ) < . , (3.8)where z BBN ∼ . For the fiducial Λ CDM cosmology, namely H Λ , we employ the parametervalues from Planck [73]. In order to obtain the joint constraints on the cosmological parameters from the aforementionedcosmological probes, we introduce the total likelihood function as L tot ( φ k ) = L S NIa × L H × L BAO . (3.9)It is easy to deduce that relevant χ is given as χ ( φ k ) = χ S NIa + χ H + χ BAO . (3.10)The involved statistical vector has k components, i.e. the ν parameters of the scenario at hand plus ν hyp hyper-parameters from the imposed datasets, namely k = ν + ν hyp . Hence, the vector containingthe free parameters of the scenaria at hand is φ k = { Ω m , h , λ, M , r d } . Note however that from astatistical point of view there is no distinction between the intrinsic hyper-parameters of a givendataset and the free parameters of a cosmological scenario.Finally, for the likelihood maximization we use an affine-invariant Markov Chain Monte Carlosampler [74], obtained in the Python package emcee [75]. We use 1000 chains (walkers) and 3500steps (states). As a prior we employ firstly the conditions 0 . < Ω m <
1, 0 . < h < . − . < M < − . − . < λ < .
8, 135 < r d < As a last step we present the standard ways in order to compare a set of cosmological scenarios,namely we apply the Akaike Information Criterion (AIC) [76], the Bayesian Information Criterion(BIC) [77], and the Deviance Information Criterion [78].The AIC criterion is based on information theory and it is an asymptotically unbiased estima-tor of the Kullback-Leibler information. Under the standard assumption of Gaussian errors, thecorresponding estimator for the AIC criterion reads [79, 80]AIC = − L max ) + k + k ( k + N t ot − k − , (3.11)with L max the maximum likelihood of the dataset(s) under consideration and N t ot the total datapoints number. It is apparent that for N t ot >> (cid:39) − L max ) + k . As it is discussed in [81], it is considered as best practise to usethe modified AIC criterion.The BIC criterion is a Bayesian evidence estimator, and it is written as [79–81]BIC = − L max ) + k l og ( N tot ) . (3.12)Finally, the DIC criterion employs both Bayesian statistics and information theory concepts [78],and it is expressed as [81] DIC = D ( φ k ) + C B . (3.13)The quantity C B is the Bayesian complexity C B = D ( φ k ) − D ( φ k ), where overlines imply thestandard mean value. Moreover, D ( φ k ) is the Bayesian Deviation, which can be expressed as D ( φ k ) = − L ( φ k )] in the case of exponential class of distributions. It is closely related to thenumber of effective degrees of freedom [78], which is actually the number of parameters that affectthe fitting. In a less strict manner, it could be considered as a measure of the “spread” of thelikelihood.In contrast with AIC and BIC criteria, instead of using just the best fit likelihood, DIC uses thewhole sample. Furthermore, AIC and BIC count and penalize all the involved parameters, whileDIC penalizes only the number of parameters that contribute to the fit in an actual way. Finally, anadditional appealing feature of DIC criterion is that its calculation is computationally light underthe MCMC samples.Given a set of scenarios that describe the same class of phenomena, our problem is to sortthe models according to their fitting efficiency in the context of the available data. We employ theaforementioned three information criteria (IC) and we calculate the relative difference of the ICvalue for the given set of models, ∆ IC model = IC model − IC min , where the IC min is the minimumIC value inside the competing models set. In order to qualify each model in terms of its relevantadequacy, we apply the Jeffreys scale [82]. Specifically, the condition ∆ IC ≤ < ∆ IC < ∆ IC ≥
10 implies strongtension.
In this section we proceed to the observational analysis of Myrzakulov gravity using the datasetsand the methods described above. For convenience we summarize the obtained results in Table 1.Additionally, in Figs. 1 and 2 we present the corresponding contour plots for Model 1 and Model2 respectively.As we can see, according to the combined analysis of CC+SNIa+BAO data we acquire ∼ σ compatibility in all cases. The dimensionless parameter λ is constrained to an interval around0, that includes Λ CDM paradigm, which was expected since as we discussed above a realisticmodified gravity should be a small deviation from general relativity. Nevertheless, note that in bothModel 1 and Model 2, the λ -contours are slightly shifted towards positive values. We mentionthat having the likelihood contours for the parameter λ allows us to extract the constraints on theparameter c through expression (2.25) for Model 1 and on the parameter c through (2.33) for– 10 –odel Ω m h λ r d M χ χ min / do f Mod. 1 0 . + . − . . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . − . + . − . Λ CDM 0 . + . − . . + . − . - 145 . + . − . − . + . − . Table 1 . Observational constraints and the corresponding χ in for the two Myrzakulov gravity models,presented previously, using CC/Scolnic/BAO data-sets. In order to allow direct comparison, the concordance Λ CDM model is also included. m = 0.4350 +0.11770.1560 . . . . h h = 0.6908 +0.01710.0169 . . . . = 0.4911 +0.38720.5337 . . . . . = 19.3829 +0.05130.0521 .
15 0 .
30 0 .
45 0 .
60 0 . m r d .
64 0 .
68 0 .
72 0 . h . . . . . . . . . r d r d = 146.1955 +3.55383.4067 Figure 1 . The σ , σ and σ likelihood contours for Model 1 of (2.22),(2.23), for all possible 2D subsets ofthe parameter space ( Ω m , h , λ, M , r d ) . Moreover, we present the mean parameter values within the σ areaof the MCMC chain. We have performed a joint analysis of CC+SNIa+BAO data. – 11 – = 0.3411 +0.09390.1222 . . . . . h h = 0.6793 +0.01690.0168 . . . . . = 0.5371 +0.40350.5502 . . . . . = 19.3962 +0.05140.0524 .
15 0 .
30 0 .
45 0 . m r d .
63 0 .
66 0 .
69 0 .
72 0 . h . . . . . . . . . . r d r d = 146.5991 +3.57663.4461 Figure 2 . The σ , σ and σ likelihood contours for Model 2 of (2.30),(2.31), for all possible 2D subsets ofthe parameter space ( Ω m , h , λ, M , r d ) . Moreover, we present the mean parameter values within the σ areaof the MCMC chain. We have performed a joint analysis of CC+SNIa+BAO data. Model 2. In particular, for 1 σ region for Model 1 we obtain c = . + . − . , while for Model 2 wefind c = . + . − . .Concerning the values of Ω m we observe that Model 1 gives a rather large value, due tothe degeneracy with λ , while for Model 2 this is not the case. Concerning the Hubble constant h , for Model 1 we find that 0 . + . − . , while for Model 2 we obtain 0 . + . − . . This impliesthat the obtained values for the present Hubble parameter H are in between the Planck estimation H = . ± . H = . ± .
74 km/s/Mpc [83], althoughcloser to the former. This implies that the scenario at hand could offer a slight alleviation to the H tension. – 12 –s a next step we proceed to the examination of the statistical significance of our fitting results,applying the AIC, BIC and DIC information criteria described in subsection 3.2. We summarizeour results in Table 2. As we observe, Model 1 is statistically equivalent with Λ CDM paradigm, andespecially the combined and more complete DIC criterion gives an almost equal value. Addition-ally, Model 2 also presents a very good fitting behavior, and according to DIC it is also statisticallyequivalent with Λ CDM paradigm, which is an interesting result since Model 2 does not contain Λ CDM scenario as a limit for any parameter value.Model AIC ∆ AIC BIC ∆ BIC DIC ∆ DICMod. 1 72.7234 2.4757 83.9675 4.6124 69.6728 0.0007Mod. 2 74.3204 4.0727 85.5645 6.2094 71.3725 1.7004 Λ CDM 70.2477 0 79.3551 0 69.6721 0
Table 2 . The information criteria AIC, BIC and DIC for the examined cosmological models, alongside therelative difference from the best-fitted model ∆ IC ≡ IC − IC min . Figure 3 . The reconstruction of the effective dark-energy equation-of-state parameter w de ( z ) as a function ofthe redshift for Model 1 given by (2.24) . We re-sampled the chains produced by emcee taking 6000 samples,and we plot all the obtained w de ( z ) curves, alongside the curve corresponding to the best fit of the parameters(red curve). We close our analysis with a discussion on the evolution of the dark energy equation-of-stateparameter. In particular, having obtained the allowed parameter values at 1 σ confidence level, wecan use them in order to extract the resulting w de ( z ) behavior given by (2.19), with the decelerationparameter given by (2.26) for Model 1 and by (2.34) for Model 2.– 13 –n Fig. 3 we depict the reconstructed mean w de ( z ) (red curve) for Model 1, alongside theallowed curves for the 1 σ allowed model parameters presented above. As we observe, the corre-sponding behavior is very close to Λ CDM scenario for every parameter values. Similarly, in Fig. 4we present the corresponding graph for Model 2. In this case the scenario resembles Λ CDM at lowredshifts, however at earlier times the mean curve presents a deviation, since this is allowed by theused datasets, nevertheless for a large region of the parameter space the individual obtained curvesresemble the Λ CDM evolution.
Figure 4 . The reconstruction of the effective dark-energy equation-of-state parameter w de ( z ) as a functionof the redshift for Model 2 given by (2.32) . We re-sampled the chains produced by emcee taking 6000samples, and we plot all the obtained w de ( z ) curves, alongside the curve corresponding to the best fit ofthe parameters (red curve). The over-populated area at the bottom corresponds to a peak within 1 σ area,nevertheless since we extract the median value of each parameter within 1 σ as the best fit, the “best” w de ( z ) curve differs. Finally, we reconstruct the decceleration parameter using random sampling of the obtainedchains. Concerning the current value q , for Model 1 using (2.27) we obtain q = − . + . − . ,while for Model 2 using (2.35) we acquire q = − . + . − . . These are in agreement with thevalues obtained using other datasets, such as supernovae, quasars and gamma-ray bursts by meansof model-independent techniques [84]. In this work we have used observational data from Supernovae (SNIa) Scolnic sample, from Bary-onic Acoustic Oscillations (BAO), and from cosmic chronometers measurements of the Hubbleparameter (CC), alongside arguments from Big Bang Nucleosynthesis (BBN), in order to extract– 14 –onstraints on Myrzakulov F ( R , T ) gravity. This is a connection-based theory belonging to theRiemann-Cartan subclass, that uses a specific but non-special connection, which then leads to extradegrees of freedom. One introduces a parametrization that quantifies the deviation of torsion andcurvature scalars form their values corresponding to the special Levi-Civita and Weitzenb¨ock con-nections, and then constructs various models by assuming specific forms for the involved functions.In all models, one obtains an effective dark-energy sector of geometrical origin.We considered two specific models, which are known to lead to interesting phenomenology.Our analysis shows that both models are capable of describing adequately the imposed datasets,namely CC+SNIa+BAO ones, obtaining ∼ σ compatibility in all cases. Concerning Model 1,which includes Λ CDM paradigm as a particular limit, we found a relatively large value for Ω m and a value for h in between the Planck and local estimation, although closer to the former. Forthe dimensionless parameter λ we found that it is constrained to an interval around 0, which corre-sponds to Λ CDM scenario, however the corresponding contours are slightly shifted towards pos-itive values. In the case of Model 2, we found smaller Ω m and h , while λ is again constrainedaround 0 with favoured positive values. Furthermore, applying the AIC, BIC and the combinedDIC criteria, we deduced that both Model 1 and Model 2 present a very efficient fitting behavior,and are statistically equivalent with Λ CDM cosmology. This is an interesting result since Model 2does not contain Λ CDM scenario as a limit for any parameter value.Finally, we used the obtained parameter region at 1 σ confidence level, and we reconstructedthe induced dark-energy equation-of-state parameter as a function of the redshift. As we saw, forModel 1 w de ( z ) is very close to Λ CDM scenario, while for Model 2 it resembles Λ CDM at lowredshifts, however at earlier times deviations are allowed.In summary, Myrzakulov F ( R , T ) gravity is in agreement with cosmological data, and it couldserve as a candidate for the description of nature. Nevertheless, one should also investigate thetheory at the perturbation level and confront it with perturbation-related data, such as the growth-index and f σ ones. Such an analysis, although both interesting and necessary, lies beyond thescope of the present work and it is left for a future project. References [1] K. A. Olive,
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