Low-Redshift Constraints on Covariant Canonical Gauge Theory of Gravity
David Benisty, David Vasak, Johannes Kirsch, Jurgen Struckmeier
LLow-Redshift Constraints on Covariant Canonical Gauge Theory of Gravity
David Benisty,
1, 2, ∗ David Vasak, † Johannes Kirsch, ‡ and Jürgen Struckmeier
1, 3, § Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Fachbereich Physik, Goethe-Universitat, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
Constraints on the Covariant Canonical Gauge Gravity (CCGG) theory from low-redshift cos-mology are studied. The formulation extends Einstein’s theory of General Relativity (GR) by aquadratic Riemann-Cartan term in the Lagrangian, controlled by a "deformation" parameter. Inthe Friedman universe this leads to an additional geometrical stress energy and promotes, dueto the necessary presence of torsion, the cosmological constant to a time-dependent function. TheMCMC analysis of the combined data sets of Type Ia Supernovae, Cosmic Chronometers and BaryonAcoustic Oscillations yields a fit that is well comparable with the Λ CDM results. The modifica-tions implied in the CCGG approach turn out to be subdominant in the low-redshift cosmology.However, a non-zero spatial curvature and deformation parameter are shown to be consistent withobservations.
I. INTRODUCTION
Dark energy, inflation, and dark matter are examplesof much disputed concepts that have been added to Ein-stein’s General Relativity (GR) in order to explain ob-servations that otherwise would not be accounted for,see e.g. [1–3]. The cosmological constant, as its valueadjusted to fit the current accelerated expansion of theuniverse is far at odds with the calculated vacuum energyof matter which it is supposed to represent. Quintessence[4–15] – and similar scalar fields invoked to generate aninitial explosive inflation of the universe and explain themeasured isotropy of cosmic radiation – lack fundamen-tal physical underpinning. Modifications of the gravityare, among other models, hand-crafted just for matchingspecific observations. The invisible dark matter, finally,necessary to explain the dynamics of galaxies, could notyet been attributed to any field theory, or a known orunknown particle, despite astronomical budgets devotedto its search [16–36].Recently a novel, rigorously derived covariant canon-ical gauge theory of gravity (CCGG) has been appliedto Friedman cosmology. CCGG is based on the covari-ant version of the canonical transformation theory withwhich all gauge theories are derived on the same footing.The difference is just the symmetry group under con-sideration delivering the appropriate minimal couplingscheme for matter fields and the dynamical space-time.Such a "universal" approach must of course be subject toa comprehensive testing against all kinds of experiments,especially as CCGG is its novel application to gravity. Inthat study [37] CCGG was shown to deliver an explana-tion of dark energy as a torsion based phenomenon Forearlier investigations on the possible cosmological role oftorsion see for example [38–45]. There a first analysis of ∗ [email protected] † vasak@fias.uni-frankfurt.de ‡ jkirsch@fias.uni-frankfurt.de § struckmeier@fias.uni-frankfurt.de the CCGG-Friedman cosmology was limited to varyingthe only new parameter beyond Λ CDM. The comparisonof the theory with the Hubble diagram indicated that themodel can deliver viable scenarios of cosmic evolution.In this paper that preliminary analysis is extended toBayesian analysis with the aim to explore further cos-mological constraints on the full parameter set, and tocompare the results with the standard Λ CDM cosmol-ogy. After a brief review of the CCGG theory and thepertinent Friedman equations we first list the observa-tional data considered with focus on low z . A discussionof the numerical analysis and the resulting figures follows.The paper concludes with a discussion of the findings. II. THE CCGG FORMULATION
Rather than following ad-hoc or trial-and-error ap-proaches for modifying GR for compatibility with experi-ments, we rely ab initio on the powers of proven compre-hensive mathematical frameworks. In analogy to pointparticle physics we apply the covariant, field theoreticalversion of the canonical transformation theory in the DeDonder-Hamiltonian formalism to imprint a given sym-metry on a system of covariant fields. In this way aconsistent interaction of gravity with matter is derived as laid out in Refs. [46–51]. This approach yields the
Covariant Canonical Gauge Gravity (CCGG), a Yang-Mills type theory in the spirit of Utiyama, Sciama, Kib-ble, Hayashi and Shirafuji, and Hehl [52–56], rooted in afew key assumptions. While Einstein’s Principle of Gen-eral Relativity translates into the requirement of diffeo-morphism invariance of the coupled dynamics of mat-ter and space-time, the Equivalence Principle is incor-porated by defining at any point of space-time an in-ertial (observers’) frame of reference. The pertinentmathematical structure is a ("Lorentzian") frame bun-dle with fibers spanned by ortho-normal bases fixedup to arbitrary (local) Lorentz transformations. Thegauge group underlying the CCGG approach is thus the SO (1 , (+) × Dif f ( M ) group. The emerging gauge fields a r X i v : . [ g r- q c ] J a n are the (spin) connection coefficients not restricted totorsion-free and/or metric compatible geometries. Thegauge field is a priori independent of the metric tensor,or equivalently of the vierbein fields, which come as fun-damental structural elements of the Lorentzian manifold.Minimal couplings are discovered in that way, not pos-tulated a priori. Of course, the structure of the dynam-ical space-time is dynamically implemented by a specificchoice of the Hamiltonian of free gravity, but that re-mains the only freedom the theory leaves open.In order to secure the existence of the action inte-gral in the Hamiltonian picture we also postulate non-degeneracy of the Lagrangian and the correspondingHamiltonian densities, which implies that the Lagrangianmust contain an at least quadratic Riemann-Cartan ten-sor concomitant [57]. A quadratic term is therefore addedas a parameter-controlled deformation to Einstein’s lin-ear ansatz, endowing space-time with kinetic energy andthus inertia. In this way the framework delivers a clas-sical, quadratic, first-order (Palatini) field theory wherethe connection coefficients emerge as independent gaugefields which, in addition to the symmetric metric tensor,determine the space-time geometry and mediate gravi-tation. The couplings of matter fields and gravity areunambiguously fixed. The so called consistence equationin CCGG is a combination of the canonical (or equiv-alently Euler-Lagrange) equations of motion, extendingthe field equation of GR: g (cid:0) R αβγµ R αβγν − g µν R αβγδ R αβγδ (cid:1) − πG (cid:104) R ( µν ) − g µν (cid:0) R + λ (cid:1)(cid:105) = T ( µν ) . (1)Here g is the dimensionless deformation parameter, G Newton’s coupling constant, and λ the "bare" cosmo-logical constant. R αβµν = γ αβν,µ − γ αβµ,ν + γ αξµ γ ξβν − γ αξν γ ξβµ (2)is the Riemann-Cartan tensor (in general built froman asymmetric connection), and T ( µν ) the symmetrizedstress-energy tensor of matter. (Our conventions are thesignature (+ , − , − , − ) of the metric, and natural units (cid:126) = c = 1 . A comma indicates a partial derivative.) III. THE CCGG-FRIEDMAN UNIVERSEA. Homogenous solution
Our aim is to establish a form of the equations govern-ing the dynamics of the universe that allow for a closecomparison with GR. In particular we require that thestress-energy tensor be covariantly conserved. This re-quirement is here not based on the Bianchi identity forthe Einstein tensor. It is invoked independently to retainthe standard scaling properties of matter and radiation.As shown in Ref. [37] this leads, in a metric compati-ble space-time, to the necessity to invoke torsion. For the CCGG version of the Friedman model that promotesthe cosmological constant to the time or scale dependentfunction Λ( a ) =: λ + P ( a ) =: Λ f ( a ) .P ( a ) is the torsion-dependant portion of the Ricci scalar. Λ( a ) reduces to the "bare" cosmological constant λ intorsion-free geometries. The Hubble function acquires, inaddition, a further geometric correction originating fromthe quadratic Riemann-Cartan gravity. We ultimatelyget E ( a ) =: H ( a ) H = ρ ( a ) ρ crit (3a) ρ ( a ) =: ρ m + ρ r + ρ Λ + ρ K + ρ geom (3b) ρ m ( a ) =: ρ crit Ω m a − (3c) ρ r ( a ) =: ρ crit Ω r a − (3d) ρ K ( a ) =: ρ crit Ω K a − = − ρ crit K a − (3e) ρ Λ ( a ) =: ρ crit Ω Λ f ( a ) = ρ crit Λ f ( a ) (3f) ρ geom ( a ) =: ρ crit Ω geom (3g) = ρ crit (1 / m + Ω Λ )(3 / m + Ω r )Ω g − / m − Ω Λ , (3h)where a is the scale factor and K the curvature parame-ter of the FLRW metric. Ω i with i = m, r, Λ , K are thestandard density constants related respectively to (dark)particle matter, radiation and dark energy. H ( a ) = ˙ a/a is the Hubble function, H ≡ H ( a = 1) the Hubble con-stant, and ρ crit ≡ H / πG ) . For convenience we use Ω g =: [32 πGH g ] − . We notice at this point that thevarious pressure terms combine to p ( a ) =: ρ r − ρ Λ − ρ K + ρ geom . (4)Obviously the cosmological function has the equation ofstate p Λ = − ρ Λ of dark energy, and the "geometric fluid"has the equation of state p geom = ρ geom , i.e. it behaveslike (dark) radiation.The normalized dark energy function, f ( a ) , is deter-mined from the ordinary first-order, non-linear differen-tial equation f (cid:48) ( a ) = 3Ω m Λ a α ( a ) β ( a ) . (5)where: α ( a ) = Ω g (cid:0) Ω m a − + Ω r a − (cid:1) − (cid:0) Ω g − Ω m a − − Ω Λ f (cid:1) (cid:0) Ω m a − + Ω Λ f (cid:1) , FIG. 1.
The stream plot for a universe with dark matter,dynamical dark energy Λ( a ) and the quadratic term from theCCGG equations. The x refers to the matter part and the y refers to the dark energy density. The figure shows thatthe matter domination (B) is an unstable point and the darkenergy domination (C) is a stable point. β ( a ) = Ω g (cid:0) Ω m a − + Ω r a − (cid:1) + (cid:0) Ω g − Ω m a − − Ω Λ f (cid:1) , with the boundary condition f (1) = 1 . By setting g = 0 (which means f (cid:48) ( a ) = 0 ) and f ( a ) ≡ we recover inEq. (3) the Einstein-Friedman equation for the Hubblefunction based on General Relativity. There are five in-dependent parameters in the CCGG model that must beoptimized, namely Ω m , Ω r , Ω Λ , Ω K and Ω geom . By solv-ing H = H ( a = 1) , we get the relation for the additional,deformation parameter: g = 12 πGH Ω K + Ω Λ + Ω m + Ω r − Λ + Ω m )(4Ω Λ + Ω m + 4Ω K − . (6)Notice that for the standard Λ CDM model (with a spatialcurvature Ω K ) the sum Ω K + Ω Λ + Ω m + Ω r gives andthus from Eq. 6 we get g = 0 . B. Stability Analysis
In order to test the stability of the model we use theautonomous system method [58, 59]. For simplicity weignore the spacial curvature and the radiation part andinclude matter, dark energy and the quadratic term thatincorporates the torsional part. In that case the correctdefinition for the dimensionless parameters reads: x = Ω m a E , y = Ω Λ ( a ) E , z = Ω geom E (7) with x + y + z = 1 . After some algebra, one can definethe evaluation equations for the system: dxdN = x (cid:0) − x − y + 1 (cid:1) , (8a) dydN = 3 x (cid:0) x + y − (cid:1) (cid:0) x + 4 y − (cid:1) y (cid:16) x + 4 x (8 y −
5) + 16 ( y − (cid:17) − y (cid:0) x + 4 y − (cid:1) , (8b)where N = log( a ) . By setting x (cid:48) ( N ) = y (cid:48) ( N ) = 0 , threesolutions for the system are discovered. In order to esti-mate the stability of those points, we evaluate the matrixthat contains the derivatives of the system.Fig 1 shows the stream plot for the system. Point A ( x = 0 , y = 0) describes domination of the quadraticterm. The eigenvalues at that point, λ , = 4 , , areboth positive, which indicates an unstable point. Point B ( x = 1 , y = 0) where matter dominates the universe isalso unstable as the eigenvalues of the point are λ , = − , +3 . In contrast, point C ( x = 0 , y = 1) with darkenergy domination and the eigenvalues λ , = − , − isstable. The solution shows the evolution from point A to C with matter domination during the evolution of theuniverse. IV. COSMOLOGICAL PROBESA. Dataset
In order to constraint our model, we deploy the follow-ing data sets:
Cosmic Chronometers (CC) exploitthe evolution of differential ages of passive galaxies atdifferent redshifts to directly constrain the Hubble pa-rameter [60]. We use uncorrelated 30 CC measurementsof H ( z ) discussed in [61–64]. As Standard Candles(SC) we use uncorrelated measurements of the PantheonType Ia supernova [65] that were collected in [66]. Theparameters of the models are adjusted to fit the theoret-ical µ thi value of the distance modulo, µ = m − M = 5 log ((1 + z ) · D M ) + µ , (9)to the observed µ obsi value. m and M are the apparentand absolute magnitudes and µ = 5 log (cid:0) H − /M pc (cid:1) +25 is the nuisance parameter that has been marginalized.The luminosity distance is defined by D L = (1 + z ) D M ,where D M = cH S k (cid:18)(cid:90) z dz (cid:48) E ( z (cid:48) ) (cid:19) , (10)and S k ( x ) = √− Ω K sinh (cid:0) √− Ω K x (cid:1) if Ω K < x if Ω K = 0 √ Ω K sin (cid:0) √ Ω K x (cid:1) if Ω K > . (11) H ( z ) [ k m / s e c / M p c ] CCGGobservations0.0 0.5 1.0 1.5 2.0 2.5z14161820222426 CCGGPantheon
FIG. 2.
The upper panel shows the CCGG best fit vs. theCosmic Chronometers dataset. The lower panel shows thethe CCGG best fit vs. 40 uncorrelated points of the Type Iasupernova dataset. The dataset is presented in red, and thebest fit is presented in blue color.
In addition, we use the uncorrelated data points fromdifferent
Baryon Acoustic Oscillations (BAO) col-lected in [67] from [68–79]. Studies of the BAO fea-tures in the transverse direction provide a measurementof D H ( z ) /r d = c/H ( z ) r d , with the comoving angular di- Parameter CCGG CCGG + R19 Λ CDM H [ kms · Mpc ] 69 . ± . . ± .
75 70 . ± . m . ± .
016 0 . ± .
014 0 . ± . Λ . ± .
09 0 . ± .
09 0 . ± . r (10 − ) 4 . ± .
96 5 . ± . - r d [ Mpc ] 147 . ± . . ± .
62 146 . ± . g (10 ) − . ± .
541 0 . ± . q − . ± . − . ± . − . ± . χ min .
89 73 .
89 74 . χ /Dof .
92 0 .
93 0 . AIC .
51 86 .
89 84 . TABLE I.
Observational constraints and the corresponding χ for the CCGG model with uniform prior and with theSH0ES prior, and Λ CDM model. Here we set Ω K = 0 . ameter distance defined in [80, 81]. In our database weuse the parameters D A = D M / (1 + z ) and D V ( z ) ≡ [ zD H ( z ) D M ( z )] / . (12)which is a combination of the BAO peak coordinates. r d is the sound horizon at the drag epoch. Finally, for veryprecise "line-of-sight" (or "radial") observations, BAOcan also measure directly the Hubble parameter [82].We use a nested sampler as it is implementedwithin the open-source packaged P olychord [83] withthe
GetDist package [84] to present the results. Theprior we choose is with a uniform distributions, where Ω r ∈ [0; 10 − ] , Ω m ∈ [0 . ; 1 . ] , Ω Λ ∈ [0 . ; 1 . ] , H ∈ [50; 100] Km/sec/M pc , r d ∈ [120; 160] M pc . When weinclude a spatial curvature we extend the prior with Ω K ∈ [ − .
1; 0 . . The measurement of the Hubble con-stant yielding H = 74 . ± . km/s ) /M pc at CL by [85] has been incorporated into our analysis asan additional prior (
R19 ). We contrast best-fit parame-ters and goodness of fit between CCGG and the standard Λ CDM with these datasets. We also compare the Akaikeinformation criteria (AIC) of the two models applied tothe data set [86–88]. In order to make a complete discus-sion, we include also the deceleration parameter in ourdiscussion, the dimensionless quantity q = − − ˙ H/H measuring the acceleration of the cosmic expansion. B. Spatially flat universe
Table I summarises the results with Ω K = 0 . In theCCGG model the quadratic term provides with the de-formation parameter an additional degree of freedom.Hence while for Λ CDM we set Ω r = 1 − Ω m − Ω Λ for theradiation part, for CCGG we have Ω geom = 1 − Ω m − Ω Λ − Ω r for the additional geometry term. The Hubble param-eter fitted for the CCGG model is . ± . km/sec/M pc for a uniform prior or . ± . km/sec/M pc with theSH0ES prior. The Hubble parameter for the Λ CDMmodel is in between these values . ± . km/sec/M pc for the uniform prior.The Ω m matter part in the CCGG model is . ± . or . ± . for the SH0ES prior, which is a bit higherthen the Λ CDM fit . ± . . The dark energy Ω Λ partis being . ± . or . ± . with the SH0ES prior,a bit lower then the Λ CDM fit . ± . .The BAO scale is set by the redshift at the drag epoch z d ≈ when photons and baryons decouple [89]. Fora flat Λ CDM, the Planck measurements yield . ± . M pc and the WMAP fit gives . ± . M pc [90].Final measurements from the completed SDSS lineage ofexperiments in large-scale structure provide r d = 149 . ± . M pc [91]. The Λ CDM model for the combined dataset we use gives . ± . M pc . However, the CCGGmodel gives . ± . M pc . For the SH0ES prior, thedistance is . ± . M pc . The quadratic term thuschanges the horizon scale in the early universe, but stillin a moderate and reasonable range. m CCGGCCGG + R19CDM
66 68 70 72 74 H ( km / sec / Mpc ) m
66 68 70 72 74 H ( km / sec / Mpc ) g
66 68 70 72 74 H ( km / sec / Mpc ) r d FIG. 3.
One- (68 % CI) and two-dimensional (68 % and 95 % CI) marginalized posterior distributions for the relevant sampledand derived CCGG parameters. The upper left panel shows the contour for Ω m vs. Ω Λ and the right upper panel shows thecontour for Ω m vs. H . The lower left panel shows the contour for g vs. H and the lower upper panel shows the contour for r d vs. H . The gray contour describes the CCGG best fit with a uniform prior. The red contour describes the CCGG best fitwith the SH0ES measurement as a prior. Finally, the blue contour describes the Λ CDM best fit with a uniform prior.
From the AIC we see that Λ CDM is still the betterfit to the late universe, since the AIC for Λ CDM model . is then the CCGG case . or with SH0ES prior . . However the Λ CDM model does not describe theinflationary epoch, which the quadratic term naturallyprovides.
C. With Ω K (cid:54) = 0 The shape of the universe is a fundamental question.The latter can be characterized by measuring the spatialcurvature of the universe K , quantifying how much thespatial geometry locally differs from that of flat space.Most models of inflation predict a universe which is ex-tremely close to being spatially flat [90, 92, 93]. Becauseof the quadratic term the spatial curvature may be larger. Fig 5 shows the spatial curvature vs. the Hubble parame-ter. For the uniform prior case the spatial curvature turnsout to be (0 . ± . · − , while for the SH0ES priorthe spatial curvature is (2 . ± . · − . The modelpredicts a positive value for Ω K but the error bar is suf-ficient large for the negative values as well. Moreover,from the AIC criteria it seems that the case for absorb-ing the spatial curvature is better since the AIC for thiscase is higher: . for the uniform prior, and . forthe SH0ES prior. V. DISCUSSION
This paper discusses the cosmological constraints onthe CCGG formulation from low-redshift observations.CCGG is a gauge theory of gravity ensuring in a covariant
66 68 70 72 74 H ( km / sec / Mpc ) q CCGGCCGG + R19CDM g ×10 q H ( k m / s e c / M p c ) FIG. 4.
One- (68 % CI) and two-dimensional (68 % and 95 % CI) marginalized posterior distributions for the the decelera-tion parameter q vs. the Hubble parameter H . The gray con-tour describes the CCGG best fit with a uniform prior. Thered contour describes the CCGG best fit with the SH0ES mea-surement as a prior. Finally, the blue contour describes the Λ CDM best fit with a uniform prior. The lower panel showsa 3D of q and the parameter g vs. the Hubble parameter H . way full diffeomorphism invariance of the system action.Using canonical transformation theory the approach un-ambiguously fixes how matter fields interact with curvedgeometry of space-time, and enforces a parameter con-trolled admixture of a quadratic Riemann-Cartan con-comitant to the Einstein-Hilbert linear term. In a pre-liminary study [37] the cosmological consequence of thatquadratic extension were examined in alignment with the Λ CDM model. Here we go a step further and test theCCGG cosmology against a comprehensive database oflow-redshift cosmological measurements that include thePantheon Type Ia supernova, Cosmic Chronometers andBaryon Acoustic Oscillations.Using the Polychord package we find a good best fitof the CCGG cosmology with data. By the AIC cri-terium the CCGG fit accuracy is comparable with that
66 68 70 72 74 H ( km / sec / Mpc ) k CCGGCCGG + R19
Parameter CCGG CCGG + R19 H [ kms · Mpc ] 69 . ± . . ± . m . ± .
052 0 . ± . Λ . ± .
017 0 . ± . r (10 − ) 5 . ± .
82 5 . ± . K (10 − ) 0 . ± .
53 2 . ± . r d [ Mpc ] 146 . ± .
72 143 . ± . g (10 ) 0 . ± .
29 1 . ± . q − . ± . − . ± . χ .
83 73 . χ /Dof .
93 0 . AIC .
84 88 . FIG. 5.
One- (68 % CI) and two-dimensional (68 % and 95 % CI) marginalized posterior distributions for the the decelera-tion parameter Ω K vs. the Hubble parameter H . The graycontour describes the CCGG best fit with a uniform prior.The red contour describes the CCGG best fit with the SH0ESmeasurement as a prior.The table shows Observational con-straints and the corresponding χ for the CCGG model withuniform prior and with the SH0ES prior with spatial curva-ture. of Λ CDM. The key density parameters are in reasonableagreement with the Λ CDM model. The new free param-eter of the theory controlling the admixture of quadraticgravity and the inflationary dynamics is of the order . However, the statistical error bars do not permitany further conclusions about its value, not even aboutits sign. Also the non-zero spatial curvature parame-ter found for the late universe is, within the error bars,consistent with zero. The deceleration parameter q , onthe other hand, is predicted to be lower then the Λ CDMbest fit: − . ± . for CCGG and − . ± . for Λ CDM.We conclude that the CCGG approach reproduceslow-redshift observation with a similar accuracy as the Λ CDM model. The novel features of CCGG, namely thepresence of torsion and the influence of quadratic cur-vature, represented by the additional deformation pa-rameter, turn out to be subdominant in this late era ofthe cosmic evolution. However, albeit this calculationdoes not conclusively determine the relative admixtureof quadratic gravity to Einsten-Cartan gravity, the dataneither excludes a non-zero deformation parameter, nora deviation from the flat geometry assumed in Λ CDM.This leaves the possibility open that the more complexspace-time geometry of CCGG applies, which naturallyinvokes inflation [94] and substantially alters the Hubbleexpansion in the early universe [37]. The model’s supe-riority thus might become obvious only when includingthe early universe data in the analysis.The discussion on whether torsion of space-time couldbe excluded by solar tests have been sparked by the Grav-ity Probe B (GPB) experiment. While Mao et al. [95]propose to use the high-precision gyroscope for detect-ing torsion, Hehl et al. [96] conclude, referring to thePoincare Gauge Gravity, that torsion can couple to parti-cle spin only, not to the gyroscope’s angular momentum,and that the accuracy needed for detecting such a cou-pling is far beyond any currently available technologies.Moreover, they also dismiss possible deviations of testparticle trajectories from the geodesic (trajectory of ex-tremal length) by postulating that it is force-free trajec-tory of particles, rather than the autoparallel ("straightestrajectory"). These discussions indicate that torsion maynot be detectable in the solar system, but they so far donot exclude its existence.While we agree with Hehl’s first conjecture, we advo-cate the autoparallel to be the correct force-free trajec- tory as the obvious generalization of Newton’s notion ofa straight line to curvilinear space-time. A propagat-ing torsion field that arises naturally in CCGG will alsodirectly interact with spin carrying particles. As in pres-ence of torsion the autoparallel and the geodesic are notidentical, a modified connection will in principle be felt bytest particles. Direct spin-torsion interactions [97] will inaddition affect the trajectories of spin-polarized as com-pared to spinless test particles. However, if the densityof the torsion field is very low in the solar system then wewill encounter similar restrictions on the detectability ofthose spin-torsion interactions. Work along these linesincluding advanced modeling of the torsion tensor is inprogress.
Public Source : The files with the dataset and the fitpackage can be found in https://github.com/benidav/CCGGcosmology2020 . ACKNOWLEDGMENTS
We thank Horst Stöcker and Denitsa Sticova forfruitful discussions. This work has been supportedby the Walter Greiner Gesellschaft zur Förderung derphysikalischen Grundlagenforschung e.V., and partiallyby the European COST actions CA15117 and CA18108.D.B., D.V. and J.K. especially thank the Fueck Stiftungfor support. [1] S. Weinberg, Rev. Mod. Phys. , 1 (1989), [,569(1988)].[2] L. Lombriser, Phys. Lett. B797 , 134804 (2019),arXiv:1901.08588 [gr-qc].[3] J. Frieman, M. Turner, and D. Huterer, Ann. Rev. As-tron. Astrophys. , 385 (2008), arXiv:0803.0982 [astro-ph].[4] C. Wetterich, Nucl. Phys. B897 , 111 (2015),arXiv:1408.0156 [hep-th].[5] B. Ratra and P. J. E. Peebles, Phys. Rev.
D37 , 3406(1988).[6] R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev.Lett. , 1582 (1998), arXiv:astro-ph/9708069 [astro-ph].[7] J. Kehayias and R. J. Scherrer, Phys. Rev. D100 , 023525(2019), arXiv:1905.05628 [gr-qc].[8] V. K. Oikonomou and N. Chatzarakis, Nucl. Phys.
B956 ,115023 (2020), arXiv:1905.01904 [gr-qc].[9] A. Chakraborty, A. Ghosh, and N. Banerjee, Phys. Rev.
D99 , 103513 (2019), arXiv:1904.10149 [gr-qc].[10] E. Babichev, S. Ramazanov, and A. Vikman,(2018), 10.1088/1475-7516/2018/11/023,[JCAP1811,023(2018)], arXiv:1807.10281 [gr-qc].[11] I. Zlatev, L.-M. Wang, and P. J. Steinhardt, Phys. Rev.Lett. , 896 (1999), arXiv:astro-ph/9807002 [astro-ph].[12] R. R. Caldwell, Phys. Lett. B545 , 23 (2002), arXiv:astro-ph/9908168 [astro-ph].[13] T. Chiba, T. Okabe, and M. Yamaguchi, Phys. Rev.
D62 , 023511 (2000), arXiv:astro-ph/9912463 [astro-ph]. [14] M. C. Bento, O. Bertolami, and A. A. Sen, Phys. Rev.
D66 , 043507 (2002), arXiv:gr-qc/0202064 [gr-qc].[15] S. Tsujikawa, Class. Quant. Grav. , 214003 (2013),arXiv:1304.1961 [gr-qc].[16] E. Di Valentino et al. , (2020), arXiv:2008.11283 [astro-ph.CO].[17] E. Di Valentino et al. , (2020), arXiv:2008.11284 [astro-ph.CO].[18] E. Di Valentino et al. , (2020), arXiv:2008.11285 [astro-ph.CO].[19] G. Efstathiou, (2020), arXiv:2007.10716 [astro-ph.CO].[20] S. Borhanian, A. Dhani, A. Gupta, K. G. Arun, and B. S.Sathyaprakash, (2020), arXiv:2007.02883 [astro-ph.CO].[21] A. Hryczuk and K. Jodłowski, Phys. Rev. D102 , 043024(2020), arXiv:2006.16139 [hep-ph].[22] A. Klypin, V. Poulin, F. Prada, J. Primack,M. Kamionkowski, V. Avila-Reese, A. Rodriguez-Puebla,P. Behroozi, D. Hellinger, and T. L. Smith, (2020),arXiv:2006.14910 [astro-ph.CO].[23] M. M. Ivanov, Y. Ali-Haïmoud, and J. Lesgour-gues, Phys. Rev.
D102 , 063515 (2020), arXiv:2005.10656[astro-ph.CO].[24] A. Chudaykin, D. Gorbunov, and N. Nedelko, JCAP , 013 (2020), arXiv:2004.13046 [astro-ph.CO].[25] K.-F. Lyu, E. Stamou, and L.-T. Wang, (2020),arXiv:2004.10868 [hep-ph].[26] G. Alestas, L. Kazantzidis, and L. Perivolaropoulos,(2020), arXiv:2004.08363 [astro-ph.CO]. [27] P. Motloch and W. Hu, Phys. Rev.
D101 , 083515 (2020),arXiv:1912.06601 [astro-ph.CO].[28] N. Frusciante, S. Peirone, L. Atayde, and A. De Fe-lice, Phys. Rev.
D101 , 064001 (2020), arXiv:1912.07586[astro-ph.CO].[29] W. Yang, E. Di Valentino, O. Mena, S. Pan, andR. C. Nunes, Phys. Rev.
D101 , 083509 (2020),arXiv:2001.10852 [astro-ph.CO].[30] E. Di Valentino, (2020), arXiv:2011.00246 [astro-ph.CO].[31] E. Di Valentino and O. Mena, (2020), 10.1093/mn-rasl/slaa175, arXiv:2009.12620 [astro-ph.CO].[32] H. B. Benaoum, W. Yang, S. Pan, and E. Di Valentino,(2020), arXiv:2008.09098 [gr-qc].[33] W. Yang, E. Di Valentino, S. Pan, and O. Mena, (2020),arXiv:2007.02927 [astro-ph.CO].[34] E. Di Valentino, E. V. Linder, and A. Melchiorri, (2020),arXiv:2006.16291 [astro-ph.CO].[35] E. Di Valentino, S. Gariazzo, O. Mena, and S. Vagnozzi,(2020), arXiv:2005.02062 [astro-ph.CO].[36] W. Yang, E. Di Valentino, O. Mena, and S. Pan, Phys.Rev.
D102 , 023535 (2020), arXiv:2003.12552 [astro-ph.CO].[37] D. Vasak, J. Kirsch, and J. Struckmeier, Eur. Phys. J.Plus , 404 (2020), arXiv:1910.01088 [gr-qc].[38] S. Capozziello, Int. J. Mod. Phys.
D11 , 483 (2002),arXiv:gr-qc/0201033 [gr-qc].[39] S. Capozziello, S. Carloni, and A. Troisi, Recent Res.Dev. Astron. Astrophys. , 625 (2003), arXiv:astro-ph/0303041.[40] S. Capozziello, F. S. N. Lobo, and J. P. Mimoso, Phys.Lett. B , 280 (2014), arXiv:1312.0784 [gr-qc].[41] H. Chen, F.-H. Ho, J. M. Nester, C.-H. Wang, and H.-J.Yo, JCAP , 027 (2009), arXiv:0908.3323 [gr-qc].[42] H. Arcos and J. Pereira, Int. J. Mod. Phys. D , 2193(2004), arXiv:gr-qc/0501017.[43] A. V. Minkevich, A. S. Garkun, and V. I. Kudin, Class.Quant. Grav. , 5835 (2007), arXiv:0706.1157 [gr-qc].[44] K.-F. Shie, J. M. Nester, and H.-J. Yo, Phys. Rev. D ,023522 (2008), arXiv:0805.3834 [gr-qc].[45] G. Unger and N. Popławski, Astrophys. J. , 78(2019), arXiv:1808.08327 [gr-qc].[46] J. Struckmeier and A. Redelbach, International Journalof Modern Physics E , 435 (2008), arXiv:0811.0508[math-ph].[47] J. Struckmeier, J. Phys. G , 015007 (2013),arXiv:1206.4452 [nucl-th].[48] J. Struckmeier, D. Vasak, and H. Stoecker, Proceedings,4th International Symposium on Strong ElectromagneticFields and Neutron Stars (STARS2015): Havana, Cuba,May 10-16, 2015 , Astron. Nachr. , 731 (2015).[49] J. Struckmeier, J. Muench, D. Vasak, J. Kirsch,M. Hanauske, and H. Stoecker, Phys. Rev.
D95 , 124048(2017), arXiv:1704.07246 [gr-qc].[50] J. Struckmeier, J. Muench, P. Liebrich, M. Hanauske,J. Kirsch, D. Vasak, L. Satarov, and H. Stoecker, Int.J. Mod. Phys.
E28 , 1950007 (2019), arXiv:1711.10333[gr-qc].[51] J. Struckmeier, D. Vasak, A. Redelbach, P. Liebrich, andH. Stöcker, (2018), arXiv:1812.09669 [gr-qc].[52] R. Utiyama, Phys. Rev. , 1597 (1956).[53] D. W. Sciama, “On the analogy between charge and spinin general relativity,” in
Recent Developments in GeneralRelativity (1962) p. 415.[54] T. Kibble, Phys. Rev. , 1554 (1967). [55] F. Hehl, P. Von Der Heyde, G. Kerlick, and J. Nester,Rev. Mod. Phys. , 393 (1976).[56] K. Hayashi and T. Shirafuji, Prog. Theor. Phys. , 318(1981).[57] D. Benisty, E. I. Guendelman, D. Vasak, J. Struck-meier, and H. Stoecker, Phys. Rev. D98 , 106021 (2018),arXiv:1809.10447 [gr-qc].[58] S. Bahamonde, C. G. Böhmer, S. Carloni, E. J. Copeland,W. Fang, and N. Tamanini, Phys. Rept. , 1(2018), arXiv:1712.03107 [gr-qc].[59] S. D. Odintsov, D. Sáez-Chillón Gómez, and G. S.Sharov, Eur. Phys. J.
C77 , 862 (2017), arXiv:1709.06800[gr-qc].[60] R. Jimenez and A. Loeb, Astrophys. J. , 37 (2002),arXiv:astro-ph/0106145 [astro-ph].[61] M. Moresco, L. Verde, L. Pozzetti, R. Jimenez, andA. Cimatti, JCAP , 053 (2012), arXiv:1201.6658[astro-ph.CO].[62] M. Moresco et al. , JCAP , 006 (2012),arXiv:1201.3609 [astro-ph.CO].[63] M. Moresco, Mon. Not. Roy. Astron. Soc. , L16(2015), arXiv:1503.01116 [astro-ph.CO].[64] M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez,C. Maraston, L. Verde, D. Thomas, A. Citro, R. To-jeiro, and D. Wilkinson, JCAP , 014 (2016),arXiv:1601.01701 [astro-ph.CO].[65] D. Scolnic et al. , Astrophys. J. , 101 (2018),arXiv:1710.00845 [astro-ph.CO].[66] F. K. Anagnostopoulos, S. Basilakos, and E. N. Sari-dakis, Eur. Phys. J.
C80 , 826 (2020), arXiv:2005.10302[gr-qc].[67] D. Benisty and D. Staicova, (2020), arXiv:2009.10701[astro-ph.CO].[68] W. J. Percival et al. (SDSS), Mon. Not. Roy. Astron. Soc. , 2148 (2010), arXiv:0907.1660 [astro-ph.CO].[69] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders, andF. Watson, Mon. Not. Roy. Astron. Soc. , 3017(2011), arXiv:1106.3366 [astro-ph.CO].[70] N. G. Busca et al. , Astron. Astrophys. , A96 (2013),arXiv:1211.2616 [astro-ph.CO].[71] L. Anderson et al. , Mon. Not. Roy. Astron. Soc. ,3435 (2013), arXiv:1203.6594 [astro-ph.CO].[72] H.-J. Seo et al. , Astrophys. J. , 13 (2012),arXiv:1201.2172 [astro-ph.CO].[73] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival,A. Burden, and M. Manera, Mon. Not. Roy. Astron.Soc. , 835 (2015), arXiv:1409.3242 [astro-ph.CO].[74] R. Tojeiro et al. , Mon. Not. Roy. Astron. Soc. , 2222(2014), arXiv:1401.1768 [astro-ph.CO].[75] J. E. Bautista et al. , Astrophys. J. , 110 (2018),arXiv:1712.08064 [astro-ph.CO].[76] E. de Carvalho, A. Bernui, G. C. Carvalho, C. P.Novaes, and H. S. Xavier, JCAP , 064 (2018),arXiv:1709.00113 [astro-ph.CO].[77] M. Ata et al. , Mon. Not. Roy. Astron. Soc. , 4773(2018), arXiv:1705.06373 [astro-ph.CO].[78] T. M. C. Abbott et al. (DES), Mon. Not. Roy. Astron.Soc. , 4866 (2019), arXiv:1712.06209 [astro-ph.CO].[79] Z. Molavi and A. Khodam-Mohammadi, Eur. Phys. J.Plus , 254 (2019), arXiv:1906.05668 [gr-qc].[80] N. B. Hogg, M. Martinelli, and S. Nesseris, (2020),arXiv:2007.14335 [astro-ph.CO]. [81] M. Martinelli et al. (EUCLID), (2020), arXiv:2007.16153[astro-ph.CO].[82] N. Benitez et al. , Astrophys. J. , 241 (2009),arXiv:0807.0535 [astro-ph].[83] W. J. Handley, M. P. Hobson, and A. N. Lasenby,Mon. Not. Roy. Astron. Soc. , L61 (2015),arXiv:1502.01856 [astro-ph.CO].[84] A. Lewis, (2019), arXiv:1910.13970 [astro-ph.IM].[85] A. G. Riess, S. Casertano, W. Yuan, L. M.Macri, and D. Scolnic, Astrophys. J. , 85 (2019),arXiv:1903.07603 [astro-ph.CO].[86] K. P. Burnham and D. R. Anderson, Socio-logical Methods & Research , 261 (2004),https://doi.org/10.1177/0049124104268644.[87] A. R. Liddle, Mon. Not. Roy. Astron. Soc. , L74(2007), arXiv:astro-ph/0701113.[88] F. K. Anagnostopoulos, S. Basilakos, and E. N.Saridakis, Phys. Rev. D100 , 083517 (2019),arXiv:1907.07533 [astro-ph.CO]. [89] E. Aubourg et al. , Phys. Rev. D , 123516 (2015),arXiv:1411.1074 [astro-ph.CO].[90] N. Aghanim et al. (Planck), (2018), arXiv:1807.06209[astro-ph.CO].[91] S. Alam et al. (eBOSS), (2020), arXiv:2007.08991 [astro-ph.CO].[92] D. Baumann, in Theoretical Advanced Study Institute inElementary Particle Physics: Physics of the Large andthe Small (2011) pp. 523–686, arXiv:0907.5424 [hep-th].[93] J. Martin, C. Ringeval, and V. Vennin, Phys. Dark Univ. , 75 (2014), arXiv:1303.3787 [astro-ph.CO].[94] D. Benisty, D. Vasak, E. Guendelman, andJ. Struckmeier, Mod. Phys. Lett.
A34 , 1950164 (2019),arXiv:1807.03557 [gr-qc].[95] Y. Mao, M. Tegmark, A. H. Guth, and S. Cabi, Phys.Rev. D , 104029 (2007), arXiv:gr-qc/0608121.[96] F. W. Hehl, Y. N. Obukhov, and D. Puetzfeld, Phys.Lett. A377