Modelindependent reconstruction of teleparallel cosmology
Abstract
We propose a modelindependent formalism to numerically solve the modified Friedmann equations in the framework of teleparallel cosmology. Our strategy is to expand the Hubble parameter around the redshift up to a given order and to adopt cosmographic bounds as initial settings to determine the corresponding function. In this perspective, we distinguish two cases: the first expansion is up to the jerk parameter, the second expansion is up to the snap parameter. We show that inside the observed redshift domain , only the net strength of is modified passing from jerk to snap, whereas its functional behavior and shape turn out to be identical. As first step, we set the cosmographic parameters by means of the most recent observations. Afterwards, we calibrate our numerical solutions with the concordance CDM model. In both cases, there is a good agreement with the cosmological standard model around , with severe discrepancies outer of this limit. We demonstrate that the effective dark energy term evolves following the testfunction: . Bounds over the set are also fixed by statistical considerations, comparing discrepancies between with data. The approach opens the possibility to get a wide class of testfunctions able to frame the dynamics of without postulating any model a priori. We thus reobtain the function through a backscattering procedure once is known. We figure out the properties of our function at the level of background cosmology, to check the goodness of our numerical results. Finally, a comparison with previous cosmographic approaches is carried out giving results compatible with theoretical expectations.
I Introduction
A challenge of modern cosmology is to determine how the universe constituents precisely affect the dynamics as the universe expands. In particular, at late times of its evolution, the universe seems to be dominated by an exotic fluid dubbed dark energy, whose origin is so far unknown uno (). A wide number of observations points out that dark energy should act as a fluid with negative pressure counterbalancing the action of gravity, and speeding up the universe after a transition epoch SNe (); Planck15 (). The standard approach which most likely aims to describe such a dynamics assumes that the source of dark energy is supplied by a nonzero cosmological constant , associated to vacuum quantum field fluctuations due (). The paradigm which makes use of is referred to as the CDM concordance model and represents the simplest approach describing the observed universe dynamics tre ().
Although wellsupported by observations, the concordance model does not give explanations towards the coincidence problem Zlatev99 () between dark matter and dark energy orders of magnitude and the finetuning issue between predictions of quantum gravity and today observational constraints on the value of quattro (). Hence, a possible alternative interpretation is that the fluid which triggers the current universe speed up may not be due to . For example, one can conclude that the effect of cosmic acceleration may be obtained in the framework of modified gravities where extensions of General Relativity (GR) are accounted. In any gravitational theorie which extends GR, additional degrees of freedom could be interpreted as dark energy sources cinque (); francaviglia ().
In the context of modified gravity, it is possible to replace the Ricci scalar in the EinsteinHilbert action by arbitrary functions of and any curvature invariant giving rise to Extended Theories of Gravity cinque (); deFelice10 (); Odin (). However a torsional formulation of GR, namely the Teleparallel Equivalent of General Relativity (TEGR) ft1 () is also possible. In this approach, the gravitational action is given by the torsion scalar and one can construct the extension of TEGR storny1 (); Teleparallelism (); Cai15 (), where is a function of . The predictions of GR turn out to coincide with TEGR, whereas the same does not happen for with respect to gravity Cai15 (). Indeed, models lead to different outcomes if compared with the corresponding approach. For these reasons, the formulation of gravity represents an interesting approach with a significative number of cosmological implications which are today object of intensive investigation Teleparallelism (); Cai15 (); storny2 (); storny3 ().
In this paper we are going to investigate a modelindependent approach to reconstruct the function from cosmological observations. In this perspective, we consider the cosmolgical Friedmann equations for a generic analytical function. Afterwards, we recast the corresponding equations in terms of the redshift and Hubble parameter . In so doing, we obtain a differential equation which can be numerically solved, once is somehow fixed by data. Even though we are interested in reconstructing , if we impose parametric forms of we would fix the model a priori, influencing our numerical analyses which turn out to be model dependent. To overcome this problem, we expand in Taylor series around , which corresponds to present time . Since cosmography or cosmokinetics lie on expansions of around , our expansions of can be related to the cosmographic parameters, as in the case of . In particular, we consider two expansions: the former considering the cosmographic parameters and , while the latter considering also the snap parameter . In such a picture, the cosmographic series does not require the assumption of any model a priori. So that, using the most recent bounds over the cosmographic parameters, we are able to modelindependently reconstruct . In fact, it is possible to demonstrate that the above set of parameters univocally defines the shapes of in the interval . Furthermore, being , it is possible to set a direct correspondence between and . The former, , can be easily matched with arbitrary functions of the redshift . Thus, we first reconstruct by means of the cosmographic data and afterwards we come back to and its evolution in terms of .
In both cases, we find a good agreement between the predicted and recovered shapes of and and is it possible to notice that curve strengths are only modified, leaving unaltered the functional behaviors of each curves. To figure out this result, we numerically analyze the obtained curves with a class of testfunctions. We conclude that the best approach is offered by a particular class of combined exponential functions. With these considerations in mind, we obtain the corresponding and study the background cosmology. In particular, it is possible to describe the effective dark energy term and its evolution in time. Finally, it is possible to fix limits over the equation of state (EoS) and achieve a form for which well adapts to cosmic evolution at late times.
The layout of the paper is the following. In Sec. II we briefly summarize the main ingredients of teleparallel gravity and cosmology. In Sec. III, the basic requirements to solve numerically the modified Friedmann equations are discussed. In particular, the standard issues of cosmography are considered and the main reasons to conclude that cosmography is a modelindependent approach to frame the universe dynamics are discussed. In Sec. IV, the method for numerically solving the Friedmann equations is implemented and solutions for and are introduced. Sec. V is devoted to the investigation of the obtained models and, in Sec. VI, we compare our results with previous ones obtained in cosmography. Sec. VII is devoted to conclusions and perspectives.
Ii gravity and cosmology
The equations of motion in gravities can be derived using as dynamical variables the veirbein fields , which form an orthonormal basis for the tangent space at each point of a generic manifold^{1}^{1}1Here, we use the capital Latin indices to denote the coordinate of the tangent spacetime, while the Greek letters indicate the coordinates of the manifold.. Introducing the dual basis , the metric tensor reads ft1 ()
(1) 
where is the Minkowski metric of tangent space. The torsion tensor can be expressed by the zerocurvature Weitzenböck connections as Cai15 ()
(2) 
A convenient choice is to introduce the tensor
(3) 
where
(4) 
is the contorsion tensor. Hence, one can write the torsion scalar, which represents the teleparallel Lagrangian density, in the compact form
(5) 
Generalizing to a generic function of means that the gravity action can be rewritten as:
(6) 
where and . We will assume throughout the paper the physical units with . is the Lagrangian density of the matter fields. The field equations are thus obtained by varying the action (6) with respect to the vierbein fields:
(7) 
where represents the energymomentum tensor of matter. In the above relations, the primes indicate derivatives with respect to .
Hence, assuming the validity of the cosmological principle Peebles93 (), we consider the homogeneous and isotropic FriedmannLemaîtreRobertsonWalker (FLRW) metric for a spatially flat universe^{2}^{2}2Generalizing from spatially flat to nonflat is straightforward. However, spatial curvature would influence the cosmographic analysis, as we will discuss in Sec. III. In this paper we limit our attention to the flat case only.,
(8) 
which corresponds to take the veirbein as: . The modified Friedmann equations are thus sar (): \cref@addtoresetequationparentequation
(9a)  
(9b) 
where and represent the matter density and pressure for a perfect fluid source. In particular, we assume:

dark matter and dark energy do not interact between them;

the matter density, , scales as in standard cosmology, i.e. with ;

the matter pressure is to guarantee a pressureless counterpart composed by baryons and cold dark matter;

radiation, neutrinos, gravitational relics and so forth are negligibly small at our time and cannot be considered in our analyses,
which turn out to be the basic demands of any cosmological theory to be consistent with current observations. Moreover, in our picture and are the torsion density and pressure respectively: \cref@addtoresetequationparentequation
(10a)  
(10b) 
They are clearly zero for . One can define an “effective” dark energy torsional component, whose EoS takes the standard form in the hydrodynamic approach to cosmology galaxy (). Since, for each species one has:
(11) 
and
(12) 
then one gets:
(13) 
where we assumed
(14) 
It can be interpreted as the torsional counterpart of giving rise to dark energy effect. Depending on the form of , it can be the origin of today observed acceleration of the Hubble flow. Using Eqs. 10b and 10a, the Friedmann equations can be rewritten as Aviles13 () \cref@addtoresetequationparentequation
(15a)  
(15b) 
where is the normalized matter density parameter whose functional form scales as: , with the current value as measured by observations. In the FLRW universe, the torsion scalar obeys the following constraint:
(16) 
which relates the torsion directly with the Hubble parameter at all stages of the universe evolution. With these considerations in mind, in the next paragraph, we rewrite the Friedmann equations as function of the redshift and finding the form, once is constrained by cosmography.
Iii cosmography
Let us highlight now some basic assumptions of cosmography in view of modeling the dynamics of cosmografia1 (). The expansion of scale factor in Taylor series around the present time cosmografia2 () is
(17) 
From this expansion, one defines
(18)  
(19) 
named Hubble, deceleration, jerk and snap parameters, respectively. The luminosity distance can be written in terms of the redshift as cosmografia3 ()
(20) 
We can expand in series Eq. 20 by simply plugging Eq. 17 in it and then, by using the cosmographic parameters Eqs. 18 to 19 evaluated at our epoch, obtaining cosmografia4 ():
(21) 
Here, is the error associated to the truncation of the series at a given order. In particular, indicating with the errors over the cosmographic coefficients, one gets:
(22) 
with the order of truncating series. The consequences of such requests are that cosmography cannot be used with an arbitrary precision to frame the universe dynamics since any expansions is jeopardized by severe degeneracies among coefficients^{3}^{3}3For additional details see cosmografia4bis (). This is not the only limit of cosmography but in this work we do not need to discriminate among models when significatively increases, but we only require cosmography to guarantee that is featured in a modelindependent way cosmografia5 (). Thus, up to the fourth order, as reported in Eq. 21, one gets:
(23)  
Hence, using Eq. 20 one can relate the Hubble parameter to the luminosity distance by means of
(24) 
and so, making use of Eq. 21, one finds out the Taylor series expansion of as a function of the cosmographic parameters:
(25) 
where , while the first three orders read cosmografia6 ()
(26)  
Now, if the values of the parameters are known, one can combine Eqs. 15b and 15a to numerically infer the function . To figure out this point, we need to convert time derivatives and derivatives with respect to into derivatives with respect to the redshift . For any redshiftdepending function , it is cosmografia7 ():
(27)  
(28) 
where in the latter equation we have used Eq. 16. We will use Eqs. 28 and 27 in what follows, in order to get the forms of the functions and .
Iv Numerical reconstructions of function
In this section we want to reconstruct the form of by using the requirements imposed in the modified Friedmann equations. To do so, let us consider Eqs. 27 and 9a which give the following differential equation in terms of and :
(29) 
with
(30) 
As one can soon notice, Eq. 29 depends upon and . Assuming the form of leads to impose the model and to force the consequent analysis. In other words, assuming a given , for example the one of the concordance CDM model, would force to be compatible with a slightly evolving dark energy term. Since we are looking for reconstructing the Hubble flow at different stages as modelindependent as possible, we make use of cosmography to take free from any assumptions. Indeed, once the form of is reconstructed by Eq. 25, we obtain a modelindependent procedure which relates Eq. 29 to only, without passing through postulating the cosmological model. In particular, Eq. 29 is a secondorder differential equation which requires two initial conditions over and . On the one hand, the initial condition on can be obtained imposing that the present value of the effective gravitational constant is set to be the Newton constant cardone (); Aviles13 ():
(31) 
On the other hand, the initial condition over comes from Eqs. 31 and 16:
(32) 
Moreover, following the strategy in delaCruz16 (), we may recast the cosmographic parameters as altro1 ():
(33)  
(34)  
(35)  
with
(36) 
(37) 
With those assumptions in mind, we adopt the indicative values reported below delaCruz16 ():
(38) 
where . These results are compatible with the current expectations over the cosmographic set of parameters using different data sets altro2 ().
Thus, we handle the aforementioned limits over the cosmographic parameters to numerically solve Eq. 29. To do so, we perform a twostep analysis. In particular, we first consider the Hubble rate expansion up to the second order in , namely up to the jerk parameter. We then show our results in Fig. 1 for different values of the set , holding to be fixed.
We then match the numerical behaviours using the following testfunctions^{4}^{4}4Here, testfunctions are used to match a posteriori the shapes of numerical curves obtained from our numerical analysis. The forms of such functions have been chosen analysing the shapes and checking any formal trends of the involved curves.: \cref@addtoresetequationparentequation
(39a)  
(39b)  
(39c)  
(39d)  
(39e)  
(39f) 
with , and free coefficients to be fixed from a direct comparison with the numerical curves. To get suitable outcomes, we find a good analytical approximation according to the following requirements.

The function has to be neither odd nor even.

The linear term, i.e. , is not favored and does not contribute significantly around the sphere .

Cosmography does not influence the functional analysis but rather it fixed the strengths of any testfunction. The limits of cosmography are evident, since enlarging the prior domains leads to anomalous results on the form of which are not compatible with current dark energy behaviour.

The results are all compatible with the concordance paradigm, but do not exclude dark energy to vary.

The numerical analyses do not depend upon adding scalar curvature and radiation.
To find, among the testfunctions, the best approximation for , we perform the test Draper98 (). Denoting by the values of obtained from the numerical solution, and by the correspondent values, we can define
(40) 
where
(41) 
and is the number of points. The statistics provides information on how well the testfunctions approximate the numerical , being the ideal case in which the testfunction agrees exactly with .
Testfunction  

0.99974  
0.99997  
0.99102  
0.99630  
0.99909  
0.99959 
As an example, we consider the red curve of Fig. 1. The test suggests that the most suitable choice corresponds to the function (39b) (see Table 1), i.e.
(42) 
with
(43) 
However, it is evident from Table 1 that very good approximations for are also the functions Eq. 39a and Eq. 39f, whose values are far from the best one by only and , respectively. The comparison between the numerical solution of and the functional form of Eq. 42 is shown in Fig. 2 .
The second step takes up to the third order, restricting the value of the snap parameter to the interval cosmosupp ()
(44) 
In this case, fixing to the bestfit values, we show in Fig. 3 the behaviour of for different values of . Consistently with what found before, the best approximation for corresponding to the bestfit value of is provided by the same function as in Eq. 42, but with slightly different free parameters, namely:
(45) 
V cosmological models
Our goal is now to reconstruct the function through a backscattering procedure. In particular, inverting the expansion of up to the snap parameter by means of Eq. 16, one finds in terms of , which can be inserted back into Eq. 42 to finally reconstruct . To take into account the uncertainties in the estimate of the cosmographic parameters that may propagate through our numerical analysis, we rescale Eq. 42 as
(46) 
where the constant will be determined from cosmological constraints. One obtains:
(47) 
and
(48) 
where
(49) 
(50) 
(51) 
(52) 
We can use the reconstructed to study and and their cosmological implications and to compare our results with cosmological models developed in the literature so far h67 (). So that, Eqs. 10b and 10a become:
(53) 
(54) 
where
(55) 
(56) 
(57) 
(58) 
(59) 
and
(60) 
(61) 
(62) 
(63) 
We notice that is in fact independent of the rescaling factor . From the condition of having an accelerating expansion today, we are able to constrain by imposing that . Thus,
(64) 
Fig. 4 shows the reconstructed