Can teleparallel 𝑓 ( 𝑇 ) models play a bridge between early and late time Universe?

The ability of Big Bang Nucleosynthesis theory to accurately predict the primordial abundances of helium and deuterium, as well as the baryon content of the Universe, is considered one of the most significant achievements in modern physics. In the present study, we consider two highly motivated hybrid 𝑓 ( 𝑇 ) models and constrain them using the observations from the Big Bang Nucleosynthesis era. In addition, using late-time observations of Cosmic Chronometers and Gamma-Ray-Bursts, the ranges of the model parameters are confined which are in good agreement with early time bounds. Subsequently, the common ranges obtained from the analysis for early and late time are summarized. Further, we verify the intermediating epochs by investigating the profiles of cosmographic parameters using the model parameter values from the common range. From this study, we find the considered teleparallel models are viable candidates to explain the primordial-intermediating-present epochs.


INTRODUCTION
The present understanding of the Universe's evolution is strongly explained on the basis of the Friedmann-Robertson-Walker (FRW) cosmological model, commonly known as the hot Big Bang model.Cosmological observations from Type Ia Supernovae (Riess et al. 1998;Perlmutter et al. 1999;Riess et al. 2004), Baryon acoustic oscillations (BAO) (Eisenstein et al. 2005;Percival et al. 2010), galaxy redshift survey (Fedeli et al. 2009), large-scale structure (LSS) (Koivisto and Mota 2006;Daniel et al. 2008;Nadathur et al. 2020), and cosmic microwave background radiation (CMBR) (Spergel et al. 2003), prove that the Universe is undergoing accelerated expansion.This phenomenon is generally attributed to the presence of a dark energy (DE) sector in the Universe, a hypothetical energy source characterized by negative pressure.The most successful model to explain this phenomenon is the one in which the cosmological constant Λ explains DE, known as the concordance model.The concrete affirmation and validity of the model comes from the very beginning of the timeline covering primordial nucleosynthesis to the present-time acceleration.Based on the cosmological principle and the high energy interactions of fundamental elements, the standard cosmological model is capable of predicting the events that can reach up to nearly 10 −43 s of the post-Big Bang era.Currently, the standard model of particle physics, represented by the  (3)  ⊗ (2)  ⊗ (1)  gauge theory of strong and electroweak interactions, presents a fundamental framework for understanding quarks and leptons which has been experimentally verified at energies up to approximately 1000 GeV.
★ E-mail: kavya.samak.10@gmail.com† E-mail: saiswagat009@gmail.com‡ E-mail: pksahoo@hyderabad.bits-pilani.ac.in § E-mail: vensmath@gmail.com Numerous observational surveys, with several teams working together, probe the evolution of the Universe at different stages, covering a wide range of observational factors.For instance, the Hubble parameter  0 , the present mass density  0 , the deceleration parameter  0 , and the distance modulus  provide insight into late-time phenomena of the Universe (Spergel et al. 2007;Komatsu et al. 2011;Demianski et al. 2017), and observations of the transition redshift   are crucial for understanding changes in the Universe's phase.However, observations of the intermediate epoch of the Universe remain under explored to date.Interestingly, there are several significant experimental observations of the primordial Universe to center on, such as the abundance of 3 He, 4 He, D, 7 Li, the baryonto-photon ratio, and the CMBR, which help us predict dynamics as early as 0.01 s after the Big Bang.Since the Universe undergoes expansion and cooling in its early stages, the rates of weak interactions, pivotal for maintaining thermal equilibrium between neutrinos and matter, decrease progressively.Likewise, the rates of reactions converting neutrons and protons decline.Eventually, these rates dip below the expansion rate, leading to a freeze-out.Simultaneously, the strong and electromagnetic nuclear reaction rate also decreases and undergoes freeze-out.These epochs contribute to the nucleosynthesis era of the Big Bang model (Fields et al. 2020;Barrow et al. 2021;Anagnostopoulos et al. 2023;Kang and Panotopoulos 2009;Mishra et al. 2024;Capozziello et al. 2017).
Though the standard cosmological model provides viable explanations for phenomena from early to late times and aligns well with observational data, it has some limitations (Joyce et al. 2015;Tsujikawa 2003;Velten et al. 2014;Astesiano and Ruggiero 2022;Katsuragawa and Matsuzaki 2017;Zaregonbadi et al. 2016;Joudaki et al. 2017).This necessitates exploring models beyond ΛCDM.While quintessence models, which incorporate scalar fields with potential, can explain cosmological data with certain adjustments, the absence of strongly motivated scalar field models from theoretical particle physics is significant.Consequently, the effects of cosmic acceleration might be better explained within the framework of modified gravity theories, where extensions of General Relativity (GR) are considered.
In this study, we explore an extension of gravitational theory known as the Teleparallel Equivalent of General Relativity (TEGR), which was initially proposed by Einstein himself in (Unzicker and Case 2005;Maluf et al. 2002;Obukhov and Pereira 2003;Ferraro and Fiorini 2007;Linder 2010).TEGR describes the gravitational field using the torsion tensor.Although many results from TEGR match those from GR, extensions of TEGR can yield different outcomes compared to extensions of GR (Cai et al. 2016).As a result, torsionbased modified theories have garnered significant attention and have been thoroughly investigated in recent years (Wu and Yu 2010;Chen et al. 2011;Dent et al. 2011;Sudharani et al. 2024;Bahamonde et al. 2022;Moreira et al. 2021;Mishra et al. 2023;Kofinas and Saridakis 2014;Gonzalez-Espinoza et al. 2021).One can also see some interesting works that adopt different techniques to constrain  () gravity models (????).Motivated by this, we study two teleparallel  () models and their influence on the dynamics of the Universe from early to late times.Article organization: In section 2 we provide a detailed geometric foundation of  () teleparallel gravity.The section 3 presents the formalism of BBN.Using BBN constraints we analyze two hybrid gravity models in section 4. The early Universe constraints are obtained from BBN, while the late time constraints are obtained from recent observational data sets of Cosmic chronometers and Gamma-Ray-Bursts.The methodology and results of this procedure are explained in section 5 and in section 6, we conclude our results.

FORMALISM OF 𝐹 (𝑇) GRAVITY
The simplest departure from the Teleparallel Equivalent of General Relativity is considering an arbitrary functional form of the torsion scalar gives rise to the so-called "  () theory".This theory uses the Weitzenböck connection instead of the usual torsion-less Levi-Civita connection of GR.Using the connection, the non-null torsion tensor can be obtained as The torsion scalar can be defined as where the superpotential tensor is defined as and the contorsion tensor is defined as Using the torsion scalar one can modify the GR action to teleparallel action as where  = √ 8 and  =  (   ) = √ −.Variation of the above action with respect to the tetrads gives rise to the motion equation Inserting the vierbein in the motion equation ( 4), the modified Friedmann equations can be obtained as where  =   +   and  =   +   .The above motion equations can be rewritten in the GR equivalent form as The dark energy components in the above equations are expressed in the context of  () gravity as

BACKGROUND OF BBN
Recent studies have suggested that BBN occurred within the initial fractions of seconds following the Big Bang, approximately around 0.01 s, extending to several hundred seconds after.During this period, the Universe was extremely hot and dense.BBN, along with the CMBR, provides firm validation for the high temperatures that characterized the primordial Universe.Here, we shall explore BBN physics in the context of  () teleparallel gravity.The occurrence of BBN takes place in the era in which radiation was dominating.
The relativistic elements (as well as mass-less radiation) in this stage have the energy density   given by Here, T is the temperature, and  * represents the effective number of degrees of freedom, which is nearly equal to 10.The abundance of neutrons is calibrated by considering the proton-to-neutron conversion rate (Kolb and Turner 1990) The total rate is given by where    (T ) is the inverse of   (T ).In view of (12), the above equation takes the form (Anagnostopoulos et al. 2023) In the above equation,  represent the neutron-proton mass difference, given by  =   −   = 1.29 × 10 −3 GeV .Additionally, A is taken as A = 1.02 × 10 −11 GeV −4 .Using the relation the primordial mass fraction of 4 He can be calibrated with   and T  being the freeze-out time and freeze-out temperature for nucleosynthesis and weak interactions, respectively, and  refers to the mean lifetime of neutrons (Barrow et al. 2021).
In the context of GR, the first Friedmann equation can be written as  2 =  2 3     (7), where     is the effective energy density of the system.Since radiation dominates in the BBN era, observational evidence indicates that any other contribution is negligible compared to radiation when the concordance model is considered.Thus, we have where  = 1/  , with   = 1.22 × 10 19 GeV being the plank mass.
Here, we provide the convention adopted in the later part of this work: to distinguish the Hubble rate in GR from that obtained in  () theory, we denote the former as   and the latter simply as .Thus, equation ( 7) along with the equation ( 16) reads Expanding the quantity within the square root to first order is justified by the dominance of dark energy density being significantly smaller than that of radiation during the radiation-dominated era and thus obtaining The equation (11) in the above equation yields Thus, the scale factor evolves as a ∼  1/2 , with  being the cosmic time.The relation between temperature and time is therefore given by 1 or in other words T () ∼ (/sec) −1/2 MeV.Central to BBN is the concept of the freeze-out temperature, T  .Neutrons and protons interconvert via weak interactions.When the temperature exceeds 1 MeV, much higher than the expansion rate, these conversions remain in equilibrium.However, as the temperature falls below 1 MeV, the neutron-to-proton ratio 'freezes out' at approximately 1/6, gradually declining due to the decay of free neutrons.The freeze-out temperature in hand is related to the specific Hubble rate  (T  ) =   (T  ) ≈   T 5  , where   = 4A4!≈ 9.8×10 −10 GeV −4 .Utilizing equation ( 19) at T = T  , we find Given  (T  ) =   T 5  , we derive Δ = 5   4  ΔT  .Substituting Δ from equation ( 18), we obtain Taking into account the observational constraint, we arrive at the expression constraining the freeze-out temperature as

CONSTRAINING 𝐹 (𝑇) MODELS
From theoretical and observational aspects of BBN, one can derive constraints on a given cosmological model.A cosmological model is presumed to be viable only if it meets the conditions set by BBN.Therefore, in this section, we shall compare various  () gravity models with BBN calculations.

Hybrid exponential model
We begin considering the Hybrid exponential  () model having the form with  being the free parameter.The model, in similar lines with (Anagnostopoulos et al. 2021), has proved its worth by providing a better fit to the observational datasets than the Λ  model.The model reduces to TEGR with the limit  = 0.The dark energy density (9) for this model can be calculated as Using the above expression in the first motion equation ( 5), we obtain where Ω 0 =  0 /3 0 2 , is the present density parameter.Again we use (25) to find the deviation in freeze out temperature from (22), where  = is considered as  0 = 1.47 × 10 −42  which is equivalent to the value from Planck 2018 results i.e.,  0 = 67.2km  −1 Mp −1 .
We depict from Figure 1 that the model satisfies the BBN bound (23) for  ≤ 7.3147 × 10 34 .The blue dashed curve represents the upper bound of the deviation ΔT  T  = 0.00047 .As the curve is asymptotic to zero towards the left, the domain we obtain for the model parameter is unbounded below.

Hybrid tangent hyperbolic model
The Hybrid tangent hyperbolic model is motivated from (Wu and Yu 2011) because it allows phantom divide crossing line for the effective equation of state (EoS) parameter and corroborates with the recent observational datasets.The model consists of a tangent hyperbolic and a power law term that has the following functional form with  and  being the free parameters.The model reduces to TEGR with the limit  = 0.The dark energy density (9) for this model reads where  = / 0 .Using the above expression in the first motion equation ( 5), we obtain Considering the present time scenario in the above equation, one can obtain the dependency between the model parameters as Using (29) in the theoretical expression for the deviation in freeze-out temperature ( 22), we get ΔT  T  = 1 where  = T  = 0.00047 .Similar to the first model, the curve is asymptotic to zero towards the left, hence the domain for the model parameter is unbounded below.

Datasets
To develop a cosmological model that viably explains the evolution of the Universe, we need to find valid ranges of space for free parameters.This is achieved through the Bayesian approach along with the Markov Chain Monte Carlo (MCMC) methodology.For this purpose, in our present work, we use Cosmic Chronometers and Gamma-Ray-Bursts datasets.

Cosmic Chronometers (CC)
Using the differential aging technique, the CC method deduces the Hubble rate by studying ancient, quiescent galaxies that are closely positioned in redshift.This approach is based on defining the Hubble rate  = − 1 1+   for a FLRW metric.The CC method is significant because it can determine the Hubble parameter independently of any specific cosmological model.Based on various surveys (Jimenez et al. 2003;Simon et al. 2005;Stern et al. 2010;Moresco et al. 2012;Zhang et al. 2014;Moresco 2015;Moresco et al. 2016;Ratsimbazafy et al. 2017), we utilized 34 CC data points in this study within the redshift range of 0.1 <  < 2.

Gamma-Ray-Bursts (GRBs)
GRBs are the most intense explosions in the Universe, having enormous energy.It is possible to observe these cosmic phenomena at comparatively higher redshifts.The correlation between   ,  , , and  iso is given by log 10 iso 1 erg =  +  log 10  , 300 keV where   represents the rest-frame spectrum peak energy,  iso is the isotropic-equivalent radiated energy (Amati et al. 2002),  and  are constants, and  , denotes the spectral peak energy in the cosmological rest frame of the GRBs. , correlates with the observer-frame quantity   through its relation being derived from   as  , =   (1 + ).  can be calibrated using the luminosity distance   and bolometric fluence  bolo which is related as In the present work, we analyze 162 data points for the redshift ranging from 0 to 9.3 (Demianski et al. 2017).This offers the possibility of studying beyond the Supernovae observation and covering a higher redshift.
To find the best-fit parameter space, we maximize the likelihood , where  2 function is defined by, Here,  denotes the covariance matrix of uncertainties, and Δ is the difference between the theoretical and observational values of the key physical quantity.For instance,  signifies the Hubble parameter for CC and the distance modulus function to that of GRBs.

Results and Interpretation
In the previous section, we restricted the model parameters of two highly motivated models through BBN constraints.This section is dedicated to test the credibility of the said models in various epochs while staying within the limits.We start by performing an MCMC analysis to constrain the free parameters using the CC and GRB datasets.It can be observed in Figure 3-6 that the model parameter ranges obtained from the sampling lie well within the BBN range which confirms that the models are suitable candidates to explain the early time as well as late-time behavior of the Universe.However, agreement with all the intermediating epochs is another essential requirement for any theory to be considered as an alternative to GR.In this context, we study the behavior of the deceleration parameter for both our models.For the hybrid exponential model, one can observe in Figure 11 that the parameter transits from the deceleration to the acceleration zone at redshifts mentioned in Table 1.On the other hand, for the hybrid tangent hyperbolic model in Figure 12, the q-profile shows a better trend in the phase transition with more accurate redshifts.The present values of the deceleration parameter and the transition redshifts summarized in Table 1 are found to be suitably aligned with the recent observational data.
Furthermore, the Hubble and distance modulus functions are confronted with the 34 data points of CC and 162 data points of GRB  datasets, respectively.The GRB data is particularly considered because of its capability to explore the higher redshifts.For both models, we find the functions are in nice agreement with the data points (See Figure 7-10).The common ranges obtained for the model parameters from considering early and late-time analysis are summarized in Table 2.

CONCLUDING REMARKS
To execute this work, we adopted a new approach that includes different evolutionary phases of the Universe starting from the nucleosynthesis era.The challenging outcomes of this work raise curiosity to  know more about the assumed teleparallel models.The presence of T term in the denominator makes both the functions approach zero in early times ( → ∞).This indicates that without any limits on the model parameters, the models reduce to TEGR at the time of the Big Bang, which solves the cosmological constant problem.A similar line of reasoning can be followed for the late-time as well, the coupling with polynomial T term makes it GR-like when  → 0.
To find a proper justification for whether the models are eligible candidates to connect the early-time and late-time epochs, we commenced from the primordial era by constraining them using BBN observations.The model parameters are restricted as  ≤ 7.3147×10 34 for the exponential model and  ≤ 1.94566 for the tangent hyperbolic model.Two widely used datasets CC and GRB have been used to constrain the models further by performing the MCMC technique.It is to be noted that the ranges of the model parameters align with the range obtained from BBN and the other cosmological parameters are in agreement with the observational data (Aghanim et al. 2020;Mandal et al. 2023).Deceleration parameter, the most important tool to show phase changes in the intermediate epochs, is then studied to observe the behavior of our models in the transitional times.We find that both models show the transition from the deceleration zone to the acceleration zone.The redshift at which the alteration is occurring is slightly higher (Rani et al. 2015;Naik et al. 2023) for the Hybrid exponential model constrained by CC+GRB, while it is around the most widely accepted value for the remaining three (Cunha and Lima 2008).For the hybrid tangent hyperbolic model, the deceleration parameter approaches 0.5 for higher redshift which coincides with the behavior of the ΛCDM model.However, for the hybrid exponential model, a definitive intermediate epoch cannot be observed as it deviates after a certain redshift.Nevertheless, in both models, the present values of the deceleration parameter confirm the ongoing accelerated expan-sion of the Universe.In addition, the constrained teleparallel  () models are found to be viable in the early phase of the Universe and they align with the data points of the CC and GRB dataset, which is verified using  () and () functions.
Thus by joining the pieces, we conclude that both the hybrid exponential and hybrid tangent hyperbolic models can fairly explain the physical aspects of the dynamics of the Universe.In particular, the hybrid tangent hyperbolic model can be considered a fine alternative to GR.
4) where   ,   are the first and second derivative of  with respect to the torsion scalar, respectively and     denotes the usual energy momentum tensor.For further discussion, we consider a spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric of the form  2 =  2 − 2 ()        with () being the scale factor in terms of time.The vierbein ansatz, corresponding to the metric, considered of the form    = (1, (), (), ()).The vierbein along with (2) gives rise to the relation  = −6 2 where the Hubble parameter  = /.

2 𝜋 2 FFigure 1 .
Figure 1. vs ΔT  /T  for the Hybrid exponential model.The blue dashed line represents the upper bound of ΔT  /T  .

FFigure 2 .
Figure 2.  vs ΔT  /T  for the Hybrid tangent hyperbolic model.The blue dashed line represents the upper bound of ΔT  /T  .
Figure 2 that the model satisfies the BBN bound (23) for  ≤ 1.94566.The values of fixed parameters are considered the same as the first model.By straightforward calculation, one can obtain the range for the other model parameter  from (31).The blue dashed curve represents the upper bound of the deviation ΔT

Figure 3 .
Figure 3. Likelihood probability distributions and 2D contours for the Hybrid exponential model obtained from MCMC analysis of the CC dataset.The dark-shaded region represents 1 CL and the light-shaded region represents 2 CL.

Figure 4 .Figure 5 .
Figure 4. Likelihood probability distributions and 2D contours for the Hybrid exponential model obtained from MCMC analysis of the CC + GRB dataset.The dark-shaded region represents 1 CL and the light-shaded region represents 2 CL.

Figure 6 .Figure 7 .
Figure 6.Likelihood probability distributions and 2D contours for the Hybrid tangent hyperbolic model obtained from MCMC analysis of the CC + GRB dataset.The dark-shaded region represents 1 CL and the light-shaded region represents 2 CL.

Figure 8 .Figure 9 .Figure 10 .Figure 11 .Figure 12 .
Figure 8.The Hubble function (constrained from CC and CC+GRB) against redshift and error bars of CC dataset for the Hybrid tangent hyperbolic model.
) create a tangent space at each point   of the manifold.The metric tensor of the manifold can be written in terms of the tetrad fields as   () =      ()

Table 1 .
Summary of the transition redshift (  ) and present value of deceleration parameter ( 0 ) obtained from  () of both models.

Table 2 .
Summary of the ranges obtained from both early and late time for the  ( ) models.In the first row, the results of model parameter  of the Hybrid Exponential model are presented while in the second row, the results of model parameter  of the Hybrid Tangent Hyperbolic model are presented.