1. Introduction
2. f(T, B) gravity with homogeneous space-time
3. Statistical fitting of H
Table 1. The three regression models and their corresponding R2. |
Regression model | R2 |
---|---|
H(z) = b0 + b1z + b2z2 | 0.9404 |
H1(z) = b0 + b1z | 0.9394 |
${H}_{2}(z)={b}_{0}{e}^{({b}_{1}z)}$ | 0.9246 |
Table 2. Summary of the best fit and the mean values of the cosmological parameters. |
Model | Par | Prior | Best fit | Mean |
---|---|---|---|---|
ΛCDM | Ωm | [0.001, 1] | ${0.312308}_{-0.00607812}^{+0.00607812}$ | ${0.312583}_{-0.00607362}^{+0.00607362}$ |
h | [0.4, 1] | ${0.678603}_{-0.00447778}^{+0.00447778}$ | ${0.678435}_{-0.00447417}^{+0.00447417}$ | |
Ωbh2 | [0.001, 0.08] | ${0.0224102}_{-0.000134672}^{+0.000134672}$ | ${0.0224002}_{-0.00013449}^{+0.00013449}$ | |
| ||||
Polynomial model | b1 | [0.001, 100] | ${55.87}_{-1.4800}^{+1.4800}$ | ${55.0}_{-1.500}^{+1.500}$ |
b2 | [0.001, 100] | ${0.961}_{-0.0171}^{+0.0171}$ | ${0.960}_{-0.017}^{+0.017}$ | |
H0 | [40, 100] | ${66.16}_{-1.2900}^{+1.2900}$ | ${66.2}_{-1.3000}^{+1.3000}$ |
Table 3. Summary of the ${\chi }_{{red}}^{2}$ and ${\chi }_{\min }^{2}$. |
Model | ${{\chi }_{{tot}}^{2}}^{\min }$ | ${\chi }_{{red}}^{2}$ |
---|---|---|
ΛCDM | 1073.9795 | 0.9816 |
Polynomial model | 1125.2628 | 1.0195 |
4. Observational constraints
R-1 | b1 | b2 | H0 |
---|---|---|---|
0.017025 | 0.01672 | 0.01692 |
Figure 1. The distance modulus behavior in terms of the redshift z. |
Figure 2. The Hubble parameter behavior in terms of the redshift z. |
Figure 3. The 1 − σ and 2 − σ confidence contours obtained from SNIa+BAO+H(z) data obtained for the statistical fitting model. |
4.1. Supernovae type Ia
4.2. Cosmic microwave background
4.3. Baryon acoustic oscillation
4.4. Hubble data
5. Some cosmographic parameters
Figure 4. Deceleration parameter $\left(q(z)\right)$ versus redshift $\left(z\right)$. |
Figure 5. Jerk parameter $\left(j(z)\right)$ versus redshift. |
6. Some specific models of f(T, B) gravity
6.1. Model-I
Figure 6. Energy density parameter of statistical f(T, B) = aT2 + cB2 gravity model $\left({\rho }_{d}\right)$ versus redshift $\left(z\right)$. |
Figure 7. Equation of state parameter of statistical f(T, B) = aT2 + cB2 gravity model $\left({\omega }_{d}\right)$ versus redshift $\left(z\right)$. |
Figure 8. Null, Dominant and Strong energy conditions of f(T, B) = aT2 + cB2 gravity models versus redshift $\left(z\right)$. |
6.2. Model-II
Figure 9. Energy density parameter of statistical f(T, B) = aT2 + cB3 gravity model $\left({\rho }_{d}\right)$ versus redshift $\left(z\right)$. |
Figure 10. Equation of state parameter of statistical f(T, B) = aT2 + cB3 gravity model $\left({\rho }_{d}\right)$ versus redshift $\left(z\right)$. |
Figure 11. Null, Dominant and Strong energy conditions of f(T, B) = aT2 + cB3 gravity models versus redshift $\left(z\right)$. |
6.3. Model-III
Figure 12. Energy density parameter of statistical f(T, B) = aT3 + cB2 gravity model $\left({\rho }_{d}\right)$ versus redshift $\left(z\right)$. |
Figure 13. Equation of state parameter of the statistical f(T, B) = aT3 + cB2 gravity model $\left({\rho }_{d}\right)$ versus redshift $\left(z\right)$. |
Figure 14. Null, Dominant and Strong energy conditions of f(T, B) = aT3 + cB2 gravity models versus redshift $\left(z\right)$. |
7. Conclusions
• | We have considered a well-known f(T, B) gravity model, namely f(T, B) = aTb + cBd, to examine other cosmological parameters. |
• | By parametrizing the Hubble parameter and estimating the best fit values of the model parameters b0, b1, and b2 imposed from Supernovae type Ia, Cosmic Microwave Background, Baryon Acoustic Oscillation, and Hubble data using the MCMC method, we propose a method to determine the precise solutions to the field equations. We then observed that the model appears to be in good agreement with the observations. |
• | The Hubble parameter H and the redshift parameter z are related in the current study’s regression equation. We have used the best regression relation feasible to construct an appropriate functional relation between the Hubble parameter H and redshift z. We have calculated R2, where ${R}^{2}=\tfrac{{SSR}}{{SST}}$, to determine the best fit. |
• | We employed supernovae from the Pantheon compilation, which consists of 1048 spectroscopically verified Type Ia Supernovae spread in the redshift range 0.01 < z < 2.26 [53], in our investigation for various dark energy models. |
• | We calculated the angular diameter distance DA(z) from BAO datasets by clustering perpendicular to the line of sight. Additionally, by seeing clustering along the line of sight, we were able to determine the Universe’s expansion rate, H(z). |
• | Two model parameters, b1 and b2, are present in the considered form of H(z), and they have been restricted for further investigation using certain observational data. For the purpose of validating our methodology, we have used a number of external datasets, including a recently compiled observational Hubble dataset with 57 data points, the Pantheon compilation of SNeIa data with 1048 data points, and a Baryonic Acoustic Oscillation dataset with six data points, to determine the best fit values for these model parameters. |
• | We have also obtained three specific models for f(T, B) gravity for different values of d and b. In Model I, b = d = 2; in Model II, b = 2, d = 3, and in Model III, b = 3, d = 2. In all three of these models, we have seen that the cosmos has an ΛCDM era for z = 0 and −1, but the early universe for z > 0 has both a barotropic and a quintessence field dark energy era. Thus, all three models of f(T, B) gravity are in good agreement with the ΛCDM era. Using information from the H(z), Pantheon samples, and BAO datasets, the model-independent constraints on the modified f(T, B) gravity have been investigated in the current study. This type of model-independent constraint on modified gravity from current data has been recently investigated [46–50] and is already mentioned in the introduction. Also of note is that in the current epoch, all models under study anticipate a violation of the strong energy condition. It is a well-known fact that applying the energy conditions to modified theories of gravity is an open investigation that ultimately leads to a conflict between theory and observation. |
• | In recent years, f(T, B) gravity has been used to study a number of cosmic aspects. [39] demonstrates how f(T, B) gravity can mimic the de Sitter universe, power law, and CDM models using cosmological reconstruction methods. [64] used this method to present the bouncing solution in this extended teleparallel gravity and to explore the singularity and little rip cosmology, and [65] carried out a stability analysis on the cosmological models that were modelled in conflict with the observational data for the accelerating universe. Four cosmological models that have shown promise in meeting late-time cosmic acceleration measurements that can generate quintessence behavior and experience transition along the phantom-divide line are presented in [66] as the observational side of this class of models. |
• | In 2023, Kadam et al [67] investigated a different type of cosmological model in f(T,B) gravity, where the Witzenb $\ddot{o}$ ck connection has been used instead of the usual levi-Civita connection by using f(T, B) = αT + βBn. A number of writers have looked into two different f(T,B) gravity model types so far: the first using the Levi-Civita connection and the second using the Witzenb $\ddot{o}$ ck connection. Kadam et al [67] and other authors obtained the cosmological models in f(T,B) gravity by using f(T, B) = αT + βBn. But in the present paper, we have investigated the cosmological models in f(T,B) gravity by using f(T, B) = aTb + cBd. It follows that our model is superior and more general than the prior one. All of the aforementioned findings give us reason to assume that an extension in the form of f(T, B) gravity could result in intriguing situations where we can think about how torsion and boundary terms combine with astrophysical evidence to shed insight on the study of the late-time accelerating universe. |