# Bulk Viscous Flat FLRW Model with Observational Constraints in f(T, B) Gravity

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## Abstract

**:**

## 1. Introduction

## 2. Basic Formalism of $f(T,B)$ Gravity

#### Bulk Viscosity

## 3. Cosmological Model with Observational Constraints

## 4. Dynamical Parameters and Their Physical Discussion for Viscous Fluid

## 5. Energy Condition

- Null energy condition (NEC) if ${\rho}_{eff}\left(t\right)+{p}_{eff}\left(t\right)\ge 0;$
- Weak energy conditions (WEC) if ${\rho}_{eff}\left(t\right)\ge 0,\phantom{\rule{3.33333pt}{0ex}}{\rho}_{eff}\left(t\right)+{p}_{eff}\left(t\right)\ge 0;$
- Strong energy conditions (SEC) if ${\rho}_{eff}\left(t\right)+3{p}_{eff}\left(t\right)\ge 0$, ${\rho}_{eff}\left(t\right)+{p}_{eff}\left(t\right)\ge 0$;
- Dominant energy conditions (DEC) if ${\rho}_{eff}\left(t\right)\ge 0$, ${\rho}_{eff}\left(t\right)\pm {p}_{eff}\left(t\right)\ge 0.$

## 6. $f(T,B)$ Tachyon Model

## 7. Conclusions

- Figure 3 shows the effective equation of state parameter involving the bulk viscous pressure ${\omega}_{eff}=\frac{{p}_{T}+\mathrm{\Pi}}{{\rho}_{m}+{\rho}_{T}}$. The evolutionary trajectory of the EoS parameter exhibits quintom-like behavior (transition from phantom to quintessence and evolution of the LCDM limit) for the limiting conditions of the viscosity parameter ${\xi}_{0}>0$ shown in (Figure 3a).
- We have seen that the viscous EoS parameter achieves a phantom-like universe for a particular choice of for ${\xi}_{0}<0$ shown in (Figure 3b). Thus, the viscous EoS parameter represents a DE-dominated universe for all the models depending upon model parameters. This behavior is compatible with the $f\left(TB\right)$ gravity model.
- Figure 4 shows the temporal evolution of the energy conditions. Keep in mind that to serve the late-time acceleration of the universe, the SEC has to violate. The SEC is the most discussed of all the energy conditions. According to recent findings from the speeding universe, the SEC must be violated on a cosmological scale. In our derived model, NEC, WEC, and DEC satisfy the criteria established from the Raychaudhuri equations. However, SEC is violated. As a consequence, our model is accurate and its solutions are physically feasible.
- Figure 5 depicts the correspondence between the scalar field and the $f(T,B)$ gravity model. We have noticed that the scalar field $\varphi $ increases and potential decreases as time increases. The corresponding potential function exhibits a positive but decreasing relationship with time. Later, the scalar field exhibits inverse proportionality to its decreasing behavior from its maximum. In the brane-world cosmology, this kind of behavior corresponds to scaling solutions.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**One-dimensional and two-dimensional marginalized confidence regions (68%CL and 95% CL) for a, ${H}_{0}$ obtained from the 46 OHD data for the $f(T,B)$ gravity model.

**Figure 3.**The variation of EoS parameter versus time t for various limiting conditions of the bulk viscosity coefficients. (

**a**) ${\xi}_{0}$ > 0; (

**b**) ${\xi}_{0}$ < 0.

**Figure 5.**Correspondence of tachyon scalar field and potential versus t for various values of bulk viscosity coefficients.

$\mathit{S}.\mathit{No}$ | Z | $\mathit{H}\left(\mathbf{Obs}\right)$ | ${\mathsf{\sigma}}_{\mathit{i}}$ | Reference | $\mathit{S}.\mathbf{No}$ | Z | $\mathit{H}\left(\mathbf{Obs}\right)$ | ${\mathsf{\sigma}}_{\mathit{i}}$ | Reference |
---|---|---|---|---|---|---|---|---|---|

1 | 0 | 67.77 | 1.30 | [77] | 24 | 0.4783 | 80.9 | 9 | [78] |

2 | 0.07 | 69 | 19.6 | [79] | 25 | 0.48 | 97 | 60 | [80] |

3 | 0.09 | 69 | 12 | [81] | 26 | 0.51 | 90.4 | 1.9 | [82] |

4 | 0.01 | 69 | 12 | [80] | 27 | 0.57 | 96.8 | 3.4 | [83] |

5 | 0.12 | 68.6 | 26.2 | [79] | 28 | 0.593 | 104 | 13 | [84] |

6 | 0.17 | 83 | 8 | [80] | 29 | 0.60 | 87.9 | 6.1 | [85] |

7 | 0.179 | 75 | 4 | [84] | 30 | 0.61 | 97.3 | 2.1 | [82] |

8 | 0.1993 | 75 | 5 | [84] | 31 | 0.68 | 92 | 8 | [84] |

9 | 0.2 | 72.9 | 29.6 | [79] | 32 | 0.73 | 97.3 | 7 | [85] |

10 | 0.24 | 79.7 | 2.7 | [86] | 33 | 0.781 | 105 | 12 | [84] |

11 | 0.27 | 77 | 14 | [80] | 34 | 0.875 | 125 | 17 | [84] |

12 | 0.28 | 88.8 | 36.6 | [79] | 35 | 0.88 | 90 | 40 | [80] |

13 | 0.35 | 82.7 | 8.4 | [87] | 36 | 0.9 | 117 | 23 | [80] |

14 | 0.352 | 83 | 14 | [84] | 37 | 1.037 | 154 | 20 | [84] |

15 | 0.38 | 81.5 | 1.9 | [82] | 38 | 1.3 | 168 | 17 | [80] |

16 | 0.3802 | 83 | 13.5 | [78] | 39 | 1.363 | 160 | 33.6 | [88] |

17 | 0.4 | 95 | 17 | [81] | 40 | 1.43 | 177 | 18 | [80] |

18 | 0.4004 | 77 | 10.2 | [78] | 41 | 1.53 | 140 | 14 | [80] |

19 | 0.4247 | 87.1 | 11.2 | [78] | 42 | 1.75 | 202 | 40 | [80] |

20 | 0.43 | 86.5 | 3.7 | [86] | 43 | 1.965 | 186.5 | 50.4 | [88] |

21 | 0.44 | 82.6 | 7.8 | [85] | 44 | 2.3 | 224 | 8 | [89] |

22 | 0.44497 | 92.8 | 12.9 | [78] | 45 | 2.34 | 222 | 7 | [90] |

23 | 0.47 | 89 | 49.6 | [91] | 46 | 2.36 | 226 | 8 | [92] |

Substance | Observation | EoS Parameter |
---|---|---|

Phantom Universe (Ph) | Lead to Big Rip, resist to weak energy condition | $\omega <-1$ |

Quintessence (Q) | 68% of the universe | ${\omega}_{T}\in (-1/3,-1)$ |

Cosmological constant | Inconsistent with observation | ${\omega}_{T}=-1$ |

Hard Universe (HU) | high densities | ${\omega}_{T}\in (1/3,1)$ |

Radiation (R) | Influential in past | ${\omega}_{T}=1$ |

Hot matter (HM) | insignificant in present time | ${\omega}_{T}=0,1/3$ |

Ekpyrotic matter (Ek-M) | Resist DEC | ${\omega}_{T}>1$ |

Pressless cold matter (PCM) | 32% of the Universe | ${\omega}_{T}=0$ |

Stiff Fluid (SF) | ${\omega}_{T}=1$ |

Range of $\mathsf{\xi}$ | ${\mathsf{\xi}}_{0}$ | ${\mathsf{\xi}}_{1}$ | ${\mathsf{\xi}}_{2}$ | ${\mathsf{\omega}}_{\mathit{eff}}$ | Behavior |
---|---|---|---|---|---|

${\xi}_{0}>0$, ${\xi}_{0}$+${\xi}_{1}<1,{\xi}_{2}<2$ | 0.9 | 0.02 | 1.5 | −8.8 ≤ ${\omega}_{eff}$ = 0 | Ph-$\Lambda $CDM-Ek-M- SF- PCM |

${\xi}_{0}>0$,${\xi}_{0}$+${\xi}_{1}>1,{\xi}_{2}<2$ | 0.45 | 0.65 | 1.5 | −8.6 ≤ ${\omega}_{eff}$ = 0 | Ph-$\Lambda $CDM-Ek-M- SF- PCM |

${\xi}_{0}>0$,${\xi}_{0}+{\xi}_{1}=1$, ${\xi}_{2}<2$ | 0.65 | 0.35 | 1.5 | −8.5 ≤ ${\omega}_{eff}$ = 0 | Ph-$\Lambda $CDM-Ek-M- SF- PCM |

${\xi}_{0}>0$,${\xi}_{0}+{\xi}_{1}=1$, ${\xi}_{2}>2$ | −0.5 | 1.45 | 2.1 | −37.9 ≤ ${\omega}_{eff}$ = 0 | Ph-$\Lambda $CDM-Ek-M- SF- PCM |

${\xi}_{0}<0$,${\xi}_{0}$+${\xi}_{1}>1$, ${\xi}_{2}>2$ | −0.5 | 2.5 | 3 | −37.3 ≤ ${\omega}_{eff}$ = 0 | Ph-$\Lambda $CDM- PCM |

${\xi}_{0}<0$,${\xi}_{0}+{\xi}_{1}>1,{\xi}_{2}>2$ | −0.5 | 1.5 | 2.17 | −38.8 ≤ ${\omega}_{eff}$ = 0 | Ph-$\Lambda $CDM- PCM |

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## Share and Cite

**MDPI and ACS Style**

Dixit, A.; Pradhan, A.
Bulk Viscous Flat FLRW Model with Observational Constraints in *f*(*T*, *B*) Gravity. *Universe* **2022**, *8*, 650.
https://doi.org/10.3390/universe8120650

**AMA Style**

Dixit A, Pradhan A.
Bulk Viscous Flat FLRW Model with Observational Constraints in *f*(*T*, *B*) Gravity. *Universe*. 2022; 8(12):650.
https://doi.org/10.3390/universe8120650

**Chicago/Turabian Style**

Dixit, Archana, and Anirudh Pradhan.
2022. "Bulk Viscous Flat FLRW Model with Observational Constraints in *f*(*T*, *B*) Gravity" *Universe* 8, no. 12: 650.
https://doi.org/10.3390/universe8120650