# Transit f(Q,T) Gravity Model: Observational Constraints with Specific Hubble Parameter

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## Abstract

**:**

## 1. Introduction

## 2. Basic Formalism in $\mathit{f}\left(\mathit{Q},\mathit{T}\right)$ Gravity

## 3. Flat FLRW Universe in $\mathit{f}\left(\mathit{Q},\mathit{T}\right)$ Cosmology

#### Specific Hubble Parameters and Analysis

## 4. Observational Constraints

#### 4.1. Hubble Dataset

#### 4.2. Supernovae SNe-Ia

## 5. Cosmic Parameters and Energy Conditions

**The deceleration parameter $\left(q\right)$ is**

**The statefinder parameters.**

**Energy conditions**

- Strong energy condition (SEC): One inequality of SEC, formulated as $\rho +3p\ge 0$, asserts that gravity should always be attractive. Another component of SEC is $\rho +p\ge 0$.
- Dominant energy condition (DEC): When an observer measures, the matter–energy density will be positive and will propagate causally, which leads to $\rho \ge \left|p\right|$, $\rho \ge 0$. DEC implies that the flow of sound energy will not exceed the speed of light.
- Weak energy condition (WEC): The matter–energy density measured by any time-like observer should be positive, $\rho \ge 0,\rho +p\ge 0$. WEC implies that the energy density should not be negative
- Null energy condition (NEC): For a perfect fluid energy–momentum tensor, NEC is given by $\rho +p\ge 0$.

## 6. Models of $\mathit{f}(\mathit{Q},\mathit{T})$ Gravity

#### 6.1. Model-I

#### 6.2. Model-II

## 7. Conclusions

- The best-fit plots based on the observational datasets are presented in Figure 1 and Figure 2. We used a hybrid model of the least squares method and gradient descent for the best fit. The ${R}^{2}$ values for the Hubble and SNe-Ia datasets are $0.9321$ and $0.9930$, respectively. SNe-Ia has 580 observations, providing the superior fit amongst the two datasets.
- We considered two functional forms of $f(Q,T)$ gravity in Section 6.1 and Section 6.2 to observe the behaviors of energy density and the EoS parameter. In the considered models, the EoS parameter traces its journey from the matter-dominated, decelerating phase during early times to the dark energy-dominated, accelerating phase in later times. The energy density remains positive in both models, subjected to the values of model parameters.
- In both functional forms of $f(Q,T)$ of Section 6.1 and Section 6.2, NEC, WEC, and DEC are satisfied, whereas SEC is violated (see Figure 8, Figure 9, Figure 12 and Figure 13).
- The cosmological redshift ($z>0$) provides insight into the evolution of the early universe and $z=0$ denotes the present universe. From the cosmological redshift ($z<0$), one may predict the future universe evolution. For the expanding universe, the relationship between the scale factor $\left(a\right)$ and cosmological redshift $\left(z\right)$ is $\frac{{a}_{0}}{a}=1+z$. For the present universe, $a={a}_{0}=1$ (by convention of the observational cosmology) [92], which will yield $z=0$. For the past universe evolution, $0<a<1$, which will yield $0<z<\infty $. The future evolution of the universe may be portrayed by $-1<z<0$ with $z\to -1$ (in the extreme future). For a detailed compilation of the cosmological scale issues, one may refer to Ref. [93]. In this sense, the present models are decelerating in the past and will approach the $\Lambda CDM$ phase in the extreme future.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The best-fit plot of the Hubble parameter versus redshift for the Hubble dataset (red); the black dotted line and blue dash-dot line show plots for the $\Lambda CDM$ model and SNe-Ia dataset. Dots denote the Hubble datasets with the error bars.

**Figure 2.**The best-fit plot for luminosity distance $\left(\mu \right)$ versus redshift for SNe-Ia datasets (red); the black dotted line and blue dash-dot line show plots for the $\Lambda CDM$ model and the Hubble dataset. Dots denote the SNe datasets with error bars.

**Figure 3.**The $1-\sigma $ and $2-\sigma $ maximum likelihood contour plots for the parameters using Hubble (

**left**) and SNe-Ia (

**right**) datasets.

**Figure 4.**Behavior of the deceleration parameter (left plot (

**a**) for Hubble and the right plot (

**b**) for SNe-Ia), with redshift for the best-fit values of $\delta $ and $\lambda $ provided in Table 1.

**Figure 5.**Behavior of the universe in the $s-r$ plane for the best-fit values of $\delta $ and $\lambda $ provided in Table 1.

**Figure 6.**Behavior of the (redshift–density) plane for the constant values ${H}_{0}$, $\delta $, and $\lambda $ provided in Table 1.

**Figure 7.**Behavior of the (redshift–EoS) plane for the constant values of ${H}_{0}$, $\delta $, and $\lambda $, provided in Table 1.

**Figure 8.**Behavior of the (redshift–energy condition) plane for the Hubble dataset for ${a}_{1}$ = −0.0125, and ${a}_{2}$ = −0.012.

**Figure 9.**Behavior of the (redshift–energy condition) plane for SNe-Ia datasets for ${a}_{1}$ = −0.0125, and ${a}_{2}$ = −0.012.

**Figure 10.**Behavior of the (redshift–density) plane for the constant values of $\u03f5$, $\delta $, and $\lambda $, provided in Table 1.

**Figure 11.**Behavior of the (redshift–EoS) plane for the constant values of ${H}_{0}$, $\delta $, and $\lambda $ provided in Table 1.

**Figure 12.**Behavior of the (redshift–energy condition) plane for the Hubble dataset for ${a}_{3}$ = −0.0125 and ${a}_{4}$ = −0.012.

**Figure 13.**Behavior of the (redshift–energy condition) plane for SNe-Ia dataset for ${a}_{3}$ = −0.0125 and ${a}_{4}$ = −0.012.

**Table 1.**Best-fit values of the model parameters ${H}_{0}$, $\delta $, and $\lambda $ for both datasets.

Datasets | ${\mathit{H}}_{0}$ (km/s/Mpc) | $\mathit{\delta}$ | $\mathit{\lambda}$ | $\mathit{\u03f5}$ |
---|---|---|---|---|

Hubble | ${64.49}_{-0.32}^{+0.33}$ | ${1.54}_{-0.02}^{+0.02}$ | ${1.14}_{-0.077}^{+0.068}$ | ${30.2}_{-0.87}^{+0.90}$ |

SNe-Ia | ${68.665}_{-2.1}^{+2.2}$ | ${1.53}_{-0.29}^{+0.28}$ | ${1.86}_{-0.34}^{+0.37}$ | ${23.954}_{-2.84}^{+3.74}$ |

z | H(z) | ${\mathit{\sigma}}_{\mathit{H}}$ | Ref. | z | H(z) | ${\mathit{\sigma}}_{\mathit{H}}$ | Ref. |
---|---|---|---|---|---|---|---|

0.07 | 69 | 19.6 | [59] | 0.9 | 69 | 12 | [60] |

0.120 | 68.6 | 26.2 | [59] | 0.170 | 83 | 8 | [61] |

0.179 | 75 | 4 | [62] | 0.2 | 72.9 | 29.6 | [59] |

0.27 | 77 | 14 | [61] | 0.28 | 88.8 | 36.6 | [59] |

0.350 | 76.3 | 5.6 | [63] | 0.38 | 83 | 13.5 | [64] |

0.4 | 95 | 17 | [61] | 0.42 | 87.1 | 11.2 | [64] |

0.44 | 92.8 | 12.9 | [64] | 0.47 | 89 | 34 | [59] |

0.48 | 97 | 62 | [65] | 0.6 | 87.9 | 6.1 | [66] |

0.68 | 92 | 8 | [62] | 0.73 | 97.3 | 7 | [66] |

0.78 | 105 | 12 | [62] | 0.87 | 125 | 17 | [62] |

0.90 | 117 | 23 | [61] | 1.037 | 154 | 20 | [62] |

1.3 | 168 | 17 | [61] | 1.363 | 160 | 33.6 | [61] |

1.430 | 177 | 18 | [61] | 1.530 | 140 | 14 | [61] |

1.750 | 202 | 40 | [61] | 1.965 | 186.5 | 50.4 | [67] |

0.24 | 79.69 | 2.99 | [68] | 0.30 | 81.7 | 6.22 | [69] |

0.31 | 78.18 | 4.74 | [70] | 0.34 | 83.8 | 3.66 | [68] |

0.35 | 87.7 | 9.1 | [71] | 0.36 | 79.94 | 3.38 | [70] |

0.38 | 81.5 | 1.9 | [72] | 0.40 | 82.04 | 2.03 | [70] |

0.43 | 86.45 | 3.97 | [68] | 0.44 | 82.6 | 7.8 | [73] |

0.44 | 84.81 | 1.83 | [70] | 0.48 | 87.79 | 2.03 | [70] |

0.51 | 90.4 | 1.9 | [72] | 0.52 | 94.35 | 2.64 | [70] |

0.56 | 93.34 | 2.3 | [70] | 0.57 | 87.6 | 7.8 | [74] |

0.57 | 96.8 | 3.4 | [75] | 0.59 | 98.48 | 3.18 | [70] |

0.6 | 87.9 | 6.1 | [73] | 0.61 | 97.3 | 2.1 | [72] |

0.64 | 98.82 | 2.98 | [70] | 0.73 | 97.3 | 7 | [73] |

2.30 | 224 | 8.6 | [76] | 2.33 | 224 | 8 | [77] |

2.34 | 222 | 8.5 | [78] | 2.36 | 226 | 9.3 | [79] |

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**MDPI and ACS Style**

Kale, A.P.; Solanke, Y.S.; Shekh, S.H.; Pradhan, A.
Transit *f*(*Q*,*T*) Gravity Model: Observational Constraints with Specific Hubble Parameter. *Symmetry* **2023**, *15*, 1835.
https://doi.org/10.3390/sym15101835

**AMA Style**

Kale AP, Solanke YS, Shekh SH, Pradhan A.
Transit *f*(*Q*,*T*) Gravity Model: Observational Constraints with Specific Hubble Parameter. *Symmetry*. 2023; 15(10):1835.
https://doi.org/10.3390/sym15101835

**Chicago/Turabian Style**

Kale, A. P., Y. S. Solanke, S. H. Shekh, and A. Pradhan.
2023. "Transit *f*(*Q*,*T*) Gravity Model: Observational Constraints with Specific Hubble Parameter" *Symmetry* 15, no. 10: 1835.
https://doi.org/10.3390/sym15101835