# Generic Modification of Gravity, Late Time Acceleration and Hubble Tension

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## Abstract

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## 1. Introduction

## 2. Hubble Tension Overview

**Early-Time Modifications**: In cosmology, the positions of the acoustic peaks in the CMB temperature and anisotropy spectra are among the most accurately measured quantities [80]. These acoustic peaks help in determining the size of the sound horizon at the recombination epoch. In order to attempt to modify the sound horizon, one needs to introduce new physics during the pre-recombination epoch that deform $H\left(z\right)$ at $z>1100$ [94,95,96,97,98,99,100,101,102,103,104]. One such modification was motivated by some string–axiverse-inspired scenarios for dark energy, in which dark energy density at early times behaves like the cosmological constant but then decays quickly. However, in this approach, the Hubble constant can, at most, shift to $1.6$ km/s/Mpc at redshift $z\simeq 1585$ [94]. Other approaches, such as modifying the standard model neutrino sector [105,106], additional radiation [107], primordial magnetic fields [108], or adjusting basic constants with the goal of lowering the sound horizon at recombination [109], are insufficient to properly answer the Hubble Tension problem. Additionally, they expect large growth of matter perturbations than reported by redshift space distortion (RSD) and weak lensing (WL) data, worsening the ${\Omega}_{0m}-{\sigma}_{8}$ tension [110].**Late-Time Modifications**: In this approach, one considers late-time alternative forms of DE [1], unified dark fluid models [2,5] where the Dark Matter (DM) and DE behave as a single fluid, alternative gravitational theories including either modified versions of GR or new gravitational theories beyond GR [57], and interacting DE models [1,2] in which DM and the DE interact with each other in a non-gravitational way [111,112], (for more, please see the Refs. [113,114,115,116,117,118]). In the latter scenario, the DE–DM interaction provides a possible solution to the cosmic coincidence problem, and can also explain the phantom DE regime without any scalar fields having a negative kinetic term. It is argued that some of these models are also not able to fully resolve the Hubble Tension problem [119].**Late-Time Transition of SnIa absolute magnitude M**: The shifting of M to a lower value by $-0.2$ at redshift ${z}_{t}\simeq 0.1$ is also a possible approach to address the Hubble Tension problem. Such a reduction in M at $z>{z}_{t}$ may be caused, for example, by a comparable transition of the effective gravitational constant ${G}_{eff}$, which would result in a rise in the SnIa intrinsic brightness at $z>{z}_{t}$. This class of models has the potential to solve the Hubble Tension problem entirely while also addressing the growth tension by slowing the growth rate of matter perturbations [89,120,121].

## 3. Coupling between Baryonic and Dark Matter Components in the Einstein Frame

#### 3.1. Disformal Coupling between Matter Components

#### 3.2. Equations of Motion for DM Field

#### 3.3. Dynamics in FRW Universe

#### 3.4. Polynomial Parametrization

#### 3.5. Exponential Parametrization

## 4. Observational Datasets

**Pantheon + MCT SnIa data:**

**Observational Hubble Data (OHD):**

- 1
- Differential age technique: In this technique, the data points are calculated using the relation between the redshift z and the rate of change of the galaxy’s age.$$\frac{dt}{dz}=\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}\frac{1}{(1+z)H\left(z\right)}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}.$$
- 2
- Galaxy clustering technique: The data points are obtained by utilising galaxy or quasar clustering that provides direct measurements of $H\left(z\right)$ from the radial peaks of baryon acoustic oscillations (BAO) [76,127].The ${\chi}^{2}$ for the Hubble parameter measurements is$${\chi}_{H}^{2}=\sum _{i}{\left[\frac{{H}_{i}^{th}-{H}_{i}^{obs}}{{\sigma}_{i}^{H}}\right]}^{2}\phantom{\rule{0.166667em}{0ex}}.$$

**Masers Data:**

## 5. Parametric Estimations

## 6. Constraints on Hubble Parameter and Dark-Energy Equation-of-State Parameter: Phantom-Crossing and Hubble Tension

## 7. Comparison with $\Lambda $CDM

## 8. Conclusions and Future Perspectives

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Note

1 | Note that the Baryonic sector only couples to the metric $\tilde{g}$, and not to g. |

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**Figure 1.**Whisker plot (following the Ref. [54]) with $68\%$ confidence limit constraints of the Hubble constant ${H}_{0}$ through several direct and indirect measurements. The yellow horizontal band corresponds to the ${H}_{0}$ value from the $SH0ES$ Team (${H}_{0}=73.2\pm 1.3$ km/s/Mpc at $68\%$ confidence limit), ref. [45] and the pink vertical band corresponds to the ${H}_{0}$ value, as reported by Planck (2018) [80] within a vanilla $\Lambda $CDM scenario.

**Figure 2.**Polynomial: $2\sigma $ contour levels between $\alpha $, $\beta $ and $\tilde{h}$ for OHD and its combinations with Pantheon + Masers [79].

**Figure 3.**Polynomial: Figures (

**a**,

**b**) depict the evolution of $\tilde{H}\left(\tilde{z}\right)/(1+\tilde{z})$ with $\tilde{z}\in [0,4]$ for the datasets OHD and OHD + Pantheon + Masers, respectively [129,130]. The dark line represents the best fit and the shaded region corresponds to the $1\sigma $ limit [79].

**Figure 4.**Polynomial: Figures (

**a**,

**b**) depict the evolution of ${\tilde{w}}_{de}\left(\tilde{z}\right)$ with $\tilde{z}\in [0,4]$ for the datasets OHD and OHD + Pantheon + Masers, respectively. The dark line represents the best fit and the shaded region corresponds to the $1\sigma $ limit [79].

**Figure 5.**Exponential: $2\sigma $ contour levels between $\alpha $ and $\tilde{h}$ for OHD and its combinations with Pantheon + Masers [79].

**Figure 6.**Exponential: Figures (

**a**,

**b**) depict the evolution of ${\tilde{w}}_{de}\left(\tilde{z}\right)$ with $\tilde{z}\in [0,4]$ for the datasets OHD and OHD + Pantheon + Masers, respectively [129,130]. The dark line represents the best fit and the shaded region corresponds to the $1\sigma $ limit [79].

**Figure 7.**Exponential: Figures (

**a**,

**b**) depict the evolution of ${\tilde{w}}_{de}\left(\tilde{z}\right)$ with $\tilde{z}\in [0,4]$ for the datasets OHD and OHD + Pantheon + Masers, respectively. The dark line represents the best fit and the shaded region corresponds to the $1\sigma $ limit [79].

**Figure 8.**Exponential: Figures (

**a**,

**b**) show the variation of the two model parameters with the equation-of-state of the dark energy (${w}_{DE}$) [79].

**Table 1.**Best fits with their $1\sigma $ levels for polynomial and exponential parametrizations, and for the $\Lambda $CDM model from the OHD and OHD + Pantheon + Masers datasets [79].

Parametrizations | |||
---|---|---|---|

Observational | Polynomial | Exponential | $\mathbf{\Lambda}$CDM |

Dataset | Best-Fit ($\pm 1\mathit{\sigma}$) | Best-Fit ($\pm 1\mathit{\sigma}$) | |

$\tilde{h}=0.{7279}_{-0.05}^{+0.05}$ | $\tilde{h}=0.{671}_{-0.029}^{+0.029}$ | $\tilde{h}=0.{6770}_{-0.030}^{+0.030}$ | |

OHD | $\alpha =-0.{101}_{-0.077}^{+0.07}$ | $\alpha =-0.{299}_{-0.042}^{+0.043}$ | ${\tilde{\Omega}}_{M}=0.{3249}_{-0.059}^{+0.064}$ |

$\beta =-0.{078}_{-0.049}^{+0.051}$ | - | ||

$\tilde{h}$ = $0.{689}_{-0.015}^{+0.015}$ | $\tilde{h}$ = $0.{677}_{-0.007}^{+0.007}$ | $\tilde{h}=0.{6683}_{-0.026}^{+0.026}$ | |

OHD + Pantheon + Masers | $\alpha =-0.{145}_{-0.051}^{+0.078}$ | $\alpha $ = $-0.{335}_{-0.017}^{+0.016}$ | ${\tilde{\Omega}}_{M}=0.{3440}_{-0.054}^{+0.061}$ |

$\beta =-0.{041}_{-0.047}^{+0.029}$ | - |

**Table 2.**The evidence in support of polynomial and exponential parametrizations for OHD and OHD + Pantheon + Masers datasets with respect to the standard $\Lambda $CDM scenario [79].

Observational Dataset | Polynomial ($\mathbf{\Delta}$BIC) | Polynomial Evidence | Exponential ($\mathbf{\Delta}$BIC) | Exponential Evidence |
---|---|---|---|---|

OHD | 9.63 | Strong | 0.62 | Not worth |

OHD + Pantheon + Masers | 8.88 | Strong | 4.01 | Positive |

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**MDPI and ACS Style**

Gangopadhyay, M.R.; Pacif, S.K.J.; Sami, M.; Sharma, M.K.
Generic Modification of Gravity, Late Time Acceleration and Hubble Tension. *Universe* **2023**, *9*, 83.
https://doi.org/10.3390/universe9020083

**AMA Style**

Gangopadhyay MR, Pacif SKJ, Sami M, Sharma MK.
Generic Modification of Gravity, Late Time Acceleration and Hubble Tension. *Universe*. 2023; 9(2):83.
https://doi.org/10.3390/universe9020083

**Chicago/Turabian Style**

Gangopadhyay, Mayukh R., Shibesh K. Jas Pacif, Mohammad Sami, and Mohit K. Sharma.
2023. "Generic Modification of Gravity, Late Time Acceleration and Hubble Tension" *Universe* 9, no. 2: 83.
https://doi.org/10.3390/universe9020083