Reconstructing $f(Q)$ in Modified Theories of Gravity

arXiv:2404.13095v1 Announce Type: new
Abstract: The increase of discrepancy in the standard procedure to choose the arbitrary functional form of the Lagrangian $f(Q)$ motivates us to solve this issue in modified theories of gravity. In this regard, we investigate the Gaussian process (GP), which allows us to eliminate this issue in a $f(Q)$ model-independent way. In particular, we use the 57 Hubble measurements coming from cosmic chronometers and the radial Baryon acoustic oscillations (BAO) to reconstruct $H(z)$ and its derivatives $H'(z)$, $H”(z)$, which resulting lead us to reconstruct region of $f(Q)$, without any assumptions. The obtained mean curve along $Lambda$CDM constant in the reconstructed region follows a quadratic behavior. This motivates us to propose a new $f(Q)$ parametrization, i.e., $f(Q)= -2Lambda+ epsilon Q^2$, with the single parameter $epsilon$, which signifies the deviations from $Lambda$CDM cosmology. Further, we probe the widely studied power-law and exponential $f(Q)$ models against the reconstructed region and can improve the parameter spaces significantly compared with observational analysis. In addition, the direct Hubble measurements, along with the reconstructed $f(Q)$ function, allow the $H_0$ tension to be alleviated.

Abstract: In this article, we address the issue of choosing an arbitrary functional form for the Lagrangian in modified theories of gravity. To eliminate this issue, we use the Gaussian process (GP) method to reconstruct the $f(Q)$ function without any assumptions. We utilize 57 Hubble measurements and Baryon acoustic oscillations (BAO) data to reconstruct the Hubble parameter $H(z)$ and its derivatives. The resulting mean curve follows a quadratic behavior, leading us to propose a new parametrization for $f(Q)$. We then compare the widely studied power-law and exponential models to the reconstructed region, improving the parameter spaces significantly. Additionally, we show that the direct Hubble measurements, combined with the reconstructed $f(Q)$ function, can alleviate the $H_0$ tension.

Future Roadmap

Challenges:

1. The Gaussian process (GP) method used to reconstruct the $f(Q)$ function may have limitations and uncertainties that need to be addressed.
2. Obtaining accurate and precise measurements of the Hubble parameter $H(z)$ and its derivatives is crucial for an accurate reconstruction of $f(Q)$.
3. Further research is needed to understand the physical implications and consequences of the proposed new parametrization for $f(Q)$.

Opportunities:

1. The use of the Gaussian process (GP) method provides a model-independent way to eliminate the issue of choosing an arbitrary functional form for the Lagrangian in modified theories of gravity.
2. By comparing different models to the reconstructed region, we can significantly improve the parameter spaces and better understand the behavior of $f(Q)$.
3. The combination of direct Hubble measurements and the reconstructed $f(Q)$ function has the potential to alleviate the tension in the measurement of the Hubble constant $H_0$.

Overall, this research provides a promising direction for addressing the issue of choosing the functional form of the Lagrangian in modified theories of gravity. However, further studies and advancements are necessary to overcome the challenges and fully explore the opportunities presented by the Gaussian process method and the proposed parametrization for $f(Q)$.
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