Exploring cosmological constraints on galaxy formation time

The Universe consists of a variety of objects that formed at different epochs, leading to variations in the formation time which represents the time elapsed from the onset of structure formation until the formation time of a particular object. In this work, we present two approaches to reconstruct and constrain the galaxy formation time $t_f(z)$ using non-parametric reconstruction methods, such as Gaussian Processes (GP) and High-performance Symbolic Regression (SR). Our analysis uses age estimates of 32 old passive galaxies and the Pantheon+ type Ia supernova sample, and considers two different values of the Hubble constant $H_0$ from the SH0ES and Planck Collaborations. When adopting the $\Lambda$CDM model and the GP reconstructions, we find $\left<t_f\right>=0.72_{-0.16}^{+0.14}$ Gyr (SH0ES) and $\left<t_f\right>=1.26_{-0.11}^{+0.10}$ Gyr (Planck). Without considering a specific cosmological model, we obtain $\left<t_f\right>=0.71 \pm {0.19}$ Gyr (SH0ES) and $\left<t_f\right> = 1.35_{-0.23}^{+0.21}$ Gyr (Planck). Similar values are obtained from the SR reconstructions, with both methods (GP and SR) indicating the same behavior regarding the time evolution of $t_f(z)$. The results also show significant differences in the formation time from SH0ES and Planck values, highlighting the impact of the $H_0$ tension on the cosmological estimates of $t_f(z)$. In particular, the different approaches used in the analysis agree with each other, demonstrating the robustness and consistency of our results. Overall, this study suggests that galaxies have different evolutionary timescales and that $t_f$ is not constant, with noticeable variations at lower redshifts ($z \lesssim 0.5$).