### Speaker

### Description

In this talk, relying on the fact that the comoving angular diameter distance to last scattering, $D_M(z_*)$, is strictly constrained almost model-independently, I will present how a dark energy (DE) density that attains negative values in the past can alleviate the $H_0$ tension along with the $S_8$, and Ly-$\alpha$ discrepancies. Observational studies suggest that matching the mean value of the Ly-$\alpha$ data requires a negative DE density, and to keep $D_M(z_*)$ unaltered compared to that of $\Lambda$CDM, this is naturally accompanied by a higher $H_0$ value. Of course, a negative DE density should transit to the positive regime to drive the present-day acceleration of the universe, and for a continuous density function, this requires that the DE density vanishes at a certain time. This vanishing point is accompanied with a singularity in the equation of state parameter of the DE, I will discuss why such a singularity is necessary, and how a negative energy density is not problematic from the point of view of fundamental physics. Finally, imposing that $D_M(z_*)$ is constant amongst different models, along with an identical pre-recombination and present-day universe, requires that any modifications to the Hubble radius of $\Lambda$CDM should be described in terms of \textit{ admissible wavelets}. The oscillatory behaviour of wavelets provides a natural way to fit a multitude of BAO data, and through their characteristics, I will discuss how the bumps found in the Hubble function and the DE density in their observational reconstructions may be fake.