Resolving the S8 tension with the Lambda Prime ($\Lambda'$) model

The $S_8$ parameter, which quantifies the amplitude of matter fluctuations on scales of $8h^{-1}$ Mpc, has been a source of tension between weak lensing surveys (e.g. KiDS, DES, HSC) and the Planck Cosmic Microwave Background (CMB) measurements. This discrepancy challenges the standard ${\Lambda}$CDM model and has become one of the most significant tensions in modern cosmology. The ${\Lambda'}$ model offers a potential resolution by introducing modifications to the cosmic growth history through alterations to the gravitational sector. The alterations involve including a Ricci soliton into Einstein's field equations which introduce a time dependent factor yielding a time varying cosmological constant ${\Lambda'}=(1-{\alpha(t)^2}{\Lambda_{DE}}\frac{{\rho}_g}{\rho_{DE}}$ and subsequently the evolution of the cosmos. The Ricci soliton is sourced from gravitational energy density. In this study we analyze results from six surveys and compare the results for $w_a$ and $w_0$ with the ${\Lambda'}$ model. We also find ${\sigma}_8={0.750}_{-0.020}^{+0.020}$ , $S_8={0.788}$. These values are closer to some low $S_8$ measurements from weak lensing surveys (e.g DES, KiDS), which report $S_8 \approx 0.76-0.78$, suggesting that the model may alleviate the $S_8$ tension. High values of ${\alpha(t)}$ in the late universe are the cause of suppressed structure formation and low values of ${\Lambda'}$. The late universe in the ${\Lambda'}$ model is effectively or apparently 5-10% younger than in ${\Lambda}$CDM which translates to $H_0={72.734}_{-1.687}^{+1.687}$ km/s/Mpc, which is in agreement with late universe probes. ${\Lambda'}$ is classified under the dynamical dark energy models, however unlike alternatives, it does not invoke exotic particles nor phantom energy.