Quantum Vacuum energy as the origin of Gravity

Abstract: We explore the idea that quantum vacuum energy $\rho_{\rm vac} $ is at the origin of Gravity as a theoretical exercise. We formulate a gravitational version of the electromagnetic Casimir effect, and provide an argument for how gravity can arise from $\rho_{\rm vac} $ by showing how Einstein's field equations emerge in the form of Friedmann's equations. This leads to the idea that Newton's constant $G_N$ is environmental, namely it depends on the total mass-energy of the Universe $M_\infty $ and its size $R_\infty $, with $G_N = c^2 R_\infty /2 M_\infty$. This leads to a new interpretation of the Gibbons-Hawking entropy of de Sitter space, and also the Bekenstein-Hawking entropy for black holes, wherein the quantum information bits are quantized massless particles at the horizon with wavelength $\lambda = 2 \pi R_\infty$. We assume a recently proposed formula for $\rho_{\rm vac} \sim m_z^4/\mathfrak{g}$, where $m_z$ is the mass of the lightest particle, and $\mathfrak{g}$ is a marginally irrelevant coupling. This leads to an effective, induced RG flow for Newton's constant $G_N$ as a function of an energy scale, which indicates that $G_N$ decreases at higher energies until it reaches a Landau pole at a minimal value of the cosmological scale factor $a(t) > a_{\rm min}$, thus avoiding the usual geometric singularity at $a=0$. We propose that this energy scale dependent $G_N$ can explain the Hubble tension and we thereby constrain the coupling constant $\mathfrak{g}$ and its renormalization group parameters. For the $\Lambda{\rm CDM}$ model we estimate $a_{\rm min} \approx e^{-1/\hat{b} }$ where $\hat{b} \approx 0.02$ based on the Hubble tension data.