$H_0$ Tension, Phantom Dark Energy and Cosmological Parameter Degeneracies

Abstract

Phantom dark energy can produce amplified cosmic acceleration at late times, thus increasing the value of $H_0$ favored by CMB data and releasing the tension with local measurements of $H_0$. We show that the best fit value of $H_0$ in the context of the CMB power spectrum is degenerate with a constant equation of state parameter $w$, in accordance with the approximate effective linear equation $H_0+30.93w-36.47=0$ ($H_0$ in $kmse{c}^{-1}Mp{c}^{-1}$). This equation is derived by assuming that both ${\Omega }_{0\mathrm{m}}{h}^2$ and $d_A={\int }_0^{z_{rec}}\frac{dz}{H(z)}$ remain constant (for invariant CMB spectrum) and equal to their best fit Planck/$\Lambda$CDM values as $H_0$, ${\Omega }_{0\mathrm{m}}$ and $w$ vary. For $w=-1$, this linear degeneracy equation leads to the best fit $H_0=67.4kmse{c}^{-1}Mp{c}^{-1}$ as expected. For $w=-1.22$ the corresponding predicted CMB best fit Hubble constant is $H_0=74kmse{c}^{-1}Mp{c}^{-1}$ which is identical with the value obtained by local distance ladder measurements while the best fit matter density parameter is predicted to decrease since ${\Omega }_{0\mathrm{m}}{h}^2$ is fixed. We verify the above $H_0-w$ degeneracy equation by fitting a $w$CDM model with fixed values of $w$ to the Planck TT spectrum showing also that the quality of fit (${\chi }^2$) is similar to that of $\Lambda$CDM. However, when including SnIa, BAO or growth data the quality of fit becomes worse than $\Lambda$CDM when $w<-1$. Finally, we generalize the $H_0-w(z)$ degeneracy equation for $w(z)=w_0+w_{1}z/(1+z)$ and identify analytically the full $w_0-w_1$ parameter region that leads to a best fit $H_0=74kmse{c}^{-1}Mp{c}^{-1}$ in the context of the Planck CMB spectrum. This exploitation of $H_0-w(z)$ degeneracy can lead to immediate identification of all parameter values of a given $w(z)$ parametrization that can potentially resolve the $H_0$ tension.

Comments: 11 pages, 7 Figures. Version published in Physical Review D. Added references and discussion on physical models. The numerical analysis files used for the production of the figures of the paper may be downloaded from https://github.com/GeorgeAlestas/H0_Tension_Data