<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> tension, phantom dark energy, and cosmological parameter degeneracies
George Alestas, Lavrentios Kazantzidis, Leandros Perivolaropoulos
Abstract
Phantom dark energy ($w<\ensuremath{-}1$) 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 ${H}_{0}+30.93w\ensuremath{-}36.47=0$ (${H}_{0}$ in $\mathrm{km}\text{ }{\mathrm{sec}}^{\ensuremath{-}1}\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$). This equation is derived by assuming that both ${\mathrm{\ensuremath{\Omega}}}_{0m}{h}^{2}$ and ${d}_{A}={\ensuremath{\int}}_{0}^{{z}_{\mathrm{rec}}}\frac{dz}{H(z)}$ remain constant (for an invariant CMB spectrum) and equal to their best fit Planck/$\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ values as ${H}_{0}$, ${\mathrm{\ensuremath{\Omega}}}_{0m}$, and $w$ vary. For $w=\ensuremath{-}1$, this linear degeneracy equation leads to the best fit ${H}_{0}=67.4\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{sec}}^{\ensuremath{-}1}\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ as expected. For $w=\ensuremath{-}1.22$, the corresponding predicted CMB best fit Hubble constant is ${H}_{0}=74\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{sec}}^{\ensuremath{-}1}\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$, which is identical with the value obtained by local-distance ladder measurements, while the best fit matter density parameter is predicted to decrease, since ${\mathrm{\ensuremath{\Omega}}}_{0m}{h}^{2}$ is fixed. We verify the above ${H}_{0}\ensuremath{-}w$ degeneracy equation by fitting a $w\mathrm{CDM}$ model with fixed values of $w$ to the Planck TT spectrum, showing also that the quality of fit (${\ensuremath{\chi}}^{2}$) is similar to that of $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$. However, when including SnIa, baryon acoustic oscillation, or growth data, the quality of fit becomes worse than $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ when $w<\ensuremath{-}1$. Finally, we generalize the ${H}_{0}\ensuremath{-}w(z)$ degeneracy equation for the parametrization $w(z)={w}_{0}+{w}_{1}z/(1+z)$ and identify analytically the full ${w}_{0}\ensuremath{-}{w}_{1}$ parameter region (straight line) that leads to a best fit ${H}_{0}=74\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{sec}}^{\ensuremath{-}1}\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ in the context of the Planck CMB spectrum. This exploitation of ${H}_{0}\ensuremath{-}w(z)$ degeneracy can lead to immediate identification of all parameter values of a given $w(z)$ parametrization that can potentially resolve the ${H}_{0}$ tension.