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

TL;DR

The paper addresses the $H_0$ tension between CMB measurements and local distance ladders by analyzing how phantom-like dark energy affects the CMB spectrum through parameter degeneracies. It derives an approximate linear $H_0$–$w$ degeneracy, $H_0 + 30.93\, w - 36.47 = 0$, and shows that $w=-1$ yields $H_0\approx 67.4\;\mathrm{km\,s^{-1}\,Mpc^{-1}}$, while $w\approx -1.22$ can push $H_0$ to $\approx 74\;\mathrm{km\,s^{-1}\,Mpc^{-1}}$, aligning with local measurements. Numerical fits with Planck TT confirm the degeneracy, but incorporating SN Ia, BAO, or growth data tends to disfavor $w<-1$ due to growth-tension and overall poorer fits than $\Lambda$CDM. Extending the analysis to time-varying $w(z)$ (e.g., CPL) identifies regions in $(w_0,w_1)$ space that could realize $H_0\approx 74$, offering a practical route to evaluating $w(z)$ forms for resolving the tension while highlighting the need for stable phantom or modified gravity models to preserve fit quality across datasets.

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 $H_0 + 30.93\; w - 36.47 = 0$ ($H_0$ in $km \; sec^{-1} \; Mpc^{-1}$). This equation is derived by assuming that both $Ω_{0 \rm m}h^2$ and $d_A=\int_0^{z_{rec}}\frac{dz}{H(z)}$ remain constant (for invariant CMB spectrum) and equal to their best fit Planck/$Λ$CDM values as $H_0$, $Ω_{0 \rm m}$ and $w$ vary. For $w=-1$, this linear degeneracy equation leads to the best fit $H_0=67.4 \; km \; sec^{-1} \; Mpc^{-1}$ as expected. For $w=-1.22$ the corresponding predicted CMB best fit Hubble constant is $H_0=74 \; km \; sec^{-1} \; Mpc^{-1}$ which is identical with the value obtained by local distance ladder measurements while the best fit matter density parameter is predicted to decrease since $Ω_{0 \rm m}h^2$ 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 ($χ^2$) is similar to that of $Λ$CDM. However, when including SnIa, BAO or growth data the quality of fit becomes worse than $Λ$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 $H_0=74\; km \; sec^{-1} \; Mpc^{-1}$ 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.

Figures (7)

• Figure 1: The predicted value of $h$ as a function of the fixed $w$ for the one parameter dark energy ($w$CDM) model. The orange line corresponds to the theoretically predicted best fit values of $h$ for different values of $w$ in the case of the $w$CDM model, whereas the dashed blue line corresponds to the linear fitting that has been made. The red points display the actual best fit values, including the errorbars, of $h$ for specific values of $w$ obtained by fitting these models to the CMB TT anisotropy via the MGCosmoMC (see Table \ref{['tab:homcosmo']}).
• Figure 2: The CMB power spectrum for $\Lambda$CDM (blue line) and $w=-1.2$ (green line). We also show the binned high-$l$ and low-$l$ Planck data (red points).
• Figure 3: The degeneracy with respect to the CMB spectrum in the parameter space $(w_0-w_1)$. The dashed lines correspond to $h=0.674$ ($\Lambda$CDM value) and to $h=0.74$ (the value of Ref. Riess:2019cxk).
• Figure 4: The evolution of $w(z)$ for various values of $(w_0,w_1)$ along the degeneracy $h=0.74$ line of Fig. \ref{['fig:cpl_contour']}. All these parameter values lead to a best fit value $h=0.74$ in the context of the CMB power spectrum. However, they do not have the same quality of fit to other cosmological data which can be used to break this model degeneracy. The common $(z, w)$ point of intersection of all the $w(z)$ plots is $(0.31, -1.22)$.
• Figure 5: The contour plots constructed with MGCosmoMC using the PlanckTT and lowP likelihoods for $\Lambda$CDM and $w$CDM models. The gray contours correspond to the $\Lambda$CDM model. The green contours correspond to $w = -1.1$, the red ones to $w = -1.2$, while the blue to $w=-1.3$. For $w=-1.1$, the best fit value of $H_0$ is close to that of the Planck/$\Lambda$CDM measurement Ade:2015xua, while the $w=-1.2$ and $w=-1.3$ values shift $h$ closer to the local distance ladder measurements Riess:2019cxk.