This is the repository that contains the Mathematica code as well as useful comments that reproduce the figures of arxiv:2004.08363.

Phantom dark energy ($w<-1$) 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 $H_0 + 30.93 w - 36.47 = 0$ ($H_0$ in $km sec^{-1} Mpc^{-1}$). This equation is derived by assuming that both $\Omega_{0 \rm m}h^2$ and $d_A=\int_0^{z_{rec}}\frac{dz}{H(z)}$ remain constant (for invariant CMB spectrum) and equal to their best fit Planck/$\Lambda$ CDM values as $H_0$, $\Omega_{0 \rm m}$ 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 $\Omega_{0 \rm m}h^2$ 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 ($\chi^2$) is similar to that of $\Lambda$ CDM. However, when including SnIa, BAO or growth data the quality of fit becomes worse than \lcdm when $w< -1$. Finally, we generalize the $H_0-w(z)$ degeneracy equation for the parametrization $w(z)=w_0+w_1 z/(1+z)$ and identify analytically the full $w_0-w_1$ parameter region (straight line) 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.

If you use any of the above codes or the figures in a published work please cite the following paper:
H₀ Tension, Phantom Dark Energy and Cosmological Parameter Degeneracies
George Alestas, Lavrentios Kazantzidis and Leandros Perivolaropoulos, arxiv:2004.08363

Any further questions/comments are welcome.

George Alestas - [email protected]
Lavrentios Kazantzidis - [email protected]
Leandros Perivolaropoulos - [email protected]