Price of shifting the Hubble constant

An anisotropic measurement of the baryon acoustic oscillation (BAO) feature fixes the product of the Hubble constant and the acoustic scale $H_0{r}_d$. Therefore, regardless of the dark energy dynamics, to accommodate a higher value of $H_0$ one needs a lower $r_d$ and so necessarily a modification of early time cosmology. One must either reduce the age of the Universe at the drag epoch or else the speed of sound in the primordial plasma. The first can be achieved, for example, with dark radiation or very early dark energy, automatically preserving the angular size of the acoustic scale in the cosmic microwave background (CMB) with no modifications to post-recombination dark energy. However, it is known that the simplest such modifications fall afoul of CMB constraints at higher multipoles. As an example, we combine anisotropic BAO with geometric measurements from strong lensing time delays from H0LiCOW and megamasers from the Megamaser Cosmology Project to measure $r_d$, with and without the local distance ladder measurement of $H_0$. We find that the best fit value of $r_d$ is indeed quite insensitive to the dark energy model and is also hardly affected by the inclusion of the local distance ladder data.

Figure 1

BAO only results. Top: In the $\mathrm{\Lambda }\mathrm{CDM}$ model, the only relevant parameters are ${\mathrm{\Omega }}_m$ and $c/(H_0{r}_d)$. The standard error ellipses are plotted, corresponding to $\mathrm{\Delta }{\chi }_{\mathrm{BAO}}^2=2.3$, 6.0, and 11.8. The Planck benchmark value and uncertainties, derived in the case of a $\mathrm{\Lambda }\mathrm{CDM}$ model, are included for comparison. Bottom: $\mathrm{\Delta }{\chi }^2$ is plotted for various values of $c/(H_0{r}_d)$ with all nuisance parameters optimized. The red, green, blue, and black curves correspond to the $H\mathrm{\Lambda }\mathrm{CDM}$-Planck, $\mathrm{\Lambda }\mathrm{CDM}$, $w\mathrm{CDM}$, and CPL models, respectively.

Figure 2

$\mathrm{\Delta }{\chi }^2$ is plotted for various values of $H_0$ with all nuisance parameters optimized. The red, green, blue, black, and brown curves correspond to the $\mathrm{\Lambda }\mathrm{CDM}$-Planck, $\mathrm{H}\mathrm{\Lambda }\mathrm{CDM}$-Planck, $\mathrm{\Lambda }\mathrm{CDM}$, $w\mathrm{CDM}$, and CPL models, respectively. Top: BAO and strong lensing time delay results. Middle: Including also megamaser data. Bottom: Including the local distance ladder measurement of $H_0$.

Figure 3

The profiled values of the nuisance parameters in the fit of BAO, lensing, and maser data. The red, green, blue, black, and brown curves correspond to the $\mathrm{\Lambda }\mathrm{CDM}$-Planck, $\mathrm{H}\mathrm{\Lambda }\mathrm{CDM}$-Planck, $\mathrm{\Lambda }\mathrm{CDM}$, $w\mathrm{CDM}$, and CPL models, respectively.

Figure 4

The standard error ellipses corresponding to the profile likelihood for $H_0$ and $r_d$, with all nuisance parameters optimized. BAO, strong lensing, and megamaser data are used, with the local distance ladder measurement of $H_0$ included in the right panels. Top: $\mathrm{\Lambda }\mathrm{CDM}$. Bottom: CPL.

Figure 5

The redshift of the drag epoch $z_d$ is shown as a function of ${\mathrm{\Omega }}_m{h}^2$ and ${\mathrm{\Omega }}_b{h}^2$ in a standard CDM cosmology. The Planck best fit value, given $\mathrm{\Lambda }\mathrm{CDM}$, is indicated with its associated error bars. Note that a shift in $z_d$ of just 10 requires a change in the other parameters at the level of tens of percent.

Figure 6

An MCMC is used to calculate the standard error ellipses in $r_d$ and $H_0$ from the marginalized likelihood in the $\mathrm{\Lambda }\mathrm{CDM}$ model (top) and CPL model (bottom) using BAO, lensing, and masers (left) plus the distance ladder (right). There is excellent agreement with the corresponding panels of Fig. 4 which were calculated using the profile likelihoods.

Figure 7

Constraints on $H_0$ and ${\mathrm{\Omega }}_M$ given by only strong lensing (left) or megamaser (right) data, assuming $\mathrm{\Lambda }\mathrm{CDM}$.

Figure 8

As in Fig. 2, but using megamaser data instead of lensing data.

Figure 9

As in Fig. 4, but using only BAO and megamaser data (left and right panels).