Hubble trouble or Hubble bubble?

TL;DR

The paper addresses the $H_0$ tension by linking the luminosity distance to the volume-averaged density contrast of local inhomogeneities and introduces a simple, metric-free inversion method to reconstruct the monopole density profile from uncorrected $D_L$ data. It demonstrates that low-$z$ distance measures are sensitive to the monopole, with a dominant velocity term $k_v(z)$, while high-$z$ distances are largely immune due to volume averaging, thereby explaining why CMB-based $H_0$ remains unaffected. Applying the inversion to a directional inhomogeneity (subregion 3) yields $\overline{\delta}\approx-0.57$, consistent with Keenan et al., and implies that about $40\%$ of Cepheids are affected, potentially biasing local $H_0$ estimates. The work provides a practical tool for density-field reconstruction from SN data and highlights the importance of depth and directionality in redshift corrections for accurate cosmological inferences.

Abstract

The recent analysis of low-redshift supernovae (SN) has increased the apparent tension between the value of $H_0$ estimated from low and high redshift observations such as the cosmic microwave background (CMB) radiation. At the same time other observations have provided evidence of the existence of local radial inhomogeneities extending in different directions up to a redshift of about $0.07$. About $40\%$ of the Cepheids used for SN calibration are directly affected because are located along the directions of these inhomogeneities. We derive a new simple formula relating directly the luminosity distance to the monopole of the density contrast, which does not involve any metric perturbation. We then use it to develop a new inversion method to reconstruct the monopole of the density field from the deviations of the redshift uncorrected observed luminosity distance respect to the $ΛCDM$ prediction based on cosmological parameters obtained from large scale observations. The inversion method confirms the existence of inhomogeneities whose effects were not previously taken into account because the $2M++$ density field maps used to obtain the peculiar velocity for redshift correction were for $z\leq 0.06$, which is not a sufficiently large scale to detect the presence of inhomogeneities extending up to $z=0.07$. The inhomogeneity does not affect the high redshift luminosity distance because the volume averaged density contrast tends to zero asymptotically, making the value of $H_0^{CMB}$ obtained from CMB observations insensitive to any local structure. The inversion method can provide a unique tool to reconstruct the density field at high redshift where only SN data is available, and in particular to normalize correctly the density field respect to the average large scale density of the Universe.

Figures (3)

• Figure 1: The peculiar velocity associated to an inhomogeneity profile as the one shown in the upper panel of fig.(2) is plotted in units of the speed of light $c$ as a function of redshift. As can be seen the effect reaches its peak around the hedge of the inhomogeneity and is then asymptotically suppressed due to volume averaging as shown in eq.(\ref{['dzOzA']}).
• Figure 2: The fractional difference of the luminosity distance $\Delta D_L/\overline{D_L}$ and the local Hubble parameter $\Delta H/\overline{H_0}$ are plotted as a function of the redshift for a compensated (top) inhomogeneity such as the one studied in Romano:2014iea and an uncompensated (bottom) void. For $\Delta D/D$ the effects of the inhomogeneity are computed with a non pertubative approach using a LTB metric (red line) and are in good agreement with the approximation (black line) given in eq.(\ref{['Dlowz']}), confirming that $k_v$ in eq.(\ref{['kvlowz']}) produces the dominant effect at low redshift. The relative fractional difference $\Delta H/H$ is computed with eq.(\ref{['dHnew']}) (black line) and with eq.(\ref{['H0locP']}) (dashed line). For the compensated case the difference is particularly important since eq.(\ref{['H0locP']}) would predict a negative variation, while eq.(\ref{['dHnew']}) gives the correct sign, in agreement with the results of a local fitting procedure shown in fig.(\ref{['fig:DH']}). The volume averaged fractional density contrast $\overline{\delta}$ defined in eq.(\ref{['dav']}) is plotted with a black line and the local density contrast $\delta$ with a dashed line.
• Figure 3: The fractional difference respect to the background $\Delta H/H$, estimated according to the same procedure used for fig.(12) in Riess:2016jrr by fitting $H_0$ from luminosity distance data in a range $z_{min}<z<z_{min}+\Delta z$, is plotted as a function of $z_{min}$. The black line is for $\Delta z=0.15$ as in Riess:2016jrr, the dashed for $\Delta z=0.03$ and the dotted line for $\Delta z=0.0015$. The luminosity distance used as input for the $H_0$ fit is the one for the compensated (top) and uncompensated (bottom) inhomogeneities shown in fig.(\ref{['fig:DLDHDC']}). For $\Delta z=0.15$ the fitting interval is much larger than the size of the inhomogeneity and consequently the fitted $H_0$ is affected at less than $\approx 0.1\%$ level for a compensated inhomogeneity because the effects of the homogeneity are smeared out, while for an uncompensated inhomogeneity the effect can be up to $\approx 2\%$. For $\Delta z=0.03$ and $\Delta z=0.0015$ the effect is clearly noticeable for both a compensated and an uncompensated inhomogeneity. This shows that, even if a compensated inhomogeneity were present, a fitting procedure such as the one used for fig.(12) in Riess:2016jrr would not be able to detect it unless a sufficiently small $\Delta z$ were used. Furthermore it should be noted that if the inhomogeneity is extending only in some direction the effect on the full sky Hubble diagram would be much smaller than in the above plots, since only some SN would affected. Consequently the above plots are just given as an example of the directional effects and should be compared to the directional Hubble diagram obtained only from SN in a given direction, not to the full sky diagram.