Directional dependence of the local estimation of H₀ and the nonperturbative effects of primordial curvature perturbations

Recent observations [1] of the cosmic microwave background (CMB) have pointed to an apparent discrepancy between the value of the Hubble parameter H₀ deduced from cosmological data and the value inferred from local astrophysical observations [1–3].

Following [4] we denote with ${H_0}^{\!\!\!\textit{app}}$ the value of the Hubble constant estimated from observational data ignoring the presence of an inhomogeneity and with $H_0^{\textit{true}}$ the value obtained taking into account the inhomogeneity. According to this notation, since both the above-mentioned estimations were based on the assumption of homogeneity, the apparent tension can be expressed as $H_{0,SN}^{\textit{app}}\approx1.09H_{0,\textit{CMB}}^{\textit{app}}$, where $H_{0,SN}^{\textit{app}}$ and $H_{0,\textit{CMB}}^{\textit{app}}$ are the values estimated from fitting respectively low-redshift supernovae and CMB observations. One possible explanation of such a difference could be the effects of local structure on the analysis of cosmological and astrophysical observations, and we will show what kind of inhomogeneity could resolve the tension, i.e. could give $H_{0,SN}^{\textit{true}}=H_{0,\textit{CMB}}^{\textit{true}}$.

The effects of a local inhomogeneity on the estimation of cosmological parameters have been studied already in different contexts such as the estimation of the equation of state of dark energy [4], where it has been shown that ignoring the presence of local inhomogeneities in data analysis can lead to the wrong conclusion of an apparent evolving dark energy, while only a cosmological constant is in fact present, or to corrections [5] to the apparent value of the cosmological constant. More recently [6] it was shown that a present-day local inhomogeneity seeded by a peak of primordial curvature perturbations could in fact not only affect the estimation of the value of the cosmological constant, but also of H₀. In particular it was shown that the effects of such an inhomogeneity on H₀ depend on the spatial gradient of the inhomogeneity, and could be important independently of the amplitudes of the curvature perturbations. This implies that this kind of inhomogeneities arise naturally from fluctuations of the primordial curvature perturbations, and require a careful investigation. In this letter we will focus on the early-time origin of an inhomogeneity able to solve the apparent tension in the H₀ estimation, showing how the inflationary scenario can easily explain the formation of such a local structure today.

Some of the previous studies of the effects of inhomogeneities on the expansion rate, such as, for example, [7–9], were based on an implicit assumption of the validity of the Hubble law and on the use of a Newtonian approximation to relate the local velocity field to H. Other approaches are based on the Hubble bubble model [10], which is the basis of the top hat spherical collapse, or on the averaging of the perturbations [11]. All these different attempts to estimate the effects of inhomogeneities are based on a Newtonian or relativistic linear perturbations approximation, and as such are not able to obtain the nonperturbative relativistic effects that the use of an exact solution of Einstein's equations allows to calculate. In fact, as shown in [12], perturbation theory is not able to fully account for the relativistic effects of an inhomogeneities for the luminosity distance, because gradient terms can have important contributions which are normally ignored in the Newtonian or relativistic perturbative calculations.

The discrepancy between apparent and true values of cosmological parameters is a consequence of the intrinsic limitation of cosmological or astrophysical observations involving redshift measurements, implying an intrinsic uncertainty [13] on the estimation of cosmological parameters. Under the assumption of homogeneity the redshift is explained exclusively as a consequence of the expansion of the Universe, while taking into account spatial inhomogeneity additional contributions to the redshift can come from the spatial variation the gravitational potential. While the effects of these variations are normally considered to be small compared to the contribution associated to the Universe expansion, it is possible that the local structure can actually induce some important corrections with respect to the apparent values, i.e. the ones obtained from observation ignoring the space variation of the metric.

From the relation between the Hubble parameter and the luminosity distance in a spatially flat FLRW Universe,

$$\begin{eqnarray} D_L(z)&=(1+z)\int^{z}_0\frac{\text{d} x}{H(x)},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} H^{\Lambda CDM}(z)&=H_0{^{\!\!app}}\sqrt{\Omega_M(1+z)^3+\Omega_{\Lambda}}, \end{eqnarray} \tag{ 2 }$$

we can see that the larger estimated value of $H_{0,SN}^{\textit{app}}$ compared to $H_{0,\textit{CMB}}^{\textit{app}}$ corresponds to the observation of supernovae closer than what is expected in a $\Lambda \text{CDM}$ model with the Hubble parameter equal to $H_{0,\textit{CMB}}^{\textit{app}}$. In the framework of homogenous cosmological models the observed brightness of low-redshift supernovae is interpreted as the result of a faster expansion of the Universe, while if the effects of a local inhomogeneity are properly taken into account there is no need to invoke a value of H₀ different from $H_{0,\textit{CMB}}^{\textit{app}}$. In the particular case of the set of observations we are interested in in this letter, we have that an appropriate local inhomogeneity does not affect the distance to the last scattering, and consequently the estimation of H₀ from CMB data, so that the assumption of homogeneity in analyzing CMB data does not give a different result with respect to the corresponding data analysis taking into account the inhomogeneity, i.e. $H^{\textit{app}}_{0,\textit{CMB}}\!\approx\! H^{\textit{true}}_{0,\textit{CMB}}$. The presence of a local inhomogeneity could instead affect the local estimation of H₀, so that $H^{\textit{app}}_{0,SN}\!>\!H^{\textit{true}}_{0,SN}$, where $H^{\textit{true}}_{0,SN}$ is the value of H₀ which, according to the above definitions, is obtained from the data taking into account the effects of the inhomogeneity. In the presence of an inhomogeneity in fact the determination of H₀ from $D_L^{\textit{obs}}(z_{SN})$, where $z_{SN}\approx 0.04$ is the mean redshift of the supernovae observations used for the local estimation of H₀ [2], should not be based on the standard $\Lambda \text{CDM}$ redshift distance relation, but another one which takes into account the local inhomogeneity, and in this way the discrepancy can be solved, i.e. $H^{\textit{app}}_{0,\textit{CMB}}\approx H^{\textit{true}}_{0,\textit{CMB}}=H^{\textit{true}}_{0,SN}$.

We will adopt the following notation for the luminosity distance: $D_L^{\textit{mod}}(z,H_0)$, where we are denoting with a superscript the cosmological model or the observed data, and H₀ is explicitly treated as an argument to avoid confusion. According to the above notation low-redshift supernovae observations are fitted by a homogeneous cosmological model according to

$$\begin{equation} D_L^{\textit{hom}}(z_{SN},{H_{0,SN}^{\textit{app}}})=D_L^{\textit{obs}}(z_{SN}), \end{equation} \tag{ 3 }$$

with $H_{0,SN}^{\textit{app}}\approx 1.09 H_{0,\textit{CMB}}^{\textit{app}}$. We will show that if the effects of primordial curvature perturbations are taken into account, a local inhomogeneity could resolve the apparent discrepancy, i.e.

$$\begin{equation} D_L^{\textit{inh}}(z_{SN},{H_{0^,\textit{CMB}}^{\textit{true}}})=D_L^{obs}(z_{SN}). \end{equation} \tag{ 4 }$$

Such an inhomogeneity can arise naturally from primordial curvature perturbations as predicted by inflation and constrained by CMB observations to have a standard deviation of about $5 \times 10^{-5}$, providing a simple mechanism for its origin.

We will model the local structure with a LTB solution of Einstein's equation with a cosmological constant term, assuming to be located around its center. This is a pressureless spherically symmetric solution which allows to take into account the nonperturbative effects due to the inhomogeneity. The central location assumption can be interpreted [14] as calculating the monopole contribution to the corrections coming from local structure. Since we are considering inhomogeneities which can be seeded by not very large fluctuations of primordial curvature perturbations, these structures can arise very naturally. Only the use of an exact solution allows to study their formation, since today they can become highly nonlinear, and the standard theory of structure formation based on a perturbative approach cannot be applied to study their full evolution.

The LTB solution is given by [15–17]

$$\begin{eqnarray} &&\text{d}s^2 = -\text{d}t^2 + \frac{\left(R,_{r}\right)^2 \text{d}r^2}{1 + 2\,E(r)}+R^2 \text{d}\Omega^2, \end{eqnarray} \tag{ 5 }$$

where R is a function of the time coordinate t and the radial coordinate r, E(r) is an arbitrary function of r, and $R_{,r}=\partial_rR(t,r)$. The Einstein's equations with a cosmological constant give

$$\begin{eqnarray} \left({\frac{\dot{R}}{R}}\right)^2&=\frac{2 E(r)}{R^2}+\frac{2M(r)}{R^3}+\frac{\Lambda}{3},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \rho(t,r)&=\frac{2M,_{r}}{R^2 R,_{r}},\end{eqnarray} \tag{ 7 }$$

where M(r) is an arbitrary function of r, $\dot{R}=\partial_t R(t,r)$ and $c=8\pi G=1$ and is assumed in the rest of the paper. We will also adopt, without loss of generality, the coordinate system in which $M(r)\propto r^3$, fix the geometry of the solution by using a function k(r) according to $2E(r)=-k(r)r^2$, and consider models with a vanishing bang function, i.e. $t_b(r)=0$.

The metric after inflation near a peak of the primordial curvature perturbation [4,6] can be written as

$$\begin{equation} \text{d}s^2=-\text{d}t^2 + a_F^2(t)e^{2\zeta({ r})}(\text{d}r^2+r^2\text{d}\Omega^2), \end{equation} \tag{ 8 }$$

where $\zeta(r)$ is the primordial curvature perturbation. The spherical symmetry approximation is justified by the property of a Gaussian random field [18] according to which large peaks of a stochastic function tend to have a spherical shape. At a sufficiently early time when both the LTB and the metric in eq. (8) are valid, they can be matched [4] leading to the following relation:

$$\begin{equation} k(r)=-\frac{1}{ r^2}[(1+r\zeta')^2-1]. \end{equation} \tag{ 9 }$$

The use of the LTB solution allows to study exactly the formation of the structure seeded by $\zeta(r)$, and in particular to go beyond the limitations of the perturbation theory. It turns out in fact that such early-time tiny curvature perturbations of the order of $10^{-5}$ can lead to a highly nonlinear structure today whose effects cannot be neglected.

Equation (9) is very important because it gives the connection between the early and the present Universe, and it can be used in two ways: given an early-Universe curvature perturbations it allows to find the present-day structure seeded by it, or vice versa, given a present-day structure, it can be used to find what primordial curvature perturbation could have originated it. Normally the first approach is adopted [4,6], but in this letter we will adopt the second one, since it is more convenient to define the model of the inhomogeneity in terms of k(r). In this way we can find a natural explanation for the origin of the present-day highly nonlinear structure whose effect cannot be captured by the perturbations theory, while the use of the LTB solution gives the nonperturbative evolution of the structure.

To compute $D_L^{\Lambda LTB}(z)$ we need to solve the radial null geodesics

$$\begin{eqnarray} \frac{\text{d}r}{\text{d}z}&=\frac{\sqrt{1+2E(r(z))}}{{(1+z)\dot {R'}[T(r(z)),r(z)]}},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} \frac{\text{d}t}{\text{d}z}&=-\,\frac{R'[T(r(z),r(z))]}{{(1+z)\dot {R'}[T(r(z)),r(z)]}}, \end{eqnarray} \tag{ 11 }$$

and then we substitute in the formula for the luminosity distance in a LTB space,

$$\begin{equation} D_L^{\textit{inh}}(z,H_{0}^{\textit{true}})=D^{\Lambda LTB}(z)=(1+z)^2 R(t(z),r(z)), \end{equation} \tag{ 12 }$$

where we set the parameters of the LTB solution so that

$$\begin{equation} H_0^{\textit{LTB}}=\frac{2}{3}\frac{\dot{R}(t_0,0)}{R(t_0,0)}+\frac{1}{3}\frac{\dot{R}'(t_0,0)}{R'(t_0,0)}=H_{0}^{\textit{true}}. \end{equation} \tag{ 13 }$$

For homogeneous cosmological models we assume a flat $\Lambda \text{CDM}$ solution according to

$$\begin{eqnarray} H(z)&=H_0^{\textit{app}}\sqrt{\Omega_M(1+z)^3+\Omega_{\Lambda}},\nonumber\\ D_L^{\textit{hom}}(z,H_0^{\textit{app}})&=(1+z)\int^{z}_0\frac{\text{d} x}{H(x)}. \end{eqnarray} \tag{ 14 }$$

The above definitions are very important since they give an explicit mathematical definition of $H_0^{\textit{app}}$ and $H_0^{\textit{true}}$, which is coherent with the notion of apparent observable adopted in this paper and which can be expressed as

$$\begin{eqnarray} &&D_L^{\textit{hom}}(z,H_0^{\textit{app}})=D_L^{\textit{inh}}(z,H_0^{\textit{true}})=D_L^{\textit{obs}}(z). \end{eqnarray} \tag{ 15 }$$

We consider solutions with the function k(r) given by a Gaussian

$$\begin{eqnarray} &&k(r)=A e^{-\left(\frac{r-r_0}{\sigma}\right)^2}, \end{eqnarray} \tag{ 16 }$$

centered approximately at the radial comoving coordinate of low-redshift supernovae used for the local estimation of H₀. Different values of r₀ can be chosen to control the shape of the inhomogeneity. This solution corresponds to a compensated inhomogeneity which is asymptotically homogeneous and, if the width of the Gaussian is sufficiently small, there is no important difference between the central and the asymptotic value of k(r). This property is quite important because it ensures that the Universe is effectively a flat $\Lambda \text{CDM}$ at the center and sufficiently far from it, making sure that the effect on the luminosity distance at the last scattering is very small.

We will adopt a system of units in which $H_{0,\textit{CMB}}^{\textit{true}}=1$, and use the fiducial value $\Omega_{\Lambda}=0.692$. The time used as initial condition at the center for the geodesics equations is obtained by integrating the Einstein equation in the scale factor from the Big Bang till today.

As shown in fig. 1, since the type of inhomogeneity we have considered is compensated, it has no effect on the luminosity distance outside it, i.e.

$$\begin{equation} D^{\textit{hom}}(H_{0,\textit{CMB}}^{\textit{true}},z_{LS})\approx D^{\textit{inh}}(H_{0,\textit{CMB}}^{\textit{true}},z_{LS}), \end{equation} \tag{ 17 }$$

where z_LS is the redshift of the last scattering surface, or any redshift sufficiently larger than z_SN. This is due to the fact that the central and the asymptotic value of the curvature function are approximately the same. According to our definitions of apparent observable, fitting CMB data with a homogeneous or inhomogeneous model is consequently giving the same estimate for $H_{0,\textit{CMB}}^{\textit{app}}\approx H_{0,\textit{CMB}}^{\textit{true}}$.

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For the case of low-redshift supernovae instead it can be seen in fig. 2 that

$$\begin{equation} D^{\textit{inh}}(H_{0,\textit{CMB}}^{\textit{true}},z_{SN})\approx D^{\textit{hom}}(H_{0,SN}^{\textit{app}},z_{SN})=D^{\textit{obs}}_{z_{SN}}, \end{equation} \tag{ 18 }$$

which implies that

$$\begin{equation} H_{0,SN}^{\textit{app}}>H_{0,SN}^{\textit{true}}=H_{0,\textit{CMB}}^{\textit{true}}. \end{equation} \tag{ 19 }$$

This shows how the effects of the inhomogeneity are able to resolve the apparent disagreement between cosmological and local estimations of H₀, since the luminosity distance is modified only around z_SN.

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As seen in figs. 1, 2 the effect of the structure seeded by this primordial curvature perturbation profile corresponds to an underdense central region followed by an overdense shell, which is what we expect for a compensated inhomogeneity, due to the fact that the function k(r) is asymptotically zero. The luminosity distance is decreased with respect to the homogeneous case both in the under- and overdense regions, a relativistic effect associated to large gradient terms which are normally neglected in Newtonian or perturbative calculations as shown, for example, in [12]. This inhomogeneity is seeded by a curvature perturbation of less than two standard deviations as shown in fig. 3, and the effect on the luminosity distance is associated to the region of transition between different values of $\zeta(r)$. Such a perturbation can occur quite naturally according the inflationary scenario, and smaller amplitude perturbations could still induce some important effects.

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The density profile shown in fig. 1 is compatible with the results of the analysis of galaxy surveys [19]. Quite remarkably the supernovae used for the local estimation of H₀ in [2] are inside the region with the largest overdensity (Subregion 2 in [19]) with a density profile compatible with our model in the region of available data, suggesting a possible directional dependence¹ of the local structure effects on the H₀ estimation which could be partially attributed to the presence in the region of the Sloan Great Wall [20].

A careful investigation of such a directional dependence goes beyond the scope of this paper, so we will not pursue it here, but we expect our calculations to provide a good estimation of the effects due to the density profile along that particular direction. It would be interesting though to check carefully if using supernovae in different directions away from the Sloan Great Wall the H₀ estimation could change significantly from the one obtained by [2].

We have shown how the apparent value of $H_{0,SN}^{app}$ obtained from analyzing the supernovae low-redshift luminosity distance data under the assumption of homogeneity can receive an important correction when the local structure is taken into proper account. The apparent disagreement is due to the different way in which the H₀ estimation is affected by the presence of the local inhomogeneity we have considered. In the case of CMB, the inhomogeneity does not affect the distance to the last scattering surface, implying that analyzing CMB data under the assumption of homogeneity does not introduce any misestimations. For low-redshift luminosity distance instead there can be an important impact which could resolve the discrepancy, or at least could be able to account for part of the difference. The type of inhomogeneity able to explain the difference can be seeded by a primordial curvature perturbation of the order of less than two standard deviations, and, as such, can arise quite naturally according to the inflationary scenario. This shows the importance of nonpertubative effects for any local observation in order to avoid this kind of apparent tension with other measurements less affected by local structure.

Quite remarkably the comparison with galaxies redshifts surveys suggests that the supernovae used for local estimation of H₀ are in a region of sky associated to a particularly high overdensity and with an inhomogeneity profile compatible with the one we have studied, which could be partially attributed to the presence in that region of the Sloan Great Wall. It would be interesting in the future to go beyond the monopole contribution and extend the study to the possible directional dependence of the local H₀ estimation. In this way it would be possible to determine indirectly the higher multipoles of the local structure which causes the apparent angular dependence of $H_0^{\textit{app}}$. The same type of analysis could be extended to other cosmological parameters, such as, for example, the effective equation of state of dark energy, which have already been shown to be affected by local inhomogeneities, but are normally analyzed under the assumption of isotropy.

AER thanks Ryan Keenan and Licia Verde for useful comments and discussions, the Department of Physics of Heidelberg University for the kind hospitality, and Luca Amendola, Valeria Pettorino and Miguel Zumalacarregui for insightful discussions. This work was supported by the European Union (European Social Fund, ESF) and Greek national funds under the "ARISTEIA II" Action, the Dedicacion exclusiva and Sostenibilidad programs at UDEA, the UDEA CODI project IN10219CE. The authors thank the anonymous referee for useful comments to improve the manuscript.