A structured analysis of Hubble tension

Over the last decade cosmological observations have significantly improved, both for early-time as well as late-time observations. Early-time measurements by the Planck collaboration give a value of $H_{0}=67.36\pm 0.54$ [1]. This value is corroborated by the results published in [2] combining Dark Energy Survey (DES), Baryon Acoustic Oscillation (BAO) and Big Bang Nucleosynthesis (BBN) constraints to give a value of $H_{0}=67.4\pm 1.1$.

For late-time measurements, results from [3, 4, 5, 6, 7] were combined in [8] to give a combined estimate of $H_{0}=73.3\pm 0.8$ (though with some caveats on the error bounds due to not fully accounting for shared contributions in the distance ladder). Furthermore, it is shown that these results, even when discarding some, still provide a $4-6\sigma$ tension between early and late-time values of $H_{0}$. There have since been further late-time measurements reported in [9, 10], but other than strengthening error bounds, these do not significantly change this picture.

This tension is large enough that it has created a need for an explanation. As such, there have been a myriad of models proposed that alleviate (part of) this tension, overviews of which are given in [11, 12].

Currently, there are few general results placing formal conditions on what a model alleviating the Hubble tension should look like. The authors are only aware of [13], providing a general result showing that a viable model needs to modify more than just the size of the sound horizon.¹¹1For the claim that a modification of the sound horizon is necessary, or variations on it, the authors are not aware of a formal argument showing all the assumptions neccessary for such a conclusion. The goal of this paper is to provide a new general (model independent) result setting conditions on models alleviating the Hubble tension.

We do this by taking a higher level view of possible alleviating models. In section 2, we provide a set of 7 assumptions based on CMB observations and direct measurements of $H_{0}$ which combine to create a space of options in which the cosmological standard model $\Lambda$CDM is an extremum. This then allows us to show that our 7 assumptions exclude any model that has the potential to alleviate the Hubble tension, implying that any such model should break at least 1 of our 7 assumptions.

We then demonstrate one application of this result in section 3 by making a categorisation of the space of existing proposals by means of the assumptions they break. This provides us with two things: A somewhat reasonable categorisation of existing proposals, and an under-explored avenue for new solutions.

We will start by making sufficient assumptions on our model of the universe and its equations of motion to be able to use the regular distance formula for the comoving angular diameter distance $D_{A}(z)$ of an object at redshift $z$.

Further assumptions are then needed to allow us to use the results from the Planck analysis of the Cosmic Microwave Background (CMB). Finally, we impose an at first glance reasonable restriction on the shape of the total matter density $\rho$, which will then result in limitations on the values for $H_{0}$ that are attainable.

Note that, with the above choice for the Einstein Equations, we absorb any cosmological constant $\Lambda$ into the stress-energy tensor $T\indices{{}_{\mu}{}_{\nu}}$.

Assuming a spatially isotropic universe, behaving according to the laws of general relativity, the metric becomes the Friedmann-Robertson-Walker metric, and can be written in spherical coordinates as

$\displaystyle\mathrm{d}s^{2}$

$\displaystyle=-\mathrm{d}t^{2}+a(t)^{2}\left[\frac{\mathrm{d}r^{2}}{1-\kappa r^{2}}+r^{2}\mathrm{d}\Omega^{2}\right].$

(1)

Here, $a(t)$ is the scale factor of the universe at time $t$, and $\kappa\in\{-1,0,1\}$ indicates whether the metric is spatially hyperbolic, spatially flat, or spatially spherical respectively.

Given this form of metric, the Einstein equations reduce to the Friedmann Equation and covariant energy conservation equation, and can be written as

	$\displaystyle H(t)^{2}$	$\displaystyle=\frac{8\pi G}{3}\rho(t)-\frac{\kappa}{a(t)^{2}},$		(2)
	$\displaystyle 0$	$\displaystyle=\partial_{0}\rho(t)+3H(t)(\rho(t)+p(t)),$		(3)

with $p(t)$ the pressure, and $H(t)=\frac{\partial_{0}a(t)}{a(t)}$ the Hubble parameter, giving as present-day value $H_{0}\equiv H(0)$.

We choose to use the covariant energy conservation equation, instead of the second Friedmann Equation, in order to more easily reason about its impact on the behaviour of $\rho$ later. It is a relatively straightforward calculation to show that both forms are in fact equivalent in this situation.

Ignoring the possibility of spatial curvature further simplifies the metric to

$\displaystyle\mathrm{d}s^{2}$

$\displaystyle=-\mathrm{d}t^{2}+a(t)^{2}\left[\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}\right].$

(4)

and removes the curvature term from the first Friedmann Equation. This in turn implies that

$\displaystyle H$

$\displaystyle=\sqrt{\frac{8\pi G}{3}\rho}$

(5)

in an expanding universe.

This assumption now allows us to use $z$ as a time variable. Combined with the other assumptions made so far, it also fully constrains the relationship between $\rho(t)$, $D_{A}$ and $z_{*}$. This can be written as

	$\displaystyle D_{A}$	$\displaystyle=\int_{0}^{z_{*}}\frac{\mathrm{d}z^{\prime}}{H(z^{\prime})}$		(6)
		$\displaystyle=\sqrt{\frac{3}{8\pi G}}\int_{0}^{z_{*}}\frac{\mathrm{d}z^{\prime}}{\sqrt{\rho(z^{\prime})}}$		(7)

With this in place, we now turn to making the assumptions needed to bring in measured values for the comoving angular diameter distance $D_{A}$, the redshift of the CMB $z_{*}$, the matter density at decoupling $\rho_{m}(z_{*})$ from Planck observations, and the directly measured value of $H_{0}$. By necessity, the assumptions leading up to these are somewhat broad:

Before we can start drawing conclusions, there is one final constraint needed on the behaviour of the total density. For this, we define the model specific density $\rho_{x}$ as

$\displaystyle\rho_{x}(z)=\rho(z)-\rho_{m}(z_{*})\left(\frac{1+z}{1+z_{*}}\right)^{3}.$

(8)

This definition splits off the non-interacting dilution of matter due to expansion from any other, model-specific, effects. Note that there is no specific meaning to $x$, rather we use $\rho_{x}$ generically to refer to the model-specific contributions to $\rho$. We then assume the following:

Intuitively, this assumption states that there is no net matter creation in $\rho_{x}$, and that there is no decay from $\rho_{m}$ to a component contributing to $\rho_{x}$ after the decoupling of the CMB.

Combining this final assumption into Equation 7, we find

	$\displaystyle D_{A}$	$\displaystyle=\sqrt{\frac{3}{8\pi G}}\int_{0}^{z_{*}}\frac{\mathrm{d}z^{\prime}}{\sqrt{\rho_{m}(z_{*})\left(\frac{1+z^{\prime}}{1+z_{*}}\right)^{3}+\rho_{x}(z^{\prime})}}$		(9)
		$\displaystyle\leq\sqrt{\frac{3}{8\pi G}}\int_{0}^{z_{*}}\frac{\mathrm{d}z^{\prime}}{\sqrt{\rho_{m}(z_{*})\left(\frac{1+z^{\prime}}{1+z_{*}}\right)^{3}+\rho_{x}(0)}}.$		(10)

As $H_{0}$ is directly related to $\rho(0)$, we find that the following holds under the assumptions made above: for a given value of $H_{0}$, the $\Lambda$CDM model has maximal comoving angular diameter distance to the CMB, and any alternative model results in a lower value for $D_{A}$. Since $D_{A}$ is fixed by measurements, this implies that for any model conforming to our assumptions with non-constant contribution $\rho_{x}$ to the total energy density of the universe, the value for $H_{0}$ would need to be lower than that predicted by extrapolating from the $\Lambda$CDM model. As we find from observations that the currently measured value for $H_{0}$ is higher than predicted from the $\Lambda$CDM assumptions in combination with the Planck observations, we can conclude that one of the assumptions 1-7 must be broken.

We can now reconsider existing models for alleviating the Hubble tension within the framework of our assumptions. This gives insight into how existing models achieve their alleviation of the Hubble tension, and provides insight into avenues of reducing the early-late time tension that are not yet as well explored.

In doing this, we will consider a variety of models from the literature in the coming pages. However, our focus here will be primarily on providing clarity in illustrating the various ways our assumptions can be broken, and as such, this will not be an exhaustive overview of theories alleviating the Hubble tension. The interested reader can find good starting points for further theories in [11].

Furthermore, in choosing these examples, we have not focused on whether such models are, or have the potential to be, consistent with further observational evidence beyond the constraints imposed by direct measurements of $H_{0}$ and measurements of the CMB. This choice was made specifically to provide a more complete illustration of the approaches already considered.

Finally, some of the examples given here could be classified under multiple of these assumptions, either because they break multiple conditions, or because one can reasonably disagree which of the conditions is actually broken. The classification here represents the opinions of the authors as to which condition(s) best reflect the way in which a model alleviates Hubble tension. Although others might classify these models differently, we believe that the resulting conclusions drawn from this classification are robust under these differing opinions.

We have shown that, for any theory to be able to solve the Hubble tension, it needs to break at least one of the Assumptions 1 through 7 defined above. This provides us with two main benefits. First of all, these conditions can be used as a tool to quickly determine whether a new proposed model has any hope of relieving the Hubble tension, and as a check on results for those models that break these conditions only in certain parts of their parameter space.

Furthermore, it allowed us to categorise the models already proposed. This showed us that breaking the standard relationship between redshift and scale factor can potentially solve the Hubble tension, but has so far not been explored in models. Exploration of mechanisms which can achieve such a changed link between redshift and scale factor could be an interesting avenue of further research.

One limitation of our work is that it constrains the theory space only from two very specific observations, the direct measurement of $H_{0}$ through distance ladders, and the interpretation of the CMB. Further research into general conditions on models resulting from other observations could provide useful tools for further exploring the space of models alleviating the Hubble tension.

The authors would like to thank M. Postma and S. Vagnozzi for their feedback on early versions of this paper.