A structured analysis of Hubble tension
aa r X i v : . [ a s t r o - ph . C O ] F e b A structured analysis of Hubble tension
Wim Beenakker a,b , David Venhoek aa Theoretical High Energy Physics, Radboud University Nijmegen,Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands b Institute of Physics, University of Amsterdam, Science Park 904, 1018 XEAmsterdam, The Netherlands
Abstract
As observations of the Hubble parameter from both early and latesources have improved, the tension between these has increased to be wellabove the 5 σ threshold. Given this, the need for an explanation of sucha tension has grown. In this paper, we explore a set of 7 assumptions,and show that, in order to alleviate the Hubble tension, a model needs tobreak at least one of these 7, providing a quick and easy to apply checkfor new model proposals. We also use this framework to make a roughcategorisation of current proposed models, and show the existence of atleast one under-explored avenue of alleviating the Hubble tension. Over the last decade cosmological observations have significantly improved, bothfor early-time as well as late-time observations. Early-time measurements bythe Planck collaboration give a value of H = 67 . ± .
54 [1]. This value is cor-roborated by the results published in [2] combining Dark Energy Survey (DES),Baryon Acoustic Oscillation (BAO) and Big Bang Nucleosynthesis (BBN) con-straints to give a value of H = 67 . ± . H = 73 . ± . − σ tension between early and late-time values of H . Therehave since been further late-time measurements reported in [9, 10], but otherthan 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 whata model alleviating the Hubble tension should look like. The authors are only1ware of [13], providing a general result showing that a viable model needs tomodify more than just the size of the sound horizon. The goal of this paperis to provide a new general (model independent) result setting conditions onmodels alleviating the Hubble tension.We do this by taking a higher level view of possible alleviating models. Insection 2, we provide a set of 7 assumptions based on CMB observations anddirect measurements of H which combine to create a space of options in whichthe cosmological standard model ΛCDM is an extremum. This then allows usto show that our 7 assumptions exclude any model that has the potential toalleviate the Hubble tension, implying that any such model should break atleast 1 of our 7 assumptions.We then demonstrate one application of this result in section 3 by makinga categorisation of the space of existing proposals by means of the assumptionsthey break. This provides us with two things: A somewhat reasonable categori-sation of existing proposals, and an under-explored avenue for new solutions. We will start by making sufficient assumptions on our model of the universeand its equations of motion to be able to use the regular distance formula forthe 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 fromthe Planck analysis of the Cosmic Microwave Background (CMB). Finally, weimpose an at first glance reasonable restriction on the shape of the total matterdensity ρ , which will then result in limitations on the values for H that areattainable. Assumption 1.
The laws of general relativity, and in particular the EinsteinEquations R µν − Rg µν = 8 πGT µν , are a good approximation of the physics ofthe universe at cosmological scales.Note that, with the above choice for the Einstein Equations, we absorb anycosmological constant Λ into the stress-energy tensor T µν . Assumption 2.
The universe can be approximated as spatially isotropic andhomogeneous at large scales.Assuming a spatially isotropic universe, behaving according to the laws ofgeneral relativity, the metric becomes the Friedmann-Robertson-Walker metric,and can be written in spherical coordinates asd s = − d t + a ( t ) (cid:20) d r − κr + r dΩ (cid:21) . (1) For 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 forsuch a conclusion. a ( t ) is the scale factor of the universe at time t , and κ ∈ {− , , } indicateswhether the metric is spatially hyperbolic, spatially flat, or spatially sphericalrespectively.Given this form of metric, the Einstein equations reduce to the FriedmannEquation and covariant energy conservation equation, and can be written as H ( t ) = 8 πG ρ ( t ) − κa ( t ) , (2)0 = ∂ ρ ( t ) + 3 H ( t )( ρ ( t ) + p ( t )) , (3)with p ( t ) the pressure, and H ( t ) = ∂ a ( t ) a ( t ) the Hubble parameter, giving aspresent-day value H ≡ H (0).We choose to use the covariant energy conservation equation, instead of thesecond Friedmann Equation, in order to more easily reason about its impact onthe behaviour of ρ later. It is a relatively straightforward calculation to showthat both forms are in fact equivalent in this situation. Assumption 3.
The universe has no significant spatial curvature.Ignoring the possibility of spatial curvature further simplifies the metric tod s = − d t + a ( t ) (cid:2) d x + d y + d z (cid:3) . (4)and removes the curvature term from the first Friedmann Equation. This inturn implies that H = r πG ρ (5)in an expanding universe. Assumption 4.
The relationship between photon redshift z and scale factor a is a = z .This assumption now allows us to use z as a time variable. Combined withthe other assumptions made so far, it also fully constrains the relationship be-tween ρ ( t ), D A and z ∗ . This can be written as D A = Z z ∗ d z ′ H ( z ′ ) (6)= r πG Z z ∗ d z ′ p ρ ( z ′ ) (7)With this in place, we now turn to making the assumptions needed to bring inmeasured values for the comoving angular diameter distance D A , the redshift ofthe CMB z ∗ , the matter density at decoupling ρ m ( z ∗ ) from Planck observations,and the directly measured value of H . By necessity, the assumptions leadingup to these are somewhat broad: 3 ssumption 5. The physics leading to the angular scale of the baryonic soundhorizon, and its size, as well as the related estimates of total matter density atdecoupling, are accurately approximated in the Planck analysis. In particular,the measured angular diameter distance to the CMB, the redshift of decoupling,and the matter density at decoupling, have no significant systematic or model-induced errors that are unaccounted for.
Assumption 6.
The physics related to distance measurements of late-timeobjects is well understood and provide an accurate way of measuring the Hub-ble parameter at late times. In particular, there are no significant systematicor model-induced errors on measured redshifts and absolute brightnesses unac-counted for.Before we can start drawing conclusions, there is one final constraint neededon the behaviour of the total density. For this, we define the model specificdensity ρ x as ρ x ( z ) = ρ ( z ) − ρ m ( z ∗ ) (cid:18) z z ∗ (cid:19) . (8)This definition splits off the non-interacting dilution of matter due to ex-pansion from any other, model-specific, effects. Note that there is no specificmeaning to x , rather we use ρ x generically to refer to the model-specific contri-butions to ρ . We then assume the following: Assumption 7. ∂ρ x ∂z ≥ z < z ∗ .Intuitively, this assumption states that there is no net matter creation in ρ x ,and that there is no decay from ρ m to a component contributing to ρ x after thedecoupling of the CMB.Combining this final assumption into Equation 7, we find D A = r πG Z z ∗ d z ′ r ρ m ( z ∗ ) (cid:16) z ′ z ∗ (cid:17) + ρ x ( z ′ ) (9) ≤ r πG Z z ∗ d z ′ r ρ m ( z ∗ ) (cid:16) z ′ z ∗ (cid:17) + ρ x (0) . (10)As H is directly related to ρ (0), we find that the following holds underthe assumptions made above: for a given value of H , the ΛCDM model hasmaximal comoving angular diameter distance to the CMB, and any alternativemodel results in a lower value for D A . Since D A is fixed by measurements, thisimplies that for any model conforming to our assumptions with non-constantcontribution ρ x to the total energy density of the universe, the value for H would need to be lower than that predicted by extrapolating from the ΛCDMmodel. As we find from observations that the currently measured value for H is higher than predicted from the ΛCDM assumptions in combination with thePlanck observations, we can conclude that one of the assumptions 1-7 must bebroken . 4 Broken conditions in existing theories
We can now reconsider existing models for alleviating the Hubble tension withinthe framework of our assumptions. This gives insight into how existing modelsachieve their alleviation of the Hubble tension, and provides insight into avenuesof 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 thecoming pages. However, our focus here will be primarily on providing clarity inillustrating the various ways our assumptions can be broken, and as such, thiswill 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 whethersuch models are, or have the potential to be, consistent with further observa-tional evidence beyond the constraints imposed by direct measurements of H and measurements of the CMB. This choice was made specifically to provide amore complete illustration of the approaches already considered.Finally, some of the examples given here could be classified under multipleof these assumptions, either because they break multiple conditions, or becauseone can reasonably disagree which of the conditions is actually broken. Theclassification here represents the opinions of the authors as to which condition(s)best reflect the way in which a model alleviates Hubble tension. Although othersmight classify these models differently, we believe that the resulting conclusionsdrawn from this classification are robust under these differing opinions. H Let us first consider the option of systematics affecting late-time measurements,breaking assumption 6. In the case of CMB observations by Planck, [1] indicatesno direction for poorly or uncontrolled sources of systematic effects from knownphysics. For late-time measurements of H , the wide variety of independentobservations ([3, 4, 5, 6, 7], since augmented with [9, 10]) also leads the authorsof [8] to consider this to be an unlikely option.However, there exist several effects that act similarly to late-time systematicswithin our framework of assumptions. This could be from effects such as photonbrightening (section IV.2 in [11], originating from ideas in [14] and [15]), itcould be the result of a local matter void [16], or originate from screened fifthforces [17]. These theories cause late-time measurements to overestimate H bydecreasing the apparent distance, reducing the tension directly that way. At early times, there is similarly the option of systematics, or systematics-likeinfluences of a new model, breaking Assumption 5. In the case of CMB observa-tions by Planck, [1] indicates no direction for poorly or uncontrolled sources ofsystematic effects from known physics. However, apparent systematics can re-sult from effects changing the scale of the sound horizon, such as neutrino mass5echanisms [18] or the effects of additional light particles [19, 1]. Such theo-ries typically modify the time of matter-radiation equality, resulting in changesto the size of the sound horizon. So-called early dark energy models [20, 21]achieve a change of the sound horizon by changing the evolution just aftermatter-radiation equality, giving similar net effects from the perspective of ourreasoning. This changed sound horizon, which typically becomes smaller, inturn decreases the angular diameter distance, resulting in higher values of H from early-time measurements.These are not the only options considered in the literature. A second ap-proach studied in [22, 23] involves modifications to the mechanism of recombina-tion, resulting in a different temperature of recombination, or a more stretchedrecombination. These modifications change the interpretation of the CMB, re-sulting in a different angular distance and/or dark matter mass estimate. A second approach to alleviating the Hubble tension is found in subvertingthe assumptions leading to the framework of flat FRW spacetimes and theirevolution equations.One option for this is modifying gravity to the point where the effects ofsuch modifications influence cosmological evolution, breaking Assumption 1.This approach has been studied for various modified gravity theories in [24, 25,26, 27].A second way this can happen is when the evolution of the universe gainssignificant effects from deviations from perfect spatial flatness or isotropy. Thiswould then break either or both of Assumptions 2 and 3. Such an approachis studied for an ”effective” curvature type contribution arising from matterclumping in [28]. Approaches based on more explicit breaking of isotropy canbe found in [29].These ideas have in common that they, directly or indirectly, modify therelation between angular diameter distance and (some of the ingredients of) H ( z ), putting the applicability of Equation 7 into doubt.A third option for breaking this would be an explicit global curvature. How-ever, the authors are not aware of attempts to resolve the hubble tension thisway, and while there seems to be some mild evidence for curvature in theCMB[1, 30], this in general is not supported by other datasets[31]. This leaves three final assumptions we have not yet discussed: Breaking therelation between redshift and scale factor (Assumption 4), having (dark) matterdecay to some slower-diluting component, or having components in the universewhose density grows with redshift (both part of Assumption 7).Both decaying (dark) matter solutions, as well as growing density-componentsolutions are very active areas of research. Because of this, we will not try togive an overview here of the candidates, but merely point to a number of recent6apers [32, 33, 34, 35, 36, 37, 38, 39] as a starting point for the interested reader,together with the more complete literature survey provided in [12]. By breakingAssumption 7, such solutions can increase H (0) relative to H ( z ) (where z > H determined from the CMB are reasonably sensitive to these effects(due to the third-power dependency on the scale factor in ΛCDM) this mightbe an interesting area for further research. The H shift in this situation arisesbecause smaller than expected values for a ∗ would give a larger range of inte-gration in Equation 7, which in turn would allow H ( z ) to be larger without D A dropping below the measured value. We have shown that, for any theory to be able to solve the Hubble tension, itneeds to break at least one of the Assumptions 1 through 7 defined above. Thisprovides us with two main benefits. First of all, these conditions can be usedas a tool to quickly determine whether a new proposed model has any hope ofrelieving the Hubble tension, and as a check on results for those models thatbreak these conditions only in certain parts of their parameter space.Furthermore, it allowed us to categorise the models already proposed. Thisshowed us that breaking the standard relationship between redshift and scalefactor can potentially solve the Hubble tension, but has so far not been ex-plored in models. Exploration of mechanisms which can achieve such a changedlink between redshift and scale factor could be an interesting avenue of furtherresearch.One limitation of our work is that it constrains the theory space only fromtwo very specific observations, the direct measurement of H through distanceladders, and the interpretation of the CMB. Further research into general con-ditions on models resulting from other observations could provide useful toolsfor further exploring the space of models alleviating the Hubble tension. The authors would like to thank M. Postma and S. Vagnozzi for their feedbackon early versions of this paper.
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