Acoustic Dark Energy: Potential Conversion of the Hubble Tension
AAcoustic Dark Energy: Potential Conversion of the Hubble Tension
Meng-Xiang Lin, Giampaolo Benevento,
2, 3, 1
Wayne Hu, and Marco Raveri Kavli Institute for Cosmological Physics, Department of Astronomy & Astrophysics,Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637, USA Dipartimento di Fisica e Astronomia “G. Galilei”,Universit`a degli Studi di Padova, via Marzolo 8, I-35131, Padova, Italy INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy
We discuss the ability of a dark fluid becoming relevant around the time of matter radiationequality to significantly relieve the tension between local measurements of the Hubble constantand CMB inference, within the ΛCDM model. We show that the gravitational impact of acousticoscillations in the dark fluid balance the effects on the CMB and result in an improved fit toCMB measurements themselves while simultaneously raising the Hubble constant. The requiredbalance favors a model where the fluid is a scalar field that converts its potential to kinetic energyaround matter radiation equality which then quickly redshifts away. We derive the requirementson the potential for this conversion mechanism and find that a simple canonical scalar with twofree parameters for its local slope and amplitude robustly improves the fit to the combined data by∆ χ ≈ . I. INTRODUCTION
The ΛCDM model of cosmology has been tested byan extensive number of independent probes, showing itsgeneral robustness and remarkable ability to explain awide range of observational data with only six parame-ters. Despite its indubitable success, ΛCDM seems to failin reconciling distance-redshift measurements when an-chored at high redshift by cosmic microwave background(CMB) anisotropies to the same measurements anchoredat low redshift by the local distance ladder.This discrepancy is commonly quantified as tensionbetween the inferences for the Hubble constant ( H ), andits statistical significance has been steadily increasingwith increasing experimental precision. The most recentlocal estimate of the Hubble constant places its valueat H = 74 . ± .
42 km s − Mpc − [1], showing a 4 . σ tension with the value inferred by the Planck 2018 CMBdata, H = 67 . ± . − Mpc − , assuming theΛCDM model [2].This tension is mainly a discrepancy between the an-chors for the absolute distance scale rather than an in-dicator of missing physics between the anchors. Onceanchored at one end, the same ladder of intermediateredshift measurements from baryon acoustic oscillations(BAOs) to supernovae Type IA (SN) predict the anchorat the other end, leaving little room for missing cosmo-logical physics in between (for a recent assessment anddiscussion, see Refs. [3, 4] and references therein). On thehigh redshift side, the anchor is the CMB sound horizon r s . Under ΛCDM, the shapes of the CMB acoustic peakscalibrate the sound speed and all of the energy densitiesof species relevant around recombination and thus deter-mine the physical scale of r s . Measurements of its angu-lar scale in the CMB then fix the remaining parameter,the cosmological constant or equivalently H . Even be-yond ΛCDM, this measurement determines the distance to recombination and anchors the inverse distance lad-der from which H can again be inferred so that even themost general dark energy or modified gravity model canonly moderately alter its value [5–8].Altering the high redshift anchor requires modifyingcosmological physics at recombination. Adding extra en-ergy density raises the expansion rate before recombina-tion and lowers the sound horizon r s . For example, anextra sterile neutrino or other dark radiation has longbeen considered as a possible solution [9–20]. Howeversuch a component would also change the driving of theacoustic oscillations and damping scale [21], which is nowdisfavored by increasingly precise CMB data, leaving lit-tle ability to raise H (see Refs. [2, 22] for recent assess-ments).The problem with the damping scale arises becausethese additions affect the background expansion like ra-diation. As pointed out in Ref. [23], this problem can beavoided by making the dark component only importanttransiently near the epoch of recombination. Specifically,Ref. [23] introduces a component of so-called “early darkenergy” (EDE) where a scalar field oscillates anharmon-ically around the minimum of a periodic potential andfinds that the Hubble tension can be efficiently relieved.On the other hand, Ref. [24] finds that for a monomialEDE model, which coincides with the periodic poten-tial [23] at the minimum, the Hubble tension is only par-tially relieved. Due to the behavior of its perturbations,these EDE scenarios change the amplitudes and phasesof the CMB acoustic peaks in complex ways, leading toquestions as to the robustness of this method for relievingthe Hubble tension.To build a more robust method to relieve the Hub-ble tension, we study the general phenomenology of per-turbations in a dark fluid which similarly becomes tran-siently important, here around matter-radiation equality.Since its impact on the CMB comes through the gravi-tational effects of its own acoustic oscillations, we call a r X i v : . [ a s t r o - ph . C O ] O c t this species acoustic dark energy (ADE) and in particu-lar uncover the critical role its sound speed plays in re-lieving the Hubble tension. We find that the sound speedmust vary with the equation of state in the backgroundin a manner consistent with the conversion of potentialto kinetic energy for a minimally coupled scalar with ageneral kinetic term [25]. Unlike the oscillatory EDEmodels, once released from Hubble drag, the scalar re-mains kinetic energy dominated until it redshifts away.Indeed for a simple canonical kinetic term, this allowsfor H = 70 . ± .
85 with a better fit than ΛCDM evenfor the CMB alone and a better total χ by 12.7 for 2extra parameters. This method is also robust and can beexactly realized in a wide class of potentials. We provideboth the required conditions on the potential and explicitexamples.This paper is organized as follows. In § II, we intro-duce the ADE fluid model, its parameters, and the datasets that we use in the analysis. In § III, we discuss thephenomenological impacts of ADE, especially its soundspeed, on acoustic driving and CMB polarization. In § IV,we show that ADE models that can relieve the Hubbletension correspond to scalars that convert potential tokinetic energy suddenly upon Hubble drag release andconstruct a canonical scalar model as proof of principle.In § V, we discuss the relation to the previous work andwe conclude in § VI.
II. METHODOLOGY
Acoustic dark energy is defined to be a perfect darkfluid and is specified by its background equation of state w ADE = p ADE /ρ ADE and rest frame sound speed c s [26].The latter is only equivalent to the adiabatic sound speed˙ p ADE / ˙ ρ ADE for a barotropic fluid so that in the generalcase the acoustic phenomenology of linear ADE soundwaves, which we shall see is crucial for relieving the Hub-ble tension, is defined independently of the background.In order to have a transiently important ADE contri-bution, we model the ADE equation of state as1 + w ADE ( a ) = 1 + w f [1 + ( a c /a ) w f ) /p ] p . (1)The ADE component therefore changes its equation ofstate around a scale factor a = a c from w ADE = − w f . Additionally, p controls the rapidity of this transi-tion, with small values corresponding to sharper transi-tions. Since this parameter does not qualitatively changeour results, we use p = 1 / § IV that this corresponds to a simplequadratic potential for scalar field ADE. This is a gen-eralization of the background of the EDE model [23],discussed in § V, where p = 1 and the fluid description isapproximate.The ADE background energy density is fully specifiedonce its normalization is fixed, since w ADE determines its evolution. Defining the ADE fractional energy densitycontribution f ADE ( a ) = ρ ADE ( a ) ρ tot ( a ) , (2)we choose f c = f ADE ( a c ) as the normalization parameter.The behavior of ADE perturbations is determined bytheir rest frame sound speed c s ( a, k ) which is, for an ef-fective fluid, a function of both time and scale [26]. Inthe context of a perfect fluid with a linear dispersion re-lation, it is scale independent. In particular this holds forK-essence scalar field models [25], when treated exactlyinstead of in a time-averaged approximation. We shallreturn to this point in § V.The equations of motion for ADE acoustic oscillationsdepend only on the value of c s , not its time derivative.Since the impact of ADE on cosmological observables isextremely localized in time due to the parametrizationof w ADE , we fix c s to be a constant, effectively its valueat a c . In § IV, we construct K-essence models where c s isstrictly constant as a proof of principle, but our analysisholds more generally if we interpret the constant c s asmatching a suitably averaged evolving one.In our most general case, the ADE model is thereforecharacterized by four parameters { w f , a c , f c , c s } once p is fixed. When varying these parameters we impose flat,range bound priors: − . ≤ log a c ≤ − .
0, 0 ≤ f c ≤ .
2, 0 ≤ w f ≤ . ≤ c s ≤ .
5. We shall latersee that the Hubble tension can be relieved by varyingjust two of these four parameters, fixing c s = w f = 1,corresponding to models where the ADE is a canonicalscalar that converts its energy density from potential tokinetic around matter radiation equality (see § IV B). Werefer to this particular ADE model as cADE.The full cosmological model also includes the sixΛCDM parameters: the cold dark matter density is char-acterized by Ω c h , baryon density by Ω b h , the angu-lar size of the sound horizon by θ s , the optical depth toreionization by τ , and the initial curvature spectrum byits normalization at k = 0 .
05 (wavenumbers throughoutare in units of Mpc − , which we drop when no confusionshould arise), A s and tilt n s . These have the usual non-informative priors. We fix the sum of neutrino massesto the minimal value (e.g., Ref. [27]). We modify theCAMB [28] and CosmoMC [29] codes to include all themodels that we discuss, following Ref. [26]. We samplethe posterior parameter distribution until the Gelman-Rubin convergence statistic [30] satisfies R − < .
02 orbetter unless otherwise stated.For the principal cosmological data sets, we use thepublicly available Planck 2015 measurements of the CMBtemperature and polarization power spectra at largeand small angular scales and the CMB lensing potentialpower spectrum in the multipole range 40 ≤ (cid:96) ≤
400 [31–33]. To expose the Hubble tension, we combine thiswith the latest measurement of the Hubble constant, H = 74 . ± .
42 (in units of km s − Mpc − here andthroughout) [1]. To these data sets we add the PantheonSupernovae sample [34] and BAO measurements fromBOSS DR12 [35], SDSS Main Galaxy Sample [36], and6dFGS [37]. These datasets prevent resolving the Hub-ble tension by modifying the dark sector only betweenrecombination and the very low redshift universe [4].Our baseline configuration thus contains: CMB tem-perature, polarization and lensing, BAO, SN and H measurements. Unless otherwise specified all of our re-sults will be for this combined data set. We include allthe recommended parameters and priors describing sys-tematic effects for these data sets.As we shall see, the CMB polarization data providean important limitation on the ability to raise H andfuture polarization data can provide a definitive test ofthe ADE models that alleviate the Hubble tension. Wetherefore also consider the joint data set without CMBpolarization data. We refer to this data set as -POL. III. ADE PHENOMENOLOGY
In this section we discuss the phenomenology and ob-servational implications of ADE and their dependence onits parameters.At the background level, the addition of ADE in-creases the total energy density before recombinationthat changes the expansion history lowering the soundhorizon r s . This changes the calibration of distance mea-sures not only for the CMB but also the whole inversedistance ladder through BAO to SN. Given the preciseangular measurements of the sound horizon θ s , the in-verse distance ladder scale is reduced and hence the in-ferred H rises.The prototypical example of this method for reliev-ing the Hubble tension is an extra sterile neutrino thatis at least mildly relativistic at recombination. Neutri-nos, however, do not provide a good global solution (e.g.,Ref. [22]) since they behave as free-streaming radiationbefore recombination and therefore change the phase ofthe CMB acoustic oscillations as well as the CMB damp-ing scale, the distance a photon random walks throughthe ionized plasma before recombination, approximatelyas λ D ∝ r / s [21]. A more general dark fluid, on theother hand, can reduce the fraction of the dark compo-nent vs. normal radiation before matter radiation equal-ity, allowing the two scales to change in a proportionalway [23].Beyond these background effects, ADE and otherdark sector candidates for relieving the Hubble tension,gravitationally drive photon-baryon acoustic oscillationschanging the amplitudes and phases of the CMB peaks(e.g., Ref. [6, 22]). ADE perturbations undergo their ownacoustic oscillations under its sound horizon, leading tonovel CMB driving phenomenology. As detailed in Ta-ble I, this modified phenomenology leads to a maximumlikelihood (ML) solution with w f ≈ c s and H = 70 . χ = − . f c is detected at 2 . σ . In the next section, we focus on the details of thesephysical effects. We then address the impact of Planckpolarization data and the ability of future polarizationdata to test the ADE solutions to the Hubble tension.Finally, note that although we do not consider mea-surements of the amplitude of local structure here,the ML and constraints for ADE are σ Ω / m =0.4623(0.4573 ± ± ± H in ΛCDM lowers σ Ω / m unlike in ADE.If the tension with weak lensing measurements of the am-plitude increases in ΛCDM in the future, it will disfavornot only ΛCDM but these ADE models as well. A. Acoustic Driving
Under the sound horizon or Jeans scale of the ADE, itsdensity perturbations acoustically oscillate rather thangrow, leading to changes in the decay of the Weyl po-tential (Ψ + Φ) /
2. This decay drives CMB acoustic os-cillations, and the ADE impact is especially importantfor modes that enter the CMB sound horizon near a c ,roughly k = 0 .
04 in the ML ADE model from Table I.The excess decay is countered by raising the cold darkmatter through Ω c h since it remains gravitationally un-stable on the relevant scales.For ADE, at the parameter level, this effect is con-trolled by the sound speed c s in conjunction with theequation of state w ADE ( a ) through w f . These two param-eters are hence strongly correlated, as shown in Fig. 1,reflecting degenerate effects on the CMB when they areraised or lowered together. Near the ML solution, thisrequires w f ≈ c s .We explore this degeneracy in Fig. 2 by showing theevolution of the Weyl potential for this mode in the MLmodel (red) from Table I relative to the same model withno ADE (ML, f c = 0) as a baseline (black). The Weylpotential is relatively suppressed at a < a c , enhanced at a ∼ a c and suppressed again at a (cid:29) a c due to ADE. Theenhancement and subsequent suppression correspond tothe first acoustic compression extremum in the ADE den-sity perturbation and the subsequent Jeans stabilizationof the perturbations. The net impact is a reduction inthe Weyl potential. This reduction is compensated byraising Ω c h . For comparison we also show the differencein reverting the value of Ω c h in the baseline model to theML ΛCDM value (cyan dashed). Since a c ∼ a eq , ADEbecomes important around the same epoch when radi-ation driving has the maximal impact on the shape ofthe CMB acoustic peaks. Along with other adjustmentsin ΛCDM parameters, in particular n s and Ω b h , theseeffects compensate for each other.This compensation leaves the CMB acoustic peaksnearly unchanged despite raising H from 68.58 to 70.81.In Fig. 3, we show the data and model residuals relative Model ΛCDM cADE ADE100 θ MC ± ± ± b h ± ± ± c h ± ± ± τ ± ± ± A s ) 3.092 (3.079 ± ± ± n s ± ± ± f c - 0.082 (0.082 ± ± a c - -3.45 (-3.46 ± ± w f - 1 (fixed) 0.87 (1.89 ± c s - 1 (fixed) 0.86 (1.07 ± H ± ± ± χ χ = −
2∆ ln L reflects the ratio between the maximum likelihood value andthat of ΛCDM for the joint data. w f c s ML ADEML cADE c s = 1 c s = w f FIG. 1. The joint marginalized distribution of the ADE pa-rameters c s and w f , obtained using our combined datasets.The darker and lighter shades correspond respectively to the68% C.L. and the 95% C.L. The markers indicate the maxi-mum likelihood values for ADE (solid circle) from Tab. I andthe intersection between canonical models c s = 1 (solid line)and models which convert potential to kinetic energy at thetransition c s = w f (dashed line), i.e. c s = w f = 1 (open circle)as in cADE. to the ΛCDM ML in Table I. As we can see, this effectdoes not exacerbate the oscillatory residuals in the data,where the acoustic peaks (vertical lines) are suppressedrelative to troughs, which would occur if H were raisedin ΛCDM. Note that the residuals are scaled to the cos-mic variance per (cid:96) -mode for the ML ΛCDM model as: σ CV = (cid:113) (cid:96) +1 C T T(cid:96) , T T ; (cid:113) (cid:96) +1 (cid:113) C T T(cid:96) C EE(cid:96) + ( C T E(cid:96) ) , T E ; (cid:113) (cid:96) +1 C EE(cid:96) , EE . (3)We can better understand the origin of the ADE ef-fects and their impact on the CMB by varying w f and c s independently. Fig 2 (upper) also shows a +0 . c s makes the ADE acoustic oscillations and Jeans stabilityoccur earlier and as a consequence also cuts into the en-hancement. Raising w f has two effects. Before the firstcompression peak and above the CMB sound horizon, thecomoving ADE density perturbation grows adiabaticallyso that its amplitude grows relative to the photons ap-proximately as δ ADE ∝ (1 + w ADE ) δ γ . For w f > / a (cid:46) a c and then an enhancementin the Weyl potential for a (cid:38) a c , especially approach-ing the first compression. A larger w f then suppresses f ( a > a c ) which also causes a relative enhancement at a > a c . Combined, these effects imply that for a fixedamount of driving of acoustic oscillations through the de-cay of the Weyl potential, raising c s should be compen-sated by raising w f . This is the leading order degeneracythat we see in Fig. 1.In terms of the residuals, shown in Fig. 3, a positivevariation in c s , when not compensated by w f , leads toa sharp TT feature around (cid:96) ∼
500 near the second TTpeak whereas along the degeneracy line the ML modelTT residuals remain small. The modes responsible forhigher multipoles are not sensitive to ADE perturbationparameters since the Weyl decay that drives them oc-curred before the ADE became important a (cid:28) a c .The degeneracy is truncated at low w f in Fig. 1. If w f < /
3, the ADE redshifts slower than the radiation andthus has a large impact on the driving of CMB acoustic . . - ( Ψ+ Φ ) a eq a c a ∗ -0.03-0.02-0.010.000.01 ∆ ( Ψ+ Φ ) k = 0 . / Mpc . . - ( Ψ+ Φ ) a eq a c a ∗ − − − − a -0.03-0.02-0.010.000.01 ∆ ( Ψ+ Φ ) k = 0 . / MpcML ADEML ADE , c s +ML ADE , f c = 0 ML ADE , w f +ML ADE , f c = 0 , Ω c h − FIG. 2. The Weyl potential evolution of the ML ADE modelfrom Table I for two modes: k = 0 .
01 and 0 .
04 Mpc − . Lowersubpanels show differences with respect to the baseline valueof Weyl potential for the ΛCDM parameters of ML ADE butwith no ADE ( f c = 0), as displayed in the upper subpan-els. Shown are ML ADE (red solid) and parameter variationsaround it: c s + (orange dashed) and w f + (dark blue dashed)mean +0 . c h − (cyan dashed) means low-ering it to the ML ΛCDM value in Table I. Relevant temporalscales (matter-radiation equality a eq , ADE transition a c andrecombination a ∗ ) are shown with vertical lines. oscillations between a c and recombination which cannotbe balanced by the same variations in c s . B. CMB Polarization
Even for the ML ADE model, the compensation be-tween Ω c h , c s , and w f is imperfect for modes that enterthe CMB sound horizon between a c and recombination.These modifications leave distinct imprints on the po-larization spectra that already limit the ability of ADEto raise H using the Planck data and, in the future,can definitively test this scenario. Polarization providesthe cleanest signatures of driving on these scales given -1-0.500.51 ∆ C TT (cid:30) / σ C V -1-0.500.51 ∆ C EE (cid:30) / σ C V
30 500 1000 1500 2000 (cid:28) -1-0.500.51 ∆ C T E (cid:30) / σ C V ML ADEML ADE , c s + ML ΛCDM FIG. 3. The CMB model and Planck data residuals withrespect to the ML ΛCDM model. Shown are the ML ADEmodel (red solid) and a ∆ c s = +0 . that it isolates the acoustic oscillations at recombinationfrom the early integrated Sachs-Wolfe that smooths outits signatures in the TT spectrum.In Fig. 2 (lower), we show the Weyl potential evolu-tion for such a mode, k = 0 .
01 in the ML model. Whilethe qualitative behavior is similar to the higher k mode,the balance between w f , c s and Ω c h changes. First, theimpact of the change in Ω c h is relatively higher sincethe ADE redshifts faster than radiation. Second, the im-pact of w f is also somewhat higher relative to c s . Thesechanges lead to uncompensated sharp features in the po-larization residuals.For the Planck data, where measurements of the EEspectrum are still noisy, this makes the TE spectrum themost informative for these features (see Fig. 3). In theΛCDM model, this sensitivity provides an important con-straint on Ω c h and hence supporting evidence for a low H from multipoles (cid:96) < (cid:96) ∼ σ low point compared to the best fitΛCDM model shown in Fig. 3 [38]. Note that in ΛCDMraising H requires lowering Ω c h which raises TE there,making the fit even worse.In the ADE ML model, the impact of raising Ω c h lowers TE in this region providing a better fit to the data.Even without direct H data, the CMB data favor theADE model (see Table II). However raising H furtherthan the ADE ML would make TE too low at (cid:96) (cid:46) H = 72 .
27 comparedwith 70 .
81 for our joint dataset.Raising c s has the impact of making the ADE moreimportant for Weyl decay and counters the effect of rais-ing Ω c h . As we can see from Fig. 3, this has the effect ofraising TE in this region and degrading the fit. Thus theTE data are also important for disfavoring a canonicalscalar field with c s = 1 if w f is too low. We shall see in § V that this explains why previously considered modelswhere a canonical scalar field oscillates in its potentialmust have its initial conditions set to avoid this region.Finally, given the sharp features in the EE model resid-uals with up to ∼ . − . per multipole , all of these cases where H is raised by dark components that also change the driv-ing of the acoustic peaks can be tested to high signifi-cance once EE measurements approach the precision ofTT measurements today. IV. POTENTIAL-KINETIC CONVERSION
The ADE phenomenology favored by the Hubble ten-sion can be concretely and exactly realized in the K-essence class of dark energy models, where the dark com-ponent is a perfect fluid represented by a minimally cou-pled scalar field φ with a general kinetic term [25]. Morespecifically, the class of constant sound speed c s modelsintroduced in § II is given by the Lagrangian density [39] P ( X, φ ) = (cid:18) XA (cid:19) − c s c s X − V ( φ ) , (4)where the kinetic term involves X = −∇ µ φ ∇ µ φ/ A is a constant density scale. For a scalar with a canonicalkinetic term c s = 1, and more generally w ADE → c s if thekinetic term dominates, whereas w ADE → − V ( φ ) dominates. The fluid correspondence holdswhen ∇ µ φ remains always timelike; then c s = δp/δρ inconstant field gauge or rest frame, where the momentumdensity of the field vanishes and the potential energy isspatially constant.The correlation shown in Fig. 1 implies that around theML ADE model from Table I, setting w f = c s providesa good fit to the combined data. Since w ADE → − a (cid:28) a c and w ADE → w f for a (cid:29) a c , this suggests that the best fitting P ( X, φ ) models are those that suddenlyconvert nearly all of their energy density from potentialto kinetic at a c . If we focus on a model that has sucha potential to kinetic energy conversion feature, we have c s = w f and the number of parameters of ADE reduces tothree. The ML of this model gives c s = 0 .
84 , H = 70 . χ = − . w f = c s = 1 whichcorresponds to the case of a canonical field, cADE. FromTable I, we see that the ML cADE model has only amarginally smaller H = 70 .
57 for a ∆ χ = − . f c of 3 . σ given the smaller set ofparameters. We shall now consider how to construct acorresponding potential V ( φ ). A. Canonical Conditions
A canonical scalar which converts its potential to ki-netic energy around a c provides a simple, concrete ex-ample of ADE that alleviates the Hubble tension. Toexplicitly construct such a model that matches require-ments on the two remaining quantities a c and f c we candetermine the equivalent requirements for the potential V ( φ ).At a (cid:28) a c , the φ field is stuck on its potential due toHubble friction and rolls according to dφdN ∼ − V (cid:48) H , (5)where N = ln a denotes e-folds. For the purposes ofthis qualitative discussion we drop factors of order unity.After a c , we want the field to be released from Hubbledrag and convert its potential energy to kinetic energyon the e-fold timescale ∆ N ∼
1. Defining φ c = φ ( a c )and linearizing the change V ( φ ) ≈ V ( φ c ) + V (cid:48) ( φ c )∆ φ. (6)Therefore, around φ c , we want (cid:18) V (cid:48) H (cid:19) (cid:38) V, (7)or in terms of f c ∼ V /ρ tot , (cid:15) V f c (cid:38) , (cid:15) V ≡ M (cid:18) V (cid:48) V (cid:19) . (8)This is the main condition for the potential to kineticconversion.For the linearization in Eq. (6) to be valid in the senseof the second order term (∆ φ ) V (cid:48)(cid:48) not preventing theconversion, we also want V (cid:48)(cid:48) ( φ c ) < − V (cid:48) ( φ c )∆ φ (9)so V (cid:48)(cid:48) (cid:46) H . (10)Putting these two criteria together, η V (cid:46) (cid:15) V , η V ≡ M V (cid:48)(cid:48) V , (11)where we have restored a factor of 2 so as to matchthe well-known condition for no tracking solution to ex-ist [40]. Tracking potentials do not work since the scalarfield follows an equation of state that is determined bythe dominant component of the total energy densityrather than the kinetic energy dominated limit. A similarderivation applies to the c s (cid:54) = 1 case with a modificationto the Hubble drag evolution (5) [39].Finally we want the field to maintain kinetic energydomination until its energy density has largely redshiftedaway. This excludes models where the field oscillatesaround a minimum and so is different from those in Refs.[23, 24] as we shall discuss in the next section. Fur-thermore, the fluid description is exact for our modelswhereas it is only approximate for oscillatory models.Thus our requirements on the potential are fairlygeneric and correspond to setting the amplitude andslope of the potential at the desired point of Hubble dragrelease, along with the condition that the field remainskinetic energy dominated until most of the energy den-sity has redshifted away. A wide class of potentials cansatisfy these requirements and we shall give concrete ex-amples next. B. Canonical Solution
To make these considerations concrete, consider theclass of potentials: V ( φ ) = (cid:40) A φ m , φ > , , φ ≤ . (12)Then for φ > (cid:15) V = (cid:18) mφ (cid:19) , η V = m ( m − φ , (13)and any m > η V < (cid:15) V . The flat potential at φ ≤ A and φ initial to give thedesired f c and a c .In Fig. 4 (upper), we show a worked example of thismatching. We fix cosmological parameters to the MLADE model in Table I and take a quadratic potentialwith m = 2. We find a good match to the form of Eq. (1)with p = 1 /
2. This motivates our fixed fiducial choice in § II.To showcase the robustness of the potential to kineticconversion mechanism for relieving the Hubble tension, f A D E cADE, p = 1 / V ( φ ), m = 2 − − − − a f A D E cADE, p = 1 V ( φ ), m = 4 FIG. 4. Scalar field potential V ( φ ) match to the fractionalADE energy density f ADE of the ML cADE parameters in Ta-ble I. Top: a locally quadratic potential with Eq. (12) com-pared with p = 1 / p = 1. we also consider a quartic potential m = 4 (Fig. 4, lower).The change in f ADE ( a ) is itself small and, once a shift in a c is absorbed, corresponds to a slight broadening of thetransition. Even this small change can be matched to thegeneral ADE form of Eq. (1) by adopting p = 1. For thelinear m = 1 case, p ≈ . p in this range provide comparable ML solutions to ourfiducial p = 1 / p and m holds for non-canonicalvalues of c s with the Lagrangian (4).These simple canonical or cADE models still providegood fits to the data as illustrated in Fig. 5 for the MLcADE model of Table I. The main difference comparedwith ML ADE is the slight lowering of H from 70.81 to70.57. The total improvement over ΛCDM for 2 extraparameters is ∆ χ = − . − . χ = +1 . χ + χ ) = − .
7. Ifcompared to ML ΛCDM fit to CMB only, the ML cADEfits CMB lensing as well and fits the TT and polarization -1-0.500.51 ∆ C TT (cid:30) / σ C V -1-0.500.51 ∆ C EE (cid:30) / σ C V
30 500 1000 1500 2000 (cid:28) -1-0.500.51 ∆ C T E (cid:30) / σ C V ML cADEML EDE ML ΛCDMML EDE , Θ i − FIG. 5. Canonical scalar field model and data residualsof ML cADE (orange solid) and ML EDE (dark blue solid)models with respect to the ML ΛCDM model as in Fig. 3.The model with ∆Θ i = − . spectrum better by ∆( χ + χ ) = − . Fig. 6 shows the parameter covariances and posteriorsin the cADE model. The centered values for the ML pa-rameters is indicative of the nearly Gaussian posteriorsand reflects the fact that the parameters are constrainedmainly by the data rather than the priors. The one excep-tion is a c since if a c → f c is equivalent to ΛCDM sothat the prior volume begins to matter. Even in this case, We have also explicitly checked that a direct solution forthe scalar field Klein Gordon equation is nearly indistinguish-able from a cADE model with the best matching parameters,e.g. ∆ χ = 1 . (cid:96) ≤ w ) ∝ a implied by Hubble frictionthrough Eq. (5) [39], which Eq. (1) does not do. ΛCDM is sufficiently disfavored so that constraints on a c are data not prior driven. Correspondingly the ADEfraction is significantly detected with f c = 0 . ± . f c and raising the CDMdensity Ω c h as well as adjusting Ω b h and n s slightlyhigher to minimize the data residuals (see Fig. 6). Thechange in θ ∗ to lower values is also notable. The modi-fications in driving make a small change in the phasingof the CMB acoustic peaks relative to its sound horizon.Note that in ΛCDM, θ ∗ drifts lower once the high multi-poles (cid:96) >
800 are included [41].Finally under the -POL data set, the ADE canonicalmodel allows a higher H = 71 . ± .
05 and ML value of71.93. This is because of the limitations the TE spectrumaround (cid:96) (cid:46)
500 places on these solutions as discussed in § III B.
V. RELATION TO PRIOR WORK
In Ref. [23], a canonical scalar field component, re-ferred to as early dark energy, with a potential V ( φ ) ∝ [1 − cos( φ/f )] n , (14)plays a similar role as our ADE. Unlike ADE, EDE oscil-lates after being released from Hubble drag and Ref. [42]finds that the time averaged background equation of statecan be modeled by Eq. (1) with p = 1 and w f = n − n + 1 . (15)As we have discussed in the previous sections, w f is a rel-evant parameter for the resolution of Hubble tension, soits adjustment should be considered a parameter varia-tion in the EDE model in spite of n taking discrete integervalues.The time-averaged behavior of perturbations is de-scribed by a fluid approximation with a rest frame soundspeed [42]: c s ( a, k ) = , a ≤ a c , a ( n − (cid:36) + k a ( n + 1) (cid:36) + k , a > a c . (16)Unlike in our case, this fluid description is approximate,especially at a c . The time dependence of (cid:36) is fixed bythe parameters a c , w f and the initial field position Θ i = φ i /f [42]: (cid:36) ( a ) = GP (cid:114) P + 28 n Θ i sin Θ i H ( a c ) a − w f . (17)Here P = H ( a c ) t and we approximate it as: P ( x = a c /a eq ) = 23 x − x + 2 √ x − x , (18) log l og log Λ CDMcADE
FIG. 6. The marginalized joint posterior of the parameters of the cADE model, obtained using our combined datasets. ΛCDMresults are also added for comparison. The 8 fundamental parameters are shown in the lower triangle whereas the implicationsfor H are shown in the upper triangle. The darker and lighter shades correspond respectively to the 68% C.L. and the 95%C.L. while G ( a c , n ) = √ π Γ( n +12 n )Γ(1 + n ) 2 − n n ( n − a − n +1 c × (cid:20) a nn +1 c + 1 (cid:21) ( n − . (19) Note that for w f > (cid:36) decreases with a , and so thesound speed evolves from 1 at a ≤ a c back to 1 at latetimes, with higher k exhibiting smaller amplitude devia-tions, and a k -dependent minimum c s ( a c , k ).The EDE model therefore has four parameters a c , f c , Θ i and n . Following Ref. [42], we choose the bestvalue n = 3, which corresponds to w f = 1 /
2, and conduct0 Θ i /π . . . . c s ( a c , k = . ) Θ i /π . . . . . c s ( a c , k = . ) . . . . . . Θ i /π H EDE ML EDE
FIG. 7. The marginalized distribution of the EDE initialphase Θ i and some other parameters, obtained using our com-bined data set. The darker and lighter shades correspondrespectively to the 68% C.L. and the 95% C.L. The orangecircle indicates the maximum likelihood values for EDE. an MCMC likelihood analysis on the remaining parame-ters. We treat a c and f c as in the ADE model and imposea flat prior on 0 ≤ Θ i /π ≤
1. Because of the large pa-rameter volume of degenerate models around ΛCDM, weonly sample the posterior until R − < .
05 which shouldgive an adequate, but not perfect, estimate of parameterconstraints out to 95% C.L. The results are comparedwith our ADE model in Table II. The EDE ML modelallows a slightly higher H = 71 .
92 and hence a betterfit to the data ∆ χ = − . c s = w f or ∆ χ = − . w f generally requires a low c s . In the EDE model, this translates into specific re-quirements for the initial phase Θ i . In Fig. 7, we showthe relationship between constraints on Θ i , H and theminimum sound speed at k = 0 . , .
04. As we can see,achieving a higher value of H requires a large initialphase and its ML value is Θ i /π = 0 .
90. The 68% con-fidence region is 0 . ≤ Θ i /π ≤ .
94 and Θ i /π < / /π ∼ .
96 [42].The reason for this preference is that Θ i controls theminimum sound speed. Following the degeneracy line inFig. 1 to w n = 1 /
2, we would expect that for a constantsound speed c s ≈ .
77. Given the effective EDE soundspeed of Eq. (16), this represents an average over therelevant timescales and wavemodes. For the ML EDEmodel c s ( a c , .
01) = 0 .
51 and c s ( a c , .
04) = 0 .
63 as theminimum value for each k -mode.The scale dependence of the sound speed also explainsthe slightly better fit to CMB data, specifically the TEdata. In Fig. 5, we compare the EDE and cADE residualsfor their respective ML models. Notice that the TT resid-uals are very similar. However, in TE, by allowing thesound speed to decrease in the k range associated with (cid:96) < (cid:96) ∼
200 from raising Ω c h as discussed abovebut now without adverse consequences elsewhere. Thisin turn better fits the low TE residuals and allows H to increase further relative to the ADE model. We alsoshow the impact of reducing Θ i in the ML EDE model,making the sound speed closer to one at all times. Themost significant effect is localized to (cid:96) <
500 and in par-ticular destroys the pattern of lower TE at (cid:96) ∼
200 vs (cid:96) ∼
400 compared with cADE.We conclude that the small improvement of the EDEover ADE fit requires a specific range in the initialphase that lowers the sound speed in a scale-dependentway. Comparing to the canonical ADE mode, this im-provement gives ∆ χ = − . w n , Θ i and is therefore marginal. In the future, po-larization measurements that approach the cosmic vari-ance limit can distinguish between the EDE and ADEclasses. For example, we forecast that with cosmic vari-ance TT,TE,EE measurements to (cid:96) ≤ χ = 22 .
4. Furthermore the ADE model provides ageneral class of exact solutions where the potential energyis converted quickly to kinetic, whereas the EDE modelrequires a specific set of initial conditions to achieve asimilar phenomenology with an approximation to an os-cillating field.Relatedly, Ref. [24] considers a model where the scalarfield oscillates in a monomial potential V ( φ ) ∝ φ n , (20)1with parameters adjusted to reproduce the EDE phe-nomenology. This coincides with Eq. (14) only near thebottom of the potential, Θ i (cid:28) π/ Model (Data) ∆
N H ∆ χ ∆ χ ∆ χ cADE 2 70.57(70.60 ± ∗ ,4 70.81(70.20 ± ± ± ∗ ,4 72.27(71.30 ± ± H results for the ML cADE, ADE and EDE mod-els and posterior constraints with the joint data set and withCMB polarization data removed (-POL). ∆ N is the number ofadditional parameters in addition to the ΛCDM ones. ∗ Notethat ML ADE in the potential conversion case where c s = w f ,is essentially the same as the general case but with ∆ N = 3 . The total ∆ χ relative to the ΛCDM model is broken downinto contributions from the Planck CMB data sets and thelocal H measurement. VI. DISCUSSION
Acoustic dark energy, appearing around the epoch ofmatter radiation equality, can substantially relieve thetension between CMB inference of H and local mea-surements, exhibited in the ΛCDM model. The presenceof extra energy density lowers the CMB sound horizonthat anchors the inverse distance ladder for BAO andSN, while its disappearance before and after equality al-lows for a good fit to CMB data in the damping tail.Furthermore by introducing ADE at equality, the grav-itational effects of raising the cold dark matter densitycan be balanced by the acoustic oscillations in the ADEitself.Our main findings regarding the Hubble tension aresummarized in Table II and Fig. 8. In all cases relievingthe Hubble tension requires ADE to be a ∼
8% contribu-tion to the total energy density around matter radiationequality, leading to at least a two parameter extension toΛCDM.In the general ADE class of models, the acoustic phe-nomenology is controlled by two additional parameters,the asymptotic equation of state w f at late times, andthe sound speed c s . The sound speed plays a crucial P / P m a x . . . . H P / P m a x -POL ADE cADE EDE ΛCDM
FIG. 8. The marginalized posterior distribution of the H parameter in the four models considered in Table II for twodifferent datasets: our combined data set (upper panel) andthe same with CMB polarization data removed, -POL (lowerpanel). The dashed vertical lines indicate the ML values fordifferent models. role in the gravitational driving of CMB acoustic oscilla-tions through its impact on the Weyl potential, leadingto a strong correlation between the two, consistent with w f = c s ≈ H = 70 .
57 vs. H = 70 .
81. In fact, the two parametermodel has the advantage of providing a more significantdetection of the ADE fraction f c = 0 . ± .
025 and thusallows less parameter volume around the ΛCDM limit,producing a posterior centered around the ML: H =70 . ± .
85 in cADE vs. the lower H = 70 . ± . H provided for an im-provement of ∆ χ = − . w f = c s )and ∆ χ = − . w f = c s = 1) re-spectively.This class of w f = c s ∼ ∼
8% fraction required by the data. Anypotential that obeys this property and efficiently convertspotential to kinetic energy until the latter redshifts awaywill satisfy these requirements. As a proof of principle,we explicitly construct an example where the potentialis locally quadratic around its release. In this model,the timing of the release to equality is not explained butthe identification of this coincidence may lead to moresophisticated models where it is.The robustness and generality of this potential-kineticconversion mechanism for relieving the Hubble tensionseparates it from similar models in the literature. InRefs. [23, 24], the EDE scalar field oscillates after Hub-ble drag release leading to an effective fluid described bytime-averaged values of w f and c s . Converting our re-quirements on the relationship between the two, we findthat in Ref. [23], where the potential is periodic, the ini-tial field must be on the concave part of the potential,and near the maximum to best relieve the Hubble ten-sion. This also explains the poorer fits in Ref. [24], wherethe potential is convex and matches the periodic poten-tial only near the minimum.The periodic EDE model [23, 24] allows for a slightlyhigher H = 71 . ± .
09 and better ∆ χ = − . H = 72 .
27 vs72 .
40 respectively. The reason is that between equalityand recombination, changes in acoustic driving between raising the CDM density and adding the dark compo-nent no longer cancel. This is in fact a beneficial fea-ture of both models since the Planck TE data show lowresiduals with respect to the ML ΛCDM model around (cid:96) ∼ χ ≈ . ACKNOWLEDGMENTS
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