Chain Early Dark Energy: Solving the Hubble Tension and Explaining Today's Dark Energy
UUTTG-02-2021, NORDITA-2021-020
Chain Early Dark Energy:Solving the Hubble Tension and Explaining Today’s Dark Energy
Katherine Freese ∗1,2,3 and Martin Wolfgang Winkler †1,21
Department of Physics, The University of Texas at Austin, Austin, 78712 TX, USA Oskar Klein Center for Cosmoparticle Physics, University of Stockholm, 10691Stockholm, Sweden Nordita, KTH Royal Institute of Technology and Stockholm University,Roslagstullsbacken 23, 10691 Stockholm, Sweden
Abstract
We propose a new model of Early Dark Energy (EDE) as a solution to the Hubble tension incosmology, the apparent discrepancy between local measurements of the Hubble constant H (cid:39) km s − Mpc − and H (cid:39) km s − Mpc − inferred from the Cosmic Microwave Background(CMB). In Chain EDE, the Universe undergoes a series of first order phase transitions, startingat a high energy vacuum in a potential, and tunneling down through a chain of every lowerenergy metastable minima. As in all EDE models, the contribution of the vacuum energy to thetotal energy density of the universe is initially negligible, but reaches ∼ around matter-radiation equality, before cosmological data require it to redshift away quickly – at least asfast as radiation. We indeed obtain this required behavior with a series of N tunneling events,and show that for N > , the phase transitions are rapid enough to allow fast percolationand thereby avoid large scale anisotropies in the CMB. We construct a specific example ofChain EDE featuring a scalar field in a quasiperiodic potential (a tilted cosine), which isubiquitous in axion physics and, therefore, carries strong theoretical motivation. Interestingly,the energy difference between vacua can be roughly the size of today’s Dark Energy (meVscale). Therefore, the end result of Chain EDE could provide a natural explanation of DarkEnergy, if the tunneling becomes extremely slow in the final step before the field reaches zero(or negative) energy. We discuss a simple mechanism which can stop the scalar field in thedesired minimum. Thus Chain EDE offers the exciting prospect to explain EDE and DarkEnergy by the same scalar field. ∗ [email protected] † [email protected] a r X i v : . [ a s t r o - ph . C O ] F e b Introduction
The cosmological standard model Λ CDM has provided dramatic insights into the evolution of theuniverse, the formation of Large Scale Structure (LSS) and the physics of the Cosmic MicrowaveBackground (CMB). However, one striking anomaly referred to as the ‘Hubble tension’ has persistedover the last years and could indicate that Λ CDM is incomplete. Supernova redshift observationsconsistently prefer a value of H (cid:39) km s − Mpc − [1, 2] significantly different from H (cid:39) km s − Mpc − as inferred from the CMB by Planck [3]. A possible resolution of this discrepancyrequires an alteration of the expansion history compared to Λ CDM. In particular, if the soundhorizon at matter-radiation decoupling is reduced, a larger value of H would be derived from theCMB [4].The challenge is that any type of physics affecting the sound horizon induces further changesin the pattern of CMB fluctuations. Simple scenarios like an extra dark radiation increase theCMB-inferred Hubble parameter at the price of degrading the cosmological fit to the higher CMBpeaks and baryon acoustic oscillation data [3, 5]. However, one particular form of energy dubbed‘Early Dark Energy’ (EDE) has been shown to resolve the Hubble tension while (so far) passing allcosmological tests [6–14]. Initially, EDE behaves as a cosmological constant and its contribution ρ EDE /ρ tot to the total energy density of the universe is negligible. But as the universe cools downthe ratio ρ EDE /ρ tot grows and reaches ∼ around matter-radiation equality, before cosmologicaldata require it to redshift away quickly – at least as fast as radiation [8].The original model of EDE involved a scalar field φ which is initially displaced from its minimum(see e.g. [8–10]). As long as the Hubble friction dominates its evolution, the field value remains(nearly) constant and the corresponding potential energy behaves as a cosmological constant. Butonce the Hubble scale approaches the effective mass of the scalar field, φ becomes dynamical. Whileit performs coherent oscillations around its minimum, the energy density ρ φ stored in φ decreases.A major drawback of the scalar field models is that ρ φ typically redshifts away more slowly thanradiation – in contrast to what is required in the EDE scenario. Only for highly non-generic choicesof the scalar field potential like V ∝ (1 − cos( φ/f )) n with n ≥ , a sufficiently fast decrease of theenergy density can be realized [8].An alternative model of EDE invokes a first order phase transition. The universe is trapped in afalse minimum, but tunnels into the true minimum around matter radiation equality. Unfortunately,this scenario suffers from a shortcoming which bears resemblance to the ‘empty universe problem’[15] in old inflation [16]: in order for EDE to stay around long enough the tunneling rate mustbe suppressed. This implies that during the phase transition bubbles of true vacuum are formedfar away from each other. They grow to macroscopic sizes before eventually colliding with aneighboring bubble and releasing the energy contained in the bubble walls. As a consequence largescale anisotropies would arise and leave undesirable footprints in the CMB.In order to avoid these anisotropy problems in EDE, one can mimic solutions to the empty uni-verse problem in Guth’s original model of ‘old’ inflation: (i) Double Field inflation [17,18]) employedan additional trigger field to make the tunneling rate time-dependent; at first the tunneling rate isslow to allow for a long enough period of inflation but then suddenly switches to rapid tunnelingso that bubbles nucleate simultaneously throughout the Universe, collide, and reheat. Ref. [11, 13]employed the same approach, using a trigger field to obtain a time-dependent tunneling rate, toconstruct a viable model of EDE. (ii) In Chain Inflation [19, 20], the Universe tunnels from a largevacuum energy through a series of minima of ever lower vacuum energy. Each tunneling event israpid enough for successful percolation and reheating, with hundreds of tunneling events requiredto add up to enough inflation.In this work we propose a new model of EDE in which the universe undergoes a series of phasetransitions instead of just one. Drawing on its similarity with Chain Inflation [19, 20], we will2ub our proposal ‘Chain Early Dark Energy’ (Chain EDE). In the simplest case, Chain EDE isrealized through a scalar field which tunnels along a series of metastable minima. The requiredquasiperiodic potential (e.g. a tilted cosine) is ubiquitous in axion physics and, therefore, carriesstrong theoretical motivation.Furthermore, in Section 2, we will argue that Chain EDE does not suffer from the anisotropyproblem which plagues the single phase transition case. Given a series of metastable vacua, therequirement of EDE to survive until matter-radiation equality can be fulfilled even if each individualvacuum is short-lived. Hence, the bubbles of new vacuum in Chain EDE are created in closeproximity and only grow for a short time before they percolate. We will show that the resultinganisotropies occur at small distance scales and do not spoil cosmological observables.In Section 3, we will trace the evolution of the Chain EDE component over the history of theuniverse. In particular, we will show that the evolution is very similar as in the best fit EDEmodels [10] invoking oscillating scalar fields. Since the latter have been proven to resolve theHubble tension via dedicated cosmological fits, the same can be claimed for Chain EDE.Section 4 presents a specific model realization of Chain EDE in form of a tilted cosine potential.We also introduce a mechanism for stopping the axion once the EDE is dissipated at z (cid:39) ,i.e. which prevents φ from further tunneling down the chain into the regime of negative vacuumenergy.Finally, in Section 5 we will show that the energy difference between vacua in Chain EDE cannaturally be of O ( meV ) . We will argue that if the tunneling field is still trapped in the lowestminimum with positive energy, it can account for the Dark Energy which dominates our universetoday. In the first step we want derive constraints on one or several first order phase transition(s) thattake place shortly before matter-radiation equality. A phase transition induces scale-dependentanisotropies which can spoil CMB and LSS observations.
We consider a scalar field φ which is initially trapped in a false vacuum. The energy density of thescalar is assumed to be subdominant at all times such that the expansion of the universe is mainlycontrolled by the radiation component. Once the age of the universe approaches the life-time ofthe false vacuum, bubbles of true vacuum are formed with the energy of the false vacuum storedin the bubble walls. The bubble walls expand approximately at the speed of light and release theirenergy upon collision (e.g. into radiation). In order to avoid dangerous energy injection into thevisible sector φ is assumed to be a dark sector field that does not couple directly to the visiblesector.If we denote the tunneling rate per volume as Γ , we can determine the mean lifetime τ of theuniverse in the false vacuum by requiring Γ V ( τ ) = 1 . Here V ( τ ) is the spacetime volume of thepast lightcone at time t = τ (at an arbitrary position) . For a radiation-dominated universe weobtain τ = (cid:18) π Γ (cid:19) / (cid:39) . × Γ − / . (1) V ( τ ) = (cid:82) τ π r ( t ) dt with r ( t ) = a ( t ) (cid:82) τt dt (cid:48) a ( t (cid:48) ) , where a denotes the scale factor of the universe.
3e can invert this equation and express Γ in terms of the redshift of the phase transition z b ≡ z ( τ ) , Γ − / (cid:39) . × yr × (cid:18) z b (cid:19) (cid:39) . kpc × (cid:18) z b (cid:19) . (2)One can easily verify that the typical distance between the bubble nucleation centers is also givenby ∼ Γ − / . It is convenient to consider the comoving bubble separation d b (cid:39) Γ − / z b (cid:39) Mpc × (cid:18) z b (cid:19) . (3)This is because d b will also correspond to the comoving size of the bubbles upon percolation(assuming that bubble walls expand at the speed of light until collision). After the bubble wallstransferred their energy, anisotropies of comoving size d b are present in the dark sector which aregravitationally transferred to the visible sector. These anisotropies can leave imprints in the CMBwhich are not observed in the data [11, 13]. Indeed if the energy density in the vacuum is 10% ofthe radiation, one might expect δρ/ρ ∼ at the time of the phase transition, growing up to O (1) at the time of the CMB epoch. In the following we will require the anisotropies to occur atsmall enough scales to be unobservable in CMB and LSS data.The anisotropies occur at an angle in the sky which can be estimated as follows [13], θ (cid:39) d b D CMB , (4)where D CMB denotes the comoving angular diameter distance of the CMB D CMB = z CMB (cid:90) dz H − ( z ) (cid:39) Gpc . (5)Current observations of the CMB temperature power spectrum by Planck and earth-bound detec-tors reach up to (cid:96) (cid:39) [3]. Even smaller scales are accessible through the Lyman- α forest whichcovers the dynamical range k (cid:39) (0 . − h Mpc − [21] corresponding to (cid:96) (cid:39) − [22]. If weuse the stronger Lyman- α constraint we need to require θ < π(cid:96) max = 0 . ◦ (6)on the angular scale of anisotropies. We use Eq. (5), (6) in Eq. (4) to find a bound on d b , whichusing Eq. (3) translates to Γ − / < . kpc × z b . (7)Combining Eq. (2) and (7) we obtain a constraint on the redshift of the phase transition z b > . × . (8)However, in order to resolve the Hubble tension EDE has to stay around until redshift z ∼ [8,10]. Therefore, EDE models with a single phase transition suffer from unacceptable anisotropies inthe CMB.As mentioned in the introduction, a caveat to this argument which employs a time-dependenttunneling rate has been pointed out in [11, 13]: if Γ is initially small, but increases around matter-radiation equality the lifetime of the universe in the false vacuum and the duration of the phasetransition (which controls the size of anisotropies) can effectively be decoupled. In particular, theanisotropies can be made compatible with CMB constraints if the increase in Γ occurs rapidly. Therequired mechanism, which includes the EDE field and an additional trigger field, has first beenintroduced in the context of double field inflation [17, 18].4 V ( ϕ ) single phase transition ϕ V ( ϕ ) series of phase transitions Figure 1:
Models of EDE in which the energy is initially stored in a scalar field. In the left panel the EDEis dissipated through a single phase transition. In the right panel a series of phase transitions occur. Wedenote the scenario in the right panel as Chain EDE.
We now turn to an alternative possibility to evade the CMB constraints. In Chain EDE the vacuumenergy is dissipated through a series of phase transitions instead of just one. As a simple realizationwe consider a scalar field which tunnels along a chain of false vacua with decreasing energy (seeFig. 1 for illustration). In order to keep the discussion simple, we assume that the tunneling rateper volume Γ remains constant for all vacuum transitions within the chain. Again, the energydensity of the scalar is assumed to be subdominant at all times such that the expansion of theuniverse is mainly controlled by the radiation component.Initially, the field sits in its highest energy minimum; in the meantime the (dominant) radiationdensity and the Hubble parameter decrease with the expansion of the Universe. Once the Hubbleparameter drops to the point where it becomes comparable to the tunneling rate, Γ /H ∼ ,the first phase transition in the chain takes place. The first transition can be treated completelyanalogously to the single phase transition case in the previous section. Thus as in the case of thesingle phase transition, it occurs when the lifetime of the universe reaches . × Γ − / (cf. (1)).Since bubbles are nucleated at every tunneling step, we need to apply the CMB constraintsto each of the vacuum transitions. Eq. (7) then implies that the strongest CMB constraints al-ways apply to the first phase transition. This can easily be understood since the correspondinganisotropies have the longest time available to grow by the expansion of the universe. The firstphase transition, again as found in the case of a single phase transition, needs to occur at redshift z > . × (see Eq. (8)), which according to Eq. (2) translates to a tunneling time Γ − / < yr (cid:39) . pc (9)(comparable to the age of the Universe at that redshift).In the case of many vacua each tunneling step reduces the EDE only by a small amount. If thereare sufficiently many transitions N along the chain the EDE can stay around until matter-radiationequality while still satisfying the constraint above. In order to find the minimal number N we needto determine the (mean) lifetime of the universe in each vacuum.We realize that for each subsequent transition the lifetime per vacuum decreases. Althoughwe have assumed constant Γ for all the transitions, the Hubble parameter decreases with redshiftduring radiation domination as H ∝ z . Thus the ratio Γ /H ∝ z − increases rapidly, so thatafter the first few phase transitions the field has reached the fast tunneling regime Γ /H (cid:29) .The number of vacuum transitions per time is obtained by counting the bubble walls hitting anobserver at a fixed point in space. The first transition occurs when the lifetime of the universe,5orresponding to / (2 H ) reaches . × Γ − / (cf. (1)). Therefore, the vacuum bubbles seeded bythe first transition are generated at a distance Γ − / = O (1 /H ) from each other and take aroundone Hubble time to collide. Their evolution is, hence, significantly affected by the expansion ofthe universe. For subsequent transitions with Γ /H (cid:29) , a large number of bubbles is created perHubble four-volume implying that the bubbles collide before they ‘realize’ the expansion.Simulations of colliding bubbles have recently been performed in [23]. Even though the simula-tions assumed an inflationary background the results in the limit Γ /H (cid:29) also apply to radiationdomination since the bubbles are not affected by the expansion as we argued. For Γ /H (cid:29) weextract the time of the universe in one vacuum i as τ i = ∆ φd (cid:104) φ (cid:105) /dt (cid:39) . − / , (10)where ∆ φ is the field-space distance between minima. Comparing (1) for the first phase transitionand (10) for later phase transitions in the series, we see that τ i decreases by an O (1) -factor alongthe chain.In practice we will apply (10) to all vacuum transitions. Since Γ /H (cid:29) is satisfied after thefirst few tunneling steps, the error we make by this assumption is negligible. The total time τ elapsed after N phase transitions can, hence, be estimated as τ = N τ i = 0 . N × Γ − / < N × . pc . (11)In the last step we applied the constraint derived from CMB anisotropies (9). Requiring that EDEremains present until redshift z ∼ leads to the following constraint on the number of vacuumtransitions N > × . (12)We emphasize that we derived this constraint by requiring that anisotropies occur on small scaleswhich are experimentally inaccessible. However, the energy dissipation through a series of phasetransitions (compared to just one) also suppresses the amplitude of EDE-induced CMB anisotropiesby a factor /N . Hence, they could be acceptable even if they fall into the observable range ofscales. While a more dedicated analysis goes beyond the scope of this work, one would expect thedensity perturbations sourced by EDE at redshift z to be bounded by ∆ ρ EDE ρ tot (cid:46) N ρ
EDE ( z ) ρ tot ( z ) , (13)where ρ EDE ( z ) /ρ tot ( z ) is the fraction of the total energy density of the universe contained in EDEat redshift z . Assuming ρ EDE /ρ tot ∼ . at z ∼ in order to resolve the Hubble tensionand logarithmic (linear) growth of perturbations during radiation (matter) domination, we candetermine the maximal amplitude of EDE-induced fluctuations as a function of scale and redshift.Requiring ∆ ρ EDE /ρ tot < − at last scattering for scales observable in the CMB and in LSS thenleads to the constraint N > × , (14)which is significantly weaker than (12).We conclude that if EDE disappears not a by single phase transition, but instead by a seriesof phase transitions, dangerous anisotropies can be avoided. The advantage of multiple phasetransitions is two-fold: (i) the amplitude of the anisotropies is reduced by the number of transitions N ; (ii) the scale of the anisotropies is smaller. For the case of one phase transition, requiring theEDE to stick around long enough leads to a large bubble size and large scale anisotropies; for the6ase of many transitions, each phase transition need only last /N of the total EDE epoch andthe faster tunneling rate produces smaller bubbles (at percolation) and smaller scale anisotropies.Requiring at least O (10 ) transitions one is definitely on the safe side since anisotropies only occurat scales which are experimentally inaccessible. Most likely, even a few hundred transitions aresufficient since the amplitude of EDE-induced fluctuations is strongly suppressed in this case. The Hubble tension consists in the apparent discrepancy between local measurements of the Hubbleconstant yielding H (cid:39) km s − Mpc − [1,2] and H (cid:39) km s − Mpc − inferred from the CMB [3].EDE resolves the discrepancy by adding an additional energy component which reduces the soundhorizon r s at recombination. In order to preserve the angular size of the first peak in the CMB,the decrease of r s needs to be compensated by a reduction of the angular diameter distance of theCMB D CMB . As apparent from (5) this in turn leads to an increase of H inferred from the CMBcompared to Λ CDM, i.e. to a resolution of the Hubble tension.Since a shorter sound horizon would also shift the position of the higher CMB peaks to larger (cid:96) and affect their amplitude, further cosmological parameters including the dark matter density,the baryon density and the scalar spectral index need to be modified in order to balance thiseffect [8–10]. Furthermore, to minimize the impact on other successful Λ CDM predictions, EDEshould redshift away at least as fast as radiation at z (cid:46) . Existing models of EDE invoke a scalar field φ in a dark sector which exhibits a potential of theform [8, 10] V = m f (cid:20) − cos (cid:18) φf (cid:19)(cid:21) n , (15)where m and f are parameters of mass dimension one and n is an integer number. Other choices,e.g. V ∝ φ n , have also been considered [9]. One can easily trace the time evolution of the EDEenergy density ρ EDE = ˙ φ / V by solving the homogeneous Klein-Gordon equation ¨ φ + 3 H ˙ φ + V (cid:48) ( φ ) = 0 . (16)The scalar field is initially frozen at the field value φ by the Hubble friction and ρ EDE remainsconstant. Once H falls below the the effective mass V (cid:48)(cid:48) ( φ ) the field starts to perform dampedoscillations around its minimum. During this period, the energy oscillates around the asymptoticsolution ρ EDE ∝ a − n/ ( n +1) . (17)where a denotes the scale factor of the universe. Since, during the oscillation period, ρ EDE mustredshift at least as fast as radiation n ≥ is required.In [10] the cosmological predictions of the EDE model were investigated in a combined fitto the Planck power spectra [24], BAO data [25–27] and supernova measurements of the Hubbleconstant [2, 28]. It was found that the cases n = 2 ( n = 3 ) reduce the total χ by 16 (20) comparedto Λ CDM – suggesting a clear preference for the EDE component. In Fig. 3 we depict the evolutionof ρ EDE for the best fit point with n = 2 . As can be seen, the Early Dark Energy fraction reachesa maximum of ρ EDE /ρ tot = 0 . at z = 3111 . The best fit points features a Hubble constant H (cid:39) . km s − Mpc − close to the value preferred by local measurements.7 .2 Chain EDE In the following we want to argue that Chain EDE is capable of resolving the Hubble tension. Wehave seen that the EDE solution requires ρ EDE to contribute significantly ( ∼ ) around matter-radiation equality, but then to disappear quickly. However, the cosmological fit is expected to beinsensitive to the details of the underlying model. In fact, a simple modeling of the EDE componentin an effective fluid approach [8] yielded very similar results compared to the full implementation ofthe oscillating scalar field model [10]. In this light we can refrain from performing a full cosmologicalfit for Chain EDE. Instead, we will show that Chain EDE is able to closely reproduce the redshift-dependence of ρ EDE ( z ) in the oscillating scalar field model.In Chain EDE, the energy density of the EDE sector ρ EDE = ρ φ + ρ wall + ρ DR consists of threecomponents, namely• the vacuum energy stored in the scalar field ρ φ ,• the energy of bubble walls ρ wall ,• the energy density ρ DS created by the collision of bubble walls which may consist of darkradiation, gravity waves and anisotropic stress [11, 13]. We will refer to this component asthe energy density of the dark sector (DS) in the following.Let us first investigate how the potential energy ρ φ evolves. Initially, all energy of the EDE sectoris stored in φ and, hence, ρ EDE = ρ φ ≡ V . With each tunneling process the potential energy isreduced by the energy difference between vacua ∆ V = V /N , where N again denotes the numberof transitions required to dissipate the energy in φ . To keep the discussion simple, we take both ∆ V and the decay rate (per volume) Γ to be constant along the entire chain of vacua.If there are sufficiently many vacua in the chain, we can approximate ρ φ to continuously decreasewith ˙ ρ φ (cid:39) − ∆ Vτ i , (18)with τ i , denoting the time spent in one vacuum as given in (10). Furthermore, since (most)transitions in Chain EDE occur quickly (compared to the Hubble time), we will approximate theenergy transfer from ρ φ to ρ DS as instantaneous and neglect ρ wall in the following.The dark sector is permanently heated by vacuum transitions of φ . At the same time the energydensity of ρ DS redshifts with the scale factor as a − w ) , where w stands for the equation-of-stateparameter. The evolution of the energy densities in the EDE sector is, hence, governed by thefollowing set of differential equations (1 + z ) d ρ φ d z (cid:39) . V Γ / H ( z ) , (1 + z ) d ρ DS d z (cid:39) − . V Γ / H ( z ) + 3(1 + w ) ρ DS , (19)where we traded the time-dependence for a redshift dependence and used Eq.(10). Note that inthe 2nd of the above equations, the first term on the right hand side (RHS) is simply the negativeof the RHS of the 1st equation (as energy is transferred from the vacuum to the dark sector), andthe second term is the redshifting of the DS component.The equation-of-state parameter depends on the underlying dissipation mechanism for the vac-uum energy. It is expected that bubble walls release their energy into small scale anisotropic stress,dark radiation and, subdominantly, gravity waves. The distribution among these forms of energy8s, unfortunately, highly model-dependent. The generation of dark radiation e.g. requires the avail-ability of light final states in the dark sector. Furthermore, while w = 1 / for dark radiation, theequation-of-state parameter for anisotropic stress is not precisely known. A number of heuristicarguments suggest that it falls in the range / < w < [13]. Luckily, the EDE solution to theHubble tension only requires w ≥ / [8] which is satisfied by all forms of energy emerging fromthe bubble collisions. We can, therefore, refrain from a more detailed investigation of the equation-of-state parameter and simply consider the limiting cases w = 1 / and w = 1 in order to bracketthe uncertainties.The evolution of the EDE sector is coupled to the visible sector through the expansion rate.We will later perform a full numerical solution of (19) taking into account the impact of EDEon H . However, in order to roughly understand how ρ EDE evolves with redshift, it is instructiveto perform an analytic estimate which neglects the (subdominant) impact of EDE on the Hubbleparameter. Since the phase transitions occur during radiation domination we can then approximate H ( z ) (cid:39) . z × − kpc − . Solving the first equation in Eq. (19), we obtain ρ φ = V (cid:32) − . × kpc Γ / N z (cid:33) , (20)where we replaced V / ∆ V by the total number of transitions N . We note that the change in thepotential scales as /z i.e. drops off linearly with time, as expected. The expression above holdsas long as ρ φ ≥ . We assume that φ settles in a stable minimum at V ∼ once the entire potentialenergy has been dissipated . In order to resolve the Hubble tension this should occur at z ∗ (cid:39) which allows us to constrain Γ , Γ / N (cid:39) . kpc − (cid:16) z ∗ (cid:17) . (21)Notice that, in terms of the background evolution, models of Chain EDE are indistinguishableas long as they feature the same Γ / /N . The absence of dangerous CMB anisotropies requires N (cid:38) transitions (see Sec. 2.2), but one is otherwise free to choose N . Using (20) the differentialequation for the dark sector energy density can also be solved analytically such that we arrive at ρ φ = V (cid:40) − (cid:0) z ∗ z (cid:1) z > z ∗ z < z ∗ ,ρ DS = 2 V w (cid:0) z ∗ z (cid:1) z > z ∗ (cid:16) zz ∗ (cid:17) w z < z ∗ ,ρ EDE = ρ φ + ρ DS = V − w w (cid:0) z ∗ z (cid:1) z > z ∗ w (cid:16) zz ∗ (cid:17) w z < z ∗ . (22)Note that we have defined the EDE component of the Universe ρ EDE to include both the chainvacuum energy ρ φ plus the dark sector ρ DS that it decays into. We thus find that ρ EDE behavesapproximately as a cosmological constant until z ∗ and then redshifts away at least as fast asradiation (since w ≥ / ). Hence, Chain EDE meets the criteria for a successful solution to theHubble tension.We have then performed a numerical solution of (19) taking into account all subdominantcontributions to the Hubble parameter and the full dynamics of the visible sector. The initial We assume tunneling into anti-de Sitter does not happen. adiationMatterCosmological ConstantTotal DensityChain EDE ( w = / ) Chain EDE ( w = ) - - z ρ E D E / ρ t o t ( π G / ) ρ i [ M p c - ] Figure 2:
Evolution of the radiation, matter, Dark Energy and EDE densities in the Chain EDE scenario.The cases w = 1 and w = 1 / bracket the uncertainties in the equation-of-state parameter of the final stateeffective fluid generated by vacuum transitions in the EDE sector. vacuum energy and the decay rate were chosen as V = 0 . eV and Γ / /N = 0 . kpc − . InFig. 2 we depict the resulting evolution of the energy densities in radiation, matter, Dark Energy andEDE (the latter essentially follows our analytic estimate (22)). The cases w = 1 / and w = 1 aredepicted separately. As can be seen, the EDE component amounts to an energy injection stronglypeaked around matter-radiation equality, while it plays virtually no role outside this window.In Fig. 2, we have plotted ρ EDE = ρ φ + ρ DS ; i.e. the EDE curve has contributions both fromthe chain vacuum energy and the dark sector it decays into. If we were to plot only the vacuumcomponent ρ φ ( z ) , (the first equation in Eq.(22)), in this log-log plot it would look similar to astep function: essentially flat for all z > z ∗ and plummeting to ρ φ =0 at z = z ∗ . As a reminder,for sufficiently many vacua, we can treat ρ φ as continuously decreasing for the purposes of thesefigures. Below z ∗ the vacuum energy has converted to ρ DS which redshifts away.In Fig. 3 we compare the evolution of ρ EDE in the Chain EDE scenario and in the oscillatingscalar field model [10] described in the previous section (which we refer to as standard EDE in thefigure). For the latter we have chosen the best fit point with n = 2 .We observe that all three cases in Fig. 3 are virtually indistinguishable for z > z ∗ . For z < z ∗ the Early Dark Energy density in the standard EDE scenario oscillates between the Chain EDEsolutions with w = 1 / and w = 1 . Since the true equation-of-state parameter of Chain EDE isexpected to lie between the two extremes, a very similar scaling of ρ EDE in Chain EDE and in thestandard EDE scenario is expected.An explicit proof that Chain EDE resolves the H -problem of Λ CDM would require a dedicatedcosmological fit including the full modeling of the EDE component at the fluctuation level. However,we have shown that Chain EDE follows almost exactly the background evolution of the standard10 tandard EDE ( n = ) Chain EDE ( w = / ) Chain EDE ( w = )
100 1000 10 ρ E D E / ρ t o t Figure 3:
Energy density in the EDE component compared to the total energy density of the universe.The orange line shows the best fit EDE solution for an oscillating scalar field ( n = 2 model from [10]). Thetwo purple lines refer to Chain EDE for the parameters stated in the text. The cases w = 1 and w = 1 / bracket the uncertainties in the equation-of-state parameter of the final state effective fluid generated bythe bubble wall collisions. EDE scenario. Since the latter has been proven to resolve the Hubble tension, we consider it almostcertain that the same is true for Chain EDE.
As a simple realization of Chain EDE, we consider an axion field φ in a quasi-periodic potential V ( φ ) = − µ φ + Λ cos (cid:18) φf (cid:19) + V , (23)where f denotes the axion decay constant, while the parameters µ and Λ control the strength ofthe shift symmetry breaking and the barrier height of individual minima. Finally, V stands fora possible constant in the potential. Without loss of generality we can take φ (cid:39) as the initialfield value such that V corresponds to the initial EDE energy density (thus matching our previousdefinition of V ). For convenience, we define the parameter x = f µ / Λ . Given that x < , thepotential features an (infinite) series of minima with decreasing vacuum energy.The tunneling rate Γ between two minima is given by Γ =
A e − S E , (24)where S E stands for the Euclidean action of the bounce solution [29], while A denotes a prefactorwhich incorporates quantum fluctuations about the classical action [30]. In a recent paper [23], wederived the following analytic approximation of the tunneling rate for the potential in Eq. (23), Γ (cid:39) Λ f (1 − x ) S E π exp (cid:18) . − . x . (cid:19) × exp ( − S E ) (25)11ith S E (cid:39) f Λ (cid:112) (1 − x ) (1 − . x ) 4 π (cid:18) x (cid:19) , x = f µ Λ . (26)The above expressions allow us to directly determine parameter combinations µ , Λ , f which giverise to successful Chain EDE. As an example we choose µ = 29 . meV , f = 13 . meV , Λ = 26 . meV , V = (0 . eV ) , (27)yielding an axion mass m φ ∼ Λ /f = 50 meV. The initial field value is set to φ = 0 as mentionedpreviously. For the parameter choice above, the tunneling rate takes the value Γ / = 4 pc − corresponding to a lifetime τ i = 0 . yr per vacuum which remains constant along the chain. Theenergy density ρ EDE follows precisely the evolution depicted in Fig. 2 until the entire vacuum energy ρ φ has been dissipated at z (cid:39) . The number of phase transitions is N = 10 such that CMBand LSS constraints (see Sec. 2.2) are easily satisfied.The only problem of the tilted cosine model (23) is that it lacks a mechanism to stop theaxion once ρ φ = 0 , i.e. which prevents φ from further tunneling down the chain into the regimeof negative vacuum energy. However, we remind the reader that the tunneling rate between twoadjacent minima is exponentially sensitive to the parameters in the potential. Hence, small changesin the energy difference or barrier height between minima can quickly change the tunneling ratefrom fast to slow, i.e. prevent φ from further tunneling.We, therefore, now extend the tilted cosine model by a stopping mechanism for the axion. Forthis purpose we consider the potential V = ( M − g M φ ) χ − g M φ + (Λ + Λ χ ) cos φf + λχ + V , (28)which has originally been motivated in the context of the relaxion mechanism [31]. However, in contrast to the relaxion mechanism, we identify χ with a scalar field in the darksector (rather than with the Higgs boson). The above potential has been argued to be radiativelystable since the breaking of the axionic shift symmetry is controlled by the (small) couplings g ∼ g .We will, furthermore, assume that M is much larger than the axion mass m φ ∼ Λ /f .Let us now look at the evolution of the two field-system starting from φ = 0 . The field χ isinitially stabilized at χ = 0 by the large mass term M and can be integrated out. We thus obtain V = − g M φ + Λ cos φf + V , (29)in the axion direction which agrees with (23) if we identify µ ≡ g / M and Λ ≡ Λ . The axiontunnels down the potential with the time spent in each vacuum remaining constant. However, onceit reaches a field value φ c (cid:39) M/g , the squared mass of χ turns negative and χ gets displacedfrom the origin. The quartic term stabilizes χ at a finite field value. As soon as χ (cid:54) = 0 , the term Λ χ cos φ/f increases the barriers in the axion potential. Therefore, the tunneling time betweenvacua increases rapidly and becomes larger than the age of the universe shortly after the axion haspassed φ c . In Fig. 4 we (schematically) depict the potential in the axion direction with χ set to its φ -dependent minimum. The approximation (25) is valid as long as gravitational corrections to the tunneling rate are negligible (whichwe explicitly verified for the parameter combinations provided in this section). We assume that there is no backreaction of the dark sector energy density ρ DS on the tunneling rate. We consider the non-QCD version of the relaxion mechanism, see Sec. III in [31] and [32]. Notice that φ c is only approximately given by M/g . This is because the term Λ χ cos φ/f yields an additionalsubdominant mass term for χ which slightly shifts the transition. c V ( ϕ ) Figure 4:
Illustration of the potential (28) in the axion direction with χ set to its respective minimum.The barrier height between minima remains constant as long as φ < φ c , but increases quickly once the axionpasses the critical value φ c . The axion tunneling rate between minima almost immediately switches fromfast to slow at φ c . In order to realize a successful EDE scenario, we consider the following parameter example, M = 1 . eV , Λ = 26 . meV , Λ = 42 . meV , f = 13 . meV , C = (0 . eV ) ,λ = 0 . , g = 1 . × − , g = 0 . × − . (30)Starting from φ = 0 the axion undergoes ∼ tunnelings until it reaches the critical field value φ c . In the field range φ = [ φ , φ c ] , the axion follows exactly the dynamics of the tilted cosinemodel in Eq. (23) (the parameters (30) were chosen to reproduce (27)). The lifetime of eachvacuum remains constant at τ i = 0 . yr. But once the axion passes φ c , the tunneling rate betweenvacua starts decreasing dramatically. For the specific example (30), only three more tunnelingevents occur after passing φ c with corresponding lifetimes τ i (cid:39) yr, yr and . Myr. The nexttransition in the chain would already take ∼ Gyr, i.e. longer than the age of the universe. Onemight worry that the handful of late transitions (at φ > φ c ) could spoil the cosmological evolution.However, this is not the case as ρ EDE remains strongly subdominant in the late universe which weexplicitly verified for the example above. We can, hence, conclude that the relaxion mechanism (28)provides a successful exit from the EDE epoch.We want to emphasize, however, that Chain EDE does not necessarily require two scalar fields.Another model with only one field is suggested in the remainder of this paragraph. A modelbuilding challenge for single-field realizations of Chain EDE consists in the prompt transition fromrapid tunneling to a (meta)stable ground state. Most of the EDE must be dissipated aroundmatter-radiation equality in order not to affect the late-time evolution of the universe. This couldhappen via a trigger mechanism like in Eq.(28). However, another simple possibility is to consider apotential in which the barrier height between minima continuously decreases along the chain . TheEDE field would tunnel quicker and quicker between minima until most of the EDE has decayedaway. In the last stage the barriers in the potential become so shallow (or disappear entirely)that φ starts rolling. If the potential features a stable minimum at V = 0 , the EDE field wouldperform coherent oscillations around the minimum. Different from the oscillating scalar field EDEmodels discussed in Sec. 3.1 most of the EDE would, however, already be dissipated in the previoustunneling stage. The remaining subdominant EDE fraction would typically redshift at least as fastas matter during the oscillation stage. Given this fraction is sufficiently small compared to the Potentials of this type have e.g. been considered in the context of modulated natural inflation [33, 34].
In this section we suggest a new model for the Dark Energy (DE) that currently dominates theenergy density of the Universe. Again we imagine a chain of tunneling events. A scalar field startssomewhere up in the potential. This time, after the field successfully tunnels through a series ofhigher energy minima, it gets stuck in a low-energy false vacuum ρ DE (cid:39) (2 meV ) with a lifetimelonger than the current age of the Universe. The energy of this false vacuum could be responsiblefor the Dark Energy today.An enthralling possibility is that the same field is responsible for the EDE and the DE simul-taneously. Comparing the EDE and DE energy densities, we have ρ EDE ρ DE (cid:39) (cid:18) . eV meV (cid:19) (cid:39) . (31)This ratio can find a striking explanation within the EDE scenario and simply correspond to thenumber of phase transitions required to dissipate the EDE.For illustration let us consider the Chain EDE model described in the previous section (seeEq. (28)): the EDE field tunnels quickly through a large number of vacua until matter-radiationequality. But once most of the EDE has been dissipated, the lifetime of individual vacua blowsup and only a few more tunneling events occur. If we set the initial EDE density to (0 . eV ) andrequire N (cid:39) phase transitions, the energy difference between individual vacua comes out as ∆ V = (2 meV ) . (32)If the EDE field settles in the lowest de Sitter minimum, the corresponding energy density is of O (∆ V ) . Hence, it could naturally account for the DE which dominates our universe today. Onlyafter a time longer than the age of the universe, the EDE field would ultimately tunnel into the nextminimum along the chain with negative energy. Far in the future our observable universe wouldthen end in a big crunch. For illustration we depict the evolution of the energy density stored inthe EDE field as a function of redshift in Fig. 5.Our chain model relates the cosmological constant to the parameters in the axion potential.Furthermore, it successfully establishes a connection between the EDE and DE energy densitiesin terms of the number of phase transitions. Besides these desirable features, it is exciting tostudy, whether it can also provide a solution of the cosmological constant problem. Within themechanism (28) the fact that the axion stops tunneling at the right moment in time (i.e. whenthe vacuum energy is small) is a coincidence and relies on a parameter choice. A full solution tothe cosmological constant problem would require a dynamical reason for the axion to stop in thedesired minimum.An intriguing idea in this direction has been formulated by Abbott in 1984 [35] who suggesteda solution to the cosmological constant problem in terms of a tunneling field – similar to ourChain Dark Energy proposal. However, he relies on gravitational corrections to the tunneling inorder to dynamically stop the tunneling field in a vacuum with small energy density. Abbott’sproposal fails since it requires an extremely flat potential of the tunneling field (in order to makegravitational corrections important) which renders the phase transitions far too slow to relaxatethe vacuum energy within the age of the universe. Nevertheless, it would be very interesting toexplore, whether we can employ a dynamical stopping mechanism for the axion in our chain (E)DE14
10 100 1000 10 - - - - ρ ϕ [ e V ] Figure 5:
Illustration of the scenario, where the same scalar field φ accounts for the EDE and the DE. De-picted is the vacuum energy stored in φ as a function of redshift. At large redshift ρ φ remains approximatelyconstant. But around matter-radiation equality ρ φ decreases quickly by fast tunneling along the chain ofvacua. The energy is dumped into dark radiation or anisotropic stress (not shown in the picture). Oncemost of ρ φ has been dissipated, the tunneling rate becomes small and only a few more vacuum transitionsoccur within the lifetime of the universe (shown as the steps in the above figure). The energy density in thefinal vacuum in which φ settles until today corresponds to the DE of our present universe. scenario which relies on gravity. For example one could try to extend the Abbott mechanismby a non-minimal coupling of the the tunneling field to gravity in order to increase gravitationalcorrections. We leave further investigation of the cosmological constant problem within our chainmodels for future work. We have suggested Chain Early Dark Energy as a solution to two problems in cosmology, the Hubbletension and today’s small value of the Dark Energy. The original idea of EDE was proposed [6, 8]to resolve the apparent discrepancy between local measurements of the Hubble constant H (cid:39) km s − Mpc − [1, 2] and H (cid:39) km s − Mpc − inferred from the CMB [3] by altering theexpansion history of the Universe right around the epoch of matter-radiation equality. The originalEDE model employed a scalar field oscillating in a potential.In Chain EDE, the Universe instead undergoes a series of first order phase transitions, startingat a vacuum with energy density ρ EDE = O ( eV ) , and tunneling down through a chain of metastableminima with decreasing energy. As in all EDE models, the contribution of the vacuum energy tothe total energy density of the universe is initially negligible, but reaches ∼ around matter-radiation equality, before it redshifts away at least as fast as radiation – as required by cosmologicaldata (see Fig. 2). As a consequence of the additional contribution to the energy density of theUniverse prior to recombination, the sound horizon at matter-radiation decoupling is reduced anda larger value of H would be derived from the CMB.In principle the bubbles formed from first order phase transitions in the early Universe couldleave dangerous imprints in the CMB or affect LSS. However, we have shown that if the lifetime ofindividual vacua does not exceed a few years, the scale of the anisotropies is below the resolution15f current experiments, i.e. all observational constraints are satisfied. Since the solution to theHubble tension requires the EDE field to stay around for about years (until matter-radiationequality) the anisotropy constraints impose N (cid:38) phase transitions. Further, since Chain EDEhas essentially the same background evolution as previously studied EDE models which were shownto resolve the Hubble tension while satisfying all cosmological constraints [10], we expect the sameto be true for Chain EDE.We have provided a concrete model of chain EDE which easily achieves the required number oftransitions and which carries strong motivation from axion physics. The model employs a scalarfield in a quasi-periodic potential (a tilted cosine). The scalar field tunnels from vacuum to vacuumat (nearly) constant rate until it reaches a critical field value, where it is stopped almost instantly,e.g. due to the backreaction from a second scalar field (employing the relaxion mechanism). Furtherpathways to successful Chain EDE models with only a single scalar field are also discussed.Interestingly, the energy difference between vacua can be of the same size as the Dark Energydensity of the present universe (meV scale). This offers the exciting prospect to explain EDE andDE by the same scalar field. The tunneling field would be subdominant to ordinary matter andradiation throughout its evolution, until it gets trapped in the last minimum of the chain beforereaching zero/negative energy. If this minimum has a lifetime longer than the age of the universe,the remaining vacuum energy of the scalar field would produce today’s Dark Energy.We end with speculation about Recurrent Chain Dark Energy. Most exciting of all would bea Chain vacuum energy model that could explain all epochs in the history of the Universe, wherethe vacuum energy dominates or becomes significant: inflation, EDE, today’s Dark Energy (andperhaps others we do not yet even know about). We imagine a Chain potential in which a fieldtunnels through a series of ever lower minima. In this Recurrent Model, the vacuum energy isinitially the dominant energy density in a Chain Inflationary epoch; after that the vacuum wouldbe mostly subdominant to radiation and matter, but occasionally raises its head to a large enoughvalue to affect the Universe evolution. It becomes important at z ∼ at the level of 10% of thetotal energy density to provide the EDE that can resolve the Hubble tension, and it is dominantagain today as the origin of the Dark Energy. Recurrent Chain Dark Energy would most likelyrequire multiple scales in the potential (rather than e.g. a single tilted cosine). The difficulty ofthis idea is that inflation must reheat to the Standard Model (SM), but EDE must reheat to a darksector (suggesting no direct coupling to visible matter) in order to avoid unacceptable modificationsto the CMB. It would be interesting to look for a successful model of Recurrent Dark Energy thatavoids this problem, e.g. inflation producing very massive DS particles that later decay to the SMparticles, but which are too massive to be created in the EDE epoch. Acknowledgments
We thank Martina Gerbino, Jon Gudmundsson, Dragan Huterer, Massimiliano Lattanzi, and SunnyVagnozzi for helping us understand bounds on anisotropies. KF also thanks Matt Johnson and JimLiu for extremely helpful discussions about the cosmological constant problem. K.F. is Jeff & GailKodosky Endowed Chair in Physics at the University of Texas at Austin, and K.F. and M.W. aregrateful for support via this Chair. K.F. and M.W. acknowledge support by the Swedish ResearchCouncil (Contract No. 638-2013-8993). 16 eferences [1] A. G. Riess et al., Astrophys. J. , 126 (2018), 1804.10655.[2] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, Astrophys. J. , 85(2019), 1903.07603.[3] Planck, N. Aghanim et al., Astron. Astrophys. , A6 (2020), 1807.06209.[4] A. G. Riess et al., Astrophys. J. , 56 (2016), 1604.01424.[5] J. L. Bernal, L. Verde, and A. G. Riess, JCAP , 019 (2016), 1607.05617.[6] T. Karwal and M. Kamionkowski, Phys. Rev. D , 103523 (2016), 1608.01309.[7] V. Poulin, T. L. Smith, D. Grin, T. Karwal, and M. Kamionkowski, Phys. Rev. D , 083525(2018), 1806.10608.[8] V. Poulin, T. L. Smith, T. Karwal, and M. Kamionkowski, Phys. Rev. Lett. , 221301(2019), 1811.04083.[9] P. Agrawal, F.-Y. Cyr-Racine, D. Pinner, and L. Randall, (2019), 1904.01016.[10] T. L. Smith, V. Poulin, and M. A. Amin, Phys. Rev. D , 063523 (2020), 1908.06995.[11] F. Niedermann and M. S. Sloth, (2019), 1910.10739.[12] J. C. Hill, E. McDonough, M. W. Toomey, and S. Alexander, Phys. Rev. D , 043507(2020), 2003.07355.[13] F. Niedermann and M. S. Sloth, Phys. Rev. D , 063527 (2020), 2006.06686.[14] T. L. Smith et al., (2020), 2009.10740.[15] A. H. Guth and E. J. Weinberg, Nucl. Phys. B , 321 (1983).[16] A. H. Guth, Phys. Rev. D , 347 (1981).[17] F. C. Adams and K. Freese, Phys. Rev. D , 353 (1991), hep-ph/0504135.[18] A. D. Linde, Phys. Lett. B , 18 (1990).[19] K. Freese and D. Spolyar, JCAP , 007 (2005), hep-ph/0412145.[20] K. Freese, J. T. Liu, and D. Spolyar, Phys. Rev. D , 123521 (2005), hep-ph/0502177.[21] S. Zaroubi, M. Viel, A. Nusser, M. Haehnelt, and T. S. Kim, Mon. Not. Roy. Astron. Soc. , 734 (2006), astro-ph/0509563.[22] Planck, Y. Akrami et al., Astron. Astrophys. , A10 (2020), 1807.06211.[23] M. W. Winkler and K. Freese, Phys. Rev. D , 043511 (2021), 2011.12980.[24] Planck, N. Aghanim et al., Astron. Astrophys. , A11 (2016), 1507.02704.[25] F. Beutler et al., Mon. Not. Roy. Astron. Soc. , 3017 (2011), 1106.3366.[26] A. J. Ross et al., Mon. Not. Roy. Astron. Soc. , 835 (2015), 1409.3242.1727] BOSS, S. Alam et al., Mon. Not. Roy. Astron. Soc. , 2617 (2017), 1607.03155.[28] D. M. Scolnic et al., Astrophys. J. , 101 (2018), 1710.00845.[29] S. R. Coleman, Phys. Rev. D , 2929 (1977), [Erratum: Phys.Rev.D 16, 1248 (1977)].[30] J. Callan, Curtis G. and S. R. Coleman, Phys. Rev. D , 1762 (1977).[31] P. W. Graham, D. E. Kaplan, and S. Rajendran, Phys. Rev. Lett. , 221801 (2015),1504.07551.[32] T. Flacke, C. Frugiuele, E. Fuchs, R. S. Gupta, and G. Perez, JHEP , 050 (2017), 1610.02025.[33] R. Kappl, H. P. Nilles, and M. W. Winkler, Phys. Lett. B , 653 (2016), 1511.05560.[34] M. W. Winkler, M. Gerbino, and M. Benetti, Phys. Rev. D , 083525 (2020), 1911.11148.[35] L. F. Abbott, Phys. Lett. B150