Reducing the $H_0$ Tension with Exponential Acoustic Dark Energy

The Hubble tension arises from different observations between the late-time and early Universe. We explore a new model with dark fluid, called the exponential Acoustic Dark Energy (eADE) model, to relieve the Hubble tension. The eADE model gives an exponential form of the equation of state (EoS) in the acoustic dark energy, which is the first time to explore an exponential form for the EoS. In this model, the gravitational effects from the acoustic oscillations of the model can impact the CMB phenomena at the matter radiation equally epoch. We give the constraints of the eADE model by the current cosmological dataset. The comparison of the phenomena with the standard model can be shown through CMB and matter power spectra. The fitting results of our model have $H_0 = 70.06^{+1.13}_{-1.09}$ in 95$\%$ C.L. and a smaller best-fit value than $\Lambda$CDM.


I. INTRODUCTION
After the accelerated expanding Universe discovered in 1998 [1,2], the ΛCDM has been one of the most important models to explain modern cosmology, and that can fit very well with many observations [3]. When considering the question "how fast is this expansion nowadays?" we need to focus on the Hubble parameter. Unfortunately, the debate about the value of the present Hubble constant (H 0 ) attracted much attention with the different precise observations in recent years. The standard candle measurements in low redshifts, such as type Ia Supernovae, get an H 0 equal to 74.03±1.42 [km/s/Mpc] [4], while the result from high redshift give an H 0 of 67.4 ± 0.5 [km/s/Mpc] in Cosmic Microwave Background's paper [5].
This phenomenon is called the "Hubble tension", which has beyond 4.4σ. The precision of these cosmological measurements is about 1%, which indicates the discrepancy can not be systematic errors. Moreover, a model-independent technique with strong gravitational lensing experiment -H0LiCOW -gives H 0 = 73.3 +1.7 −1.8 [km/s/Mpc] [6], that confirms the measurements from Supernovae, but the Baryon Acoustic Oscillations (BAO) result agrees with the CMB from Planck [7]. From these results, the tension even increased to 5.3σ by combining all of the H 0 values with different origins.
One popular approach to release the H 0 tension is to add extra dark radiation or another sterile neutrino in the epoch before recombination. These additional components can make the cosmological expanding faster with a lower sound horizon r s , which lead to a shorter history of the Universe before recombination and a larger H 0 value . However, this approach will change the oscillation of acoustic and damping scale [32], which has been constrained by the precise observation of CMB. So this approach is unable to raise H 0 drastically [5,33].
In another approach, the model named the "early dark energy (EDE)" provided a dark component only becomes effective around the recombination time. The EDE model can avoid the previous problem and release the Hubble tension manifest [34][35][36][37]. However, the EDE model will also change the CMB acoustic peaks and amplitudes by its perturbation. In the ADE model, the sound speed varies with the EoS in the background. This energy density becomes important in the matter radiation equally epoch and impacts the CMB through the gravitational effects of its acoustic oscillations [40].
We explore a new exponential form of EoS for the ADE model in this paper. The new model has the same advantages as ADE but has fewer free parameters called the Exponential Acoustic Dark Energy (eADE) model. In this study, we will introduce this eADE model and its fitting results from observational data. In particular, we use the CAMB [41] and CosmoMC [42] packages with the Markov chain Monte Carlo (MCMC) method to give the constraints of the model. This paper is organized as follows. In Sec. II, we introduce our eADE model, and derive the evolution equations for the dark fluid part in the linear perturbation theory. In Sec. III, we present our numerical calculations. In particular, we show the CMB power spectra, matter power spectra and constrain the model parameters from several cosmological observation datasets. Finally, we will present our conclusion in Sec. IV.

II. EXPONENTIAL ACOUSTIC DARK ENERGY MODEL
The eADE is defined to be a perfect dark fluid. Its EoS w eADE and the sound speed c 2 s give the main mark of this model. We can relieve the Hubble tension by acoustic phenomenology of the linear sound waves from the background and the perturbation in the eADE model.
We define the EoS w eADE to be where a c is the moment for critical redshift z c (z c = 1/a c − 1), which epoach make eADE becomes dominant. For the value of w eADE equal to −1 when a ≪ a c , w eADE equal to √ 2−1 when a = a c , and w eADE equal to 1 when a = 1. Since the value of a c is around O(10 −4 ), we regard a c as a c ≪ 1 for the present time. The development of this EoS is the same as that in ADE model [38], which can be seen in the Table I. Moreover, this EoS in our eADE model has one less free parameter than that in the ADE model (parameter p at Eq.1 of ref [38]).
And our model also has a transiently important contribution to the energy density around a c then decays quickly. So comparing with the ADE model [38], the eADE model not only has a simpler form but also takes a new exponential term in EoS.
In Tab.I, we also compared the evolution of the EoS in the EDE model [34] with the parameter n = 2 and n = 3, which shows the different evolutions of EoS in ADE and EDE. Value of EoS a ≪ a c a = a c a = 1 The eADE density evolves as [43], where f c = ρ eADE (ac) ρtot(ac) means the contribution of eADE at a c . The pressure of the eADE can be given as P eADE = w eADE ρ eADE . From Eq. 1 and Eq. 2, the P eADE can be written naturally as where the c 2 s is the sound speed which is defined as The parametrization of w eADE can get from the cosmological observation. The value of c 2 s close to 1, and the parameter was fixed by 1 in many previous work [38,39]. So we will also consider c 2 s to be 1 in the following calculation. In this way, the Ω eADE and P eADE can be rewritten as, Since the eADE model has a main effect in the early Universe around a c , we can consider the Ω eADE back to zero when a equals 1 in the present time.
The pressure term can be shown as, where ξ(a) = 2 − ac 2a . It can give a polynomial of the ξ, which order in pressure can be shown as a group with 3, 2, 1, and 0 naturally. And when a is 1, the pressure value of the eADE is equal to zero. It shows the effect of the P eADE is the same as the matter in the late time Universe. Now we discuss the perturbation part of this model. Theρ eADE can be calculated as δρ eADE = δ eADE ρ eADE , where δ eADE is the density contrast. Thus, the δP eADE can be given by We consider The quantity δw eADE is the spatial fluctuations in the EoS, so the pressure fluctuation in Eq.7 can be rewritten as The EoS fluctuations can arise from temporal variations in the background EoS through a general coordinate.
We are working on the synchronous gauge and consider the corresponding velocities of the dark fluid as θ eADE . The conservation equations then becomė where k is the k-space unit vector, h is the metric perturbation in the Fourier space [44] and θ eADE = 0 in the rest frame of this dark fluid.

III. FITTING RESULT OF THE EXPONENTIAL ACOUSTIC DARK ENERGY
In this section, we present our Markov Chain Monte Carlo (MCMC) fitting results by the public code of CAMB [41] and CosmoMC program [42]. The prior of the free parameters were set in Table II. Note we limit f c in the range from 0 to 1. And for the parameter a c , we used the form of log(a c ) in the global fitting.  In addition, the type Ia Supernovae data in the fitting comes from the latest Pantheon result with 1048 data points [52]. Without proper interdependence consideration, the covariance and light curve parameters in SH0ES and Pantheon observation can not valid the constraints simultaneously and independently. That leads to a sudden or rapid change in H(z) at z < 0.1, which would impact both constraints for the Hubble flow and absolute peak magnitude M B . One method is to use the SNIa distance ladder to calibrate Hubble flow, that constrains and covariance will be contained in the SNIa sample. This approach will use in SH0ES + Pantheon's future data release and cosmological models' constrain [53].
First of all, we set the sound speed c 2 s as a free parameter and fit the eADE model with Planck 2018 data. The result of Ω m , log(a c ), f c , H 0 , and other parameters are shown in Table III. Especially, we give the contour result of c 2 s and H 0 in Fig.1, which shows the Hubble parameter equals to 69.62 +1.06 −2.09 in 95% C.L. and can release the H 0 tension effectly. The sound speed has the best-fit value of c 2 s = 0.907 +0.027 −0.024 and close to 1. So in the next simulation, we set c 2 s is 1 and focus on the value of log(a c ) and f c .
To test the effect of the eADE model in different observations, we consider the combina-   [54,55].
Tab. III gives the fitting result of the matter density parameter Ω m and the χ 2 in the ΛCDM and eADE model. In the epoch of a c ≈ a eq , the acoustic peak of CMB is impacted by the eADE. At the background level, because the additional dark energy can enlarge the value of the total energy density before recombination, it will lead to a smaller sound horizon r s and a shorter expanding history at this time. In general, the sound horizon can be written as CMB last-scattering as where the t ⋆ and z ⋆ correspond to the end of baryon drag epoch, so we has r drag ≈ r s . In this paper, we can see the difference of sound horizon at the two model in Fig.2 and Tab.III, that shows the r drag in eADE is smaller than the result in ΛCDM. In this way, our model has possible to release the H 0 tension. The angular size on the last-scattering surface θ M C will also have a smaller value in the eADE model. It can give a reduced inverse distance ladder scale, which will lead to the increase of H 0 . This distance measurement calibration modification is not only for CMB observation but also for the BAO and SN through the inverse distance ladder.
Since there are two additional free parameters a c and f c in the eADE model, we consider the Akaike Information Criterion (AIC) [56] to statistical analysis and comparison of the two models. The AIC estimator can be expressed by where L max ≡ p(d|θ max , M) is the maximum likelihood value, and k is the number of free parameters. Since χ 2 equal to −2 ln L max , the χ 2 value of the eADE model in CMB+BAO+SN In Fig.3, we can see the difference of CMB power spectra between the eADE model and ΛCDM in TT mode and EE mode. The data points of CMB in blue color come from Planck 2018 [45], the theoretical value of ΛCDM (orange dash line) and eADE (grey blue line) comes from fitting results in Planck data. In the TT mode, the eADE result is closer to the Planck data points in the first peak, which shows the advantages of the eADE model.
Meanwhile, the EE mode resulting from Planck still has too much noise in the high l, so the eADE model has a smaller error with some data points than the standard model but is hard to distinguish with the ΛCDM.
In Fig. 4, we can compare the eADE and ΛCDM in matter power spectra with different observations. The Luminous Red Galaxy (LRG) data come from SDSS DR7 [58], which is in the redshift between 0.6 and 1.0. The data points are closer to the eADE model prediction than the ΛCDM especially at high wavenumber k [h/Mpc]. On the small scale, we compared the two models with the eBOSS DR14 Ly-α forest data [59]. The center value of the Ly-α data points also prefers eADE in high k [h/Mpc]. Since the error bar of the Ly − α forest dataset are still too large, we can not distinguish the eADE model and the ΛCDM in matter power spectra.

IV. CONCLUSIONS
In this work, we have introduced a new additional perfect dark fluid, named the eADE model, as a candidate to release the H 0 tension. The eADE model was defined with a special EoS as an exponential function of the sound speed. This model is different from ΛCDM at the matter radiation equally time.
We The fitting result of eADE shows a smaller χ 2 than that in ΛCDM. The AIC analysis with different datasets also confirms this result. Finally, we compared the eADE model with ΛCDM in the CMB and the matter power spectra. The observational data points prefer our eADE model to ΛCDM at a small scale structure but are hard to distinguish in present precision.