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Reducing the H0 Tension with Exponential Acoustic Dark Energy

Lu Yin 2, ∗ Center for Quantum Spacetime, Sogang University, Seoul 04107, Korea Department of Physics, Sogang University, Seoul 04107, Korea Abstract We explored a new dark fluid model in the early universe to relieve the Hubble tension significantly, that tension is the different result of the Hubble constant derived from CMB inference and local measurements within the ΛCDM. We discussed an exponential form of the equation of state in the acoustic dark energy for the first time, in which gravitational effects from its acoustic oscillations can impact the CMB phenomena at the epoch of matter radiation equivalent, called the exponential acoustic dark energy (eADE). And the constraints were given by the current cosmological data and the comparison of the phenomena with the stander model was shown through CMB and matter power spectra. The result shows our model has the H0 = 71.65 +1.62 −4.4 in 68% C.L. with a smaller best fitted χ2 value than that in ΛCDM.


I. INTRODUCTION
After the accelerated expanding Universe was discovered in 1998 [1,2], the ΛCDM has been one of the most important models to explain the modern Universe and can be fitted with many observations very well [3]. When consider "how fast is this expanding nowadays?", we need to use the Hubble parameter. But in recent years, the debate about the value of present Hubble parameter H 0 was much attention with the more precise measurements of different techniques. The standard candle measurements from type Ia Supernovae at low redshift get an H 0 value of 74.03 ± 1.42 [km/s/Mpc] [43], while the standard ruler measurements from the CMB at high redshift give an H 0 of 67.4 ± 0.5 [km/s/Mpc] [35]. This implacable result was called the "Hubble tension" and the tension is beyond 4.4σ. The precision of these cosmological measurements is about 1%, which indicates that such a large discrepancy cannot be explained by systematic errors. Moreover, an independent strong gravitational lensing experiment H0LiCOW gives H 0 = 73.3 +1.7 −1.8 [km/s/Mpc] [6], so that it confirms the measurements from supernovae, whereas the BAO result from BOSS agrees with that of Planck [7]. As a result, the tension is even increased to 5.3σ by combining all of the H 0 values of different origins.
One popular approach to release this tension is to add some extra dark radiation or another sterile neutrino in the epoch before the recombination, so that the cosmological expansion becomes faster with a lower sound horizon r s , resulting in a shorter expanding history of the Universe and a larger H 0 value [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. However, this approach would change for the acoustic oscillations and damping scale [22], which has been constrained by the observation of CMB precisely. So this approach is scant to raise H 0 drastically [23,35].
In another approach, a dark component only becomes effective around the epoch of recombination, which named "the early dark energy (EDE)", can avoid the previous problem and can release the Hubble tension manifest [30]. But the EDE model will also change the CMB acoustic peaks at phases and amplitudes by its perturbation. A species Acoustic Dark Energy (ADE) model can avert this trouble of EDE and increase the value of H 0 [25,26].
The ADE model can be considered as a dark fluid component and the equation of state w ADE is much different from that in EDE.
In the ADE model, the sound speed varies with the equation of state in the background that becomes important in the time of matter-radiation equivalent and it impacts the CMB through the gravitational effects of its own acoustic oscillations at this epoch. The ADE also can be considered as a scalar field that conversion of potential to kinetic energy [27].
We explored a new equation of state of the ADE model, which is expanded into an exponential form. It has the same advantages as the previous model but has fewer free parameters. This model can increase the value of H 0 . In this study, we plan to not only explain our Exponential Acoustic Dark Energy (eADE) model but also fit this model by using the most recent observational data. In particular, we use the CAMB [33] and CosmoMC [34] packages with the Markov chain Monte Carlo (MCMC) method to give then constrains of the model. This paper is organized as follows. In Sec. II, we introduce our eADE model. We also derive the evolution equations for the dark fluid part of eADE in the linear perturbation theory. In Sec. III, we present our numerical calculations. In particular, we show the CMB and matter power spectra and the constraints on the model-parameters from several cosmological observation datasets. Finally, we will present our conclusion in Sec. IV.

II. EXPONENTIAL ACOUSTIC DARK ENERGY MODEL
Exponential acoustic dark energy (eADE) is defined to be a perfect dark fluid and its equation of state (EoS) w eADE and the sound speed c 2 s give the main remark of this model. In the eADE, we can relieve the Hubble tension by acoustic phenomenology of the linear sound waves from the background and the perturbation.
We defined the equation of state w eADE to be where a c is the moment for critical redshift z c (z c = 1/a c − 1) and in this epach eADE  Table I). Moreover, this EoS in our eADE model has one less free parameter than the ADE model (parameter p at Eq.1 in [25]). And the eADE model also has a transiently important contribution around a c then decays quickly. So comparing with the equation of state in ADE model [25], our eADE model is not only considered a more simple form of the equation of state but also take a new form which is given by exponential.
We also take the evolution of this EoS in the Early Dark Energy (EDE) model [30] into comparison, where the EDE model with the condition of parameter n = 2 and n = 3 are shown in Tab.I. Value of EoS a a c a = a c a = 1 The exponential acoustic dark energy density evolves as [28], where f c = ρ eADE (ac) ρtot(ac) means the contribution of eADE at a c . The pressure of the eADE can be given from P eADE = w eADE ρ eADE . From Eq. 1 and Eq. 2, the P eADE can be written naturaly as The c 2 s is the sound speed , definited as Since the only parametrization of w eADE can be given by the cosmological observation and the value of c 2 s ≈ 1, we fix c 2 s to be 1 in the next. So the Ω eADE and P eADE can be rewritten as, We will use global fitting to give the numerical value of this parameter later. Since the eADE model has a main effect in the early Universe around a c , we can see the Ω eADE is equal to zero when a = 1 in nowadays.
The pressure term can be shown as, where ξ(a) = 2 − ac 2a . It can give a polynomial of the ξ, and the order of ξ in pressure can be shown as a group with 3, 2, 1, and 0 naturally. And when a = 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 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 equation of state, so the pressure fluctuation in Eq.7 can be rewritten as We present the next calculation in 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 [29] and θ eADE = 0 in the rest frame of this dark fluid.

III. PHENOMENA OF THE EXPONENTIAL ACOUSTIC DARK ENERGY
In this section, we present our global fitting results by running the Markov Chain Monte Carlo (MCMC). We modified the public code of CAMB [33] and CosmoMC program [34] with the model and the data of CMB + BAO + H 0 + SN. For the dataset of  [41,42] in our global fitting. The Hubble Space Telescope (HST) observational result [43] also was used as the single data point of the Hubble constant H 0 = 74.03 +1.42 −1. 42 . In addition, the 1048 type Ia supernovae data points in our fitting come from the latest Pantheon dataset in 2018 [44].
The prior of the free parameters were set in Table II. We opened the bound of parameter f c from 0 to 1. For the limitation of the parameter a c , we select log(a c ) to do the global fitting.
We present our fitting result of the marginalized 1D and 2D posterior distributions in The Table. III also gives the fitting result of the Baryon density parameter Ω b and the cold dark matter density parameter Ω c , the χ 2 results of the ΛCDM and the eADE model. Since a c ≈ a eq , the eADE plays an important role in the CMB acoustic peaks. In the time of a c , the CMB was impacted by eADE and radiation maximally. And later the eADE redshifts faster than the radiation, it makes the value of Ω c h 2 higher than the normal case.
For the same reason, the eADE model has a larger value of tilt n s and Ω b compared with ΛCDM.
The χ 2 value of eADE is 3853.93, which is smaller than the result in ΛCDM (3858.21).
The difference between the χ 2 is 4.28. Since there are two additional free parameters for eADE model, we consider the Akaike Information Criterion (AIC) [46] to the statistical analyse and compare the two models.
The AIC estimator can be expressed by AIC ≡ −2 ln L max + 2k (12) where L max ≡ p(d|θ max , M ) is the maximum likelihood value and k at here is the number of free parameters for model. Since χ 2 = −2 ln L max and our eADE model just has two  Table.III. The eADE model is illustrated by the red line, which almost coincides with the black one from ΛCDM, while the Planck 2018 dataset with blue points are closer to the line of eADE than ΛCDM.
additional free parameters, we can get AIC = AIC ΛCDM − AIC eADE = 0.28 in the CMB+H 0 +BAO+SN dataset. The value for AIC shows a preference for the eADE model compared to the ΛCDM model [47], and consistent with our conclusion from the difference of χ 2 in the global fitting.
There is a remarkable result for the Hubble parameter give a smaller value of the sound horizon r s and a shorter expanding history before this recombination. Once the precise angular given by CMB measurements, the inverse distance ladder scale will be reduced and this will lead to the rise of H 0 .
We can see the difference between eADE and ΛCDM for TT mode and EE mode of the CMB power spectra in Fig. 2. The observational data points of CMB come from Planck 2018 [36]. In the TT mode, eADE has a smaller value in the first peak and the second peak before l < 500, same with the ADE result. The EE mode from Planck measurement is still too noisy in the high l. The eADE can fit the data much better than ΛCDM in l < 1500.
In the Fig. 3, we can compare the eADE and ΛCDM in matter power spectra with different observations. The Luminous Red Galaxy (LRG) data come from SDSS DR7 [48], which in the redshift between 0. 6    We also did the comparison with ΛCDM and different measurements in the CMB power spectra and the matter power spectra. The result also prefers our eADE model to ΛCDM in small scale structure.

ACKNOWLEDGMENTS
Thanks for the support from the CQUeST, Sogang University.
Thank Prof. G. Tasinato for the inspirer. Thanks to Prof. X. Zhang and Dr. S. Ni in the Northeastern University for part of the numerical calculations running by the workstation of their group. Thanks to my laptop, it took on many MCMC jobs that did not belong to it in the isolation yea