Constraining Holographic Dark Energy and Analyzing Cosmological Tensions

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We investigate cosmological constraints on the holographic dark energy (HDE) using the stateof-the-art cosmological datasets: Planck CMB angular power spectra and weak lensing power spectra, Atacama Cosmology Telescope (ACT) temperature power spectra, baryon acoustic oscillation (BAO) and redshift-space distortion (RSD) measurements from six-degree-field galaxy survey and Sloan Digital Sky Survey (DR12 & DR16) and the Cepheids-Supernovae measurement from SH0ES team (R22).We also examine the HDE model and ΛCDM with and without N eff (effective number of relativistic species) being treated as a free parameter.We find that the HDE model can relieve the tensions of H0 and S8 to certain degrees.With "Planck+ACT+BAO+RSD" datasets, the constraints are H0 = 69.70 ± 1.39 km s −1 Mpc −1 and S8 = 0.823 ± 0.011 in HDE model, which brings down the Hubble tension down to 1.92σ confidence level (C.L.) and the S8 tension to (1−2)σ C.L. By adding the R22 data, their values are improved as H0 = 71.86 ± 0.93 km s −1 Mpc −1 and S8 = 0.813 ± 0.010, which further brings the Hubble tension down to 0.85σ C.L. and relieves the S8 tension.We also quantify the goodness-of-fit of different models with Akaike information criterion (AIC) and Bayesian information criterion (BIC), and find that the HDE agrees with the observational data better than the ΛCDM and other extended models (treating N eff as free for fitting).
Different theoretical models beyond ΛCDM have been proposed to physically explain the H 0 and σ 8 tensions, but solving the problems "all in one go" is not easy because the existing observational data already highly constrain most alternative models.For instance, changing the physics prior to the recombination to shrink the acoustic horizon (r d ) may boost the CMB-measured H 0 to a higher value (i.e., reducing the Hubble tension), but it may also increase the projected value of S 8 and make it more incompatible with the large-scale structure measurements.This type of solutions includes the primordial magnetic field (PMF [22]) and early dark energy (EDE [23]), which have been proven unable to relieve the S 8 tension [22,24].There is also some current research on EDE extension models [25][26][27], show-ing that they can alleviate the S 8 tension by introducing additional conditions or parameters.Another solution is to have self-interacting neutrinos [28][29][30], which can delay the onset of neutrino free-streaming until close to the radiation-matter equality and thus accommodates a larger Hubble constant [31].A recent such proposal added a fourth "self-interacting" neutrino, but this model requires N eff ∼ 4 which is much higher than the constraints set by Planck and ACT [7,18,32].Other studies that intend to solve the H 0 and S 8 include, the interacting dark energy model [33], the small-scale baryon inhomogeneities (i.e.clumping) [34], and the mirror twin Higgs model with three extra parameters [35] etc.For other proposed solutions and theoretical guide, please refer to [36][37][38].
Different from other dark energy (DE) models, the holographic dark energy (HDE) was invoked from the idea that in quantum field theory, the infrared cut-off should exclude all microscopic states lying within the Schwarzschild radius [39].Hence, by relating the ultraviolet (UV) cut-off at a short distance to the infrared (IR) cut-off at a large distance, one can formulate the dark energy with only one extra free parameter (c).In this holographic dark energy model, the IR cutoff is taken as the future event horizon of the Universe [40], and the cosmic coincidence problem can be solved by inflation in this scenario by assuming the minimum number of e-foldings.Earlier research confronted the HDE model with the then observational data and showed that it can fit the CMB, Type-Ia supernovae and BAO data slightly better than ΛCDM [41][42][43][44].In recent years, the HDE fitting results from Dai, Ma and He (2020) [45] showed that the HDE model can reconcile the Hubble tension: with Planck 2018 CMB data and SDSS-DR12 (Sloan Digital Sky Survey Data Release 12) BAO data, the best-fitting Hubble constant is H 0 = 71.54± 1.78 km s −1 Mpc −1 , which is consistent with the local H 0 measurement of 2019 LMC Cepheids data (R19) [46] at 1.4σ confidence limit.
In this work, we update the previous constraints by incorporating recent observational data from ACT, SDSS-DR12, SDSS-DR16 (denoted as BAO and RSD), and also the refined determination of H 0 from 75 Milky Way Cepheids with Hubble Space Telescope (HST) photometry and Gaia EDR3 parallaxes recalibrated distance ladders (denoted as R22 [16]).We will also treat the effective number of relativistic species (N eff ) as a free parameter in our likelihood analysis and examine whether the fitting results can be improved or not.We want to inspect whether the HDE model can play the following three roles altogether: • to reconcile the Hubble tension between the recent CMB data (Planck and ACT) and BAO from galaxy surveys (SDSS-DR16), and the distanceladder measured H 0 value (R22); • to reconcile the S 8 tension between the CMB and BAO measured values and the large-scale structure measured values (galaxy and CMB lensing); • to allow a slightly larger value of N eff that can accommodate the standard model prediction (N eff = 3.046 [47][48][49][50][51]).
The rest of the paper is organised as follows.In Sec.II, we present the theoretical formulation and predictions of HDE model.In Sec.III, we show the observational data we will use and explain the methodology of data fitting.In Sec.IV, we present our results and discuss the implications.Finally, we conclude in Sec.V.

II. HOLOGRAPHIC DARK ENERGY
Holographic dark energy (HDE) was proposed in light of the holographic principle (HP) [52], which states that all the information contained in a space of with scale L can be described by a theory at the boundary of the space [53].This principle is regarded as one of the most important fundamental principles of quantum gravity, and holds the promise to provide the correct description of the quantum gravity at microscopic scales.But, the conventional local quantum field theory does not agree with HP when including black hole into the system or at the scale of its Schwarzschild radius because most quantum states would collapse.Cohen, Kaplan and Nelson (1999) avoided this catastrophe by suggesting that the energy within a space of the Schwarzschild radius L should not exceed the mass of a black hole of the same size, which means Pl [39], where Λ represents the UV cutoff and M Pl = (8πG) −1/2 is the reduced Planck mass.Thus, a UV cutoff scale Λ is related to an IR cutoff L due to the limit set by avoiding the formation of a black hole.This requires the enclosed vacuum energy to satisfy: Pl L −2 .Taking the entire universe into account and regarding the vacuum energy as dark energy, Li (2004) suggested that the holographic dark energy density is given by saturating the above inequality up to a coefficient c as free parameter [40]: where 3c 2 is a numerical factor for the convenience of calculation, as introduced in Li (2004) [40].L is the future event horizon, which is an assumption chosen to obtain an accelerated expansion of the universe with a = a(t) and a ′ = a(t ′ ) .
Refs. [40,42] further showed that Eq. (2) leads to the DE equation of the state (EoS) being close to −1 .Then, from Friedmann equations we have The energy conservation requires that the covariant derivative acting on the energy-momentum tensor should vanish, and this leads to the following conditions, dρ de dt + 3H (ρ de + P de ) = 0 .
One can solve this dynamic system completely.Utilizing w de ≡ P de /ρ de , and pl Ω de , we can substitute them and obtain a complete set of differential equations for Ω de (z) and H(z) [42,43]: The EoS parameter of the dark energy can be solved in the following form: As can be seen, at the early universe when Ω de ≪ 1, we have w de → −1/3 which is greater than −1 .In this case the universe experiences a quintessential acceleration [54][55][56].But at the late stage when HDE is dominant, Ω de → 1 , and thus w de → −1/3−2/3c .As long as c is positive, the expansion of the universe would accelerate according to the second Friedmann equation, ä/a = −4πGρ (1 + 3w)/3 , which agrees with the current observations.Hence, we can fit the parameter c and other cosmological parameters with the observational data of the CMB angular power spectra, galaxy BAOs and BBN to find the best-fitting parameters, and then compare the fitting results with ΛCDM.As shown in Dai et al. (2020) [45], the then Planck+BAO12+R19 data [57] gave the best-fitting value as c = 0.51 ± 0.02 and H 0 = 73.12± 1.14 km s −1 Mpc −1 , which resolves the Hubble tension completely.And also, we can observe that 0 < c < 1 due to the constraints of observations, i.e., w de < −1 in the late universe, indicating that it crosses the phantom divide.The physical reason is that, HDE can cause the Universe to have a smaller acceleration earlier on, and a faster acceleration at the later stage, but still keeps the total angular diameter distance to the last-scattering surface unchanged.Therefore, this delayed-and-catch up acceleration can make the model still fit the CMB angular power spectrum, but enhance the local expansion rate (local H 0 ) to match the R19 result as in Dai et al. (2020) [45].In the following sections, we will analyze the model by using the most up-to-date datasets and present the fitting results.

III. DATA AND METHODOLOGY
We modify the Boltzmann code camb [58] to embed the HDE model [Eq.( 5)] as the background expansion model to compute the angular power spectra of CMB (see also Fig. 5).We have only considered the effect of the HDE on the background expansion, i.e., we have replaced the components of DE (ρ de ) in the background density (ρ) of the perturbation equations as well as its EoS w de , without considering its perturbation term (δ de = 0. [59] We then utilize the Markov Chain Monte Carlo (MCMC) technique from the public code CosmoMC [60] to constrain the parameters of models by using the latest Planck and ACT CMB data, SDSS DR16 BAO and RSD data and also the R22 data.The sampling method we used in CosmoMC is a modified Metropolis algorithm, specifically a fast-slow dragging algorithm [61,62].This approach is suitable for likelihood with numerous fast nuisance parameters like Planck.The benchmark models we fit are ΛCDM and HDE.We also add N eff (the effective number of relativistic species) as a free parameter to the fit of each model.So we have four models (ΛCDM, HDE, ΛCDM+N eff , and HDE+N eff ) for the present analysis.The ΛCDM model has six cosmological parameters Ω b h 2 , Ω c h 2 , 100θ MC , τ , ln(10 10 A s ), and n s , which correspond to the fractional baryon and dark matter densities, the angular size to the last-scattering surface, the optical depth, the amplitude, and the tilt of primordial curvature perturbations, respectively.The HDE model has one more free parameter c as shown in the Eq. ( 1).
Then, we explain each dataset used in our fitting: • Planck : We use the CMB high-ℓ TT, TE and EE plik likelihood, low-ℓ temperature Commander likelihood, low-ℓ SimAll EE polarization from the Planck 2018 data release [64], and Planck 2018 CMB lensing power spectrum [65].In the following, we use Planck to denote the combination of all Planck data mentioned here.
• ACT: We use the likelihood computed from the CMB power spectrum on scales of 350 < ℓ < 4000 measured by the ACTPol instrument of the Atacama Cosmology Telescope, derived from seasons 2013-2016 of temperature and polarization data release 4 (DR4) [17].
• BAO: We use the Baryon Acoustic Oscillation measurements at different redshifts as the standard ruler to constrain cosmological parameters.The Six-degree Field Galaxy Survey (6dF) measures D V at z eff = 0.106 [63] and the "Main Galaxy Sample" (MGS) from Sloan Digital Sky Survey Data Release 12 (SDSS-DR12) measures D V at z eff = 0.15 [66].
We also add D M (z eff )/r d and D H (z eff )/r d measurements at z eff = 0.38 and 0.51 from SDSS-DR12 (BOSS CMASS galaxies) [67] and same quantities at z eff = 0.698 from SDSS-IV DR16 eBOSS Luminous Red Galaxies (LRG) [12].The five samples  [12]).We use "BAO" to denote the BAO-only dataset at these five effective redshifts (Table I).
The r d = 147.8Mpc mentioned above is the sound horizon at the drag epoch, where the value is set by the fiducial cosmological parameters in the spatially-flat ΛCDM model [68].D M (z) is the co-moving angular diameter distance, and , where the powers of 2/3 and 1/3 approximately account for two transverse and one radial dimension and the extra factor of z is a conventional normalization [12].
• BAO+RSD: We also adopt the BAO measurements with additional measurement of f σ 8 (z) from redshift-space-distortion (RSD).The RSD arises from the peculiar velocities of distant galaxies due to the inhomogeneous distribution of matters.f σ 8 (z) is the amplitude of the velocity power spectrum, where σ 8 (z) is the amplitude of linear matter fluctuations on the comoving scale of 8 h −1 Mpc, and f (z) ≡ ∂ ln D/∂ ln a is the linear growth rate of structure (a = 1/(1 + z) is the scale factor).The RSD measurement provides direct measurement on f σ 8 (z) [12,67,69].
Hence, we use the joint likelihood of D M (z eff )/r d , D H (z eff )/r d , and f σ 8 (z eff ) at z eff = 0.38, 0.51 and 0.698 as determined from the combined BAO and RSD likelihoods [70], and also the two 6dF and MGS samples (Table I).The measurements at z eff = 0.38 and 0.51 are taken from the data presented in Table 8 of SDSS-DR12 [67], and the measurement at z eff = 0.698 is taken from SDSS-DR16 [12].Our CosmoMC likelihoods include the full covariance matrix of • R22: We use the measurement of Hubble constant from SH0ES team as H 0 = 73.04 ± 1.04 km s −1 Mpc −1 by Riess et al. [16].This value is the baseline result from the Cepheid-SN Ia sample including systematic uncertainties and lies near the median of all analysis variants.
To compare the effects of different datasets on the cosmological parameter constraints, we use the following dataset combinations: "Planck +BAO", "Planck +ACT+BAO", "Planck +ACT+BAO+RSD", "Planck +ACT+BAO+RSD+R22".We also compare the results of the constraints for four different models ΛCDM, HDE, ΛCDM+N eff , and HDE+N eff (the latter two treat N eff as free parameter in the fits), and use the two datasets "Planck +ACT+BAO+RSD" and "Planck +ACT+BAO+RSD+R22" for the model comparison.
In this work, we do not use the Pantheon Type-Ia supernovae samples which comprise 1048 data spanning the redshift range 0.01<z < 2.3 [71].The reason is the possible inconsistency of the parameter constraints due to the large correlation between high-z (z > 0.2) and low-  [45]).In Ref. [72], it shows that "Planck+BAO" and "Planck+Pantheon" give inconsistent results by more than 95% C.L., also suggesting that there might be some uncounted systematics in the Pantheon dataset.
Currently, the new Pantheon+ data has been released, including 1701 light curves of 1550 distinct Ia supernovae covering a redshift range from z = 0.001 to 2.26 [73], which represents a greater number of Ia SN measurements compared to Pantheon.We present the findings of constraints resulting in the Appendix from a combination of Pantheon+ data with Planck, ACT, and BAO+RSD utilized in our work.

IV. RESULTS AND DISCUSSIONS
In this section, we present the fitting results for the four cosmological models by using the latest observational data.Furthermore, we analyze the effects of different models, and the combination of different datasets on the problem of cosmic tensions.We will also compare the fitting results by treating N eff as a free parameter.There are eight free parameters in our fitting of the HDE and ΛCDM models where the priors are listed in Table III.
Figure 1 shows the marginalized constraints on cosmological parameters N eff , c, H 0 , Ω m , σ 8 and S 8 of model HDE+N eff .Here the first two parameters are the base parameters, and the H 0 , Ω m , σ 8 and S 8 are the derived parameters.In this analysis, we focus on the effect of the four different dataset combinations.We can see from the contours of different colours that, the more data is included, the more precise the constraint becomes.Regarding parameter N eff in Fig. 1, the green contour (Planck+BAO) lies on the right of the other contours which contain the ACT data, which suggests that the value of N eff tends to be lower if including the ACT dataset.Further information about the constraints on N eff can be found in Sec.IV C.The HDE parameter c does not change significantly with these four datasets in Fig. 1.For the parameters related to cosmic tensions, H 0 and S 8 , the local Hubble constant measurement by Riess et al. [16] (R22, blue contours) plays an important role.When compared to other datasets like ACT, BAO, and RSD, the inclusion of R22 data leads to a significant increase of H 0 and a slight decrease of S 8 .This can be seen from the posterior distribution of H 0 and S 8 on the far right of Fig. 1 and the corresponding contours, because the blue curve and contours show a notable change compared to other colors.Regarding S 8 , the addition of ACT, BAO and RSD had little impact on its constraints.In the following, we discuss the resolution of Hubble tension, S 8 tension, and the constraints of N eff respectively.

A. The Hubble Tension
We first examine the HDE model's capability of reconciling Hubble tension with different dataset combina-   [21]) and the R22 result (H 0 = 73.04 ± 1.04 kms −1 Mpc −1 [16]) are shown as the horizontal and vertical gray bands.The closer the constrained contour is towards the bands, the more reconciliation that the model can do to match the R22 and LSS's S 8 measurements.One can see from both panels of Fig. 2 that, by switching from ΛCDM model (green) to HDE model (red), the contours move towards the right lower corner, indicating that the HDE model prefers a larger H 0 value and lower S 8 value (see also Table II).Certainly, the reconciliation of HDE does better if R22 data is included (right panel).But even without the R22, such trend is still obvious in the left panel.In both panels, we also include the fitting results of the models including N eff as a free parameter (ΛCDM+N eff and HDE+N eff ), showing as the gray and blue contours.There is no significant effect to relieve the H 0 tension, but we notice that HDE+N eff (blue contours) still provide a better fit than ΛCDM+N eff (grey contours).2 of ACT DR6 lensing [18].In this plot, Planck CMB only refers to the Planck data mentioned in Sec.III above without lensing, ACT CMB lensing refers to the ACT lensing measurement from ACT DR6 lensing, and Planck+ACT CMB lensing refers to lensingalone data from Planck and ACT DR6.The gray bands are the 1σ and 2σ confidence limits on the model HDE+N eff by using the dataset Planck +ACT+BAO+RSD+R22, (i.e., S8 = 0.807 ± 0.011 (1σ C.L.) in the bottom line, right column of Table II).
We plot the marginalized posterior distribution of H 0 for the four models and two datasets in Fig. 3 (D1 = Planck +ACT+BAO+RSD, D2 = Planck +ACT+ BAO+RSD+R22).With D1 dataset, H 0 is measured to be 69.70 ± 1.39 km s −1 Mpc −1 in the HDE model and 67.55 ± 0.39 km s −1 Mpc −1 in the ΛCDM model (see also Table II).Thus, HDE brings the H 0 measurement closer to R22, with a tension of only 1.92σ instead of the 4.94σ tension in the ΛCDM model [74].By including the R22 in the likelihood, the tension in the HDE model is further reduced.The H 0 is measured to be 71.86 ± 0.93 km s −1 Mpc −1 , which brings the tension further down to 0.85σ, showing a perfect consistency between the CMB measurements and R22.This comparison is also evident in Fig. 3, where H 0 's posteriors of the HDE (red curves) are always on the right-hand side of ΛCDM (green curves), and are much closer to the R22 measurement (gray vertical band).
After further including R22 data, the HDE+N eff model predicts the S 8 value as 0.807 ± 0.011, N eff = 2.89 ± 0.15, and the parameter c as 0.52 ± 0.03, where the S 8 value is even closer to the low-level galaxy lensing measurements.Although this change does not entirely resolve the S 8 tension, it still indicates that the HDE partially alleviates the S 8 tension, and the overall fit is still slightly better than the standard ΛCDM model.

C. The N eff Constraints
The consideration of N eff as a varying parameter stems from the substantial deviation between the derived value of N eff = 2.74 ± 1.3 [17] obtained from fitting ACT data with the ΛCDM+N eff model and the theoretical value of 3.046.Our primary aim is to explore whether replacing ΛCDM with HDE can mitigate this disparity, while also assessing its impact on cosmological discrepancies.
We further examine the effect of including N eff as a free parameter in the joint likelihood and the changes in the CMB power spectra.What affects the CMB power spectra is the total radiation density: where in natural units, are the energy densities of photons and neutrinos.The factor 7/8 arises from the Fermi-Dirac distribution, N eff is the effective number of neutrino species, which does not need to be an integer [3,4,75,76].In fact, N eff = 3.046 in the standard model of particle physics [47][48][49][50][51]. Thus, because neutrinos contribute to the total energy density and change the anisotropic stress of the baryon fluid, to increase N eff can enhance the expansion rate, reduce the CMB damping tail, and also shift the peak and trough positions of the CMB angular power spectra [75,77].Hence, the accurate measurements of CMB power spectra can essentially constrain the N eff value to a high precision, which is a primary science target for this and next generations of CMB experiments [8,17,78,79].
In Fig. 5, we plot the CMB temperature (TT) and Emode (EE) power spectra in the left and right columns, for both the power spectra measurements and the model predictions, and the relative difference with respect to the fiducial ΛCDM model.In the upper panel, the gray and red data are the unbinned and binned power spectra respectively.We plot the four models' best-fitting power spectra (Table II) with combined "Planck +ACT+BAO+RSD+R22" dataset and that of ΛCDM with Planck 2018 best-fitting value (i.e.Planck TT+TE+EE+lowE+lensing) in different color and stylish lines.The upper panel shows that all four cosmological models almost overlay on each other, and their fittings to Planck data are almost indistinguishable.The lower panel of Fig. 5 shows the fractional differences of the four models with respect to the Planck best-fitting ΛCDM model.One can see that the HDE model (blue solid line) matches the data slightly better than the standard model ΛCDM (black solid line), because the additional wiggles with the suitable amplitude appeared in the HDE model can better match the upand-down features of the binned angular power spectra (red data dots) than the ΛCDM.With the additional N eff , the HDE model is boosted further up (blue dashed line), and the ΛCDM model is pushed downwards (black dashed line), making the fits less good than that without the N eff parameter.This situation is also evident in Table IV, where we use the χ 2 , Akaike information criterion (AIC) and Bayesian information criterion (BIC) to quantify the "goodness of fit".The third column is the minimal χ 2 of each model, and ∆χ 2 (the fourth column) is the relative difference to the best-fitting ΛCDM model.The total χ 2 of the combined dataset can be written as χ 2 = χ 2 Planck + χ 2 ACT + χ 2 BAO+RSD + χ 2 H0 .One can see that, by switching from ΛCDM model to the HDE model, the χ 2 min drops by a factor of −9.352, which is the result from the better matching of the CMB angular power spectra in Fig. 5.In addition, one can see that adding N eff does not alter the goodness of fit significantly.Be- cause HDE has one additional parameter (c in Eq. ( 1)) than the base ΛCDM, we also use the AIC and BIC as metrics to compare the models, as they compensate for models with fewer parameters.Here we only need to focus on the relative values, which are defined as ∆AIC = ∆χ 2 min + 2∆k and ∆BIC = ∆χ 2 min + ∆k ln N .Here, ∆χ 2 min and ∆k represent the difference in the minimum χ 2 value and the additional number of free parameters compared to the standard model, which is unity in our case.The symbol N denotes the number of data points, for dataset "Planck+ACT+BAO+RSD+R22", N = 10520 [80].One can see that, even for AIC and BIC, the HDE model provides a better fit compared to the ΛCDM model, because the AIC drops by a factor of −7.352 and the BIC drops by a factor of −0.091.It's worth noting that BIC penalizes model parameters more for larger data volumes compared to AIC.Adding the N eff makes the fit less good because it turns to increase the AIC and BIC values.
We now compare the constraints on N eff value.We compare the fitting results of N eff in Fig. 6 with different combined datasets and two extension models.The gray vertical dashed line with the band is the result of constraint from Planck+ACT data on ΛCDM+N eff model  [17]): N eff = 2.74 ± 1.3, where the gray solid line is the prediction of 3.046 from the standard model of particle physics (SMPH) [47][48][49][50][51].In Fig. 6, the change in the N eff value is insignificant for HDE relative to ΛCDM (red and grey curves), without taking into account the R22 measurement.Once R22 is added, N eff is shifted from 2.78 ± 0.14 (orange dashed curve) to 3.26 ± 0.12 (green dashed curves).But for HDE, N eff increases from 2.80±0.16(orange solid curve) to 2.89±0.15(green solid curves), which is a much smaller shift than the ΛCDM.Nonetheless, for all these combined datasets and theoretical models, the resultant constraints on N eff are consistent with SMPH's prediction within 2σ range.

V. CONCLUSIONS
In this work, we revisited the cosmological constraints on the holographic dark energy model and compared its results with the standard ΛCDM model.To constrain the models, we combined the state-of-the-art cosmological datasets including the Planck 2018 CMB temperature, polarisation and lensing power spectra data with the ACT-DR4 power spectrum, Baryon Acoustic Oscillations (BAO) and Redshift Space Distortion (RSD) from 6dF and SDSS survey, and the Cepheids-SN measurement of H 0 (R22).We also treat the effective number of relativistic species (N eff ) as a free parameter of fit in the likelihoods to examine whether it alters the fitting results.In comparison with the ΛCDM, we found that the HDE model can relieve the H 0 and S 8 tensions to a certain degrees.Hence, to respond the three questions raised in Sec.I, we conclude as follows: • Using the datasets "Planck +BAO+RSD+ACT", we derived constraints on H 0 in the ΛCDM model to be 67.55 ± 0.39 km s −1 Mpc −1 , which is 4.94σ lower than the R22 measurement on H 0 ( = 73.04 ± 1.04 km s −1 Mpc −1 ).With the same dataset, the HDE model brings the value of H 0 down to 69.70 ± 1.39 km s −1 Mpc −1 , which reduces the H 0 tension down to the level of 1.92σ and thus mainly solves the Hubble tension (Figs. 2 and 3).
After including the R22 dataset in the likelihood, the tension in the HDE model is further reduced.The H 0 of the HDE model is measured to be 71.86±0.93km s −1 Mpc −1 , which further brings the tension between the HDE prediction and the R22 measurement down to 0.85σ, leading to a perfect agreement with R22.
• By using "Planck+ACT+BAO+RSD" dataset in the HDE+N eff model, the value of S 8 is reduced down to 0.816 ± 0.012 ; hence the S 8 tension is relieved to the 1σ-2σ level as compared to the lowredshift gravitational lensing measurement (Fig. 4).This tension is further relieved after including the R22 dataset (Fig. 4).
• By treating N eff as a free parameter for fit in the likelihood chain, the posterior distribution of N eff is around 2.7-2.8,depending on the models and the datasets used (Fig. 6).This constraint is consistent with the standard model value 3.046 within 2σ confidence limit, and is on the lower side.
• Overall, by using the AIC criterion of the goodnessof-fit, we find the HDE model can fit the joint datasets "Planck+ACT+BAO+RSD+R22" to the best extent.
In the near future, many ongoing surveys such as the Vera C. Rubin Observatory (LSST) [81], Simons Observatory [82], and Square Kilometre Array (SKA) [83] will provide more fruitful and precise data on galaxies, CMB, and large-scale structure.We will reexamine the HDE model soon with these advanced datasets.

RESULTS OF ADDING PANTHEON+ DATA
Here we present the constraints on four models ΛCDM, HDE, ΛCDM+N eff and HDE+N eff with dataset "Planck +ACT+BAO+RSD+Pantheon+".In Fig. 7, the posterior contours of H 0 and S 8 were presented.It is apparent that with the inclusion of the Pantheon+ data, particularly in the H 0 comparisons, the models in the HDE scenario do not exhibit a significant advantage over the ΛCDM.They yield similar results (see Table V) and slightly larger values of H 0 in the ΛCDM scenario: H 0 = 67.44 ± 0.37 in ΛCDM compared to H 0 = 66.07 ± 0.77 in HDE; and H 0 = 65.70 ± 1.01 in ΛCDM+N eff compared to H 0 = 65.44±1.00 in HDE+N eff .When comparing the constraint on S 8 in the ΛCDM scenario, including Pan-theon+ data has little effect on its value.However, in the HDE and extended model, there is an increasing gap with the galaxy lensing observations, as shown in Figs.2(a) and 7, and Tables II and V, which leads to a similar result in the HDE as in the LCDM with the addition of Pantheon+ data.
Overall, the addition of the Pantheon+ data brings the HDE closer to the ΛCDM in terms of the constraints on H 0 and S 8 .As discussed in the last part of Section III and in Dai et al. (2020) [45] large difference between partial and full redshifts on the constraints on H 0 in HDE, with the Pantheon samples having a strong correlation between high and low redshifts.Pantheon+ data likely have a similar effect, and whether there is a unique theoretical reason for this variation in the Pantheon-related data on the constraints of H 0 and S 8 for the HDE model deserves to be studied and explored in more detail further in the future.

FIG. 1 .
FIG. 1. Posterior distributions of HDE+N eff model for selection parameters for four of the dataset combinations, at 68% C.L. (1σ) and 95% C.L. (2σ).The first two parameters (N eff and c) are from the model's base parameter set (TableII), and the last four parameters (H 0 , Ωm, σ8 and S 8 ) are the derived parameters.

FIG. 4 .
FIG.4.Marginalized constraints on S8 (68% C.L.) of different weak lensing, CMB and BAO measurements, and their combinations.Here we compare our results (red) with the constraints (blue) taken from Table2of ACT DR6 lensing[18].In this plot, Planck CMB only refers to the Planck data mentioned in Sec.III above without lensing, ACT CMB lensing refers to the ACT lensing measurement from ACT DR6 lensing, and Planck+ACT CMB lensing refers to lensingalone data from Planck and ACT DR6.The gray bands are the 1σ and 2σ confidence limits on the model HDE+N eff by using the dataset Planck +ACT+BAO+RSD+R22, (i.e., S8 = 0.807 ± 0.011 (1σ C.L.) in the bottom line, right column of TableII).

FIG. 5 .
FIG.5.CMB TT (left) and EE (right) power spectrum with parameters constraints by Planck +ACT+BAO+RSD+R22 data combination, which illustrates the effects of ΛCDM, ΛCDM+N eff , HDE, and HDE+N eff on the phase and amplitude of the power spectra.Measurements from the Planck 2018 data release are included, and the grey solid line is the best-fitting ΛCDM CMB power spectra from the baseline Planck TT, TE, EE+lowE+lensing (2 ⩽ ℓ ⩽ 2508).

ACKNOWLEDGMENTS
This research is funded by the research program "New Insights into Astrophysics and Cosmology with Theoretical Models Confronting Observational Data" of the National Institute for Theoretical and Computational Sciences of South Africa.YZM acknowledges the support from National Research Foundation of South Africa with Grant No. 150580, No. 159044, No. CHN22111069370 and No. ERC23040389081.HJH was supported in part by the National NSFC (under grants 12175136 and 11835005).

TABLE I .
[12] et al. (2021)in the two datasets BAO and BAO+RSD for the present analysis.The 6dF DV measurement is fromBeutler et al. (2011)[63]and the other measurements are all from Table III ofAlam et al. (2021)[12].

TABLE II .
Constraints of Planck +ACT+BAO+RSD and Planck +ACT+BAO+RSD+R22 datasets on the ΛCDM, ΛCDM+N eff , HDE, and HDE+N eff models.For each dataset, the upper block is for the constraints on the fundamental parameters, and the lower block is for the derived parameters.The quoted error is given at the 68% C.L.

TABLE III .
Flat priors of cosmological parameters in MCMC.

TABLE IV .
Comparison of χ 2 fits for different models.

TABLE V .
Constraints of Planck +ACT+BAO+RSD+Pantheon+ dataset on the ΛCDM, ΛCDM+N eff , HDE, and HDE+N eff models.The upper block is for the constraints on the fundamental parameters, and the lower block is for the derived parameters.The quoted error is given at the 68% C.L.