1. Introduction
Although scrutinized by a plethora of observational data across various scales in the past, the Lambda cold dark matter (ΛCDM) model is currently facing skepticism regarding its internal inconsistencies. The Hubble tension, i.e. disagreement between direct [1] and indirect measurements [2] of the Hubble constant H0, is a major factor contributing to this outcome. Although, up to this point, we cannot exclude the possibility of systematics as the origin of the Hubble tension, many cosmologists are beginning to address the tension by introducing new physics. Generally speaking, solutions for resolving the Hubble tension in a theoretical manner can be categorized into two types: early-time solutions and late-time solutions, based on the introduction of new physics before and after recombination. Early-time solutions, such as early dark energy (EDE) models [3–12], which introduce an exotic dark sector that acts as a cosmological constant before a critical redshift zc around 3000 but whose density then dilutes faster than radiation, and dark radiation models [13–18], which include extra relativistic degrees of freedom that do not interact with photons and baryons, are considered to play a important role in resolving the Hubble tension. Nevertheless, as pointed out in [19], early-time new physics alone will always fall short of fully solving the Hubble tension. On the other hand, late-time solutions, such as late dark energy (DE) models [20–26] and interacting DE models [27–39], are also considered to be incapable of fully resolving the Hubble tension, since consistency with baryon acoustic oscillation (BAO) and uncalibrated Type Ia supernovae (SN Ia) data requires new physics to be introduced before recombination, in order to reduce the sound horizon by ∼7% [40–44].
Among all the early-time solutions, EDE models may be the most promising category. However, similar to other early-time solutions, EDE models are incapable of fully resolving the Hubble tension. One of the primary reasons for this is that consideration of a non-zero EDE fraction fEDE(zc) near the matter–radiation equality would increase the dark matter (DM) density ωcdm compared to that of ΛCDM. A higher value of ωcdm would lead to a higher value of the ${S}_{8}={\sigma }_{8}\sqrt{{{\rm{\Omega }}}_{{\rm{m}}}/0.3}$ parameter (where Ωm is the matter density parameter and σ8 is the matter fluctuation amplitude on scales of 8h−1Mpc) if the value of σ8 undergoes little variation, leading to a more significant S8 tension between cosmic microwave background (CMB) data and weak lensing (WL) as well as large-scale structure (LSS) data [45–62]. However, it is still possible that an extension of an EDE model would mitigate the need for an increase in the ωcdm or, alternatively, compensate the associated increase in the amplitude of fluctuations. In fact, various extensions of EDE models have been proposed by researchers guided by this idea to simultaneously alleviate the H0 tension and the S8 tension. See e.g. [63–70]. These models employ various methods to supplement or modify the CDM paradigm, including the introduction of the total neutrino mass Mν as a free parameter [63, 65, 70], the inclusion of ultra-light axions that comprise five percent of DM [64], the proposal to replace CDM with decaying DM [69] and the consideration of DM coupled with EDE [66–68]. Since the nature of DM remains mysterious, it is appropriate to explore phenomenological models and observe how they align with data. The simplest phenomenological modification to the CDM paradigm, i.e. considering a constant DM equation of state (EoS) as a free parameter, has not yet been explored in the existing literature related to the extension of EDE. Nevertheless, it is well known that a negative value of DM EoS has the capability to reduce the value of the σ8 parameter compared to a CDM setting. Therefore, in this paper, we will extend the axion-like EDE model by replacing CDM with DM characterized by a constant EoS (referred to as WDM hereafter), and examine whether this extension can simultaneously alleviate the H0 tension and the S8 tension.
The rest of this paper is organized as follows. In section 2 , we review the dynamics of axion-like EDE and WDM, and discuss their cosmological implications, specifically on the CMB and matter power spectra. In section 3 , we describe the observational datasets and the statistical methodology. In section 4 , we present the results of a Markov Chain Monte Carlo analysis applied on a combination of CMB, BAO and SN Ia data. In the last section, we present a brief conclusion for this paper.
2. Model overview
Our model consists of two modifications to ΛCDM. The first is the inclusion of the axion-like EDE, and the second is the replacement of CDM with WDM. Therefore, we refer to this model as EDE+WDM hereafter. In this section, we will give a brief summary of the dynamics of EDE+WDM.
In EDE+WDM, EDE is represented by a canonical scalar field [7], while WDM is represented by an ideal fluid. Their energy density and pressure affect the dynamics of other species through Einstein's equation. At the homogeneous and isotropic level, the expansion rate of the Universe can be written as:
$\begin{eqnarray}H={H}_{0}\sqrt{{{\rm{\Omega }}}_{\mathrm{dm}}(a)+{{\rm{\Omega }}}_{b}(a)+{{\rm{\Omega }}}_{r}(a)+{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{\phi }(a)},\end{eqnarray}$
where a is the scale factor, ΩX ≡ ρX/ρcrit and ${\rho }_{\mathrm{crit}}=3{H}_{0}^{2}{M}_{P}^{2}$, with ${M}_{P}\equiv {(8\pi G)}^{-\tfrac{1}{2}}$ being the reduced Planck mass. The energy density and pressure of the scalar field at the background level are given by: $\begin{eqnarray}{\rho }_{\phi }=\displaystyle \frac{1}{2}{\dot{\phi }}^{2}+V(\phi ),\end{eqnarray}$
$\begin{eqnarray}{p}_{\phi }=\displaystyle \frac{1}{2}{\dot{\phi }}^{2}-V(\phi ),\end{eqnarray}$
where the dot indicates a derivative with respect to cosmic time. The potential of the axion-like EDE scalar field is chosen in the following form: $\begin{eqnarray}V(\phi )={m}^{2}{f}^{2}{[1-\cos (\phi /f)]}^{3},\end{eqnarray}$
where m and f are the EDE mass and decay constant. The evolution of the scalar field φ is described by the Klein–Gordon (KG) equation: $\begin{eqnarray}\ddot{\phi }+3H\dot{\phi }+{V}_{,\phi }=0.\end{eqnarray}$
As described in the literature [7], initially, the scalar field is frozen due to Hubble friction at a position displaced from the minimum of its potential, and its energy density is sub-dominant. It is only after the Hubble rate drops below the effective mass of the scalar field that the field becomes dynamical, starts rolling down and oscillates around the minimum of its potential. This leads to a faster dilution of its energy density compared to radiation. Given the evolutionary behavior of EDE, the fundamental particle physics parameters m and f can be related to the phenomenological parameters ${\mathrm{log}}_{10}{z}_{c}$ and fEDE(zc). Here, fEDE(zc) is the maximum fraction of the total energy density in the scalar field, and zc is the redshift at which this fraction reaches its maximum. Therefore, the EDE component is governed by three parameters: ${\mathrm{log}}_{10}{z}_{c}$, fEDE(zc) and the initial misalignment angle θi = φi/f, with φi being the initial field value. As shown in [7], m largely controls the value of zc, while f controls the value of fEDE(zc). And the approximate equations relating m to zc and f to fEDE(zc) are: $\begin{eqnarray}{m}^{2}| {(1-\cos {\theta }_{i})}^{2}{(2+3\cos {\theta }_{i})| \simeq 3H({z}_{c})}^{2},\end{eqnarray}$
$\begin{eqnarray}{f}_{\mathrm{EDE}}{({z}_{c})\simeq \displaystyle \frac{{m}^{2}{f}^{2}}{{\rho }_{{\rm{tot}}}({z}_{c})}(1-\cos {\theta }_{i})}^{3}.\end{eqnarray}$
It can be inferred from the above equations that for a fixed θi, a value of m determines zc and a value of f determines fEDE(zc). According to [71], θi, once other EDE parameters are fixed, is a parameter whose value controls the oscillation frequency of the background field. It is tightly constrained by the small-scale polarization measurements: in fact, the full Planck 2015 dataset, excluding the region θi < 1.8 at 95% confidence level.The evolution of the energy density of WDM, however, is described by a simpler equation, namely, the continuity equation of WDM:
$\begin{eqnarray}{\dot{\rho }}_{\mathrm{dm}}+3H(1+{w}_{\mathrm{dm}}){\rho }_{\mathrm{dm}}=0.\end{eqnarray}$
Here, wdm is the EoS of WDM, and it is a constant. After solving this equation, we obtain ${\rho }_{\mathrm{dm}}={\rho }_{\mathrm{dm}0}{a}^{-3(1+{w}_{\mathrm{dm}})}$ (without losing any generality, we set the current value of the scale factor a0 to be unity). According to this formula, WDM dilutes either faster or slower than CDM, depending on the sign of wdm.The EDE+WDM model is determined by ten parameters: six basic parameters shared with ΛCDM, plus four new parameters, namely, ${\mathrm{log}}_{10}{z}_{c}$, fEDE(zc), θi and wdm.
At the linear perturbation level, the perturbation of the scalar field φ (in the Fourier space) is governed by the linearized KG equation:
$\begin{eqnarray}\delta {\phi }_{k}^{{\prime\prime} }+2{ \mathcal H }\delta {\phi }_{k}^{{\prime} }+({k}^{2}+{a}^{2}{V}_{,\phi \phi })\delta {\phi }_{k}=-{h}^{{\prime} }{\phi }^{{\prime} }/2\end{eqnarray}$
where the prime denotes the derivative with respect to conformal time, ${ \mathcal H }$ is the conformal Hubble parameter, h is the metric potential in synchronous gauge and k is the magnitude of the wavenumber $\vec{k}$.On the other hand, the linear perturbation equations of WDM consist of the continuity and Euler equations of this component, i.e.:
$\begin{eqnarray}\begin{array}{l}{\delta }_{\mathrm{dm}}^{{\prime} }=-(1+{w}_{\mathrm{dm}})\left({\theta }_{\mathrm{dm}}+\displaystyle \frac{1}{2}{h}^{{\prime} }\right)-3{ \mathcal H }{\delta }_{\mathrm{dm}}({c}_{{\rm{s}},\mathrm{dm}}^{2}-{w}_{\mathrm{dm}})\\ -9(1+{w}_{\mathrm{dm}})({c}_{{\rm{s}},\mathrm{dm}}^{2}-{c}_{{\rm{a}},\mathrm{dm}}^{2}){{ \mathcal H }}^{2}\displaystyle \frac{{\theta }_{\mathrm{dm}}}{{k}^{2}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}{\theta }_{{\rm{dm}}}^{{\rm{{\prime} }}}=-(1-3{c}_{{\rm{s}},{\rm{dm}}}^{2}){ \mathcal H }{\theta }_{{\rm{dm}}}\\ +\displaystyle \frac{{c}_{{\rm{s}},{\rm{dm}}}^{2}}{1+{w}_{{\rm{dm}}}}{k}^{2}{\delta }_{{\rm{dm}}},\end{array}\end{eqnarray}$
where δdm and θdm are the relative density and velocity divergence perturbations of DM, and cs,dm and ${c}_{{\rm{a}},\mathrm{dm}}\,={\dot{p}}_{\mathrm{dm}}/{\dot{\rho }}_{\mathrm{dm}}$ denote the sound speed and adiabatic sound speed of DM. The sound speed of DM describes its micro-scale properties and needs to be provided independently; in this article, we consider ${c}_{{\rm{s}},\mathrm{dm}}^{2}=0$ to understand the extent to which modifying the DM EoS alone can alleviate the S8 tension. Having presented the equations above, the background and perturbation dynamics of the EDE+WDM model are clearly understood.At the end of this section, we present some implications regarding the impacts of EDE+WDM on the CMB TT and matter power spectra. For a simplified demonstration of these effects, we display, in figure 1, the CMB TT and matter power spectra of EDE+WDM, along with their residuals compared to a baseline ΛCDM model. We define our benchmark ΛCDM model with the following cosmological parameters: the peak scale parameter 100θs = 1.041783, the baryon density today ωb = 0.02238280, the DM density today ωdm = 0.1201705, the optical depth τreio = 0.0543082, the amplitude of the primordial scalar As = 2.100549 × 10−9, and the spectral index of the primordial scalar ns = 0.9660499. These values are extracted from the Planck 2018 + lowE + lensing dataset [2]. The six basic parameters of EDE+WDM are fixed to the values of their counterparts in the baseline ΛCDM model. Other parameters, namely, ${\mathrm{log}}_{10}{z}_{c}$, fEDE(zc), θi and wdm, are set to the following values for different choices: 3.5, 0, 2.5, 0.001; 3.5, 0, 2.5, 0; 3.5, 0, 2.5, −0.001; and 3.5, 0.05, 2.5, 0. Of course, we point out that the values of six basic parameters of EDE+WDM extracted from the Planck 2018 + lowE + lensing dataset are different from the six corresponding values provided above. In fact, EDE+WDM has a larger value of ωcdm compared to ΛCDM due to the wdm-ωdm and EDE-ωdm degeneracy, and this will lead to different CMB TT and matter power spectra. However, to consider the individual impact of changing the values of wdm and fEDE(zc) on the CMB TT and matter power spectra, we set the values of the six basic parameters for both EDE+WDM and ΛCDM to be consistent.
Figure 1. The CMB TT and matter power spectra are computed for different values of the parameters fEDE(zc) and wdm, while keeping six basic parameters fixed at the values of their counterparts in the baseline ΛCDM model, as extracted from the Planck 2018 + lowE + lensing data. Additionally, ${\mathrm{log}}_{10}{z}_{c}$ and θi are set to 3.5 and 2.5. |
Focusing on the CMB TT power spectrum and fixing the fEDE(zc) parameter, we observe that the amplitudes of the acoustic peaks in the CMB predicted by EDE+WDM with a negative wdm are increased. This is attributed to the fact that negative values of the parameter wdm will postpone the moment of matter–radiation equality when other model parameters are fixed. On larger scales where l < 10, the curves are depressed when the value of the parameter wdm is negative due to the integrated Sachs–Wolfe effect. In addition, when the values of the parameter wdm turn positive, while keeping the values of other relevant cosmological parameters unchanged, the resulting effects are opposite to those observed with negative values of wdm. On the other hand, when fixing the wdm parameter, an increase in the amplitudes of the acoustic peaks in the CMB is observed in EDE+WDM when fEDE(zc) is non-zero. Additionally, the curves show an increase on larger scales under these circumstances. These effects differ from those arising from a positive or negative wdm.
Concentrating on the matter power spectrum with a fixed fEDE(zc) parameter, it becomes evident that a negative value of wdm results in a decrease, while positive values of wdm lead to an increase in the matter power spectrum; this is because the former situation results in a delay of matter and radiation equality, while the latter situation leads to an advance of matter and radiation equality. However, when fixing the wdm parameter, a non-zero value of fEDE(zc) leads to a different outcome, i.e. an increase in the large scale and a decrease in the small scale of the matter power spectrum. From these effects, one can infer that a non-zero fEDE(zc) itself is not the direct reason for the worsening S8 tension, considering it leads to a decrease in the small scale of the power spectrum. The problem arises from the EDE-ωdm degeneracy; it counteracts the effects of a non-zero fEDE(zc), and we hope that a negative wdm, in addition to a non-zero fEDE(zc), will solve this problem3(3It is worth mentioning that, if both the values of wdm and fEDE(zc) are non-zero, then it is not difficult to infer that, as long as both values are small enough, the observed results in the power spectrum will be a combination of the outcomes corresponding to only one of the two values being zero.).
3. Datasets and methodology
To extract the mean values and confidence intervals of the model parameters, we utilize the recent observational datasets described below.
Pantheon: the Pantheon catalogue of Supernovae Type Ia, comprising 1048 data points in the redshift region z ∈ [0.01, 2.3], is also considered.
Hubble constant: the latest local measurements of the Hubble constant obtained by Riess et al [1], i.e. H0 = 73.04 ± 1.04 are also included; we denote it as R22 hereafter.
S8 parameter: in addition to the above datasets, we include a Gaussianized prior on S8, i.e. S8 = 0.766 ± 0.017, chosen according to the joint analysis of KID1000+BOSS+2dLenS. (The use of a prior as an approximation for the full WL likelihoods has been demonstrated to be justified in the context of EDE models [77]. Nevertheless, in the EDE+WDM model, a comprehensive assessment of the likelihoods necessitates dedicated treatment of nonlinearities. Due to the absence of such tools, we restrict our analysis to the linear power spectrum, and make the assumption that the incorporation of this S8 prior correctly captures the constraints from the KID1000+BOSS+2dLenS likelihoods on EDE+WDM.)
To constrain the EDE+WDM model, we run a Markov Chain Monte Carlo using the public code MontePython-v3 [78, 79], interfaced with a modified version of the CLASS_EDE code [77], which is an extension to the CLASS code [80, 81]. We perform the analysis with a Metropolis–Hastings algorithm and consider chains to be converged using the Gelman–Rubin [82] criterion R − 1 < 0.03, assuming flat priors in the following parameter space:
$\begin{eqnarray}{ \mathcal P }=\{{\omega }_{b},{\omega }_{{\rm{dm}}},{\theta }_{s},{A}_{s},{n}_{s},{\tau }_{{\rm{reio}}},{{\rm{log}}}_{10}{z}_{c},{f}_{{\rm{EDE}}}({z}_{c}),{\theta }_{i},{w}_{{\rm{dm}}}\}.\end{eqnarray}$
We also adopt the Planck collaboration convention and model free-streaming neutrinos as two massless species and one massive with Mv = 0.06 eV.4. Results and discussion
In this section we constrain EDE+WDM, EDE4(4‘EDE' here represents the axion-like early dark energy model instead of early dark energy; the specific meaning of ‘EDE' will no longer be explicitly stated in the following content. Readers are encouraged to infer its significance based on the context.), ΛWDM and ΛCDM using CMB, BAO and Pantheon datasets, as well as R22 and the Gaussianized prior on S8 that we showed in the previous section in order to perform a statistical comparison between these models with the aim to focus on the tension on both H0 and S8.
In table 1, we present the observational constraints on EDE+WDM, EDE, ΛWDM and ΛCDM based on the CMB+BAO+Pantheon+H0+S8 dataset. Figure 2 shows the one-dimensional posterior distributions and two-dimensional joint contours at 68% and 95% confidence levels for the most relevant parameters of EDE+WDM, EDE, ΛWDM and ΛCDM. From table 1, one can see that the values of parameters wdm of both EDE+WDM and ΛWDM are very close to 0. This does not come as a surprise, since otherwise the LSS cannot be correctly formed. In addition, we find that the fitting results for the H0 parameter in EDE+WDM, i.e. ${71.11}_{-1.0}^{+0.88}$ at 68% confidence level, are similar to that of EDE, i.e. 71.10 ± 0.90 at 68% confidence level. It still maintains the ability to alleviate the Hubble tension to ∼1.4 σ, which is a notable improvement compared to that of ΛWDM, i.e. 69.95 ± 0.60 at 68% confidence level (with Hubble tension ∼2.6 σ) and that of ΛCDM, i.e. 68.79 ± 0.37 at 68% confidence level (with Hubble tension ∼3.9 σ). However, unfortunately, we find that the fitting result for the S8 parameter in EDE+WDM, i.e. ${0.814}_{-0.012}^{+0.011}$ at 68% confidence level, is also similar to that of EDE, i.e. 0.815 ± 0.011 at 68% confidence level, showing a ∼2.3 σ tension on the S8 parameter, which is still worse than that of ΛWDM, i.e. ${S}_{8}={0.8043}_{-0.0084}^{+0.0093}$ 68% confidence level (with S8 tension ∼2 σ) and that of ΛCDM, i.e. S8 = 0.7994 ± 0.0086 68% confidence level (with S8 tension ∼1.8 σ). We attribute the failure of the EDE+WDM model to alleviate the S8 tension to the lack of a positive(negative) correlation between the S8 parameter and the wdm parameter, as well as a sufficiently large negative(positive) value of wdm in this model. Although, as expected, there is a positive correlation between the wdm parameter and the σ8 parameter in this model, the negative correlation between the wdm parameter and Ωm leads to the lack of correlation between the S8 parameter and the wdm parameter. This is somewhat different from the situation in ΛWDM. Although in ΛWDM, the wdm parameter is also positively correlated with the σ8 parameter while being negatively correlated with the Ωm parameter, it still results in a slight positive correlation between the wdm parameter and the S8 parameter. We note that even though there is a slight positive correlation between the wdm parameter and the S8 parameter in the ΛWDM case, such a model still exacerbates the S8 tension compared to ΛCDM due to the positive value of wdm. To visually illustrate the tension among the fitting results of EDE+WDM, EDE, ΛWDM, ΛCDM and the two priors included in the datasets (R22 and the Gaussianized prior on S8), figure 3 reproduces the S8-H0 contours, incorporating the boundaries corresponding to one standard deviation for these two priors. It is worth mentioning that the above result is still obtained when using both H0 prior and S8 prior simultaneously. If we were to use only one of these priors, or none at all, the EDE+WDM model would face even greater Hubble and S8 tensions.
Figure 2. One-dimensional posterior distributions and two-dimensional joint contours at 68% and 95% confidence levels for the most relevant parameters of EDE+WDM, EDE, ΛWDM and ΛCDM are presented using the CMB+BAO+Pantheon+H0+S8 dataset. |
Figure 3. The S8-H0 contours at 68% and 95% confidence levels for EDE+WDM, EDE, ΛWDM and ΛCDM regarding the CMB+BAO+Pantheon+H0+S8 dataset, as well as the boundaries corresponding to one standard deviation for R22 and the Gaussianized prior. |
Table 1. The mean values and 1, 2 σ errors of the parameters for EDE+WDM, EDE, ΛWDM and ΛCDM are provided for the CMB+BAO+Pantheon+S8+H0 dataset, along with the AIC values for these four models. |
| Model | ΛCDM | ΛWDM | EDE | EDE+WDM |
|---|---|---|---|---|
| Dataset | CMB + BAO + Pantheon + S8 + H0 | |||
| 100 ωb | $2.265\pm {0.013}_{-0.025}^{+0.025}$ | $2.248\pm {0.014}_{-0.027}^{+0.027}$ | $2.286\pm {0.021}_{-0.040}^{+0.041}$ | $2.274\pm {0.024}_{-0.045}^{+0.048}$ |
| ωcdm | $0.11707\pm 0.000\,{81}_{-0.0016}^{+0.0016}$ | $0.11586\pm 0.000\,{89}_{-0.0017}^{+0.0018}$ | $0.1255\pm {0.0033}_{-0.0062}^{+0.0063}$ | ${{0.1233}_{-0.0060}^{+0.0027}}_{-0.0088}^{+0.020}$ |
| 100 θs | $1.04218\pm 0.000\,{29}_{-0.00058}^{+0.00055}$ | $1.04202\pm 0.000\,{28}_{-0.00056}^{+0.00055}$ | $1.04163\pm 0.000\,{38}_{-0.00073}^{+0.00071}$ | ${{1.04163}_{-0.00031}^{+0.00042}}_{-0.0012}^{+0.00082}$ |
| $\mathrm{ln}({10}^{10}{A}_{s})$ | $3.050\pm {0.015}_{-0.028}^{+0.030}$ | $3.045\pm {0.014}_{-0.028}^{+0.028}$ | $3.057\pm {0.015}_{-0.029}^{+0.029}$ | $3.053\pm {0.015}_{-0.030}^{+0.032}$ |
| ns | $0.9725\pm {0.0037}_{-0.0072}^{+0.0071}$ | $0.9701\pm {0.0036}_{-0.0072}^{+0.0071}$ | $0.9859\pm {0.0065}_{-0.013}^{+0.013}$ | ${{0.9818}_{-0.010}^{+0.0068}}_{-0.018}^{+0.022}$ |
| τreio | $0.0598\pm {0.0075}_{-0.014}^{+0.016}$ | ${{0.0551}_{-0.0075}^{+0.0066}}_{-0.014}^{+0.015}$ | $0.0571\pm {0.0075}_{-0.015}^{+0.015}$ | $0.0556\pm {0.0073}_{-0.014}^{+0.015}$ |
| ${\mathrm{log}}_{10}{z}_{c}$ | — | — | $3.65\pm {0.14}_{-0.22}^{+0.29}$ | $3.67\pm {0.22}_{-0.37}^{+0.54}$ |
| fEDE(zc) | — | — | ${{0.082}_{-0.025}^{+0.028}}_{-0.054}^{+0.049}$ | ${{0.066}_{-0.047}^{+0.025}}_{-0.071}^{+0.13}$ |
| θi | — | — | ${{2.71}_{-0.069}^{+0.23}}_{-0.47}^{+0.34}$ | ${{2.48}_{+0.027}^{+0.53}}_{-1.6}^{+0.62}$ |
| wdm | — | $0.00098\pm 0.000\,{41}_{-0.00079}^{+0.00079}$ | − | ${{0.00042}_{-0.00043}^{+0.00069}}_{-0.0014}^{+0.0013}$ |
| H0 | $68.79\pm {0.37}_{-0.70}^{+0.73}$ | $69.95\pm {0.60}_{-1.2}^{+1.2}$ | $71.10\pm {0.90}_{-1.8}^{+1.8}$ | ${{71.11}_{-1.0}^{+0.88}}_{-1.9}^{+1.9}$ |
| Ωm | $0.2966\pm {0.0046}_{-0.0091}^{+0.0090}$ | $0.2841\pm {0.0065}_{-0.013}^{+0.013}$ | ${{0.2948}_{-0.0048}^{+0.0042}}_{-0.0087}^{+0.0096}$ | ${{0.2901}_{-0.0097}^{+0.0064}}_{-0.016}^{+0.019}$ |
| σ8 | $0.8039\pm {0.0057}_{-0.011}^{+0.011}$ | $0.827\pm {0.011}_{-0.023}^{+0.022}$ | $0.8224\pm {0.0094}_{-0.018}^{+0.018}$ | $0.828\pm {0.014}_{-0.026}^{+0.025}$ |
| S8 | $0.7994\pm {0.0086}_{-0.017}^{+0.017}$ | ${{0.8043}_{-0.0084}^{+0.0093}}_{-0.017}^{+0.017}$ | $0.815\pm {0.011}_{-0.021}^{+0.021}$ | ${{0.814}_{-0.012}^{+0.011}}_{-0.022}^{+0.025}$ |
| | ||||
| ${\chi }_{\min }^{2}$ | 3840.76 | 3834.62 | 3827.58 | 3827.56 |
| ${\rm{\Delta }}{\chi }_{\min }^{2}$ | 0 | −6.14 | −13.18 | −13.20 |
| k | 28 | 29 | 31 | 32 |
| AIC | 3896.76 | 3892.62 | 3889.58 | 3891.56 |
| ΔAIC | 0 | −4.14 | −7.18 | −5.2 |
We also consider the Akaike information criterion (AIC) for model comparison among the EDE+WDM, EDE, ΛWDM and ΛCDM models. The AIC is defined as ${\chi }_{\min }^{2}+2k$, where k denotes the number of cosmological parameters. In practice, we are primarily interested in the relative values of AIC between two different models, denoted as ΔAIC = ${\rm{\Delta }}{\chi }_{\min }^{2}+2{\rm{\Delta }}k$. A model with a smaller AIC value is considered to be more favorable. In this work, the ΛCDM model serves as the reference model. From table 1, we find that the ΔAIC values for EDE+WDM, EDE and ΛWDM are −5.2, −7.18 and −4.14, respectively. Therefore, for the CMB+BAO+Pantheon+H0+S8 dataset, EDE is the most favorable model among the four models analyzed in this work.
Generally speaking, the observational data forces the wdm of EDE+WDM to be very close to 0, resulting in EDE+WDM not differing much from EDE. Additionally, a negative wdm lacks a common physical explanation. Also, as shown above, EDE+WDM is not favored by the observational data compared to EDE, as its AIC is worse than that of EDE. Furthermore, EDE+WDM cannot alleviate the Hubble tension and the S8 tension. In short, adding a new degree of freedom wdm is not a good choice.
5. Concluding remarks
The Hubble tension remains a complex issue in cosmology. The current consensus within the scientific community suggests that, if systematics are not the origin of the Hubble tension, modifications are necessary both in the early and late stages compared to ΛCDM [19]. Early-time modifications are deemed necessary to simultaneously increase the Hubble constant and reduce the sound horizon. However, such modifications fail to address the Hubble tension without exacerbating other tensions, such as the S8 tension. One of the most prominent early-time solutions is the axion-like EDE model, which considers a non-zero EDE fraction fEDE(zc) near the matter–radiation equality. This increases the DM density and exacerbates the S8 tension. Given that the axion-like EDE model is a typical early-time solution and a negative DM EoS is conducive to reduce the value of the σ8 parameter, we extend the axion-like EDE model by replacing the CDM in this model with DM characterized by a constant EoS wdm. We then impose constraints on this axion-like EDE extension model, i.e. EDE+WDM, along with three other models: the axion-like EDE model, i.e. EDE, ΛWDM and ΛCDM. These constraints are extracted from a comprehensive analysis incorporating data from the Planck 2018 CMB, BAO and the Pantheon compilation, as well as R22 and the Gaussianized prior on S8 determined through the joint analysis of KID1000+BOSS+2dLenS. Our findings indicate that, while EDE+WDM maintains EDE's ability to alleviate the Hubble tension to ∼1.4σ, it still moderately exacerbates the S8 tension. In fact, a non-zero DM EoS not only affects the value of σ8 but also shifts the value of Ωm; the combination of these two effects leads to the value of S8 remaining close to that of the CDM case. Finally, we employ AIC to make a model comparison between EDE+WDM, EDE, ΛWDM and ΛCDM regarding the CMB+BAO+Pantheon+H0+S8 dataset. Our analysis reveals that the EDE model is the most supported model among these four models regarding this dataset. Since EDE+WDM is not favored by the observational data compared to EDE, and EDE+WDM cannot alleviate both the Hubble tension and the S8 tension, we conclude that adding a new degree of freedom wdm is not a good choice.