Kinetically Coupled Scalar Fields Model and Cosmological Tensions

In this paper, we investigate the kinetically coupled early dark energy (EDE) and scalar field dark matter to address cosmological tensions. The EDE model presents an intriguing theoretical approach to resolving the Hubble tension, but it exacerbates the large-scale structure tension. We consider the interaction between dark matter and EDE, such that the drag of dark energy on dark matter suppresses structure growth, which can alleviate large-scale structure tension. We replace cold dark matter with scalar field dark matter, which has the property of suppressing structure growth on small scales. We employed the Markov Chain Monte Carlo method to constrain the model parameters, our new model reveals a non-zero coupling constant of $0.030 \pm 0.026$ at a 68% confidence level. The coupled model yields the Hubble constant value of $72.38^{+0.71}_{-0.82}$ km/s/Mpc, which resolves the Hubble tension. However, similar to the EDE model, it also obtains a larger $S_8$ value compared to the $\Lambda$CDM model, further exacerbating the large-scale structure tension. The EDE model and the new model yield the best-fit values of $0.8316$ and $0.8146$ for $S_8$, respectively, indicating that the new model partially alleviates the negative effect of the EDE model. However, this signature disappears when comparing marginalised posterior probabilities, and both models produce similar results. The values obtained from the EDE model and the new model are $0.822^{+0.011}_{-0.0093}$ and $0.819^{+0.013}_{-0.0092}$, respectively, at a 68% confidence level.


INTRODUCTION
The standard ΛCDM model established over the past decades has received significant support from cosmological data.However, with the continuous improvement of experimental measurements, inconsistencies between some observational results and the predicted results of the ΛCDM model have become apparent.The most well-known inconsistencies are the Hubble tension and the large-scale structure tension.The Hubble tension refers to the discrepancy in the value of the Hubble constant derived from early universe observations based on the ΛCDM model and late-time measurements that are independent of the model Verde et al. (2019).Specifically, the best-fit of the ΛCDM model to the Planck 2018 cosmic microwave background (CMB) data gives 67.37 ± 0.54 km / s / Mpc Planck Collaboration et al. (2020), whereas the SH0ES collaboration, which uses the distance ladder method at low redshifts, gives 73.04 ± 1.04 km / s / Mpc Riess et al. (2022), with a statistical error of 4.8 between them.
Currently, it is unclear whether these two tensions stem from unknown statistical errors or inappropriate calibrations Blanchard & Ilić (2021); Mörtsell et al. (2022), for instance, the potential presence of crowding effects in the Cepheid photometry Freedman & Madore (2023), as well as the inaccuracy in the colour/dust correction of type Ia supernovae within calibration galaxies Wojtak & Hjorth (2022); or herald new physics beyond the standard model.In any case, it is still meaningful to explore alternative models to the ΛCDM model and investigate dark matter, dark energy, or dark radiation models that can alleviate these tensions.
Numerous approaches have been proposed to address the Hubble tension, which can be mainly classified into two categories: modifying the physics of the late Universe Guo et al. (2019); Li & Shafieloo (2019); Zhou et al. (2022); Liu et al. (2023d) and introducing new physics prior to recombination Karwal & Kamionkowski (2016); Poulin et al. (2018); Alexander & McDonough (2019); Lin et al. (2019); Berghaus & Karwal (2020); Ghosh et al. (2020); Liu et al. (2023a,b).However, these methods are confronted with various challenges.For instance, the late-time models are competitively constrained by independent observations at low redshift and generally cannot account for SH0ES measurements Benevento et al. (2020); Raveri (2020); Alestas & Perivolaropoulos (2021).Early-time models that achieve relative success in increasing  0 often exacerbate the tension in large-scale structure Hill et al. (2020); McDonough et al. (2022).Despite this, the relative success in relieving Hubble tension through modifications to early universe has stimulated further investigation into such models.In this paper, we will focus on one of the most absorbing cases in early models, namely early dark energy (EDE) model Karwal & Kamionkowski (2016);Lin et al. (2019); Poulin et al. (2019), and address its associated issues.
EDE is composed of an ultra-light scalar field, which only make a significant contribution during the epoch approaching matterradiation equality, and is negligible during other epochs.The addition of a new component decreases the sound horizon at recombination, thereby allowing an increase in the value of  0 while keeping the sound horizon angular scale unchanged.
The phenomenological parameter  c is generally used to represent the redshift at which the energy density of EDE reaches its peak, and  EDE is used to denote the ratio of EDE energy density to the total energy density at this redshift.The value of  EDE reaches approximately 10% can resolve the Hubble tension.
However, obtaining a larger value of  0 through the EDE scenario comes at the cost of inducing changes in other cosmological parameters, such as the density of dark matter  c , the scalar spectral index  s , and the amplitude of density perturbations  8 McDonough et al. (2022).This exacerbates the tension between the model and the large-scale structure data.
One natural idea to address these issues is to introduce the interaction between EDE and dark matter.The drag of dark energy on dark matter can inhibit structure growth, alleviate the large-scale structure tension exacerbated by EDE.
Previous studies have investigated the interaction between EDE and dark matter, such as utilising the Swampland Distance Conjecture Ooguri & Vafa (2007); Klaewer & Palti (2017); Palti (2019) to consider the exponential dependence of the EDE scalar on the mass of dark matter McDonough et al. (2022);Lin et al. (2023); Liu et al. (2023c).In this work, inspired by string theory, we consider the kinetic coupling between EDE and scalar field dark matter (SFDM).
SFDM, consisting of a light scalar field with a mass of approximately 10 −22 eV, is a viable alternative to cold dark matter (CDM) Ferreira (2021); Téllez-Tovar et al. (2022).SFDM forms condensates on small scales, which suppress structure growth, while on large scales it exhibits behavior consistent with that of CDM.Fig. 1 illustrates the equation of state evolution with respect to scale factor for SFDM of varying masses.The remaining cosmological parameters adopt the results of the Planck 2018 best-fitting ΛCDM model.In the early universe, SFDM behaves like a cosmological constant, then undergoes oscillations before ultimately evolving similarly to CDM, and the mass of the scalar field affects the onset time of the oscillation process.
The kinetic coupling between two scalar fields has been previously investigated, particularly in relation to the axio-dilaton models Alexander & McDonough (2019); Alexander et al. (2023).In this paper, we introduce a EDE scalar-dependent function to the kinetic term of SFDM, allowing for energy exchange between the two scalar fields.The specific Lagrangian is written as where  and  represent the EDE scalar and SFDM, respectively.The kinetic term of SFDM is multiplied by the coupling function  ().
We conducted a detailed study of the evolution equations for the kinetically coupled scalar fields (KCS) model, including both the background and perturbation parts.We incorporated various commonly used cosmological data, including the Planck 2018 CMB  , respectively, at a 68% confidence level.The structure of this paper is as follows: Section 2 introduces the KCS model, discussing its dynamics in both the background and perturbation levels, as well as its modifications to the original EDE model.In Section 3, we present the numerical results of the new model, including its impact on the evolution of the Hubble parameter and large-scale structure.Section 4 provides an introduction to the various cosmological data used in the MCMC analysis, along with the resulting parameter constraints.Finally, we summarise our findings in Section 5.

KINETICALLY COUPLED SCALAR FIELDS
Considering the Lagrangian shown in equation ( 1), we employed the potential form of EDE as follows Hill et al. (2020); Smith et al. (2020), where   represents the axion mass,   denotes its decay constant, and  Λ plays the role of a cosmological constant.For SFDM, we assume a simple quadratic potential, with   representing the mass of the SFDM.As for the coupling function, a natural choice is to assume an exponential form, where  represents a dimensionless constant that characterises the strength of the interaction, and   is the reduced Planck mass.

Background Equations
The equations of motion for the two scalar fields in a flat Friedmann-Robertson-Walker (FRW) metric can be expressed as where the dot denotes the derivative with respect to cosmic time,  is the Hubble parameter, and   represents the partial derivative of the EDE potential with respect to .It is easy to see that for a non-trivial kinetic coupling function  (), the EDE field has a source term proportional to  2 , while the SFDM field has a friction term proportional to .Therefore, if  is not equal to zero and the coupling constant  is positive, energy will be transferred from SFDM to EDE.
The energy density and pressure of SFDM can be expressed as where we abbreviate  () as . where 2 represents the fraction of the dark matter density.Combining with the Friedmann Equation, where   represents the total equation of state, defined as the ratio of the total pressure to the total energy density, the evolution equation for the new variable is thus formulated as, If we consider the energy density and pressure of the EDE scalar field, combined with equation ( 6) and using the equations of motion for both fields, we obtain the following continuity equations,  2017), which guarantees that the total stress tensor is covariantly conserved.Fig. 2 illustrates the evolution of the EDE scalar (left panel) and the fraction of the EDE energy density to the total energy density (right panel) with respect to the scale factor.We fix the mass of SFDM at 10 −22 eV and vary the coupling constant, while the other cosmological parameters, including the EDE parameter, are given by equation ( 23).Slight variations in the amplitude and phase of the evolution of the EDE scalar field are induced by different coupling constants.The sign of the coupling constant determines the direction of energy transfer.Positive coupling constants indicate energy flow from dark matter to dark energy, resulting in an increase in the energy density fraction of EDE, as demonstrated in the right panel of the figure.

Perturbution Equations
We employ the synchronous gauge to compute the perturbation equations for both EDE and SFDM.The line element is defined as, The perturbed Klein-Gordon equation in the Fourier mode are given by where    represents the second order partial derivative of the EDE potential with respect to .According to Ferreira & Joyce (1998); Hu (1998), the density perturbation, pressure perturbation, and velocity divergence of SFDM can be expressed as, We employ some new variables to compute the perturbation equations of SFDM Cedeño et al. (2017), One can derive the evolution equation for the new variable, where The relationship between the density perturbations, pressure perturbations, and velocity divergence of SFDM and the new variables can be deduced from equation ( 15),

Initial Conditions
In the early universe, Hubble friction in the scalar fields dominated and both EDE and SFDM were effectively frozen, undergoing a slow-roll process.The initial value of  can be set to zero, the term in the EDE equation containing  can be neglected during that period.Therefore, the equations of motion for EDE and SFDM simplify to an uncoupled form (the equation for the new variable  is an exception, and we will address this point later).We introduce the ratio of the initial value of the EDE scalar to the axion decay constant,   =   /   as the model parameter Hill et al. (2020); Smith et al. (2020).We refer to the initial conditions of uncoupled SFDM and make modifications, where Ω  represents the energy density fraction of all radiation components at present, and   denotes the initial value of the scale factor.
It should be noted that due to coupling, the initial value of  is multiplied by an additional factor, exp . For the derivation of the initial conditions of the original SFDM model, please refer to Ureña-López & Gonzalez-Morales (2016); Cedeño et al. (2017).Based on the current value of the dark matter energy density, we employ the widely used shooting method in the Boltzmann code CLASS1 Blas et al. (2011); Lesgourgues (2011) to obtain the initial value of Ω  .
For the perturbation equations of EDE and SFDM, we employ adiabatic initial conditions, with detailed descriptions provided in Smith et al. (2020) and Cedeño et al. (2017).

NUMERICAL RESULTS
Based on the description provided in the previous section, we modified the publicly available Boltzmann code CLASS Blas et al. (2011); Lesgourgues (2011) to incorporate the new model.
We present numerical results using the cosmological parameters borrowed from Table IV in  We present in Fig. 3 the evolution of the the differences in the Hubble parameter between the KCS model and the ΛCDM model for various coupling constants with redshift.The ΛCDM model is depicted by the black dotted line, while the results for the KCS model with coupling parameters 0, -0.03, and 0.03 are represented by the blue solid, orange dashed, and green dash-dotted lines, respectively.
The presence of an early dark energy component in the KCS models leads to higher values of the Hubble parameter compared to the ΛCDM model.Different coupling constants further influence the evolution of the Hubble parameter, with positive coupling constants indicating a transfer of energy density from dark matter to dark energy, resulting in an increase in the EDE energy density fraction (as shown in Fig. 2), leading to a larger Hubble parameter near the critical redshift  c .Conversely, negative coupling constants have the opposite effect, reducing the Hubble parameter.
Fig. 4 illustrates the redshift evolution of   8 .The 63 observed Redshift Space Distortion   8 () data points are gathered from Kazantzidis & Perivolaropoulos (2018).The KCS models yield higher   8 compared to the ΛCDM model, exacerbating the existing  8 tension.This primarily stems from the effects of the EDE component in the KCS model.However, when the coupling constant is non-zero, indicating an interaction between dark matter and dark energy, this result can change.In the case of a positive coupling constant (green dash-dotted line), the value of   8 is less than the result when the coupling constant is zero (blue solid line), while the effect is opposite when the coupling constant is negative (orange dashed line).
In Fig. 5, we present the linear matter power spectra for various models (top panel) and the differences between the power spectra of KCS model relative to the ΛCDM model (bottom panel).
Similar to previous discussions, the EDE component in the KCS model amplifies the matter power spectrum on small scales, exacerbating the large-scale structure tension.However, the interaction   In the KCS model, the impact of the EDE component leads to a higher value of   8 compared to the results obtained within the ΛCDM model.However, when the coupling constant is non-zero (indicating the existence of interaction between dark matter and dark energy), it serves to amend this effect.A positive value of the coupling constant mitigates the adverse effects introduced by EDE.The EDE component in the KCS model increases the matter power on small scales, but the interaction between dark matter and dark energy modifies this result, with positive coupling constants reducing this effect.
between dark matter and dark energy can rectify this outcome.A positive coupling constant leads to the flow of dark matter energy density towards dark energy, suppressing the growth of matter structure and resulting in a smaller power spectrum on small scales.

DATA AND METHODOLOGY
We conducted the Markov Chain Monte Carlo (MCMC) analysis using MontePython2 Audren et al. ( 2013); Brinckmann & Lesgourgues ( 2018) to obtain the posterior distribution of the model parameters.The MCMC chains were analysed using GetDist3 Lewis (2019).

Datasets
To perform the MCMC analysis, we used the following datasets: 1. CMB: The temperature and polarization power spectra obtained from the Planck 2018 low-ℓ and high-ℓ measurements, as well as the CMB lensing power spectrum Aghanim et al. (2020a,b);Planck Collaboration et al. (2020).
By combining CMB and BAO data, acoustic horizon measurements can be made at multiple redshifts, breaking geometric degeneracies and constraining the physical processes between recombination and the redshift at which BAO is measured.The supernova data obtained from the Pantheon sample significantly constrains late-time new physic within its measured redshift range.

SH0ES:
According to the latest SH0ES measurement, the value of the Hubble constant is estimated to be 73.04 ± 1.04 km / s / Mpc Riess et al. (2022).

DES-Y3:
The Dark Energy Survey Year-3 has yielded valuable data on weak lensing and galaxy clustering, from which the parameter  8 has been measured to be 0.776 ± 0.017 Abbott et al. (2022).
We employ the  0 measurements from SH0ES to alleviate the prior volume effect Smith et al. (2021) and assess the ability of the novel model to mitigate the tension between  0 local measurement and CMB inference result.Additionally, we incorporate the  8 data from DES-Y3 to investigate the efficacy of the model in easing the large-scale structure tension.

Results
The results of parameter constraints are presented in Table 1, where we utilised a comprehensive dataset comprising CMB, BAO, SNIa, SH0ES, and  8 from DES-Y3 data to individually constrain the ΛCDM, EDE, and KCS models.The upper segment of the table represents the parameters subjected to MCMC sampling, while the lower segment displays the derived parameters.
Firstly, it is noteworthy that the KCS model constrains the coupling constant  to be 0.030 ± 0.026 at a 68% confidence level, with a bestfit value of 0.0112.This indicates an interaction between dark matter and dark energy, specifically the conversion of dark matter to dark energy.
From the perspective of the Hubble constant, the EDE model and KCS model yield  0 values of 72.46 ± 0.86 km / s / Mpc and 72.38 +0.71  −0.82 km / s / Mpc, respectively, at a 68% confidence level, both of which exceed the value of 68.71 +0.35  −0.41 km / s / Mpc obtained by the ΛCDM model.Therefore, both the EDE model and KCS model demonstrate the capacity to address the Hubble tension.
However, the performance of both models on  8 is suboptimal.The best-fit values of  8 for the EDE and KCS models are 0.8316 and 0.8146, respectively, whereas ΛCDM yields a result of0.8016.Both models exacerbate the preexisting  8 tension.The result obtained from the KCS model is smaller than that from the EDE model, indicating that the new model partially alleviates the negative effect Table 1.The table illustrates the best-fit parameters and 68% confidence level marginalised constraints for the ΛCDM, EDE, and KCS models.These constraints are derived from comprehensive data sets including CMB, BAO, SNIa, SH0ES, and  8 measurements obtained from DES-Y3.The upper portion of the table consists of the cosmological parameters that were explored using MCMC sampling, while the lower portion presents the derived parameters. of the EDE model.However, when comparing the marginalised posterior probabilities, this feature vanishes, both models yield similar results.The constrained values of  8 obtained from the EDE model and the KCS model are 0.822 +0.011 −0.0093 and 0.819 +0.013 −0.0092 , respectively, at a 68% confidence level.

Model
These discussions can be visually depicted in the marginalised posterior distributions of various models, as illustrated in Fig. 6.The complete posterior distributions can be found in Fig. A1 in the Appendix section.
The penultimate row of Table 1 displays the  2 tot values of the different models.The discrepancies of the EDE and KCS models compared to the ΛCDM model are -11.74 and -12.78, respectively.Both models exhibit significantly reduced  2 tot values compared to the ΛCDM model, primarily due to the contribution from the SH0ES data.Furthermore, the KCS model demonstrates a lower  2 tot value than the EDE model, attributed to its smaller  8 value, which better aligns with the DES-Y3 data.
We also calculated the Akaike Information Criterion (AIC) for model comparison Akaike (1974), where  represents the number of fitting parameters.The results of different models compared to the ΛCDM model are presented in the last row of Table 1.We find that the EDE model has the lowest AIC, followed by the KCS model, both of which yield smaller values than that of the ΛCDM model.This indicates that both models outperform the ΛCDM model.Furthermore, compared to the EDE model, although the KCS model has a smaller  2 tot value, its introduction of additional parameters makes its performance inferior to that of the EDE model from the perspective of AIC.
Additionally, we conducted a renewed MCMC analysis after ex- cluding the SH0ES data to investigate its impact on the EDE parameters.Our constrained parameter results are illustrated in Table 2, where the utilised data only consist of CMB, BAO, SNIa, and  8 measurements obtained from DES-Y3.We observed that in the absence of SH0ES data, the phenomenological parameters  EDE obtained from the EDE model and KCS model constraints are very small, with best-fit values of 0.0548 and 0.0311, respectively, which are consistent with previous findings Smith et al. (2021); Poulin et al. (2023).It is noteworthy that in our MCMC analysis, we sampled over the axion mass   and decay constant   , while the EDE parameters  EDE and  c are derived parameters.As a result, our findings may differ from those of other studies.
In Fig. 7, we present the contour plots of the constraints on the  0 ,  8 , and EDE parameters for the KCS model and EDE model, with and without the inclusion of SH0ES data.
The impact of the SH0ES data on the EDE parameter is clearly evident.Excluding the SH0ES data leads to a reduction in the EDE parameter  EDE and an associated higher critical redshift  c , indicating a subtle signal of the existence of EDE.The results obtained from the KCS model and the EDE model are consistent.
In order to quantify the level of tension with the SH0ES data, we computed the following tension metric (in units of Gaussian ) Raveri & Hu (2019); Schöneberg et al. (2022), which calculates the difference in  2 when considering with and without SH0ES data.This metric effectively captures the non-Gaussianity of the posterior.The tension metric yields results of 4.4, 2.1, and 1.8 for the ΛCDM model, EDE model, and KCS model, respectively (it is noted that our data includes  8 from DES-Y3, hence yielding slightly different results from previous studies).
According to this criterion, we consider both the EDE model and the KCS model to be superior to the ΛCDM model.

CONCLUSIONS
In this paper, we investigate the interaction between early dark energy (EDE) and scalar field dark matter (SFDM), proposing a kinetically coupled scalar fields (KCS) model to alleviate cosmological tensions.The EDE model offers a resolution to the Hubble tension, but exacerbates the  8 tension.
In light of this, we propose an interaction between dark matter and dark energy, aiming to alleviate the  8 tension through the drag exerted by dark energy on dark matter.
In particular, motivated by the ability of SFDM to suppress structure growth on small scales, we replace cold dark matter with SFDM.Inspired by string theory, we consider a kinetic coupling between two scalar fields, where the kinetic term of SFDM is multiplied by an EDE scalar-dependent function.
We derived the evolution equations of the KCS model at both the background and perturbation levels, and investigated its impact on the evolution of the Hubble parameter, as well as on the structure growth and matter power spectrum.By using cosmological data from various sources, including the CMB, BAO, SNIa, the SH0ES measurement of the Hubble constant, and the DES-Y3 observations, we conducted the MCMC analysis.
We obtained a non-zero coupling constant  of 0.030 ± 0.026 at a 68% confidence level.This suggests the interaction between dark matter and dark energy, with energy transferring from dark matter to dark energy.The values of  0 of the EDE model and KCS model are 72.46 ± 0.86 km / s / Mpc and 72.38 +0.71  −0.82 km / s / Mpc, respectively, while the result of ΛCDM model is 68.71 +0.35  −0.41 km / s / Mpc.Both models alleviate the Hubble tension.
However, the corresponding cost is that both the EDE model and KCS model yield larger values of  8 , with their best-fit values being 0.8316 and 0.8146, respectively, which are greater than the result of the ΛCDM model, 0.8016.Compared to the EDE model, the result from the KCS model is smaller, indicating that the new model partially mitigates the negative effect of the original EDE model.However, this feature disappears when comparing the marginalised posterior probabilities, as both models yield similar results.The constrained results for  8 from the EDE model and the KCS model are 0.822 +0.011 −0.0093 and 0.819 +0.013 −0.0092 , respectively, at a 68% confidence level.
We computed the Δ 2 tot values of the EDE model and KCS model relative to the ΛCDM model, which are -11.74 and -12.78, respectively.This is primarily attributed to the contribution of SH0ES data.The  2 tot value of the KCS model is slightly smaller than that of the EDE model, mainly due to the fact that the KCS model yields a smaller  8 , which aligns better with the DES-Y3 data.We further calculated the AIC to compare the models, and the results indicate that the EDE model has the lowest AIC, followed by the KCS model.The introduction of additional parameters in the KCS model leads to its inferior performance compared to the EDE model.
We also briefly investigated the impact of the SH0ES data on the EDE parameter.When excluding the SH0ES data, the constrained  EDE parameters from both the EDE and KCS models are smaller, suggesting a weak signal of the presence of EDE, consistent with previous findings in the literature.
Building upon the EDE model, we considered the coupling between dark matter and dark energy to mitigate the negative effect inherent in the EDE model.However, the coupled model still falls short of resolving all cosmological tensions, and it is not fully supported by the data.Further efforts are still required to address these challenges comprehensively.
Figure1.The evolution of the equation of state of scalar field dark matter with respect to the scale factor  is investigated.The evolution of SFDM exhibits behavior similar to a cosmological constant in the early universe, then experiences oscillations, and finally evolves similarly to that of cold dark matter.

Figure 3 .
Figure 3.The evolution of the difference in the Hubble parameter between the KCS model and the ΛCDM model with redshift.The inclusion of an early dark energy component in the KCS models results in higher values of the Hubble parameter compared to the ΛCDM model.Furthermore, distinct coupling constants further influence the evolution of the Hubble parameter.Negative coupling constants decrease the Hubble parameter, while positive coupling constants increase it.

Figure 4 .
Figure 4.The redshift evolution of   8 for the various models is examined.In the KCS model, the impact of the EDE component leads to a higher value of   8 compared to the results obtained within the ΛCDM model.However, when the coupling constant is non-zero (indicating the existence of interaction between dark matter and dark energy), it serves to amend this effect.A positive value of the coupling constant mitigates the adverse effects introduced by EDE.

Figure 5 .
Figure 5.The linear matter power spectra of various models (top panel) and their deviations from the ΛCDM model (bottom panel) are presented.The EDE component in the KCS model increases the matter power on small scales, but the interaction between dark matter and dark energy modifies this result, with positive coupling constants reducing this effect.
The evolution of the EDE scalar (left panel) and the EDE energy density fraction (right panel) with respect to the scale factor is presented.The amplitude and phase of the EDE scalar are modulated by different coupling constants.Positive coupling constants indicate that energy flows from dark matter to dark energy, thereby increasing the energy density fraction of EDE, while negative coupling constants lead to a decrease in the energy density fraction of EDE.
The marginalised posterior distributions of the three models are presented.Both the EDE and KCS models yield larger values of  0 and hence, a larger  8 compared to the ΛCDM model.However, the KCS model partially alleviates the  8 tension when compared to the EDE model.The marginalised posterior distributions of the parameters  0 ,  8 , and EDE parameters for the KCS and EDE models are derived with and without the inclusion of SH0ES data.After the exclusion of SH0ES data, both models yield very small values for  EDE .

Table 2 .
After excluding SH0ES data, the best-fit values and 68% confidence level marginalised constraints on the parameters of the ΛCDM model, the EDE model, and the KCS model, using only CMB, BAO, SNIa, and  8 measurements obtained from DES-Y3 data.