Testing dark energy after pre-recombination early dark energy

In the studies on pre-recombination early dark energy (EDE), the evolution of Universe after recombination is usually regarded as ${\Lambda}CDM$-like, which corresponds that the equation of state of dark energy responsible for current accelerated expansion is $w=-1$. However, in realistic models, $w$ might be evolving. We consider the parametrizations of $w$ with respect to the redshift $z$ in Axion-like EDE and AdS-EDE models, respectively. We performed the Monte Carlo Markov chain analysis with recent cosmological data, and found that the bestfit $w(z)$ is compatible with $w_0=-1,w_a=0$ (the cosmological constant) and the evolution of $w$ is only marginally favored, which so has little effect on lifting the bestfit value of ${H_0}$.

is suppressed, where z * is the redshift at recombination, which naturally brings a higher H 0 (noting that CMB and BAO have fixed the angular scales where D * A is the angular diameter to last scattering surface), without spoiling fit to CMB and baryon acoustic oscillations (BAO) data, see also [32][33][34][35] for combined Planck+SPT dataset and [36,37] for Planck+ACT dataset. In particular, an Anti-de Sitter (AdS) phase around recombination can allow H 0 73km/s/Mpc, so the corresponding AdS-EDE model [20,21] can be 1σ consistent with local H 0 measurements. In Ref. [38], it has been found that the prerecombination solutions of the Hubble tension implies a scale-invariant Harrison-Zeldovich spectrum of primordial scalar perturbation, i.e. n s = 1 for H 0 ∼ 73km/s/Mpc.
The beyond-ΛCDM modifications after recombination have also been proposed e.g. [39][40][41][42][43][44][45][46][47][48][49][50], see also [51,52] for recent reviews. The current accelerated expansion of our Universe suggests the existence of dark energy at present, with the equation of state (EOS) w = p/ρ −1. In the studies on pre-recombination EDE, the evolution of Universe after recombination is usually regarded as ΛCDM-like, which corresponds to w = −1. However, in realistic models, w might be evolving, see e.g. [51] for a review, so having a consistency check for the ΛCDM model after recombination is significant.
In this paper, we will consider different parametrization of w in Axion-like EDE and AdS-EDE models, respectively, and perform the Markov Chain Monte Carlo (MCMC) analysis with PlanckCMB, BAO, Pantheon and H 0 dataset. This paper is organised as follows. In sect-II, we outline the parametrizations of w of current dark energy. In sect-III, we perform the MCMC analysis and present our results. We conclude in sect-IV.

II. PARAMETRIZATIONS OF DARK ENERGY
In ΛCDM model, the dark energy activates as a cosmological constant with w = −1.
Here, it is convenient to work with the parametrizations of w(z).

III. MCMC RESULTS FOR EDE MODELS
In this section we will confront the Axion EDE and AdS-EDE models (with CPL and oscillating parametrizations, respectively) with recent cosmological data. We modified the Montepython-3.3 [77,78] and CLASS [79,80] codes to perform the MCMC analysis.
Here, we will set the SH0ES result [1] as the Gaussian prior. The datasets consist of Planck2018 high-l and low-l TTTEEE as well as Planck lensing likelihoods [81], the BOSS DR12 [82] with its full covariant matrix for BAO as well as the 6dFGS [83] and MGS of SDSS [84] for low-z BAO, the Pantheon data [85]. We consider chains to be converged when the Gelman-Rubin statistic [68] satisfies R − 1 < 0.05.

A. Axion-like EDE model 1
In Axion-like EDE model [15], an oscillating axion field with the potential V (φ) ∝ (1 − cos[φ/f ]) n , naturally arising in the string theory, is responsible for EDE. At the critical resdshift, EDE starts to oscillate and dilutes away like a fluid with w = (n − 1)/(n + 1). It is noted that n = 3 is better for a higher best-fit H 0 [15,22].
We perform the MCMC analysis on the parameters set {ω b , ω cdm , H 0 , ln(10 10 A s ), n s , τ reio , log 10 a c , f ede , φ i , w 0 , w a }, where φ i is the initial value of EDE field, a c is when the field starts rolling and f ede is the energy fraction of EDE at a c . We also set n = 3 [15,22]. In Table-   In Fig.1, we see that the evolving w has little effect on H 0 and n s , compared with the model with w = −1 in Ref. [15]. However, the EDE parameters are constrained more tightly 1 We follow the name in Ref. [37]. in AdS-EDE model (see Table-II), the amplitude σ 8 of matter density fluctuation at low redshift is larger than local measurements [86,87], which, however, may be pulled lower by new physics beyond cold dark matter [88][89][90] We also follow the Ref. [18] and plot the difference ∆C l = C l,model − C l,ref in units of the cosmic variance per multipole for both parametrizations in CMB TT, EE and TE spectrum in Fig.2, where C l,ref is that for the ΛCDM model. Compared with the results in Ref. [15], the residual oscillations caused by 2 Here, we will not involve it. panel is that for the TT spectrum, the right one is for the EE and the bottom one is for the TE spectrum.

B. AdS-EDE model
In AdS-EDE model [20], we consider the potential as V (φ) = V 0 ( φ Mp ) 4 − V ads , which is glued to V (φ) = 0 at φ = ( V ads V 0 ) 1/4 M p by interpolation, where V ads is the depth of AdS well. The existence of AdS phase enables the density ρ ede of field dilutes away faster, and so allows a larger EDE fraction but without spoiling fit to CMB and BAO data, which makes AdS-EDE possible have a higher H 0 .
We perform the MCMC analysis on the parameters set {ω b , ω cdm , H 0 , ln(10 10 A s ), n s , τ reio , ln(1 + z c ), f ede , α ads , w 0 , w a }, where z c is the redshift when the field starts rolling, f ede is the energy fraction of EDE at z c , and α ads corresponds to V ads by V ads = α ads (ρ m (z c ) + ρ r (z c )), which will be fixed to 3.79 × 10 −4 , see [20]. In   In Table-II and Fig.3, we see again that the evolving w has little effect on H 0 and n s , compared with AdS-EDE model with w = −1 in Ref. [20]. It is also clear that the parameterizations of dark energy hardly affect the EDE parameters. The result on w(z) is still compatible with the cosmological constant (w 0 = −1, w a = 0), and only marginally favors the evolution of w. The difference ∆C l /σ CV is plotted in Fig.4. Compared with the results in Ref. [20], the parameterizations of dark energy not only maintain the character of oscillating in both TT and EE spectrum, but also strengthen the going-upwards of amplitude with l in TT spectrum and the bump at l ∼ 200 in EE spectrum. The residuals of the TT spectrum are within the cosmological variance for large scales (l 1500) and become comparable to the σ CV as l grows. However, the residuals of the EE and TE spectrum are larger than the σ CV in the l 1300 multipoles for CPL and l 850 for oscillating one, which may be detected.

C. Discussion
We present the H 0 −r * s contours for the CPL and oscillating parametrizations, respectively, with colored scatters as w 0 in Fig.5. We see that w 0 at 1σ contour is closed to -1, which is consistent with the cosmological constant. As expected, we have H 0 73 and w 0 −1 for AdS-EDE model.
We list the χ 2 of all datasets for different models in Table-III, respectively. We find that all best-fit models are improved over the best-fit ΛCDM model by ∆χ 2 ∼ −20. We see that both parameterizations reduce the χ 2 of Axion EDE model markedly, but slightly reduce that of the AdS-EDE model. This suggests that with the evolving w of current dark energy, Axion model seems to be favored over the AdS model. However, here since the AdS parameter α ads (relevant with the depth of AdS well) is fixed as 3.79×10 −4 , releasing α ads might gives better fit for the AdS-EDE model. It is also noted that although the χ 2 CM B in Axion EDE model is reduced, its fit to BAO dataset is worsened seriously.
To test whether the smoothing effect of lensing to the CMB power spectrum is consistent with that measured by the lensing reconstruction, the lensing potential is often scaled by a consistency parameter A L , theoretically A L = 1 [91]. It has been pointed out that Planck data seems favor a closed universe [92], while flat universe suggests A L = 1.180 ± 0.065   (Planck TT,TE,EE+lowE) [6], which is called the lensing anomaly. The oscillating parameterization of dark energy might help to explain this problem.
We set A L as a MCMC parameter, and show the posterior distribution of parameters set {H 0 , n s , w 0 , w a , A L } in Axion-like EDE model with the oscillating parametrization in Fig.6.
We see that the bestfit of H 0 is consistent with that in sect-III.A, and A L = 1.0421 +0.036 −0.050 , but with a slightly smaller Ω m , see Fig.7. Thus it is possible to seek for certain oscillating parametrizations to alleviate the lensing anomaly.

IV. CONCLUSIONS
In the studies on pre-recombination EDE, the evolution of Universe after recombination is usually regarded as ΛCDM-like, which corresponds to w = −1. However, in realistic models, w might be evolving or oscillating. Here, we investigate the effects of different parametrization of w in Axion-like EDE and AdS-EDE models, respectively.
We performed the MCMC analysis with recent cosmological data, and found that bestfit  [38], which suggests a scale-invariant Harrison-Zeldovich spectrum (n s = 1) for H 0 ∼ 73km/s/Mpc, see also Table-I,II. We also show ∆C l /σ CV for both parametrizations in CMB TT, EE and TE spectrum, and found that compared with the results in Refs. [15,20], the parameterization of w(z) basically maintains the shape of spectrum, but slightly amplifies the residual oscillations caused by EDE at small scale, which might be detectable. In addition, we also found that the oscillating parametrization could alleviate the lensing anomaly. Thus it is interesting to test other parametrizations of current dark energy in EDE cosmologies.