# Constraining Holographic Dark Energy and Analyzing Cosmological Tensions

Xin Tang1,2 Yin-Zhe Ma3,4 Corresponding author: Y.-Z. Ma<EMAIL_ADDRESS>Wei-Ming Dai5 Hong-Jian He6,7 1Purple Mountain Observatory, CAS, No. 10
Yuanhua Road, Qixia District, Nanjing 210034, China
2School of Astronomy and Space Science, University of Science and Technology
of China, Hefei, Anhui 230026, China
3Department of Physics, Stellenbosch University, Matieland 7602, South Africa
4National Institute for Theoretical and Computational Sciences (NITheCS),
Stellenbosch, Matieland, 7602, South Africa
5School of Physical Science and Technology, Ningbo University, Ningbo 315211,
China
6Tsung-Dao Lee Institute & School of Physics and Astronomy, Key Laboratory for
Particle Astrophysics and Cosmology, Shanghai Jiao Tong University, Shanghai,
China
7Physics Department & Institute of Modern Physics, Tsinghua University,
Beijing, China

###### Abstract

We investigate cosmological constraints on the holographic dark energy (HDE)
using the state-of-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 $\Lambda$CDM with and
without $N_{\rm eff}$ (effective number of relativistic species) being treated
as a free parameter. We find that the HDE model can relieve the tensions of
$H_{0}$ and $S_{8}$ to certain degrees. With “Planck+ACT+BAO+RSD” datasets,
the constraints are $H_{0}\\!=\\!69.70\pm 1.39\ \mathrm{km\ s^{-1}Mpc^{-1}}$
and $S_{8}\\!=\\!0.823\pm 0.011$ in HDE model, which brings down the Hubble
tension down to $1.92\sigma$ confidence level (C.L.) and the $S_{8}$ tension
to $(1\\!-\\!2)\sigma$ C.L. By adding the R22 data, their values are improved
as $H_{0}=71.86\pm 0.93\,\mathrm{km\ s^{-1}Mpc^{-1}}$ and $S_{8}=0.813\pm
0.010$, which further brings the Hubble tension down to $0.85\sigma$ C.L. and
relieves the $S_{8}$ 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 $\Lambda$CDM and other extended models
(treating $N_{\rm eff}$ as free for fitting).

††preprint: APS/123-QED

##  I Introduction

The standard $\Lambda$ Cold-Dark-Matter ($\Lambda\mathrm{CDM}$) cosmological
model has successfully explained the observations of the cosmic microwave
background radiation (CMB Spergel et al. (2003, 2007); Komatsu et al. (2009,
2011); Hinshaw et al. (2013); Planck Collaboration et al. (2014, 2016, 2021)),
large-scale distributions of the galaxies Cole et al. (2005); Beutler et al.
(2011); Tegmark et al. (2004); Alam et al. (2021), and also the relic
abundance of the primordial elements formed during the Big-Bang
Nucleosynthesis (BBN) Cooke et al. (2016, 2018). However, it still has several
outstanding problems to be solved. One of the major observational
contradictions in recent years is the Hubble tension, which states that the
measured $H_{0}$ value from the CMB and the $\Lambda\mathrm{CDM}$ model is
lower than the distance-ladder measurements by more than $5\sigma$ confidence
level (C.L.) Verde et al. (2019). For local measurements, the value of $H_{0}$
is obtained using distances and redshifts measured from Cepheid in Type Ia
supernovae (SN Ia) hosts Riess et al. (2022), with no dependence on
cosmological models. The recent Cepheids-determined distance ladder from
“SH0ES” team gives $H_{0}\\!=\\!73.30\pm 1.04\,{\rm km}\,{\rm s}^{-1}\,{\rm
Mpc}^{-1}$ Riess et al. (2022). In constrast, Planck CMB experiment Planck
Collaboration et al. (2021) and the Atacama Cosmological Telescope (ACT Aiola
et al. (2020)) gave $H_{0}\\!=\\!67.4\pm 0.5\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$
and $67.9\pm 1.5\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ respectively, which is more
than $5\sigma$ C.L. smaller than the CMB-measured values. Another tension is
called “$S_{8}$ tension”, in which $S_{8}=\sigma_{8}\sqrt{\Omega_{\rm
m}/0.3}$, where $\sigma_{8}$ is the rms (root mean square) density
fluctuations on scale of $8\,h^{-1}{\rm Mpc}$, and $\Omega_{\rm m}$ is the
fractional matter density. Here $\sigma_{8}$ and $\Omega_{\rm m}$ are derived
parameters which rely on a specific cosmological model, both in CMB and weak
lensing measurements. The Planck+ACT measured $S_{8}$ as $S_{8}\\!=\\!0.831\pm
0.023$ Madhavacheril et al. (2023), which is systematically higher than the
weak lensing measurements, such as the Dark Energy Survey Year-3 Results
(DES-Y3, with $S_{8}\\!=\\!0.759^{+0.025}_{-0.023}$ Amon et al. (2022)), Kilo
Degree Survey (KiDS-1000, with $S_{8}\\!=\\!0.759^{+0.024}_{-0.021}$ Asgari,
Marika et al. (2021)), and Subaru Hyper Suprime-Cam Year-3 Results (HSC-Y3,
with $S_{8}\\!=\\!0.769^{+0.031}_{-0.034}$ Li et al. (2023)). Hence, the CMB-
derived quantity is systematically higher than that measured by using the
large-scale-structure datasets.

Different theoretical models beyond $\Lambda$CDM have been proposed to
physically explain the $H_{0}$ and $\sigma_{8}$ 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 ($r_{\rm
d}$) 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 Jedamzik and
Pogosian (2023)) and early dark energy (EDE Poulin et al. (2019)), which have
been proven unable to relieve the $S_{8}$ tension Jedamzik and Pogosian
(2023); Jedamzik et al. (2021). There is also some current research on EDE
extension models Gómez-Valent et al. (2021); Reeves et al. (2023); Cruz et al.
(2023), showing that they can alleviate the $S_{8}$ tension by introducing
additional conditions or parameters. Another solution is to have self-
interacting neutrinos Cyr-Racine and Sigurdson (2014); Kreisch et al. (2020);
He et al. (2020), which can delay the onset of neutrino free-streaming until
close to the radiation-matter equality and thus accommodates a larger Hubble
constant 111We also notice that Ref. He et al. (2020) provided an attractive
resolution to the Hubble tension by consistently realizing the neutrino self-
interactions, which respects the electroweak gauge symmetry and avoids
producing a too large $N_{\rm eff}$ by having $\Delta N_{\rm eff}\simeq
0.2\,$.. A recent such proposal added a fourth “self-interacting” neutrino,
but this model requires $N_{\rm eff}\\!\\!\sim\\!4$ which is much higher than
the constraints set by Planck and ACT Planck Collaboration et al. (2016);
Madhavacheril et al. (2023); Rida Khalife et al. . Other studies that intend
to solve the $H_{0}$ and $S_{8}$ include, the interacting dark energy model Di
Valentino et al. (2020), the small-scale baryon inhomogeneities (i.e.
clumping) Rashkovetskyi et al. (2021), and the mirror twin Higgs model with
three extra parameters Bansal et al. (2022) etc. For other proposed solutions
and theoretical guide, please refer to Knox and Millea (2020); Heisenberg et
al. (2022); Cardona and Sabogal (2023).

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 Cohen et al. (1999). 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 Li (2004), 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 $\Lambda$CDM Huang and Gong (2004); Ma et al.
(2009, 2010); Li et al. (2013). In recent years, the HDE fitting results from
Dai, Ma and He (2020) Dai et al. (2020) 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 $H_{0}=71.54\pm 1.78\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$, which is
consistent with the local $H_{0}$ measurement of 2019 LMC Cepheids data (R19)
Riess et al. (2019) at 1.4$\sigma$ 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 Riess et al. (2022)). We will also treat the
effective number of relativistic species ($N_{\rm eff}$) 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:

  * $\bullet$

to reconcile the Hubble tension between the recent CMB data (Planck and ACT)
and BAO from galaxy surveys (SDSS-DR16), and the distance-ladder measured
$H_{0}$ value (R22);

  * $\bullet$

to reconcile the $S_{8}$ tension between the CMB and BAO measured values and
the large-scale structure measured values (galaxy and CMB lensing);

  * $\bullet$

to allow a slightly larger value of $N_{\rm eff}$ that can accommodate the
standard model prediction ($N_{\rm eff}=3.046$ Mangano et al. (2005); de Salas
and Pastor (2016); Akita and Yamaguchi (2020); Froustey et al. (2020); Bennett
et al. (2021)).

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) ’t Hooft (1993), 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 Susskind (1995). 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
$\,L^{3}\Lambda^{4}\\!\leqslant\\!LM^{2}_{\rm Pl}\,$ Cohen et al. (1999),
where $\Lambda$ represents the UV cutoff and $M_{\rm Pl}=(8\pi G)^{-1/2}$ is
the reduced Planck mass. Thus, a UV cutoff scale $\Lambda$ 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: $\rho_{\rm
de}\\!\\!\sim\\!\Lambda^{4}\\!\leqslant\\!M^{2}_{\rm Pl}L^{-2}$. 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 Li
(2004):

$\displaystyle\rho_{\rm de}=3c^{2}M^{2}_{\rm Pl}L^{-2},$ (1)

where $3c^{2}$ is a numerical factor for the convenience of calculation, as
introduced in Li (2004) Li (2004). $L$ is the future event horizon, which is
an assumption chosen to obtain an accelerated expansion of the universe

$\displaystyle L=R_{\rm eh}=a\int_{t}^{\infty}\\!\frac{{\rm
d}t^{\prime}}{\,a(t^{\prime})\,}=a\int_{a}^{\infty}\\!\\!\frac{{\rm
d}a^{\prime}}{\,H(a^{\prime})a^{\prime 2}~{}}\,,$ (2)

with $a\\!=\\!a(t)$ and $a^{\prime}\\!=\\!a(t^{\prime})\,$.

Refs. Li (2004); Ma et al. (2009) further showed that Eq. (2) leads to the DE
equation of the state (EoS) being close to $-1$ . Then, from Friedmann
equations we have

$\displaystyle 3M^{2}_{\rm pl}H^{2}$ $\displaystyle=$ $\displaystyle\rho_{\rm
m}+\rho_{\rm de}\,,$ (3a) $\displaystyle\frac{{\rm d}H}{{\rm d}t}$
$\displaystyle=$ $\displaystyle-\frac{1}{2M^{2}_{\rm Pl}}\left(\rho_{\rm
m}+\rho_{\rm de}+\mathcal{P}_{\rm de}\right).~{}~{}~{}~{}$ (3b)

The energy conservation requires that the covariant derivative acting on the
energy-momentum tensor should vanish, and this leads to the following
conditions,

$\displaystyle\frac{{\rm d}\rho_{\rm m}}{{\rm d}t}+3H\rho_{\rm m}$
$\displaystyle=$ $\displaystyle 0\,,$ (4a) $\displaystyle\frac{{\rm
d}\rho_{\rm de}}{{\rm d}t}+3H\left(\rho_{\rm de}+\mathcal{P}_{\rm de}\right)$
$\displaystyle=$ $\displaystyle 0\,.$ (4b)

One can solve this dynamic system completely. Utilizing $w_{\rm
de}\equiv\mathcal{P}_{\rm de}/\rho_{\rm de}$, and $\rho_{\rm
de}=3H^{2}M^{2}_{\rm pl}\Omega_{\rm de}$, we can substitute them and obtain a
complete set of differential equations for $\Omega_{\rm de}(z)$ and $H(z)$ Ma
et al. (2009, 2010):

$\displaystyle\frac{{\rm d}\Omega_{\rm de}(z)}{{\rm d}z}+\frac{\Omega_{\rm
de}}{1+z}\left[\left(1-\Omega_{\rm de}\right)\left(1+\frac{2\sqrt{\Omega_{\rm
de}}}{c}\right)\right]\\!\\!$ $\displaystyle=$ $\displaystyle\\!0\,,$ (5a)
$\displaystyle\frac{{\rm d}H}{{\rm
d}z}+\frac{H(z)}{1+z}\left[\frac{\Omega_{\rm
de}}{2}\left(1+\frac{2}{c}\sqrt{\Omega_{\rm
de}}\right)-\frac{3}{2}\right]\\!\\!$ $\displaystyle=$
$\displaystyle\\!0\,.~{}~{}~{}~{}$ (5b)

The EoS parameter of the dark energy can be solved in the following form:

$\displaystyle w_{\rm de}=-\frac{1}{3}-\frac{2}{3}\frac{\rho^{1/2}_{\rm
de}}{\sqrt{3\,}cM_{\rm Pl}H\,}=-\frac{1}{3}-\frac{2}{3}\frac{\sqrt{\Omega_{\rm
de}(z)}\,}{c}.~{}~{}~{}$ (6)

As can be seen, at the early universe when $\Omega_{\rm de}\\!\ll\\!1$, we
have $w_{\rm de}\\!\to\\!-1/3$ which is greater than $-1\,$. In this case the
universe experiences a quintessential acceleration Ratra and Peebles (1988);
Wetterich (1988); Carroll (1998). But at the late stage when HDE is dominant,
$\Omega_{\rm de}\\!\rightarrow\\!1\,$, and thus $w_{\rm
de}\\!\rightarrow\\!-1/3\\!-\\!2/3c\,$. As long as $c$ is positive, the
expansion of the universe would accelerate according to the second Friedmann
equation, $\ddot{a}/a\\!=\\!-4\pi G\rho\,(1\\!+\\!3w)/3$ , 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 $\Lambda$CDM. As shown in Dai et al. (2020)
Dai et al. (2020), the then Planck+BAO12+R19 data 222BAO12 refers to the SDSS
data release 12 (DR12) of the measurements of $D_{\rm M}r_{\rm fid,d}/r_{\rm
d}$ at $z_{\rm eff}=0.38,0.51$ and $0.61$ Alam et al. (2017). R19 refers to
the Riess et al. (2019) Riess et al. (2019)’s measurement of $H_{0}$ by using
Large Magellanic Cloud Cepheid Standards. gave the best-fitting value as
$c=0.51\pm 0.02$ and $H_{0}=73.12\pm 1.14\,{\rm km}\,{\rm s}^{-1}{\rm
Mpc}^{-1}$, which resolves the Hubble tension completely. And also, we can
observe that $0<c<1$ due to the constraints of observations, i.e., $w_{\rm
de}<-1$ 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)
Dai et al. (2020). 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 Lewis et al. (2000) 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
($\rho_{\rm de}$) in the background density ($\rho$) of the perturbation
equations as well as its EoS $w_{\rm de}$, without considering its
perturbation term ($\delta_{\rm de}=0$.333Notice that perturbation of HDE
itself and its interactions are considered in Wang et al. (2017), and Chunshan
Lin (2021) Lin (2021) introduced a covariant local field theory for the HDE
model, providing a method to study its perturbations.

We then utilize the Markov Chain Monte Carlo (MCMC) technique from the public
code CosmoMC Lewis and Bridle (2002) 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 algorithmNeal (2005); Lewis
(2013). This approach is suitable for likelihood with numerous fast nuisance
parameters like Planck. The benchmark models we fit are $\Lambda$CDM and HDE.
We also add $N_{\mathrm{eff}}$ (the effective number of relativistic species)
as a free parameter to the fit of each model. So we have four models
($\Lambda$CDM, HDE, $\Lambda$CDM+$N_{\rm eff}$, and HDE+$N_{\rm eff}$) for the
present analysis. The $\Lambda$CDM model has six cosmological parameters
$\Omega_{\rm b}h^{2}$, $\Omega_{\rm c}h^{2}$, $100\theta_{\mathrm{MC}}$,
$\tau$, $\ln(10^{10}A_{\rm s})$, and $n_{\rm s}$, 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).

Table 1: Measurements used in the two datasets BAO and BAO+RSD for the present analysis. The 6dF $D_{\rm V}$ measurement is from Beutler et al. (2011) Beutler et al. (2011) and the other measurements are all from Table III of Alam et al. (2021) Alam et al. (2021). | 6dF | MGS | BOSS galaxy (DR12) | BOSS galaxy (DR12) | eBOSS LRG (DR16)
---|---|---|---|---|---
$z_{\mathrm{eff}}$ | 0.106 | 0.15 | 0.38 | 0.51 | 0.698
Dateset: BAO
$r_{\rm d}/D_{\rm V}(z_{\rm{eff}})$ | $0.336\pm 0.015$ |  |  |  |
$D_{\rm V}(z_{\rm{eff}})/r_{\rm d}$ |  | $4.47\pm 0.17$ |  |  |
$D_{\rm M}(z_{\rm{eff}})/r_{\rm d}$ |  |  | $10.23\pm 0.17$ | $13.36\pm 0.21$ | $17.86\pm 0.33$
$D_{H}(z_{\rm{eff}})/r_{\rm d}$ |  |  | $25.00\pm 0.76$ | $22.33\pm 0.58$ | $19.33\pm 0.53$
Dateset: BAO+RSD
$r_{\rm d}/D_{\rm V}(z_{\rm{eff}})$ | $0.336\pm 0.015$ |  |  |  |
$D_{\rm V}(z_{\rm{eff}})/r_{\rm d}$ |  | $4.47\pm 0.17$ |  |  |
$D_{\rm M}(z_{\rm{eff}})/r_{\rm d}$ |  |  | $10.27\pm 0.15$ | $13.38\pm 0.18$ | $17.65\pm 0.30$
$D_{H}(z_{\rm{eff}})/r_{\rm d}$ |  |  | $24.89\pm 0.58$ | $22.43\pm 0.48$ | $19.78\pm 0.46$
$f\sigma_{8}(z_{\rm{eff}})$ |  |  | $0.497\pm 0.045$ | $0.459\pm 0.038$ | $0.473\pm 0.041$
Figure 1: Posterior distributions of HDE+$N_{\mathrm{eff}}$ model for
selection parameters for four of the dataset combinations, at 68% C.L.
($1\sigma$) and 95% C.L. ($2\sigma$). The first two parameters ($N_{\rm eff}$
and $c$) are from the model’s base parameter set (Table 2), and the last four
parameters ($H_{0}$, $\Omega_{\rm m}$, $\sigma_{8}$ and $S_{8}$) are the
derived parameters.

(a) Data set: Planck+ACT+BAO+RSD (b) Data set: Planck+ACT+BAO+RSD+R22

Figure 2: 2D marginalized posterior contours in the plane of $H_{0}$ and $S_{8}$ (68% C.L. and 95% C.L.), by using two data combinations “Planck+ACT+BAO+RSD” [panel-(a)] and “Planck+ACT+BAO+RSD+R22” [panel-(b)] in the four models $\Lambda$CDM, $\Lambda$CDM+$N_{\mathrm{eff}}$, HDE, and HDE+$N_{\mathrm{eff}}$. The gray horizontal and vertical bands are the $1\sigma$ and $2\sigma$ regions of the measurements $S_{8}=0.769^{+0.031}_{-0.034}$ (at $1\sigma$ level) Li et al. (2023) and $H_{0}=73.04\pm 1.04\,\mathrm{km\ s^{-1}Mpc^{-1}}$ (at $1\sigma$ level) Riess et al. (2022). Table 2: Constraints of Planck+ACT+BAO+RSD and Planck+ACT+BAO+RSD+R22 datasets on the $\Lambda$CDM, $\Lambda$CDM+$N_{\rm eff}$, HDE, and HDE+$N_{\rm eff}$ 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. Parameters |  $\Lambda$CDM |  $\Lambda$CDM+$N_{\mathrm{eff}}$ | HDE | HDE+$N_{\mathrm{eff}}$
---|---|---|---|---
Dateset: Planck+ACT+BAO+RSD
$\Omega_{\rm b}h^{2}$ | $0.02236\pm 0.00012$ | $0.02215\pm 0.00017$ | $0.02242\pm 0.00012$ | $0.02220\pm 0.00019$
$\Omega_{\rm c}h^{2}$ | $0.11962\pm 0.00090$ | $0.11545\pm 0.00238$ | $0.11876\pm 0.00098$ | $0.11521\pm 0.00248$
$100\,\theta_{\rm MC}$ | $1.04111\pm 0.00028$ | $1.04159\pm 0.00038$ | $1.04122\pm 0.00027$ | $1.04163\pm 0.00039$
$\tau$ | $0.0543\pm 0.0070$ | $0.0537\pm 0.0070$ | $0.0569\pm 0.0076$ | $0.0555\pm 0.0074$
$\ln(10^{10}A_{\rm s})$ | $3.051\pm 0.013$ | $3.037\pm 0.015$ | $3.055\pm 0.014$ | $3.041\pm 0.017$
$n_{\rm s}$ | $0.9684\pm 0.0034$ | $0.9581\pm 0.0064$ | $0.9700\pm 0.0036$ | $0.9601\pm 0.0074$
$c$ | - | - | $0.58484\pm 0.04158$ | $0.56174\pm 0.04594$
$N_{\mathrm{eff}}$ |  $3.046$ (fixed) | $2.78\pm 0.14$ |  $3.046$ (fixed) | $2.80\pm 0.16$
$H_{0}[\mathrm{km\,s^{-1}\,Mpc^{-1}}]$ | $67.55\pm 0.39$ | $65.84\pm 0.99$ | $69.70\pm 1.39$ | $68.78\pm 1.52$
$\Omega_{\rm de}$ | $0.6874\pm 0.0054$ | $0.6810\pm 0.0064$ | $0.7077\pm 0.0118$ | $0.7079\pm 0.0122$
$\Omega_{\rm m}$ | $0.3126\pm 0.0054$ | $0.3190\pm 0.0064$ | $0.2923\pm 0.0118$ | $0.2921\pm 0.0122$
$\sigma_{8}$ | $0.8139\pm 0.0056$ | $0.8007\pm 0.0089$ | $0.8337\pm 0.0119$ | $0.8269\pm 0.0129$
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{\rm m}/0.3}$ | $0.831\pm 0.010$ | $0.826\pm 0.010$ | $0.823\pm 0.011$ | $0.816\pm 0.012$
Dateset: Planck+ACT+BAO+RSD+R22
$\Omega_{\rm b}h^{2}$ | $0.02249\pm 0.00012$ | $0.02264\pm 0.00014$ | $0.02241\pm 0.00012$ | $0.02227\pm 0.00019$
$\Omega_{\rm c}h^{2}$ | $0.11820\pm 0.00084$ | $0.12179\pm 0.00217$ | $0.11900\pm 0.00097$ | $0.11684\pm 0.00237$
$100\,\theta_{\rm MC}$ | $1.04131\pm 0.00027$ | $1.04092\pm 0.00034$ | $1.04119\pm 0.00028$ | $1.04143\pm 0.00038$
$\tau$ | $0.0587\pm 0.0072$ | $0.0580\pm 0.0072$ | $0.0553\pm 0.0072$ | $0.0538\pm 0.0073$
$\ln(10^{10}A_{\rm s})$ | $3.056\pm 0.014$ | $3.065\pm 0.015$ | $3.052\pm 0.014$ | $3.042\pm 0.017$
$n_{\rm s}$ | $0.9716\pm 0.0033$ | $0.9786\pm 0.0051$ | $0.9696\pm 0.0035$ | $0.9634\pm 0.0072$
$c$ | - | - | $0.53625\pm 0.02700$ | $0.51616\pm 0.03197$
$N_{\mathrm{eff}}$ |  $3.046$ (fixed) | $3.26\pm 0.12$ |  $3.046$ (fixed) | $2.89\pm 0.15$
$H_{0}[\mathrm{km\,s^{-1}\,Mpc^{-1}}]$ | $68.23\pm 0.37$ | $69.41\pm 0.75$ | $71.86\pm 0.93$ | $71.70\pm 0.93$
$\Omega_{\rm de}$ | $0.6963\pm 0.0049$ | $0.6989\pm 0.0050$ | $0.7248\pm 0.0076$ | $0.7280\pm 0.0080$
$\Omega_{\rm m}$ | $0.3037\pm 0.0049$ | $0.3011\pm 0.0050$ | $0.2752\pm 0.0076$ | $0.2720\pm 0.0080$
$\sigma_{8}$ | $0.8119\pm 0.0058$ | $0.8225\pm 0.0083$ | $0.8489\pm 0.0096$ | $0.8473\pm 0.0097$
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{\rm m}/0.3}$ | $0.817\pm 0.010$ | $0.824\pm 0.010$ | $0.813\pm 0.010$ | $0.807\pm 0.011$
Figure 3: Marginalized posterior distributions for $H_{0}$ using two data set
constraints D1 and D2, where D1 is Planck+ACT+BAO+RSD (solid curves) and D2 is
Planck+ACT+BAO+RSD+R22 (dashed curves). The vertical grey line is the R22
measurements of $H_{0}=73.04\pm 1.04\ \mathrm{km\ s^{-1}\ Mpc^{-1}}$, and the
bands response its regions allowed at 1$\sigma$ (dark grey) and 2$\sigma$
(light grey) C.L. Figure 4: Marginalized constraints on $S_{8}$ ($68\%$ 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 Table
2 of ACT DR6 lensing Madhavacheril et al. (2023). 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 lensing-alone data from Planck and ACT
DR6. The gray bands are the $1\sigma$ and $2\sigma$ confidence limits on the
model HDE+$N_{\mathrm{eff}}$ by using the dataset Planck+ACT+BAO+RSD+R22,
(i.e., $S_{8}=0.807\pm 0.011$ ($1\sigma$ C.L.) in the bottom line, right
column of Table 2).

Then, we explain each dataset used in our fitting:

  * $\bullet$

${\it Planck}$: We use the CMB high-$\ell$ TT, TE and EE plik likelihood,
low-$\ell$ temperature Commander likelihood, low-$\ell$ SimAll EE polarization
from the Planck 2018 data release N. Aghanim et al. (2020a), and Planck 2018
CMB lensing power spectrum N. Aghanim et al. (2020b). In the following, we use
Planck to denote the combination of all Planck data mentioned here.

  * $\bullet$

ACT: We use the likelihood computed from the CMB power spectrum on scales of
$\,350\\!<\\!\ell\\!<\\!4000$ measured by the ACTPol instrument of the Atacama
Cosmology Telescope, derived from seasons 2013-2016 of temperature and
polarization data release 4 (DR4) Aiola et al. (2020).

  * $\bullet$

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 $D_{\rm V}$ at
$z_{\mathrm{eff}}=0.106$ Beutler et al. (2011) and the “Main Galaxy Sample”
(MGS) from Sloan Digital Sky Survey Data Release 12 (SDSS-DR12) measures
$D_{\rm V}$ at $z_{\mathrm{eff}}=0.15$ Ross et al. (2015). We also add $D_{\rm
M}(z_{\mathrm{eff}})/r_{\rm d}$ and $D_{H}(z_{\mathrm{eff}})/r_{\rm d}$
measurements at $z_{\mathrm{eff}}=0.38$ and $0.51$ from SDSS-DR12 (BOSS CMASS
galaxies) Alam et al. (2017) and same quantities at $z_{\mathrm{eff}}=0.698$
from SDSS-IV DR16 eBOSS Luminous Red Galaxies (LRG) Alam et al. (2021). The
five samples at corresponding redshifts are shown Table 1. The two SDSS-DR12
samples spanning $\,0.2\\!<\\!z\\!<\\!0.6\,$ are correlated due to overlapping
in $z$, independent of the DR16 results occupying $\,0.6\\!<\\!z\\!<\\!1.0\,$.
Hence, we include the full covariance matrix of $D_{\rm
M}(z_{\mathrm{eff}})/r_{\rm d}$ and $D_{H}(z_{\mathrm{eff}})/r_{\rm d}$ in our
CosmoMC likelihoods (see also Table III in Alam et al. (2021)). We use “BAO”
to denote the BAO-only dataset at these five effective redshifts (Table 1).

The $r_{\rm d}=147.8\,{\rm Mpc}$ mentioned above is the sound horizon at the
drag epoch, where the value is set by the fiducial cosmological parameters in
the spatially-flat $\Lambda$CDM model 444The detailed values are listed in the
first row of Table I in Ref. Gil-Marín et al. (2020). $D_{\rm M}(z)$ is the
co-moving angular diameter distance, and $D_{H}(z)=c/H(z)$ is the Hubble
radius. $D_{\rm V}(z)$ is the spherically averaged distance defined as $D_{\rm
V}(z)=\big{(}zD_{H}(z)D^{2}_{\rm M}(z)\big{)}^{1/3}$, 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 Alam et
al. (2021).

  * $\bullet$

BAO+RSD: We also adopt the BAO measurements with additional measurement of
$f\sigma_{8}(z)$ from redshift-space-distortion (RSD). The RSD arises from the
peculiar velocities of distant galaxies due to the inhomogeneous distribution
of matters. $f\sigma_{8}(z)$ is the amplitude of the velocity power spectrum,
where $\sigma_{8}(z)$ is the amplitude of linear matter fluctuations on the
comoving scale of $8\,h^{-1}{\rm Mpc}$, and $f(z)\equiv\partial\ln
D/\partial\ln a$ is the linear growth rate of structure ($a=1/(1+z)$ is the
scale factor). The RSD measurement provides direct measurement on
$f\sigma_{8}(z)$ Kaiser (1987); Alam et al. (2017, 2021).

Hence, we use the joint likelihood of $D_{\rm M}(z_{\mathrm{eff}})/r_{\rm d}$,
$D_{H}(z_{\mathrm{eff}})/r_{\rm d}$, and $f\sigma_{8}(z_{\mathrm{eff}})$ at
$z_{\mathrm{eff}}\\!=\\!0.38$, $0.51$ and $0.698$ as determined from the
combined BAO and RSD likelihoods Gil-Marín et al. (2020), and also the two 6dF
and MGS samples (Table 1). The measurements at $z_{\mathrm{eff}}\\!=\\!0.38$
and $0.51$ are taken from the data presented in Table 8 of SDSS-DR12 Alam et
al. (2017), and the measurement at $z_{\rm eff}\\!=\\!0.698$ is taken from
SDSS-DR16 Alam et al. (2021). Our CosmoMC likelihoods include the full
covariance matrix of $D_{\rm M}(z_{\mathrm{eff}})/r_{\rm d}$,
$D_{H}(z_{\mathrm{eff}})/r_{\rm d}$ and $f\sigma_{8}(z)$, as $f\sigma_{8}(z)$
correlates with the BAO measurement at the same redshift but the two BAO SDSS-
DR12 results correlate in between redshifts. We use BAO+RSD to denote the
$D_{\rm M}(z_{\mathrm{eff}})/r_{\rm d}$, $D_{H}(z_{\mathrm{eff}})/r_{\rm d}$,
and $f\sigma_{8}(z_{\mathrm{eff}})$ measurements combined with the two $D_{\rm
V}$ measurements from 6dF and MGS/SDSS-DR7 (five samples in total), which are
presented at Table 1.

  * $\bullet$

R22: We use the measurement of Hubble constant from SH0ES team as
$H_{0}\\!=\\!73.04\pm 1.04$ km $\mathrm{s^{-1}Mpc^{-1}}$ by Riess et al. Riess
et al. (2022). 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 $\Lambda$CDM,
HDE, $\Lambda$CDM+$N_{\mathrm{eff}}$, and HDE+$N_{\mathrm{eff}}$ (the latter
two treat $N_{\rm eff}$ 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
$0.01\\!\\!<\\!\\!z\\!<\\!2.3$ Scolnic et al. (2018). The reason is the
possible inconsistency of the parameter constraints due to the large
correlation between high-$z$ ($z\\!>\\!0.2$) and low-$z$ ($z\\!<\\!0.2$)
samples of Pantheon subsamples. As a result, the joint constraints on HDE’s
$w_{\rm de}(z\\!=\\!0)$ and $H_{0}$ from individual subsamples and the full
Pantheon sample become inconsistent with each other (as shown by Fig. 3 of Dai
et al. (2020) Dai et al. (2020)). In Ref. Di Valentino et al. (2021), 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 $2.26$Scolnic et al. (2022), 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

Figure 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 $\Lambda$CDM, $\Lambda$CDM+$N_{\mathrm{eff}}$, HDE, and HDE+$N_{\mathrm{eff}}$ 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 $\Lambda$CDM CMB power spectra from the baseline Planck TT, TE, EE+lowE+lensing ($2\leqslant\ell\leqslant 2508$). Figure 6: Constraints on the parameter $N_{\rm eff}$ in $\Lambda$CDM+$N_{\mathrm{eff}}$ model (dashed curves) and HDE+$N_{\mathrm{eff}}$ model (solid curves) from the three combined data sets. The gray vertical dashed line with the band is the constraint from Planck+ACT data on $\Lambda$CDM+$N_{\rm eff}$ model (ACT DR4 Aiola et al. (2020)): $N_{\mathrm{eff}}=2.74\pm 1.3\,$. The gray solid line indicates $N_{\mathrm{eff}}=3.046$ as computed in the standard model of particle physics Mangano et al. (2005); de Salas and Pastor (2016); Akita and Yamaguchi (2020); Froustey et al. (2020); Bennett et al. (2021). Table 3: Flat priors of cosmological parameters in MCMC. Parameters | Prior range
---|---
$c$ | $[0.2,2]$
$\Omega_{\rm b}h^{2}$ | $[0.005,0.1]$
$\Omega_{\rm c}h^{2}$ | $[0.001,0.99]$
$100\theta_{\mathrm{MC}}$ | $[0.5,10]$
$\tau$ | $[0.01,0.8]$
$\ln(10^{10}A_{\rm s})$ | $[1.61,3.91]$
$n_{\rm s}$ | $[0.8,1.2]$
$N_{\rm eff}$ | $[0.05,10]$

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 $N_{\rm eff}$ as a free parameter. There are eight free parameters in
our fitting of the HDE and $\Lambda$CDM models where the priors are listed in
Table 3.

Figure 1 shows the marginalized constraints on cosmological parameters
$N_{\mathrm{eff}}$, $c$, $H_{0}$, $\Omega_{\rm m}$, $\sigma_{8}$ and $S_{8}$
of model HDE+$N_{\mathrm{eff}}$. Here the first two parameters are the base
parameters, and the $H_{0}$, $\Omega_{\rm m}$, $\sigma_{8}$ 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 $N_{\mathrm{eff}}$ 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 $N_{\mathrm{eff}}$ tends to be lower if
including the ACT dataset. Further information about the constraints on
$N_{\mathrm{eff}}$ can be found in Sec. IV.3.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. Riess et al. (2022) (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 $N_{\rm eff}$ respectively.

###  IV.1 The Hubble Tension

We first examine the HDE model’s capability of reconciling Hubble tension with
different dataset combinations. To compare it with $\Lambda$CDM model, we plot
the posterior distributions in the $H_{0}\\!-\\!S_{8}$ plane for the four
models $\Lambda$CDM, $\Lambda$CDM+$N_{\mathrm{eff}}$, HDE and
HDE+$N_{\mathrm{eff}}$ with two data set combinations “Planck+ACT+BAO+RSD”
(left panel) and “Planck+ACT+BAO+RSD+R22” (right panel) in Fig. 2. In Fig. 2,
the measurements of $S_{8}$ from HSC-Y3 real-space galaxy correlation function
($S_{8}\\!=\\!0.769^{+0.031}_{-0.034}$ Li et al. (2023)) and the R22 result
($H_{0}\\!=\\!73.04\pm 1.04\,{\rm km}{\rm s}^{-1}{\rm Mpc}^{-1}$ Riess et al.
(2022)) 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 $\Lambda$CDM 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 2). 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 $N_{\mathrm{eff}}$ as a free parameter ($\Lambda$CDM+$N_{\rm
eff}$ and HDE+$N_{\rm eff}$), showing as the gray and blue contours. There is
no significant effect to relieve the $H_{0}$ tension, but we notice that
HDE+$N_{\mathrm{eff}}$ (blue contours) still provide a better fit than
$\Lambda$CDM+$N_{\mathrm{eff}}$ (grey contours).

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 $69.70\pm 1.39\
\mathrm{km\,s^{-1}Mpc^{-1}}$ in the HDE model and $67.55\pm
0.39\,\mathrm{km\,s^{-1}Mpc^{-1}}$ in the $\Lambda$CDM model (see also Table
2). Thus, HDE brings the $H_{0}$ measurement closer to R22, with a tension of
only $1.92\sigma$ instead of the $4.94\sigma$ tension in the $\Lambda$CDM
model 555We calculate the distance in the best-fitting value by quantifying
the difference of the best-fits with respect to the square root of quadrature
sum of each errors, i.e.,
$(73.04\\!-\\!69.70)/\sqrt{1.04^{2}\\!+\\!1.39^{2}}\\!=\\!1.92\sigma$ and
$(73.04-67.55)/\sqrt{1.04^{2}\\!+\\!0.39^{2}}=4.94\sigma$.. By including the
R22 in the likelihood, the tension in the HDE model is further reduced. The
$H_{0}$ is measured to be $71.86\pm 0.93\,\mathrm{km\,s^{-1}Mpc^{-1}}$, which
brings the tension further down to $0.85\sigma$, 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 $\Lambda$CDM (green curves), and are much closer to the R22
measurement (gray vertical band).

###  IV.2 The $S_{8}$ Tension

We present in Fig. 4 the constraints on $S_{8}$ in the four models using
different weak lensing and CMB datasets, where blue horizontal bars are the
$\pm 1\sigma$ measurements we take from Madhavacheril et al. (2023), and red
bars are the various constraints by this work. We can see that the $S_{8}$
values are significantly less than $0.8$ by different galaxy lensing
observations (e.g., HSC-Y3 Li et al. (2023), KiDS Asgari, Marika et al.
(2021), and DES-Y3 Amon et al. (2022)Madhavacheril et al. (2023)), but the
projected value of the $\Lambda$CDM is $0.831\pm 0.010$ (Planck+ACT+BAO+RSD),
which is significantly higher than the lensing measurements. If we replace
$\Lambda$CDM model by the HDE model, then the $S_{8}$ can be brought down to
$0.823\pm 0.011$ (see Table 2). Therefore, with respect to the KiDS-1000
Asgari, Marika et al. (2021) ($S_{8}=0.759^{+0.024}_{-0.021}$), DES-Y3 Amon et
al. (2022) ($S_{8}\\!=\\!0.759^{+0.025}_{-0.023}$), and Hyper Suprime-Cam
Year-3 Results (HSC-Y3) Li et al. (2023) ($S_{8}\\!=0.769^{+0.031}_{-0.034}$),
the discrepancy can be lowered down to $2.56\sigma$, $2.42\sigma$, and
$1.57\sigma$, respectively, which considerably reduces the $S_{8}$ tension.

After further including R22 data, the HDE+$N_{\mathrm{eff}}$ model predicts
the $S_{8}$ value as $0.807\pm 0.011$, $N_{\mathrm{eff}}\\!=\\!2.89\pm 0.15$,
and the parameter $c$ as $0.52\pm 0.03$, 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 $\Lambda$CDM model.

###  IV.3 The $N_{\mathrm{eff}}$ Constraints

The consideration of $N_{\mathrm{eff}}$ as a varying parameter stems from the
substantial deviation between the derived value of $N_{\rm eff}=2.74\pm 1.3$
Aiola et al. (2020) obtained from fitting ACT data with the
$\Lambda\mathrm{CDM}$+$N_{\mathrm{eff}}$ model and the theoretical value of
3.046. Our primary aim is to explore whether replacing $\Lambda\mathrm{CDM}$
with HDE can mitigate this disparity, while also assessing its impact on
cosmological discrepancies.

We further examine the effect of including $N_{\rm eff}$ 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:

$\displaystyle\rho_{\rm r}\,=\,\rho_{\gamma}+\rho_{\nu}\,,$ (7)

where in natural units,

$\displaystyle\rho_{\gamma}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{15}T^{4}_{\gamma}\,,$ (8a)
$\displaystyle\rho_{\nu}$ $\displaystyle=$
$\displaystyle\frac{7}{8}\frac{\pi^{2}}{15}N_{\rm eff}T^{4}_{\nu}\,,$ (8b)

are the energy densities of photons and neutrinos. The factor $7/8$ arises
from the Fermi-Dirac distribution, $N_{\mathrm{eff}}$ is the effective number
of neutrino species, which does not need to be an integer Komatsu et al.
(2009, 2011); Aich et al. (2020); Dodelson and Schmidt (2020). In fact,
$N_{\mathrm{eff}}=3.046$ in the standard model of particle physics Mangano et
al. (2005); de Salas and Pastor (2016); Akita and Yamaguchi (2020); Froustey
et al. (2020); Bennett et al. (2021).

Thus, because neutrinos contribute to the total energy density and change the
anisotropic stress of the baryon fluid, to increase $N_{\mathrm{eff}}$ can
enhance the expansion rate, reduce the CMB damping tail, and also shift the
peak and trough positions of the CMB angular power spectra Hou et al. (2013);
Aich et al. (2020). Hence, the accurate measurements of CMB power spectra can
essentially constrain the $N_{\rm eff}$ value to a high precision, which is a
primary science target for this and next generations of CMB experiments Planck
Collaboration et al. (2021); Aiola et al. (2020); Dutcher et al. (2021);
Abazajian et al. (2019).

In Fig. 5, we plot the CMB temperature (TT) and E-mode (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
$\Lambda$CDM 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 2) with combined “Planck+ACT+BAO+RSD+R22” dataset and
that of $\Lambda$CDM 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 $\Lambda$CDM model. One can see that the HDE model (blue
solid line) matches the data slightly better than the standard model
$\Lambda$CDM (black solid line), because the additional wiggles with the
suitable amplitude appeared in the HDE model can better match the up-and-down
features of the binned angular power spectra (red data dots) than the
$\Lambda$CDM. With the additional $N_{\mathrm{eff}}$, the HDE model is boosted
further up (blue dashed line), and the $\Lambda$CDM model is pushed downwards
(black dashed line), making the fits less good than that without the $N_{\rm
eff}$ parameter.

This situation is also evident in Table 4, where we use the $\chi^{2}$, Akaike
information criterion (AIC) and Bayesian information criterion (BIC) to
quantify the “goodness of fit”. The third column is the minimal $\chi^{2}$ of
each model, and $\Delta\chi^{2}$ (the fourth column) is the relative
difference to the best-fitting $\Lambda$CDM model. The total $\chi^{2}$ of the
combined dataset can be written as $\chi^{2}=\chi^{2}_{\it
Planck}+\chi^{2}_{\rm ACT}+\chi^{2}_{\mathrm{BAO+{RSD}}}+\chi^{2}_{H_{0}}$.
One can see that, by switching from $\Lambda$CDM model to the HDE model, the
$\chi^{2}_{\rm min}$ 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 $N_{\rm eff}$ does not alter the goodness of fit
significantly. Because HDE has one additional parameter ($c$ in Eq. (1)) than
the base $\Lambda$CDM, 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
$\Delta\mathrm{AIC}=\Delta\chi^{2}_{\rm min}+2\Delta k$ and
$\Delta\mathrm{BIC}=\Delta\chi^{2}_{\rm min}+\Delta k\ln N$. Here,
$\Delta\chi^{2}_{\rm min}$ and $\Delta k$ represent the difference in the
minimum $\chi^{2}$ 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”, $N=10520$
666$N=N_{\it Planck}+N_{\rm ACT}+N_{\rm BAO+RSD}+N_{\rm R22}$. For $N_{\it
Planck}$, data points combine high-$\ell$ TT ($30\leq\ell\leq 2508$), TE
($30\leq\ell\leq 1996$) and EE ($30\leq\ell\leq 1996$), low-$\ell$ temperature
($0\leq\ell\leq 29$), low-$\ell$ EE ($2\leq\ell\leq 27$) and lensing
($8\leq\ell\leq 400$) spectrum, so $N_{\it
Planck}=2478+1966+1966+29+27+392=6858$; ACT data points from CMB power
spectrum on scales of $350<\ell<4000$, so $N_{\rm ACT}=3650$; $N_{\rm
BAO+RSD}=11$ verified as in Table 1 and $N_{\rm R22}=1$. Therefore,
$N=6858+3650+11+1=10520$.. One can see that, even for AIC and BIC, the HDE
model provides a better fit compared to the $\Lambda$CDM 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 $N_{\rm eff}$ makes the fit less good
because it turns to increase the AIC and BIC values.

Table 4: Comparison of $\chi^{2}$ fits for different models. Date Set | Model | $\chi^{2}_{\mathrm{min}}$ | $\Delta\chi^{2}_{\mathrm{min}}$ | $\Delta\mathrm{AIC}$ | $\Delta\mathrm{BIC}$
---|---|---|---|---|---
Planck+ACT +BAO+RSD +R22 | $\Lambda\mathrm{CDM}$ | 3058.77 | 0 | 0 | 0
$\Lambda$CDM+$N_{\mathrm{eff}}$ | 3057.66 | -1.107 | 0.893 | 8.154
HDE | 3049.42 | -9.352 | -7.352 | -0.091
HDE+$N_{\mathrm{eff}}$ | 3049.63 | -9.142 | -5.142 | 0.119

We now compare the constraints on $N_{\rm eff}$ value. We compare the fitting
results of $N_{\mathrm{eff}}$ 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 $\Lambda$CDM+$N_{\rm eff}$ model
(ACT DR4 Aiola et al. (2020)): $N_{\rm eff}=2.74\pm 1.3$, where the gray solid
line is the prediction of $3.046$ from the standard model of particle physics
(SMPH) Mangano et al. (2005); de Salas and Pastor (2016); Akita and Yamaguchi
(2020); Froustey et al. (2020); Bennett et al. (2021). In Fig. 6, the change
in the $N_{\mathrm{eff}}$ value is insignificant for HDE relative to
$\Lambda$CDM (red and grey curves), without taking into account the R22
measurement. Once R22 is added, $N_{\mathrm{eff}}$ is shifted from $2.78\pm
0.14$ (orange dashed curve) to $3.26\pm 0.12$ (green dashed curves). But for
HDE, $N_{\mathrm{eff}}$ increases from $2.80\pm 0.16$ (orange solid curve) to
$2.89\pm 0.15$ (green solid curves), which is a much smaller shift than the
$\Lambda$CDM. Nonetheless, for all these combined datasets and theoretical
models, the resultant constraints on $N_{\rm eff}$ are consistent with SMPH’s
prediction within $2\sigma$ range.

##  V Conclusions

In this work, we revisited the cosmological constraints on the holographic
dark energy model and compared its results with the standard $\Lambda$CDM
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 ($N_{\mathrm{eff}}$) as a free
parameter of fit in the likelihoods to examine whether it alters the fitting
results. In comparison with the $\Lambda$CDM, 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:

  * $\bullet$

Using the datasets “Planck+BAO+RSD+ACT”, we derived constraints on $H_{0}$ in
the $\Lambda$CDM model to be $67.55\pm 0.39\,\mathrm{km\,s^{-1}Mpc^{-1}}$,
which is $4.94\sigma$ lower than the R22 measurement on $H_{0}$
($\,=\\!73.04\pm 1.04\,{\rm km}\,{\rm s}^{-1}{\rm Mpc}^{-1}$). With the same
dataset, the HDE model brings the value of $H_{0}$ down to $69.70\pm 1.39\
\mathrm{km\ s^{-1}Mpc^{-1}}$, which reduces the $H_{0}$ tension down to the
level of $1.92\sigma$ 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
$71.86\pm 0.93\,\mathrm{km\,s^{-1}Mpc^{-1}}$, which further brings the tension
between the HDE prediction and the R22 measurement down to $0.85\sigma$,
leading to a perfect agreement with R22.

  * $\bullet$

By using “Planck+ACT+BAO+RSD” dataset in the HDE+$N_{\mathrm{eff}}$ model, the
value of $S_{8}$ is reduced down to $0.816\pm 0.012$ ; hence the $S_{8}$
tension is relieved to the $1\sigma$-$2\sigma$ level as compared to the low-
redshift gravitational lensing measurement (Fig. 4). This tension is further
relieved after including the R22 dataset (Fig. 4).

  * $\bullet$

By treating $N_{\mathrm{eff}}$ as a free parameter for fit in the likelihood
chain, the posterior distribution of $N_{\rm eff}$ 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 $2\sigma$ confidence
limit, and is on the lower side.

  * $\bullet$

Overall, by using the AIC criterion of the goodness-of-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) LSST Dark Energy Science Collaboration (2012), Simons Observatory Ade
et al. (2019), and Square Kilometre Array (SKA) Square Kilometre Array
Cosmology Science Working Group et al. (2020) 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.

###### Acknowledgements.

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).

## Results of adding Pantheon+ data

Table 5: Constraints of Planck+ACT+BAO+RSD+Pantheon+ dataset on the $\Lambda$CDM, $\Lambda$CDM+$N_{\rm eff}$, HDE, and HDE+$N_{\rm eff}$ 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. Parameters |  $\Lambda$CDM |  $\Lambda$CDM+$N_{\mathrm{eff}}$ | HDE | HDE+$N_{\mathrm{eff}}$
---|---|---|---|---
$\Omega_{\rm b}h^{2}$ | $0.02235\pm 0.00012$ | $0.02212\pm 0.00017$ | $0.02246\pm 0.00012$ | $0.02235\pm 0.00018$
$\Omega_{\rm c}h^{2}$ | $0.11986\pm 0.00086$ | $0.11542\pm 0.00245$ | $0.11801\pm 0.00098$ | $0.11604\pm 0.00241$
$100\,\theta_{\rm MC}$ | $1.04108\pm 0.00027$ | $1.04160\pm 0.00039$ | $1.04132\pm 0.00027$ | $1.04155\pm 0.00037$
$\tau$ | $0.0535\pm 0.0070$ | $0.0535\pm 0.0069$ | $0.0617\pm 0.0081$ | $0.0610\pm 0.0079$
$\ln(10^{10}A_{\rm s})$ | $3.050\pm 0.014$ | $3.037\pm 0.016$ | $3.063\pm 0.015$ | $3.055\pm 0.017$
$n_{\rm s}$ | $0.9678\pm 0.0034$ | $0.9574\pm 0.0065$ | $0.9717\pm 0.0036$ | $0.9665\pm 0.0070$
$c$ | - | - | $0.69551\pm 0.04075$ | $0.68549\pm 0.03387$
$N_{\mathrm{eff}}$ |  $3.046$ (fixed) | $2.76\pm 0.15$ |  $3.046$ (fixed) | $2.91\pm 0.15$
$H_{0}[\mathrm{km\,s^{-1}\,Mpc^{-1}}]$ | $67.44\pm 0.37$ | $65.70\pm 1.01$ | $66.07\pm 0.77$ | $65.44\pm 1.00$
$\Omega_{\rm de}$ | $0.6858\pm 0.0051$ | $0.6798\pm 0.0062$ | $0.6767\pm 0.0086$ | $0.6752\pm 0.0076$
$\Omega_{\rm m}$ | $0.3141\pm 0.0051$ | $0.3202\pm 0.0062$ | $0.3233\pm 0.0086$ | $0.3248\pm 0.0076$
$\sigma_{8}$ | $0.8141\pm 0.0057$ | $0.8006\pm 0.0092$ | $0.8071\pm 0.0082$ | $0.8021\pm 0.0098$
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{\rm m}/0.3}$ | $0.833\pm 0.010$ | $0.827\pm 0.010$ | $0.837\pm 0.011$ | $0.834\pm 0.011$
Figure 7: 2D marginalized posterior contours in the plane of $H_{0}$ and
$S_{8}$ (68% C.L. and 95% C.L.), by using data combination
“Planck+ACT+BAO+RSD+Pantheon+” in the four models $\Lambda$CDM,
$\Lambda$CDM+$N_{\mathrm{eff}}$, HDE, and HDE+$N_{\mathrm{eff}}$.

Here we present the constraints on four models $\Lambda$CDM, HDE,
$\Lambda$CDM+$N_{\rm eff}$ and HDE+$N_{\rm eff}$ 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 $\Lambda$CDM. They
yield similar results (see Table 5) and slightly larger values of $H_{0}$ in
the $\Lambda$CDM scenario: $H_{0}=67.44\pm 0.37$ in $\Lambda$CDM compared to
$H_{0}=66.07\pm 0.77$ in HDE; and $H_{0}=65.70\pm 1.01$ in
$\Lambda$CDM+$N_{\rm eff}$ compared to $H_{0}=65.44\pm 1.00$ in HDE+$N_{\rm
eff}$. When comparing the constraint on $S_{8}$ in the $\Lambda$CDM scenario,
including Pantheon+ 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 2 and 5, 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
$\Lambda$CDM 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)Dai et al. (2020),
Pantheon data show a 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.

## References

  * Spergel et al. (2003) D. N. Spergel, L. Verde, H. V. Peiris, and et al., The Astrophysical Journal Supplement Series 148, 175 (2003), eprint astro-ph/0302209.
  * Spergel et al. (2007) D. N. Spergel, R. Bean, O. Doré, M. R. Nolta, and et al., The Astrophysical Journal Supplement Series 170, 377 (2007), eprint astro-ph/0603449.
  * Komatsu et al. (2009) E. Komatsu, J. Dunkley, M. R. Nolta, and et al., The Astrophysical Journal Supplement Series 180, 330 (2009), eprint 0803.0547.
  * Komatsu et al. (2011) E. Komatsu, K. M. Smith, J. Dunkley, and et al., The Astrophysical Journal Supplement Series 192, 18 (2011), eprint 1001.4538.
  * Hinshaw et al. (2013) G. Hinshaw, D. Larson, E. Komatsu, and et al., The Astrophysical Journal Supplement Series 208, 19 (2013), eprint 1212.5226.
  * Planck Collaboration et al. (2014) Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, and et al., Astronomy and Astrophysics 571, A16 (2014), eprint 1303.5076.
  * Planck Collaboration et al. (2016) Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, and et al., Astronomy and Astrophysics 594, A13 (2016), eprint 1502.01589.
  * Planck Collaboration et al. (2021) Planck Collaboration, Aghanim, N., Akrami, Y., Ashdown, M., and et al., Astron. Astrophys. 652, C4 (2021), URL https://doi.org/10.1051/0004-6361/201833910e.
  * Cole et al. (2005) S. Cole, W. J. Percival, J. A. Peacock, and et al., Monthly Notices of the Royal Astronomical Society 362, 505 (2005), eprint astro-ph/0501174.
  * Beutler et al. (2011) F. Beutler, C. Blake, M. Colless, and et al., Monthly Notices of the Royal Astronomical Society 416, 3017 (2011), eprint 1106.3366.
  * Tegmark et al. (2004) M. Tegmark, M. A. Strauss, M. R. Blanton, and et al., Physical Review D 69, 103501 (2004), eprint astro-ph/0310723.
  * Alam et al. (2021) S. Alam, M. Aubert, S. Avila, and et al., Phys. Rev. D 103, 083533 (2021), URL https://link.aps.org/doi/10.1103/PhysRevD.103.083533.
  * Cooke et al. (2016) R. J. Cooke, M. Pettini, K. M. Nollett, and R. Jorgenson, The Astrophysical Journal 830, 148 (2016), eprint 1607.03900.
  * Cooke et al. (2018) R. J. Cooke, M. Pettini, and C. C. Steidel, The Astrophysical Journal 855, 102 (2018), eprint 1710.11129.
  * Verde et al. (2019) L. Verde, T. Treu, and A. G. Riess, Nature Astronomy 3, 891 (2019), eprint 1907.10625.
  * Riess et al. (2022) A. G. Riess, W. Yuan, L. M. Macri, and et al., The Astrophysical Journal Letters 934, L7 (2022), URL https://dx.doi.org/10.3847/2041-8213/ac5c5b.
  * Aiola et al. (2020) S. Aiola, E. Calabrese, L. Maurin, and et al., Journal of Cosmology and Astroparticle Physics 2020, 047 (2020), URL https://dx.doi.org/10.1088/1475-7516/2020/12/047.
  * Madhavacheril et al. (2023) M. S. Madhavacheril, F. J. Qu, B. D. Sherwin, and et al., _The atacama cosmology telescope: Dr6 gravitational lensing map and cosmological parameters_ (2023), eprint 2304.05203.
  * Amon et al. (2022) A. Amon, D. Gruen, M. A. Troxel, and et al. (DES Collaboration), Phys. Rev. D 105, 023514 (2022), URL https://link.aps.org/doi/10.1103/PhysRevD.105.023514.
  * Asgari, Marika et al. (2021) Asgari, Marika, Lin, Chieh-An, Joachimi, Benjamin, and et al., A&A 645, A104 (2021), URL https://doi.org/10.1051/0004-6361/202039070.
  * Li et al. (2023) X. Li, T. Zhang, S. Sugiyama, and et al., _Hyper suprime-cam year 3 results: Cosmology from cosmic shear two-point correlation functions_ (2023), eprint 2304.00702.
  * Jedamzik and Pogosian (2023) K. Jedamzik and L. Pogosian, _Primordial magnetic fields and the hubble tension_ (2023), eprint 2307.05475.
  * Poulin et al. (2019) V. Poulin, T. L. Smith, T. Karwal, and M. Kamionkowski, Phys. Rev. Lett. 122, 221301 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.122.221301.
  * Jedamzik et al. (2021) K. Jedamzik, L. Pogosian, and G.-B. Zhao, Communications Physics 4, 123 (2021), eprint 2010.04158.
  * Gómez-Valent et al. (2021) A. Gómez-Valent, Z. Zheng, L. Amendola, V. Pettorino, and C. Wetterich, Phys. Rev. D 104, 083536 (2021), URL https://link.aps.org/doi/10.1103/PhysRevD.104.083536.
  * Reeves et al. (2023) A. Reeves, L. Herold, S. Vagnozzi, B. D. Sherwin, and E. G. M. Ferreira, Monthly Notices of the Royal Astronomical Society 520, 3688–3695 (2023), ISSN 1365-2966, URL http://dx.doi.org/10.1093/mnras/stad317.
  * Cruz et al. (2023) J. S. Cruz, F. Niedermann, and M. S. Sloth, Journal of Cosmology and Astroparticle Physics 2023, 033 (2023), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2023/11/033.
  * Cyr-Racine and Sigurdson (2014) F.-Y. Cyr-Racine and K. Sigurdson, Physical Review D 90, 123533 (2014), eprint 1306.1536.
  * Kreisch et al. (2020) C. D. Kreisch, F.-Y. Cyr-Racine, and O. Doré, Physical Review D 101, 123505 (2020), eprint 1902.00534.
  * He et al. (2020) H.-J. He, Y.-Z. Ma, and J. Zheng, Journal of Cosmology and Astroparticle Physics 2020, 003 (2020), eprint 2003.12057.
  * Note (1) Note1, we also notice that Ref.+.1667emHe et al. (2020) provided an attractive resolution to the Hubble tension by consistently realizing the neutrino self-interactions, which respects the electroweak gauge symmetry and avoids producing a too large $N_{\rm eff}$ by having $\Delta N_{\rm eff}\simeq 0.2\tmspace+{.1667em}$.
  * (32) A. Rida Khalife, M. Bahrami Zanjani, S. Galli, S. Günther, J. Lesgourgues, and K. Benabed (????), eprint 2312.09814.
  * Di Valentino et al. (2020) E. Di Valentino, A. Melchiorri, O. Mena, and S. Vagnozzi, Physics of the Dark Universe 30, 100666 (2020), ISSN 2212-6864, URL http://dx.doi.org/10.1016/j.dark.2020.100666.
  * Rashkovetskyi et al. (2021) M. Rashkovetskyi, J. B. Muñoz, D. J. Eisenstein, and C. Dvorkin, Physical Review D 104 (2021), ISSN 2470-0029, URL http://dx.doi.org/10.1103/PhysRevD.104.103517.
  * Bansal et al. (2022) S. Bansal, J. H. Kim, C. Kolda, M. Low, and Y. Tsai, Journal of High Energy Physics 2022 (2022), ISSN 1029-8479, URL http://dx.doi.org/10.1007/JHEP05(2022)050.
  * Knox and Millea (2020) L. Knox and M. Millea, Phys. Rev. D 101, 043533 (2020), URL https://link.aps.org/doi/10.1103/PhysRevD.101.043533.
  * Heisenberg et al. (2022) L. Heisenberg, H. Villarrubia-Rojo, and J. Zosso, Physical Review D 106 (2022), ISSN 2470-0029, URL http://dx.doi.org/10.1103/PhysRevD.106.043503.
  * Cardona and Sabogal (2023) W. Cardona and M. Sabogal, Journal of Cosmology and Astroparticle Physics 2023, 045 (2023), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2023/02/045.
  * Cohen et al. (1999) A. Cohen, D. Kaplan, and A. Nelson, Phys. Rev. L 82, 4971 (1999).
  * Li (2004) M. Li, Physics Letters B 603, 1 (2004), ISSN 0370-2693, URL https://www.sciencedirect.com/science/article/pii/S0370269304014388.
  * Huang and Gong (2004) Q.-G. Huang and Y. Gong, Journal of Cosmology and Astroparticle Physics 2004, 006 (2004), eprint astro-ph/0403590.
  * Ma et al. (2009) Y.-Z. Ma, Y. Gong, and X. Chen, European Physical Journal C 60, 303 (2009), eprint 0711.1641.
  * Ma et al. (2010) Y.-Z. Ma, Y. Gong, and X. Chen, European Physical Journal C 69, 509 (2010), eprint 0901.1215.
  * Li et al. (2013) M. Li, X.-D. Li, Y.-Z. Ma, X. Zhang, and Z. Zhang, Journal of Cosmology and Astroparticle Physics 2013, 021 (2013), eprint 1305.5302.
  * Dai et al. (2020) W.-M. Dai, Y.-Z. Ma, and H.-J. He, Physical Review D 102, 121302 (2020), eprint 2003.03602.
  * Riess et al. (2019) A. G. Riess, S. Casertano, W. Yuan, and et al., The Astrophysical Journal 876, 85 (2019), URL https://dx.doi.org/10.3847/1538-4357/ab1422.
  * Mangano et al. (2005) G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti, and P. D. Serpico, Nuclear Physics B 729, 221 (2005), eprint hep-ph/0506164.
  * de Salas and Pastor (2016) P. F. de Salas and S. Pastor, Journal of Cosmology and Astroparticle Physics 2016, 051 (2016), eprint 1606.06986.
  * Akita and Yamaguchi (2020) K. Akita and M. Yamaguchi, Journal of Cosmology and Astroparticle Physics 2020, 012 (2020), eprint 2005.07047.
  * Froustey et al. (2020) J. Froustey, C. Pitrou, and M. C. Volpe, Journal of Cosmology and Astroparticle Physics 2020, 015 (2020), eprint 2008.01074.
  * Bennett et al. (2021) J. J. Bennett, G. Buldgen, P. F. de Salas, and et al., Journal of Cosmology and Astroparticle Physics 2021, 073 (2021), eprint 2012.02726.
  * ’t Hooft (1993) G. ’t Hooft, _Dimensional reduction in quantum gravity_ (1993), eprint gr-qc/9310026.
  * Susskind (1995) L. Susskind, Journal of Mathematical Physics 66, 6377 (1995).
  * Ratra and Peebles (1988) B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988), URL https://link.aps.org/doi/10.1103/PhysRevD.37.3406.
  * Wetterich (1988) C. Wetterich, Nuclear Physics B 302, 668 (1988), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/0550321388901939.
  * Carroll (1998) S. M. Carroll, Phys. Rev. Lett. 81, 3067 (1998), eprint astro-ph/9806099.
  * Note (2) Note2, bAO12 refers to the SDSS data release 12 (DR12) of the measurements of $D_{\rm M}r_{\rm fid,d}/r_{\rm d}$ at $z_{\rm eff}=0.38,0.51$ and $0.61$ Alam et al. (2017). R19 refers to the Riess et al.+.1667em(2019) Riess et al. (2019)’s measurement of $H_{0}$ by using Large Magellanic Cloud Cepheid Standards.
  * Lewis et al. (2000) A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538, 473 (2000), URL https://dx.doi.org/10.1086/309179.
  * Note (3) Note3, notice that perturbation of HDE itself and its interactions are considered in Wang et al. (2017), and Chunshan Lin (2021) Lin (2021) introduced a covariant local field theory for the HDE model, providing a method to study its perturbations.
  * Lewis and Bridle (2002) A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002), URL https://link.aps.org/doi/10.1103/PhysRevD.66.103511.
  * Neal (2005) R. M. Neal, _Taking bigger metropolis steps by dragging fast variables_ (2005), eprint math/0502099, URL https://arxiv.org/abs/math/0502099.
  * Lewis (2013) A. Lewis, Physical Review D 87 (2013), ISSN 1550-2368, URL http://dx.doi.org/10.1103/PhysRevD.87.103529.
  * Beutler et al. (2011) F. Beutler, C. Blake, M. Colless, and et al., Monthly Notices of the Royal Astronomical Society 416, 3017 (2011), URL https://doi.org/10.1111%2Fj.1365-2966.2011.19250.x.
  * N. Aghanim et al. (2020a) N. Aghanim, Y. Akrami, M. Ashdown, and et al., Astron. Astrophys. 641, A5 (2020a), URL https://doi.org/10.1051/0004-6361/201936386.
  * N. Aghanim et al. (2020b) N. Aghanim, Y. Akrami, M. Ashdown, and et al., Astron. Astrophys. 641, A8 (2020b), URL https://doi.org/10.1051%2F0004-6361%2F201833886.
  * Ross et al. (2015) A. J. Ross, L. Samushia, C. Howlett, and et al., Monthly Notices of the Royal Astronomical Society 449, 835 (2015), URL https://doi.org/10.1093%2Fmnras%2Fstv154.
  * Alam et al. (2017) S. Alam, M. Ata, S. Bailey, and et al., Monthly Notices of the Royal Astronomical Society 470, 2617 (2017), eprint 1607.03155.
  * Note (4) Note4, the detailed values are listed in the first row of Table I in Ref. Gil-Marín et al. (2020).
  * Kaiser (1987) N. Kaiser, Monthly Notices of the Royal Astronomical Society 227, 1 (1987).
  * Gil-Marín et al. (2020) H. Gil-Marín, J. E. Bautista, R. Paviot, and et al., Monthly Notices of the Royal Astronomical Society 498, 2492 (2020), URL https://doi.org/10.1093/mnras/staa2455.
  * Scolnic et al. (2018) D. M. Scolnic, D. O. Jones, A. Rest, and et al., The Astrophysical Journal 859, 101 (2018), URL https://dx.doi.org/10.3847/1538-4357/aab9bb.
  * Di Valentino et al. (2021) E. Di Valentino, A. Melchiorri, and J. Silk, Astrophysical Journal Letters 908, L9 (2021), eprint 2003.04935.
  * Scolnic et al. (2022) D. Scolnic, D. Brout, A. Carr, and et al., The Astrophysical Journal 938, 113 (2022), ISSN 1538-4357, URL http://dx.doi.org/10.3847/1538-4357/ac8b7a.
  * Note (5) Note5, we calculate the distance in the best-fitting value by quantifying the difference of the best-fits with respect to the square root of quadrature sum of each errors, i.e., $(73.04\tmspace-{.1667em}-\tmspace-{.1667em}69.70)/\sqrt{1.04^{2}\tmspace-{.1667em}+\tmspace-{.1667em}1.39^{2}}\tmspace-{.1667em}=\tmspace-{.1667em}1.92\sigma$ and $(73.04-67.55)/\sqrt{1.04^{2}\tmspace-{.1667em}+\tmspace-{.1667em}0.39^{2}}=4.94\sigma$.
  * Aich et al. (2020) M. Aich, Y.-Z. Ma, W.-M. Dai, and J.-Q. Xia, Physical Review D 101, 063536 (2020), eprint 1912.00995.
  * Dodelson and Schmidt (2020) S. Dodelson and F. Schmidt, _Modern Cosmology_ (2020).
  * Hou et al. (2013) Z. Hou, R. Keisler, L. Knox, M. Millea, and C. Reichardt, Phys. Rev. D 87, 083008 (2013), URL https://link.aps.org/doi/10.1103/PhysRevD.87.083008.
  * Dutcher et al. (2021) D. Dutcher, L. Balkenhol, P. A. R. Ade, Z. Ahmed, E. Anderes, A. J. Anderson, M. Archipley, J. S. Avva, K. Aylor, P. S. Barry, et al., Physical Review D 104, 022003 (2021), eprint 2101.01684.
  * Abazajian et al. (2019) K. Abazajian, G. Addison, P. Adshead, Z. Ahmed, S. W. Allen, D. Alonso, M. Alvarez, A. Anderson, K. S. Arnold, C. Baccigalupi, et al., arXiv e-prints arXiv:1907.04473 (2019), eprint 1907.04473.
  * Note (6) Note6, $N=N_{\it Planck}+N_{\rm ACT}+N_{\rm BAO+RSD}+N_{\rm R22}$. For $N_{\it Planck}$, data points combine high-$\ell$ TT ($30\leq\ell\leq 2508$), TE ($30\leq\ell\leq 1996$) and EE ($30\leq\ell\leq 1996$), low-$\ell$ temperature ($0\leq\ell\leq 29$), low-$\ell$ EE ($2\leq\ell\leq 27$) and lensing ($8\leq\ell\leq 400$) spectrum, so $N_{\it Planck}=2478+1966+1966+29+27+392=6858$; ACT data points from CMB power spectrum on scales of $350<\ell<4000$, so $N_{\rm ACT}=3650$; $N_{\rm BAO+RSD}=11$ verified as in Table 1 and $N_{\rm R22}=1$. Therefore, $N=6858+3650+11+1=10520$.
  * LSST Dark Energy Science Collaboration (2012) LSST Dark Energy Science Collaboration, arXiv e-prints arXiv:1211.0310 (2012), eprint 1211.0310.
  * Ade et al. (2019) P. Ade, J. Aguirre, Z. Ahmed, S. Aiola, A. Ali, D. Alonso, M. A. Alvarez, K. Arnold, P. Ashton, J. Austermann, et al., Journal of Cosmology and Astroparticle Physics 2019, 056 (2019), eprint 1808.07445.
  * Square Kilometre Array Cosmology Science Working Group et al. (2020) Square Kilometre Array Cosmology Science Working Group, D. J. Bacon, R. A. Battye, P. Bull, S. Camera, P. G. Ferreira, I. Harrison, D. Parkinson, A. Pourtsidou, M. G. Santos, et al., Publications of the Astron. Soc. of Australia 37, e007 (2020), eprint 1811.02743.
  * Hawking (1975) S. W. Hawking, Communications in Mathematical Physics 43, 199 (1975), URL https://api.semanticscholar.org/CorpusID:55539246.
  * Bekenstein (1973) J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973), URL https://link.aps.org/doi/10.1103/PhysRevD.7.2333.
  * Riess et al. (2021) A. G. Riess, S. Casertano, W. Yuan, J. B. Bowers, L. Macri, J. C. Zinn, and D. Scolnic, The Astrophysical Journal Letters 908, L6 (2021), URL https://dx.doi.org/10.3847/2041-8213/abdbaf.
  * Hsu (2004) S. D. Hsu, Physics Letters B 594, 13 (2004), ISSN 0370-2693, URL https://www.sciencedirect.com/science/article/pii/S0370269304007658.
  * Lin (2021) C. Lin, Journal of Cosmology and Astroparticle Physics 2021, 003 (2021), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2021/07/003.
  * Wang et al. (2017) S. Wang, Y. Wang, and M. Li, Physics Reports 696, 1–57 (2017), ISSN 0370-1573, URL http://dx.doi.org/10.1016/j.physrep.2017.06.003.


††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Homogeneous Linear Ion Crystal in a Hybrid Potential

Ming-shen Li  School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai,
519082, China  Yang Liu  School of Physics and Astronomy, Sun Yat-Sen
University, Zhuhai, 519082, China  Center of Quantum Information Technology,
Shenzhen Research Institute of Sun Yat-sen University, Nanshan Shenzhen
518087, China  Xin-Xin Rao  School of Physics and Astronomy, Sun Yat-Sen
University, Zhuhai, 519082, China  Peng-Fei Lu  School of Physics and
Astronomy, Sun Yat-Sen University, Zhuhai, 519082, China  Zhao Wang  School of
Physics and Astronomy, Sun Yat-Sen University, Zhuhai, 519082, China  Feng Zhu
<EMAIL_ADDRESS>School of Physics and Astronomy, Sun Yat-Sen
University, Zhuhai, 519082, China  Center of Quantum Information Technology,
Shenzhen Research Institute of Sun Yat-sen University, Nanshan Shenzhen
518087, China  Le Luo<EMAIL_ADDRESS>School of Physics and
Astronomy, Sun Yat-Sen University, Zhuhai, 519082, China  Center of Quantum
Information Technology, Shenzhen Research Institute of Sun Yat-sen University,
Nanshan Shenzhen 518087, China  School of Physics and Astronomy, Sun Yat-sen
University, Zhuhai 519082, China

###### Abstract

We investigate the properties of a linear ion crystal in a combination of
quadratic and quartic potentials. Both the discrete and the continuous model
are employed to explore the homogeneity of a linear ion crystal by controlling
the proportional parameter between the quadratic and quartic components. It is
found that a uniform ion distribution in such a hybrid potential can be made
larger than that in a purely quadratic or quartic potential. The zigzag
transition is also investigated in the hybrid potential. Its critical
condition and phase diagram are determined numerically, which agrees well with
previous theoretical and experimental results. This paves the way for
experimental investigation of phase transition in a large linear coulomb
crystal.

###### pacs:

313.43

##  I Introduction

Quantum computers have the potentials to tackle many problems hard to be
solved or even unsolvable using classical computers.kaye2007introduction ;
james1998quantum  Owing to the long-lived internal qubit coherence, the ion
trap becomes one of the most attractive candidates for the realization of a
quantum computer. Today’s prototype quantum computers containing tens of
qubits have been demonstrated in several systems including ion traps. To
realize the promise of quantum computing, intensive efforts had been paid to
construct quantum computer with a large scale of ion qubits.cirac1995quantum ;
sorensen1999quantum ; milburn2000ion  Different types of ion traps had been
investigated, such as the linear ion trap designed to trap a long ion chain
with even space,johanning2016isospaced ; xie2017creating  the ion trap on
chips graham2014system  to facilitate ion transportation in different areas,
the collection of small-scale ion traps integrated by optical
methods,blain2006microfabricated  etc.

A linear chain of trapped ions in a harmonic potential has been demonstrated
as a quantum computer of tens of qubits. dubin1997minimum ; bastin2017ion ;
johanning2016isospaced ; ohira2020confinement  However, in the harmonic
potential it is difficult to address single ion due to the unequal spacing.
The zigzag transition will occur as the number of ions increases.
schiffer1993phase  Therefore, it is requisite to realize the uniform
distribution of a long ion chain for a large-scale quantum information system.
The uniform ion chain is critical for the single and multiple qubit processing
and gate operation. It would also provide an ideal test-bed for the study of
many-body effects such as topological phase transition. gutierrez2020defect
In addition, under the same radial confinement, the evenly spaced linear ion
chain can accommodate more ions than in the harmonic potential before the
zigzag transition occurs. lin2009large

To achieve a uniform distribution of a long chain, ion distributions in
anharmonic potentials have been studied.bastin2017ion  The results showed that
the distribution of the ions in an anharmonic potential is more uniform at the
center than that in a harmonic potential, but it changes quickly at both
edges. Additionally, the combination of the harmonic and anharmonic potentials
was proposed as the trapping potential,lin2009large  in order to scale up the
qubits and to operate the entangling gates more efficiently. Recently, using
anharmonic axial potentials, 100 ions in a linear configuration have been
realized and up to 44 ions have nearly equidistant spacings
pagano2018cryogenic  Prospects for even larger system sizes look quite
optimistic in the near future.

In this paper, we present a homogenous crystal with large number of ions
trapped in a hybrid potential, consisting of harmonic and anharmonic
potentials with a dimensionless ratio. First, we use the discrete model to
calculate the requisites for making a homogenious ion chain by using the
hybrid potential. Then, we test our calculations using a universal variance
criterion. To compare with the discrete model, we further apply variational
method to the hybrid potential by using the continuous model. Almost identical
conditions are obtained, which confirms the accuracy of our calculations and
the demanding of a well-designed hybrid potential in order for a large
homogenious ion chain. At the end, we discuss the zigzag transition in the
hybrid potential.

##  II Discrete model

The discrete model has been applied to the linear ion chain in a purely
quadratic(i.e. harmonic) or quartic(i.e. anharmonic) potential.
dubin1997minimum ; james1998quantum ; bastin2017ion  The potential energy of
the system in a quadratic potential can be described as: dubin1997minimum ;
james1998quantum

$\displaystyle
e=\sum_{i=1}^{N}\dfrac{1}{2}m\omega^{2}x_{i}^{2}+\sum_{i,j=1;i<j}^{N}\dfrac{q^{2}}{\left|x_{i}-x_{j}\right|}$
(1)

where $m$ and $q$ are the mass and charge of an ion, $\omega$ is the axial
trap frequency, $x_{i}$ is the axial position of each ion, $N$ is the total
number of ions. When replacing $m\omega^{2}x_{i}^{2}/2$ with $ax_{i}^{4}/4$,
the confinement becomes a purely quartic potential, in which $a$ is the
coefficient.

Here, we first use the discrete model to calculate the ground state
distribution of the ion chain in the hybrid potential. The energy of the
system can be written in the following form:lin2009large

$\displaystyle
e=\sum_{i=1}^{N}\frac{\alpha_{4}}{4}x^{4}_{i}-\sum_{i=1}^{N}\frac{\alpha_{2}}{2}x^{2}_{i}+\sum_{i,j=1,i<j}^{N}\frac{q^{2}}{|x_{i}-x_{j}|}$
(2)

This equation can be rescaled by $z_{i}=x_{i}/l$ with $x_{i}$ with a length
scale $l^{5}=q^{2}/\alpha_{4}$. Through a dimensionless processing, Eq.(2) can
be rewritten as:

$\displaystyle
E=\sum_{i=1}^{N}\frac{1}{4}z^{4}_{i}-\sum_{i=1}^{N}\frac{D}{2}z^{2}_{i}+\sum_{i,j=1,i<j}^{N}\frac{1}{|z_{i}-z_{j}|},$
(3)

using a dimensionless ratio
$D=|\alpha_{2}/q^{2}|^{2/5}(\alpha_{2}/\alpha_{4})^{3/5}$, which is introduced
to characterize the strength of the quadratic potential relative to the
quartic potential.

The ground state positions of the ions can be determined by the following
conditions

$\displaystyle\frac{\partial E}{\partial{z}_{i}}=0$ (4)

By solving totally $N$ nonlinear differential equations, we obtain the
equilibrium position of each ion. The positions of $20$ ions in different
potentials are compared in Fig.1. The ions in the hybrid potential are much
more evenly distributed across the chain compared to the quadratic or quartic
potential.

Figure 1: Positions of $20$ ions in (a) quadratic, (b) quartic and (c) the
hybrid potentials. Notice that the ions in the hybrid potential are more
uniformly distributed. In these plots, the position z has been rescaled to be
dimensionless.

By defining the ion density as follows:

$\displaystyle n(\tilde{z_{i}})=\frac{1}{|z_{i+1}-z_{i}|},$ (5)

where ${z_{i}}$ is the coordinate of the $i$th ion, $\tilde{z_{i}}$ is the
midpoint between ${z_{i}}$ and ${z_{i+1}}$, the resultant ion density
distribution is shown in Fig.2 for different $D$ parameters when $N=1000$. It
can be seen from Fig.2 that the density in the middle part of the ion chain
decreased with the increase of the $D$ parameters. This is because when the
parameter $D$ increases, the extreme point of the hybrid potential deviates
farther from the center, resulting in an increase of the ion spacing at the
center.

Figure 2:  The ground state ion density $n$ in hybrid potentials for different
$D$ in discrete model, when the total number of ions is $N=1000$ . In the
middle part of the ion chain (i.e. around $z=0$), the density decreases with
the increase of $D$. A represents the midpoint of the ion chain, B represents
the point where the density reaches the maximum, and C resprents the point
where the density is the same as point A. Both $n$ and $z$ are rescaled to be
dimensionless.

In order to find an appropriate combination of the quadratic and quartic
potentials to maximize the uniform distribution of the ions, the variance
$s(z)=\frac{1}{N}\sum_{i=1}^{N}(\Delta z_{i}-\overline{\Delta z_{i}})^{2}$ of
$\Delta z$ in $[0,z_{0}$] is introduced. Fig.3(a) shows the variance as a
function of the position $z_{0}$. The optimal distribution is set by the
criterion of $s(z)\leq\eta$. The optimization of $D$ parameter is shown for
$\eta=6\times 10^{-7}$ and $N=1000$. We define the optimal number of ions that
satisfy the optimization criterion to be $N^{\prime}$. As shown in Fig.3(b),
the optimal number of ions fluctuate little for $10<D<18$ since these curves
intersect with the threshold in a small range. The optimal ion numbers of
$N^{\prime}$ have a maximum of $888$ when $D\in[11.5,15.5]$. Under the same
conditions, $N^{\prime}$ is $396$ or $844$ in the quadratic or the quartic
potential respectively from our caculation, which is significantly lower than
the optimal ion number in the hybrid potential.

Figure 3:  (a) Variance criterion in discrete model, as a function of $z_{0}$
with different $D$ parameters for $N=1000$. The dashed line represents the
threshold $\eta=6\times 10^{-7}$, and we choose the right intersection point
between the dashed line and the curve for determining the optimal length of
the homogenious ion chain for a $D$ parameter, which in turn corresponds to an
optimal ion number $N^{\prime}$ for this chain. Here, $z_{0}$ are rescaled to
be dimensionless. (b) The optimal number of ions as a function of $D$
parameter for the same $\eta$. When $D>18$, a sharp reduce in $N^{\prime}$
appears. This is due to a smaller $z_{0}$ with a larger $D$ in (a).

##  III Continuous model

Although the discrete model is effective in obtaining requsites for a
homogenious distribution of an ion chain, it is very time-consuming especially
when the number of ions is hundreds or thousands. Thus, in order to solve the
properties of the ion chain more efficiently and also to confirm our
calculations with discrete model, we implement continuous model based on
Dubin’s variational method dubin1997minimum , which revolves around the
assumption of the local density approximation, hohenberg1964inhomogeneous
where the distribution of each ion in an ion chain is approximated by a
continuous density function $n(z)$. Time consumption using this method does
not relate to the scale of the total ion number $N$, thus making the
calculation much faster and more flexible.

The first term in the integral is the energy due to the quadratic trapped
potential. When replacing ${\frac{1}{2}z^{2}}$ with ${\frac{1}{4}z^{4}}$ it
becomes the energy due to the quartic potential. The other term is the Coulomb
energy of the ion chain and $\gamma\approx 0.57721$ is Euler’s constant. The
principle of minimum energy is used to minimize Eq.(7), and the density
function $n(z)$ can be obtained.The variational method was extended to an
quartic potential recently.bastin2017ion

According to the local density approximation, the density function of the ions
in a hybrid potential can be expressed as
$n(z)=Az^{4}+Bz^{2}+C$.dubin1997minimum ; bastin2017ion  By substituting the
boundary condition and the normalization condition, it becomes:

$n(z)=-\frac{5(-2C+6CL-3N)}{2L^{4}(-5+3L)}z^{4}+\frac{3(8CL-5N)}{2L^{2}(-5+3L)}z^{2}+C,$
(6)

which is shown in Fig.4. It is worth to notice that both $L$ and $C$ are
functions of $N$ and $D$, since the distribution of the ions is determined by
$N$ and $D$ according to the principle of the minimum energy. It is found that
these parameters have extremely complicated representations with $N$ and $D$
as variables. However, we find simple expressions for these two parameters, by
using numerical methods such as gradient search algorithm, such that the
potential energy of of the system bastin2017ion

$\displaystyle E[n]=$
$\displaystyle\int_{-\infty}^{\infty}dz\\{\frac{1}{2}z^{2}n(z)+\gamma
n(z)^{2}$ (7)
$\displaystyle-\frac{1}{2}n(z)\int_{0}^{\infty}dy\ln{[yn(z)]}\frac{d}{dy}[n(z-y)+n(z+y)]\\}$

reaches its minimum. Notice here, a quartic term of $\frac{1}{4}z^{4}$
representing anharmonic potential shall be added into the integral while using
this expression. When $N$ is in the order of hundreds or thousands, the $C$
and $L$ can be formulated as:

$\displaystyle C=0.26N^{0.80}-0.36D^{1.23}+3.80$ (8) $\displaystyle
L=0.70N^{0.30}+0.04D^{1.13}+2.16$ (9)

Consequently, the density function is determined, as shown in Fig.4.

Figure 4:  The ground state density functions calculated by the variational
method when $N=1000$. The shapes of the curves are identical to the
corresponding ones calculated by the discrete model in Fig.2. Both $n$ and $z$
are rescaled to be dimensionless.

Using global density function , it is convenient to analyze the whole
distribution uniformity to find the number of the evenly distributed ions. In
order to demonstrate the characteristic of the general distribution, we define
the variance criterion as:

$\displaystyle
s(z_{0})=\frac{1}{z_{0}}\int_{0}^{z_{0}}[\Delta(z)-\frac{1}{z_{0}}\int_{0}^{z_{0}}\Delta(z)dz]^{2}dz$
(10)

where $\Delta(z)=\frac{1}{n(z)}$ represents the distance between the adjacent
ions. The variance criterion is a function of position $z_{0}$, as shown in
Fig.5(a).

Figure 5: (a) Variance criterion as a function of $z_{0}$ with different $D$
parameters for $N=1000$. The dashed line represents the threshold of
$\eta=6\times 10^{-7}$, and the intersections between the dashed line and the
curves determine the available lengths of the ion chains for different $D$
parameters, which can be converted into ion numbers by the density function.
(b) The optimal number of ions as a function of $D$ parameters with the
$\eta=6\times 10^{-7}$.

With the continuous model, we have $N(z_{0})=\int_{0}^{z_{0}}n(z)dz$, where
$N(z_{0})$ represents the particle number in the range of $[0,z_{0}]$. Once we
determine the appropriate $z_{0}$, we get the number of particles. The
variance criterion of the density function can have different requirements
restricted by the design of the experiment. Fig.5(b) show, the number of ions
with homogenious distribution depends heavily on $D$ parameter, same as the
trend shown in Fig.3(b). If the variance criterion $s\leq 6\times 10^{-7}$, we
have $N^{\prime}=872$ when $D\in[10,12]$, while $N^{\prime}=796$ for quartic
potential and $380$ for quadratic potential from our caculation. We also
notice there are small differences between critical $D$ value between
continuous model and discrete model, which is also shown in bastin2017ion  and
is due to the differences introduced by the integration by the variational
method.

##  IV Zigzag phase transition

During the discussion above, we assume the motion in the radial direction is
frozen out by a tight radial potential which is much stronger than the axial
potential, so we ignore the effect of radial potential. However, when the
radial confinement is not that strong, it is energetically favorable for the
ions to be displaced in the radial direction, and the zigzag transition
occurs,dubin1993theory ; nigmatullin2016formation ; pedregosa2020defect  which
is proven to be a second-order phase transition. With the results of the
continuous model, we analyze the zigzag transition under the condition that
$x$-potential is much weaker than $y$-potential, so the $y$-potential can be
ignored. We use $\beta$ as the strength of the $x$-potential relative to
$z$-potential, and the anisotropy of the trap can be approximated by the
$\alpha(z\to 0)=\frac{\omega_{x}^{2}}{\omega_{z}^{2}}(z\to
0)\approx\beta^{2}/D$, because it is always the case the phase transition
first occurs at the center of the ion chain($z=0$). The potential energy of
the system is then:

$\displaystyle e=$
$\displaystyle\sum_{i=1}^{N}(\frac{\beta^{2}}{2}x_{i}^{2}+\frac{1}{4}z^{4}_{i}-\frac{D}{2}z_{i}^{2})$
(11)
$\displaystyle+\sum_{i,j=1;i<j}^{N}\frac{1}{|(x_{i}-x_{j})^{2}+(z_{i}-z_{j})^{2}|^{\frac{1}{2}}}$

The motion of the ions can be described by the Lagrangian,james1998quantum ;
enzer2000observation ; kielpinski2000sympathetic where the Hessian matrix can
be expressed as:bastin2017ion

$\displaystyle
A_{mn}=\left\\{\begin{array}[]{c}(\beta^{2}-\sum_{p\not=n}\dfrac{1}{|z_{p}-z_{n}|^{3}}),\
\ \ \ m=n,\\\ \\\ \ \ \ \ \ \ \ \ \ \ \ \ \dfrac{1}{|z_{m}-z_{n}|^{3}},\ \ \ \
\ \ \ \ \ m\not=n\end{array}\right.$ (15)

The eigenvectors and eigenvalues of the real, symmetric, positive-definite
matrix $A_{mn}$ define the normal modes of oscillation of the ions along the
$z$ direction. The eigenvectors $b_{m}^{(p)}$ are defined by
$\sum_{m=1}^{N}A_{mn}b_{m}^{(p)}=\mu_{p}b_{n}^{(p)}$, where eigenvalue
$\mu_{p}\geq 0$ and $p=1,\cdots,N$ are the mode index. The eigenvectors are
normalized and enumerated in order of increasing eigenvalue.

Figure 6:  The impact of different $D$ parameters on $k$ and $\lambda$. When
$D=0$, the potential corresponds to an quartic potential with $k=0.47$ and
$\lambda=1.15$, which is in agreement with Ref.bastin2017ion .

In order to find the critical value $\beta_{c}$ at which the zigzag transition
occurs, we search when the minimum eigenvalue $A=0$ as $\beta$ decreases, thus
obtaining the critical point $\beta_{c}$ for the phase transition as a
function of given $N$. Then we use a well-known simple power form
relationship: schiffer1993phase ; dubin1993theory ; hughes1998alamos ;
enzer2000observation ; morigi2004dynamics ; lin2009large

$\beta_{c}=kN^{\lambda}$ (16)

for fitting, since this relationship exist in both purely quadratic and purely
quartic potential. During the calculation, the location of each particle
$z_{i}$ is determined from the continuous model by using the reciprocal
function of the integration of the density function. In Fig.6, we demonstrate
the impact of $D$ parameters on $k$ and $\lambda$.

With this relationship, we further map out the phase diagram for this phase
transition, which is shown in Fig.7. In Fig.7(a), the dependences of critical
anisotropy $\alpha_{cr}$ on the $D$ and $N$ parameter are shown. Clearly,
$\alpha_{cr}$ is much larger at large $N$ and small $D$. While it is much
smaller at small $N$ and large $D$. Fig.7(b) shows $\alpha_{cr}$ as a function
of $N$ when $D=10$ for the hybrid potential(solid black line), pure quadratic
potential(solid red line),schiffer1993phase ; dubin1993theory ;
hughes1998alamos ; enzer2000observation  and pure quartic potential(solid blue
line).bastin2017ion  The function obtained from experimental data
enzer2000observation  is also shown in solid green line. Fig.7(c) shows
$\alpha_{cr}$ as a function of $D$ when $N=800$ for the hybrid potential. As
it is shown, $\alpha_{cr}$ increases with $N$ in a power form, more
importantly, $\alpha_{cr}$ in our cases lies between that of pure quadratic
and quartic potential, and reach closely to the value inferred from
experimental data enzer2000observation  for large $N$. Such a good agreements
verifies the accuracy and the uo niversality of our proposal.

Figure 7:  (a)The phase diagram showing the dependence of critical anisotropy
$\alpha_{cr}$ on both $D$ and $N$ parameter. This critical anisotropy
$\alpha_{cr}$ determins when the phase transition from linear chain to zigzag
phase occurs. (b)When $D=10$, the critical anisotropy $\alpha_{cr}$ as a
function of $N$ for the hybrid potential(solid black line), pure quadratic
potential(solid red line), schiffer1993phase ; dubin1993theory ;
hughes1998alamos ; enzer2000observation and pure quartic potential(solid blue
line), bastin2017ion as well as the function(solid green line) obtained from
experimental data.enzer2000observation  (c) When $N=800$, the critical
anisotropy $\alpha_{cr}$ as a function of $D$ for the hybrid potential.

##  V Conclusion

We have studied the properties of the one-dimensional ion crystal in a hybrid
potential. Both the density distribution and the optimal parameter $D$ are
attained by both the discrete model and the continuous model. Large
homogeneous ion chain with ion number more than 800 have been achieved, which
has never been realized before. In principle, the number of ions can be
increased to tens of thousands, even millions in such hybrid potential, if the
trapping space allowed. Based on the result from continuous model, we
calculate the critical condition for the zigzag transition. When the zigzag
transition occurs, the $\beta_{c}$ can be controlled by adjusting the
parameter $D$, as Fig.6 illustrates.

Such potential can accommodate more ions uniformly distrubuted than both
quadartic potential and quartic potential, avoiding zigzag transitions at the
center of the chain. Additionally, it can prevent the ions in the center
coming too close, thus reducing cross-talk in ion state detection and enabling
single ion manipulation with focused laser beams. Particularly, it offers
unique advantages in performing coherent operations on large ion
chain.lin2009large ; lin2016sympathetic  It can also be used to investigate
the effects of the quartic term in the hybrid potential on the normal mode
structure.home2011normal ; johanning2016isospaced  As the spin-spin
interaction approximately scales to the second power of the distance, a
homogenious ion chain can be used for quantum simulation of Ising
models.porras2004effective

Moreover, the large ratio $\alpha_{cr}$ of the radial potential and the axial
potential provided by the homogenious ion chain, given the required hybrid
potential, would enable significant improvement on the gate fidelity, since
the thermal ion motion can be reduced by a factor ranging from
$\alpha_{cr}^{2}$ to $\alpha_{cr}^{3}$.zhu2006trapped  Our proposal is also a
promising route for scaling up towards a large-scale quantum computer in a
single harmonic trap, and could potentially open up a new avenue for the study
of large-scale quantum information processing.

##  VI Acknowledgement

This work was supported by the Natural Science Foundation of Guangdong
Province of China (Grants No. 2017A030310452, 2020A1515010864,
2020A1515011159), the National Natural Science Foundation of China (Grants
No.11904423, 11974434), and Fundamental Research Funds for the Central
Universities of Education of China(Grants No.17lgpy27, 191gpy276).

## References

  * (1) Kaye P,Laflamme R, Mosca M, et al. 2007 Oxford university press
  * (2) Daniel FV James, 1998 App. Phy. B. 66(2):181
  * (3) Cirac J I and Zoller P 1995 Phys. Rev. Lett. 74 4091
  * (4) Sørensen A and Mølmer K 1999 Phys. Rev. Lett. 1971, 1999 82(9)
  * (5) GJ Milburn, S Schneider, and DFV James Fortschritte der Physik 2000 Progress of Physics. 48(9-11):801-810
  * (6) Johanning M 2016 App. Phy. B. 122(4):71
  * (7) Xie Y, Zhang XF, Ou BQ, Chen T, Zhang J, Wu CW, Wu W, and Chen PX 2017 Phys. Rev. A. 95(3):032341
  * (8) Graham R D, Chen S-P, Sakrejda T, Wright J, Zhou ZC, and Blino B B 2014 AIP Advan. 4(5):057124
  * (9) Blain M G and James G F December 26 2006. US Patent 7,154,088
  * (10) Daniel HE Dubin 1997 Phys. Rev. E. 55(4):4017
  * (11) Bastin S R and Tony E L 2017 J. App. Phys. 121(1):014312
  * (12) Ohira R, Kume , Takayama K, Muralidharan S, Takahashi H, and Toyoda K 2020 arXiv preprint arXiv:2003.03962
  * (13) Schiffer J P 1993 Phys. Rev. Lett 70(6):818
  * (14) Gutierrez, Jofre Pedregosa and Mukherjee, Manas 2020 New. J. Phys. 22(7):073044
  * (15) Lin GD, Zhu SL, Rajibul I, Kim K, Chang MS, Korenblit S, Monroe C, and Duan LM 2009 Europhys. Lett. 86(6):60004
  * (16) Pagano, Guido and Hess, PW and Kaplan, HB and Tan, WL and Richerme, Phil and Becker, Patrick and Kyprianidis, Antonis and Zhang, Jiehang and Birckelbaw, Eric and Hernandez, MR and others 2018 Quantum Science and Technology 4(1):014004
  * (17) Hohenberg, Pierre and Kohn, Walter 1964 Phys. Rev. 136(3B):B864
  * (18) Shimshoni, Efrat and Morigi, Giovanna and Fishman, Shmuel 2011 Phys. Rev. Lett. 106(1), 010401
  * (19) Daniel HE Dubin 1993 Phys. Rev. Lett. 71(17):2753
  * (20) Nigmatullin R, Campo A D, Chiara G D, Morigi G, Plenio M B, and Retzker A 2016 Phys. Rev. B. 93(1):014106
  * (21) J Pedregosa-Gutierrez and M Mukherjee 2020 arXiv preprint arXiv:2001.09586
  * (22) Enzer, DG and Schauer, MM and Gomez, JJ and Gulley, MS and Holzscheiter, MH and Kwiat, PG and Lamoreaux, SK and Peterson, CG and Sandberg, VD and Tupa, D and others 2000 Phys. Rev. Lett 85(12):2466
  * (23) Kielpinski, D and King, BE and Myatt, CJ and Sackett, CA and Turchette, QA and Itano, Wayne M and Monroe, C and Wineland, David J and Zurek, WH 2000 Phys. Rev. A 61(3):032310
  * (24) Hughes, Richard J and James, Daniel FV and Gomez, JJ and Gulley, MS and Holzscheiter, MH and Kwiat, Paul G and Lamoreaux, SK and Peterson, CG and Sandberg, VD and Schauer, MM and others 1998 Fortschritte der Physik: Progress of Physics 46(4-5):329–361
  * (25) Morigi, Giovanna and Fishman, Shmuel 2004 Phys. Rev. E 70(6):066141
  * (26) Lin, Guin-Dar and Duan, L-M 2016 Quantum information processing 15(12):5299
  * (27) Home, Jonathan P and Hanneke, David and Jost, John D and Leibfried, Dietrich and Wineland, David J 2011 New Journal of Physics 13(7):073026
  * (28) Johanning, Michael 2016 Applied Physics B 122(4):71
  * (29) Porras, Diego and Cirac, J Ignacio 2004 Phys. Rev. Lett. 92(20):207901
  * (30) Zhu SL, Monroe C, and Duan LM 2006 Phys. Rev. Lett. 97(5):050505