# Testing the consistency of early and late cosmological parameters with BAO
and CMB data

Guanlin Liu  0009-0000-9225-7169<EMAIL_ADDRESS>CAS Key
Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy,
University of Science and Technology of China, Chinese Academy of Sciences,
Hefei, Anhui 230026, China  School of Astronomy and Space Sciences, University
of Science and Technology of China, Hefei 230026, China  Yu Wang  CAS Key
Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy,
University of Science and Technology of China, Chinese Academy of Sciences,
Hefei, Anhui 230026, China  School of Astronomy and Space Sciences, University
of Science and Technology of China, Hefei 230026, China  Wen Zhao
0000-0002-1330-2329<EMAIL_ADDRESS>CAS Key Laboratory for Researches in
Galaxies and Cosmology, Department of Astronomy, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei, Anhui 230026, China
School of Astronomy and Space Sciences, University of Science and Technology
of China, Hefei 230026, China

###### Abstract

The recent local measurements of the Hubble constant $H_{0}$, indicate a
significant discrepancy of over 5$\sigma$ compared to the value inferred from
Planck observations of the cosmic microwave background (CMB). In this paper,
we try to understand the origin of this tension by testing the consistency of
early and late cosmological parameters in the same observed data. In practice,
we simultaneously derive the early and late parameters using baryon acoustic
oscillation (BAO) measurements, which provide both low and high-redshift
information. To resolve parameter degeneracy, the complementary data from CMB
observations are included in the analysis. By using the parameter
$\omega_{m}=\Omega_{m}h^{2}$, we introduce ${\rm ratio}(\omega_{m})$, defined
as the ratio of $\omega_{m}$ which are constrained from high and low-redshift
measurements respectively, to quantify the consistency between early and late
parameters. We obtained a value of ${\rm ratio}(\omega_{m})=1.0069\pm 0.0070$,
indicating there is no tension between early parameters and late parameters in
the framework of $\Lambda$CDM model. As a result, the Hubble tension may arise
from the differences of datasets or unknown systematic errors in the current
data. In addition, we forecast the future BAO measurements of ${\rm
ratio}(\omega_{m})$, using several galaxy redshift surveys and 21 cm intensity
mapping surveys, and find that these measurements can significantly improve
the precision of cosmological parameters.

##  I Introduction

The standard cosmological model, $\Lambda$CDM, has withstood the rigorous
scrutiny due to significant advances in observations of the cosmic microwave
background (CMB) Aghanim  _et al._ (2020); Balkenhol  _et al._ (2021); Aiola
_et al._ (2020), Type Ia supernovae (SNIa) Scolnic  _et al._ (2018); Brout
_et al._ (2022) and large-scale structure Alam  _et al._ (2017, 2021); Abbott
_et al._ (2018) in recent years. However, tensions have emerged for several
parameters when they are inferred from distinct observations within the
$\Lambda$CDM. The most significant tension is related to determinations of the
Hubble constant $H_{0}$. The representative difference is between the SH0ES
111Supernovae and $H_{0}$ for the Equation of State. Riess  _et al._ (2022)
determination of $H_{0}$ and the value inferred from Planck Aghanim  _et al._
(2020) for the $\Lambda$CDM cosmology, which has increased to $5\sigma$ level
Verde  _et al._ (2019); Abdalla  _et al._ (2022a). Actually, the tension
exists in not only these two values, but also in two sets of measurements
Abdalla  _et al._ (2022a).

One set contains indirect estimates at early times which are cosmological
model dependent, such as CMB and Baryon Acoustic Oscillation (BAO) Cuceu  _et
al._ (2019); Schöneberg  _et al._ (2022). As mentioned above, the Planck 2018
release Aghanim  _et al._ (2020) predicts $H_{0}=67.36\pm 0.54\ \rm km\
s^{-1}\ Mpc^{-1}$ in combination with the CMB lensing. Moreover, the final
combination of the BAO measurements from galaxies, quasars, and Lyman-$\alpha$
forest (Ly-$\alpha$) from Baryon Oscillation Spectroscopic Survey (BOSS)
prefers $H_{0}=67.64^{+0.94}_{-1.03}\ \rm km\ s^{-1}\ Mpc^{-1}$ Schöneberg
_et al._ (2022) involving a prior on the physical density of the baryons
$\Omega_{b}h^{2}$ derived from the latest Big Bang Nucleosynthesis (BBN)
measurements of the primordial deuterium Cooke  _et al._ (2018); Mossa  _et
al._ (2020). In contrast, another set includes direct late time cosmological
model-independent measurements, such as distance ladders Riess  _et al._
(2022); Jang and Lee (2017); Freedman  _et al._ (2020); Huang  _et al._
(2020); Kourkchi  _et al._ (2020); Santos  _et al._ (2017) and strong lensing
Wong  _et al._ (2020). The determination of $H_{0}$ is based on calibration of
the SNIa using the Cepheids, Miras and Tip of the Red Giant Branch (TRGB),
which prefers a higher value. The latest SH0ES measurement gives
$H_{0}=73.04\pm 1.04\ \rm km\ s^{-1}\ Mpc^{-1}$ using Cepheid
period–luminosity relations to calibrate distances to Type Ia SNe host
galaxies Riess  _et al._ (2022).

There are various potential solutions proposed for the Hubble tension. One
kind of viewpoint thinks that some unknown systematic errors affect either
early-universe or late-universe observation Di Valentino  _et al._ (2021).
However, the other viewpoint is related to the new physics beyond the standard
cosmological model. The late time modifications of the $\Lambda$CDM expansion
history have been showed that they cannot resolve the Hubble tension through a
model independent method in Lemos  _et al._ (2019); Efstathiou (2021).
Therefore, many solutions to the Hubble tension is to introduce new physics
about early universe that increases the value of the Hubble constant inferred
from the CMB. An popular early-time resolution Karwal and Kamionkowski (2016);
Mörtsell and Dhawan (2018); Poulin  _et al._ (2019) is an exotic early dark
energy (EDE) that behaves like a cosmological constant at early time and
dilutes faster than radiation at later time. It addresses the Hubble tension
by decreasing the sound horizon while leaving the later evolution of the
Universe unchanged.

In order to understand the Hubble tension, it is important to ask the
question: Whether or not the inconsistency of the cosmological parameter
exists in the same observed data? In this paper, we achieve the simultaneous
constraints on the early and late cosmological parameters with the help of BAO
measurements that contain both low and high-redshift universe information.
This approach enables us to investigate if any tension exists between early
and late epochs within the $\Lambda$CDM cosmology. If the Hubble tension is
caused by the new physics beyond $\Lambda$CDM model, in principle we should
discover the different values of $H_{0}$ from only early and only late
information of BAO data. However, if this kind of difference is not
discovered, as we present in this article, it would prefer to the solution of
Hubble tension with some unsolved systematics in the cosmological measurement.
In practice, two distinct $\Lambda$CDM cosmological parameter sets are
employed to construct the expansion history of the early and late universe
separately. However, due to the degeneracy, CMB data is combined with BAO data
for breaking it and $H_{0}$ is replaced by a new parameter ${\rm
ratio}(\omega_{m})$, which will be introduced detail in Section II. This new
parameter can help us test the consistency of cosmological parameters and
determine the origin of the Hubble tension. As a result, we find no evidence
of new physics beyond $\Lambda$CDM cosmology.

This article is organized as follows. In Section II, we introduce the methods
and datasets used in this analysis. The results and related discussions are
presented in Section III. We simulate future BAO experiments and analyze the
results in Section IV. Finally, Section V presents our conclusions in this
article.

##  II Methodology

In the standard Friedmann-Lemaitre-Robertson-Walker (FLRW) universe, the
Hubble parameter $H(z)$ on the basis of flat $\Lambda$CDM is given by:

$H(z)=H_{0}\sqrt{{\Omega}_{r}(1+z)^{4}+{\Omega}_{m}(1+z)^{3}+{\Omega}_{\Lambda}},$
(1)

where ${\Omega}_{r}$, ${\Omega}_{m}$ and ${\Omega}_{\Lambda}$ are the
fractional densities of radiation, matter and dark energy at redshift $z=0$
and constrained by ${\Omega}_{r}+{\Omega}_{m}+{\Omega}_{\Lambda}=1$.

As previously indicated regarding the Hubble tension, the 5$\sigma$ difference
between early time and late time inspires us to investigate whether its reason
is physical. In practice, the BAO data which contains both low and high-
redshift universe information will be analyzed in the framework of the flat
$\Lambda$CDM cosmology. We employ two distinct independent cosmological
parameter sets to describe the expansion history of the early and late
universe separately. Therefore, the consistency of these two cosmological
parameter sets can help us distinguish the cause of the Hubble tension. In
order to realize it, the parameters in the standard cosmological model can be
generalized as follows:

  1. (i)

Parameter set undergoes a transition from
$\\{H_{0},\Omega_{m}\\}$ to
$\\{\tilde{H}_{0},\tilde{\Omega}_{m},H_{0},\Omega_{m}\\}$. Note that, here and
afterwards, the parameters with tilde label that their values are derived from
early cosmological information and we call them as “early cosmological
parameters” in this article, while the parameter without tilde label that
their values are derived from late cosmological information and we call them
as “late cosmological parameters”.

  2. (ii)

Recombination redshift $z_{*}$ serves as the dividing line between the early
and late universe. Namely the early parameters
$\left(\tilde{H}_{0},\tilde{\Omega}_{m}\right)$ are determined from the
measurements whose redshift is larger than the recombination redshift $z_{*}$,
while the late parameters remain similar. Therefore we realize constructing
the early and late expanding history of the Universe individually.

  3. (iii)

In order to test $\Lambda$CDM model through the constraint of both early and
late parameters simultaneously, we can compare whether early parameters
$\left(\tilde{H}_{0},\tilde{\Omega}_{m}\right)$ and
$\left(H_{0},\Omega_{m}\right)$ are consistent. We are able to determine
whether the tension between early and late parameters of $\Lambda$CDM exists
by the combination of BAO and CMB data.

More details will be discussed in the data analysis part.

###  II.1 BAO data

In the primordial plasma of the early universe, photon-matter interactions
give rise to substantial heat and outward pressure. This pressure drives
baryons and photons out of the overdensity area at a speed of sound $c_{s}$.
As a result, the spherical sound waves propagate baryon density fluctuations
in the Universe until the recombination epoch. The distance traveled by sound
waves between the end of inflation and the decoupling of baryons from photons
after recombination is referred to as the sound horizon $r_{s}$. Dark matter
remains at the center of acoustic wave since it is influenced solely by
gravity. Finally, the baryons and dark matter resulted in a structure that
included matter overdensities both at the original site and at the sound
horizon scale. Thus, we can anticipate observing a more significant number of
galaxy pairs separated by the sound horizon distance than other length scales.
In other words, the BAO method relies on the imprint left by propagation of
acoustic density waves to provide a cosmic standard ruler which has a comoving
scale $r_{s}$.

The comoving sound horizon $r_{s}(z)$ is given by Eisenstein and Hu (1998):

$\displaystyle
r_{s}(z)=\frac{c}{H_{0}}\int_{z}^{\infty}\frac{c_{s}}{E(z^{\prime})}\mathrm{d}z^{\prime},$
(2)

where $c$ is the speed of light, $H_{0}=100h\rm\ km\ s^{-1}Mpc^{-1}$ with $h$
the dimensionless Hubble constant, the sound speed is
$c_{s}=1/\sqrt{3(1+R_{b}a)}$, with $R_{b}a=3\rho_{b}/(4\rho_{r})$, and
$R_{b}=31500\omega_{b}(T_{CMB}/2.7K)^{-4}$ with $\omega_{b}=\Omega_{b}h^{2}$
and the CMB temperature $T_{CMB}=2.7255K$. It is necessary to emphasize that
the radiation component cannot be ignored when calculating the comoving sound
horizon $r_{s}$. It can be described by the matter fraction $\omega_{m}$
applying matter-radiation equality relation
$\Omega_{r}=\Omega_{m}/(1+z_{eq})$, and $z_{eq}=2.5\times
10^{4}\omega_{m}(T_{CMB}/2.7K)^{-4}$, where $\omega_{m}=\Omega_{m}h^{2}$.

The BAO feature appears in both the line-of-sight direction and the transverse
direction and provides a measurement of $D_{H}(z)/r_{d}$ and $D_{M}(z)/r_{d}$
individually:

$\displaystyle\frac{D_{H}(z)}{r_{d}}$ $\displaystyle=$
$\displaystyle\frac{c}{H(z)r_{d}},$ (3) $\displaystyle\frac{D_{M}(z)}{r_{d}}$
$\displaystyle=$ $\displaystyle\frac{c}{H_{0}r_{d}}\Gamma(z),$ (4)

where $r_{d}$ is the sound horizon at the drag epoch $r_{s}(z_{d})$ and
$D_{M}$ is the same as transverse comoving distance Hogg (1999). The
transverse comoving distance, which depends on FLRW metric to an object at
redshift $z$, is given by

$D_{M}(z)=r(z)=\frac{c}{H_{0}}S_{k}(\Gamma(z)),$ (5)

where $\Gamma(z)=\int_{0}^{z}{\mathrm{d}z^{\prime}}/{E(z^{\prime})}$,
$E(z)=H(z)/H_{0}$. Note that the angular diameter distance $D_{A}(z)$ has
relation with the comoving angular diameter distance $r(z)$:
$D_{A}(z)=r(z)/(1+z)$. Because $S_{k}(x)=x$ for flat $\Lambda$CDM model, we
have $r(z)=c\Gamma(z)/H_{0}$ in the following analysis.

The BAO measurements were also historically summarized by a single quantity
representing the spherically averaged distance

$D_{V}(z)\equiv[zD_{M}^{2}(z)D_{H}(z)]^{1/3}.$ (6)

The redshift of the drag epoch can be approximated as Hu and Sugiyama (1996)

$z_{d}=\frac{1345\omega_{m}^{0.251}}{1+0.659\omega_{m}^{0.828}}[1+b_{1}\omega_{b}^{b_{2}}],$
(7)

where

$b_{1}=0.313\omega_{m}^{-0.419}[1+0.607\omega_{m}^{0.674}],b_{2}=0.238\omega_{m}^{0.223}.$
(8)

It is obvious that BAO measurements provide simultaneous information about the
early and late universe through the comoving sound horizon $r_{d}$ and Hubble
parameter $H(z)$, which enables us to determine the physics in the early and
late universe simultaneously.

To calculate the sound horizon $r_{d}$ and complete the set of cosmological
parameters, we must also include information on baryon density $\Omega_{b}$ as
an early parameter not only because it derives from CMB observation, but also
it is just entailed to compute sound horizon $r_{d}$. Carrying over the
remaining parameters introduced above, the initial full parameter set is now:

$\displaystyle\centering\\{\tilde{\Omega}_{b},\tilde{\Omega}_{m},\tilde{H}_{0},\Omega_{m},H_{0}\\}.\@add@centering$

Then, the BAO measurements can be expressed in this parameter set:

$\frac{D_{H}(z)}{r_{d}}=\frac{\left[(1+z)^{3}+\Omega_{l}-1\right]^{-\frac{1}{2}}}{\int_{z_{d}}^{\infty}\left[3\left(1+\frac{R_{b}}{1+z^{\prime}}\right)(1+z^{\prime})^{3}f(\tilde{\omega}_{m},\omega_{m})\right]^{-\frac{1}{2}}\mathrm{d}z^{\prime}},$
(9)

where $\Omega_{l}\equiv 1/\Omega_{m}$, and

$f(\tilde{\omega}_{m},\omega_{m})\equiv\frac{\tilde{\omega}_{m}}{\omega_{m}}\left(1+\frac{1+z}{1+z_{eq}}\right).$
(10)

The Hubble constant $H_{0}$ and matter density fraction $\Omega_{m}$ are fully
degenerate with the form $\omega_{m}$. We can also find in Eq.(10) the ratio
of $\omega_{m}$ is degenerate with the late cosmological parameter
$\omega_{m}$ in the form of a sum due to $z_{eq}\propto\tilde{\omega}_{m}$. We
will see the CMB data can break this degenerate relation in the next section.
In addition, constraining this ratio help us compare the early and late
cosmological parameters directly. Therefore, we introduce ${\rm
ratio}(\omega_{m})\equiv\tilde{\omega}_{m}/\omega_{m}$ as a new parameter. The
analysis of transverse measurement $D_{M}(z)/r_{d}$ is similar to
$D_{H}(z)/r_{d}$.

The BAO measurements used in this work are taken from the 6dF Galaxy Survey
(6dFGS,Beutler  _et al._ (2011)), the complete Sloan Digital Sky Survey (SDSS)
data release, including SDSS-Main Galaxy Sample (MGS,Ross  _et al._ (2015)),
the BOSS and the extended Baryon Oscillation Spectroscopic Survey (eBOSS) DR16
data Alam  _et al._ (2021); Des Bourboux  _et al._ (2020). A specific
description of BAO data is shown in TABLE 1.

The likelihood of BAO $\mathcal{L}_{BAO}$ have different forms for different
datasets. The covariance matrix of SDSS DR12 and eBOSS Quasar data, and the
full (non-Gaussian) likelihoods for the Ly$\alpha$ forest data Des Bourboux
_et al._ (2020) and emission line galaxies (ELG) analysis De Mattia  _et al._
(2021), are from the public SDSS
repository222https://svn.sdss.org/public/data/eboss/DR16cosmo/tags
/v1_0_0/likelihoods/..

$z_{eff}$ | Measured Quantity | Value | Dataset | Reference
---|---|---|---|---
0.106 | $r_{d}/D_{V}$ | $0.336\pm 0.015$ | 6dFGS | Beutler  _et al._ (2011)
0.15 | $D_{V}/r_{d}$ | $4.47\pm 0.17$ | SDSS DR7 | Ross  _et al._ (2015)
0.38 | $D_{H}/r_{d}$ | $25.00\pm 0.76$ | SDSS DR12 | Alam  _et al._ (2021)
0.38 | $D_{M}/r_{d}$ | $10.23\pm 0.17$ | SDSS DR12 | Alam  _et al._ (2021)
0.51 | $D_{H}/r_{d}$ | $22.33\pm 0.58$ | SDSS DR12 | Alam  _et al._ (2021)
0.51 | $D_{M}/r_{d}$ | $13.36\pm 0.21$ | SDSS DR12 | Alam  _et al._ (2021)
0.70 | $D_{H}/r_{d}$ | $19.33\pm 0.53$ | eBOSS LRG | Alam  _et al._ (2021)
0.70 | $D_{M}/r_{d}$ | $17.86\pm 0.33$ | eBOSS LRG | Alam  _et al._ (2021)
0.85 | $D_{V}/r_{d}$ | $18.33^{+0.57}_{-0.62}$ | eBOSS ELG | Alam  _et al._ (2021)
1.48 | $D_{H}/r_{d}$ | $13.26\pm 0.55$ | eBOSS Quasar | Alam  _et al._ (2021)
1.48 | $D_{M}/r_{d}$ | $30.69\pm 0.80$ | eBOSS Quasar | Alam  _et al._ (2021)
2.33 | $D_{H}/r_{d}$ | $8.93\pm 0.28$ | eBOSS Ly$\alpha\times$ Ly$\alpha$ | Des Bourboux  _et al._ (2020)
2.33 | $D_{M}/r_{d}$ | $37.6\pm 1.9$ | eBOSS Ly$\alpha\times$ Ly$\alpha$ | Des Bourboux  _et al._ (2020)
2.33 | $D_{H}/r_{d}$ | $9.08\pm 0.34$ | eBOSS Ly$\alpha\times$ Quasar | Des Bourboux  _et al._ (2020)
2.33 | $D_{M}/r_{d}$ | $37.3\pm 1.7$ | eBOSS Ly$\alpha\times$ Quasar | Des Bourboux  _et al._ (2020)
Table 1: Datasets measuring the BAO peak that are used in our cosmological
analysis.

###  II.2 CMB data

CMB observation contribution in the constraint of cosmological parameter can
be converted to CMB shift parameters for breaking degeneracy of parameters,
instead of using the full CMB power spectrumWang and Mukherjee (2007); Wang
and Wang (2013):

$\displaystyle R$ $\displaystyle=$
$\displaystyle\sqrt{\Omega_{m}H^{2}_{0}}r(z_{*})/c,$ (11) $\displaystyle
l_{a}$ $\displaystyle=$ $\displaystyle\pi r(z_{*})/r_{s}(z_{*}).$ (12)

These parameters can be considered as the CMB measurements which are extracted
from CMB likelihood, where $z_{*}$ is the redshift of recombination. The
sufficient information of Planck data release Aghanim  _et al._ (2020) for
constraint is given by CMB shift parameters with baryon density
$\omega_{b}=\Omega_{b}h^{2}$.

The redshift $z_{*}$ can be calculated by the fitting formula Hu and Sugiyama
(1996):

$z_{*}=1048[1+0.00124\omega^{-0.738}_{b}][1+g_{1}\omega^{g_{2}}_{m}],$ (13)

where

$g_{1}=\frac{0.0783\omega^{-0.238}_{b}}{1+39.5\omega^{0.763}_{b}},g_{2}=\frac{0.560}{1+21.1\omega^{1.81}_{b}}.$
(14)

The final result, which is extracted from Planck 2018 TT,TE,EE+lowE chains,
can be transformed in terms of a data vector $\bm{v}=(R,l_{a},\omega_{b})^{T}$
333Actually, spectral index of the primordial power spectrum $n_{s}$ is
entailed for completeness, however we neglect $n_{s}$ because it doesn’t
relate to our constraint. and their covariance matrix. For a flat universe
Zhai and Wang (2019), this data vector is

$\bm{v}=\begin{pmatrix}1.74963\\\ 301.80845\\\ 0.02237\\\ \end{pmatrix}$ (15)

and the covariance matrix is

$\bm{C_{v}}=10^{-8}\times\begin{pmatrix}1598.9554&17112.007&-36.311179\\\
17112.007&811208.45&-494.79813\\\
-36.311179&-494.79813&2.1242182\end{pmatrix}$

The compression of CMB data forms the so-called distance priors Wang and
Mukherjee (2007), which can help us break the degeneracy of the BAO data:

$l_{a}=\frac{\pi
R}{\int_{z_{*}}^{\infty}\left[3\left(1+\frac{R_{b}}{1+z^{\prime}}\right)(1+z^{\prime})^{3}\left(1+\frac{1+z^{\prime}}{1+z_{eq}}\right)\right]^{-\frac{1}{2}}\mathrm{d}z^{\prime}}.$
(16)

Apparently, $\tilde{\omega}_{m}$ is the only cosmological parameter to be
determined in the CMB shift parameter $l_{a}$. Namely, combining the CMB data
can give a constraint on $\tilde{\omega}_{m}$ and break the degeneracy to
release ${\rm ratio}(\omega_{m})$. Therefore the final parameter set is now

$\displaystyle\centering\\{\tilde{\omega}_{b},\tilde{\omega}_{m},\Omega_{l},{\rm
ratio}(\omega_{m})\\}.\@add@centering$

In summary, the baryon density $\tilde{\omega}_{b}$ and the matter density
$\tilde{\omega}_{m}$ serve as early parameters to compute sound horizon
$r_{s}$. $\Omega_{l}$ describes the evolution of late universe and ${\rm
ratio}(\omega_{m})$ is a bridge between them.

The loglikelihood function for CMB can be written in the form:

$\mathcal{L}_{CMB}=-\frac{1}{2}\chi^{2}_{CMB}=-\frac{1}{2}(\bm{v}-\bm{v}_{th})^{T}\bm{C_{v}}^{-1}(\bm{v}-\bm{v}_{th}),$
(17)

and the complete likelihood for our cosmological analysis is the following
$\mathcal{L}=\mathcal{L}_{\rm CMB}+\mathcal{L}_{\rm BAO}$. We run a Markov
Chain Monte Carlo (MCMC) with the sampler emcee Foreman-Mackey  _et al._
(2013) to constrain parameters, where the priors chosen for our Bayesian
calculations regarding the cosmological parameters are uniform distributions:
$\tilde{\omega}_{b}\in(0.001,0.3)$, $\tilde{\omega}_{m}\in(0.05,0.5)$,
$\Omega_{l}\in(1,10)$, ${\rm ratio}(\omega_{m})\in(0.1,10)$.

##  III Results and Discussion

(a)

(b)

Figure 1: Posterior likelihoods (68% and 95% confidence level) for the initial
parameter set (a) and final parameter set (b) considering the full samples of
CMB and BAO.

The limitations of the parameters using BAO+CMB are illustrated in FIG.1, and
the results are also summarized in TABLE 2. Notably, the parameters of initial
set show complex degeneracy compared to those of the final set, particularly
for early parameters such as $\tilde{\Omega}_{b}$, $\tilde{\Omega}_{m}$ and
$\tilde{H}_{0}$. The fundamental reason for the degeneracy was elaborated in
the previous section.

Parameters | Value
---|---
$\tilde{\omega}_{b}$ | $0.02237\pm 0.00014$
$\tilde{\omega}_{m}$ | $0.1430\pm 0.0011$
$\Omega_{l}$ | $3.287\pm 0.087$
${\rm ratio}(\omega_{m})$ | $1.0069\pm 0.0070$
Table 2:  Cosmological constraints on early and late parameters
simultaneously.

###  III.1 Comparison with separate constraints

There are early parameters and late parameters incorporated in our model,
enabling the simultaneous constraint of both. Note that, there are substantial
cosmological parameters estimations obtained from high-redshift or low-
redshift data alone. Therefore, we firstly compare the results presented in
this paper with other results which are constrained separately before
discussing on the ${\rm ratio}(\omega_{m})$ and cosmological models.

Initially, we concentrate on the physical density of the early universe,
denoted by $\tilde{\omega}_{m}=0.1430\pm 0.0011$. Another recent estimation of
early cosmological parameters from CMB observations almost independently of
the late-time evolution yields $\tilde{\omega}_{m}=0.1414\pm 0.0013$ Lemos and
Lewis (2023). They accomplish it by constructing a CMB likelihood that does
not require a cosmological model to model the late-time effects. This
indicates that the estimation of early parameters obtained independently is
consistent with the simultaneous constraint.

In contrast of degeneracy of early parameters, we can analyze late parameters
$H_{0}$ and $\Omega_{m}$ individually. Their posteriors are derived from
$\tilde{\omega}_{m}$, $\Omega_{l}$ and ${\rm ratio}(\omega_{m})$ showed in
TABLE 2, and given by

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle 68.34\pm 0.97\rm\ km\
s^{-1}Mpc^{-1},$ $\displaystyle\Omega_{m}$ $\displaystyle=$ $\displaystyle
0.3042\pm 0.0081.$ (18)

We can compare them with these values constrained by low-redshift
observations, such as Supernovae. At first, we concentrate on the matter
density $\Omega_{m}$ and adopt $\Omega_{m}=0.298\pm 0.022$ from the Pantheon
analysis Scolnic  _et al._ (2018). It is apparent that the Pantheon result is
in agreement with ours. We also notice that constraint of $\Omega_{m}$ in this
work is tighter than the result Pogosian  _et al._ (2020)
$\Omega_{m}=0.294^{+0.014}_{-0.016}$ constrained by BAO with a prior of
$\omega_{m}$.

We divide the BAO measurements into early and late portions to limit the value
of the Hubble constant, $H_{0}$, using information from the late universe.
Therefore it is interesting that compare it with $H_{0}$ measurements which
are derived from local observation of SNIa, such as the SH0ES Team’s analysis.
They measured via a three-step distance ladder employing a single,
simultaneous fit between geometric distance measurements to standardized
Cepheid variables, standardized Cepheids and colocated SNIa in nearby
galaxies, and SNIa in the Hubble flow. Namely the analysis employs a direct
and cosmological model-independent measurement, refers $H_{0}=73.04\pm
1.04\rm\ km\ s^{-1}Mpc^{-1}$. The $3.3\sigma$ tension with constraint in this
paper indicates that the Hubble tension is not only between high-redshift and
local measurements, but also low-redshift measurements. However, we have found
that replacing Cepheid variables with TRGB method Freedman (2021) to calibrate
SNIa yields a consistent value of $H_{0}=69.8\pm 1.7\rm\ km\ s^{-1}Mpc^{-1}$.
We expect the local measurements of $H_{0}$ will be more precise in the
future.

Figure 2: Fractional errors on $D_{M}(z)/r_{d}$ and $D_{H}(z)/r_{d}$ for
galaxy surveys, as a function of redshift. Figure 3: Constraints (68% and 95%
confidence level) on ${\rm ratio}(\omega_{m})$ by using CMB+Nancy Roman,
CMB+4MOST, CMB+CSST and CMB+Euclid.

###  III.2 Test of $\Lambda$CDM model

As mentioned above, differences in early or late cosmological parameters
solely depend on the redshift of the measurements used in the constraint.
Namely the two values of a cosmological parameter should be identical, or else
the expansion history described by the $\Lambda$CDM model is not compatible.
We take $\omega_{m}$ as a representative to analyze the potential difference
between the early and late universe due to parameter degeneracy. We find that

${\rm ratio}(\omega_{m})=1.0069\pm 0.0070,$ (19)

with the unit falling within the $1\sigma$ range. This result indicates there
is no discrepancy between the early and late epochs of the universe in
$\Lambda$CDM model. Therefore, from these analysis, we cannot find the tension
between early and late universe in the BAO and CMB data, which hints that
Hubble tension may originate from the differences of datasets or the
systematic error in the current data.

###  III.3 Test of EDE model

In recent years, the EDE model has emerged as a potential solution for
addressing the Hubble tension. The theory involves an additional scalar field
operating in the early universe, which is frozen and acts like a cosmological
constant before a critical redshift $z_{c}$ (around matter-radiation equality
$z_{eq}$). But its energy density then dilutes faster than radiation and
almost vanishes after recombination. This addresses the Hubble tension by
increasing the early expansion rate while leaving the later evolution of the
universe unchanged. Therefore, it is natural to expect that EDE have a
different effect on the early universe (at $z>z_{*}$) and the late universe
(at $z<z_{*}$). The difference can be manifested as the value of ${\rm
ratio}(\omega_{m})$ which is induced from an EDE model. Namely, this value
will deviate from the unit and we will see how large the deviation of ${\rm
ratio}(\omega_{m})$ can be.

Parameters |  | Parameters |
---|---|---|---
$H_{0}\rm[\ km\ s^{-1}Mpc^{-1}]$ | $71.01^{+1.1}_{-1}$ | $\Omega_{m}$ | $0.3022^{+0.0053}_{-0.0054}$
$100\omega_{b}$ | $2.28^{+0.021}_{-0.022}$ | $\omega_{\rm cdm}$ | $0.1289\pm 0.0039$
$f_{\rm EDE}(z_{c})$ | $0.097^{+0.035}_{-0.029}$ | ${\rm Log}_{10}(z_{c})$ | $3.61^{+0.13}_{-0.054}$
$r_{s}(z_{*})\rm[Mpc]$ | $139.4^{+1.7}_{-1.9}$ |  |
Table 3:  The EDE parameters Murgia  _et al._ (2021) used in our fiducial EDE
cosmology are reconstructed from the combined analysis of various data sets
(Planck+Pantheon+SH0ES+BAO).

In this analysis, we fix the fiducial cosmology to the EDE model constraint
results from Ref.Murgia  _et al._ (2021) and the parameters are listed in
TABLE 3. Note that ${\rm ratio}(\omega_{m})$ is constrained from the fiducial
EDE parameters directly but analysed assuming $\Lambda$CDM. As mentioned
above, the additional component, early dark energy, works mainly in the early
universe to reduce the sound horizon and vanishes in the late universe to
maintain other properties invariant. And all early information that we derive
from measurements are provided by the sound horizon, therefore, the parameter
$\tilde{\omega}_{m}$ for the early universe can be derived through their sound
horizon constraint $r_{s}(z_{*})=139.4^{+1.7}_{-1.9}\ \rm Mpc$. Furthermore
the late universe parameter $\omega_{m}$ can be obtained directly because EDE
model regresses to $\Lambda$CDM after the field become dynamical. Then we can
test EDE by comparing the ${\rm ratio}(\omega_{m})$ induced from EDE with the
${\rm ratio}(\omega_{m})$ constrained by the realistic data. If these two
values of ${\rm ratio}(\omega_{m})$ are not consistent, it means there are no
differences which are contributed by EDE in the realistic observation. In
other words, the deviation of the induced ${\rm ratio}(\omega_{m})$ is a
signal of supporting EDE and our work is to find whether this signal exists in
realistic data. The complete test steps can be generalized as follows:

  1. (i)

Deriving $\tilde{\omega}_{m}$ from $r_{s}(z_{*})$ constraint of the fiducial
EDE model.

  2. (ii)

Deriving $\omega_{m}$ from the fiducial EDE model directly.

  3. (iii)

Comparing ${\rm ratio}(\omega_{m})$ inferred from the EDE fiducial model with
the realistic one.

As a result, we find that the induced ${\rm ratio}(\omega_{m})=1.075\pm
0.056$, which serves as a signal of EDE. This value is about $1.2\sigma$
disagreement with the result constrained by realistic data. Note that the
authors of Ref.Farren  _et al._ (2022) implement a similar test for EDE by
using the difference of EDE between the equality scale (at $z\approx 3600$)
and the sound-horizon scale (at $z\approx 1050$), similarly, they find EDE is
not supported.

Figure 4: Fractional errors on $D_{A}(z)$ and $H(z)$ for 21 cm IM surveys, as
a function of redshift. Figure 5: Constraints (68% and 95% confidence level)
on ${\rm ratio}(\omega_{m})$ by using CMB+BINGO, CMB+MeerKAT, CMB+FAST,
CMB+CHIME, CMB+SKA and CMB+Tianlai.

##  IV Forecasts for the future surveys

We aim to analyze the surveys that measures BAO features with higher accuracy,
assessing their potential to enhance precison in our cosmological parameter
estimates ${\rm ratio}(\omega_{m})$. In this section, we consider two kinds of
potention observations on BAO: the galaxy surveys and the neutral hydrogen
surveys.

The method with galaxy surveys has been discussed above, which catalog
individual galaxies in angle and redshift to construct a map of matter
distribution. While the method with neutral hydrogen surveys is proposed in
Chang  _et al._ (2008), which detects structures in the universe traced by the
redshifted diffuse 21 cm hyperfine transition line of neutral hydrogen ($\rm
H_{I}$) using the intensity mapping (IM) technique. The idea is to measure the
$\rm H_{I}$ brightness temperature field to obtain information on the high
levels of hydrogen abundance. This will result in a hydrogen map that can
serve as a distinct tracer from galaxies, aiding in the measurement of the BAO
scale. In both cases, the fiducial $\Lambda$CDM flat cosmological model with
parameters is inferred by Planck Aghanim  _et al._ (2020) is used in the
analysis.

###  IV.1 Galaxy Surveys

We use the analytic fitting formula developed by Blake  _et al._ (2006) to
depict the standard ruler accuracies in terms of the galaxy redshift survey
parameters. Their analysis is based on Monte Carlo realizations Blake and
Glazebrook (2003); Glazebrook and Blake (2005) and the covariance matrix
evaluated in Peacock and Dodds (1994). Three configuring parameters are
required for spectroscopic redshift surveys: central redshift $z$, total
volume $V$, average number density of galaxies $n$. The formula is

$x=x_{0}\sqrt{\frac{V_{0}}{V}}\left(1+\frac{n_{eff}}{n}\frac{D(z_{0})^{2}}{b(z)^{2}D(z)^{2}}\right),$
(20)

where $x$ is the fractional error, i.e. if $A$ is a measurement thus
$x\equiv\sigma_{A}/A$, and normalized parameter $x_{0}$, fiducial survey
volume $V_{0}$ and linear growth factor $D(z_{0})$ defined in Carroll  _et
al._ (1992) are listed in Table 1 of Blake  _et al._ (2006) along with other
best-fitting coefficients.

Four surveys with distinct objectives are considered, including 4MOST
444https://4MOST (4-meter Multi-Object Spectroscopic Telescope) Richard  _et
al._ (2019), Roman 555http://roman.gsfc.nasa.gov (Nancy Grace Roman Space
Telescope) Spergel  _et al._ (2013), CSST 666http://www.nao.cas.cn/csst/ (The
China Space Station Telescope) Zhan (2011, 2021) and Euclid
777https://www.euclid-ec.org Laureijs  _et al._ (2011); Blanchard  _et al._
(2020). The configuration parameters are listed in Table 4.

| 4MOST | N.Roman | CSST | Euclid
---|---|---|---|---
$z_{\rm min}$ | 0.15 | 1.0 | 0 | 0.6
$z_{\rm max}$ | 2.2 | 2.8 | 1.6 | 2.1
$S_{\rm area}[\rm deg^{2}]$ | 7500(1000) | 2000 | 17500 | 15000
$n$ | Richard  _et al._ (2019) | Font-Ribera  _et al._ (2014) | Ding  _et al._ (2023) | Font-Ribera  _et al._ (2014)
Table 4: The configurations of spectroscopic galaxy surveys used in this
paper. $n$ is the number density of galaxies. Note that 4MOST observe
different sources in different sky coverage area.

By using the fitting formula Eq.(20), we can obtain the fractional errors for
both transverse measurement $D_{M}(z)/r_{d}$ and line-of-sight measurement
$D_{H}(z)/r_{d}$. FIG 2 shows the features of different surveys: 4MOST covers
a large redshift range and CSST performs optimally for observing low-redshift
objects $(z<1)$. However, in redshift range $1<z<2$ Euclid has advantages and
Roman are precise enough in the measurement of high redshift galaxies $(z>2)$.

We perform a Markov Chain Monte Carlo analysis using our simulation data
generated with the CMB distance prior to constrain cosmological parameters.
The forecast results using different surveys are plotted in FIG 3 and list
$1\sigma$ errors of ${\rm ratio}(\omega_{m})$ in TABLE 5. We can find that
Roman and 4MOST are similar in constraining the ratio of $\omega_{m}$. Thus,
either Roman $\sigma({\rm ratio}(\omega_{m}))=0.0075$ or 4MOST $\sigma({\rm
ratio}(\omega_{m}))=0.0072$ alone have comparable precision with the
combination of current BAO probes $\sigma({\rm ratio}(\omega_{m}))=0.0070$.
CSST and Euclid can tighten the constraints due to their large sky cover area.
Euclid gives the best constraints in galaxy redshift surveys, $\sigma({\rm
ratio}(\omega_{m}))=0.0050$, which means Euclid alone can achieve about 30
percent improvement in precision.

Surveys | $\sigma({\rm ratio}(\omega_{m}))$
---|---
Nancy Roman | 0.0075
4MOST | 0.0072
CSST | 0.0054
Euclid | 0.0050
Table 5: Forecast $1\sigma$ errors of ${\rm ratio}(\omega_{m})$ derived from
future galaxy redshift surveys combined with Planck CMB data.

###  IV.2 21 cm IM Surveys

We use the public code RadioFisher Bull  _et al._ (2015) to obtain the Fisher
matrix for $D_{A}(z)$ and $H(z)$, allowing us to invert it and derive the
covariance matrix for observations of future IM surveys. The core of code is
the construction and calculation of the Fisher matrix. For a cosmological
parameter set $\\{p\\}$, the Fisher matrix can be expressed as Seo and
Eisenstein (2007); Bull  _et al._ (2015); Wu and Zhang (2022)

$F_{ij}=\frac{V_{\rm
bin}}{8\pi^{2}}\int_{-1}^{1}\mathrm{d}\mu\int_{k_{min}}^{k_{max}}k^{2}\mathrm{d}k\frac{\partial\ln
C^{T}}{\partial p_{i}}\frac{\partial\ln C^{T}}{\partial p_{j}},$ (21)

where $V_{\rm bin}$ is the physical volume, $\mu$ is the cosine of the angle
between the line of sight and the direction of $k$. The parameter set
$\\{p\\}$ of this analysis includes the angular diameter distance $D_{A}(z)$,
Hubble parameter $H(z)$, redshift-space distortion (RSD) observable
$[f\sigma_{8}](z)$, $\rm H_{I}$ bias parameter $[b_{HI}\sigma_{8}](z)$ and
non-linear dispersion scale parameter $\sigma_{\rm NL}$. And $C^{T}$ is the
total covariance of 21 cm emission line measurements, including the
contribution of signal $C^{S}$, noise $C^{N}$ and foreground $C^{F}$. More
details are discussed in Sec.2 of Ref.Wu and Zhang (2022).

We determine several 21 cm IM surveys, including single-dish surveys BINGO
888https://bingotelescope.org (Baryon Acoustic Oscillations from Integrated
Neutral Gas Observations) Battye  _et al._ (2013); Wuensche  _et al._ (2020);
Abdalla  _et al._ (2022b), FAST 999https://fast.bao.ac.cn (Five-hundred-meter
Aperture Spherical Telescope) Nan  _et al._ (2011); Smoot and Debono (2017)
and dish interferometer surveys MeerKAT radio telescope 101010https://meerkat
Jonas (2009); Santos  _et al._ (2015); Bacon  _et al._ (2020), SKA
111111https://www.skatelescope.org (Square Kilometre Array) Dewdney  _et al._
(2013) and cylinder interferometer surveys CHIME 121212https://chime-
experiment.ca (Canadian Hydrogen Intensity Mapping Experiment) Newburgh  _et
al._ (2014); Bandura  _et al._ (2014), Tianlai 131313http://tianlai.bao.ac.cn
Chen (2012); Li  _et al._ (2020). The configuration parameters of 21 cm IM
surveys include the receiver instrument noise temperature $T_{\rm inst}$, the
number of dishes $N_{d}$ and beams $N_{b}$, diameter of the dish $D_{d}$ and
the survey area $S_{\rm area}$, which are listed in Table 6.

| BINGO | FAST | MeerKAT | SKA | CHIME | Tianlai
---|---|---|---|---|---|---
$N_{d}$ | $1$ | $1$ | $64$ | $190$ | $1280$ | $2048$
$N_{b}$ | $50$ | $20$ | $1$ | $1$ | $1$ | $1$
$T_{\rm inst}\ [\rm K]$ | 50 | 20 | 29 | 28 | 50 | 50
$D_{\rm dish}\ [\rm m]$ | 40 | 300 | 13.5 | 15 | 20 | 15
$z_{\rm min}$ | 0.13 | 0 | 0.4 | 0.35 | 0.77 | 0.49
$z_{\rm max}$ | 0.45 | 0.35 | 1.45 | 3 | 2.55 | 2.55
$S_{\rm area}[\rm deg^{2}]$ | 3000 | 20000 | 25000 | 25000 | 20000 | 20000
Table 6: The configurations of 21 cm IM surveys used in this paper is mainly
from Ref.Bull  _et al._ (2015).

The fractional errors of the BAO measurements are showed in FIG. 4 and it is
clear Tianlai have the most precise measurements. We perform a Markov Chain
Monte Carlo analysis using our simulation data generated with the CMB distance
prior to constrain cosmological parameters. From the contours displayed in
FIG. 5 and $1\sigma$ errors summarized in TABLE 7, BINGO and FAST provide weak
constraints on ${\rm ratio}(\omega_{m})$ due to their narrow redshift range.
Additionally, the constraint given by MeerKAT is between them because it
covers a small sky area in contrast to FAST or SKA. However, CHIME, SKA and
Tianlai performs tighter constraint than the combination of current BAO
probes, with $\sigma({\rm ratio}(\omega_{m}))=0.0063,\ 0.0058,\ 0.0047$
respectively, for their wide redshift ranges and high spatial resolution.
Especially Tianlai enhances the ability of constraining ${\rm
ratio}(\omega_{m})$ significantly, which means we can investigate the
potential discrepancy between early and late universe more accurately.

Surveys | $\sigma({\rm ratio}(\omega_{m}))$
---|---
BINGO | 0.0110
MeerKAT | 0.0098
FAST | 0.0082
CHIME | 0.0063
SKA | 0.0058
Tianlai | 0.0047
Table 7: Forecast $1\sigma$ errors of ${\rm ratio}(\omega_{m})$ derived from
future 21 cm IM surveys combined with Planck CMB data.

##  V Conclusions

In recent years, the Hubble tension between the indirect model dependent
estimates at early time and the direct late time model-independent
measurements have increased to $5\sigma$. This huge discrepency prompt us to
investigate its origin. Therefore, we desire to know whether the inconsistency
of the cosmological parameters exists in the same observed data. We utilize
two discrete sets of parameters describing the expansion history of the
Universe in its early and late stages respectively. The recombination redshift
$z_{*}$ is chosen as the dividing line to distinguish between the early and
late epochs. Because of the degeneracy, we convert the complete parameter set
from
$\\{\tilde{\Omega}_{b},\tilde{\Omega}_{m},\tilde{H}_{0},\Omega_{m},H_{0}\\}$
to $\\{\tilde{\omega}_{b},\tilde{\omega}_{m},\Omega_{l},{\rm
ratio}(\omega_{m})\\}$.

The combination of BAO and CMB data gives constraints on both early and late
parameters at the same time, allowing for a comparison with early and late
parameters that are constrained separately. We find that our constraints are
almost all consistent with separate constraints except $H_{0}$ derived from
SH0ES. The most important result is that the ratio of $\tilde{\omega}_{m}$ to
$\omega_{m}$ can help us judge the consistency between early and late
cosmological parameters. The result ${\rm ratio}(\omega_{m})=1.0069\pm 0.0070$
implicates the early epoch and late epoch of the Universe described by
$\Lambda$CDM are consistent. It means that Hubble tension may arise from the
difference of datasets or the systematic error in the current data, rather
than new physics beyond $\Lambda$CDM cosmology.

To assess the level of precision to which ${\rm ratio}(\omega_{m})$ can be
constrained, simulation data of future BAO observations has been generated.
This includes galaxy redshift surveys as well as 21 cm Intensity Mapping (IM)
surveys. In galaxy surveys, we find Euclid gives the best constraint with a
$30\%$ improvement in precision, $\sigma({\rm ratio}(\omega_{m}))=0.0050$. In
21 cm IM surveys, SKA and Tianlai significantly tighten the constraints with
$\sigma({\rm ratio}(\omega_{m}))=0.0058$ and $\sigma({\rm
ratio}(\omega_{m}))=0.0047$. Taking into account new future CMB observations,
the consistency of cosmological parameters will face a challenging assessment.
We hope that these powerful future observations can shed light on the origin
of the Hubble tension.

###### Acknowledgements.

We appreciate the helpful discussion with Philip Bull and Salvatore
Capozziello. This work is supported by the National Key R&D Program of China
(Grant No. 2021YFC2203102 and 2022YFC2204602), Strategic Priority Research
Program of the Chinese Academy of Science (Grant No. XDB0550300), the National
Natural Science Foundation of China (Grant No. 12325301 and 12273035), the
Fundamental Research Funds for the Central Universities (Grant No.
WK2030000036 and WK3440000004), the Science Research Grants from the China
Manned Space Project (Grant No.CMS-CSST-2021-B01), the 111 Project for
”Observational and Theoretical Research on Dark Matter and Dark Energy” (Grant
No. B23042).

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# A unified hoop conjecture for black holes and horizonless compact stars

Yan<EMAIL_ADDRESS>
1 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong
273165, China
2 Center for Gravitation and Cosmology, College of Physical Science and
Technology, Yangzhou University, Yangzhou 225009, China

###### Abstract

Abstract

We propose a unified version of hoop conjecture valid for various black holes
and horizonless compact stars. This conjecture is expressed by the mass to
circumference ratio $4\pi M_{in}/C\leqslant 1$, where C is the circumference
of the smallest ring that can engulf the object in all azimuthal directions
and $M_{in}$ is the mass within the engulfing sphere.

###### pacs:

11.25.Tq, 04.70.Bw, 74.20.-z

##  I Introduction

The famous hoop conjecture introduced almost five decades ago asserts that the
existence of black hole horizons is characterized by the mass and
circumference relation $4\pi\mathcal{M}/C\geqslant 1$ hc1 ; hc2 . Here C is
the circumference of the smallest ring that can engulf the black hole in all
azimuthal directions and $\mathcal{M}$ is usually interpreted as the
asymptotically measured total ADM mass of black holes hc3 -ahc9 .

For horizonless compact stars, the curved spacetimes should be characterized
by the relation $4\pi\mathcal{M}/C<1$ hc1 ; hc2 . However, if $\mathcal{M}$ is
still interpreted as the total ADM mass, this relation $4\pi\mathcal{M}/C<1$
is violated in the background of horizonless charged compact objects hc19 ;
hc20 . In fact, Thorne has not provided an explicit definition for the mass
term $\mathcal{M}$ in the mass to circumference ratio hc1 . And it has been
proved that the relation $4\pi\mathcal{M}/C<1$ holds in horizonless spacetimes
when $\mathcal{M}$ is interpreted as the mass contained within the engulfing
sphere (not as the total ADM mass) hc21 ; hc22 .

It is natural to assume that a unified hoop conjecture may be
$4\pi\mathcal{M}/C\geqslant 1$ for black holes and $4\pi\mathcal{M}/C<1$ for
horizonless compact stars, where $\mathcal{M}$ is the total ADM mass or the
mass contained in the engulfing sphere hc23 . In the background of horizonless
stars, the calculations in hc21  imply that the term $\mathcal{M}$ should be
the mass contained in the sphere and cannot be the total ADM mass. However, in
the black hole spacetimes, Hod found that the hoop conjecture holds if
$\mathcal{M}$ is the ADM mass and cannot be the mass contained in the sphere
hc23 . So it seems that there is no unified hoop conjecture valid for both
black holes and compact stars.

In this paper, we propose another version of unified hoop conjecture, which
holds for Schwarzschild black holes, neutral Kerr black holes, RN black holes,
Kerr-Newman black holes and horizonless charged compact stars. It may be a
general property for both black holes and horizonless compact stars.

##  II Test the unified hoop conjecture

In the background of horizonless charged compact stars, it has recently been
proved that the relation $4\pi\mathcal{M}/C<1$ is violated when $\mathcal{M}$
is interpreted as the total ADM mass and the relation holds if
$\mathcal{M}=M_{in}$, where $M_{in}$ is the mass contained within the
engulfing sphere hc19 ; hc20 ; hc21 ; hc22 . In addition, the numerical data
in hc23  implies that $4\pi M_{in}/C\leqslant 1$ may be a general property in
black hole spacetimes. With these facts, we propose a unified version of hoop
conjecture

$\displaystyle 4\pi M_{in}/C\leqslant 1,$ (1)

where C is the circumference of the smallest ring that can engulf the object
in all azimuthal directions and $M_{in}$ is the mass within the engulfing
sphere. The relation (1) is natural according to the idea that the ring of the
engulfing sphere is related to the mass within the sphere (not the mass in the
total spacetime).

In the following, we give examples supporting the bound (1) following studies
in hc19 ; hc20 ; hc21 ; hc22 ; hc23 . A Kerr-Newman black hole spacetime is
described by the curved line element hc23

$\displaystyle ds^{2}$
$\displaystyle=-\frac{\Delta-a^{2}sin^{2}\theta}{\rho^{2}}dt^{2}+\frac{\rho^{2}}{\Delta^{2}}dr^{2}-\frac{2asin^{2}\theta(2Mr-Q^{2})}{\rho^{2}}dtd\phi+\rho^{2}d\theta^{2}+\frac{(r^{2}+a^{2})^{2}-a^{2}\Delta
sin^{2}\theta}{\rho^{2}}sin^{2}\theta d\phi^{2},$ (2)

where M is the ADM mass, Q is the electric charge, $J=Ma$ is the angular
momentum, $\Delta=r^{2}-2Mr+Q^{2}+a^{2}$ and
$\rho^{2}=r^{2}+a^{2}cos^{2}\theta$. Black hole horizons are expressed as

$\displaystyle r_{\pm}=M\pm(M^{2}-Q^{2}+a^{2})^{1/2},$ (3)

which are determined by the zeros of the metric function $\Delta(r)=0$.

Setting $dt=dr=d\theta=0,\theta=\frac{\pi}{2}$, we get the relation

$\displaystyle ds=\frac{r_{+}^{2}+a^{2}}{r_{+}}d\phi.$ (4)

Also taking $\Delta\phi=2\pi$, we obtain the equatorial circumference

$\displaystyle C=2\pi\frac{r_{+}^{2}+a^{2}}{r_{+}}.$ (5)

For Schwarzschild black holes, there is $Q=0$ and $a=0$. The mass to
circumference ratio is

$\displaystyle\frac{4\pi M_{in}}{C}=\frac{4\pi M}{C}=\frac{4\pi M}{2\pi
r_{+}}=\frac{4\pi M}{2\pi(2M)}=1.$ (6)

For RN black holes with $Q\neq 0$ and $a=0$, the ratio satisfies

$\displaystyle\frac{4\pi M_{in}}{C}=\frac{4\pi(M-\frac{Q^{2}}{2r_{+}})}{2\pi
r_{+}}=\frac{4Mr_{+}-2Q^{2}}{2r_{+}^{2}}=\frac{4M(M+\sqrt{M^{2}-Q^{2}})-2Q^{2}}{2(M+\sqrt{M^{2}-Q^{2}})^{2}}=1.$
(7)

The neutral Kerr black holes correspond to $Q=0$ and $a\neq 0$. It yields the
relation

$\displaystyle\frac{4\pi M_{in}}{C}=\frac{4\pi
M}{2\pi\frac{r_{+}^{2}+a^{2}}{r_{+}}}=\frac{2Mr_{+}}{r_{+}^{2}+a^{2}}=\frac{2M(M+\sqrt{M^{2}-a^{2}})}{(M+\sqrt{M^{2}-a^{2}})^{2}+a^{2}}=1.$
(8)

In the background of Kerr-Neuman black holes, both charge and rotating
parameters are nonzero as $Q\neq 0$ and $a\neq 0$. The equatorial
circumference is $C=2\pi\frac{r_{+}^{2}+a^{2}}{r_{+}}$ and the mass contained
within the black hole horizon is
$M_{in}=M-\frac{Q^{2}}{4r_{+}}[1+\frac{r_{+}^{2}+a^{2}}{ar_{+}}arctg(\frac{a}{r_{+}})]$
mass . In this case of Kerr-Neuman black holes, numerical data in hc23
implies an upper bound on the ratio

$\displaystyle 4\pi M_{in}/C\leqslant 1.$ (9)

The horizonless charged compact star satisfies the inequality hc21 ; hc22

$\displaystyle 4\pi M_{in}/C<1.$ (10)

According to relations (6-10), we propose the unified hoop conjecture (1),
which may be a general property for both black holes and horizonless compact
stars.

##  III Conclusions

We proposed a unified hoop conjecture, which is valid for various black holes
and horizonless compact stars. Our conjecture is that the mass and
circumference relation $4\pi M_{in}/C\leqslant 1$ holds, where C is the
circumference of the smallest ring that can engulf the object in all azimuthal
directions and $M_{in}$ is the gravitating mass within the engulfing sphere.
Our statement is in accordance with the idea that the ring of the engulfing
sphere should be related to the mass within the sphere (not the mass in the
total spacetime).

###### Acknowledgements.

This work was supported by the Shandong Provincial Natural Science Foundation
of China under Grant No. ZR2018QA008. This work was also supported by a grant
from Qufu Normal University of China under Grant No. xkjjc201906.

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