# A model-independent determination of the sound horizon using recent BAO
measurements and strong lensing systems

Tonghua Liu  School of Physics and Optoelectronic, Yangtze University,
Jingzhou 434023, China;  Shuo Cao<EMAIL_ADDRESS>School of Physics and
Astronomy, Beijing Normal University, Beijing 100875, China;  Institute for
Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing
102206, China;  Jieci Wang<EMAIL_ADDRESS>Department of Physics, and
Collaborative Innovation Center for Quantum Effects and Applications, Hunan
Normal University, Changsha 410081, China;

###### Abstract

We propose an improved method to determine the sound horizon in a cosmological
model-independent way by using the latest observations of BAO measurements
from DES, BOSS/eBOSS, and DESI surveys and gravitationally time-delay lensed
quasars from H0LiCOW collaboration. Combining the 6$D_{\Delta t}$ plus
4$D_{d}$ measurements and the reconstructed BAO datasets, we obtain a model-
independent result of $r_{d}=139.7^{+5.2}_{-4.5}$ Mpc, with the precision at
the $\sim 3.7\%$ level, which is in agreement with the result of Planck 2018
within $\sim 1.7\sigma$ uncertainty. Our method is independent of cosmological
parameters such as the Hubble constant, dark energy, (and, more importantly,
does not involve the cosmic curvature when using the $D_{d}$ measurements of
the lenses, and also avoids the obstacle of mass-sheet degeneracy in
gravitational lensing). Meanwhile, it does not need to consider the Eddington
relation with concerning the transformation of distance. Since only two types
of data are considered, the contribution of each can be clearly understood.
Our results also highlight the Hubble tension and may give us a better
understanding of the discordance between the datasets or reveal new physics
beyond the standard model.

##  I Introduction

In the early universe, baryon acoustic oscillation (BAOs) are sound waves
produced by gravitational interactions between photon-baryon fluids and
inhomogeneity Wald (1984); Weinberg (1972). During the drag period, baryons
decouple from photons and freeze at a scale equal to the acoustic horizon at
the drag period redshift. This scale, known as the sound horizon scale, is a
standard ruler embedded in the distribution of galaxies. It is a crucial
theoretical prediction of the cosmological model, dependent on the speed of
sound in the baryon-photon plasma and the rate of expansion of the early
Universe before matter and radiation decoupled. The Sloan Digital Sky Survey
(SDSS) Eisenstein et al. (2005) and the 2dF Galaxy Redshift Survey (2dFGRS)
Cole et al. (2005) first detected BAO signal and demonstrated the power of the
BAO as a standard ruler for cosmology. Subsequent BAO surveys including the
Six-degree Field Galaxy Survey (6dFGS) Beutler et al. (2011), the Baryon
Oscillation Spectroscopic Survey (BOSS) Alam et al. (2017), the extended
Baryon Oscillation Spectroscopic Survey (eBOSS) Alam et al. (2021), and the
WiggleZ Survey Blake et al. (2012) aimed to achieve more precise cosmic
distance measurements at a percentage level. These measurements could offer a
highly statistically significant study of the current tension problems (please
see e.g. Di Valentino et al. (2021); Abdalla et al. (2022); Perivolaropoulos &
Skara (2022) and references therein for a more comprehensive discussion).
Currently, the Dark Energy Spectroscopic Instrument (DESI) collaboration has
presented Data Release 1 (DR1) of BAO measurements and showed more than
2$\sigma$ evidence for the dynamical dark energy DESI Collaboration et al.
(2024a, b, c). However, before using the BAO as a standard ruler for cosmology
and as a powerful cosmological probe, one needs to know the comoving length of
this ruler, i.e., the sound horizon $r_{d}$ at the radiation drag epoch.

The sound horizon $r_{d}$ is usually calibrated at $z\approx 1100$ relying on
the CMB observations. Since the length of the scale is not known, BAO can only
give a relative measurement of the expansion history. This is similar to the
type of Ia supernovae (SNe Ia) acting as cosmological standard candles. If the
value of absolute magnitude is not known, then SNe Ia can only provide
relative distances. This implies that the Hubble constant $H_{0}$ and the
sound horizon $r_{d}$ are closely related, that there is strong degeneracy
between them, and that they link late and early cosmology. Planck cosmic
microwave background (CMB) anisotropy (both temperature and polarization) data
reported the $r_{d}=147\pm 0.30$ Mpc with assuming cosmological constant plus
cold dark matter ($\Lambda$CDM) model Planck Collaboration et al. (2020). An
alternative approach is to combine BAO measurements with other low-redshift
observations. Combining standard clocks and the local $H_{0}$ measurement to
the SNe and BAO, the work Heavens et al. (2014) obtained the $r_{d}=142.8\pm
3.7$ Mpc and consisted of the results derived from Planck data. Similarly,
considering the same data type, the work Bernal et al. (2016) got
$r_{d}=136.80\pm 4.0$ Mpc using the spline interpolation method for
reconstruction of expansion history, when curvature $\Omega_{K}$ as a free
parameter. Subsequently, under the assumption that the universe is flat, Verde
et al. (2017) inferred sound horizon $r_{d}=143.9\pm 3.1$ Mpc. See Refs.
Macaulay et al. (2019); L’Huillier & Shafieloo (2017); Shafieloo et al.
(2018); Camarena & Marra (2020); Zhang & Huang (2021); Giarè et al. (2024) for
more works about the sound horizon. It is important to emphasize here that
most of the work to determine sound horizon is cosmological model dependent.
If one considers the data from standard clocks to reconstruct the expansion
history (eliminating the assumptions of the cosmological background), then one
needs to have an assumption about the curvature of the universe or use it as a
free parameter. Considering the current critical situation for the
measurements of the Hubble constant in astronomical observations, in
particular the assumptions of cosmological models, it is very necessary to
realize the calibration or determination of $r_{d}$ in the cosmological model-
independent way.

As one of the most ubiquitous phenomena in astronomy, strong gravitational
lensing by elliptical galaxies directly provides absolute distance, and is a
powerful tool to study the velocity dispersion function of early-type galaxies
(Matsumoto & Futamase, 2008; Geng et al., 2021; Chae, 2007), the distribution
of dark matter (Cao et al., 2022; Mellier et al., 1993; Newman et al., 2009),
and cosmological parameters (Suyu et al., 2014; Bonvin et al., 2017; Chen et
al., 2019). In particular, gravitational lensing systems with time-delay
measurements between multiple images provide a valuable opportunity for the
determination of $H_{0}$. In representative work of Wong et al. (2020), the
$H_{0}$ Lenses in COSMOGRAIL’s Wellspring (H0LiCOW) collaboration combined the
six gravitationally lensed quasars with well-measured time delays to constrain
the $H_{0}$. The full dataset consists of six lenses, five of which were
analyzed blindly, and four of which have both time-delay distance $D_{\Delta
t}$ and the angular diameter distance to the lens $D_{d}$ measurements.
Current state-of-the-art lensing programs for time-delay with lensed quasars
have great progress, such as TDCOSMO collaboration111http://tdcosmo.org Millon
et al. (2020, 2020) (formed by members of H0LiCOW Wong et al. (2020),
COSMOGRAIL Eigenbrod et al. (2005), STRIDES Treu et al. (2018), and SHARP.
Recently, TDCOSMO Birrer et al. (2020) based on a new hierarchical Bayesian
approach with the original seven lenses, six of which are from H0LiCOW, and
reported $H_{0}=67.4^{+4.1}_{-3.5}$ $\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$.
Previously, there has been some work considering the use of time-delay
gravitational lensing to calibrate the sound horizon Aylor et al. (2019);
Wojtak & Agnello (2019); Arendse et al. (2019). However, these works were
either cosmological model dependent or introduced data like standard clock.
Using these data together, it is not possible to disentangle and determine the
contribution of these different probes to calibrating the sound horizon in the
BAO.

Inspired by the above, this work will use time-delay strong gravitational
lensing systems combined with the recent BAO measurements from DES,
BOSS/eBOSS, and DESI, to calibrate the sound horizon $r_{d}$ in a model-
independent method. The combination of these two probes has a number of
advantages: first, it is independent of the cosmological model and is
independent of the early universe; second, it is independent of cosmological
parameters such as the Hubble constant, dark energy, (and even more no cosmic
curvature is involved when using the $D_{d}$ of the lenses, and also avoids
the obstacle of mass-sheet degeneracy in gravitational lensing); and third, it
does not need to take into account the Eddington relation with respect to the
transformation of distance. Fourth, only two types of data are considered,
enabling a clear understanding of the contributions of each data.

This paper is organized as follows: in Section 2, we present the data used in
this work and the methodology of calibration of the sound horizon. In Section
3, we give our results and discussion. Finally, the main conclusions are
summarized in Section 4.

##  II Data and Methodology

###  II.1 BAO angular scale measurements

The clustering of matter created by BAO provides a “standard scale” of length
in cosmology. The length of this standard scale (roughly 150 Mpc at present)
can be measured by astronomical surveys looking at the large-scale structure
of matter, thus constraining the cosmological parameters (especially the
density of baryonic matter), and further understanding the nature of the dark
energy that causes the accelerated expansion of the Universe. However, when
using BAO for cosmological studies, it is important to know the length of this
standard ruler. During the drag epoch, baryons decoupled from photons and
“froze in” at a scale equal to the sound horizon at the drag epoch redshift
$z_{d}$, i.e., $r_{s}\equiv r_{d}(z_{d})$, if $r_{s}$ is interpreted as the
sound horizon at radiation drag, then

$r_{d}=\int^{\infty}_{z_{d}}dzc_{s}(z)/H(z),$ (1)

where $z_{d}$ being the redshift of the drag epoch, $c_{s}(z)$ is the sound
speed, and $H(z)$ is the Hubble parameter. The angular BAO scale
$\theta_{BAO}$ can be written as

$\theta_{BAO}=\frac{r_{d}}{(1+z)D^{A}},$ (2)

in terms of the angular diameter distance $D^{A}$. In this work, we consider
the 15 transverse BAO angular scale measurements (denoted as 2D-BAO)
summarized in Table 1 of Nunes et al. (2020). These values were obtained using
public data releases (DR) of the Sloan Digital Sky Survey (SDSS), namely: DR7,
DR10, DR11, DR12, and DR12Q (quasars), without assuming any fiducial
cosmological model. It is important to note that because these transverse BAO
measurements are performed using cosmology-independent methods, their
uncertainties are larger than those obtained using the fiducial cosmological
method. For anisotropic BAO (denoted as 3D-BAO), we considered two sources of
data, a dataset from DES Y6 and BOSS/eBOSS Alam et al. (2017, 2021), and other
from recent DESI DR1 data DESI Collaboration et al. (2024c). The 3D-BAO
angular scale measurements can be found in Table 1 of Favale et al. (2024).
These BAO angular scale measurements including the 2D-BAO and 3D-BAO are shown
in Fig. 1.

As mentioned above, the joint use of other low redshift observations is
necessary if a cosmological model-independent $r_{d}$ calibration is to be
realized. Here we consider observations from the H0LiCOW analysis of six
lensed quasars with good lens modeling. The realization of a gravitational
lens calibrated sound horizon requires that the BAO be able to provide the
corresponding cosmological information both at the redshifts of the sources
and lenses. However, both BAO and gravitational lensing data are very sparse.
Therefore, we consider in this work a cosmological model-independent data
reconstruction method, Gaussian Process Regression (GPR) (Holsclaw et al.,
2010b; Joudaki et al., 2018; Shafieloo et al., 2012), to reconstruct the
angular scale measurements of the observed BAO. We generate samples of
reconstructed angular scale BAO measurements from the posteriors of 2D and 3D
BAO datasets. The posterior sampling of GPR is realized with the code
GPHist222https://github.com/dkirkby/gphist. (Kirkby & Keeley, 2017). GPR is a
completely data-driven reconstruction method and performs in an infinite
dimensional function space without overfitting problem (Joudaki et al., 2018).
GPR works by generating large samples of functions $\gamma(z)$ determined by
the covariance function. The covariance between these functions can be
described by the kernel function. We use here the most general and commonly
used squared exponential kernel to parameterize the covariance

$\langle\gamma(z_{1})\gamma(z_{2})\rangle=\sigma_{f}^{2}\,\exp\\{-[s(z_{1})-s(z_{2})]^{2}/(2\ell^{2})\\},$
(3)

where $\sigma_{f}$ and $\ell$ are hyperparameters and are marginalized over.
The $\gamma(z)$ is a random function inferred from the distribution defined by
the covariance, and we adopt $\gamma(z)=\ln(\theta_{BAO}(z))$ to generate more
angular scale measurements by using BAO dataset. The 1000 reconstructed curves
of angular scale measurements from the BAO dataset are shown in Fig. 1. It
shows the shape of the angular scale-redshift relation of BAO data very well.
It should be noted that the redshift of the BAO dataset well covers the range
of H0LiCOW lensing system redshifts, we needn’t extrapolate the redshift range
of the reconstructed BAO dataset.

###  II.2 Strong gravitational lensing systems from H0LiCOW

Let us briefly outline the standard procedure for $D_{\Delta t}$ and $D_{d}$
measurements used in the H0LiCOW procedure. The distances of a gravitational
lens involve only the angular diameter distances. For a given strong lensing
system, quasars act as a background source at redshift $z_{s}$, which is
lensed by foreground elliptical galaxies (at redshift $z_{d}$), and multiple
bright images of active galactic nuclei (AGN) are formed along with arcs of
their host galaxies. The subscripts $d$ and $s$ stand for the lens galaxy and
the source, respectively. The lensing time delay between any two images is
determined by the geometry of the universe and the gravity field of the
lensing galaxy (Shapiro, 1964; Refsdal, 1964)

$\Delta t_{AB}=D_{\Delta
t}\left[\phi(\theta_{A},\beta)-\phi(\theta_{B},\beta)\right]=D_{\Delta
t}\Delta\phi_{AB}(\xi_{lens}),$ (4)

where $\phi(\theta,\beta)=\left[{(\theta-\beta)^{2}}/{2}-\psi(\theta)\right]$
is the Fermat potential at images, $\beta$ is the source position, $\psi$ is
lensing potential obeying the Poisson equation $\nabla^{2}\psi=2\kappa$, where
$\kappa$ is the surface mass density of the lens in units of critical density
$\Sigma_{crit}=D_{s}/(4\pi D_{d}D_{ds})$, and $\xi_{lens}$ denotes the lens
model parameters. The cosmological background is reflected in the “time delay
distance”

$D_{\Delta t}=(1+z_{d})\frac{D_{d}D_{s}}{D_{ds}}=\frac{\Delta
t_{AB}}{\Delta\phi_{AB}(\xi_{lens})}.$ (5)

The variability of the AGN light curve can be monitored to measure the time
delay between multiple images. The key point here is that the Fermat potential
difference $\Delta\phi_{AB}(\xi_{lens})$ can be reconstructed by high-
resolution lensing imaging from space telescopes.

Figure 1:  The points and error bars come from observed BAO angular scale
measurements. The light blue lines are the GPR reconstruction and consist of
1000 curves.

On the other hand, assuming an explicit model for the lens, such as the
simplest singular isothermal sphere (SIS) model (not limited to the SIS model,
and denoted as lens model parameter $\xi_{{lens}}$), plus observations on
stellar kinematics, such as the lens galaxy such as the light profile
$\xi_{{light}}$, the line of sight (LOS) projected stellar velocity dispersion
of the lens galaxy $\sigma_{v}$, the anisotropy distribution of the stellar
orbits $\beta_{{ani}}$, one can yield the absolute distance measure of $D_{d}$
at the lens (Birrer et al., 2019)

$\displaystyle D_{d}=\frac{1}{1+z_{d}}\frac{c\Delta
t_{AB}}{\Delta\phi_{AB}(\xi_{{lens}})}\frac{c^{2}J(\xi_{{lens}},\xi_{{light}},\beta_{{ani}})}{\sigma_{v}^{2}}\,,$
(6)

where the function $J$ captures all model components calculated from the
lensing image and the photometrically weighted projected velocity dispersion
(from spectroscopy). For more details on the modelling problem for the
function $J$, see Section 4.6 of (Birrer et al., 2019).

The posterior distribution of these lenses (namely RXJ1131-1231 (Suyu et al.,
2013, 2014), PG1115+080 (Chen et al., 2019a), B1608+656333Except for this len,
other lenses were analyzed blindly with respect to the cosmological
parameters. (Suyu et al., 2010; Jee et al., 2019), J1206+4332 (Birrer et al.,
2019), WFI2033-4723 (Rusu et al., 2020), HE0435-1223 (Wong et al., 2017; Chen
et al., 2019a)) including the time delay distances and the angular diameter
distances of the lenses can be found at H0LiCOW
website444http://www.h0licow.org.. The redshifts of both lenses and sources,
time delay distances and angular diameter distances to lenses for systems are
summarized in Table 2 of Wong et al. (2020).

Although TDCOSMO recently considered different strategies for analyzing these
six lenses, compared to the results of H0LiCOW’s constraints on the Hubble
constant, the results of the latest $H_{0}$ released by TDCOSMO have a large
reduction in precision. This is mainly due to the release of the choice of
parameterization of the lens mass profile in H0LiCOW lens modeling, which is
the so-called mass-sheet transformation (MST) Falco et al. (1985). To counter
this increase in uncertainty, the TDCOSMO team obtained stellar kinematics
from the Sloan Lens ACS (SLACS) catalogue for constraining the MST. The
TDCOSMO IV likelihood products are all publicly available on this
repository555https://github.com/TDCOSMO. Birrer et al. (2020). However, it is
necessary to emphasize that TDCOSMO uses the assumptions of the
$\rm{\Lambda}CDM$ model and the cosmological a priori from SNe Ia in its
lensing modeling. Until TDCOSMO realizes a fully cosmological model-
independent analysis, we still use the posterior distributions of distances
publicly released by H0LiCOW. Although the lensing data are not used the
latest, the data from the BAO are the latest, and this is the first work to
use gravitational lensing to calibrate the sound horizon of the BAO in a
cosmological model-independent way.

##  III Results and Discussion

Let’s emphasize that calibration of BAO sound horizon $r_{d}$ using
observations of the strong lensing system is straightforward, we do not make
assumptions about cosmological models and other cosmological parameters.
Combing the Eq. (2) and (5), the BAO standard ruler $r_{d}$ can be rewritten
as

$r_{d}=D_{\Delta t}(\theta_{BAO}(z_{d})-\theta_{BAO}(z_{s})),$ (7)

where $\theta_{BAO}(z_{d})$ and $\theta_{BAO}(z_{s})$ are the angular BAO
scale at redshifts of source and lens, and we assume a spatially flat universe
and use the standard distance relation to obtain the angular diameter distance
between the lens and the source $D_{ds}=D_{s}-[(1+z_{d})/(1+z_{s})]D_{d}$
(Weinberg, 1972). When we combine the angular diameter distances to lenses
with Eq. (2), one can obtain

$r_{d}={(1+z_{d})\theta_{BAO}D_{d}}.$ (8)

The 1000 curves $\theta_{BAO}$ reconstructed by GPR contain information about
its uncertainty. For given a redshift point, we have a distribution of 1000
measurements on the BAO angular scale. Based on these distributions, and
combining Eqs. (7) and (8), respectively, we bring them into the skewed log-
normal distributions of the time-delay and angular diameter distances provided
by gravitational lensing (as published by H0LiCOW) to obtain the Probability
Distribution Functions (PDFs) of $r_{d}$ for the combination of lenses and
reconstructed BAO data. Strong lensing systems are in principle uncorrelated,
so we obtain the individual distributions of the $r_{d}$ given by each lens.

Figure 2: The probability density functions of $r_{d}$ using GPR reconstructed BAO measurements and time delay distances provided by H0LiCOW.  Table 1:  Summary of the constraints on the sound horizon $r_{d}$ using recent BAO dataset and H0LiCOW lenses. Comparison with previous works is also provided. Case | 6$D_{\Delta t}$+4$D_{d}$ | 6$D_{\Delta t}$ | 4$D_{d}$ | Planck18 Planck Collaboration et al. (2020) | Aylor et al. Aylor et al. (2019) | Wojtak et al. Wojtak & Agnello (2019) | Zhang et al. Zhang & Huang (2021)
---|---|---|---|---|---|---|---
$r_{d}$ [Mpc] | $139.7^{+5.2}_{-4.5}$ | $139.5^{+5.2}_{-4.4}$ | $126.3^{+13.3}_{-10.5}$ | $147.05^{+0.3}_{-0.3}$ | $139.3^{+4.8}_{-4.8}$ | $137.0^{+4.5}_{-4.5}$ | $148.0^{+3.6}_{-3.6}$

Using time-delay distances from H0LiCOW, we obtain the final result obtained
by combining all lenses is the following: $r_{d}=139.5^{+5.2}_{-4.4}$ Mpc
(median value with the $16^{th}$ and $84^{th}$ percentiles around it), and are
displayed in Fig. 2. This result is incompatible with the result of Planck
2018 within 1$\sigma$ uncertainty, but consistent within $\sim 1.7\sigma$
uncertainty. This is quite reasonable. First of all, H0LiCOW based on these 6
time-delay gravitational lenses gives $H_{0}=73.3^{+1.7}_{1.8}$
$\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$ in the framework of the $\rm{\Lambda}CDM$
model. Planck CMB observation showed their result $H_{0}=67.4\pm 0.5$
$\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$, there is 3.1$\sigma$ tension with
H0LiCOW result. Moreover, the Hubble constant and the sound horizon are
directly correlated with each other, and there is a strong degeneracy between
them and a negative correlation. Our results also highlight the Hubble tension
problem to some extent. To understand the relative contributions, we constrain
$r_{d}$ with each lens in the model-independent manner and also show that Fig.
2. For each of the individual lens results, they are all compatible with
Planck’s results within $1\sigma$ uncertainties due to the large
uncertainties, except for the lens RXJ1131, its result is
$r_{d}=131.5^{+8.1}_{-6.4}$ Mpc, which is agreement within the $\sim
1.7\sigma$ observational uncertainty. The uncertainty in the GPR reconstructed
BAO data is almost equal to the uncertainty in the observed data, and our
error budget in the measurement of $r_{d}$ is clearly dominated by the
uncertainty from the observations of the strong lenses, we do not expect any
improvement in the precision, as shown by our results. However, such a
combination already constrains the low-redshift standard ruler scale $r_{d}$
at the $\sim 3.7\%$ level.

Figure 3: The probability density functions of $r_{d}$ using GPR reconstructed
BAO measurements and angular diameter distances provided by H0LiCOW.

Combination of the posterior distributions of angular diameter distances of
four H0LiCOW lenses and reconstructed 1000 curves from the BAO dataset, the
final PDFs for $r_{d}$ are reported in Table 1 and displayed in Fig. 3.
Compared with the results of time delay distance, we see that the mean values
of $r_{d}$ for four individual lenses have large changes and large
uncertainties. For a combination of four lenses $D_{d}$, the corresponding
constraint result is $r_{d}=126.3^{+13.3}_{-10.5}$ Mpc, the constrained
precision on $r_{d}$ is only the $\sim 10.5\%$ level, but it is also in
agreement with the results of Planck 2018 within $\sim 1.5\sigma$ uncertainty.
However, it is worth pointing out that, despite the poor precision of the
constraints on $r_{d}$, the method using $D_{d}$ provided by gravitational
lensing has two great benefits. First, the method does not require assumptions
about the curvature of the universe, whereas using time-delayed distances
requires assumptions to be taken about the curvature. As suggested in work
Aylor et al. (2019); Wojtak & Agnello (2019), taking different values for the
curvature of the universe might have some impact on the significance of
results for measuring $r_{d}$. In addition, the method circumvents the mass-
sheet degeneracy obstacle in gravitational lensing.

Figure 4: The probability density function of $r_{d}$ using GPR reconstructed
BAO measurements and full data set consisting of 6$D_{\Delta t}$ and 4 $D_{d}$
measurements from H0LiCOW. Comparison with previous works is also provided.

For the full data set consisting of 6 lenses, 4 of which have both $D_{\Delta
t}$ and $D_{d}$ measurements, our model-independent constrained result is
$r_{d}=139.7^{+5.2}_{-4.5}$ Mpc and shown in Fig. 4. Compared with results
obtained by using 6$D_{\Delta t}$ alone, we see that the mean values of
$r_{d}$ for the full dataset have a few changes (though not significant). As
pointed out in the work of Millon et al. (2020), the addition of $D_{d}$ (or
dispersion measurements) does not play a significant role in the $H_{0}$
estimation of the H0LiCOW analysis. However, as we mentioned above,
considering different types of data for gravitational lensing has its own
advantages. In order to highlight the potential of our method, it is necessary
to compare our results with previous works. Compared our results with previous
works, Wojtak & Agnello (2019) used the $D_{d}$ of three lenses from the
H0LiCOW collaboration, combined relative distances from SNe Ia and BAOs with a
prior of $H_{0}$ from H0LiCOW, their reported $r_{d}=137\pm 4.5$ Mpc by using
cosmography and global fitting method. The work Zhang & Huang (2021) used the
measurements of BAO, observational $H(z)$ data, GW170817, and SNe Ia to
constrain the $H_{0}$ and $r_{d}$, and obtained the results of $H_{0}=68.58\pm
1.7$ $\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$ and $r_{d}=148.0\pm 3.6$ Mpc. These
works are either cosmological model-dependent or involve more than two or more
types of data, which prevents us from clearly understanding the contribution
of each type of data. However, it should be stressed that, compared with
assuming a specific model, a combination of strong lensing and current
astronomical probes such as BAO can reduce possible bias in our work. More
importantly, there is no significant increase in uncertainty. As we mentioned
earlier (see Introduction), our approach effectively avoids these problems,
which suggests that our approach has great potential to provide more precise
and accurate measurements of $r_{d}$ in the future, further precision
cosmology research.

##  IV Conclusion

In this work, we propose an improved model-independent method to calibrate
sound horizon $r_{d}$ by using the latest observations of BAO measurement from
DES, BOSS/eBOSS, and DESI surveys and gravitationally time-delay lensed
quasars from H0LiCOW collaboration. We adopt the non-parameterized method of
GPR to reconstruct observed BAO angular scale measurements. Our approach has
several significant advantages. First, it is independent of the model of the
universe and early universe; second, it is independent of cosmological
parameters such as the Hubble constant, dark energy, (and, more importantly,
does not involve the curvature of the universe when using the $D_{d}$ of the
lenses, and also avoids the obstacle of mass-sheet degeneracy in gravitational
lensing); third, it avoids the transformation of distances by the Eddington
relation. Fourth, it involves only two types of data, allowing a clear
understanding of the contribution of data. Such combinations of time-delay
strong lensing systems and current astronomical probes such as BAO can reduce
possible bias in the $r_{d}$ estimation of our analysis.

Combining the 6$D_{\Delta t}$ measurements and reconstructed BAO dataset, our
model-independent result is $r_{d}=139.5^{+5.2}_{-4.4}$ Mpc with the precision
at $\sim 3.7\%$ level. This result changes to $r_{d}=126.3^{+13.3}_{-10.5}$
Mpc when combining 4$D_{d}$ datasets with the reconstructed BAO dataset.
Despite the poor precision of the constraints on $r_{d}$ in this case, the
method using $D_{d}$ provided by gravitational lensing does not require
assumptions about the curvature of the universe and avoids the obstacle of
mass-sheet degeneracy, whereas using time-delayed distances requires
assumptions to be taken about the curvature. For the full dataset of
6$D_{\Delta t}$ and 4$D_{d}$ measurements, we obtain the result
$r_{d}=139.7^{+5.2}_{-4.5}$ Mpc, which is in agreement with the results of
Planck 2018 within $\sim 1.7\sigma$ uncertainty, the addition of $D_{d}$ (or
dispersion measurements) does not play a significant role in our analysis.
Considering the correlation between $r_{d}$ and $H_{0}$, our results also
highlight the Hubble tension and may give us a better understanding of the
discordance between the datasets or reveal new physics beyond the standard
model.

As a final remark, we are also looking forward to other new cosmological
probes, such as gravitational waves as standard sirens. Recently, the work
Giarè et al. (2024) forecast a relative precision of $\sigma_{r_{d}}/r_{d}\sim
1.5\%$ from forthcoming surveys such as LISA gravitational wave (GW) standard
sirens and DESI or Euclid angular BAO measurements. Focusing on the available
data before the true gravitational wave era, it has been possible to constrain
$r_{d}$ to up to 3.7$\%$ precision using our method. The methodology of this
paper also applies to the $r_{d}$ estimation of joint GW and BAO measurements.
This shows that our method has great potential to provide more precise and
accurate measurements of $r_{d}$ in the future, further precision cosmology
research.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
under Grant No. 12203009, 12122504 and 12035005; the Chutian Scholars Program
in Hubei Province (X2023007); and the Hubei Province Foreign Expert Project
(2023DJC040).

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