# 2D BAO vs 3D BAO: Hints for new physics?

Ruchika1
1Physics Department and INFN, Università di Roma “La Sapienza”, Ple Aldo Moro
2, 00185, Rome, Italy e-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In the era of tensions, when precision cosmology is blooming, numerous new
theoretical models are emerging. However, it’s crucial to pause and question
the extent to which the observational data we rely on are model-dependent. In
this work, we study the comoving position of the acoustic peak, a cornerstone
standard ruler in cosmology. We considered BAO observational datasets from two
distinct teams and calculated the product $hr_{d}$ with the help of each BAO
data set along with SN I-a data from the Pantheon Plus sample. Our conclusion
at present is that 2D and 3D BAO datasets are compatible with each other.
Considering, no systematics in BAO, interpreting $\Omega_{m0}-hr_{d}$ plane
may require physics beyond $\Lambda$CDM not just while using observational BAO
data but also while observing it.

###### keywords:

Baryon Acoustic Oscillations, Cosmological Tensions, Absolute Magnitude SNe
I-a

††pubyear: 24

##  1 Introduction

Since the pivotal supernova (SN) discovery of 1997, indicating the onset of
the accelerated phase of the universe, cosmologists have diligently pursued
measurements of cosmic expansion utilizing various observational probes.
Notably, the cosmic microwave background (CMB) (Ade et al., 2016a, b; Aghanim
et al., 2018), Type Ia supernovae (SN Ia) (Betoule et al., 2014; Perlmutter et
al., 1997; Riess et al., 1998), and Baryon Acoustic Oscillations (BAO)
(Beutler et al., 2011, 2012; Blake et al., 2012; Lauren et al., 2013, 2014)
have emerged as key instruments for gauging the Hubble expansion or Hubble
Constant. Precision cosmology hinges upon the accurate calibration and
interpretation of observational data obtained from diverse probes, including
the CMB and the standard distance ladder. Anomalies within these datasets have
garnered significant attention, as they may herald breakthroughs in our
understanding of fundamental cosmological processes.

Despite general agreement among most probes within a two to three-sigma range,
recent analyses, particularly the Planck 2018 (Aghanim et al., 2018) and SH0ES
2022 datasets (Riess et al., 2022), have revealed tensions exceeding five
sigmas. Several works have been dedicated to careful examination of CMB sky
and standard distance ladder.

Hubble Hunter’s Guide (Knox et al., 2020) lists out many departures from
$\Lambda$CDM to solve the cosmological tensions and singles out a solution
which increases Hubble expansion rate before recombination by modifying sound
horizon at drag epoch $r_{d}$. Other efforts have been made to identify
anomalies in CMB polarization data (Matteo et al., 2023) with the help of JWST
datasets. Studies like G-Transition hypothesis(Ruchika et al., 2022), Planck
Mass transition (Kable et al., 2023) assess the requisite signatures within
the standard distance ladder. By dissecting the underlying assumptions and
methodologies, these studies seek to contextualize these anomalies within the
broader framework of precision cosmology.

We recall that Baryon acoustic oscillations are considered to be one of the
most powerful probes for measuring the undergoing phase of accelerated
expansion of the universe. When combined with the Planck satellite results,
one finds that the Universe is spatially flat (Aghanim et al., 2018). However,
we stress that the traditional methods or the official BAO distances have an
injection of cosmological information while estimating distances and the
standard assumptions hold only for models that are not too dissimilar from
$\Lambda$CDM (Thepsuriya & Lewis, 2015). This may be the serious limitation of
the standard BAO approach and therefore while using them to quantify tensions
among various cosmological probes, one needs to be very cautious. For BAO to
infer cosmological distance and later use them to understand the fundamentals
of our universe, estimates should be as much as possible model-independent,
allowing one to perform multiple consistency checks of data and theory. It is
still unclear if we can use all sorts of exotic models such as Ruchika et al.
(2022); Kable et al. (2023); Dutta et al. (2020) to infer cosmological
distances from standard BAO observations.

Recent solutions such as the Purely-Geometric-BAO (Anselmi et al., 2024) avoid
the undesired injection of cosmological information into distances inferred
from the BAO and may work better for studying cosmological tensions. Such
analyses are particularly relevant in light of the current cosmological
tensions.

In this work, we will analyze anomalies within Baryon Acoustic Oscillations.
Before questioning the 6-parameter standard model of cosmology like everyone
else, we address the question of how standard are the measurements for the
standard ruler i.e. Baryon Acoustic Oscillations. We compared two different
datasets from BAO. We argue that using different datasets to estimate
cosmological information from the standard ruler should not bring significant
change in the final inference of cosmological parameters. Keeping the model
fixed to the simplest $\Lambda$CDM model, we focused on the inherent tensions
between BAO measurements. This paper scrutinizes the discordance among two
independent BAO datasets and explores the implications for our understanding
of cosmic expansion. We also incorporated the SN-Ia dataset in our study.
Taking just low redshift observational probes like SN I-a, we question here if
BAO can be used as a model-independent probe to study the evolution of the
universe and if it can be at all used to check different exotic models that
are being proposed to solve cosmological tensions.

This paper is organized as follows. In section 2 we briefly review the physics
of the BAOs. We describe the data used and the method of SNe calibration in
section 3. We also discuss that standard BAO distance measurements are
obtained through fitting approaches where cosmological parameters were kept
fixed to a fiducial flat-$\Lambda$CDM cosmological model or by building the
whole data analysis pipeline with flat-$\Lambda$CDM survey mocks and Nbody
simulations close to the Planck best-fit cosmology. Results comparing 2D and
3D datasets are given in section 4. And, we put our comments on Early Dark
Energy in section 5. In section 6, We incorporated the analysis with the DESI
dataset. Section 7 is where we have an overall discussion of BAO 2D, BAO 3D
and DESI BAO Dataset. In section 8, we conclude the final remarks.

##  2 Baryon Acoustic Oscillation as a standard ruler

Baryon Acoustic Oscillations (BAO) serve as a crucial cosmological standard
ruler in the field of cosmology. They provide valuable information for
understanding the large-scale structure of the universe and inferring
cosmological parameters from observational data. The comoving position of the
acoustic peak, which is a characteristic feature of the BAO, is particularly
targeted by the cosmological community. This peak arises as a result of
acoustic waves travelling through the early universe, leaving a distinct
imprint on the distribution of matter. By measuring the characteristic scale
of these oscillations in the large-scale structure of the universe,
researchers can use BAO as a standard ruler to infer cosmological parameters
such as the expansion rate of the universe and the amount of dark energy. In
essence, BAO offers a powerful tool for cosmologists to probe the underlying
cosmology of the universe by studying the clustering of galaxies and other
cosmic structures. This allows them to better understand the nature of dark
energy, dark matter, and the overall geometry and evolution of the universe.

In the standard cosmological description, in the early universe before the
recombination epoch, baryons and photons were tightly coupled to each other
and there was a formation of acoustic waves within the primordial photon-
baryon plasma. As the universe expanded and cooled, reaching the epoch of
decoupling, known as the drag epoch, the propagation of these acoustic waves
ceased. At this pivotal moment, the baryon distribution retained imprints of
the acoustic oscillations, manifesting as overdensities separated by a
distinct length scale ($r_{d}\sim 150$ Mpc) known as the sound-horizon
comoving length at the drag epoch. As this signature imprinted on matter
distribution is governed by early universe physics before and around
recombination, it is treated as a standard ruler. In fact, it is also well
calibrated by CMB observations to very high accuracy (Aghanim et al., 2018).
Similarly, Eisenstein & Hu (1998) found a bao peak in large-scale correlation
function around the same comoving galaxy separation (100 $h^{-1}$) Mpc. And
that is why BAO is used to constrain dark energy behaviour and helps in
breaking the residual degeneracies with CMB observations.

The evolution of matter on large scales, including dark matter and baryons, is
primarily influenced by gravity. This gravitational interaction leaves a
distinctive feature in the 2-point correlation functions (CF) of matter and
its observed tracers, such as galaxies. This characteristic scale, a
consequence of the physics governing the early universe, serves as a
fundamental cosmological standard ruler.

Since the so-called acoustic peak was very closely identified with the
correlation function (CF) feature in BAO, it sparked the initial notion of
measuring a comparable length scale across both the early and late universe.
The intention behind this approach was to leverage the consistency of this
scale throughout cosmic history, thereby harnessing its cosmological
implications. However, in the present era of precision cosmology, this
excellent intuition is encountering several challenges.

We now evidently know that the primordial sound horizon at drag epoch $r_{d}$
estimated from Cosmic Microwave Background is cosmological model dependent and
depends on the modelling of late-time physics. On the other hand, while
calculating the $r_{d}$ from late-time physics, one needs to model scale-
dependent galaxy bias in galaxy correlation function non-linearities which can
be time as well as model-dependent.

To fit the observational anisotropic power spectrum data from CMB sky, we
mostly assume a standard cosmological model (e.g. flat $\Lambda$CDM), and we
can calculate the value of $r_{d}$ as the derived parameter. Using late-time
probes such as the galaxy correlation function, $r_{d}$ needs to be derived
again from cosmology-dependent multi-parameter fit.

###  2.1 Parameteriszing the Comoving Distance Scale

Using late-time probes, there are two ways of utilising the 2-point
correlation function to determine the BAO feature in literature. One is via
two-point correlation (3D BAO Dataset) (Alam et al., 2016) $\xi$($s$) in
configuration and Fourier space and the other is using 2-point angular
correlation function (2D BAO Dataset) (Carvalho et al., 2016)
$\omega$($\theta$) where $s$ and $\theta$ are comoving radial and angular
separation respectively. Below we describe both of them in detail.

####  2.1.1  Two Point Correlation Function

Among several 3D estimators in literature, the most commonly used is the Two
Point Correlation Function (2PCF)

$\xi(s)=\frac{DD(s)-2DR(s)+RR(s)}{RR(s)},$ (1)

where DD(s) and RR(s) represent the number of galaxy pairs in real-real and
random-random catalogues respectively. The comoving distance $s$ is chosen
assuming a fiducial cosmology. The comoving distance between two galaxies at
redshift $z_{1}$ and $z_{2}$ is given by

$s=\sqrt{r^{2}(z_{1})+r^{2}(z_{2})-2r(z_{1})r(z_{2})\cos\theta_{12}},$ (2)

where $\theta_{12}$ is the angular distance between pair of galaxies at
redshift $z_{1}$ and $z_{2}$. Choosing the expression for $r$ is what makes it
cosmological model dependent. For flat $\Lambda CDM$, $H_{0}$ and $\Omega_{m}$
being the Hubble parameter at present and matter density parameter, $r$ is
expressed as:

$r(z_{i})=\frac{c}{H_{0}}\int_{0}^{z_{i}}\frac{dz}{\sqrt{\Omega_{m}(1+z)^{3}+(1-\Omega_{m})}}.$
(3)

However, rather than studying the full 3D, the information contained in the
initial few Legendre multipoles or in clustering wedges statistics is
utilised. The Legendre multipoles in configuration space and the power
spectrum multipoles are defined respectively as

$\displaystyle\xi_{\ell}(s)\equiv\frac{2\ell+1}{2}\int_{-1}^{1}L_{\ell}(\mu)\xi(\mu,s)\,d\mu,$
(4) $\displaystyle
P_{\ell}(k)\equiv\frac{2\ell+1}{2}\int_{-1}^{1}L_{\ell}(\mu)P(\mu,k)\,d\mu,$
(5)

where $\xi(\mu,s)$ and $P(\mu,k)$ are two dimensional correlation function and
power spectrum respectively. Here variable $\mu$ is the cosine of the angle
between the line of sight direction and separation vector $s$.

The relationship between two-dimensional Legendre multipoles $\xi_{l}(s)$ and
$P_{l}(k)$ is described as

$\xi_{\ell}(s)\equiv\frac{i^{\ell}}{2\pi^{2}}\int_{0}^{\infty}P_{\ell}(k)j_{\ell}(ks)k^{2}\,dk,$
(6)

$j_{\ell}$ here describes the $\ell^{th}$ order spherical Bessel function. In
standard analysis, information just from monopole, quadrupole and hexadecapole
($\ell=0,2$ and $4$) is used as they have a nearly complete description of the
$\mu$ dependence of $\xi(\mu,s)$ in distant observer approximation in the
linear regime.

Using the power spectrum and two-point correlation function described above,
the BAO scale is measured in redshift space. The observable is the shift in
the BAO peak position with respect to fiducial cosmology. The parallel and
perpendicular shifts to the line of sight give bounds on Hubble expansion rate
$H(z)$, comoving and angular diameter distance relative to the sound horizon
at drag epoch $r_{d}$ parameter.

$\displaystyle\alpha_{\perp}=\frac{D_{M}(z)r_{d,fid}}{D_{M}^{fid}(z)r_{d}};\quad\alpha_{\parallel}=\frac{H^{fid}(z)r_{d,fid}}{H(z)r_{d}}$
(7)

To illustrate how comoving distance $D_{M}(z)$ and the Hubble parameter $H(z)$
are measured, consider the preferred angular separation of galaxies,
$\Delta\theta$, within an ensemble of galaxy pairs oriented perpendicular to
the line of sight. The comoving distance at this redshift is then determined
by $D_{M}(z)=r_{d}/\Delta\theta.$ When examining the separation vector
parallel to the line of sight, if a preferred redshift separation $\Delta z$
is observed, the corresponding equivalent distance is
$D_{H}=c/H(z)=r_{d}/\Delta z,$ thereby inferring the Hubble expansion
parameter at that redshift. For certain redshift bins with low signal-to-noise
ratios, and when both the transverse and line-of-sight components are present,
isotropic BAO (Baryon Acoustic Oscillation) measurements are obtained using
$D_{V}(z)$, where

$\displaystyle D_{V}(z)=\left[(1+z)^{2}D_{A}^{2}(z)zD_{H}\right]^{1/3}.$ (8)

Here, $D_{A}(z)=D_{M}(z)/(1+z)$ is the angular diameter distance, and
$D_{V}(z)$ represents the average of the distances measured perpendicular and
parallel to the line of sight of the observer (Eisenstein et al., 2005).

The BOSS Survey (Baryon Oscillation Spectroscopic Survey) (Alam et al., 2016)
assumes the fiducial cosmology as a flat $\Lambda$CDM model with the following
parameters: dimensionless Hubble constant $h=0.676$, fluctuation amplitude
$\sigma_{8}=0.8$, baryon density $\Omega_{b}h^{2}=0.022$, optical depth
$\tau=0.078$ and spectral tilt $n_{s}=0.97$. For this model, the sound horizon
at the drag epoch parameter is $r_{d}=147.78$ Mpc. These parameter values are
within $1\sigma$ of Planck 2015 values from CMB.

Even when the study was extended beyond $\Lambda$CDM models such as
$ow_{0}w_{a}$CDM model or $0w$CDM model with extra relativistic species and by
incorporating SN Type I-a dataset, the obtained Hubble Constant value was
$H_{0}=67.3\pm 1.0$ km s-1 Mpc-1and $H_{0}=67.8\pm 1.2$ km s-1
Mpc-1respectively not shifting from the mean value of standard $\Lambda$CDM
model ($H_{0}=67.6\pm 0.5$ km s-1 Mpc-1(Planck alone)). And, hence keeping the
tension with the standard distance ladder SH0ES’22 alive.

####  2.1.2  Two Point Angular Correlation Function

The two-point angular correlation function (2PACF) is defined as the excess
probability of finding two-point sources in two solid angles $d\Omega_{1}$ and
$d\Omega_{2}$ with angular separation $\theta$ as compared to a homogeneous
Poisson distribution (Carvalho et al., 2016). To avoid the contribution of
radial signal, only narrow redshift shells of very small width $\delta z$ are
considered.

$\omega(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$ (9)

where where $\theta=\theta_{FIT}$ is supposed to be the true bump for the
transversal BAO signal. Along with the true BAO signal, many bumps can be
observed from 2PACF due to various systematic effects, which can be removed by
taking the expected 2PCF. The expected 2PACF can be easily calculated using
the expected 2PCF.

$\displaystyle\omega_{E}(\theta,\bar{z})=\int_{0}^{\infty}dz_{1}\,\phi(z_{1})\int_{0}^{\infty}dz_{2}\,\phi(z_{2})\,\xi_{E}(s,\bar{z}),$
(10)
$\displaystyle\xi_{E}(s,z)=\int_{0}^{\infty}\frac{dk}{2\pi^{2}}k^{2}\leavevmode\nobreak\
j_{0}(ks)\leavevmode\nobreak\ b^{2}P_{m}(k,z),$ (11)

where $\bar{z}$ is the average redshift of $z_{1}$ and $z_{2}$, $j_{0}$ is the
zeroth order Bessel Function and $P_{m}$ is the matter power spectrum
calculated using the fiducial cosmological model $\Lambda$CDM with parameters
set to $w_{b}h^{2}=0.0226$, $w_{c}h^{2}=0.112$, $100\Theta=1.04$, $\tau=0.09$,
$A_{s}e^{9}=2.2$, and $n_{s}=0.96$ for SDSS DR10 galaxies ((Carvalho et al.,
2016)).
Once $\Theta_{FIT}$ is estimated, it can be used to infer cosmological
distances and therefore put bounds on cosmology. The relation describing the
true bump of the BAO signal and angular diameter distance is given by

$\theta_{BAO}=\frac{r_{d}}{(1+z)D_{A}(z)}$ (12)

where $D_{A}(z)$ is the angular diameter distance and $r_{d}$ is the sound
horizon at the drag epoch.

##  3 Data and Methodology

In the subsequent sections, we describe the observational datasets utilized in
this study.
We commence by detailing the Baryon Acoustic Oscillation (BAO) data, which
serves as a basis for comparison. Subsequently, we elucidate the Supernova
Type Ia (SN Ia) data, strategically employed to mitigate degeneracies with the
BAO measurements. Given the ongoing tension between the Hubble constant
($H_{0}$) or the absolute magnitude ($M_{B}$) used for calibrating SN Ia, we
adopt a comprehensive approach. Rather than relying solely on the $M_{B}$
derived from local distance ladder measurements, we compile a diverse array of
possible $M_{B}$ values from various probes, facilitating a complete analysis.
For the sake of brevity, we designate the theta measurements of BAO, as
tabulated in Table 1 simply as ”BAO Data - $2D$” and data given in Table 2 as
”BAO Data - $3D$”. Furthermore, upon the inclusion of the BAO Dark Energy
Spectroscopic Instrument (DESI) dataset, we explicitly refer to it as ”BAO
Data: $DESI$”, despite its categorization within the BAO Data-3D framework.

BAO Data - 2D: We employed a dataset comprising 12 Baryon Acoustic Oscillation
(BAO) measurements, denoted as $\theta_{\textrm{BAO}}(z)$, obtained under the
assumption of a fiducial cosmology derived from Planck observations. The
determination of the BAO angular scale involved conducting a two-point
correlation function analysis between pairs of data points. This measurement
of the BAO angular scale serves to impose constraints on the absolute scale of
BAO, denoted as $r_{d}$, once the angular diameter distance ($D_{A}$) to the
respective redshift is known. The relationship between
$\theta_{\textrm{BAO}}$, $D_{A}$, and $r_{d}$ is delineated by Equation (12).
Table 1 presents the compiled BAO dataset, encapsulating these measurements
for further analysis.

Table 1: BAO measurements from angular separation of pairs of galaxies (denoted as BAO Dataset: 2D throughout the analysis). $\bar{z}$ | $\theta_{\textrm{BAO}}(z)[^{\circ}]$ | Reference | $\bar{z}$ | $\theta_{\textrm{BAO}}(z)[^{\circ}]$ | Reference
---|---|---|---|---|---
0.45 | 4.77 $\pm$ 0.17 | SDSS DR10 [(Carvalho et al., 2016)] | 0.57 | 4.59 $\pm$ 0.36 | SDSS DR11 [(Carvalho et al., 2017)]
0.47 | 5.02 $\pm$ 0.25 | SDSS DR10 [(Carvalho et al., 2016)] | 0.59 | 4.39 $\pm$ 0.33 | SDSS DR11 [(Carvalho et al., 2017)]
0.49 | 4.99 $\pm$ 0.21 | SDSS DR10 [(Carvalho et al., 2016)] | 0.61 | 3.85 $\pm$ 0.31 | SDSS DR11 [(Carvalho et al., 2017)]
0.51 | 4.81 $\pm$ 0.17 | SDSS DR10 [(Carvalho et al., 2016)] | 0.63 | 3.90 $\pm$ 0.43 | SDSS DR11 [(Carvalho et al., 2017)]
0.53 | 4.29 $\pm$ 0.30 | SDSS DR10 [(Carvalho et al., 2016)] | 0.65 | 3.55 $\pm$ 0.16 | SDSS DR11 [(Carvalho et al., 2017)]
0.55 | 4.25 $\pm$ 0.25 | SDSS DR10 [(Carvalho et al., 2016)] | 2.225 | 1.77 $\pm$ 0.31 | SDSS QS [(de Carvalho et al., 2018)]
Table 2: BAO measurements from volumetric measurements (denoted as BAO Dataset: 3D throughout the analysis). ${z}$ | Anisotropic Constraint | Reference | ${z}$ | Isotropic Constraint | Reference
---|---|---|---|---|---
0.38 |  $\frac{D_{A}(0.38)}{r_{d}}$ = 7.42 | BOSS DR 12 [(Alam et al., 2017)] | 0.106 |  $\frac{D_{V}(0.106)}{r_{d}}$ = 2.98 $\pm$ 0.13 | 6dF [(F. Beutler et al, 2011)]
0.38 |  $\frac{D_{H}(0.38)}{r_{d}}$ = 24.97 | BOSS DR 12 [(Alam et al., 2017)] | 0.15 |  $\frac{D_{V}(0.15)}{r_{d}}$ = 4.47 $\pm$ 0.17 | MGS [(Ross et al., 2015)]
0.51 |  $\frac{D_{A}(0.51)}{r_{d}}$ = 8.85 | BOSS DR 12 [(Alam et al., 2017)] | 1.52 |  $\frac{D_{V}(1.52)}{r_{d}}$ = 26.1 $\pm$ 1.1 | eBOSS quasars [(Ata et al., 2017)]
0.51 |  $\frac{D_{H}(0.51)}{r_{d}}$ = 22.31 | BOSS DR 12 [(Alam et al., 2017)] |  |  |
0.61 |  $\frac{D_{A}(0.61)}{r_{d}}$ = 9.69 | BOSS DR 12 [(Alam et al., 2017)] |  |  |
0.61 |  $\frac{D_{H}(0.61)}{r_{d}}$ = 20.49 | BOSS DR 12 [(Alam et al., 2017)] |  |  |
2.4 |  $\frac{D_{A}(2.4)}{r_{d}}$ = 10.76 | BOSS DR 12 [(Bourboux et al., 2017)] |  |  |
2.4 |  $\frac{D_{H}(2.4)}{r_{d}}$ = 8.94 | BOSS DR 12 [(Bourboux et al., 2017)] |  |  |

BAO Data - 3D: For the three-dimensional (3D) Baryon Acoustic Oscillations
(BAO) data, we amalgamate isotropic BAO measurements obtained from various
surveys. These include the 6dF galaxy survey at redshift $z=0.106$ (Beutler et
al., 2011), the SDSS DR7-MGS survey at an effective redshift $z=0.15$ (Ross et
al., 2011), and measurements from the SDSS DR14-eBOSS quasar samples at
redshift $z=1.52$ (Ata et al., 2017). Additionally, BAO measurements using
Lyman-alpha samples in conjunction with quasar samples at redshift $2.4$ from
the SDSS DR12 are incorporated (Bourboux et al., 2017). Furthermore,
anisotropic BAO measurements from the BOSS DR12 galaxy sample at redshifts
$0.38$, $0.51$, and $0.61$ along with the covariance matrix (Evslin et al.,
2018) are utilised. The compiled BAO dataset is presented in Table 2.
Henceforth, we collectively refer to these datasets as ”$3D$ BAO” data.

SN Type-I:In addition to the Baryon Acoustic Oscillations (BAO) data, we used
the expansive ”Pantheon Plus Sample” of Type-I Supernovae (SN-Ia), comprising
a total of 1701 supernovae spanning the redshift range from 0.01 to 2.26
(Scolnic et al., 2018). This comprehensive dataset incorporates the SH0ES
distance anchors, utilizing the host cepheid galaxies for calibration (Table
for SN-Ia dataset, 2018).

In our analysis, we adopted a range of Hubble Constant values obtained from
different techniques. One such instance involves the adoption of a fixed value
for the absolute magnitude, $M_{B}$ = -19.214 $\pm$ 0.037 magnitudes. This
determination arises from a synthesis of geometric distance estimates derived
from Detached Eclipsing Binaries in the Large Magellanic Cloud (LMC)
(Pietrzy´nski et al., 2019), the MASER NGC4258 (Reid et al., 2019), and recent
parallax measurements of 75 Milky Way Cepheids with Hubble Space Telescope
(HST) photometry (Riess et al., 2021) GAIA Early Data Release 3 (EDR3)
(Lindegren et al., 2020a, b). Notably, this represents the most precise and
contemporary model-independent assessment of the Absolute Magnitude to date.
Other adopted values of $M_{B}$ used in our analysis are given in Table 3.

Utilizing the Absolute Magnitude of standard candles, such as Type-Ia
Supernovae (SNe-Ia), one can readily deduce their distance given their
observed apparent magnitude or flux. The relationship between the apparent
magnitude of SNe-Ia and their relative distance modulus is expressed by the
equation:

${D}_{L}=10^{(\mu-25)/5}.$ (13)

Here, $\mu=m_{b}-M_{B}$ represents the distance modulus, where $m_{b}$ denotes
the apparent magnitude of SNe-Ia and $M_{B}$ signifies the absolute magnitude
of SNe-Ia.

###  3.1 Model and Methodology

We aim to compare the constraints on cosmological parameters while using 3D
and 2D Baryon Acoustic Oscillations datasets. To keep the analysis as simple
as possible and focus on comparing the datasets of BAO, we intentionally
assumed a single model of cosmology which is the simplest six-parameter
$\Lambda$CDM model. We used data combination (BAO + Pantheon Plus) as
mentioned in Section 3 to fit the cosmological parameters.

In the $\Lambda$CDM cosmology framework, within a spatially flat FLRW
universe, the present scale factor $a_{0}$ is normalized to 1.0. The Hubble
parameter at redshift $z$ is given by:

$\frac{H^{2}(z)}{H_{0}^{2}}=\Omega_{m0}(1+z)^{3}+(1-\Omega_{m0}),$ (14)

where the subscript $0$ denotes present-day values and $\Omega_{m0}$
represents the present-day matter density. Other relevant quantities are
defined as follows:

$\displaystyle D_{M}(z)=\int_{0}^{z}\frac{c\leavevmode\nobreak\
d\bar{z}}{H(z)},$ (15) $\displaystyle D_{A}(z)=(1+z)^{-1}\leavevmode\nobreak\
D_{M}(z),$ (16) $\displaystyle D_{L}(z)=(1+z)\leavevmode\nobreak\ D_{M}(z),$
(17) $\displaystyle D_{V}(z)=((1+z)\leavevmode\nobreak\
D_{A}(z))^{2/3}\left(\frac{c\leavevmode\nobreak\ z}{H(z)}\right)^{1/3}$ (18)

where $D_{M}$, $D_{A}$ and $D_{L}$ are comoving distance, angular diameter
distance and luminosity distance respectively.
We applied a uniform prior distribution on $\Omega_{m0}$, $h$, and $r_{d}$ as
outlined in Table 3. Rather than relying solely on a single calibration for
the absolute magnitude $M_{B}$ of the SN sample, we opted to consider multiple
values of $M_{B}$ (or equivalently, $H_{0}$), a practice also observed in
various studies such as Lemos et al. (2023). For conversion, we employed the
relation:

$M_{B}=5(\textrm{log}\leavevmode\nobreak\ H_{0}-\alpha_{B}-5),$ (19)

where $\alpha_{B}$ is the observed intercept of B band apparent magnitude-
redshift relation for SNe in the Hubble flow. Presently, we assumed its value
to be $0.71273\pm 0.00176$ obtained independently of CMB and BAO. We employed
Gaussian priors on $M_{B}$ derived from various measurements of the Hubble
Constant, including those from ACT+WMAP CMB, Planck CMB + Lensing, SH0ES
collaborations, values derived from Masers, the Tully Fisher Relation, and
BOSS DR12+BBN, as detailed in Table 3. For generating Markov Chain Monte Carlo
(MCMC) chains based on the aforementioned dataset, we utilized the publicly
available code EMCEE (ForemanMackey et al., 2013). The obtained results are
presented in Table 4.

###  3.2 Cosmological Chisq Analysis

To infer information about the cosmological parameters within the framework of
the flat $\Lambda$CDM model, we conducted a Chisq analysis utilizing the
observables from the BAO 3D and BAO 2D datasets.

For the BAO 2D data, the observable is $\theta_{\textrm{BAO}}(z)$ as defined
in Equation (12). The corresponding chi-squared function is given by:

$\chi^{2}_{\textrm{2D
BAO}}(H_{0},\Omega_{m0},r_{d})=\sum_{i}\left[\frac{{\theta_{\textrm{BAO}}(z_{i})}^{\textrm{obs}}-\theta_{\textrm{BAO}}(z_{i})^{\textrm{model}}}{\sigma_{(\theta_{\textrm{BAO}}(z_{i}))}}\right]^{2},$
(20)

For the BAO 3D data, we have both anisotropic and isotropic observables.
Equation (21) describes the anisotropic component, where $D_{i}$ represents
the best fit of observables as $D_{A}(z_{i})/r_{d}$ and $D_{M}(z_{i})/r_{d}$
respectively. The covariance matrix for this dataset can be found in Table 1
of (Evslin et al., 2018).

$\chi^{2}_{\textrm{3D\leavevmode\nobreak\
BAO}}=\sum_{i,j}[D_{i}^{\rm{obs}}-D(z_{i})^{\rm{model}}]\text{C}ov^{-1}_{i,j}[D_{j}^{\rm{obs}}-D(z_{j})^{\rm{model}}],$
(21)

Further, for the isotropic component of the BAO 3D data, the observable is
$D_{V}(z_{i})/r_{d}$, where $D_{V}(z_{i})$ is the volume-averaged distance.
The Chi-squared definition for this component is provided below in Equation
(22):

$\chi^{2}_{\textrm{3D\leavevmode\nobreak\
BAO}}=\sum_{i}\left[\frac{{D_{V}(z_{i})/r_{d}}^{\rm{obs}}-{D_{V}(z_{i})/r_{d}}^{\rm{model}}}{\sigma_{({D_{V}(z_{i})/r_{d}})}}\right]^{2},$
(22)

For minimising the supernova cosmology parameters, the $\chi^{2}$ function is
defined as in Equation (23), where $\mu_{i}$ represents each supernova
distance, with $i$ ranging from 1 to 1701. $\mu_{b,i}^{\textrm{model}}$
denotes the predicted distance modulus, estimated using cosmological
parameters describing the expansion history (here, $\Lambda$CDM).

$\chi^{2}_{\textrm{SNe-
Ia}}(H_{0},\Omega_{m0},M_{B})=\sum_{i,j}[\mu_{b,i}^{\rm{obs}}-\mu_{b,i}^{\rm{model}}]\text{C}ov^{-1}_{i,j}[\mu_{b,j}^{\rm{obs}}-\mu_{b,j}^{\rm{model}}],$
(23)

And, $\text{Cov}^{-1}_{i,j}$ describes the covariance matrix for supernova
data. The total chi-square is then defined as the sum of the chi-square
contributions from the Pantheon Plus sample and the BAO sample, where ”BAO”
encompasses both BAO datasets used for comparison.

$\chi^{2}_{\textrm{Total}}(H_{0},\Omega_{m0},M_{B},r_{d})=\\\
\chi^{2}_{\textrm{SNe-
Ia}}(H_{0},\Omega_{m0},M_{B})+\chi^{2}_{\textrm{BAO}}(H_{0},\Omega_{m0},r_{d}).$
(24)

##  4 Results and Discussion

###  4.1 Tension between BAO 3D Data and BAO 2D Data

####  4.1.1 Effect on the sound horizon at the drag epoch: The Standard Ruler

In this section, we present the results of our analysis and discuss the
tension observed between the two BAO datasets: 3D and 2D. Specifically, we
focus on the cosmological parameter $r_{d}$, which represents the sound
horizon at the drag epoch. Previous studies have confirmed a high correlation
between $H_{0}$ and $r_{d}$. For instance, from Planck observations, the
derived $H_{0}$ is $67.27\pm 0.60$ km s-1 Mpc-1, corresponding to $r_{d}$ of
$147.05\pm 0.30$ Mpc. However, other studies such as (Evslin et al., 2018)
have found that when $H_{0}$ is constrained from low-redshift studies, its
value tends to be around $\sim 73$ km s-1 Mpc-1, requiring $r_{d}$ to be $\sim
137$ Mpc, leading to a tension of more than $2.5$ sigmas with Planck,
regardless of the behaviour of dark energy.

It is crucial to note that this discrepancy arises when using the 3D BAO data.
We got similar results as one can see in Table 4. While using 3D BAO data and
Pantheon Plus with SH0ES $2021_{a}$ calibration, we got constraints on $r_{d}$
as $134.30\pm 2.41$ Mpc corresponding to $h$ constraints $0.75\pm 0.012$.
Conversely, when utilizing the 2D BAO data, we obtained $h$ as $0.745\pm
0.0129$, corresponding to $r_{d}$ of $142.22\pm 3.37$ Mpc, which is compatible
with the Planck value of $r_{d}$ within one sigma (see Figure 2).

Parameter | Prior
---|---
$\Omega_{m0}$ | [0.1, 0.9]
$h$ | [0.50, 0.90]
$r_{d}$ | [130, 170]
Measurements of $M_{B}$ | Prior (Gaussian($\mathcal{N}(\mu,\sigma^{2})$))
Planck CMB + Lensing | $\mathcal{N}(-19.422,0.019^{2})$
ACT + WMAP CMB | $\mathcal{N}(-19.414,0.036^{2})$
BOSS DR12 + BBN | $\mathcal{N}(-19.385,0.070^{2})$
SH0ES 2021a | $\mathcal{N}(-19.214,0.039^{2})$
SH0ES 2021 | $\mathcal{N}(-19.241,0.040^{2})$
Masers | $\mathcal{N}(-19.220,0.089^{2})$
SH0ES 2019 | $\mathcal{N}(-19.217,0.042^{2})$
Tully Fisher | $\mathcal{N}(-19.159,0.075^{2})$
Table 3: Parameters used and their prior.

We have extensively examined this dataset to further understand the
cosmological tensions and the impact of the BAO dataset. To facilitate a
comprehensive discussion, we have divided our analysis into two subsections:

| Panth Plus + BAO Data : 3D | Panth Plus + BAO Data : 2D
---|---|---
Measurements |  $r_{d}$ (Mpc) | $h$ | $\Omega_{m0}$ |  $r_{d}$ (Mpc) | $h$ | $\Omega_{m0}$
SH0ES 2021a | $134.30\pm 2.41$ | $0.75\pm 0.012$ | $0.314\pm 0.0132$ | $142.22\pm 3.37$ | $0.745\pm 0.0129$ | $0.331\pm 0.0179$
SH0ES 2021 | $135.78\pm 2.73$ | $0.74\pm 0.01$ | $0.315\pm 0.0133$ | $143.93\pm 3.56$ | $0.736\pm 0.013$ | $0.3315\pm 0.0181$
Masers | $136.53\pm 4.46$ | $0.73\pm 0.02$ | $0.314\pm 0.0133$ | $143.026\pm 6.22$ | $0.742\pm 0.029$ | $0.3306\pm 0.0178$
SH0ES 2019 | $134.58\pm 2.67$ | $0.74\pm 0.01$ | $0.314\pm 0.0133$ | $142.359\pm 3.622$ | $0.744\pm 0.014$ | $0.3308\pm 0.018$
Tully Fisher | $134.05\pm 3.04$ | $0.75\pm 0.02$ | $0.313\pm 0.0132$ | $139.05\pm 4.91$ | $0.762\pm 0.024$ | $0.331\pm 0.0179$
Planck CMB + Lensing | $147.44\pm 1.89$ | $0.67\pm 0.01$ | $0.315\pm 0.0134$ | $156.36\pm 2.90$ | $0.677\pm 0.006$ | $0.330\pm 0.0178$
BOSS DR12 + BBN | $145.02\pm 4.82$ | $0.69\pm 0.022$ | $0.314\pm 0.0135$ | $153.92\pm 5.55$ | $0.688\pm 0.022$ | $0.331\pm 0.0182$
ACT + WMAP CMB | $146.89\pm 2.84$ | $0.68\pm 0.01$ | $0.315\pm 0.0133$ | $155.86\pm 3.64$ | $0.679\pm 0.011$ | $0.3312\pm 0.0179$
Table 4: Constraints on parameters for $\Lambda$CDM for both BAO Data Sets and
various Panthoen Plus calibrations $M_{B}$. The error bars quoted are at
$1\sigma$ confidence interval.

a) Pantheon Plus Calibration with $M_{B}$ Compatible with Low-Redshift
Experiments

Our analysis involved utilizing five different measurements of $H_{0}$ from
various low-redshift studies. We estimated the absolute magnitude of Type-Ia
Supernovae ($M_{B}$) from these measurements to calibrate our Pantheon Plus
sample. As illustrated in Table 4, the upper-left portion, utilizing the
Pantheon Plus and BAO Data: 3D, yielded an estimated $r_{d}\sim 135$ Mpc,
exhibiting more than three sigma tension with Planck’s estimation of $r_{d}$.
Conversely, when employing the 2D BAO Data alongside the same Pantheon Plus
sample and the same calibration of $M_{B}$, we obtained $r_{d}$ elevated to
around $142$ Mpc, aligning with Planck’s $r_{d}$ within one sigma (see Figure
2).

From this analysis, we infer that while the tension in $H_{0}$ between Planck
and SH0ES is confirmed, there is no tension in $r_{d}$ when using the 2D data.
With $H_{0}$ around $73$ km s-1 Mpc-1, we achieve an $r_{d}$ of roughly $142$
Mpc, compatible with Planck. To ensure the robustness of our results
independent of the calibration of the Pantheon Plus sample, we consider
calibrations of $M_{B}$ from various experiments. Notably, our findings from
all low-redshift experiments converge, indicating mutual consistency.

b) Pantheon Plus Calibration with $M_{B}$ Compatible with High-Redshift
Experiments

We calibrated the Pantheon Plus sample using $M_{B}$ calibration from three
$H_{0}$ measurements for high redshifts, namely Planck CMB + Lensing, BOSS
DR12 + BBN, and ACT+WMAP CMB. We found that with the Pantheon Plus calibrated
to high-redshift experiments, we obtained $h$ value around $0.67$ and $r_{d}$
close to $147$ Mpc while using the BAO 3D data. However, when utilizing the
BAO 2D data with the same calibration, we obtained the same $H_{0}$ value but
$r_{d}$ as $156$ Mpc, which exhibits tension with Planck’s value of $r_{d}$.

In the previous subsections, we talked about $r_{d}$ tension between 2D and 3D
BAO, and we found our results depend a lot on which SNe calibration $M_{B}$ we
used. We need to understand here, that since 2D BAO measures a higher product
of $hr_{d}$ than 3D BAO (Equation 26 and 25), a higher $rd$ corresponding to
higher $h$ can be allowed. For example, while using 2D BAO and SH0ES 2021a
calibration, h of 0.745 $\pm 0..0129$ is allowed corresponding to $r_{d}$
142.22 Mpc. On the other hand, in 3D BAO and SH0ES 2021a calibration, h of
0.75 $\pm 0.012$ allows lower value $r_{d}$ 134.30 $\pm$ 2.41 Mpc. From here
we conclude that from BAO data, we should not conclude on $h$ and $r_{d}$
individually but the product $r_{d}h$.

####  4.1.2 Cosmological constraints on the product $r_{d}*h$

Figure 4 confirms that the result we obtained is completely in alignment with
the fact that the product of Hubble Constant and BAO acoustic scale
$H_{0}r_{d}$ is constant irrespective of calibration of SNe Ia used. We can
also say that it is the artefact of the fact that BAO measurements put bounds
on $H(z)r_{d}$ rather than individually putting constraints on $r_{d}$ or
$H_{0}$ (Bernel et al., 2016). Also in CMB measurements, we measure the
angular acoustic scale $r_{d}/D_{A}(z_{d})$ where $z_{d}$ is the drag epoch
redshift. In Figure 4, all the $H_{0}r_{d}$ values while using BAO Data: 3D
agree within one sigma irrespective of the $M_{B}$ (SNe calibration) used. The
same is true for the BAO 2D Dataset. We also plotted green and orange bands
using CMB temperature, polarization, and lensing. ($r_{d}h=101.8\pm 1.3$ Mpc )
and from DESI which is $r_{d}h=98.82\pm 0.82$ Mpc. The mean value of
$H_{0}r_{d}$ from PP + 3D BAO data coincides with the CMB band except for the
cases when Pantheon Plus utilises Tully Fisher or Masers for calibrating the
sample. DESI Band does not overlap with the CMB band within one sigma. But
within two sigmas, products from both BAO datasets 2D and 3D are consistent
(Equation 25 and Equation 26). DESI BAO uses the same methodology as our
denoted BAO: 3D dataset. BAO 2D dataset also shows tension with CMB obtained
$H_{0}r_{d}$ results. We obtained the following results assuming SH0ESa
calibration for Pantheon Plus along with BAO datasets. We mentioned the BAO
dataset we used and the results stated are in 68 $\%$ credible interval.

$\displaystyle\left.\begin{aligned} &\Omega_{m0}=0.314\pm 0.0132&\\\
&r_{d}h=100.72\pm 2.42\textrm{ Mpc}&\\\ \end{aligned}\right\\}\text{ BAO : 3D
+ PP}$ (25)

$\displaystyle\left.\begin{aligned} &\Omega_{m0}=0.331\pm 0.018&\\\
&r_{d}h=105.95\pm 3.11\textrm{ Mpc}&\\\ \end{aligned}\right\\}\text{ BAO : 2D
+ PP}$ (26)

Figure 1: This plot shows different $H_{0}$ values obtained from various
measurements - high (left) and low redshifts (right) on the x-axis. On the Y
axis are the constraints on the $r_{d}$ (absolute scale of Baryon Acoustic
Oscillations). The green band in both plots shows constraints on $r_{d}$ from
the CMB Planck experiment.

Figure 2: This plot shows $H_{0}-r_{d}$ contours and the correlation of
$H_{0}$ and $r_{d}$ while using two different BAO Datasets and two different
calibrations $M_{B}$ for Pantheon Plus sample. The horizontal green band
represents CMB Planck $r_{d}$.

Figure 3: This plot shows $H_{0}-\Omega_{m0}$ contours and the correlation of
$H_{0}$ and $\Omega_{m0}$ while using two different BAO Datasets and two
different calibrations $M_{B}$ for Pantheon Plus sample. The horizontal green
band represents CMB Planck $\Omega_{m0}$.

Figure 4: Left: This plot shows the values of the product of the Hubble
parameter and sound horizon at the drag epoch $r_{d}h$ obtained while
utilising various calibrations for the Pantheon Plus sample and two different
BAO datasets. Right: This plot shows the contours of $r_{d}h$ and
$\Omega_{m0}$ matter density at present obtained for various calibrations for
the Pantheon Plus sample when combined with two different BAO datasets. The
orange and light green bands used for comparison are the results from DESI
($r_{d}h=101.8\pm 1.3$ Mpc) and CMB temperature, polarization and lensing
($r_{d}h=98.82\pm 0.82$ Mpc and $\Omega_{m0}=0.315\pm 0.007$) respectively. We
used the notation $r_{d}h$ $\equiv$ $H_{0}r_{d}$/(100 km s-1 Mpc-1) in both
plots.

##  5 Comments on Cosmological Model Used

While keeping in mind that BAO measures the product of $H_{0}r_{d}$ and not
$H_{0}$ and $r_{d}$ individually, the presented results (Figure 2-left plot)
and the last subsection reveal around 1.3$\sigma$ tension within two
independent BAO datasets: 2D BAO and 3D BAO datasets and less than two sigmas
tension between CMB and DESI estimated $H_{0}r_{d}$. But if we look at contour
$H_{0}r_{d}$-$\Omega_{m0}$ (Figure 2-right plot), the tension in
$H_{0}r_{d}-\Omega_{m}0$ contours is more than two sigma. We can clearly see
that there is no tension between $\Omega_{m0}$ of 2D BAO, 3D BAO and CMB
estimated $\Omega_{m0}$. This tension in $H_{0}r_{d}$ from 2D and 3D BAO
appeared because of $\Omega_{m0}$-$H_{0}r_{d}$ correlation. Now this tension
in $H_{0}r_{d}$ from 2D and 3D BAO in $\Omega_{m0}$-$H_{0}r_{d}$ plane calls
for questioning the relation between $\Omega_{m0}$ and $H_{0}r_{d}$ which
comes from assuming a cosmological model. Here the assumed cosmological model
is $\Lambda$CDM. As can be seen from Figure 2 (left plot), there is no tension
(around 1 $\sigma$) between 2D and 3D BAO datasets, $\Lambda$CDM is in trouble
(beyond 2$\sigma$) assuming there is no systematic error in BAO.

##  6 Analysis with DESI data

###  6.1 Model, Methodology, and Dataset

In our analysis with DESI data, we adhered to the standard cosmological model,
$\Lambda$CDM, to compare with the results obtained in the previous sections.
However, to highlight DESI’s main findings, we extended our analysis to
include the Chevallier-Polarski-Linder (CPL) model, which introduces a
redshift-dependent equation of state parameter
$w(z)=w_{0}+w_{a}(1-a)=w_{0}+w_{a}\frac{z}{1+z}$, providing a more nuanced
characterization of cosmic dynamics.

In the CPL model, the Hubble parameter $H(z)$ is expressed as:

$\frac{H^{2}(z)}{H_{0}^{2}}=\Omega_{m0}(1+z)^{3}+(1-\Omega_{m0})f(z),$ (27)

where $f(z)=\exp\left(3\int^{z}\frac{(1+w(x))}{1+x}dx\right)$.

Tracer | $z$ | DM/rd | DH/rd | DV/rd
---|---|---|---|---
BGS | 0.30 | - | - | $7.93\pm 0.15$
LRG | 0.51 | $13.62\pm 0.25$ | $20.98\pm 0.61$ | -
LRG | 0.71 | $16.85\pm 0.32$ | $20.08\pm 0.60$ | -
LRG+ELG | 0.93 | $21.71\pm 0.28$ | $17.88\pm 0.35$ | -
ELG | 1.32 | $27.79\pm 0.69$ | $13.82\pm 0.42$ | -
QSO | 1.49 | - | - | $26.07\pm 0.67$
Lya QSO | 2.33 | $39.71\pm 0.94$ | $8.52\pm 0.17$ | -
Table 5: DESI Data Release 1 BAO Measurement used

Figure 5: This plot shows three cosmological parameters $h$, $r_{d}$,
$\Omega_{m0}$ along with the product of Hubble parameter and sound horizon at
the drag epoch $H_{0}r_{d}$ obtained using various calibration for Pantheon
Plus sample combined with two different BAO datasets. In the $h$ subplot, we
show green and cyan bands from Planck 2018 and Reiss ’22 for comparison. In
the $r_{d}$ and $\Omega_{m0}$ subplots, we show Planck corresponding values.
Further in the extreme right subplot of $H_{0}r_{d}$, the orange and light
green bands used for comparison are the results from DESI ($r_{d}h=101.8\pm
1.3$ Mpc) and CMB temperature, polarization and lensing ($r_{d}h=98.82\pm
0.82$ Mpc). We used the notation $r_{d}h$ $\equiv$ $H_{0}r_{d}$/(100 km s-1
Mpc-1).

Figure 6: Left : This plot shows how the $w_{0}-w_{a}$ contour plot shifts
when we replace 3D dataset to 2D dataset. It also shows that the BAO 3D and
DESI BAO datasets have contours that are nearly overlapping. Right: This plot
shows three $h-r_{d}$ contours for the combination of Pantheon Plus + CMB +
BAO dataset. The three contours are for three different data combinations. It
clearly agrees with the fact that the product $hr_{d}$ for BAO: 2D is greater
than that for BAO : 3D or BAO DESI. Keeping $h$ constant, $r_{d}$ estimated
values from Panth Plus + BAO: 2D data combination agrees with Planck $r_{d}$
within one sigma while Planck $r_{d}$ is more than 2 $\sigma$ away from Panth
Plus + BAO 3D estimate. In both the plots, it is evident that BAO 3D has less
constraining power hence giving bigger contours than the BAO 3D dataset.

Figure 7: Both the plots have BAO DESI datasets. These plots show how the
results change when changing calibration $M_{B}$ of the Pantheon Plus sample.
Left: This plot shows that there is no change in the $w_{0}-w_{a}$ plane when
we change the calibration of Pantheon Plus. It also shows how the constraints
get better as we add CMB data. Right: We can see the shrink in the $h-r_{d}$
contours when we added the CMB data set to the Panth Plus + BAO: DESI dataset.
We can also see the parallel shift in both the parameters which is a decrease
in $h$ and an increase in $r_{d}$ while we changed the calibration on the
Pantheon Plus sample from SH0ES21a to Planck CMB + Lensing. When data
combination Panth + ($M_{B}$ = SH0ES21a) + BAO: DESI + CMB is used, the
estimated $h$ value is higher because of chosen $M_{B}$ calibration and
estimated $h$ values are between SHOES and CMB and preferring lower $r_{d}$
than Planck. However, when CMB data is removed and calibration of Pantheon
Plus is changed to PLanck CMB + Lensing, the lower $h$ is preferred and
$r_{d}$ is then pushed to a higher value keeping the product $hr_{d}$
constant.

Other than the Pantheon Plus dataset and DESI BAO dataset as provided in Table
5, we also incorporated CMB data. In particular, we utilised the Planck 2018
compressed likelihood for TT, TE, EE + lowE as obtained by Chen et al
(Chen:2018dbv) (for the detailed method for obtaining the compressed
likelihood (see Ade et al. (Planck:2015bue) )). We applied the same
methodology and definitions used for conducting the Chisq analysis of the 3D
BAO dataset to analyze the DESI data. This includes the Equation (21) and
Equation (22) used for calculating the Chisq values for the DESI dataset. The
results obtained for the $\Lambda$CDM model are presented in Table 6. Our
study involved a comparison of data while maintaining the same model. Table 4,
Table 6, and Figure 5 depict the comparison of three BAO datasets and their
impact on cosmological parameters under the assumption of cosmology as
standard $\Lambda$CDM. Additionally, Figure 6 and Figure 7 illustrate this
comparison using the CPL model.

| Panth Plus + BAO Data : DESI
---|---
Measurements |  $r_{d}$ (Mpc) | $h$ | $\Omega_{m0}$
SH0ES 2021a | $134.33\pm 2.46$ | $0.75\pm 0.011$ | $0.318\pm 0.0124$
SH0ES 2021 | $135.83\pm 2.74$ | $0.74\pm 0.013$ | $0.318\pm 0.0125$
Masers | $136.56\pm 4.44$ | $0.73\pm 0.02$ | $0.317\pm 0.0126$
SH0ES 2019 | $134.68\pm 2.67$ | $0.74\pm 0.01$ | $0.318\pm 0.0123$
Tully Fisher | $134.26\pm 3.12$ | $0.75\pm 0.02$ | $0.311\pm 0.0120$
Planck CMB + Lensing | $147.44\pm 1.95$ | $0.68\pm 0.01$ | $0.319\pm 0.0126$
BOSS DR12 + BBN | $145.10\pm 4.87$ | $0.69\pm 0.022$ | $0.318\pm 0.0127$
ACT + WMAP CMB | $146.95\pm 2.82$ | $0.68\pm 0.01$ | $0.318\pm 0.0127$
Table 6: Constraints on parameters for $\Lambda$CDM for BAO Data DESI data.
The error bars quoted are at $1\sigma$ confidence interval.

##  7 Discussion of 3D, 2D and DESI BAO Results

Here, we delve into the results and comparison between the 3D and 2D BAO
datasets, alongside the new DESI data release. Figure 5 shows the comparison
of three cosmological parameters $h$, $r_{d}$, $\Omega_{m0}$ along with the
product of Hubble parameter and sound horizon at the drag epoch $H_{0}r_{d}$
obtained while using various calibrations for Pantheon Plus sample and
combined with three BAO datasets 2D, 3D and BAO. In the $h$ subplot, we show
green and cyan bands from Planck 2018 and Reiss ’22 for comparison. In the
$r_{d}$ and $\Omega_{m0}$ subplots, we show Planck corresponding values.
Further in the extreme right subplot of $H_{0}r_{d}$, the orange and light
green bands used for comparison are the results from DESI ($r_{d}h=101.8\pm
1.3$ Mpc) and CMB temperature, polarization and lensing ($r_{d}h=98.82\pm
0.82$ Mpc). We used the notation $r_{d}h$ $\equiv$ $H_{0}r_{d}$/(100 km s-1
Mpc-1). And, since to distinguish among BAO and conclude irrespective of
Pantheon calibration, one should focus on the quantity which BAO measures, the
product of $hr_{d}$. As we can see here, within 2 $\sigma$ both 2D and 3D BAO
agree with DESI and CMB obtained $hr_{d}$.

In Figure 6, we present the $w_{0}-w_{a}$ and $h-r_{d}$ planes. Both plots
feature the Pantheon Plus sample calibrated with SH0ES21a along with CMB data,
and three BAO datasets: 2D, 3D, and DESI Release.

In the left contour plot, showcasing the $w_{0}-w_{a}$ plane, it’s notable
that the (-1.0,0) point lies on the edge of the two-sigma contour across all
three datasets. The DESI and 3D BAO datasets exhibit comparable constraining
power, while the 2D dataset displays larger contours. Moreover, the contour in
the one-sigma region extends towards lower values of $w_{0}$ and higher values
of $w_{a}$ for the 2D dataset, while maintaining the correlation. In the right
plot, a particularly noteworthy observation emerges. By maintaining
consistency with the other two datasets and solely altering the BAO datasets,
a significant finding surfaces: when utilizing the BAO 2D dataset, our
estimated $r_{d}$ aligns with Planck’s $r_{d}$ within a one-sigma region.
However, for the other two datasets, 3D and DESI Release, the constraints on
$r_{d}$ deviate from Planck’s $r_{d}$ by more than two sigma.

In Figure 7, the orange contour is the same as in Figure 6 and is provided for
comparison and reference. In the left plot, we observe that the contours
expand notably when the CMB dataset is removed. Interestingly, altering the
calibration of the Pantheon Plus sample does not induce significant changes in
the behaviour of the $w_{0}-w_{a}$ contours. However, it does lead to
substantial shifts in the values of both parameters $h$ and $r_{d}$ in the
right plot.

This outcome was obtained while employing the CPL parameterization and
utilizing the DESI dataset. It’s worth highlighting that similar results were
obtained for the $\Lambda$CDM model and the BAO: 3D dataset, as outlined in
Table 4. This consistency across different models underscores the robustness
and reliability of the findings, indicating that our results are independent
of dark energy behaviour.

This finding underscores the unique power of the BAO dataset in providing
cosmological constraints that closely align with Planck’s measurements. Such
alignment underscores the dataset’s robustness and its potential to enhance
our understanding of the universe’s fundamental parameters and break residual
degeneracies with CMB. Observations from BAO 2D and BAO 3D provide a promising
avenue for resolving discrepancies and refining our understanding of
cosmological parameters. Understanding $\Omega_{m0}-H_{0}r_{d}$ plane (in
Figure 4) and decreasing tension between the contours while using the 2D and
3D BAO dataset, may require physics beyond $\Lambda$CDM not just while
inferring cosmological information from cosmological data but also in the
assumption of fiducial cosmology in observational data such as parameter
$\alpha$ (Equation 7).

##  8 Conclusions

Through a critical examination and detailed analysis of 2D and 3D data of BAO
and using the theoretical framework of the standard model of cosmology, we aim
to check consistency in two BAO datasets. Through careful analysis, we found
out that the product of $hr_{d}$ in both 2D and 3D BAO datasets is consistent.
However, more than two sigma tension in $\Omega_{m0}-H_{0}r_{d}$ plane while
using two independent BAO datasets may advocate for the need for beyond
$\Lambda$CDM physics not just while inferring cosmological information from
observational data but also while observing BAO datasets and injecting
standard assumptions not too different from $\Lambda$CDM.

## Acknowledgements

The author Ruchika would like to thank Alessandro Melchiorri, M. M. Skeikh
Jabbari, Nils Schöneberg, Jalison Alcaniz, Thais Lemos and Anjan Ananda Sen
for useful discussions. We acknowledge IUCAA, Pune, India for the use of their
computational facilities. We acknowledge financial support from TASP,
iniziativa specifica INFN.

##  Appendix A BAO Data for redshift less than 1

To assess the potential bias introduced by BAO data at redshifts greater than
one, we removed BAO data for redshift $z>1$ and performed the analysis again.
The results are similar to those obtained when all BAO redshift measurements
were included (Figure 81). However, the contours obtained while using 3D BAO
data shift toward a higher value of $\Omega_{m0}$. Despite this shift, our
conclusion that there is more than two-sigma tension in $H_{0}r_{d}$ in
$\Omega_{m0}-H_{0}r_{d}$ plane remains unchanged.

Figure 8:  This plot shows the contours of $r_{d}h$ and $\Omega_{m0}$ matter
density at present obtained for various calibrations for the Pantheon Plus
sample when combined with two different BAO datasets (Teal and Black contours
are for full BAO dataset whereas blue and red contours are for BAO Dataset
(z$<1$)). The orange and light green bands used for comparison are the results
from DESI ($r_{d}h=101.8\pm 1.3$ Mpc) and CMB temperature, polarization and
lensing ($r_{d}h=98.82\pm 0.82$ Mpc and $\Omega_{m0}=0.315\pm 0.007$)
respectively. We used the notation $r_{d}h$ $\equiv$ $H_{0}r_{d}$/(100 km s-1
Mpc-1) in the plot.

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to exhaustively complete the set of isochrone potentials. We have also
provided a geometrical way of finding the mass in a given sphere, associated
with a given isochrone potential.

After this analysis on isochrone potentials, we turned to the study of
isochrone orbits per se in Sec. IV. We first derived a generic and analytic
formula for the radial period $T$ and the apsidal angle $\Theta$ of any
isochrone orbit in terms of the parabola parameters $(a,b,c,d)$ and the
particles properties ($\xi$ for $T$ and $\Lambda$ for $\Theta$). We provided
various geometrical interpretations of these laws and given a recipe for
computing both $T$ and $\Theta$ in the Hénon plane, as depicted in Fig. 12.
The final results involve the solution of the orbital differential equation
itself. Based on all our previous findings, we devised and completed a
geometrical program in the Hénon plane that allowed to find an analytic
parameterization for any isochrone orbit.222222We note that this procedure
could be repeated for any family of curves in the Hénon plane that is stable
under affine transformations, provided that one of them is already
analytically known. This might be a fruitful and interesting academic
exercise, perhaps by replacing parabolae by other conics, or algebraic curves
of higher degree. Last but not least, we have computed numerically and plotted
various isochrone orbits in order to check the consistency of our analytic
results in the orbital plane. We described and classified these orbits
according to the isochrone potential $\psi_{i}$ to which they belong and to
the parameter $\lambda$, that is essentially the only responsible for the
different shapes of the orbits. In particular, we have seen that orbits in the
Bounded potential have peculiar properties at apoapsis and understood these
properties analytically.

Isochrony has much to offer when it comes to useful and fruitful problems of
mathematical physics: It contains the two most fundamental potentials of
gravitation, the Keplerian and the harmonic ones, and exhibits interesting
geometrical and algebraic properties (generalized Kepler’s third laws, affine
transformations of parabolae in the Hénon plane), and on top of that,
everything can be found analytically, from the shape of the potentials ((41)
through (45)), the radial period and apsidal angle ((62) and (61)) to the
parameterization of orbits therein ((74), (80) and (82)). Last but not least,
it seems to be of physical interest in the evolution of cluster dynamics, as
suggested in Simon-Petit  _et al._ (2019) and in ongoing works. We encourage
the interested reader to take inspiration out of the geometrical methods and
results of this paper to devise exercises material that serve academic
purposes. It appears that mathematical physics problems such as isochrony are
rarely seen in the literature nowadays, and the authors are convinced that
such problems are of great interest as both academic and research material.

###### Acknowledgements.

PR is grateful to M. Langer and A. Le Tiec for helpful discussions,
suggestions and comments; and to the Centro Brasileiro de Pesquisas Fisìcas
for its hospitality, where part of this work was done.

##  Appendix A Dynamical system

In order to draw the orbit, we write the equations of motion as a three-
dimensional dynamical system. Although, in general, a generic three-
dimensional motion in classical mechanics involves 6 degrees of freedom,
namely the three coordinates and their associated momenta, the spherical
symmetry here at play reduces this number to three. Moreover, the radial
motion is decoupled from the polar one. To see this, differentiate Eq. (2)
with respect to $r$ to obtain a second-order ODE for $r(t)$, or equivalently,
a two-dimensional dynamical system for the radial motion in $(r,\dot{r})$. To
get the polar motion, and thus the full orbit $(r(t),\theta(t))$, one may
simply use the definition of the angular momentum $\Lambda=r^{2}\dot{\theta}$,
which gives $\theta(t)$ directly from $r(t)$. These three pieces together give
the following three-dimensional dynamical system in $(r,\dot{r},\theta)$

$\frac{\mathrm{d}r}{\mathrm{d}t}=\dot{r}\,,\quad\frac{\mathrm{d}\dot{r}}{\mathrm{d}t}=\frac{\Lambda^{2}}{r^{3}}-\psi^{\prime}(r)\quad\text{and}\quad\frac{\mathrm{d}\theta}{\mathrm{d}t}=\frac{\Lambda}{r^{2}}\,.$
(90)

The system (90) is sufficient to compute the trajectory of any particle in any
central potential $\psi(r)$. In particular, once $\psi(r)$ is plugged into
Eqs. (90) and some initial conditions $(r(0),\dot{r}(0),\theta(0))$ are
provided, the motion can be solved using, e.g., a classical Runge-Kutta
numerical method. Since we are interested in periodic, bounded orbits, we
must, however, choose the initial conditions carefully. In order to find these
orbits more easily, we choose to express $(r(0),\dot{r}(0))$ in terms of the
two constants of motion $\xi$ and $\Lambda$, and take $\theta(0)=0$, as the
latter does not change the periodic nature of an orbit. Since the set of
$(\xi,\Lambda)$ that produces periodic orbits is precisely the one we found in
Sec. III.3 depicted in Figs. 9, 10 and 11, this procedure allows for an easy
picking of initial conditions and allows us to draw any periodic orbit in any
isochrone potential. This has been used to draw the orbits in Figs. 15 through
17, and to check the validity of all our analytic isochrone formulae.

##  Appendix B Alternative form of the third laws

We have seen that the Hénon’s formula (21) gives the period $T(\xi)$ of an
orbit $(\xi,\Lambda)$ in any isochrone potential. For any value of $\xi$,
there exists a unique value $\Lambda_{C}$ such that the orbit is circular,
corresponding to the line $\mathscr{L}_{C}:y=\xi x-\Lambda^{2}_{C}$ being
tangent to the isochrone parabola. Using the notations $h$ and $L(h)$
introduced in Sec. II.2, this circular limit corresponds to $h\rightarrow 0$.
Taking this well-defined limit in Eq. (25) gives

$T^{2}=\frac{\pi^{2}}{16(1+\xi^{2})^{3/2}}\lim_{h\rightarrow
0}\frac{L(h)^{2}}{h}\,.$ (91)

As it can be intuited from the discussion of Sec. II.2.4, it turns out that
the limit on the right-hand side of Eq. (91) is independent of the global
aspect of the curve. In fact, this limit is simply eight times the radius of
curvature $R_{C}$ at the point $C$ corresponding to the circular
orbit.232323The intuition comes from the following remark: The information on
the period should be encoded somewhere on the curve, but be independent of
$\Lambda$ and thus of the height of the line $\mathscr{L}$. By varying
$\Lambda$ we see that the only place that is not altered by this translation
is the point $C$. In particular, the slope of the tangent encodes $\xi$, and
the curvature at that point encodes $T$. In other words, we have

$T^{2}=\frac{\pi^{2}}{2}\frac{R_{C}}{(1+\xi^{2})^{3/2}}\,.$ (92)

Equation (92) provides a geometrical way to find the period of any given orbit
in an isochrone potential, without any algebraic reference to the parabola
itself. First take a line $\mathscr{L}$ intersecting an isochrone parabola
$\mathscr{P}$ at $P$ and $A$, and then, perform a translation of this line to
construct $\mathscr{L}_{C}$, tangent to $\mathscr{P}$ at $C$. The curvature
radius $r_{C}$ of the parabola at the tangency point $C$ gives the period, via
Eq. (92). This is the local version of the result given in Eq. (21).

In a similar fashion, the law for the apsidal angle can also be written in
terms of curvature, albeit for the effective potential. If
$\Psi_{e}(u)=\psi_{e}(r)$ with $u=1/r$, we have

$\Theta^{2}=\frac{4\pi^{2}\Lambda^{2}}{\Psi_{e}^{\prime\prime}{(u_{C})}}\,.$
(93)

This law provides a way to compute the apsidal angle in the effective
potential $\Psi_{e}$ in the Binet variable $u=1/r$, or in the real effective
potential $\psi_{e}$, using
$\Psi_{e}^{\prime\prime}(u_{C})=r_{C}^{4}\psi_{e}^{\prime\prime}(r_{C})$.

##  Appendix C Hénon’s formula for $\Theta$

We detail the computation of the integral (7) for $\Theta$, with the method
used to derive the Hénon formula (21) for $T$. According to the dictionary in
Table. 1, this time we use the Binet variable $u:=1/r$ and define a potential
$\Psi_{e}(u)$ by $\psi_{e}(r)=\Psi_{e}(u)$. Inserting these notations in (7)
readily gives

$\Theta=\sqrt{2}\Lambda\int_{u_{A}}^{u_{P}}\frac{\mathrm{d}u}{\sqrt{D_{\Theta}(u)}}\,,\quad\text{with}\quad
D_{\Theta}(u):=\xi-\Psi_{e}(u)\,.$ (94)

This is the equivalent for $\Theta$, of Eq. (14) for $T$. The bounds of the
integral are $u_{A}:=1/r_{A}$ and $u_{P}=1/r_{P}\geq u_{A}$. In the $(u,y)$
plane, the quantity $D_{\Theta}(u):=\xi-\Psi_{e}(u)$ appearing in Eq. (94) is
the vertical distance between the curve $\mathscr{C}:y=\Psi_{e}(u)$ and the
line $\mathscr{L}:y=\xi$. Once again, the fact that $D_{\Theta}(u)\geq 0$
follows from the requirement $\xi-\psi_{e}(r)\geq 0$, cf. (4).

Figure 18: Illustration of the geometrical analysis involved in the
computation of $\Theta$, according to the dictionary of Table. 1. $D_{\Theta}$
is the vertical distance between the straight line $\mathscr{L}:y=\xi$ and a
generic curve $\mathscr{C}:y=\Psi_{e}(u)$. The line
$\mathscr{L}_{C}:y=\xi_{C}$ is the unique line both parallel to $\mathscr{L}$
and tangent to $\mathscr{C}$.

Next we rewrite the distance $D_{\Theta}$ as
$D_{\Theta}(u)=\varepsilon^{2}-z(u)^{2}$, with $\varepsilon^{2}:=\xi-\xi_{C}$
the vertical distance between the two lines $\mathscr{L}$ and
$\mathscr{L}_{C}$, and $z(u)^{2}:=\Psi_{e}(u)-\xi_{C}$, as depicted in Fig.
18. As we did for $T$, we may conveniently choose $z(u)$ to be negative on
$[u_{A},u_{C}]$ and positive on $[u_{C},u_{P}]$. The formula for $\Theta$
becomes

$\Theta=\sqrt{2}\Lambda\int_{u_{A}}^{u_{P}}\frac{\mathrm{d}u}{\sqrt{\varepsilon^{2}-z(u)^{2}}}\,.$
(95)

As we argued for $T$, the function $u\mapsto z(u)$ is by construction
monotonically increasing so that we can perform the change of variables
$u\rightarrow z(u)$ and introduce $f$ such that $u=f(z)$. Since
$z(u_{A})=-\varepsilon$ and $z(u_{A})=\varepsilon$, the integral becomes

$\Theta=\sqrt{2}\Lambda\int_{u_{A}}^{u_{P}}\frac{f^{\prime}(z)\mathrm{d}z}{\sqrt{\varepsilon^{2}-z^{2}}}=\sqrt{2}\Lambda\int_{-\pi/2}^{\pi/2}f^{\prime}(\varepsilon\sin\phi)\mathrm{d}\phi\,,$
(96)

where the last equality follows from the change of variables
$z\rightarrow\varepsilon\sin\phi$, with $\phi$ varying between $-\pi/2$ and
$\pi/2$ when $z\in[-\varepsilon,\varepsilon]$. Now we assume for $f^{\prime}$
a Taylor expansion around $0$ of the form $f(z)=a_{0}+\sum_{n\geq
1}a_{n}z^{n}$, and integrate term by term to get

$\Theta=\sqrt{2}\pi\Lambda a_{0}+2\sqrt{2}\Lambda\sum_{n\geq
1}a_{2n}W_{2n}\varepsilon^{2n}\,,$ (97)

with $W_{n}$ the Wallis integral as given in Eq. (18), and the odd terms
vanishing by integration over the symmetric interval
$[-\varepsilon,\varepsilon]$. Now if $\Theta$ is to be independent of $\xi$,
then it must also be independent of $\varepsilon$, since $\xi_{C}$ depends
only on $\Lambda$. Therefore, we must have $a_{2n}=0$ for all $n\geq 1$. We
thus obtain the formula $\Theta=\sqrt{2}\pi\Lambda a_{0}$ and the Taylor
expansion of $f^{\prime}$ therefore writes

$f^{\prime}(z)=\frac{\Theta}{\sqrt{2}\pi\Lambda}+\sum_{n\geq
1}a_{2n+1}z^{2n+1}\,.$ (98)

Integrating this equation over the interval
$[z(u_{A}),z(u_{P})]=[-\varepsilon,\varepsilon]$, we can make the same remarks
as we did in the paragraph below Eq. (20), except that in this case
$f(z_{A})=u_{A}$ and $f(z_{P})=u_{P}$. In the end, we obtain Eq. (22), which
is the equivalent to Eq. (21) for $T$. The right-hand side of that equation is
independent of $\xi$, even though the quantities $u_{P},u_{A}$ depend
explicitly on $\xi$.

##  Appendix D Analysis of $\Theta(\Lambda)$

Let $\psi_{i}$ be a potential in one of the four families $i\in\\{1,2,3,4\\}$
as defined in Sec. II.2. In this appendix, we study the properties of the
function $\Theta_{i}(\Lambda)$ defined in Eq. (61). These formulae are used in
Sec. IV.3 to classify the orbits in each isochrone potential $\psi_{i}$. The
claims of Sec. IV.3 regarding each function
$\Lambda\mapsto\Theta_{i}(\Lambda)$ follow easily from the mathematical
analysis detailed here, with the Latin parameters $(a,b,c,d,e)$ replaced by
the Greek ones $(\omega,\varepsilon,\lambda,\mu,\beta)$.

####  D.0.1 Harmonic and Kepler family, $\Theta_{1}(\Lambda)$ and
$\Theta_{4}(\Lambda)$

For the Harmonic potentials $\psi_{1}$, the analysis of
$\Theta_{1}(\Lambda)=\pi\Lambda(\Lambda^{2}+\lambda)^{-1/2}$ is
straightforward. For $\lambda<0$, it is strictly decreasing and $\Theta$
varies in $]\pi,+\infty[$ when $\Lambda\in]\sqrt{-\lambda},+\infty[$. For
$\lambda=0$, $\Theta=\pi$ for all $\Lambda\in\mathbb{R}$. For $\lambda>0$, it
is strictly increasing and $\Theta$ varies in $]0,\pi[$ when
$\Lambda\in]0,+\infty[$. For the Kepler family $\psi_{4}$ the analysis is also
straightforward since we have for any $\Lambda$ and $\lambda$ the identity
$\Theta_{4}(\Lambda)=2\Theta_{1}(\Lambda)$ by direct examination of Eq. (61)
when $b=0$ (harmonic) and $d^{2}=4b^{2}e$ (Kepler).

####  D.0.2 Bounded family, $\Theta_{2}(\Lambda)$

For Bounded potentials $\psi_{2}$, the analysis is more involved. For any
$\mu>0$ and $\beta>0$ we write $\alpha:=\lambda/(\lambda+4\mu\beta)$. We also
define a function $f(\Lambda,\lambda)$ of the real variables $\Lambda,\lambda$
by the formula

$f(\Lambda,\lambda):=\frac{\Lambda}{(\Lambda^{2}+\lambda)^{1/2}}-\frac{\Lambda}{(\Lambda^{2}+\lambda+4\mu\beta)^{1/2}}\,.$
(99)

With these notations, we have $\Theta_{2}(\Lambda)=\pi f(\Lambda,\lambda)$
(cf. Eq. (87)). We want to study the three cases $\lambda>0$, $\lambda=0$ and
$\lambda<0$, used to classify the orbits in Sec. IV.3.

• Case $\lambda=0$. In this case, we simply plug $\lambda=0$ in Eq. (99) and
we see that $f(\Lambda,0)\in[0,1]$. Moreover, we have easily
$\partial_{\Lambda}f<0$. Therefore, $\Theta(\Lambda)$ is strictly decreasing
and varies $[0,\pi]$.

• Case $\lambda>0$. In this case, $0<\alpha<1$ and for a fixed $\lambda$, we
have
$\partial_{\Lambda}f(\Lambda,\lambda)=\lambda(\Lambda^{2}+\lambda)^{-3/2}-(\lambda+4\mu\beta)(\Lambda^{2}+\lambda+4\mu\beta)^{-3/2}$.
Then, a few algebraic manipulation show that
$\partial_{\Lambda}f(\Lambda,\lambda)$ vanishes for a value $\Lambda_{o}$
given by

$\Lambda_{o}^{2}=\lambda\frac{\alpha^{1/3}-1}{\alpha-\alpha^{1/3}}\quad\Rightarrow\quad
f(\Lambda_{o},\lambda)=\frac{\Lambda_{o}(1-\alpha^{1/3})}{(\Lambda_{o}^{2}+\lambda)^{1/2}}\,.$
(100)

Since $0<\alpha<1$ and $0<\Lambda_{o}<(\Lambda_{o}+4\mu\beta)^{1/2}$, we have
readily $0<f(\Lambda_{o},\lambda)<1$. Now, for any fixed $\lambda>0$,
$\Lambda\mapsto f(\Lambda,\lambda)$ is continuous, $\partial_{\Lambda}f$
vanishes only once at $\Lambda_{o}$ and furthermore
$0<f(\Lambda_{o},\lambda)<1$. Furthermore, it is clear that
$f(\Lambda,\lambda)$ goes to $0$ as $\Lambda\rightarrow 0$ and
$\Lambda\rightarrow+\infty$. With all these results, the general shape of the
curve $\Lambda\mapsto f(\Lambda,\lambda)$ can be easily inferred.

• Case $\lambda<0$. In this case, $\Lambda\mapsto f(\Lambda;\lambda)$ is
defined only when $\Lambda^{2}>-\lambda$. First subcase: $\lambda<0$ and
$\lambda+4\mu\beta<0$. Then, this is the same as in the $\lambda>0$ case,
where we saw that $\partial_{\Lambda}f(\Lambda,\lambda)>0$. Second subcase:
$\lambda<0$ but $\lambda+4\mu\beta\geq 0$, then setting
$g(\Lambda,\lambda):=\lambda(\Lambda^{2}+\lambda)^{-3/2}$, we have for any
$\Lambda^{2}>-\lambda$

$\frac{\partial
g}{\partial\lambda}(\Lambda,\lambda)=\frac{2\Lambda^{2}-\lambda}{2(\Lambda^{2}+\lambda)^{5/2}}\,.$
(101)

Now, since $\Lambda^{2}>-\lambda$, the right-hand side of Eq. (101) is
strictly positive, and therefore $g$ is an increasing function of $\lambda$.
In particular, we have $\lambda+4\mu\beta>\lambda\Rightarrow
g(\Lambda,\lambda+4\mu\beta)>g(\Lambda,\lambda)$ and by definition of $g$, the
latter is exactly $\partial_{\Lambda}f(\Lambda,\lambda)>0$. To conclude, in
the $\lambda<0$ case, $\Lambda\mapsto f(\Lambda,\lambda)$ is strictly
decreasing. Furthermore, it is clear that $f(\Lambda,\lambda)$ goes to
$+\infty$ as $\Lambda\rightarrow(-\lambda)^{1/2}$, and to $0$ as
$\Lambda\rightarrow+\infty$. With all these results, the general shape of the
curve $\Lambda\mapsto f(\Lambda,\lambda)$ can be easily inferred.

####  D.0.3 Hénon family, $\Theta_{3}(\Lambda)$

For Hénon potentials $\psi_{3}$, the analysis is similarly more involved. As
for the Bounded potentials we fix $\mu>0$ and $\beta>0$ and write
$\alpha:=\lambda/(\lambda+4\mu\beta)$. This time we define a function
$h(\Lambda,\lambda)$ of the real variables $\Lambda,\lambda$ by the formula

$h(\Lambda,\lambda):=\frac{\Lambda}{(\Lambda^{2}+\lambda)^{1/2}}+\frac{\Lambda}{(\Lambda^{2}+\lambda+4\mu\beta)^{1/2}}\,.$
(102)

With these notations, we have $\Theta_{3}(\Lambda)=\pi
h(\Lambda,\lambda-2\mu\beta)$ (cf. Eq. (89)). The analysis follows the same
lines as what was done for $\Theta_{2}(\Lambda)$. We want to study the three
cases $\lambda>0$, $\lambda=0$ and $\lambda<0$, used to classify the orbits in
Sec. IV.3.

• Case $\lambda>0$. Then we have $0<\alpha<1$ and there is no problem in
showing that $\Lambda\mapsto h(\Lambda,\lambda)$ is strictly increasing and
that $0<h(\Lambda,\lambda)<1$.

• Case $\lambda=0$. Once again, there is no problem in showing that
$\Lambda\mapsto h(\Lambda,\lambda)$ is strictly increasing and that
$0<h(\Lambda,\lambda)<2$.

• Case $\lambda<0$. In this case, $h$ is only defined when
$\Lambda^{2}>-\lambda$. For any such $(\Lambda,\lambda)$, we have

$\frac{\partial
h}{\partial\Lambda}(\Lambda,\lambda)=\frac{\lambda}{(\Lambda^{2}+\lambda)^{3/2}}+\frac{\lambda+4\mu\beta}{(\Lambda^{2}+\lambda+4\mu\beta)^{3/2}}\,.$
(103)

There are two subcases. First subcase: $\lambda<0$ and $\lambda+4\mu\beta<0$.
Then from Eq. (103), $\partial_{\Lambda}h(\Lambda,\lambda)<0$. Furthermore,
$h(\Lambda,\lambda)$ goes to $+\infty$ as
$\Lambda\rightarrow(-\lambda)^{1/2}$, and to $2$ as
$\Lambda\rightarrow+\infty$. Second subcase: $\lambda<0$ and
$\lambda+4\mu\beta<0$. If $|\alpha|<1$, then there is a value $\Lambda_{o}$
that makes $\partial_{\Lambda}h(\Lambda,\lambda)$ vanish. It is given by

$\Lambda_{o}^{2}=\lambda\frac{|\alpha|^{1/3}+1}{|\alpha|-|\alpha|^{1/3}}\quad\Rightarrow\quad
f(\Lambda_{o},\lambda)=\frac{\Lambda_{o}(1+|\alpha|^{1/3})}{(\Lambda_{o}^{2}+\lambda)^{1/2}}\,.$
(104)

In this case, the function $\Lambda\mapsto h(\Lambda,\lambda)$ decreases on
$[(-\lambda)^{1/2},\Lambda_{o}]$ and increases on $[\Lambda_{o},+\infty[$. The
value $h(\Lambda_{o},\lambda)$ is always strictly between $1$ and $2$. If
$|\alpha|\geq 1$, then the function $f$ is strictly decreasing. (It can be
seen as the limit $\Lambda_{o}\rightarrow+\infty$.) The value
$h(\Lambda_{o},\lambda)$ is in this case always above $2$.

##  Appendix E Proof that $c+d\xi<0$ for isochrone orbits around finite
central mass

In Sec. IV.2 we used the fact that $a+b\xi<0$ and $c+d\xi<0$ for isochrone
orbits in order to prove that our formula (82) covers all isochrone orbits.
The former identity follows from the generalized Kepler’s third law, and here,
we prove the latter identity. By assumption, we have a particle
$(\xi,\Lambda)$ on an isochrone orbit in a potential with finite central mass
whose parabola $\mathscr{P}:(ax+by)^{2}+cx+dy=0$ verifies all hypotheses
$(H_{i})$ of Sec. III.2. First we can check easily that $cx+dy=0$ is an
equation for the tangent to $\mathscr{P}$ at the origin. geometrically, since
two intersections exist between $\mathscr{L}$ and $\mathscr{P}$, the slope of
$\mathscr{L}$ must be bigger than that of this tangent, i.e., we must have
$\xi>-c/d$. We just have to show that $d\leq 0$ and the result will follow.
First, if $b=0$ (harmonic case), then we necessarily have $d<0$ (top-oriented
parabola). Second, if $b\neq 0$, then since $\lambda=0$ ($\mathscr{P}$ crosses
the origin) we have by Eq. (36) the equality $-d=\sqrt{d^{2}}$, which implies
$d\leq 0$. Therefore, we always have $d\leq 0$ and thus $c+d\xi<0$.

##  Appendix F Peaks of orbits in Bounded potentials

Let an arbitrary orbit be given by a polar equation $r(\theta)$, and compute
the value of $|\mathrm{d}r/\mathrm{d}\theta|$. The latter is a measure of the
change of $\mathrm{d}r$ when moving from $\theta$ to
$\theta+\mathrm{d}\theta$. It vanishes for circles $r=\text{cst}$ and is
infinite for straight lines $\theta=\text{cst}$. With the help of Eq. (2) and
$\Lambda=r^{2}\dot{\theta}$, we obtain easily
$|\mathrm{d}r/\mathrm{d}\theta|^{2}=2r^{4}(\xi-\psi_{e}(r))/\Lambda^{2}$.
Using a Taylor expansion of $\psi(r)$ and Eq. (3), we can linearize this
equation around the apoapsis $r_{A}$. We then obtain

$\biggl{|}\frac{\mathrm{d}r}{\mathrm{d}\theta}\biggr{|}^{2}=\frac{2r_{A}^{4}}{\Lambda^{2}}\biggl{(}\psi^{\prime}(r_{A})-\frac{\Lambda^{2}}{r_{A}^{3}}\biggr{)}(r_{A}-r)+o(r_{A}-r)\,.$
(105)

Examining Eq. (105), we see that as $r\rightarrow r_{A}$ the right-hand side
goes to zero as every term is finite in front of $(r_{A}-r)$. The orbit is
therefore smooth and differentiable around the apoapsis. However, the quantity
$\psi^{\prime}(r_{A})$ turns out to be very large for the Bounded family, in
general. This is because the slope of a Bounded potential increases to
infinity as $r$ grows toward $\beta$ from below, as can be seen readily on Eq.
(86). Therefore, a line $\mathscr{L}$ can intersect $\mathscr{C}$ such that
$r_{A}$ is very close to $\beta$, and it is clear from Eq. (86) that
$\psi_{2}^{\prime}(r)\rightarrow\infty$ as $r\rightarrow\beta$. As a
conclusion, before the apoapsis, the term $(r_{A}-r)$ does not yet compensate
the $\psi^{\prime}(r_{A})$ which is large for the Bounded potential, making
$\frac{\mathrm{d}r}{\mathrm{d}\theta}$ large and the curve resembles a
$\theta=\text{cst}$ line. This is why we see such abrupt and pointy turns in
Fig. 16.

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