# Effects of anisotropy in an anisotropic extension of $w$CDM model

Vikrant Yadav<EMAIL_ADDRESS>School of Basic and Applied Sciences, Raffles
University, Neemrana - 301705, Rajasthan, India  Santosh Kumar Yadav
<EMAIL_ADDRESS>School of CS & AI, SR University, Warangal-506371,
Telangana, India  Rajpal<EMAIL_ADDRESS>School of Basic and Applied
Sciences, Raffles University, Neemrana - 301705, Rajasthan, India

###### Abstract

In this paper, we derive observational constraints on an anisotropic $w$CDM
model from observational data including Baryonic Acoustic Oscillations (BAOs),
Cosmic Chronometer (CC), Big Bang Nucleosynthesis (BBN), Pantheon Plus (PP)
compilation of Type Ia supernovae, and SH0ES Cepheid host distance anchors. We
find that anisotropy is of the order $10^{-13}$, and its presence in the
$w$CDM model reduces $H_{0}$ tension by $\sim 2\sigma$ and $\sim 1\sigma$ in
the analyses with BAO+CC+BBN+PP and BAO+CC+BBN+PPSH0ES data combinations,
respectively. In both analyses, the quintessence form of dark energy is
favored at 95% CL.

##  I Introduction

Our Universe is expanding with accelerated rate of expansion as observed from
type Ia supernovae (SNe Ia) observations [1, 2]. Later on, various other
observations such as the large scale structure (LSS), the Baryonic Acoustic
Oscillations (BAOs) and the cosmic microwave background (CMB) supported this
observation. A number of cosmological models have been proposed/investigated
in the literature to help us comprehend the dynamics of the universe. But,
among all cosmological models, $\Lambda$CDM has proven to be the simplest
mathematical model which has widely been accepted by the cosmological
community and referred to as the standard cosmological model of cosmology.
Basically, this model is composed of two main parts: cold (non-relativistic)
dark matter (CDM) which is the reason behind the structure formation, and dark
energy (DE) in the form of cosmological constant ($\Lambda$) which causes the
late time accelerated expansion of Universe. The $\Lambda$CDM model provides
an excellent fit across a broad range of scales and epochs [3, 4, 5, 6, 7],
and successfully describes late-time accelerated expansion of the Universe [8,
9]. Despite the excellent fit to the current available cosmological
observations, the model faces several theoretical and observational
challenges. For instance, the nature of DE is not known accurately so far, and
within the $\Lambda$CDM paradigm, DE is regarded as the cosmological constant
in its most basic form, lacking any solid physical foundation. Except for its
typical gravitational interactions with other components, the exact nature of
dark matter remains unknown. Also, there is no concrete explanation of
‘coincidence problem’ stating why, despite having very distinct cosmic
evolutions, do the dark matter densities and the DE have the same order? Is
this coincidence indicating the possibility for interaction between the dark
sector components? Up to what extent the cosmological principle has been
tested? Is the Universe homogeneous and isotropic at cosmic scales?

Further, a few indications in the observations point towards the need to
expand the $\Lambda$CDM model in order to account for the growing conflicts
between measurements made at early (high redshifts) and late (low redshifts)
Universe [10, 11]. The Hubble constant, $H_{0}$, which represents the
Universe’s current rate of expansion, has the greatest statistically
significant tension. The Hubble tension appears when we compare the value of
$H_{0}$ predicted by CMB measurements within the $\Lambda$CDM framework and
the direct local distance ladder measurements, that is, the one estimated by
the Cepheid calibrated supernovae SNe Ia. In particular, the Hubble tension is
referred to as the disagreement at $5\sigma$ between the latest SH0ES
(Supernovae and $H_{0}$ for the equation of state (EoS) of DE) collaboration
[12] constraint, $H_{0}^{\rm R22}=(73.04\pm 1.04){\rm\,km\,s^{-1}\,Mpc^{-1}}$
at 68% confidence level (CL), based on the supernovae calibrated by Cepheids,
and the Planck collaboration [3] value, $H_{0}=\left(67.27\pm
0.60\right){\rm\,km\,s^{-1}\,Mpc^{-1}}$ at 68% CL. The Hubble tension may be
defined as a difference between the observed value of $H_{0}$ from two sets of
observations: (i) all of the direct late time $\Lambda$CDM independent
measurements, and (ii) all of the indirect model dependent estimations at
early times. In general, the value of $H_{0}$ obtained at late times is
greater than the one obtained at early times.

This disparity could indicate the existence of novel physics outside of the
$\Lambda$CDM cosmology [13, 14, 15, 16, 17, 18]. In the literature, a number
of extensions of $\Lambda$CDM have been suggested to resolve the $H_{0}$
tension. These extensions include but not limited to: DM-DE interactions [19,
20, 21, 22, 23, 24, 25]; decaying DM [26, 27, 28, 29]; introducing Early DE
[30, 31, 32, 33, 34]; and introducing a sign-switching DE at intermediate
redshifts ($z\sim 2$) [35, 36, 37, 38, 39]. The current status on $H_{0}$
tension and possible solutions can be found in recent review articles [40, 41,
42, 43]

The Cosmological Principle (CP), which asserts that the Universe is
statistically homogenous and isotropic in space and matter on vast scales
$(100\rm Mpc)$, is one of the fundamental tenets of $\Lambda$CDM cosmology.
Mathematically, such a universe is described by the Friedmann-Lemaître-
Robertson-Walker (FLRW) space-time metric, in which all three of the metric’s
spatial orthogonal components are functions of cosmic time ($t$) exclusively.
This is the primary space-time metric that facilitates the creation of the
effective and remarkably predictive standard model of cosmology. However,
observational data suggest that there are slight fluctuations in CMB
intensities originating from various directions of sky [44]. Gamma Ray Bursts
(GRBs), Quasars, Galaxies, and SNe Ia are among the recent findings that
suggest the Universe may be anisotropic and provide substantial observational
evidence (see [45] and references therein). There are other interesting
observational evidences, which have questioned the validity of CP, for
instance, a strong evidence for a violation of the CP of isotropy is found by
the authors in [46] after analysing the Planck Legacy temperature anisotropy
data. This violation is likely to represent a statistical fluctuation of order
$\sim 10^{-9}$. Also, see the recent and interesting review [47] where the
authors emphasise differences and synergies that truly stimulate additional
research in this field as they detail existing observational clues for
departures from the predictions of CP. The spatially homogeneous and
anisotropic Universe is expressed mathematically by homogeneous and
anisotropic metrics and corresponding models are commonly known as Bianchi
type models [48, 49]. The simplest of the various Bianchi type models is the
Bianchi type I model. The empirical limitations on the Bianchi type I model
might offer a platform for evaluating the precision of FLRW models
characterising the current era of the universe. Various anisotropic
generalizations of the standard $\Lambda$CDM model have been investigated in
recent years. In [50], the authors analyze an anisotropic model with the CMB
and BAO data by fixing the drag redshift as $z_{\rm d}=1059.6$ and the last
scattering redshift as $z_{*}=1089.9$ and constrain the anisotropy parameter,
$\Omega_{\rm\sigma 0}\lesssim 10^{-15}$. A similar investigation is done in
[51] by considering anisotropic expansion and curvature together from
different data sets and discovered very little present-day expansion
anisotropy. Other anisotropic models are also investigated by using different
observational data in different contexts, see [52, 53, 54, 55, 56, 57, 58]. In
[59], a theoretical approach of spatially homogeneous universes with late-time
anisotropy is discussed. Most recently, in [60] an anisotroic model,
$\Lambda$CDM + $\Omega_{\rm\sigma 0}$ is investigated with the aim to derive
CMB independent constraints on $H_{0}$ with BAO, BBN, CC, Pantheon Plus and
SH0ES data. Following [60], we are motivated here to derive observational
constraints on two simplest extensions of vanilla $\Lambda$CDM model: (i) an
isotropic extension by taking a constant EoS of DE instead of cosmological
constant, namely $w$CDM model; (ii) an anisotropic extension, namely $w$CDM +
$\Omega_{\rm\sigma 0}$ by adding expansion anisotropy to $w$CDM model. It is
well known that the nature of DE could be investigated through its EoS
parameter. As stated above, the fundamental nature of DE is unknown and there
is no strong physical basis for assuming DE in the form of cosmological
constant whose density remains constant even in an expanding background. In
this paper, we have assumed a constant EoS parameter of DE. Our primary goal
in this work is to analyze the possible effects of anisotropy on the constant
EoS parameter of DE and the Hubble constant $H_{0}$. For this, we derive
observational constraints using the recent observational data and compare the
results of $w$CDM and $w$CDM + $\Omega_{\rm\sigma 0}$ models with the results
obtained in [60] for $\Lambda$CDM and $\Lambda$CDM + $\Omega_{\rm\sigma 0}$,
respectively.

The structure of the paper is as follows: In Section II, we describe the
details of the considered models. In Section III, we provide a brief overview
of the data sets and methodology used for the observational analysis of the
models. In Section IV, we present the observational constraints on the models,
and discuss the results. In Section V, we conclude with the main findings of
our study.

##  II GOVERNING EQUATIONS AND MODELS

We investigate an extension of FLRW spacetime, the Bianchi type I metric, with
three orthogonal directions of different scale factors, given by

$\text{d}s^{2}=-\text{d}t^{2}+A^{2}\text{d}x^{2}+B^{2}\text{d}y^{2}+C^{2}\text{d}z^{2},$
(1)

where $A,B,$ and $C$ are functions of cosmic time $t$ only and represent the
scale factors along the principal axes $x,y$ and $z$, respectively. Further,
we define the average expansion scale factor: $a=(ABC)^{\frac{1}{3}}$, and the
average Hubble parameter:
$H=\frac{\dot{a}}{a}=\frac{\left(H_{x}+H_{y}+H_{z}\right)}{3}$, where the
corresponding directional Hubble parameters along the principal axes $x,y$,
and $z$ are being defined as $H_{x}=\frac{\dot{A}}{A}$,
$H_{y}=\frac{\dot{B}}{B}$, and $H_{z}=\frac{\dot{C}}{C}$ respectively.

The Einstein field equations in GR read as

$G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi GT_{\mu\nu},$ (2)

where the left side of the above expression shows the Einstein tensor
$G_{\mu\nu}$, the Ricci tensor $R_{\mu\nu}$, the Ricci scalar $R$, and the
metric tensor $g_{\mu\nu}$. Additionally, the right side shows the Newton’s
gravitational constant $G$ and the energy momentum tensor $T_{\mu\nu}$.
Further, for a perfect fluid with energy density $\rho$ and pressure $p$,
$T_{\mu\nu}$ takes the form

$T^{\nu}_{\,\,\mu}=\text{diag}[-\rho,p,p,p].$ (3)

As a result of the twice-contracted Bianchi identity
($G^{\mu\nu}_{\;\;\;;\nu}=0$), the Einstein field equations (2) satisfy the
conservation equation

$T^{\mu\nu}_{\;\;\;;\nu}=0.$ (4)

In case of the perfect fluid matter distribution, it reduces to

$\dot{\rho}+3H(\rho+p)=0,$ (5)

where the dot denotes its cosmic time($t$) derivative. The evolution of the
energy density $\rho_{i}$ of a perfect fluid $i$ with pressure $p_{i}$, energy
density $\rho_{i}$, and constant EoS $w_{i}=p_{i}/\rho_{i}$ is provided by the
continuity equation (5), which is as follows:

$\rho_{i}=\rho_{i0}a^{-3(1+w_{i})},$ (6)

where $\rho_{i0}$ stands for the current value of $\rho_{i}$, that is, at
$a=a_{0}=1$. Here, $a=a_{0}=1$ signifies the present day value of the cosmic
scale factor $a$. From this point on, every quantity with the subscript 0
indicates its value in the present day universe. We consider that the universe
is made up of the standard cosmic fluids, namely the radiation (photons and
neutrinos) whose EoS is $w_{\rm r}=p_{r}/\rho_{r}=\frac{1}{3}$, pressureless
fluid (baryonic and cold dark matter) whose EoS is $w_{\rm
m}=p_{m}/\rho_{m}=0$, and DE fluid with a constant EoS $w_{\rm de}$, to be
fixed by observations in the analysis. Further, assuming only gravitational
interaction between these energy components, the continuity equation (5) being
satisfied by each component separately, and in view of (6), this gives

$\rho\equiv\rho_{\text{r}}+\rho_{\text{m}}+\rho_{\rm
de}=\rho_{\text{r}0}a^{-4}+\rho_{\text{m}0}a^{-3}+\rho_{de0}a^{-3(1+w_{\rm
de0})}$ (7)

Further, for the Bianchi type I metric (1), the Einstein field equations (2)
result into to the following set of differential equations:

$\displaystyle\frac{\dot{A}}{A}\frac{\dot{B}}{B}+\frac{\dot{B}}{B}\frac{\dot{C}}{C}+\frac{\dot{A}}{A}\frac{\dot{C}}{C}$
$\displaystyle=8\pi G\rho,$ (8)
$\displaystyle-\frac{\ddot{B}}{B}-\frac{\ddot{C}}{C}-\frac{\dot{B}}{B}\frac{\dot{C}}{C}$
$\displaystyle=8\pi Gp,$ (9)
$\displaystyle-\frac{\ddot{A}}{A}-\frac{\ddot{C}}{C}-\frac{\dot{A}}{A}\frac{\dot{C}}{C}$
$\displaystyle=8\pi Gp,$ (10)
$\displaystyle-\frac{\ddot{A}}{A}-\frac{\ddot{B}}{B}-\frac{\dot{A}}{A}\frac{\dot{B}}{B}$
$\displaystyle=8\pi Gp.$ (11)

The shear scalar may be expressed as follows in terms of the directional
Hubble parameters:

$\sigma^{2}=\frac{(H_{x}-H_{y})^{2}+(H_{y}-H_{z})^{2}+(H_{z}-H_{x})^{2}}{6}.$
(12)

The equations (8)-(11) can be recast as follows:

$\displaystyle 3H^{2}-\sigma^{2}=8\pi G\rho,$ (13)
$\displaystyle\dot{H}_{x}-\dot{H}_{y}+3H(H_{x}-H_{y})=0,$ (14)
$\displaystyle\dot{H}_{y}-\dot{H}_{z}+3H(H_{y}-H_{z})=0,$ (15)
$\displaystyle\dot{H}_{z}-\dot{H}_{x}+3H(H_{z}-H_{x})=0.$ (16)

We derive the following shear propagation equation using these equations
(14)-(16) and the time derivative of $\sigma^{2}$ provided in Eq. (12):

$\dot{\sigma}+3H\sigma=0.$ (17)

Its integration further leads to

$\sigma^{2}=\sigma_{0}^{2}a^{-6}.$ (18)

Using equations (7) and (18) into (13), we obtain

$\frac{H^{2}}{H_{0}^{2}}=\Omega_{\rm\sigma
0}a^{-6}+\Omega_{\text{r}0}a^{-4}+\Omega_{\text{m}0}a^{-3}+\Omega_{\rm
de0}a^{-3(1+w_{\rm de0})},$ (19)

where $\Omega_{\sigma 0}=\frac{\sigma_{0}^{2}}{3H_{0}^{2}}$, $\Omega_{\rm
r0}=\frac{8\pi G}{3H_{0}^{2}}\rho_{\rm r0}$, $\Omega_{\rm m0}=\frac{8\pi
G}{3H_{0}^{2}}\rho_{\rm m0}$ and $\Omega_{\rm de0}=\frac{8\pi
G}{3H_{0}^{2}}\rho_{\rm de0}$, respectively denote the expansion anisotropy,
radiation, matter, and DE density parameters, satisfying $\Omega_{\rm\sigma
0}+\Omega_{\text{r}0}+\Omega_{\text{m}0}+\Omega_{\rm de0}=1$. Here, the
expansion anisotropy parameter $\Omega_{\rm\sigma 0}$ is purely a geometric
term which is non-negative.

Furthermore, the generalized Friedmann equation (19) depicts the spatially
flat and homogeneous but probably non-isotropic Universe (because of the
presence of the expansion anisotropy). We denote this model by
$w$CDM+$\Omega_{\rm\sigma 0}$ and the set of free baseline parameters for this
model is given by

$\mathcal{P}_{w\rm CDM+\Omega_{\rm\sigma 0}}=\left\\{\omega_{\rm
b},\,\omega_{\rm c},\,H_{0},\,\Omega_{\sigma 0},\,w_{\rm de0}\right\\}.$

Here $\omega_{\rm b}=\Omega_{\rm b}h^{2}$ and $\omega_{\rm c}=\Omega_{\rm
c}h^{2}$ are physical density parameters of baryons and CDM, respectively, in
the present Universe, $H_{0}$ is the Hubble constant and $\Omega_{\sigma 0}$
is the expansion anisotropy parameter.

In the absence of expansion anisotropy, equation (19) reduces to

$\frac{H^{2}}{H_{0}^{2}}=\Omega_{\text{r}0}a^{-4}+\Omega_{\text{m}0}a^{-3}+\Omega_{\rm
de0}a^{-3(1+w_{\rm de0})}.$ (20)

It is an isotropic extension of $\Lambda$CDM cosmology. We denote this model
by $w$CDM and the set of free baseline parameters for this model is given by

$\mathcal{P}_{w\rm CDM}=\left\\{\omega_{\rm b},\,\omega_{\rm
c},\,H_{0},\,w_{\rm de0}\right\\}.$

We follow standard neutrino scheme with normal hierarchy which has three
species of neutrino, out of which two are massless and one is massive neutrino
with the standard number of effective neutrino species, $N_{\rm eff}=3.046$
and minimum allowed mass, $m_{\nu}=0.06\,\rm eV$.

##  III Data and methodology

The following data sets have been used in this work:

Baryon Acoustic Oscillation (BAO): Updates on BAO measurements utilising
galaxies, quasars, and Lyman-$\alpha$ (Ly$\alpha$) for completed experiments
are provided by the Sloan Digital Sky Survey (SDSS) [61]. These experiments
involve the compilation of data from BOSS, eBOSS, SDSS, and SDSS-II, making
accessible, as indicated in Table 1, independent BAO measurements of angular-
diameter distances and Hubble distances relative to the sound horizon from
eight separate samples.

Table 1: Clustering measurements for the BAOs from Ref. [61].

Parameter | $\bm{\,\,z_{\rm eff}\,\,}$ | $\bm{\,\,D_{V}(z)/r_{\rm d}\,\,}$ | $\bm{\,\,D_{M}(z)/r_{\rm d}\,\,}$ | $\bm{\,\,D_{H}(z)/r_{\rm d}\,\,}$
---|---|---|---|---
MGS | 0.15 | $4.47\pm 0.17$ | — | —
BOSS Galaxy | $0.38$ | — | $10.23\pm 0.17$ | $25.00\pm 0.76$
BOSS Galaxy | $0.51$ | — | $13.36\pm 0.21$ | $22.33\pm 0.58$
eBOSS LRG | $0.70$ | — | $17.86\pm 0.33$ | $19.33\pm 0.53$
eBOSS ELG | $0.85$ | $18.33_{-0.62}^{+0.57}$ | — | —
eBOSS Quasar | $1.48$ | — | $30.69\pm 0.80$ | $13.26\pm 0.55$
Ly$\alpha$-Ly$\alpha$ | $2.33$ | — | $37.6\pm 1.9$ | $8.93\pm 0.28$
Ly$\alpha$-Quasar | $2.33$ | — | $37.3\pm 1.7$ | $9.08\pm 0.34$

At the drag redshift ($z_{\rm d}$), the comoving size of the sound horizon
($r_{\rm s}$) is $r_{\rm d}$, and it is given by

${r_{\rm d}=r_{\rm s}(z_{\rm d})=\int_{z_{\rm d}}^{\infty}\frac{c_{\rm
s}\text{d}z}{H(z)},}$ (21)

where $c_{\rm s}=\frac{c}{\sqrt{3(1+\mathcal{R})}}$ is the sound speed of the
baryon-photon fluid; $\mathcal{R}=\frac{3\Omega_{\rm b0}}{4\Omega_{\rm\gamma
0}(1+z)}$ with $\Omega_{\rm b0}=0.022h^{-2}$ being the present-day physical
density of baryons and $\Omega_{\gamma 0}=2.469\times 10^{-5}h^{-2}$ being the
present-day physical density of photons [62, 63].

Direct constraints on the values $D_{H}(z)/r_{\rm d}$ and $D_{M}(z)/r_{\rm d}$
are provided by the BAO measurements. We calculate the Hubble distance as
follows:

$D_{H}(z)=\frac{c}{H(z)}.$ (22)

For flat cosmology, we calculate the comoving angular diameter distance
($D_{M}(z)$) as follows:

$D_{M}(z)={c\over H_{0}}\int_{0}^{z}\text{d}z^{\prime}{H_{0}\over
H(z^{\prime})}.$ (23)

The spherically averaged distance ($D_{V}(z)$) is defined as follows:

$D_{V}(z)\equiv\left[zD^{2}_{M}(z)D_{H}(z)\right]^{1/3}.$ (24)

We have opted $d_{1}$ for $D_{V}(z)/r_{\rm d}$, $d_{2}$ for $D_{M}(z)/r_{\rm
d}$, and $d_{3}$ for $D_{H}(z)/r_{\rm d}$. Then, the chi-squared function for
each measurement in Table 1 is written as follows:

$\displaystyle\chi^{2}_{\rm B_{1}}=$
$\displaystyle\displaystyle\sum_{i=1}^{2}\left(\frac{d_{1}^{\text{obs}}(z_{i})-d_{1}^{\text{th}}(z_{i})}{\sigma_{d_{1}^{\text{obs}}(z_{i})}}\right)^{2},$
$\displaystyle\chi^{2}_{\rm B_{2}}=$
$\displaystyle\displaystyle\sum_{j=1}^{6}\left(\frac{d_{2}^{\text{obs}}(z_{j})-d_{2}^{\text{th}}(z_{j})}{\sigma_{d_{2}^{\text{obs}}(z_{j})}}\right)^{2},$
$\displaystyle\chi^{2}_{\rm B_{3}}=$
$\displaystyle\displaystyle\sum_{j=1}^{6}\left(\frac{d_{3}^{\text{obs}}(z_{j})-d_{3}^{\text{th}}(z_{j})}{\sigma_{d_{3}^{\text{obs}}(z_{j})}}\right)^{2}.$
(25)

Here, $d^{\rm obs}$ denotes the observed distance value as shown in Table 1,
whereas $d^{\rm th}$ denotes the theoretical value computed for the models
under consideration. The effective redshifts for MGS and eBOSS ELG samples in
the $D_{V}(z)/r_{\rm d}$ scenario are denoted as $z_{i}$ ($i=1,2$). For
$D_{M}(z)/r_{\rm d}$ and $D_{H}(z)/r_{\rm d}$, the effective redshifts for the
six measurements are labelled as $z_{j}$ ($j=1,2,3,4,5,6$) which correspond to
BOSS Galaxy, eBOSS Galaxy, eBOSS LRG, eBOSS Quasar, Ly$\alpha$-Ly$\alpha$, and
Ly$\alpha$-Quasar, respectively.

For the BAO measurements, the cumulative chi-squared expression, denoted as
$\chi^{2}_{\rm BAO}$, is therefore formulated as follows:

$\chi^{2}_{\rm BAO}=\chi^{2}_{\rm B_{1}}+\chi^{2}_{\rm B_{2}}+\chi^{2}_{\rm
B_{3}}.$ (26)

Cosmic Chronometer (CC): The Cosmic Chronometer (CC) uses a reliable technique
to monitor the universe’s historical expansion through the analysis of $H(z)$
measurements. 33 $H(z)$ measurements from CC have been compiled, covering
redshift values from 0.07 to 1.965. Table 2 provides a thorough description of
these measurements and the corresponding references. The idea behind these
measurements was first presented in [64], whereby a fundamental connection was
established between the Hubble parameter $H(z)$, redshift $z$, and cosmic time
$t$:

$H(z)=-\frac{1}{1+z}\frac{\text{d}z}{\text{d}t}.$ (27)

Table 2: Compilation of CC measurements of $H(z)$. $z$ | $H(z)$ | $\sigma_{H(z)}$ | Ref.
---|---|---|---
0.07 | 69.0 | 19.6 | [65]
0.09 | 69 | 12 | [66]
0.12 | 68.6 | 26.2 | [65]
0.17 | 83 | 8 | [66]
0.179 | 75 | 4 | [67]
0.199 | 75 | 5 | [67]
0.20 | 72.9 | 29.6 | [65]
0.27 | 77 | 14 | [66]
0.28 | 88.8 | 36.6 | [65]
0.352 | 83 | 14 | [67]
0.38 | 83 | 13.5 | [68]
0.4 | 95 | 17 | [66]
0.4004 | 77 | 10.2 | [68]
0.425 | 87.1 | 11.2 | [68]
0.445 | 92.8 | 12.9 | [68]
0.47 | 89.0 | 49.6 | [69]
0.4783 | 80.9 | 9 | [68]
0.48 | 97 | 62 | [70]
0.593 | 104 | 13 | [67]
0.68 | 92 | 8 | [67]
0.75 | 98.8 | 33.6 | [71]
0.781 | 105 | 12 | [67]
0.8 | 113.1 | 15.1 | [72]
0.875 | 125 | 17 | [67]
0.88 | 90 | 40 | [70]
0.9 | 117 | 23 | [66]
1.037 | 154 | 20 | [67]
1.3 | 168 | 17 | [66]
1.363 | 160 | 33.6 | [73]
1.43 | 177 | 18 | [66]
1.53 | 140 | 14 | [66]
1.75 | 202 | 40 | [66]
1.965 | 186.5 | 50.4 | [73]

We formulate the chi-squared function $\chi^{2}_{\rm CC}$ associated with
these measurements as follows:

$\chi^{2}_{\rm
CC}=\sum_{i=1}^{33}\frac{[H^{\text{obs}}(z_{i})-H^{\text{th}}(z_{i})]^{2}}{\sigma^{2}_{H^{\text{obs}}(z_{i})}},$
(28)

where $H^{\text{obs}}(z_{i})$ signifies the value of the observed Hubble
parameter, along with its corresponding standard deviation
$\sigma^{2}_{H^{\text{obs}}(z_{i})}$ from the table provided. The equivalent
$H^{\text{th}}(z_{i})$ represents the Hubble parameter value in theory
obtained from the considered cosmological model.

Big Bang Nucleosynthesis (BBN): An independent method of inferring the density
of baryons is to examine BBN, which serves as a probe of the early Universe’s
dynamics. With the help of BBN, we can look at the limitations regarding the
anisotropic extensions of the conventional cosmological model [74, 75, 76].
For all of our calculations, we use an updated estimate of the physical baryon
density, $\omega_{b}$ (where $\omega_{b}\equiv\Omega_{b}h^{2}$), which comes
from Big Bang Nucleosynthesis (BBN) and has a value of $0.02233\pm 0.00036$.
This computation makes use of revised data obtained from experimental nuclear
physics at the INFN Laboratori Nazionali del Gran Sasso in Italy, at the
Laboratory for Underground Nuclear Astrophysics (LUNA) [77].

Pantheon Plus and SH0ES: Historically, Type Ia supernovae (SNe Ia) have played
crucial role in developing the conventional model of the universe. These
supernovae yield useful distance moduli measurements, which constrain the
late-time expansion rate or the uncalibrated luminosity distance
$H_{0}d_{L}(z)$. For a supernova at redshift $z$, the theoretical apparent
magnitude $m_{B}$ is determined by the following equation:

$\displaystyle m_{B}=5\log_{10}\left[\frac{d_{L}(z)}{1\rm
Mpc}\right]+25+M_{B},$ (29)

where $M_{B}$ stands for the absolute magnitude. Consequently, the distance
modulus ($\mu(z)$) can be written as $\mu(z)=m_{B}-M_{B}$. In a flat
cosmology, the luminosity distance is determined as follows:

$d_{L}(z)=(1+z)\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}.$ (30)

In the present study, we make use of SNe Ia distance modulus measurements from
the Pantheon+ sample [78]. We name the collection as PP, which comprises 1701
light curves that correspond to 1550 unique supernovae Ia within the
$z\in[0.001,2.26]$ redshift range. We also include constraints on $H_{0}$ and
$M_{B}$ by including SH0ES Cepheid host distance anchors [78] into our
analysis; this dataset is referred to as PPSH0ES.

We utilise PP data to minimise the equation that follows to constrain the
cosmological parameters:

$\chi^{2}_{\rm PP}=\Delta D^{T}\leavevmode\nobreak\ C_{\rm
stat+syst}^{-1}\leavevmode\nobreak\ \Delta D,$ (31)

where $D$ represents the vector of 1701 SN distance modulus residuals
calculated as

$\Delta D_{i}=\mu_{i}-\mu_{{\rm model}}(z_{i}),$ (32)

while comparing the observed supernova distance ($\mu_{i}$) with the predicted
model distance ($\mu_{{\rm model}}(z_{i})$) by utilising the measured
supernova/host redshift. Here, $C_{\rm stat+syst}=C_{\rm stat}+C_{\rm syst}$,
with $C_{\rm stat}$ and $C_{\rm syst}$ denote the statistical and the
systematic covariance matrices respectively.

While SNe analysis alone suffers from degeneracy between the parameters
$M_{B}$ and $H_{0}$, we get over this restriction by adding the recently
reported SH0ES Cepheid host distance anchors (R22) into the likelihood. With
this changement, we are able to confine $M_{B}$ as well as $H_{0}$. The
following adjustments are made to the SN distance residuals after including
SH0ES Cepheid host distances:

$\Delta D^{\prime}_{i}=\begin{cases}\mu_{i}-\mu_{i}^{{\rm
Cepheid}}&i\in\text{Cepheid hosts}\\\ \mu_{i}-\mu_{{\rm
model}}(z_{i})&\text{otherwise},\end{cases}$ (33)

where $\mu_{i}^{{\rm Cepheid}}$ stands for the Cepheid-calibrated host-galaxy
distance from SH0ES. After incorporating this data with the SH0ES Cepheid
host-distance covariance matrix ($C^{\rm Cepheid}_{\rm stat+syst}$) from R22,
the likelihood function is modified as follows:

$\chi^{2}_{\rm PPSH0ES}=\Delta D{{}^{\prime}}^{T}\leavevmode\nobreak\ (C^{\rm
SN}_{\rm stat+syst}+C_{\rm stat+syst}^{\rm Cepheid})^{-1}\leavevmode\nobreak\
\Delta D^{\prime},$

where $C^{\rm SN}_{\rm stat+syst}$ represents the SN covariance. Further
information is available in the reference [78].

The baseline free parameters of the model $w$CDM+$\Omega_{\sigma 0}$ and
$w$CDM are as discussed in section II. We have assumed three neutrino species,
approximated as two massless states and a single massive neutrino of mass
$m_{\nu}=0.06\,\rm eV$. We have used uniform priors: $\omega_{\rm
b}\in[0.01,0.03]$, $\omega_{\rm c}\in[0.05,0.25]$, $\,H_{0}\in[60,80]$,
$\Omega_{\sigma 0}\in[0,0.001]$, $w_{\rm de0}\in[-2,0]$ and the publicly
available Boltzmann code CLASS [79] with the parameter inference code Monte
Python [80] to obtain correlated Monte Carlo Markov Chain (MCMC) samples. We
analyze the MCMC samples using the python package
GetDist111https://getdist.readthedocs.io/en/latest/intro.html.

##  IV Results and Discussion

In Table 3, we summarize the observational constraints on free parameters and
some derived parameters of the $w$CDM and $w$CDM+$\Omega_{\rm\sigma 0}$ models
at $68\%$CL from two data combinations: BAO+CC+BBN+PP and BAO+CC+BBN+PPSH0ES.
The second row over each parameter (mentioned in blue color) represents the
constraints at $68\%$CL on $\Lambda$CDM and $\Lambda$CDM+$\Omega_{\rm\sigma
0}$ models from same set of data combinations. The upper bounds on
anisoptropic expansion parameter, $\Omega_{\rm\sigma 0}$, at $95\%$ CL are of
order $10^{-13}$ for $w$CDM+$\Omega_{\rm\sigma 0}$ model with both the data
combinations. The upper bounds of $\Omega_{\rm\sigma 0}$ for this model read
as: $\Omega_{\rm\sigma 0}<2.3\times 10^{-13}$ and $\Omega_{\rm\sigma
0}<4\times 10^{-13}$ from BAO+CC+BBN+PP and BAO+CC+BBN+PPSH0ES, respectively.
On other hand, the upper bounds on anisoptropic expansion parameter
$\Omega_{\rm\sigma 0}$ for $\Lambda$CDM+$\Omega_{\rm\sigma 0}$ model are:
$\Omega_{\rm\sigma 0}<2.0\times 10^{-14}$ and $\Omega_{\rm\sigma 0}<6.5\times
10^{-14}$ from BAO+CC+BBN+PP and BAO+CC+BBN+PPSH0ES combinations,
respectively. Thus, we see that the $w$CDM+$\Omega_{\rm\sigma 0}$ model
provides weaker upper bounds on anisotropy parameter by one order of magnitude
as compared to $\Lambda$CDM+$\Omega_{\rm\sigma 0}$ model with both the data
combinations. We observe that anisotropic parameter $\Omega_{\rm\sigma 0}$ has
a strong positive correlation with DE EoS parameter from both the data
combinations. This means that larger amount of anisotropy in the Universe
results larger value of DE EoS, see Figure 1.

Table 3: Constraints at 68% CL on the free and some derived parameters of the
$w$CDM and $w$CDM+$\Omega_{\rm\sigma 0}$ models from BAO+CC+BBN+PP and
BAO+CC+BBN+PPSH0ES data. The upper bounds on $\Omega_{\rm\sigma 0}$ at 95% CL
are displayed. The Hubble constant $H_{0}$ is measured in the unit of $\rm
km\,s^{-1}\rm Mpc^{-1}$. The entries in second row over each parameter (in
blue colour) represent the constraints on $\Lambda$CDM and
$\Lambda$CDM+$\Omega_{\rm\sigma 0}$ model parameters.

Data set | BAO+CC+BBN+PP | BAO+CC+BBN+PP+SH0ES
---|---|---
Model |  $w$CDM |  $w$CDM+$\Omega_{\rm\sigma 0}$ |  $w$CDM |  $w$CDM+$\Omega_{\rm\sigma 0}$
|  $\Lambda$CDM | $\Lambda$CDM+$\Omega_{\rm\sigma 0}$ |  $\Lambda$CDM |  $\Lambda$CDM+$\Omega_{\rm\sigma 0}$
$10^{-2}\omega_{\rm b}$ | $2.237^{+0.035}_{-0.035}$ | $2.231^{+0.036}_{-0.036}$ | $2.248^{+0.035}_{-0.035}$ | $2.231^{+0.036}_{-0.036}$
| $2.233^{+0.035}_{-0.035}$ | $2.230^{+0.036}_{-0.036}$ | $2.258^{+0.035}_{-0.035}$ | $2.232^{+0.036}_{-0.036}$
$\Omega_{\rm\sigma 0}$ | 0 | $<2.3\times 10^{-13}\;(95\%\;\text{CL})$ | 0 | $<4\times 10^{-13}\;(95\%\;\text{CL})$
| 0 | $<2.0\times 10^{-14}\;(95\%\;\text{CL})$ | 0 | $<6.5\times 10^{-14}\;(95\%\;\text{CL})$
$w_{\rm de0}$ | $-0.930^{+0.045}_{-0.045}$ | $-0.853^{+0.054}_{-0.048}$ | $-1.062^{+0.041}_{-0.041}$ | $-0.881^{+0.053}_{-0.053}$
| $-1$ | $-1$ | $-1$ | $-1$
$H_{\rm 0}$ | $66.4^{+1.3}_{-1.3}$ | $68.4^{+1.6}_{-1.6}$ | $71.51^{+0.83}_{-0.83}$ | $72.24^{+0.82}_{-0.82}$
| $68.02^{+0.84}_{-0.84}$ | $70.1^{+1.2}_{-1.5}$ | $70.79^{+0.69}_{-0.69}$ | $72.67^{+0.85}_{-0.85}$
$M_{B}$ | $-19.461^{+0.040}_{-0.040}$ | $-19.395^{+0.047}_{-0.047}$ | $-19.317^{+0.023}_{-0.023}$ | $-19.282^{+0.023}_{-0.023}$
| $-19.418^{+0.028}_{-0.028}$ | $-19.356^{+0.039}_{-0.044}$ | $-19.330^{+0.020}_{-0.022}$ | $-19.283^{+0.024}_{-0.024}$
$\Omega_{\rm m0}$ | $0.258^{+0.015}_{-0.015}$ | $0.230^{+0.018}_{-0.018}$ | $0.287^{+0.012}_{-0.012}$ | $0.233^{+0.018}_{-0.018}$
| $0.318^{+0.011}_{-0.011}$ | $0.309^{+0.012}_{-0.012}$ | $0.324^{+0.011}_{-0.011}$ | $0.302^{+0.012}_{-0.012}$
$z_{\rm d}$ | $1059.3^{+1.1}_{-1.1}$ | $1058.7^{+1.1}_{-1.1}$ | $1061.99^{+0.88}_{-0.88}$ | $1059.8^{+1.1}_{-1.1}$
| $1060.09^{+0.95}_{-0.95}$ | $1060.37^{+0.98}_{-0.98}$ | $1061.75^{+0.85}_{-0.85}$ | $1060.99^{+0.90}_{-0.90}$
$r_{\rm d}$ | $149.0^{+2.7}_{-2.7}$ | $144.4^{+3.1}_{-3.1}$ | $140.7^{+1.8}_{-1.8}$ | $137.7^{+1.9}_{-1.9}$
| $146.1^{+1.8}_{-1.8}$ | $142.0^{+2.7}_{-2.7}$ | $142.1^{+1.5}_{-1.5}$ | $137.7^{+1.9}_{-1.9}$
$\chi^{2}_{\rm min}$ | $1436.2$ | $1434.08$ | $1332.1$ | $1319.9$
| $1438.62$ | $1438.58$ | $1334.4$ | $1322.94$

Figure 1: Two-dimensional marginalized confidence regions at 68% and 95% CL
for $\Omega_{\rm\sigma 0}$ and $w_{\rm de0}$ for $w$CDM+$\Omega_{\rm\sigma 0}$
model from BAO+CC+BBN+PP and BAO+CC+BBN+PPSH0ES data combinations.

For the $w$CDM+$\Omega_{\rm\sigma 0}$ model, it can be observed from Table 3
that the DE EoS parameter does not prefer the cosmological constant form of DE
(having $w_{\rm de0}=-1$) with both data combinations. Also, at $95\%$ CL, the
DE EoS parameter ($w_{\rm de0}=-0.853^{+0.096}_{-0.100}$ for BAO+CC+BBN+PP
data combination and $w_{\rm de0}=-0.88^{+0.10}_{-0.11}$ for
BAO+CC+BBN+PPSH0ES data combination) does not prefer the cosmological constant
form of DE with both data combinations. For this model, the mean values of DE
EoS parameter (at $68\%$ CL as well as $95\%$ CL range), $w_{\rm de0}$, lie in
quintessence region $(w_{\rm de0}>-1)$, pointing towards quintessence
behaviour of DE with both data combinations. Also, for the $w$CDM model, the
mean value of $w_{\rm de0}$ at $68\%$ CL, as shown in Table 3, lies in
quintessence region with BAO+CC+BBN+PP data combination whereas, it touches
the phantom barrier at $95\%$ CL with the same data combination ($w_{\rm
de0}=-0.930^{+0.086}_{-0.090}$ for BAO+CC+BBN+PP data combination). Again, for
the same model with BAO+CC+BBN+PPSH0ES data combination, the mean value of
$w_{\rm de0}$ lies in phantom region $(w_{\rm de0}<-1)$ at $68\%$ CL as well
as $95\%$ CL. At $95\%$ CL, $w_{\rm de0}=-1.062^{+0.078}_{-0.082}$ for
BAO+CC+BBN+PPSH0ES data combination. Also, we observe from Figure 2 that
$w_{\rm de0}$ finds a strong negative correlation with $H_{0}$ in case of
$w$CDM model whereas slight negative correlation in case of
$w$CDM+$\Omega_{\rm\sigma 0}$ model from both the data combinations.

Figure 2: Two-dimensional marginalized confidence regions at 68% and 95% CL of
$H_{0}$ and $w_{\rm de0}$ for $w$CDM and $w$CDM+$\Omega_{\rm\sigma 0}$ models
from BAO+CC+BBN+PP and BAO+CC+BBN+PPSH0ES data combinations. The vertical
yellow band represents $H_{\rm 0}^{\rm R22}=73.04\pm 1.04\leavevmode\nobreak\
{\rm km\,s^{-1}\,Mpc^{-1}}$ from SH0ES measurement. The horizontal black line
represents $w_{\rm de0}=-1$.

Figure 3: One-dimensional marginalized distributions of $\Omega_{\rm m0}$ for
$\Lambda$CDM, $\Lambda$CDM+$\Omega_{\rm\sigma 0}$, $w$CDM and
$w$CDM+$\Omega_{\rm\sigma 0}$ models from BAO+CC+BBN+PP (left panel) and
BAO+CC+BBN+PPSH0ES (right panel) data combinations.

One of the most well known and intriguing puzzle nowadays in cosmology is
discrepancy between Planck CMB data within the baseline $\Lambda$CDM model and
direct local measurements of the current rate of expansion of the Universe
(that is, the value of Hubble constant, $H_{0}$), as discussed in Section I.
Now, we discuss the constraints on Hubble constant for the considered models
and compare the results with the previous work [60]. For BAO+CC+BBN+PP data
combination, the constraint on $H_{0}$ for $w$CDM ($\Lambda$CDM) model reads:
$H_{\rm 0}=66.4\pm 1.3\,(68.02\pm 0.84)\leavevmode\nobreak\ {\rm
km\,s^{-1}\,Mpc^{-1}}$, and for $w$CDM+$\Omega_{\rm\sigma 0}$
($\Lambda$CDM+$\Omega_{\rm\sigma 0}$) model, it reads: $H_{\rm
0}=68.4^{+1.6}_{-1.6}\,(70.1^{+1.2}_{-1.5})\leavevmode\nobreak\ {\rm
km\,s^{-1}\,Mpc^{-1}}$ at 68% CL. For BAO+CC+BBN+PP data combination, we have
obtained lower mean values of $H_{0}$ for $w$CDM and $w$CDM+$\Omega_{\rm\sigma
0}$ models as compared to $\Lambda$CDM and $\Lambda$CDM+$\Omega_{\rm\sigma 0}$
models, respectively. For BAO+CC+BBN+PPSH0ES data combination, the constraint
on $H_{0}$ for $w$CDM ($\Lambda$CDM) model reads: $H_{\rm 0}=71.51\pm
0.83\,(70.79\pm 0.69)\leavevmode\nobreak\ {\rm km\,s^{-1}\,Mpc^{-1}}$, and for
$w$CDM+$\Omega_{\rm\sigma 0}$ ($\Lambda$CDM+$\Omega_{\rm\sigma 0}$) model, it
reads: $H_{\rm 0}=72.24\pm 0.82\,(72.67\pm 0.85)\leavevmode\nobreak\ {\rm
km\,s^{-1}\,Mpc^{-1}}$ at 68% CL. Thus, from BAO+CC+BBN+PPSH0ES data $w$CDM
model provides higher mean value of $H_{0}$ as compared to $\Lambda$CDM model,
whereas $w$CDM+$\Omega_{\rm\sigma 0}$ and $\Lambda$CDM+$\Omega_{\rm\sigma 0}$
models provide more or less the same mean values of $H_{0}$. The highest value
of Hubble constant $H_{0}$ obtained in the present analysis is $72.24\pm
0.82\leavevmode\nobreak\ {\rm km\,s^{-1}\,Mpc^{-1}}$ from BAO+CC+BBN+PPSH0ES
data combinations which is consistent with $H_{0}^{\rm R22}=73.04\pm
1.04\leavevmode\nobreak\ {\rm km\,s^{-1}\,Mpc^{-1}}$ from SH0ES measurement.
Quantifying the $H_{0}$ tension with SH0ES measurement for BAO+CC+BBN+PP data
combination, we found that there is $3.9\sigma$ tension on $H_{0}$ in the
$w$CDM model and there is $2.4\sigma$ tension on $H_{0}$ in
$w$CDM+$\Omega_{\rm\sigma 0}$ model. Also, for BAO+CC+BBN+PPSH0ES data
combination, $1.1\sigma$ tension on $H_{0}$ is observed in $w$CDM model and
$0.6\sigma$ tension is observed on $H_{0}$ in $w$CDM+$\Omega_{\rm\sigma 0}$
model. So, for BAO+CC+BBN+PP data combination $H_{0}$ tension is relieved by
$1.5\sigma$ in presence of anisotropy of the order $10^{-13}$ with constant
EoS parameter of DE. Also, for BAO+CC+BBN+PPSH0ES data combination $H_{0}$
tension is relieved by $0.5\sigma$ in presence of anisotropy of the order
$10^{-13}$ with constant EoS parameter of DE. See the Figure 2 where vertical
yellow band represents $H_{\rm 0}^{\rm R22}=73.04\pm 1.04\leavevmode\nobreak\
{\rm km\,s^{-1}\,Mpc^{-1}}$ from SH0ES measurement and constraints on $H_{0}$
in Table 3. Also, from Figure 4, we observe a positive correlation of $H_{0}$
with $\Omega_{\rm\sigma 0}$ with both the data combinations. So, in order to
get a larger value of $H_{0}$, a larger amount of anisotropy is required.

Figure 4: One and two dimensional marginalized confidence regions at 68% and
95% CL for some selected parameters of $w$CDM and $w$CDM+$\Omega_{\rm\sigma
0}$ models from BAO+CC+BBN+PP and BAO+CC+BBN+PPSH0ES data combinations.

Now, we discuss how $w_{\rm de0}$ and $\Omega_{\rm\sigma 0}$ affect the
present day matter density parameter, $\Omega_{\rm m0}$. Firstly, we observe
that the $w$CDM model provides significantly lower mean values of $\Omega_{\rm
m0}$ as compared to $\Lambda$CDM model for both data combinations. Second,
adding anisotropy to $\Lambda$CDM model slightly reduces the mean value of
$\Omega_{\rm m0}$ for both the data combinations.Also, inclusion of anisotropy
into the $w$CDM model significantly reduces the mean values of $\Omega_{\rm
m0}$ for both the data combinations. The constraints on $\Omega_{\rm m0}$ for
$w$CDM model are: $\Omega_{\rm m0}=0.258\pm 0.015$ and $\Omega_{\rm
m0}=0.287\pm 0.012$ from the data combinations BAO+CC+BBN+PP and
BAO+CC+BBN+PPSH0ES, respectively. The constraints on $\Omega_{\rm m0}$ for
$w$CDM+$\Omega_{\rm\sigma 0}$ model are: $\Omega_{\rm m0}=0.230\pm 0.018$ for
the data combination BAO+CC+BBN+PP and $\Omega_{\rm m0}=0.233\pm 0.018$ for
the data combination BAO+CC+BBN+PPSH0ES. We can observe the deviations in the
distribution of $\Omega_{\rm m0}$ for considered models in one dimensional
marginalized probability distribution as shown in Figure 3 (also, see table 3)
for both the data combinations. We also observe that parameters,
$\Omega_{\rm\sigma 0}$ and $\Omega_{\rm m0}$ finds a negative correlation with
both data combinations, see $\Omega_{\rm\sigma 0}-\Omega_{\rm m0}$ parametric
space in the Figure 4.

The considered $w$CDM and $w$CDM+$\Omega_{\rm\sigma 0}$ models provides higher
values of $r_{\rm d}$ as compared to $\Lambda$CDM and
$\Lambda$CDM+$\Omega_{\rm\sigma 0}$ models with BAO+CC+BBN+PP data
combination. Also, from BAO+CC+BBN+PPSH0ES data combination, we obtain the
result that the $w$CDM model prefers a lower value of $r_{\rm d}$ than
$\Lambda$CDM model, but $\Lambda$CDM+$\Omega_{\rm\sigma 0}$ and
$w$CDM+$\Omega_{\rm\sigma 0}$ provide equivalent mean values on $r_{\rm d}$
with similar error bars. Further, we notice that the addition of anisotropy
reduces the value of $r_{\rm d}$ in both the cases. We can observe from Figure
4, that the parameters $\Omega_{\rm\sigma 0}$ and $r_{\rm d}$ are negatively
correlated with both the data combinations, which means that for a small
increment in $\Omega_{\rm\sigma 0}$ values, there would be corresponding
decrement in $r_{\rm d}$ values.

We have observed that different model parameters are influenced due to
anisotropy. For instance, we observe a negative correlation between $H_{0}$
and $w_{\rm de0}$ which can be seen in Figure 2. Note that parameters
influence each other while fitting the multi-parameter space simultaneously
[81, 82]. Also, the other parameters of the models under consideration (with
constant EoS of DE) are affected by anisotropy in such a way that results into
the relieving of the $H_{0}$ tension up to $\sim 1\sigma$. It is important to
emphasize that in equation (19), the scaling $a^{-6}$ has relevance for the
early Universe while it becomes negligible in the late Universe. Thus, the
anisotropic model under consideration behaves like the Early DE model. For
more about Early DE models, we refer the readers to see the recent and
interesting review [33] and references therein.

##  V Concluding Remarks

In the present work, we have derived CMB independent observational constraints
on two simplest extensions of $\Lambda$CDM model, namely $w$CDM and
$w$CDM+$\Omega_{\rm\sigma 0}$ models from recent data sets including BAO, CC,
BBN, PP and PPSH0ES in two combinations: BAO+CC+BBN+PP and BAO+CC+BBN+PPSH0ES.
We have obtained that DE EoS parameter favours quintessence form of DE for
$w$CDM+$\Omega_{\rm\sigma 0}$ model from both the data combinations at $68\%$
CL as well as $95\%$ CL. For the $w$CDM model, the mean value of $w_{\rm de0}$
at $68\%$ CL lies in quintessence region with BAO+CC+BBN+PP data combination
whereas, it touches the phantom barrier at $95\%$ CL with the same data
combination. Again, for the same model with BAO+CC+BBN+PPSH0ES data
combination, the mean value of $w_{\rm de0}$ lies in phantom region $(w_{\rm
de0}<-1)$ at $95\%$ CL. Also, we notice a strong positive correlation between
$w_{\rm de0}$ and anisotropy parameter ($\Omega_{\rm\sigma 0}$) with both data
combinations, see Figure 1. The upper bounds on $\Omega_{\rm\sigma 0}$ are of
the order $10^{-13}$ from both the data combinations. The constant EoS of DE
significantly affects present day matter density parameter and provides lower
values for both the models (as compared to models investigated in [60] with
cosmological constant DE) from both the data combinations.
We have obtained higher values of Hubble constant with $H_{\rm 0}=71.51\pm
0.83$ and $H_{\rm 0}=72.24\pm 0.82$ $\rm km\,s^{-1}\,Mpc^{-1}$ at 68% CL for
$w$CDM and $w$CDM+$\Omega_{\rm\sigma 0}$ models, respectively with
BAO+CC+BBN+PPSH0ES data. The obtained value of $H_{0}$ in both the considered
models from BAO+CC+BBN+PPSH0ES are consistent with SH0ES measurements
($H_{0}^{\rm R22}$), and thus $H_{0}$ tension is relieved. Further, we notice
that inclusion of anisotropy parameter in the $w$CDM model reduces the $H_{0}$
tension by $\sim 2\sigma$ and $\sim 1\sigma$ with BAO+CC+BBN+PP and
BAO+CC+BBN+PPSH0ES data combinations, respectively. Overall, we have shown
that how the anisotropic extensions of standard cosmological model can
influence the estimation of $H_{0}$, and contribute significantly to the
resolution of the $H_{0}$ tension.

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# Autotuning Symbolic Optimization Fabrics for Trajectory Generation

Max Spahn* and Javier Alonso-Mora *This research was supported by Ahold
Delhaize. All content represents the opinion of the author(s), which is not
necessarily shared or endorsed by their respective employers and/or
sponsors.The authors are with the Department of Cognitive Robotics, Delft
University of Technology, 2628 CD, Delft, The Netherlands {m.spahn,
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present an automated parameter optimization method for
trajectory generation. We formulate parameter optimization as a constrained
optimization problem that can be effectively solved using Bayesian
optimization. While the approach is generic to any trajectory generation
method, we showcase it using optimization fabrics. Optimization fabrics are a
geometric trajectory generation method based on non-Riemannian geometry. By
symbolically pre-solving the structure of the tree of fabrics, we obtain a
parameterized trajectory generator, called symbolic fabrics. We show that
autotuned symbolic fabrics reach expert-level performance in a few trials.
Additionally, we show that tuning transfers across different robots, motion
planning problems and between simulation and real world. Finally, we
qualitatively showcase that the framework could be used for coupled mobile
manipulation.

Code github.com/maxspahn/optuna_fabrics

##  I Introduction

Mobile manipulation is the field of robotics concerned with highly capable
robots characterized by their locomotion and manipulation ability. Such robots
are getting ever more attention as they will be deployed to human-shared
environments, like households or warehouses. In such dynamic environments,
fast trajectory generation is crucial to avoid collisions and react quickly to
changing goal definitions.

Trajectory generation is often addressed by solving an optimization problem
that consists of a scalar objective function – the dynamics or transition
function – and several constraints. As the degrees of freedom and number of
constraints increase, solving that problem in real-time becomes challenging.
This is especially limiting in the case of mobile manipulation [1].
Optimization fabrics represent a different approach to the problem, as they
formulate trajectory generation as the shortest-geodesic-problem in a manifold
of the configuration space [2].

With optimization fabrics, different components, or desired behaviors, such as
collision avoidance and joint limit avoidance, are combined using Riemannian
metrics. As the structure of the resulting trajectory generation methods
remains unchanged across all time steps, it can be composed before runtime,
thus saving computational costs during executing. Optimization fabrics, but
also their predecessor Riemannian Motion Policies (RMPs), have shown
impressive results for several manipulator applications, including dynamic and
crowded environments [3, 4, 5]. However, despite their theoretical properties
of inherent collision avoidance and convergence, these methods require
expertise and intuition to tune individual components to generate smooth and
well-behaving trajectories.

Figure 1:  Overview of one trial in the tuning pipeline for symbolic
optimization fabrics. The objective function is evaluated after an entire
trial run is simulated. Using Bayesian optimization, a new parameter set is
suggested based on the history of trials. The best parameter set is extracted
from all trials.

Contributions: To address this issue, we formulate optimization fabrics as a
symbolic trajectory generation method. Precisely, the combination of the
individual components (joint limit avoidance, goal reaching, collision
avoidance, etc.) is performed in a parameterized way before runtime.
Separating composition and evaluation allows for changing the individual
parameters at runtime while achieving low computational costs. Additionally,
this allows formulating parameter-tuning as a constrained optimization
problem. Solving this problem effectively automates the tuning process
systematically using Bayesian optimization. We show that automated tuning
requires only few trials to achieve similar performance to an expert in the
field, and systematically outperforms a randomized parameter setting.
Moreover, we show that one parameter tuning generalizes across different
robots, to some extent, across different tasks and between simulation and real
world. Finally, we demonstrate how coupled mobile manipulation with a
differential drive can be achieved using autotuned optimization fabrics for
in-store order-picking integrating visual servoing.

##  II Related Works

###  II-A Geometric control for trajectory generation

Operational space control was the first control method that imposed a desired
dynamical system onto a robotic system [6, 7]. The concept was an important
step toward naturally controlling kinematically redundant robots. The concept
was formalized in the field of geometric control, where the study of
differential geometry leads to stable and converging behavior under geometric
conditions [8]. More recently, RMPs for manipulation tasks offered a highly
reactive trajectory generation method [9, 10]. This method achieves composable
behavior by introducing a split between the importance metric and the forcing
term. Using the pullback and pushforward operator to change between manifolds
of the configuration space, individual components, such as collision avoidance
and goal attraction, can be designed iteratively. However, RMPs require
intuition and experience when being designed, and convergence can only be
proven conditionally [11]. Later, optimization fabrics were introduced that
are able to completely decouple importance metrics and the defining geometry.
Under simple construction rules for these two components, convergence can be
easily guaranteed [11, 4, 12, 13]. In our prior work on optimization fabrics,
the framework was first applied to mobile manipulation and generalized to more
dynamic environments. [5].

Figure 2: Two different parameter sets for optimization fabrics given the same
problem. While the greedy tuning is more aggressive (purple), the more
conservative tuning results in a smoother trajectory (green).

###  II-B Autotuning for trajectory generation

Autotuning can be beneficial for trajectory generation when using model
predictive control. In [14], an autotuned model predictive controller has
outperformed a manual tuned controller of the same kind by 25%. Jointly
optimizing parameters and the model of the controller, AutoMPC showed the
benefit of parameter tuning in the context of simultaneous system
identification and control [15]. These methods are explicitly formulated for
model predictive control and do not transfer easily to other trajectory
generation methods. In contrast, we propose a generic parameter optimization
approach to trajectory generation.

###  II-C Hyperparameter tuning in machine learning

Within the machine-learning field, hyperparameter tuning has shown to be
highly important for all different kinds of applications [16, 17, 18].
Parameter optimization aims to minimize training costs while achieving the
best possible performance. Hyperparameter tuning is most valuable in extremely
costly applications such as reinforcement learning [19]. Generally, two
different search algorithms have been investigated: grid search and random
search [20]. Current state-of-the-art methods for parameter search are based
on random search with a Bayesian optimizer [18, 21]. While the machine-
learning community has largely agreed on the importance of parameter tuning,
systematic tuning of trajectory generation methods are not well established.
In this paper, we showcase, with the example of optimization fabrics, how
important parameter tuning is and how trajectory generation can benefit from
it.

##  III Overview

In this paper, we first recall very briefly the theory of optimization fabrics
and the steps to use it for trajectory generation (Section IV). Then, we
formulate optimization fabrics as a symbolic trajectory generator, so that
combining of individual components is only performed once (Section V). Then,
we formulate parameter tuning for trajectory generation as a constrained
optimization problem and propose Bayesian optimization for effective
autotuning (Section VI). As an example, we apply this autotuning to symbolic
optimization fabrics (Section VII), but it is generally independent of the
trajectory generator at hand.

##  IV Background

In this section, we very briefly introduce the concepts required for
trajectory generation with optimization fabrics. For a more in-depth
introduction to optimization fabrics and its foundations in differential
geometry, the reader is referred to [11, 5, 3].

###  IV-A Configurations and task variables

We denote $\bm{q}\in\mathcal{Q}\subset\mathbb{R}^{n}$ a configuration of the
robot with $n$ its degrees of freedom; $\mathcal{Q}$ is the configuration
space of the generalized coordinates of the system. Generally, $\bm{q}(t)$
defines the robot’s configuration at time $t$, so that $\dot{\bm{q}}$,
$\ddot{\bm{q}}$ define the instantaneous derivatives of the robot’s
configuration. Similarly, we assume that there is a set of task variables
$\bm{x}_{j}\in\mathcal{X}_{j}\subset\mathbb{R}^{m_{j}}$ with variable
dimension $m_{j}\leq n$. The task space $\mathcal{X}_{j}$ defines an arbitrary
manifold of the configuration space $\mathcal{Q}$ in which a robotic task can
be represented. Further, we assume that there is a differential map
$\phi_{j}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m_{j}}$ that relates the
configuration space to the $j^{th}$ task space. For example, when a task
variable is defined as the end-effector position, then $\phi_{j}$ is the
positional part of the forward kinematics. On the other hand, if a task
variable is defined to be the joint position, then $\phi_{j}$ is the identity
function. In the following, we drop the subscript $j$ in most cases for
readability when the context is clear.

We assume that $\phi$ is in $\mathcal{C}^{1}$ so that the Jacobian is defined
as

$\bm{J}_{\phi}=\frac{\partial\phi}{\partial\bm{q}}\in\mathcal{R}^{m\times n},$
(1)

or $\bm{J}_{\phi}=\partial_{\bm{q}}\phi$ for short. Thus, we can write the
total time derivatives of $\bm{x}$ as $\dot{\bm{x}}=\bm{J}_{\phi}\dot{\bm{q}}$
and
$\ddot{\bm{x}}=\bm{J}_{\phi}\ddot{\bm{q}}+\dot{\bm{J}}_{\phi}\dot{\bm{q}}$.

###  IV-B Spectral semi-sprays

Inspired by simple mechanics (e.g., the simple pendulum), the framework of
optimization fabrics designs motion policies as second-order dynamical systems
$\ddot{\bm{x}}=\pi(\bm{x},\dot{\bm{x}})$ [22, 11]. The motion policy is
defined by the differential equation $\bm{M}\ddot{\bm{x}}+\bm{f}=0$, where
$\bm{M}(\bm{x},\dot{\bm{x}})$ and $\bm{f}(\bm{x},\dot{\bm{x}})$ are functions
of position and velocity. Besides, $\bm{M}$ is symmetric and invertible. We
denote such systems as $\mathcal{S}=\left(\bm{M},\bm{f}\right)_{\mathcal{X}}$
and refer to them as spectral semi-sprays, or specs for short. When the space
of the task variable is clear from the context, we drop the subscript.

###  IV-C Operations on specs

Complex trajectory generation is composed of multiple components, such as
collision avoidance, joint limits avoidance, etc. The power of optimization
fabrics lies in the metric-weighted sum to combine multiple components from
different manifolds. These operations are derived from operations on specs and
are briefly recalled here.

Given a differential map $\phi:\mathcal{Q}\rightarrow\mathcal{X}$ and a spec
$\left(\bm{M},\bm{f}\right)_{\mathcal{X}}$, the pullback is defined as

$\textrm{pull}_{\phi}{\left(\bm{M},\bm{f}\right)_{\mathcal{X}}}={\left(\bm{J}^{T}_{\phi}\bm{M}\bm{J}_{\phi},\bm{J}^{T}_{\phi}(\bm{f}+\dot{\bm{J}}_{\phi}\dot{\bm{q}})\right)}_{Q}.$
(2)

The pullback allows converting between two distinct manifolds (e.g. a spec
could be defined in the robot’s workspace and pulled into the robot’s
configuration space using the pullback with $\phi$ being the forward
kinematics).

For two specs,
$\mathcal{S}_{1}={\left(\bm{M}_{1},\bm{f}_{1}\right)}_{\mathcal{X}}$ and
$\mathcal{S}_{2}={\left(\bm{M}_{2},\bm{f}_{2}\right)}_{\mathcal{X}}$, their
summation is defined by:

$\mathcal{S}_{1}+\mathcal{S}_{2}={\left(\bm{M}_{1}+\bm{M}_{2},\bm{f}_{1}+\bm{f}_{2}\right)}_{\mathcal{X}}.$
(3)

Additionally, a spec can be energized by a Lagrangian energy. Effectively,
this equips the spec with a metric. Specifically, given a spec of form
$\mathcal{S}_{\bm{h}}=(\bm{I},\bm{h})$ and an energy Lagrangian
$\mathcal{L}_{e}$ with the derived equations of motion
$\bm{M}_{\mathcal{L}_{e}}\ddot{\bm{x}}+\bm{f}_{\mathcal{L}_{e}}=0$, we can
define the operation

$\begin{split}S_{\bm{h}}^{\mathcal{L}_{e}}&=\text{energize}_{\mathcal{L}_{e}}\\{S_{\bm{h}}\\}\\\
&=(\bm{M}_{\mathcal{L}_{e}},\bm{f}_{\mathcal{L}_{e}}+\bm{P}_{\mathcal{L}_{e}}[\bm{M}_{\mathcal{L}_{e}}\bm{h}-\bm{f}_{\mathcal{L}_{e}}]),\end{split}$
(4)

where
$\bm{P}_{\mathcal{L}_{e}}=\bm{M}_{\mathcal{L}_{e}}\left(\bm{M}_{\mathcal{L}_{e}}^{-1}-\frac{\dot{\bm{x}}\dot{\bm{x}}^{T}}{\dot{\bm{x}}^{T}\bm{M}_{\mathcal{L}_{e}}\dot{\bm{x}}}\right)$
is an orthogonal projector. The resulting spec is an energized spec and we
call the operation energization.

With spectral semi-sprays and the presented operations, avoidance behavior,
such as joint limit avoidance, collision avoidance or self-collision
avoidance, can be realized.

###  IV-D Optimization fabrics

In the previous subsection, we explained how different avoidance behaviors can
be combined. Spectral semi-sprays can additionally be forced by a potential,
denoted as the forced variant of form
$\mathcal{S}_{\bm{\psi}}=\left(\bm{M},\bm{f}+\partial_{\bm{x}}\bm{\psi}\right)$.
This forcing term clearly changes the behavior of the system. Optimization
fabrics introduce construction rules to make sure that the solution path
$\bm{x}(t)$ of $\mathcal{S}_{\bm{\psi}}$ converges towards the minimum of
$\bm{\psi}(\bm{x})$. Then, the potential is designed in such a way that its
minimum represents a goal state of the motion planning problem.

First, the initial spec that represents an avoidance component is written in
the form $\ddot{\bm{x}}+\bm{h}(\bm{x},\dot{\bm{x}})=0$, such that $\bm{h}$ is
homogeneous of degree 2:
$\bm{h}(\bm{x},\alpha\dot{\bm{x}})=\alpha^{2}\bm{h}(\bm{x},\dot{\bm{x}})$
(Creation). Secondly, the geometry is energized (Eq. 4) with a Finsler
structure [11, Definition 5.4] (Energization). The property of homogeneity of
degree 2 and the energization with the Finsler structure guarantees, according
to [11, Theorem 4.29], that the energized spec forms a frictionless fabric. A
frictionless fabric is defined to optimize the forcing potential $\bm{\psi}$
when being damped by a positive definite damping term [11, Definition 4.4].
Thirdly, all avoidance components are combined in the configuration space of
the robot using the pullback and summation operation (Combination). Note, that
both operations are closed under the algebra designed by these operations,
i.e. every pulled optimization fabric or the sum of two optimization fabrics
is, itself, an optimization fabric. In the last step, the combined spec is
forced by the potential $\bm{\psi}$ with the desired minimum and damped with a
positive definite damping term (Forcing). This resulting system of form
$\bm{M}\ddot{\bm{q}}+\bm{f}(\bm{q},\dot{\bm{q}})+\partial_{\bm{q}}\bm{\psi}+\beta\dot{\bm{q}}=0$
is solved to obtain the trajectory generation policy in acceleration form
$\ddot{\bm{q}}=\pi(\bm{q},\dot{\bm{q}})$.

##  V Symbolic fabrics

A trajectory generator that is based on optimization fabrics is composed of
several components, such as collision avoidance, joint limit avoidance, goal
attraction, etc. Each component contributes to the resulting optimization
fabric through the metric-weighted summation that creates the tree of fabrics.
The trajectory generator is parameterized by the individual terms of the
components. Here, we lay out the parameterization for collision avoidance,
joint limit avoidance, self-collision avoidance, and speed-control. In our
framework, the tree of fabrics is generated before runtime as a symbolic
expression, to which the parameters are set at runtime. Note that the approach
of symbolic pre-solving results in much higher planning frequencies. In the
following, we explain the individual parameters that we exposed symbolically.
The form of the individual terms is adapted from [11, 4, 3] but written in a
symbolic form.

#### Basic inertia

The final tree of fabrics is equipped with a basic inertia metric that
indicates how reactive the entire motion is. This basic inertia metric is
derived from the symbolic Finsler structure:
$\mathcal{L}_{e}=0.5m_{\textrm{base}}\dot{\bm{q}}^{T}\bm{I}\dot{\bm{q}}$.

#### Collision avoidance

For collision avoidance, the task manifold $\mathcal{X}$ is defined by the
distance function between an obstacle and a robot link. The differential map
used is defined as

$\phi_{\textrm{i}}(\bm{q})=\frac{\left\lVert\textrm{fk}_{\textrm{i}}(\bm{q})-\bm{x}_{\textrm{obst}}\right\rVert}{r_{\textrm{obst}}+r_{\textrm{i}}}-1,$

where $\textrm{fk}_{\textrm{i}}(\bm{q})$ is the positional forward kinematics
for link $i$ in a configuration $\bm{q}$, $r_{\textrm{obst}}$ and
$r_{\textrm{i}}$ are the radii of the englobing spheres for the obstacle and
the link respectively. While this mapping between configuration space and task
manifold is different for each obstacle and each collision link of the robot,
the geometry and metric are the same for all of them. For the geometry
$\ddot{\bm{x}}+\bm{h}(\bm{x},\dot{\bm{x}})=\bm{0}$, we use the parameterized
forcing term

$\bm{h}(\bm{x},\dot{\bm{x}})=\frac{-k_{\textrm{geo,col}}}{\bm{x}^{\beta_{\textrm{geo,col}}}}\dot{\bm{x}}^{2},$
(5)

where $k_{\textrm{geo,col}}$ and $\beta_{\textrm{geo,col}}$ are parameters of
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avoidance is parameterized as

$\mathcal{L}_{e}(\bm{x},\dot{\bm{x}})=\frac{k_{\textrm{fin,col}}}{\bm{x}^{\beta_{\textrm{fin,col}}}}\left(-0.5(\textrm{sgn}(\dot{\bm{x}})-1)\right)\dot{\bm{x}}^{2},$
(6)

where $\textrm{sgn}(\dot{\bm{x}})$ is the signum-operator returning the sign
of $\dot{\bm{x}}$.

#### Self-collision avoidance

For self-collision avoidance, the task manifold $\mathcal{X}$ is defined
similarly to collision avoidance:

$\phi_{i,j}=\frac{\left\lVert\textrm{fk}_{i}(\bm{q})-\textrm{fk}_{j}(\bm{q})\right\rVert}{r_{i}+r_{j}}-1,$

where $\textrm{fk}_{i}(\bm{q})$ and $\textrm{fk}_{j}(\bm{q})$ are the
positional forward kinematics of the two links for a self-collision pair and
$r_{i}$ and $r_{j}$ are the radii for both englobing spheres. The geometries
are defined analogously

$\bm{h}(\bm{x},\dot{\bm{x}})=\frac{-k_{\textrm{geo,self}}}{\bm{x}^{\beta_{\textrm{geo,self}}}}\dot{\bm{x}}^{2}.$
(7)

The Finsler structure for collision avoidance is parameterized as

$\mathcal{L}_{e}(\bm{x},\dot{\bm{x}})=\frac{k_{\textrm{fin,self}}}{\bm{x}^{\beta_{\textrm{fin,self}}}}\left(-0.5(\textrm{sgn}(\dot{\bm{x}})-1)\right)\dot{\bm{x}}^{2}.$
(8)

#### Joint limit avoidance

For joint-limit avoidance, two simple differential maps denoting the distance
to the joint limits are used, specifically

$\displaystyle\phi_{\textrm{limit,i,lower}}(\bm{q})$
$\displaystyle=\bm{q}_{i}-\bm{q}_{\textrm{min,i}},\forall i\in(1,\dots,n)$
$\displaystyle\phi_{\textrm{limit,i,upper}}(\bm{q})$
$\displaystyle=\bm{q}_{\textrm{max,i}}-\bm{q}_{i},\forall i\in(1,\dots,n).$

Similar to collision avoidance, we use the parameterized forcing term

$\bm{h}(\bm{x},\dot{\bm{x}})=\frac{-k_{\textrm{geo,limit}}}{\bm{x}^{\beta_{\textrm{geo,limit}}}}\dot{\bm{x}}^{2}$
(9)

and the Finsler structure

$\mathcal{L}_{e}(\bm{x},\dot{\bm{x}})=\frac{k_{\textrm{fin,limit}}}{\bm{x}^{\beta_{\textrm{fin,limit}}}}\left(-0.5(\textrm{sgn}(\dot{\bm{x}})-1)\right)\dot{\bm{x}}^{2}.$
(10)

#### Speed control

As the root of the tree of fabrics is a frictionless fabric, it only converges
if damped [11]. Constant damping is sufficient to achieve the theoretical
properties that are needed for trajectory generation. However, [11, 3, 23]
proposed enhanced damping under the name of speedcontrol. We employ the same
damping strategy while adding parameterization. The technique is based on a
dynamic damping modification based on the distance to the goal. Specifically,
the final optimization fabric is damped according to

$\ddot{\bm{q}}=-\bm{h}_{2}-\bm{M}^{-1}\partial_{\bm{q}}\bm{\psi}+\alpha_{\textrm{ex}}\dot{\bm{q}}-\beta\dot{\bm{q}},$

where $\bm{h}_{2}$ is the sum of all pulled forcing terms, $\bm{M}$ is the sum
of all metrics of the individual geometries, $\partial_{\bm{q}}\bm{\psi}$ is
the goal attraction term pulled in the configuration space,
$\alpha_{\textrm{ex}}$ is a weighted sum of $\alpha_{\textrm{ex}}^{0}$ that
maintains constant execution energy without goal attraction and
$\alpha_{\textrm{ex}}^{\bm{\psi}}$ that maintains constant execution energy
with goal attraction:

$\alpha_{\textrm{ex}}={\bm{s}_{\eta}(\mathcal{L}_{ex})}\alpha_{\textrm{ex}}^{0}+(1-{\bm{s}_{\eta}(\mathcal{L}_{ex})})\alpha_{\textrm{ex}}^{\bm{\psi}}.$

Then, $\beta$ is the damping term, computed as:

$\beta={\bm{s}_{\beta}(\bm{q})}\bm{B}_{\textrm{max}}+\bm{B}_{\textrm{min}}+\max(0,\alpha_{\textrm{ex}}-\alpha_{\mathcal{L}_{e}}),$

where $\bm{B}_{\textrm{max}}$ and $\bm{B}_{\textrm{min}}$ are the upper and
lower damping values and $\alpha_{\mathcal{L}_{e}}$ is the energization
coefficient maintaining constant system energy (not execution energy) without
goal attraction. The switching functions ${\bm{s}_{\beta}(\bm{q})}$,
${\bm{s}_{\eta}(\bm{q})}$ are further parameterized as

$\displaystyle{\bm{s}_{\beta}(\bm{q})}$
$\displaystyle=0.5(\tanh{-\alpha_{\beta}(\lVert\bm{q}\rVert-r_{shift}))+1}$
$\displaystyle{\bm{s}_{\eta}(\mathcal{L}_{ex})}$
$\displaystyle=0.5(\tanh{\left(-0.5\mathcal{L}_{ex}(1-v_{ex})-0.5\right)}+1),$

where $r_{shift}$ determines the distance to the goal at which the switch
between $\bm{B}_{\textrm{min}}$ and $\bm{B}_{\textrm{max}}$ occurs,
$\alpha_{\beta}$ is the steepness of that switching, $\mathcal{L}_{ex}$ is the
user-defined execution energy (usually a simple kinetic energy in joint space)
and $v_{ex}$ is the execution energy factor, i.e. it determines the desired
speed of motion. For a detailed discussion on speed control with optimization
fabrics, we refer to previous works on optimization fabrics [3, 11].

We group all parameters resulting from the symbolic fabrics defined here into
a vector of parameters $\bm{\Theta}$. All parameters are listed in Table I.

##  VI Parameter tuning as an optimization problem

We define parameter tuning as a constrained optimization problem:

$\bm{\Theta}^{\ast}=\arg\min_{\bm{\Theta}}c(\bm{\Theta}),\ \textrm{s.t}\
\bm{\Theta}_{\textrm{min}}<\bm{\Theta}<\bm{\Theta}_{\textrm{max}},$ (11)

where $\bm{\Theta}_{\textrm{max}}$ and $\bm{\Theta}_{\textrm{min}}$ are the
upper and lower bounds of the parameters. The objective $c(\bm{\Theta})$ is a
function of the parameters specifying the tree of fabrics and can be evaluated
after one trajectory planning problem has finished. We call the evaluation of
one parameter set a trial. Next, we propose an objective function that is
flexible as different scenarios may require different parameter tuning.

###  VI-A Objective

The objective function $c(\bm{\Theta})$ is a weighted sum of several metrics,
that are invariant to the robot:

$c(\bm{\Theta})=w_{\textrm{distance}}c_{\textrm{distance}}+w_{\textrm{path}}c_{\textrm{path}}+w_{\textrm{clearance}}c_{\textrm{clearance}}.$
(12)

$c_{\textrm{distance}}$ accounts for the normalized, summed distance to the
goal over one trial and is defined as

$c_{\textrm{distance}}=\frac{\sum_{i=0}^{T}\left\lVert\bm{x}_{i}-\bm{\bm{x}_{\textrm{goal}}}\right\rVert}{\left\lVert\bm{x}_{0}-\bm{\bm{x}_{\textrm{goal}}}\right\rVert},$
(13)

where $i\in[0,T]$ are the discretized time steps and
$\bm{\bm{x}_{\textrm{goal}}}$ is the goal of the motion planning problem.
$c_{\textrm{path}}$ accounts for the normalized path length over one trial and
is defined as

$c_{\textrm{path}}=\frac{\sum_{i=1}^{T}\left\lVert\bm{x}_{i}-\bm{x}_{i-1}\right\rVert}{\left\lVert\bm{x}_{0}-\bm{\bm{x}_{\textrm{goal}}}\right\rVert}.$
(14)

$c_{\textrm{clearance}}$ accounts for the average clearance to obstacles over
one trial and is defined as

$c_{\textrm{clearance}}=\frac{1}{T}\sum_{i=1}^{T}\min_{o^{j}}\left\lVert\bm{x}_{i}-o^{j}_{i})\right\rVert,$
(15)

where $o^{j}_{i}$ is the position of obstacle $j$ at time step $i$. Each of
these terms is evaluated after an entire trial that was obtained by a specific
set of parameters.

1Formulate trajectory generator with parameters $\bm{\Theta}$

2 Define parameter space by
$\bm{\Theta}_{\textrm{min}},\bm{\Theta}_{\textrm{max}}$

3 Formulate objective $c(\bm{\Theta})$

4 Initialize objective function estimate $\tilde{c}(\bm{\Theta})$

5 for  _i = 0 to N_ do

6 Suggest parameter $\bm{\Theta}_{i}$ based on $\tilde{c}(\bm{\Theta})$

7 for  _t = 0 to T_ do

8 Compute action with parameter set $\bm{\Theta}_{i}$

9 Apply action to robot

10 Store observation relevant for metrics

11

12 end for

13 Evaluate $c(\bm{\Theta}_{i})$

14 Update $\tilde{c}(\bm{\Theta})$

15

16 end for

17Extract the best parameter set $\bm{\Theta}_{\textrm{best}}$

Algorithm 1 Autotuning for trajectory generators

###  VI-B Bayesian optimization

In the tuning phase, the problem specification for the investigated scenario,
e.g., the goal and obstacle positions, across all trials during tuning remains
the same while $\bm{\Theta}$ are optimized according to the objective. To
solve the Bayesian optimization we employ the Tree-structured Parzen Estimator
as it has shown improved performance over grid-search and conventional random
search in machine learning applications [24, 21]. To deploy this technique we
used Optuna, a hyperparameter optimization framework initially designed for
machine learning applications [18]. The general setup for one trial is shown
in Fig. 1 and the procedure is summarized in Algorithm 1.

##  VII Experimental Results

We showcase our parameter optimization method for symbolic fabrics. The search
space for the parameters is summarized in Table I.

Parameter | boundaries | type | distribution | manual
---|---|---|---|---
$m_{\textrm{base}}$ | $[0,1]$ | float | uniform | 0.2
$k_{\textrm{geo,col}}$ | $[0.01,1]$ | float | log | 0.03
$k_{\textrm{geo,limit}}$ | $[0.01,1]$ | float | log | 0.3
$k_{\textrm{geo,self}}$ | $[0.01,1]$ | float | log | 0.03
$k_{\textrm{fin,col}}$ | $[0.01,1]$ | float | log | 0.03
$k_{\textrm{fin,limit}}$ | $[0.01,1]$ | float | log | 0.05
$k_{\textrm{fin,self}}$ | $[0.01,1]$ | float | log | 0.03
$\beta_{\textrm{geo,col}}$ | $[1,5]$ | int | uniform | 3
$\beta_{\textrm{geo,limit}}$ | $[1,5]$ | int | uniform | 2
$\beta_{\textrm{geo,self}}$ | $[1,5]$ | int | uniform | 3
$\beta_{\textrm{fin,col}}$ | $[1,5]$ | int | uniform | 3
$\beta_{\textrm{fin,limit}}$ | $[1,5]$ | int | uniform | 3
$\beta_{\textrm{fin,self}}$ | $[1,5]$ | int | uniform | 3
$\alpha_{\beta}$ | $[0,1]$ | float | uniform | 0.5
$\bm{B}_{\textrm{min}}$ | $[0,1]$ | float | uniform | 0.01
$\bm{B}_{\textrm{max}}$ | $[5,20]$ | float | uniform | 6.5
$r_{shift}$ | [0.01, 0.1] | float | uniform | 0.05
$v_{ex}$ | [1.0, 30] | float | uniform | 15.0
TABLE I: Search space for parameters. Some parameters are restricted to
integers, and for some a log-distribution is applied.

We first analyze the importance of tuning for optimization fabrics on the
performance of trajectory generation. Then, we investigate how tuned
parameters can be transferred across different robots (Section VII-C),
different scenarios (Section VII-D), and between simulation and real world
(Section VII-E).

###  VII-A Experimental setup

The method was tested in simulation and in the real world on a Panda robot and
a mobile manipulator composed of a Clearpath Boxer and a Panda robot. The
simulation uses the pybullet physics engine with an interface through OpenAI-
gym [25]. The different motion planning goals evaluated in this paper are: (a)
reaching an end-effector pose inside a ring of obstacles (Fig. 3(a)) (similar
to the experiment in [3]) and (b) reaching an end-effector pose above a
surface with random obstacles (Fig. 3(b)). The two scenarios will be referred
to as reaching-in-ring and reaching-on-table, see Fig. 3(b). Unless stated
otherwise, the weights are set to
$w_{\textrm{path}}=0.1,w_{\textrm{clearance}}=0.2,w_{\textrm{distance}}=0.7$.
We also use this weighted sum as the performance metric. While these weights
are chosen arbitrarily in this work to demonstrate the usefulness of
autotuning, they should be derived from a human evaluator in a more realistic
scenario. We refer with manual to an expert-tuning, see Table I for specific
parameters. During testing, the trial was randomized with changing obstacles
and goals. For autotuning on the robotic arms, we consistently used $N=60$
trials, although the best parameter set is usually reached earlier, see Fig.
4.

(a) reaching-in-ring

(b) reaching-on-table

Figure 4: Optimization history for simulation (left) and real world (right)
for panda robot in reaching-in-ring scenario.

###  VII-B Importance of tuning

We compare the autotuned parameters with seven random parameter sets from the
search space and a manually tuned parameter set that we obtained through
expertise in previous works like [5]. In this experiment, tuning and testing
are performed on the test scenario reaching-in-ring. Tuning is crucial for
optimization fabrics, as the performance with a random parameter set cannot
compete with tuning, Fig. 5. This result was expected and should only
demonstrate that the right parameter set is required to deploy this method.
Autotuned parameters reach a similar performance to the expert. This result
highlights the importance of tuning for optimization fabrics and shows that
autotuning is an effective way to obtain parameter sets for novice users of
optimization fabrics.

Figure 5: Evaluation for scenario reaching-in-ring autotuned parameters and
compared to random parameter selection and manual tuning. Autotuning is able
to systematically outperform random parameter sets and reach expert level
tuning. Figure 6:  The autotuned for the panda robot in simulation for the
reaching-in-ring scenario on modified scenarios (blue) is compared to
autotuned parameter sets obtained on these scenarios directly (green).
Exchanging the robot (ur5, iiwa) and changing the scenario (reaching-on-table)
results in a very small loss in performance, while the loss is higher when
parameters are transferred between simulation and real world (real-world).

###  VII-C Cross validation: Transfer across robots

Without any retuning, we deploy the symbolic optimization fabrics planner
tuned on the Panda robot on two other robots with similar specifications (Kuka
LBR IIwa 7, Universal Robot UR5) and compare the performance with tuning
performed on the respective robot. Specifically, we do not change the leaf
geometries and energies but change differential maps according to relevant
collision links on the robot at hand. From Fig. 6, we conclude that tuning is
independent of the robot. This can be explained by the fact, that optimization
fabrics are a purely geometric approach to trajectory generation and the
different dimension of the robots do not change the dynamical system enforced
onto the robot.

###  VII-D Cross Validation: Transfer across scenarios

In the third experiment, we evaluate how well an autotuned parameter set
transfers to a different scenario. In the specific example, we use the tuning
obtained from the reaching-in-ring case and test it on reaching-on-table.
Performance can be transferred smoothly if the objective remains the same, see
Fig. 6. However, note that different scenario might require generally slower
motion because of a more crowded environment. Such a step would require to
retune the parameters according to the new objective.

###  VII-E Cross Validation: Transfer real world

As optimization fabrics are a geometric method [3], they should be independent
of the robot embodiment. Relying on the low-level controller. In this paper,
we investigate how the performance is affected by the transfer from the
simulation environment to the real world. Performance benefits from tuning in
the real world highlight that low-level controller differences affect the
behavior, see Fig. 6. Specifically, the accumulated distance to the goal is
increased ($0.14$m tuned in the real world vs $0.16$m tuned in simulation)
when tuning is transferred between simulation and real world. Thus, there is
added value in tuning in the real world. Our framework offers to quickly tune
fabrics in the real-world using the fabrics-ros-bridge. With relaxed
performance requirements, it is sufficient to tune in simulation.

###  VII-F Cross Validation: Transfer mobile manipulator

Finally, we qualitatively test the performance of the tuning method on a real
mobile manipulator with 10 degrees of freedom. After only $N=30$ trials, the
robot was able to perform coupled mobile manipulation based on a visual
servoing approach [26]. Symbolic optimization fabrics are especially suited
for visual servoing as their symbolic character allows them to constantly
update the position of the goal. A video of this experiment is attached to the
paper.

Figure 7: Trajectory generation with optimization fabrics for mobile
manipulator using visual serving for product picking.

##  VIII Conclusion

We formulated parameter tuning for trajectory generation as a constrained
optimization problem. Additionally, we introduced symbolic optimization
fabrics that implement optimization fabrics in a parameterized way, for which
the general structure is pre-solved. The trajectory generator obtained with
this technique is parameterized and achieves low computational costs at
runtime. We showed that parameter tuning for symbolic optimization fabrics can
be effectively solved using Bayesian optimization. Additionally, we have shown
that the tuning generalized across different robots, tasks, and between
simulation and the real world. Finally, we qualitatively demonstrated that the
method applies to mobile manipulators. While we aim at developing a method-
agnostic autotuning framework for motion generation, symbolic optimization
fabrics were selected as an example in this work.

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