# Constraints on the Transition Redshift using Hubble Phase Space Portrait

Darshan Kumar1 Deepak Jain2 Shobhit Mahajan1 Amitabha Mukherjee1 Akshay Rana3
1Department of Physics and Astrophysics, University of Delhi, Delhi 110007,
India
2Deen Dayal Upadhyaya College, University of Delhi, Dwarka, New Delhi 110078,
India
3St. Stephen’s College, University of Delhi, Delhi 110007, India

(Received: date / Accepted: date)

###### Abstract

One of the most significant discoveries in modern cosmology is that the
universe is currently in a phase of accelerated expansion after a switch from
a decelerated expansion. The redshift corresponding to this epoch is commonly
referred to as the transition redshift $z_{t}$. In this work we put
constraints on the transition redshift with both model-independent and model-
dependent approaches. We consider the recently compiled database of 32 Hubble
parameter measurements and the Pantheon sample of Type Ia Supernovae (SNe). In
order to include the possible systematic effects in this analysis, we use the
full covariance matrix of systematic uncertainties for the Hubble parameter
measurements. We plot a Hubble Phase Space Portrait (HPSP) between
$\dot{H}(z)$ and $H(z)$ in a model-independent way. From this HPSP diagram, we
estimate the transition redshift as well as the current value of the equation
of state parameter $\omega_{0}$ in a model-independent way. By considering
H(z) measurements, we find the best fit value of
$z_{t}=0.591^{+0.332}_{-0.332}$ and $\omega_{0}=-0.677^{+0.238}_{-0.238}$. We
obtain the best fit value of $z_{t}=0.849^{+0.117}_{-0.117}$ and
$\omega_{0}=-0.870^{+0.013}_{-0.013}$ using the Pantheon database. Further, we
also use a model dependent approach to determine $z_{t}$. Here, we consider a
non-flat $\Lambda$CDM model as a background cosmological model. We reconstruct
the cosmic triangle plot among $\log(\Omega_{m0})$, $-\log(2\Omega_{\Lambda
0})$ and $3\log(1+z_{t})$ where the constraints of each parameter are
determined by the location in this triangle plot. Using $\Omega_{m0}$ and
$\Omega_{\Lambda 0}$ values, we find the best value of the transition redshift
$z_{t}=0.619^{+0.580}_{-0.758}$, which is in good agreement with the Planck
2018 results at $1\sigma$ confidence level. We also simulate the observed
Hubble parameter measurements in the redshift range $0<z<2$ and perform the
same analysis to estimate the transition redshift.

##  1 Introduction

It is now widely believed that we live in a phase of accelerated expansion of
the universe, which has been corroborated by several independent observations.
The most direct and obvious evidence of the present accelerating stage of the
universe comes from the luminosity distance versus redshift relation
measurements using Type Ia supernovae (SNe) which are confirmed by two
different research groups [1, 2]. Over the decades, the number of SNe Type Ia
catalogues have grown substantially. Even now, the SNe Type Ia observations
provide the most direct evidence for the current cosmic acceleration ([3, 4,
5, 6, 7, 8, 9]). This discovery is also supported by complementary
observations such as the Cosmic Microwave Background (CMB) radiation
measurements ([10, 11, 12]), Baryonic Acoustic Oscillations (BAO) measurements
([13, 14, 15, 16, 17]) and Hubble parameter (H(z)) measurements ([18, 19, 20,
21, 22, 23]). From a theoretical point of view, the simplest and most popular
$\Lambda$CDM model supports such an accelerating phase, with free parameters
being tightly constrained by Planck Collaboration et al. (2020) [12]. In the
standard model of cosmology, there was an era of decelerated expansion due to
matter ([24, 25, 26]).

Given the transition from a decelerated to an accelerated expansion of the
universe, it is important to precisely determine the redshift at which the
universe started to accelerate from a decelerated phase. The parameter
corresponding to such a transition from decelerated to an accelerated phase is
referred as the transition redshift $z_{t}$ and it may be treated as a new
cosmic parameter along with the present deceleration parameter $q_{0}$ and the
Hubble constant $H_{0}$. It is now commonly accepted that the upper bound on
the transition redshift is less than unity, which has been confirmed by
multiple independent investigations in both model-dependent and model-
independent approaches ([19, 20, 27, 28, 29, 30, 31, 22, 32, 33, 34]). Shapiro
and Turner (2006) in their analysis adopted a model-dependent approach and
using SNe Type Ia observations, they put a constraint on the deceleration
parameter $q(z)$ and transition redshift $z_{t}$ [28] . By considering
different dynamical dark energy models, A. Melchiorri et al. (2007) [35] put
constraints on $z_{t}$. Similarly, D. Rapetti et al. (2007) developed a new
kinematic method to estimate $z_{t}$ using the measurements of X-ray cluster
gas mass fraction [36]. On the other hand, people have also studied the
transition redshift in modified gravity models ([37]).

In addition, by considering the dynamical nature of $q(z)$, several authors
have explored constraints on the transition redshift. For example, Cunha and
Lima (2008) have used a linear parametrization of the deceleration parameter
i.e. $q(z)=q_{0}+q_{1}z$ [29]. In their work, the value of transition redshift
is $z_{\mathrm{t}}=0.43_{-0.05}^{+0.09}$ which was obtained by using $182$ SNe
Type Ia ([4]). By using the SNLS data set, Astier et al. (2006) find the value
to be $z_{\mathrm{t}}=0.61_{-0.21}^{+3.68}$ [3]. Rani et al. (2015) also used
the linear parametrization of the deceleration parameter and by using a joint
analysis of age of galaxies, strong gravitational lensing and SNe Ia data,
they found $z_{\mathrm{t}}\approx 0.98$ [30]. Along the same lines, other
researchers have updated the study by introducing additional data sets and
modifying the parametrizations for $q(z)$. For the parametrization
$q(z)=q_{0}+q_{1}z/(1+z)$ and using 307 SNe Type Ia together with BAO and
$H(z)$ data, Xu et al. (2009) found $z_{\mathrm{t}}=0.609_{-0.070}^{+0.110}$
[38]. This type of parametrization has also been investigated by Ø. Elgarøy,
and T. Multamäki (2006) [39]. Using different datasets such as Gamma Ray
Bursts (GRBs), Hubble Observational Data $H(z)$, BAO, CMB, Galaxy Clusters,
lookback time etc., many authors have reconstructed the deceleration parameter
and put constraints on the transition redshift ([40, 41, 38, 29, 42, 43, 44,
20, 45, 46, 47]). These authors have considered different types of
parametrizations for $q(z)$, besides the aforementioned two parametrizations.
Recently, by considering $\Lambda$CDM model as a background cosmological
model, Planck Collaboration et al. (2020) put a constraint on the transition
redshift $z_{t}=0.632\pm 0.018$ [12].

The determination of the deceleration parameter $q$ and hence transition
redshift $z_{t}$, as one may see, is an important and relevant subject in
modern cosmology along with the determination of the current value of Hubble
parameter $H_{0}$. The determination of these parameters plays a vital role in
the understanding of the universe. An alternate and innovative approach for
accessing cosmic parameters is a model-independent one by means of the study
of Hubble parameter estimated from the Cosmic Chronometers (CC) approach.

In the literature, a statistical technique of obtaining cosmic parameters has
been developed by Seikel et al. (2012) [48]. This method is called Gaussian
Process (GP). In cosmology, using this statistical and non-parametric method,
one can reconstruct redshift dependent cosmological functions such as the
expansion rate, luminosity distance etc. Using cosmological observations
corresponding to these functions, one can put constraints on different
cosmological parameters. By adopting this model-independent approach, Jesus et
al. (2020) found the transition redshift as $z_{t}=0.59_{-0.11}^{+0.12}$ from
$H(z)$ measurements [49]. On the other hand, using the recent and largest up-
to-date Pantheon compilation of SNe Type Ia, Scolnic et al. (2018) obtained
$z_{t}=0.683_{-0.082}^{+0.110}$ in a non-parametric way [50]. Further, using
the GP approach, the analysis of Lin et al. (2019) yields
$z_{t}=0.59_{-0.05}^{+0.05}$ [51]. In their analysis, they reconstructed the
equation of state of dark energy and cosmic expansion using Gaussian Process
from the Pantheon data and $H(z)$.

In the present work we use the updated Hubble parameter measurements and
observations of Type Ia SNe. We obtain the transition redshift in both a
model-independent and model-dependent way. Firstly, we carry out the analysis
in a model-independent way. With the help of Gaussian Process, we reconstruct
the Hubble parameter $H(z)$ and its time-derivative $\dot{H}(z)$ from Hubble
parameter measurements and Type Ia SNe observations. We plot a diagram between
$\dot{H}(z)$ versus $H(z)$ which is generally referred to as the Hubble Phase
Space Portrait (HPSP). By analysing this diagram, we put constraints on
$z_{t}$ as well as on the present value of the equation of state parameter
$\omega_{0}$. By introducing the updated cosmic triangle plot and Hubble Phase
Space Portraits, we propose that these representations will help to visualise
and estimate the parameters in a unique way. We also simulate the observed
Hubble parameter measurements in the redshift range $0<z<2$ to estimate the
transition redshift.

In a model-dependent approach analysis, we consider a non-flat $\Lambda$CDM
model as a background cosmological model. After marginalization over $H_{0}$
using a uniform prior, we put constraints on the cosmological density
parameters i.e. $\Omega_{m0}$, $\Omega_{k0}$, and $\Omega_{\Lambda 0}$. Using
the sum rule of these three cosmological density parameters i.e.
$\Omega_{m0}+\Omega_{k0}+\Omega_{\Lambda 0}=1$, we represent them by an
equilateral triangle and the so-called ternary plot, where, each parameter
runs parallel to each of the edges on the equilateral triangle. The sum rule
must be satisfied at every location where lines of constant $\Omega_{m0}$,
$\Omega_{k0}$, and $\Omega_{\Lambda 0}$ intersect. This is similar to the
‘cosmic triangle’ described by Bahcall et al. (1999) [52]. Further, using a
relation between $z_{t}$, $\Omega_{m0}$ and $\Omega_{\Lambda 0}$, we update
the ternary plot and estimate the transition redshift.

The outline of the paper is as follows. In section 2, we discuss the Hubble
parameter measurements and Type Ia Supernovae data for determining the
comoving distance and obtaining the Hubble parameter and its derivatives.
Section 3 introduces the background of ‘Hubble Phase Space Portrait’, ‘Cosmic
Triangle’ and the methodology used to obtain the constraints on the transition
redshift. In section 4, we discuss the results. Finally, we summarize the
conclusions in Section 5.

##  2 Observational Datasets

###  2.1 Hubble Parameter Measurements

The Hubble parameter $H(z)$ which reflects the dynamical features of the
universe like its expansion rate and evolution history is a crucial parameter
in cosmology. It is also useful to investigate the nature of the dark energy
since this parameter can be inferred directly from astrophysical observations
which do not depend on any underlying cosmological model. There are other
probes which do not directly measure $H(z)$, but rather integrated quantities
such as luminosity distances. Currently, the most common methods for obtaining
$H(z)$ data are (i) using the differential ages of passively evolving
galaxies, generally referred to as ‘Cosmic Chronometers (CC)’ [53, 54] (ii)
measurements of peaks of acoustic oscillations of baryons (BAO) [13, 14, 55,
56, 57] (iii) Redshift Drift [58, 59].

In this work, we consider the $H(z)$ measurements obtained by using Cosmic
Chronometers. The Hubble parameter $H(z)$ in terms of redshift $z$ can be
expressed as

$H(z)=-\dfrac{1}{1+z}\dfrac{dz}{dt}$ (1)

However, when employing this approach to calculate $H(z)$, significant caution
must be exercised in the selection of galaxies. Stars are continually born in
early developing galaxies and the emission spectra will be dominated by the
young stellar population. As a result, passively developing red galaxies are
utilised to properly estimate the differential ageing of the universe since
their light is dominated by the elderly star population [53]. For measurements
of $H(z)$, the term $dt$ is to be estimated using the differential age
evolution of the universe in a given interval of redshift $dz$. For the
estimation of $dt$, Moresco et al. (2012) suggested the use of a direct
spectroscopic observable (4000Å break) which is known to be linearly related
to the age of the stellar population of a galaxy at fixed metallicity [60].
Thus, the measurement of $H(z)$, made purely by using spectroscopic
observations, is independent of the cosmological model and has proved to be a
strong probe to constrain cosmological models and assumptions. The most recent
data compilation of Hubble parameter measurements by Magaña et al. (2018) has
$31$ measurements of $H(z)$ which are obtained by using the differential ages
of passively evolving galaxies [61]. Here we have added one additional
datapoint compiled by Borghi et al. (2022) at $z=0.75$ [62]. Thus 32 $H(z)$
measurements used in this analysis are listed in Table 2.1, spanning the
redshift range $0.07\leq z\leq 1.965$.

Most recent compilation of updated Hubble parameter measurements.  $z$ $H(z)$
[$\rm{km\ sec^{-1}\ Mpc^{-1}}$] Reference 0.07 $69.0\pm 19.6$ [63] 0.09
$69.0\pm 12.0$ [54] 0.12 $68.6\pm 26.2$ [63] 0.17 $83.0\pm 8.0$ [54] 0.179
$75.0\pm 4.0$ [60] 0.199 $75.0\pm 5.0$ [60] 0.2 $72.9\pm 29.6$ [63] 0.27
$77.0\pm 14.0$ [54] 0.28 $88.8\pm 36.6$ [63] 0.352 $83.0\pm 14.0$ [60] 0.3802
$83.0\pm 13.5$ [22] 0.4 $95.0\pm 17.0$ [54] 0.4004 $77.0\pm 10.2$ [22] 0.4247
$87.1\pm 11.2$ [22] 0.4497 $92.8\pm 12.9$ [22] 0.47 $89.0\pm 50.0$ [64] 0.4783
$80.9\pm 9.0$ [22] 0.48 $97.0\pm 62.0$ [18] 0.593 $104.0\pm 13.0$ [60] 0.68
$92.0\pm 8.0$ [60] 0.75 $\mathbf{98.8\pm 33.6}$ [62] 0.781 $105.0\pm 12.0$
[60] 0.875 $125.0\pm 17.0$ [60] 0.88 $90.0\pm 40.0$ [18] 0.9 $117.0\pm 23.0$
[54] 1.037 $154.0\pm 20.0$ [60] 1.3 $168.0\pm 17.0$ [54] 1.363 $160.0\pm 33.6$
[21] 1.43 $177.0\pm 18.0$ [54] 1.53 $140.0\pm 14.0$ [54] 1.75 $202.0\pm 40.0$
[54] 1.965 $186.5\pm 50.4$ [21]

###  2.2 Type Ia Supernovae Measurements

To estimate luminosity distance, we use the Pantheon dataset of Type Ia
Supernovae. This dataset has 1048 Type Ia SNe datapoints in the redshift range
$0.01\leq z\leq 2.26$ [50]. In Type Ia SNe analysis, the standard observable
quantity is the distance modulus, $\mu_{SNe}$ which is basically the
difference between the apparent and absolute magnitude of the Type Ia SNe. The
relation which gives the observable measurement is defined as

$\mu_{\mathrm{SNe}}=m_{\mathrm{B}}(z)+\alpha\cdot
X_{1}-\beta\cdot\mathcal{C}-M_{\mathrm{B}}.$ (2)

where $m_{\mathrm{B}}$ is the rest frame B-band peak magnitude, $M_{B}$
represents absolute B-band magnitude of a fiducial Type Ia SNe with $X_{1}=0$
and $C=0$, $X_{1}$ and $C$ represent the time stretch of light curve and
supernova colour at maximum brightness respectively. For the Pantheon sample,
the stretch-luminosity parameter $(\alpha)$ and the colour-luminosity
parameter $(\beta)$ are calibrated to zero. Thus we are left with two
parameters and hence the distance modulus can be defined as

$\mu_{\mathrm{SNe}}=m_{\mathrm{B}}(z)-M_{\mathrm{B}}.$ (3)

Further, once we know the distance modulus, luminosity distance $(d_{L})$ and
uncertainty in the luminosity distance $(\sigma_{d_{L}})$ for each Type Ia SNe
can be estimated as

$d_{\mathrm{L}}(z)=10^{\left(m_{\mathrm{B}}-M_{\mathrm{B}}-25\right)/5}~{}~{}~{}(\mathrm{Mpc}),\quad\sigma_{d_{\mathrm{L}}}=\frac{\ln(10)}{5}d_{\mathrm{L}}\sigma_{m_{\mathrm{B}}}~{}~{}~{}(\mathrm{Mpc})$
(4)

From Eq. 4, it is clear that once we estimate the value of $M_{B}$, we can
find the luminosity distance at a given redshift. Recent studies seem to
indicate that with redshift there is no evolution of luminosity (or absolute
magnitude) of Type Ia supernovae [65, 66]. So, it is generally accepted that
the Type Ia supernovae sample is normally distributed with a mean absolute
magnitude of $M_{B}=-19.22$ [67]. Finally using the following relation of
luminosity distance and comoving distance, we estimate the comoving distance
for each Type Ia SNe as

$d_{co}=\dfrac{d_{L}}{1+z}$ (5)

##  3 Background and Methodology

###  3.1 Hubble Phase Space Portrait (HPSP)

For a spatially flat, homogeneous and isotropic universe, the Friedmann
equations are

$\left(\dfrac{\dot{a}}{a}\right)^{2}=\dfrac{8\pi G}{3}\rho_{T}$ (6)

$2\dfrac{\ddot{a}}{a}+\left(\dfrac{\dot{a}}{a}\right)^{2}=-8\pi Gp_{T}$ (7)

In these equations, $a(t)$ is scale factor, $\rho_{T}$ and $p_{T}$ denote the
total energy density and total pressure respectively. By considering
$H(t)=\dfrac{\dot{a}}{a}$ and so $\dot{H}+H^{2}=\dfrac{\ddot{a}}{a}$ and
dividing equation (7) by equation (6), we get

$\dot{H}=-\dfrac{3}{2}(1+\omega)H^{2}$ (8)

where we consider an equation of state parameter $\omega$ which relates the
total energy density to total pressure of a barotropic fluid via
$p_{T}=\omega\rho_{T}$. Depending on the content of the universe, $\omega$
takes values in the range $-1\leq\omega\leq\dfrac{1}{3}$. Thus we can
reconstruct Friedmann’s equations into a single equation i.e.
$\dot{H}=\textit{F}(H)$. Using this equation, by visualizing the trajectories
in the $(H,~{}\dot{H})$ phase space, one can analyze different cosmic models
in a clear and transparent way. Equation (8) is referred to as the phase space
portrait and its solution is the phase trajectory. For more details about
phase space portrait, please refer [68, 69, 70]. Corresponding to any theory,
the phase-space portrait can be drawn in $(H,~{}\dot{H})$ phase-space. For
example, a theoretical reconstruction of the Hubble phase space portrait for
different cosmological models is shown in Figure 1.

Figure 1: The theoretical reconstruction of Hubble phase space portrait.

In this framework, we classify the different phases of Figure 1 as follows:

  1. 1.

In $(H,~{}\dot{H})$ phase space, the left half represents a contracting
universe since this portion has $H<0$. On the other hand, the right half has
$H>0$, which represents an expanding universe.

  2. 2.

For different values of the equation of state parameter, $\omega$, we can draw
lines in the $(H,~{}\dot{H})$ phase space. For example, for a matter dominated
universe we draw the phase trajectory corresponding to $\omega=0$. Similarly,
we plot lines for radiation dominated, curvature dominated and cosmological
constant dominated universes corresponding to $\omega=1/3$, $\omega=-1/3$ and
$\omega=-1$ respectively.

  3. 3.

The line corresponding to $w=-1/3$ is generally referred to as “zero
acceleration curve" since the deceleration parameter for this line is zero
i.e. $q=0$. This line splits the $(H,~{}\dot{H})$ phase space into two
regions. The upper region represents the accelerated universe where we have
$\omega<-1/3$ and $q<0$. On the other hand, the lower region shows the
decelerated universe. In this region, the equation of state parameter is
$\omega>-1/3$ and deceleration parameter is $q>0$.

  4. 4.

Therefore, the full $(H,~{}\dot{H})$ phase space is divided into four
different regions. The upper left region represents an accelerated contraction
of universe since $H<0$ and $q<0$. The upper right displays an accelerated
expansion of universe since $H>0$ and $q<0$. The lower left represents a
decelerated contraction of universe where $H<0$ and $q>0$. And the lower right
region represents a decelerated expansion of universe since in this region, we
have $H>0$ and $q>0$.

As we know, the universe passed from a radiation dominated $(\omega=1/3)$ era
to a matter dominated $(\omega=0)$ one in the past. The lower regions
characterized the usual FLRW models. As a result we expect that the
observations will lie in that region which will show a transition from a
decelerated expansion era to an accelerated expansion era by crossing the
zero-acceleration $(q=0)$ line. The value of the Hubble parameter
corresponding to this crossing point is referred to as the transition Hubble
parameter value and the corresponding redshift is called the transition
redshift. By analyzing the $(H,~{}\dot{H})$ phase space, one can easily
estimate the transition Hubble parameter value and transition redshift value.
Hence, in this analysis, we focus only on this inner right region of the
$(H,~{}\dot{H})$ phase space.

###  3.2 The Cosmic Triangle

A triangle or ternary diagram is a graphical depiction of systems via an
equilateral triangle having three-components which sum to a constant. A
ternary diagram has some significant advantages over other graphical
representations where the actual distribution of three-components is displayed
as points within an equilateral triangle. The obvious advantage of this
representation is that by presenting the three-variable data in a two-
dimensional plot, it is easy to analyze the correlation among the three
variables simultaneously. Using a ternary diagram, Bahcall et al. (1999)
proposed the ‘cosmic triangle’ to visualize the cosmological parameters [52].
In this cosmic triangle they plotted the matter, curvature and cosmological
constant density parameters; all these density parameters are related to one
another since they follow the sum rule i.e.
$\Omega_{m0}+\Omega_{k0}+\Omega_{\Lambda 0}=1$. In their analysis, they showed
that the constraints on the cosmological parameters using cosmic triangle
representation preferred the (now standard) flat $\Lambda$CDM model.

For example, suppose we have a non-linear relation between any two independent
components (say $a,~{}b$ and $ab$). By taking the logarithm of each component
in the triad one can propose a ternary plot where, a linear relationship
condition is satisfied through $\log_{10}(a)+\log_{10}(b)-\log_{10}(ab)=0$. In
this context, recently, Bernal et al. (2021) propose a new cosmic triangle to
visualize cosmological parameters like $H_{0}$, $t_{U}$ (age of the
universe)[71].

If we have a power-law like relation between three components i.e.
$a=(bc)^{n}$, then to reconstruct a linear relation for such power-law
relation, we take logarithms on both sides and thus can re-plot a ternary
diagram between $\log_{10}(a)$, $-n\log_{10}(b)$ and $-n\log_{10}(c)$ and
every point on this ternary plot satisfies the condition
$\log_{10}(a)-n\log_{10}(b)-n\log_{10}(c)=0$. Thus, by knowing any two out of
the three parameters, one can put constraints on the third parameter. In this
work, we focus on a power-law relation between cosmological parameters and put
constraints on these parameters. We show that such a representation represents
the cosmic constraints in an intuitive and illustrative manner.

###  3.3 Statistical Methods

In this work, we use the recent compilation of Hubble parameter measurements
and Type Ia SNe measurements. We apply Gaussian process to reconstruct the
Hubble parameter and its derivative using these observations. For constructing
the cosmic triangle we assume a non-flat $\Lambda$CDM model as a background
cosmological model. By minimizing the Chi-square, we put constraints on the
parameters.

####  3.3.1 Gaussian Process

The aim of the Gaussian Process (GP) method is to reconstruct a function from
data without assuming a parametrized form of the function. For example, we
have values $H(z)$ from a set of measurements i.e. $H(z_{i})\pm\sigma_{H}$,
where the value of $H(z_{i})$ follows a Gaussian distribution at every point
of $z_{i}$. Suppose we want the value of the function at an unknown point
$z^{\prime}$. For this a covariance or kernel function $k(z,z^{\prime})$ is
needed which indicates that the value of the function at $z$ is not
independent of its value at $z^{\prime}$ but is correlated by
$k(z,z^{\prime})$.

Thus, the Gaussian Process technique is non-parametric since it is dependent
on the choice of the covariance function rather than a set of parameters in an
assumed model or functional form. In general, the covariance function depends
only on the distance between the points $|z-{z}^{\prime}|$. Here, we consider
the Squared Exponential (or Gaussian) kernel function ([60, 72]) since this
function has the characteristic of being infinitely differentiable, which is
important for reconstructing a function’s derivative. The Squared Exponential
kernel function is

$k(z,{z}^{\prime})=\sigma_{f}^{2}\exp\left(-\frac{(z-{z}^{\prime})^{2}}{2\ell^{2}}\right)$
(9)

where, $\sigma_{f}$ and $\ell$ are referred as the GP hyperparameters. These
basically regulate the correlation-strength of the function value and the
length scale of the correlation in $z$ respectively. By minimizing a log
marginal likelihood function, one can calculate the value of $\sigma_{f}$ and
$\ell$ using the observed data. For maximization, we use flat priors for
$\sigma_{f}$ and $\ell$ for a particular choice of kernel function.

This method can also be used to reconstruct the derivative of the function
i.e. $f^{\prime}(z)$. While reconstructing the derivative of a function the
covariance between the observational points remains unchanged. However, we
also need a covariance between the function and its derivative as well as the
covariance between the derivatives.

Finally, the Gaussian Process for $f$ and for its derivatives $f^{\prime}(z)$
and $f^{\prime\prime}$ are

$\displaystyle f(z)$ $\displaystyle\sim\mathcal{GP}(\mu(z),k(z,{z}^{\prime}))$
(10) $\displaystyle f^{\prime}(z)$
$\displaystyle\sim\mathcal{GP}\left(\mu^{\prime}(z),\frac{\partial^{2}k(z,{z}^{\prime})}{\partial
z\partial{z}^{\prime}}\right)$ $\displaystyle f^{\prime\prime}(z)$
$\displaystyle\sim\mathcal{GP}\left(\mu^{\prime\prime}(z),\frac{\partial^{4}k(z,{z}^{\prime})}{\partial
z^{2}\partial{{z}^{\prime}}^{2}}\right)$

where $\mu(z)$ is the mean value of a given function $f$.

In this analysis, we use equation (10) to reconstruct the Hubble parameter
$H(z)$ and the derivative of the Hubble parameter $H^{\prime}(z)$ using $32$
datapoints of Hubble parameter measurements as well as the comoving distance
$d_{co}$ and the derivatives of comoving distance $d_{co}^{\prime}$,
$d_{co}^{\prime\prime}$ using 1048 measurements of Type Ia SNe.

####  3.3.2 Chi-square Analysis

The cosmological parameters of the assumed background cosmological model are
determined by maximizing the likelihood
$\mathcal{L}\sim\exp\left(-\chi^{2}/2\right)$, where Chi-square $\chi^{2}$ is

$\chi^{2}\left(\mathbf{P}\right)=\left[\mathbf{{H}_{th}}\left(z_{i};\mathbf{P}\right)-\mathbf{{H}_{obs}}\left(z_{i}\right)\right]^{T}\operatorname{\mathbf{Cov}_{ij}}^{-1}[\mathbf{{H}_{th}}(z_{j};\mathbf{P})-\mathbf{{H}_{obs}}(z_{j})]$
(11)

Here, $\mathbf{P}$ represents the cosmological parameters i.e. $H_{0}$,
$\Omega_{m0}$, $\Omega_{k0}$ and $\Omega_{\Lambda 0}$, whereas $H_{th}$ and
$H_{obs}$ are the theoretical Hubble parameter and observed Hubble parameter
respectively.

Here $\operatorname{Cov}_{ij}$ is the total uncertainty in the Hubble
parameter which includes the effect of both statistical as well as systematic
errors. Thus in our analysis, we consider the full covariance matrix
$\operatorname{Cov}_{ij}$ as

$\operatorname{Cov}_{ij}=\operatorname{Cov}_{ij}^{\text{stat
}}+\operatorname{Cov}_{ij}^{\mathrm{syst}},$ (12)

where $\operatorname{Cov}_{ij}^{\mathrm{stat}}$ denote the contributions to
the covariance due to statistical errors. The contribution due to systematic
errors is given by $\operatorname{Cov}_{ij}^{\mathrm{syst}}$. To compute the
systematic errors carefully we refer to the analysis carried out by Moresco et
al. (2020) where, we have considered the four main contributions in the
covariance matrix, i.e. initial mass function, stellar library, metallicity,
and stellar population synthesis models [73]. For a detailed discussion of the
full covariance matrix please refer to Section 3.1.4 of Moresco et al. (2022)
[74].

To remove the dependency of $H_{0}$ on the other cosmological parameters, and
by considering a flat prior for $H_{0}$, we marginalize the Chi-square over
$H_{0}$ and thus equation (11) becomes

$\tilde{\chi}^{2}={A}-\frac{{B}^{2}}{{C}}+\ln\frac{{C}}{{2}\pi}$ (13)

Here,
${A}=\mathbf{H_{obs}}^{T}(z_{i})\operatorname{\mathbf{Cov}}_{ij}^{-1}\mathbf{H_{obs}}(z_{j})$,
${B}=\mathbf{H_{obs}}^{T}(z_{i})\operatorname{\mathbf{Cov}}_{ij}^{-1}\mathbf{E}(z_{j};\mathbf{p})$
&
${C}=\mathbf{E}(z_{i};\mathbf{p})^{T}\operatorname{\mathbf{Cov}}_{ij}^{-1}\mathbf{E}(z_{j};\mathbf{p})$
where,
$E(z;\Omega_{m0},\Omega_{k0})=\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{k0}(1+z)^{2}+1-\Omega_{m0}-\Omega_{k0}}$.

####  3.3.3 Generation of simulated $H(z)$ data

It seems natural that future observations will enlarge the present dataset of
the Hubble parameter. To study the impact of a potentially enlarged dataset,
we use a mock dataset of the Hubble parameter measurements and study its
impact on the cosmic triangle and Hubble phase space portrait. To simulate the
Hubble parameter data, we use the simulation method as explained by Ma and
Zhang (2011) [75]. The simulation steps are as follows:

  1. 1.

We choose the flat $\Lambda$CDM model as a fiducial cosmological model and
adopt the best fit values of $H_{0}$, $\Omega_{m0}$ from [12]. Using these
values, we calculate the fiducial value of Hubble parameter i.e. $H_{fid}(z)$.

  2. 2.

To update the errors of observed Hubble parameter measurements, we plot their
errorbars versus redshift and based on the fact that the errors of $H(z)$ seem
to increase almost linearly. with respect to $z$, we identify $7$ datapoints
as outliers and exclude them.

  3. 3.

By using the remaining Hubble parameters points, to estimate the errors of the
simulated data, we draw upper and lower lines which are
$\sigma^{+}(z)=16.577z+18.440$ and $\sigma^{-}(z)=7.402z+2.675$ respectively.
To predict the mean error for future observations, the midline of the errors
is $\sigma^{0}(z)=11.990z+10.558$.

  4. 4.

By assuming a Gaussian distribution of the simulated datapoints, the error of
the simulated $H(z)$ is a random number drawn from a Gaussian distribution
with mean $\sigma^{0}(z)$ and variance $\epsilon(z)$ i.e.
$\tilde{\sigma}(z)=\mathcal{N}[\sigma^{0}(z),\epsilon(z)]$, where we define
$\epsilon(z)=\dfrac{\sigma^{+}(z)-\sigma^{-}(z)}{4}$. The parameter
$\epsilon(z)$ is set so that there is a $95.4\%$ probability of
$\tilde{\sigma}(z)$ lying within the strip.

  5. 5.

The values of the simulated Hubble parameter are defined as
$H_{sim}=H_{fid}+\Delta H$, where $\Delta H$ can be derived from a Gaussian
distribution of the form $\mathcal{N}[0,\tilde{\sigma}(z)$].

Using the above steps, we have generated $128$ simulated values of the Hubble
parameter as well as the uncertainties in the Hubble parameter.

##  4 Results

In this paper, our aim is to put constraints on the transition redshift
$z_{t}$ from Hubble parameter measurements and Type Ia SNe measurements. We
reconstruct the Hubble Phase Space Portrait using Gaussian Process and put
constraints on the transition redshift as well as on the equation of state
parameter in a model-independent way. Further, we propose a cosmic triangle
plot to put constraints on the transition redshift. For this purpose, we
consider a non-flat $\Lambda$CDM model as a background cosmology and put
constraints on its cosmological parameters.

###  4.1 Hubble Phase Space Portrait (HPSP)

The equation for the phase space portrait is

$\dot{H}(z)=-\dfrac{3}{2}(1+\omega)H^{2}(z)$ (14)

To plot this phase space portrait for $H$, we need to calculate the Hubble
parameter $H(z)$ and its time-derivative $\dot{H}(z)$. For this, we use GaPP,
a Python based Gaussian Process tool developed by Seikel et al. (2012) [48]
which provides the reconstruction of $H(z)$ as shown in Figure 2.

Figure 2: Hubble Parameter reconstruction using Gaussian Process.

The Gaussian Process, as operated by GaPP, reconstructs not only $H(z)$ from
data, but also helps us to reconstruct its derivative and the associated
uncertainties. For the phase space portrait, we need to estimate the time-
derivative of the Hubble parameter i.e. $\dot{H}(z)$. The relation between
$H^{\prime}(z)$ and $\dot{H}(z)$ is

$\dot{H}(z)=-(1+z)H(z)H^{\prime}(z)$ (15)

Thus, by using the reconstructed $H^{\prime}(z)$, we can estimate $\dot{H}(z)$
and uncertainties in $\dot{H}(z)$ from formula,

$\sigma_{\dot{H}}^{2}(z)=\left(\dfrac{\partial\dot{H}}{\partial
H}\right)^{2}\sigma_{H}^{2}+\left(\dfrac{\partial\dot{H}}{\partial
H^{\prime}}\right)^{2}\sigma_{H^{\prime}}^{2}+2\left(\dfrac{\partial\dot{H}}{\partial
H}\right)\left(\dfrac{\partial\dot{H}}{\partial
H^{\prime}}\right)\sigma_{HH^{\prime}}$

.

Using $H(z)$ and $H^{\prime}(z)$, we have drawn the Hubble Phase Space
Portrait (HPSP) between $\dot{H}(z)$ versus $H(z)$ as shown in Figure 3.

Figure 3: The Hubble Phase Space Portrait obtained by applying Gaussian
process on observed $H(z)$ datapoints.

In this diagram, as expected the best fit line (in red colour) is
reconstructed using Gaussian process which evolves from the decelerated
expansion region to the accelerated expansion region. The point where the best
fit line crosses the zero- acceleration curve is generally referred to as the
transition point and the redshift corresponding to it turns out to be
$z_{t}=0.591\pm 0.332$, which is in good agreement with the recent results of
Planck Collaboration et al. (2020) [12]. Further, using equation (14) and by
following the reconstructed $H(z)$ and $\dot{H}(z)$, we obtain the present
value of equation of state is $\omega_{0}=-0.677\pm 0.238$.

Similarly, for the Type Ia SNe, we estimate the $H(z)$ and $\dot{H}(z)$ by
using the derivatives of $d_{co}$ as

$\displaystyle H(z)$ $\displaystyle=\dfrac{c}{d_{co}^{\prime}}$ (16)
$\displaystyle H^{\prime}(z)$
$\displaystyle=-\dfrac{c}{\left(d_{co}^{\prime}\right)^{2}}\cdot
d^{\prime\prime}_{co}=-H(z)\dfrac{d_{co}^{\prime\prime}}{d_{co}^{\prime}}$
$\displaystyle\dot{H}(z)$ $\displaystyle=-(1+z)H(z)H^{\prime}(z)$

Thus using Eq. (16), we can draw the Hubble Phase Space Portrait (HPSP)
between $\dot{H}(z)$ versus $H(z)$ as shown in Figure 4.

Figure 4: The Hubble Phase Space Portrait obtained by applying Gaussian
process on observed Type Ia SNe datapoints.

In this case, the value of transition redshift comes out to be $z_{t}=0.848\pm
0.177$. Further, using equation (14), we obtain the present value of equation
of state as $\omega_{0}=-0.870\pm 0.013$.

###  4.2 Results from the simulated data sets

So far, we have discussed the results of the Hubble Phase Space Portrait using
$32$ datapoints of Hubble parameter measurements and Type Ia SNe measurements.
Due to the limited number of datapoints for Hubble parameter measurements, we
find large uncertainties and hence a wide region in both the cosmic triangle
and HPSP diagrams. In order to estimate the impact of future enhancement of
Hubble parameter data, we use a mock data set of the Hubble parameter
measurements only and study the potential of using this type of data. In
principle, one can generate a mock data set of the Type Ia SNe observations
also. However, due to the large number of datapoints and small errorbars at
low redshift of Type Ia SNe database, we find the value of transition redshift
which has small uncertainty as compared to the value obtained from Hubble
parameter measurements. Since the number of H(z) data points are less as
compared to SNe data points in the given redshift range. Hence we simulate
only the H(z) data points to study the constraints on the transition redshift.

To simulate the Hubble parameter data we use the simulation method as
described by Ma and Zhang (2011) [75]. As explained in Section 3, we have
generated $128$ simulated values of the Hubble parameter. The simulated
datapoints are shown in Figure 5 along with the observed Hubble parameter
measurements.

Figure 5: The simulated datapoints of $H(z)$ in the redshift $0.0<z<2.0$. The
blue points represent the simulated datapoints while the observed $H(z)$ are
represented by the red colour.

Using the simulated Hubble parameter dataset, we update the Hubble phase space
diagram as shown in Figure 6. The best fit value of the transition redshift in
this case comes out to be $z_{t}=0.574\pm 0.193$ which is consistent with the
Planck Collaboration et al. (2020) at $1\sigma$ confidence level [12] . In
this case, the present value of equation of state is $\omega_{0}=-0.750\pm
0.175$. As expected, the errors have reduced because of the increased number
of datapoints.

Figure 6: The Hubble Phase Space Portrait reconstructed by using 128 simulated
Hubble parameter datapoints.

For the completeness, we also perform a model-dependent analysis of estimating
transition redshift using Cosmic Triangle technique. For this we consider the
non-flat $\Lambda$CDM model

$H(z)=H_{0}\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{k0}(1+z)^{2}+\Omega_{\Lambda
0}}$ (17)

where, $\Omega_{\Lambda 0}=1-\Omega_{m0}-\Omega_{k0}$ and $H_{0}$ is the
Hubble constant.
In this model, we are left with three free parameters i.e. $H_{0}$,
$\Omega_{m0}$ and $\Omega_{k0}$. To reduce the number of free parameters, we
marginalise over $H_{0}$ and using Hubble parameter measurements we put
constraints on $\Omega_{m0}$ and $\Omega_{k0}$ by minimising the Chi-square.
The best fit values of the parameters obtained from Hubble parameter
measurements are given in Table 4.2.

The best fit values of $\Omega_{m0}$, $\Omega_{k0}$ and $\Omega_{\Lambda 0}$
with $1\sigma$ confidence level obtained by using 32 $H(z)$ measurements.
Parameter Best fit value [$1\sigma$ C.L.] $\Omega_{m0}$
$0.380^{+0.318}_{-0.375}$ $\Omega_{k0}$ $-0.186^{+1.137}_{-0.864}$
$\Omega_{\Lambda 0}$ $0.806^{+0.544}_{-0.806}$

The best fit value of $\Omega_{k0}=-0.186^{+1.137}_{-0.864}$ as shown in Table
4.2 suggests a spatially flat universe at $1\sigma$ confidence level which is
in good agreement with Planck Collaboration et al. (2020) [12].

Figure 7: The cosmic triangle plot among
$\log_{10}(\Omega_{m0}),~{}-\log_{10}(2\Omega_{\Lambda 0})$ and
$3\log_{10}(1+z_{t})$ with their $1\sigma$ confidence level, where the red dot
denotes the best fit value of the parameters.

For the transition epoch, we equate the deceleration parameter $q(z)$ to zero
at redshift $z_{t}$. Then using the Friedmann equation, the value of the
transition redshift in terms of cosmological parameters is given by

$z_{t}=\left(\dfrac{2\Omega_{\Lambda
0}}{\Omega_{m0}}\right)^{1/3}-1,~{}~{}~{}~{}\text{where},~{}~{}\Omega_{m0}+\Omega_{k0}+\Omega_{\Lambda
0}=1$ (18)

Thus using the constraints of $\Omega_{m0}$, $\Omega_{k0}$ and
$\Omega_{\Lambda 0}$, the best fit value of transition redshift comes out to
be $z_{t}=0.619^{+0.580}_{-0.758}$. To plot a triangle plot we require a
linear relation among its three components.Taking the logarithm of equation
(18):

$\log_{10}(\Omega_{m0})-\log_{10}(2\Omega_{\Lambda 0})+3\log_{10}(1+z_{t})=0$
(19)

Using this linear relation, we reconstruct a triangle plot among
$\log_{10}(\Omega_{m0}),~{}\log_{10}(2\Omega_{\Lambda 0})$ and
$3\log_{10}(1+z_{t})$ as shown in Figure 7.

We re-plot the cosmic triangle diagram as shown in Figure 8 by using $128$
simulated datapoints of Hubble parameter. We find the transition redshift as
$z_{t}=0.561^{+0.332}_{-0.379}$ and as expected the contour has shrunk
substantially.

Figure 8: The cosmic triangle diagrams using $128$ simulated Hubble parameter
datapoints, where the red dot denotes the best fit value of the parameters.

The summary of different values of transition redshift obtained from different
surveys is shown in Table 4.2.

Constraints on $z_{t}$ from different cosmological observations.  Dataset
Methodology Best fit value of $z_{t}$ Reference BAO, CMB, SNe Kinematic:
$\omega(z)$ ${\displaystyle\sim}$0.7 [76] Age of Galaxies, Strong Lensing, SNe
Kinematic: $q(z)$ <1 [30] H(z): CC $\Lambda$CDM Model $0.64^{+0.11}_{-0.07}$
[22] H(z): CC+BAO $\Lambda$CDM Model $0.72^{+0.05}_{-0.05}$ [23] H(z): CC+BAO,
SNe Kinematic Quadratic $H(z)$ $0.870^{+0.063}_{-0.063}$ [32] Planck-2018
$\Lambda$CDM Model $0.632^{+0.018}_{-0.018}$ [12] SNe: Pantheon Gaussian
Process $0.683^{+0.110}_{-0.082}$ [51] H(z): CC+BAO Gaussian Process
$0.59^{+0.12}_{-0.11}$ [49] H(z): CC+BAO, SNe $\Lambda$CDM Model
$0.69^{+0.25}_{-0.25}$ [33] H(z): CC+BAO Gaussian Process
$0.637^{+0.165}_{-0.175}$ [77] Updated H(z): CC Observed HPSP
$\mathbf{0.591^{+0.332}_{-0.332}}$ This work Simulated H(z): CC Simulated HPSP
$\mathbf{0.574^{+0.193}_{-0.193}}$ This work Updated H(z): CC Observed Cosmic
Triangle $\mathbf{0.619^{+0.580}_{-0.758}}$ This work Simulated H(z): CC
Simulated Cosmic Triangle $\mathbf{0.561^{+0.332}_{-0.379}}$ This work SNe:
Pantheon Observed HPSP $\mathbf{0.848^{+0.117}_{-0.117}}$ This work

##  5 Discussion and Conclusions

That the universe is in a phase of accelerated expansion is now well
established. Given this, it is an interesting exercise to determine the epoch
of the transition from a decelerated phase to the accelerated phase. In this
work, we use the Pantheon database and recent data for the Hubble parameter
and full covariance matrix for systematic uncertainties to put constraints on
the transition redshift both in a model-independent and model-dependent way.
The main conclusions are as follows:

We consider the observed value of the Hubble parameter measurements and Type
Ia SNe observations to draw a phase space $(H,~{}\dot{H})$ portrait. Our main
conclusions are listed below-

  1. 1.

We emphasize that the analysis with HPSP provides a way to estimate the
transition redshift as well as the current value of the equation of state
parameter without solving any equations for these cosmological parameters.

  2. 2.

In the HPSP, the reconstructed best fit line lies in the matter-dominated era
at higher redshift, which indicates that the universe was in decelerated
phase. Further, as expected, the best fit line crosses the transition boundary
from deceleration to acceleration at a redshift, $z_{t}$. This characteristic
of HPSP is in agreement with the standard model of cosmology. The
reconstructed best fit line in HPSP is in agreement with the thermal history
of the universe and can be used to estimate the transition redshift. The
result obtained using HPSP is also in agreement with the results obtained from
different observables as listed in Table 4.2.

  3. 3.

From HPSP, the constraint obtained on the present value of the equation of
state parameter $(\omega_{0})$ shows agreement with the Planck Collaboration
et al. (2020) [12].

  4. 4.

Using the mock datapoints of Hubble parameter measurements, we reconstruct the
HPSP and estimate the transition redshift value. This result is highly
consistent with the results obtained from 32 datapoints of Hubble parameter
measurements.

Finally, in order to check the value of the transition redshift by using
model-dependent Cosmic Triangle technique, we consider a non-flat $\Lambda$CDM
model as a background cosmological model. By marginalizing over $H_{0}$ and
minimizing the Chi-square, we put constraints on cosmological parameters i.e.
$z_{t}$, $\Omega_{m0}$, and $\Omega_{\Lambda 0}$. Further, using $z_{t}$,
$\Omega_{m0}$, and $\Omega_{\Lambda 0}$, we reconstruct the cosmic triangle
which allows us to visualize the confidence region in an intuitive and
illustrative way. The best fit value of the cosmic curvature parameter
$(\Omega_{k0})$ suggests an closed universe but a spatially flat universe can
be accommodated at $1\sigma$ confidence level. Thus our results are compatible
with the Planck Collaboration et al. (2020) at $1\sigma$ confidence level
[12]. Using an updated cosmic triangle plot among $\log_{10}(\Omega_{m0})$,
$-\log_{10}(2\Omega_{\Lambda 0})$ and $3\log_{10}(1+z_{t})$ we put constraints
on the transition redshift, which is in agreement with the the result obtained
from the different observables as listed in Table 4.2. We generate a mock
dataset of $128$ datapoints by adopting a fiducial flat $\Lambda$CDM model.
Using this simulated dataset, we find that the results are consistent with the
Planck Collaboration et al. (2020) [12]. As expected, with the increase in the
number of datapoints, the confidence region between these parameters reduces
substantially. For the completeness, we have performed the analysis of
Gaussian process by using six different kernel functions. The value of the
transition redshift obtained from the HPSP diagram with different kernel
functions is tabulated in Table 1.

Kernel Function | Best Fit Value of $z_{t}$ [68% CL]
---|---
Square Exponential | $\mathbf{0.591\pm 0.332}$
Matérn 3/2 | $0.574\pm 0.766$
Matérn 5/2 | $0.578\pm 0.433$
Matérn 7/2 | $0.581\pm 0.380$
Matérn 9/2 | $0.588\pm 0.361$
Cauchy | $0.571\pm 0.377$
Table 1:  The best fit value of $z_{t}$ from Hubble Phase Space Portrait
Diagram with six different GP kernel functions.

From Table 1, we can conclude that the results are not very sensitive to the
choice of the kernel function.

## Acknowledgements

We would like to thank the referee for making some very valuable suggestions.
Authors are grateful to Michele Moresco and Eoin Ó Colgáin for their
constructive and useful suggestions that helped to improve the work. Darshan
thanks Jose Luis Bernal for valuable discussion. Darshan is supported by an
INSPIRE Fellowship under the reference number: IF180293 [SRF], DST India.
Darshan acknowledges the facilities provided by the IUCAA Centre for Astronomy
Research and Development (ICARD), University of Delhi. This research work made
use of the free Python packages Numpy [78], Matplotlib [79], GaPP [48], and
Python-Ternary [80].

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