text:
\section{Introduction}

\begin{figure*}
\centering
\includegraphics*[width=9cm,angle=-90]{F1.ps}
\caption{The block diagram of Alpha-tag source with the Alpha Module
and the electronics for generating Alpha pulse output for tagging the
X-rays.}
\label{fig1}
\end{figure*}

The characterization and measurement of X-ray emission from the astrophysical
sources require well calibrated X-ray detectors. Parameterizing the energy
scale and monitoring the variations in the performance of X-ray detectors
can be done by using an internal calibration source which emits continuous
calibration lines at certain energies. The overall performance can further
be verified by analyzing the data recorded from the observations of any
standard X-ray source in the sky. One commonly used radio active source
is Americium-241 (Am$^{241}$), which  provides a continuous source of
calibration lines with energies between 13 and 60 keV. For example,
the calibration of the Proportional Counter Arrays (PCA) onboard the
Rossi X-ray Timing Explorer (RXTE) is done using a  Am$^{241}$ source
embedded in a proportional counter [1,2]. Similarly, a tagged Am$^{241}$
source is used to calibrate and stabilize the gain in the High Energy
X-ray Timing Explorer (HEXTE) of the RXTE satellite [3,4]. The
calibration source establishes the energy scale and its variation
with time and other parameters along with the relative variation of
the efficiency. The broadband performance and the flux calibration,
however, is usually done by observing a few standard X-ray sources in
sky such as Crab Nebula  which provide a bright stable broadband source
with a well-characterized spectrum.

Americium is a human-made radio active element with atomic number of 95.
Americium-241 (Am$^{241}$) can be produced by bombarding plutonium-234
with alpha particles. Am$^{241}$, with a half-life of 432.2 years, decays
primarily by alpha particle emission (at 5485 keV - 84.5\% and 5443 keV -
13\%) to Neptunium-237 which has  a half-life of 2.144$\times$10$^{6}$
years. These decays are accompanied by low energy gamma radiation with
the 59.5 keV gamma emission (35.9\%) being the most important one along
with 26.35 keV (2.4\%) and 13.9 keV (42\%) emission. The radio active
source Am$^{241}$ is commonly used for testing and/or calibrating
the majority of hard X-ray detectors specifically because of the presence
of the prominent 59.54 keV X-ray emission.

To detect the alpha particles simultaneous to the X-ray emission, the
Am$^{241}$ source is embedded in a radiation detector which detects the
alpha particles but is transparent to the X-rays. It can be done by using
proportional counters (as was done for the PCA detectors [2]) or thin
scintillators viewed by a Photo-multiplier tube (PMT), as was done
for HEXTE [3]. Both these methods, however, require the generation and
use of high voltages (a few hundred volts) which is cumbersome and heavy.
For the Astrosat satellite [5], a large area hard X-ray detector of area
1000 cm$^2$ using an array of Cadmium Zinc Telluride (CZT) detectors is
being developed with a overall mass budget of 50 kg. To calibrate these
detectors (called CZT-Imager or CZTI), we have developed a compact low
mass alpha-tagged X-ray source.

In this paper, we present the design of a new method of using Am$^{241}$
as an alpha-tagged X-ray source and discuss its efficiency as a
calibration source for the CZTI of the Astrosat satellite.

\section{Design of the alpha-tag X-ray source }

We have designed a very simple compact system for detecting the
alpha particle coming from the Am$^{241}$ radio active source. In
addition to the detection of alpha particles, the system also transmits
the 59.54 keV X-ray photons emitted by Am$^{241}$ which can be used for
the purpose of calibration of any radiation detector. In this detector
system, the principal components used are (i) the radio active source
Am$^{241}$ which emits the alpha particles as well as the 59.54 keV
X-rays, (ii) a CsI(TI) crystal which detects the alpha particles and
also transmits the 59.54 keV X-rays, (iii) a Si-PIN photo-diode for the
detection of the light output from the CsI(TI) crystal (due to the
interaction of the alpha particle) iv) a charge sensitive pre-amplifier
(CSPA) mounted right behind the  photo-diode. This whole system, called
the Alpha Module, is light weight ($\sim$3 gm) and requires only
a low voltage supply (+12 V) with a power dissipation of 130 mW.

\begin{figure}
\centering
\includegraphics*[width=5.5cm,angle=-90]{F2.ps}
\caption{The pulse height distribution from the Alpha-tag source.
The peak corresponds to the 5.5 MeV alpha particles.}
\label{fig2}
\end{figure}

\begin{figure*}
\centering
\includegraphics*[width=10cm,angle=-90]{F3.ps}
\caption{The experimental set-up to tests the alpha tagging efficiency of the
the Alpha-tag source.}
\label{fig3}
\end{figure*}

The Alpha Module (see Figure~\ref{fig1}) is a small 10$\times$10$\times$0.3
mm$^3$ of CsI (Tl) crystal (wavelength = 550 $nm$) with Teflon reflective
material coupled to a Si-PIN photo-diode of the same area. It contains a
small pallet of Am$^{241}$ with a low strength of about 2000 Bq which is
mounted on a 0.15 mm steel template. This radioactive source is kept in
contact with a 0.3 mm CsI(Tl) crystal that detects the alpha particles and
also transmits the 59.54 keV X-rays with a transparency of 35\%. The Si-PIN
photo-diode  detects the light output from the CsI(Tl) crystal (due to the
interaction of the alpha particle). This photo-diode has a broadband
response in visible spectrum (average efficiency of 50\%) with higher
sensitivity and low dark current. Incident energy from the alpha particle
is converted into light in CsI(Tl) whereas the light seen by the
photo-diode is converted into an electrical pulse, which is amplified by
the charge sensitive pre-amplifier (CSPA). The CsI(Tl) along with
photo-diode and CSPA, as a whole structure, is called  the Alpha Module,
which is rugged, compact, reliable and does not need a high voltage supply
for biasing purposes. This whole compact system is procured from M/s Scionix,
Holand. Since the photo-diode output impedance is high (10 Mega ohms) and
signal level is of the order of a few milli volts, a low noise CSPA is used
to process the signal. This CSPA is placed just behind the photo-diode, so
as to reduce noise. The pre-amplifier is wired on a 10$\times$10 mm$^2$
PCB (printed circuit board) and it is an integral part of the Alpha Module.

We have designed a simple electronics circuit to read out the pulses from
the Alpha Module. The signal from the CSPA is passed through a pre-amplifier
circuit with appropriate integration to achieve bipolarity with a gain of
900. The sharp pulses (rise time ~0.5 $\mu$s) resulting from the detection
of ionizing radiation  (signal from the CsI(Tl) crystal through the
photo-diode) are processed in the amplifier to have suitable shape and
amplification. This amplified signal is passed through a comparator having
a fixed LLD (lower-level discriminator) bias to cut off noise. This whole
scheme is shown in Figure 1, along with the Alpha Module. The module can be
used as a source of X-rays from Am$^{241}$ and the Alpha pulse output from
this device can be used as a tag for the X-rays. The Alpha Module, mounted
on a PCB, along with the electronics, is called the Alpha-tag source and it
is used in the CZTI payload of Astrosat for in-flight calibration.

The amplifier output is fed to a Multi Channel Analyzer (MCA) and the
results are shown in Figure~\ref{fig2}. A peak corresponding to the 5.5 MeV
alpha particles is clearly seen in the figure. A count rate corresponding to
720 counts per second is noted.

\section{X-rays from the Alpha-tag source and the alpha-tagging methodology }

We made an experimental set up to study the alpha-tagged X-ray source
characteristics. Our experiment consists of a NaI(Tl)/CsI(Na) Phoswich
scintillator detector on the top of which the Alpha-tag source was mounted.
The  scintillators are coupled to a  PMT which multiplies  the current
produced by the incident light by as much as 100 million times (i.e. 160 dB),
in multiple dynode stages, enabling individual photons to be detected. The
current signal from PMT for every incident energy is converted to voltage
and then passed  through an amplifier having a gain of around 20.

The alpha particles from the Americium source in the Alpha Module generates
pulses in the Alpha-tag source electronics (the circuit diagram of which is
shown in Figure~1 along with the Alpha Module) and the 59.54 keV X-rays, if
they interact with the Phoswich detector, generates signal in the PMT
electronics. The signal from PMT was then gated with the Alpha pulse and
plotted on computer using a pulse height analyzer (PHA) software. It was
found that there was a delay of approximately 5 to  15 $\mu$s in the Alpha
pulse with respect to the X-ray pulse, due to the different electronics
characteristics. This delay was accommodated by adding a peak detector
circuit in the path of the PMT signal. The  amplified signal was then
connected directly to a MCA and/or an oscilloscope. The pocket MCA (MCA
8000A from AMPTEK) was used in the experiment, which was again connected
to a computer for data acquisition and storage. The experimental set-up is
shown in Figure~\ref{fig3}.

\begin{figure}
\centering
\includegraphics*[width=5.5cm,angle=-90]{F4.ps}
\caption{The background spectrum from the Phoswich detector.}
\label{fig4}
\end{figure}

\begin{figure}
\centering
\includegraphics*[width=5.5cm,angle=-90]{F5.ps}
\caption{The spectrum from the Phoswich detector with the alpha-source
kept on it.}
\label{fig5}
\end{figure}

To reduce external noise, necessary care was taken to block extraneous
light entering the PMT. A thick lead sheet with a small hole at the center
was used as a collimator on the top of the Phoswich detector to block
X-rays coming from other directions to reduce the  background. An
arrangement was also made to block the X-rays from the Alpha-tag source
so that background could be measured from the Phoswich detectors without
actually removing the  Alpha-tag source. With this set-up, the data were recorded by blocking the X-rays from the alpha source to measure the ambient
background. The MCA was calibrated by shining a bright Am$^{241}$ source
directly onto the Phoswich detector and measuring the peak corresponding
to the 60 keV X-rays. The Alpha-tag source was then kept on the Phoswich
set-up and the spectrum  was taken by repeating the above procedure. Excess
counts were seen near the 60 keV peak and we note a count rate of  23
counts/s due to the Am$^{241}$ source. The 2000 Bq source gives about 720
count/s and (we estimate about 50\% efficiency for the detection of alpha
particles in the CsI detector, based on the fact that the Am$^{241}$ source
is mounted on a steel template). The expected number of 60 keV counts are
reduced by a fraction  $\alpha$ where $\alpha = \epsilon \Omega/(2 \pi) t $
where $\epsilon$ is the fraction of 60 keV X-rays emitted for each alpha
particle ($\sim$0.35), $\Omega$/(2 $\pi$) is the fractional solid angle
subtended by the Phoswich detector to the alpha source ($\sim$0.3) and
$t$ is the transmission of  CsI detector for the 60 keV X-rays ($\sim$0.35)
giving $\alpha$ to be 0.036 which agrees with the observed number of X-rays.

For the background estimation, the alpha-tag source was kept out of the
experimental set-up and necessary care was taken to avoid any X-ray photons
from the alpha-tag source entering the PMT. The photon spectra were accumulated
for 100 seconds of exposure and collected using the MCA. This count spectrum
are considered as the background spectrum. This is shown in Figure~\ref{fig4}.
After collecting the background spectra, the alpha-tag source was kept on the
Phoswich set-up and the corresponding spectra were taken. This  photon spectra
are shown in Figure~\ref{fig5}.

\begin{figure}
\centering
\includegraphics*[width=5.5cm,angle=-90]{F6.ps}
\caption{The Phoswich detector output gated with Alpha-tag pulse.}
\label{fig6}
\end{figure}

Following these observations, the Alpha-tag detector was placed on the PMT
with the necessary experimental connections as shown above (Figure~\ref{fig3}).
The observations were repeated several times and the count spectrum for one
such observation is shown in Figure~\ref{fig6}. We can see that the X-ray
peak is enhanced considerably. Comparing the 80 -- 120 keV count rates in
Figure~5 and Figure~6, we estimate that less than 10\% of background
X-rays are registered due to the tagging process. The total counts in
$<$ 80 keV region in Figure 6 is 27 s$^{-1}$. If we assume that the
background in this region too is reduced by 90\%, we can estimate that
by the alpha-tagging method, more than 95\% of the 60 keV X-rays impinging
on the Phoswich detector are registered. This is somewhat higher than the
50\% estimated earlier for the detection of alpha particles in the CsI
detector. This discrepancy could be the result of a) the pallet of
Am$^{241}$ could have a shape which enables a large fraction of alpha
particles to impinge on the CsI crystal and/or b) the reduction of background
in the $<$ 80 keV region being higher than 10\% due to some threshold
effects. This experiment is done at somewhat higher background conditions
(a total background of 177 s$^{-1}$). This alpha-tagged X-ray calibration
source is proposed to be used in the Astrosat satellite where the onboard
background is expected to be only about 20 counts/s and hence negligible
leakage is expected.

\section{Conclusions}

        We have developed a new alpha-tagged X-ray source for
space application which is rugged, cheap and easy to operate.
Though the count rate is low (23 counts/s), it is sufficient to make
onboard calibration with an integration time of about 100 s. Currently,
the leakage of background counts is about 10\%. This would be improved
in a low background environment and better tuning of the gate-width for
the alpha detection. The alpha-tagging method selects only the
calibration X-rays along with negligible contribution from the
background, thus enabling the calibration with a low count rate
source. Further, even if a small fraction of these X-rays enter the
detector without getting tagged (a few count per second), they
will have negligible impact on the overall background rates.

\section*{\bf Acknowledgment}

We thank Dr. K. Jahoda and an anonymous referee for useful suggestions
which improved the presentation of the paper. SN and RPSKK would like
to acknowledge the hospitality provided by the Department of Astronomy
and Astrophysics, Tata Institute of Fundamental Research during their
visit to carry out the present work. The research work at Physical
Research Laboratory  is funded by the Department of Space, Government
of India.


text:
\section{Introduction}
\label{sec:intro}

The standard cosmological model, $\Lambda$CDM, is parametrically the best model of the Universe \cite{Peebles:2002gy,Copeland:2006wr}, providing a reliable picture of the evolution \cite{Riess:1998cb,Perlmutter:1998np} and the growth and dynamics of matter in it \cite{Guth:1980zm,Linde:1981mu, Baudis:2016qwx,Bertone:2004pz}. However, it glosses over several theoretical issues such as the fine tuning of the cosmological constant \cite{RevModPhys.61.1} and the coincidence \cite{Dalal:2001dt} problem, and recently it was recognized that it caters a tension in its cosmological parameters \cite{DiValentino:2020vhf,DiValentino:2020zio,DiValentino:2020vvd}. The Hubble tension \cite{Schoneberg:2021qvd} -- a three to five sigma deviation between the local distance anchor measurements \cite{Riess:2021jrx} and cosmic microwave background \cite{Aghanim:2018eyx} estimates of the Hubble constant -- raises serious considerations to noncanonical statistical methods \cite{Escamilla-Rivera:2020fxq, Briffa:2020qli, LeviSaid:2021yat, Mukherjee:2022yyq, Benisty:2022psx, Grandon:2022gdr}, dynamical dark energy \cite{Cardenas:2014jya, Grandon:2021nls, Bernardo:2021cxi}, and even cosmological models with additional degrees of freedom in the matter and gravitational sectors \cite{DiValentino:2019ffd, Peirone:2019aua, Frusciante:2018jzw, Capozziello:2011et, Bahamonde:2019shr, Bernardo:2021qhu}. We take the former route in this paper.

The Approximate Bayesian Computation (ABC) is a parameter estimation algorithm that brings forth the concept of natural evolution to the physical parameters in a model \cite{10.1093/genetics/162.4.2025, 2009arXiv0901.1925T}. The main novelty of this method is that it bypasses the concept of a likelihood, and rather instead depends on a distance function. This is a useful feature particularly in model selection where the likelihood between different models is often unknown or at best intractable. Together with a sequential Monte Carlo spin that lets only the finest of a population proceed to the next generation, the ABC becomes even more powerful, resulting to only an ever sharpening constraints on the models and parameters during the course of the evolution \cite{2009arXiv0901.1925T}. The algorithm has been utilized for model selection and parameter estimation in various scientific problems \cite{10.3389/fbuil.2017.00052, 10.2307/27896228, 10.1093/bioinformatics/bty361} and in astrophysics and cosmology \cite{2013ApJ...764..116W, Akeret:2015uha, Ishida:2015wla, 2017A&C....19...16J, Bernardo:2021mfs, Bernardo:2022pyz}.

In this work, we make use of the ABC to answer an interesting question in the background of the Hubble tension: ``Which Hubble constant is preferred by the present data, SH$0$ES \cite{Riess:2021jrx} or Planck \cite{Aghanim:2018eyx}?'' In this endeavor, we consider the standard model, $\Lambda$CDM, together with cosmological observations of the late Universe. We anchor $\Lambda$CDM on two Hubble constant priors that reflect the Hubble tension, and let the ABC tell which of the two resulting models is preferred by natural selection. This provides independent estimates of the cosmological parameters of the standard model, but more importantly openly views the Hubble tension in a new light. With such newfound statistical perspective, we find the results of the ABC even more appealing as one Hubble constant emerges throughout, regardless of the priors and the observations in use. We also briefly discuss the results of the traditional Markov chain Monte Carlo (MCMC) analysis \cite{Trotta:2008qt, Lewis:2019xzd, 2020arXiv200505290T} for the same data sets, setting a baseline for the comparison of ABC as a statistics tool for cosmology.

The rest of this paper is as follows. We provide an overview of the ABC (Section \ref{sec:abc}) and then a brief description of the data sets and how they were used in the analysis (Section \ref{sec:data_sets}). We then present our main results using MCMC and ABC (Section \ref{sec:results}) and draw our conclusions (Section \ref{sec:conclusions}).

\section{The ABC method}
\label{sec:abc}

Approximate Bayesian computation, in short ABC \cite{10.1093/genetics/162.4.2025}, is applicable to finding the posterior probability $P\left(\theta|D\right)$ for the parameters $\theta$ given the data $D$, by combining the prior $P(\theta)$ and observed data to train the model so that the constrained model parameters can be obtained based on Bayes' theorem. This overcomes the traditional MCMC based Bayesian inference, which relies pragmatically on the likelihood function $P(D|\theta)$ between the observation and data, which is often challenging or intractable especially for complex stochastic systems such as in population genetics. The ABC was produced to provide an alternative likelihood-free route of statistically analyzing systems, thereby simplifying the process and improving its efficiency, and making it applicable to a wider range of scientific problems such as in biology \cite{2009arXiv0901.1925T, 10.3389/fbuil.2017.00052, 10.2307/27896228, 10.1093/bioinformatics/bty361} and physics \cite{2013ApJ...764..116W, Akeret:2015uha, Ishida:2015wla, 2017A&C....19...16J, Bernardo:2021mfs, Bernardo:2022pyz}.

The main idea behind ABC is to approximate the posterior, $P(\theta|D) \propto f\left(D^*, D\right) P(\theta)$, by building on distance function $f\left(X, Y\right)$ measurements of a population, where each member is drawn based on the model, providing its own simulated data $D^*$. Through large population sizes and an acceptance criterion, $f\left(D^*, D\right) < \epsilon$ for some sufficiently small $\epsilon$, the ABC is able to obtain a competitive estimate of the model parameters that rival those from MCMC. The algorithm receives an even more significant power up when given a sequential Monte Carlo (SMC) twist \cite{2009arXiv0901.1925T}. This makes an overall process that estimates the parameters in an inevitably tightening sequence of generations. Since the ABC is likelihood free, all these can be put together in the larger space that covers even models as statistical points, that is a bird eye view that sees models with their own probability, while also measuring their parameters. As a consequence, the resulting ABC-SMC algorithm is a natural model selection and parameter estimation statistical tool. Needless to say, it is an attractive one, free from the restrictions that a likelihood incorporates into the analysis. For this reason it is understandably applied to a variety of problems transcending scientific fields \cite{10.1093/genetics/162.4.2025, 10.2307/27896228, 10.1093/bioinformatics/bty361, 2013ApJ...764..116W, Akeret:2015uha, Ishida:2015wla, 2017A&C....19...16J, Bernardo:2021mfs, Bernardo:2022pyz}.

Building on several work that has now used ABC in astrophysics and cosmology, we want to see ABC's take on the Hubble tension in the standard cosmological model by making use of two values that represent it as a parameter prior. To accomplish this, we utilize a python package `pyABC' \cite{pyabc}, which is known for its ABC-SMC feature, allowing model selection and parameter estimation by simply inputting the models, observed data, the parameter priors, and a distance function, taking the place of the likelihood in order to sequentially approximate the posterior throughout the evolution. In the end, we have only one model we deem naturally selected.

\section{Cosmological data sets}
\label{sec:data_sets}

We consider background cosmological data from cosmic chronometers \cite{Moresco:2016mzx, Moresco:2015cya, 2014RAA....14.1221Z, 2010JCAP...02..008S, 2012JCAP...08..006M,Ratsimbazafy:2017vga}, supernovae \cite{Scolnic:2021amr, Brout:2022vxf, Brout:2021mpj}, and baryon acoustic oscillations \cite{BOSS:2014hwf, 2012MNRAS.425..405B, Chuang:2013hya, BOSS:2013igd, Bautista:2017zgn, Gaz34, Oka37, Wang33, Chuang28, Alam38, Anderson32, Busca36}. The expansion rate data is compiled in the Appendix of \cite{Bernardo:2021cxi} for convenience. We also consider two independent Hubble constant measurements that represent the Hubble tension \cite{Riess:2021jrx, Aghanim:2018eyx}. In addition, we take in the growth data from redshift space distortions for cosmological perturbations compiled in \cite{Kazantzidis:2018rnb}. We provide a brief description of each of these data and how we used it for our analysis below.

\begin{itemize}
\item Cosmic chronometers (CC). The cosmic chronometers which act as standard clocks of our universe provide a cosmology model independent way of measuring the expansion rate and the Hubble constant. This is based on observations of the age and metalicity of temporally adjacent passive galaxies such that the expansion rate at a redshift $z_{\rm CC}$ can be approximated by $H_{\rm CC}(z) \approx -\left(\Delta z_{\rm CC}/\Delta t\right)/\left(1 + z_{\rm CC}\right)$. CC observations at different redshifts therefore give a direct measurement of the expansion history which we use to constrain cosmology. We consider the CC in the redshift range $0.07 \lesssim z \lesssim 1.97$ from various sources \cite{Moresco:2016mzx, Moresco:2015cya, 2014RAA....14.1221Z, 2010JCAP...02..008S, 2012JCAP...08..006M,Ratsimbazafy:2017vga} that is compiled in \cite{Bernardo:2021cxi} for a quick reference. In this work, we consider the standard $\Lambda$CDM cosmological model in which the expansion rate is provided as
\begin{equation}
    E\left(z \right)^2 = \Omega_{m0} \left(1 + z \right)^3 + \left(1 - \Omega_{m0}\right)
\end{equation}
where $E(z) = H(z)/H_0$ is the normalized Hubble expansion rate at redshift $z$, $H_0$ is the Hubble constant, and $\Omega_{m0}$ is the fraction of nonrelativistic matter at redshift $z = 0$. We compare the predicted expansion rate with the CC at different redshifts by the chi-squared distance
\begin{equation}
\chi^2_{\rm CC} = \sum_{z_{\rm CC}} \left( \dfrac{H\left(z_{\rm CC}\right) - H_{\rm CC}\left(z_{\rm CC}\right)}{\sigma_{\rm CC}\left(z_{\rm CC}\right)} \right)^2 \,,
\end{equation}
where the quantities with the subscript CC refer to the CC measurements of the expansion rate, for example, at redshift $z_{\rm CC}$ the expansion rate is $H_{\rm CC}\left(z_{\rm CC}\right) \pm \sigma_{\rm CC}\left(z_{\rm CC}\right)$. We consider uncorrelated CC observations.

\item Supernovae (SNe). We consider supernovae type Ia, so called standard candles of the Universe as they are intrinsically very bright that they serve as cosmic lampposts that can be used to estimate the distance from the observer. Flux measurements of SNe supplemented with redshift observations thus provide a measure of the expansion rate. We understand this in terms of the distance redshift ladder relation,
\begin{equation}
    \mu(z) - M = 5 \log_{10}\left( d_L\left(z\right) \right) \,,
\end{equation}
where $d_L(z)$ is the luminosity distance given by
\begin{equation}
    d_L(z) = \dfrac{c}{H_0} \left(1 + z\right) \int_0^z \dfrac{dz'}{E(z')} \,.
\end{equation}
We use the data set released by the Pantheon$+$ team \cite{Scolnic:2021amr, Brout:2022vxf, Brout:2021mpj} where there are $1701$ data points from type Ia supernovae. This gives apparent magnitude observations at redshift ranges $0.01 \lesssim z \lesssim 2.3$ together with a covariance matrix to take into account systematic influences. We compare the model predictions with SNe observations by treating the chi-squared measure as a distance function
\begin{equation}
    \chi^2_{\rm SNe} = \sum_{z_{\rm SNe}} \left( \mu(z_{\rm SNe}) - \mu_{\rm SNe}(z_{\rm SNe}) \right) C^{-1}_{\rm SNe} \left( \mu(z_{\rm SNe}) - \mu_{\rm SNe}(z_{\rm SNe}) \right) \,,
\end{equation}
where subscripts SNe stand for the Pantheon$+$ supernovae apparent magnitude measurements $\mu_{\rm SNe}$ with the covariance matrix $C_{\rm SNe}$.

\item Baryon acoustic oscillations (BAO).
The baryon acoustic oscillations, which represent acoustic fluctuations in the early Universe, serve as standard rulers in the sky that is the distance the wave travels in the plasma before recombination. However, at a redshift $z$, this only directly measures the parameter combination $r_{\rm D}H(z)$ where the sound horizon radius $r_{\rm D}$ of the drag epoch is a cosmology independent quantity. Nonetheless since we are only concerned with $\Lambda$CDM this is given by $r_{\rm D} = 147.74$ Mpc, which then provides indirectly BAO measurements of the expansion rate at various redshifts \cite{BOSS:2014hwf, 2012MNRAS.425..405B, Chuang:2013hya, BOSS:2013igd, Bautista:2017zgn, Gaz34, Oka37, Wang33, Chuang28, Alam38, Anderson32, Busca36}. This supplements $26$ more data points to the cosmological expansion rate in the redshifts between $0.24 < z < 2.36$ (summarized in \cite{Bernardo:2021cxi}). As with CC, we measure the deviation of our predictions from observation by means of the distance function
\begin{equation}
\chi^2_{\rm BAO} = \sum_{z_{\rm BAO}} \left( \dfrac{H\left(z_{\rm BAO}\right) - H_{\rm BAO}\left(z_{\rm BAO}\right)}{\sigma_{\rm BAO}\left(z_{\rm BAO}\right)} \right)^2 \,,
\end{equation}
where the BAO measurements are given as $H_{\rm BAO}\left(z_{\rm BAO}\right) \pm \sigma_{\rm BAO}\left(z_{\rm BAO}\right)$, after dividing by the drag sound horizon radius.

\item Hubble constant measurements. In the background of the Hubble tension, we make use of the Hubble constant estimates by the SH$0$ES ($H_0^\text{R22} = 73.30 \pm 1.04$ km s$^{-1}$ Mpc$^{-1}$ \cite{Riess:2021jrx}) and Planck ($H_0^\text{P18} = 67.4 \pm 0.5$ km s$^{-1}$ Mpc$^{-1}$ \cite{Aghanim:2018eyx}) collaborations. We take these values as priors in the parameter space of $\Lambda$CDM, thus effectively organizing two models for the ABC to differentiate.

\item Redshift space distortions (RSD). RSD enables measurements of anisotropies in a galaxy distribution often due to the differences between galaxies peculiar velocities and the Hubble flow. This contains vital information about the growth of structure and consequently a blueprint of the formation and evolution of the perturbation, $\delta(z)$. This data is reported in terms of $f \sigma_8(z)$, which is a product of the growth rate $f(z) = d \ln \delta (z)/d \ln a(z)$ and the matter power spectrum $\sigma_8$ on scales of $8 h^{-1} {\rm Mpc}$. In standard cosmology, the matter density perturbation can be expressed analytically as  \cite{Nesseris:2015fqa}
\begin{equation}
\label{eq:matterpert_lcdm}
    \delta(a) = a \,_2F_1\left( \dfrac{1}{3},1, \dfrac{11}{6}; a^3 \left(1 - \Omega_{m0}^{-1}\right)  \right) \,.
\end{equation}
For a variable dark energy equation of state, a more general analytical expression is also available  \cite{Nesseris:2015fqa}. With this, the growth rate can be expressed as \cite{LeviSaid:2021yat}
\begin{equation}
\label{eq:growth_lcdm}
    f \sigma_8(z) = - \sigma_{8} (1 + z) \dfrac{\delta'(z)}{\delta(z = 0)} \,.
\end{equation}
We compare this prediction with the RSD measurements compiled in \cite{Kazantzidis:2018rnb} by considering the distance function
\begin{equation}
    \chi^2_{\rm RSD} = \sum_{z_{\rm RSD}} \left( f \sigma_8(z_{\rm RSD}) - f \sigma_{8, {\rm RSD}}(z_{\rm RSD}) \right) C^{-1}_{\rm RSD} \left( f \sigma_8(z_{\rm RSD}) - f \sigma_{8, {\rm RSD}}(z_{\rm RSD}) \right) \,,
\end{equation}
where the subscripts RSD stand for the observed values, and $C_{\rm RSD}$ is the covariance matrix.
\end{itemize}

In the next section, we use MCMC and ABC together with three splices of the full data [(i) SNe $+$ CC (ii) SNe $+$ CC $+$ BAO (iii) SNe $+$ CC $+$ BAO $+$ RSD] to see how the results are affected, if it does, when considering the differing measurements.

\section{Results}
\label{sec:results}

We start by recollecting the two models we intend to compare and briefly discuss the MCMC results in Section \ref{subsec:mcmc}. We present the results of the ABC with the full and subsets of the data in Sections \ref{subsec:sne_cc}, \ref{subsec:sne_cc_bao}, and \ref{subsec:sne_cc_bao_rsd}.

\subsection{MCMC: P18 and R22}
\label{subsec:mcmc}

We emphasize that we consider only one cosmological model ($\Lambda$CDM) throughout. From this, we build two competing models by taking into account the Planck and SH$0$ES measurements of the Hubble constant as independent priors, while the rest of the cosmological parameters take initial flat priors. We refer to the models as P18 (with the prior $H_0^{\rm P18}$) and R22 (with the prior $H_0^{\rm R22}$) for brevity.

To set a baseline for the ABC, we first look at the same models and data sets from the perspective of the traditional MCMC analysis. This brings us to the posterior shown in Figure \ref{fig:mcmctriangle} obtained using the full (background and perturbation) data set and Table \ref{tab:mcmc} which shows the constraints anchored on the full and subsets of the data. These results were obtained with a reasonable Gelman-Rubin convergence criterion of $R - 1 = 10^{-2}$.

\begin{figure}[h!]
    \centering
    \includegraphics[width = 0.5 \textwidth]{figs/Triangles80.pdf}
    \caption{MCMC constraints on the $\Lambda$CDM cosmological parameters ($\Omega_{m0}, H_0, \sigma_8$) with the Hubble constant priors from the SH$0$ES ($H_0^\text{R22} = 73.30 \pm 1.04$ km s$^{-1}$ Mpc$^{-1}$ \cite{Riess:2021jrx}) and Planck ($H_0^\text{P18} = 67.4 \pm 0.5$ km s$^{-1}$ Mpc$^{-1}$ \cite{Aghanim:2018eyx}) teams.}
    \label{fig:mcmctriangle}
\end{figure}

Figure \ref{fig:mcmctriangle} explicitly shows the prior's influences on the background parameters ($H_0$ and $\Omega_{m0}$). This is pronounced for the R22 model, which takes in a high $H_0 \sim 73$ km s$^{-1}$Mpc$^{-1}$ prior, which is reflected in its posterior giving a slightly bimodal structure with a peak at $H_0 \sim 70$ km s$^{-1}$Mpc$^{-1}$ and a false maximum at $H_0 \sim 73$ km s$^{-1}$Mpc$^{-1}$. This understandably also translates to a low $\Omega_{m0} \sim 0.24$ peak in addition to the more reasonable matter density $\Omega_{m0} \sim 0.27$ since $H_0$ and $\Omega_{m0}$ are anti correlated through the high redshift CC points, $H(z)^2 \propto \Omega_{m0} h^2$ where $h = H_0/(100 \ {\rm km \ s}^{-1}{\rm Mpc}^{-1})$ \cite{Moresco:2016mzx}. A closer look at the P18 posterior on the Hubble constant also shows this, except in this case it is not too transparent as the prior turns out close to the estimate. It is worth noticing that the Hubble constant priors representative of the Hubble tension do not reflect as an independent tension in the matter power spectrum parameter $\sigma_8$, or in other words the $\sigma_8$ estimates significantly overlap (Figure \ref{fig:mcmctriangle}). This makes sense considering that the growth data fixes $\sigma_8$ as well as the matter density $\Omega_{m0}$ while the expansion history pins down $H_0$ and $\Omega_{m0}$. In Table \ref{tab:mcmc} we also take a look at the cosmological constraints when using MCMC and subsets of the full data.

\begin{table}[h!]
    \centering
    \caption{Cosmological constraints on the $\Lambda$CDM model given Hubble constant priors by the Planck (P18) \cite{Aghanim:2018eyx} and the SH$0$ES (R22) \cite{Riess:2021jrx} values, obtained via the traditional MCMC analysis.}
    \begin{tabular}{|c|c|c|c|c|}
    \hline
    Data set & \phantom{g} Model \phantom{g} & \phantom{g} $H_0$ [km s$^{-1}$Mpc$^{-1}$] \phantom{g} & \phantom{gggg} $\Omega_{m0}$ \phantom{gggg} & \phantom{gggg} $\sigma_8$ \phantom{gggg} \\ \hline \hline
    \multirow{2}{*}{SNe $+$ CC} &
    P18 & $67.4 \pm 0.5$ & $0.33 \pm 0.02$ & $-$ \\
    & R22 & $72.1^{+1.1}_{-0.9}$ & $0.31 \pm 0.02$ & $-$ \\ \hline
    \multirow{2}{*}{SNe $+$ CC $+$ BAO} &
    P18 & $67.6 \pm 0.4$ & $0.31 \pm 0.01$ & $-$ \\
    & R22 & $70.3^{+0.4}_{-1.0}$ & $0.27^{+0.02}_{-0.01}$ & $-$ \\ \hline
    \multirow{2}{*}{\phantom{g} SNe $+$ CC $+$ BAO $+$ RSD \phantom{g}} &
    P18 & $67.6^{+0.4}_{-0.5}$ & $0.31 \pm 0.01$ & $0.75 ^{+0.02}_{-0.01}$ \\
    & R22 & $70.7 ^{+2.5}_{-1.5}$ & $0.27^{+0.02}_{-0.01} $ & $0.78^{+0.03}_{-0.01}$ \\ \hline
    \end{tabular}
    \label{tab:mcmc}
\end{table}

An immediate observation is that without the BAO the matter density constraints turn out peculiarly high with P18. This is nonetheless consistent with the Planck value at $68\%$ confidence. The addition of the BAO can be seen to have made the matter sector more consistent with P18 \cite{Aghanim:2018eyx, DiValentino:2019ffd}. In terms of the $H_0$, it is clear that the BAO compels the Hubble constant to a lower value, consistent with other results. This is the case with both P18 and R22 models. However, with the BAO, we find the matter fraction estimate in R22 to be quite low, but notably still consistent with the Planck value at $95\%$ confidence. The inclusion of the RSD growth observations, adding cosmological perturbations in the mix, now better constrains the matter sector \cite{Kazantzidis:2018rnb}. Interestingly, we find the matter sector constraints to be consistent with and without the RSD as long as the BAO is considered. Having fixed the matter fraction, the RSD now is able to further constrain one more parameter, that is, the smoothed matter perturbation amplitude, $\sigma_8$. We find reasonable results for the matter sector including the perturbation. The Hubble constant value also remained consistent with the Planck measurement regardless of the prior used.

We find Hubble constants consistent with the Planck value when considering the expansion rate data with the BAO. Without the BAO, as a matter of fact, only the R22 model produces an $H_0$ that is in tension with the Planck estimate. This may be because the background data is inconsistent with the larger $H_0$ prior that it considers. Regardless, the addition of the BAO lowers $H_0$ while we see that the RSD fixes the matter sector in addition to the perturbations.

We lastly comment on which model, given a data set, MCMC prefers. This can be assessed using various metrics, with the Bayesian evidence being the more rigorous one. However, since we are dealing with only models with the same number of parameters, we may simply resort to a comparison of the likelihoods. With this in mind, we find that in all cases, the P18 model produces a better fit than R22. This is intriguing in light of the Hubble tension which sees a significant statistical deviation between local cosmological physics and the cosmic microwave background. We find this is supported by the ABC, this work's main theme, as we proceed to show.

\subsection{ABC: SNe + CC}
\label{subsec:sne_cc}

The base of our data set consists of brightness observations of supernovae and expansion rate measurements from cosmic chronometers. Both provides indirect and direct independent observations of the cosmological background. Figure \ref{fig:modelpost_SNeCC} shows the evolution of the model posterior and the Hubble constant in each model per generation.

\begin{figure}[h!]
    \centering
	\subfigure[]{
		\includegraphics[width = 0.48 \textwidth]{figs/ABCjoint_SNeCC.pdf}}
    \subfigure[]{
		\includegraphics[width = 0.48 \textwidth]{figs/ABCH0generational_SNeCC.pdf}}
    \caption{(a) Model posterior ABC evolution with SNe and CC and (b) the generational Hubble constant estimates. The pink and blue bands in (b) correspond to $68\%$ and $95\%$ confidence limits of the Planck ($H_0 = 67.4 \pm 0.5$ km s$^{-1}$Mpc$^{-1}$ \cite{Aghanim:2018eyx}) and SH$0$ES ($H_0^\text{R22} = 73.30 \pm 1.04$ km s$^{-1}$Mpc$^{-1}$ \cite{Riess:2021jrx}) estimates of the Hubble constant.}
    \label{fig:modelpost_SNeCC}
\end{figure}

We find here that the model taking in the Planck Hubble constant as a prior overtakes the one considering the SH$0$ES estimate. The same result can be obtained with varying size of the population in each generation. Figure \ref{fig:modelpost_SNeCC}(a) shows that the P18 model takes over the competition by the $6$th generation and completely dominates by the $13$th generation, when the R22 model's population becomes extinct. To understand what occurred during the course of the evolution, we take a look at the generational evolution of the Hubble constant in Figure \ref{fig:modelpost_SNeCC}(b). This shows that as the selection becomes more competitive, as the ABC narrows down the acceptable distance function values, the Hubble constant estimate in the R22 model falls outside of its prior to be able to keep up; however, at the same time, going away from its prior increases the overall distance and so its population shrinks as the Hubble constant falls down. We can see this happened more clearly after the $9$th generation in Figure \ref{fig:modelpost_SNeCC}(b), which is after the $6$th generation when the P18 model has taken control of the competition.

The takeaway from this is that as far as ABC is concerned the Planck Hubble constant estimate is more consistent with the late time observations (SNe + CC) than with the SH$0$ES value. This could be taken as a surprising result considering that we are basing this on late time observations. However, we are about to find out that this is robust even if we consider other data sets.

We note that as a model selection tool only the parameters estimated in the surviving model after the ABC run are meaningful. In this case, we get the $H_0$ estimate consistent with the Planck data, as also illustrated in Figure \ref{fig:modelpost_SNeCC} where the P18 generational $H_0$ values coincide with the confidence contour provided by the Planck measurement.

\subsection{ABC: SNe + CC + BAO}
\label{subsec:sne_cc_bao}

Now we consider expansion rate measurements from BAO. These additional measurements can be considered more stringent as the error bars are smaller. However, we keep in mind that the BAO depend on a prescription of the physics during the Hubble drag through the sound horizon radius. We present our results with this more constraining data combination, i.e., $H_{\rm BAO}\left(z_{\rm BAO}\right)$ have relatively lower error bars compared with CC measurements.

\begin{figure}[h!]
    \centering
	\subfigure[]{
		\includegraphics[width = 0.48 \textwidth]{figs/ABCjoint_SNeCCBAO.pdf}}
    \subfigure[]{
		\includegraphics[width = 0.48 \textwidth]{figs/ABCH0generational_SNeCCBAO.pdf}}
    \caption{(a) Model posterior ABC evolution with SNe, CC, and BAO compiled data sets and (b) the generational Hubble constant estimates. The pink and blue bands in (b) correspond to $68\%$ and $95\%$ confidence limits of the Planck ($H_0 = 67.4 \pm 0.5$ km s$^{-1}$Mpc$^{-1}$ \cite{Aghanim:2018eyx}) and SH$0$ES ($H_0^\text{R22} = 73.30 \pm 1.04$ km s$^{-1}$Mpc$^{-1}$ \cite{Riess:2021jrx}) estimates of the Hubble constant.}
    \label{fig:modelpost_SNeCCBAO}
\end{figure}

Perhaps the most interesting thing observation here is that nearly the same picture emerged with the addition of the BAO. It took the same number of generations for the ABC to come out with one model, P18, and that the natural selection became one sided after about six generations. This is clearly shown in Figure \ref{fig:modelpost_SNeCCBAO}, which if one looks away may appear just like Figure \ref{fig:modelpost_SNeCC} without the BAO. This strengthens the earlier result of the Planck $H_0$ being statistically preferred by ABC and the data. We see that the Hubble constant estimates of P18 also remained consistent throughout the evolution, all the while the R22 values drop to keep up as the competition tightens, but eventually loses its population.

Thus we see that even with BAO the ABC shows that the Planck Hubble constant estimate \textit{as a prior} is more consistent with observations. Understandably we have to be skeptical about this as the Planck value is measured from the cosmic microwave background rather than local physics. But next we also take a look at the perturbations to see if this influences the result.

\subsection{ABC: SNe + CC + BAO + RSD}
\label{subsec:sne_cc_bao_rsd}

We consider this time not only background cosmological data (SNe $+$ CC $+$ BAO) but also the leading order, linear perturbations of the matter density through RSD. The results now take the input from the structure formation. The relevant ABC results are shown in Figure \ref{fig:modelpost_SNeCCBAORSD}.

\begin{figure}[h!]
    \centering
	\subfigure[]{
		\includegraphics[width = 0.48 \textwidth]{figs/ABCjoint_SNeCCBAORSD.pdf}}
    \subfigure[]{
		\includegraphics[width = 0.48 \textwidth]{figs/ABCH0generational_SNeCCBAORSD.pdf}}
    \caption{(a) Model posterior ABC evolution with SNe, CC, BAO, and RSD background and perturbation data and (b) the generational Hubble constant estimates. The pink and blue bands in (b) correspond to $68\%$ and $95\%$ confidence limits of the Planck ($H_0 = 67.4 \pm 0.5$ km s$^{-1}$Mpc$^{-1}$ \cite{Aghanim:2018eyx}) and SH$0$ES ($H_0^\text{R22} = 73.30 \pm 1.04$ km s$^{-1}$Mpc$^{-1}$ \cite{Riess:2021jrx}) estimates of the Hubble constant.}
    \label{fig:modelpost_SNeCCBAORSD}
\end{figure}

We find that the addition of the growth data has made the selection drag for a longer period, which we may expect since the algorithm needs extra time to compare the simulations with bigger data. This time, it took $18$ generations before the P18 model completely overcome the R22 model. After this, the cosmological parameters in P18 were being estimated just like in MCMC. The main takeaway is that even with the growth data constraining the matter sector, the $\Lambda$CDM model anchored on the Planck Hubble constant turns out to be more consistent with the observations.

Looking at the Hubble constant estimates per generation, we see that eventually the R22 model slips out of the competition when it has to compromise with an $H_0$ that is away from its prior. Eventually to be able to keep up with P18 its Hubble constant estimate deviates too far outside the SH$0$ES value (R22's $H_0$ prior), making the overall distance measurements of the members of its population too large. This shrinks the population's size, leading to its collapse. All the while, it can be seen that the Hubble constant estimate in the P18 model remained convincingly consistent throughout, thereby stifling its competitor.

Since we are considering growth data, it is beneficial to also look at the generational estimates of the physical quantity $\sigma_8$. We present the ABC evolution in Figure \ref{fig:sigma8ABC}.

\begin{figure}[h!]
    \centering
    \includegraphics[width = 0.48\textwidth]{figs/ABCs80generational_SNeCCBAORSD.pdf}
    \caption{Generational estimates of the smoothed amplitude of the matter power spectrum, $\sigma_8$, given the SNe $+$ CC $+$ BAO $+$ RSD data set. The inset shows a zoomed-in view beginning at generation $11$. The pink and green bands show the $68\%$ and $95\%$ confidence limits of the corresponding cosmic microwave background ($\sigma_8 = 0.811 \pm 0.006$ \cite{Aghanim:2018eyx}) and galaxy clustering ($0.69 \pm 0.04$ \cite{Philcox:2021kcw}) reference values.}
    \label{fig:sigma8ABC}
\end{figure}

The large initial error bars reflect the initial flat priors on the matter sector. The evolution eventually gets to reasonable values as it populations compete for their survival. Interestingly, the results show $\sigma_8$ estimates that are consistent with galaxy clustering (green band in Figure \ref{fig:sigma8ABC}). However, we take this with a grain of salt as the ABC estimates fall toward to edge of the galaxy clustering confidence interval. This nonetheless puts ABC's view of the $\sigma_8$ tension given the data sets we consider here.

\section{Conclusions}
\label{sec:conclusions}

We find that the Approximate Bayesian Computation naturally favors the Planck value of the Hubble constant. This holds when considering both cosmological background and matter perturbation data. Our takeaway from this is that the late Universe estimate of the Hubble constant is consistent with that of Planck. Of course, we get this from a noncanonical approach and so must be viewed with skepticism. An interesting result is that the traditional MCMC analysis agrees with this, in the sense that the analysis based on the Planck Hubble constant prior lead to a better fit compared with the one coming from SH$0$ES. It is also worthwhile to point out that the ABC's preference for a low Hubble constant consistent with the Planck value goes beyond the scope of the standard cosmological model \cite{Bernardo:2022pyz}. This motivates the use of some other independent statistical methods for later cosmological analysis. The ABC is only one such approach that looks into cosmology with fresh eyes, but there are also others that may be more mainstream such as neural networks \cite{Grandon:2022gdr, Mukherjee:2022yyq} and genetic algorithms \cite{Bogdanos:2009ib, Alestas:2022gcg}. It will be worth the time looking into all these together.

We emphasize that we do not solve or propose a physics solution to the Hubble tension. Rather, the ABC brings into the table a fresh new insight that the Planck's Hubble constant is consistent with late time cosmology, contrary to the Hubble tension's premise. Granted, the reliability of the ABC in cosmological model selection and parameter estimation remains to be established further. One angle is it leads to perhaps unreasonably too narrow parameter estimates when the number of generations is taken too large \cite{Bernardo:2022pyz}. Model selection in ABC also does not take into account, at least superficially, the number of parameters of the models and therefore has no safeguard for overfitting. For the meantime, it is always nice and worthwhile to try alternative statistical methods, like we do with cosmological models. An assessment tool that transcends different statistical approaches will be useful to establish in this regard. With openness to new methods, we may just find the Universe to be more exciting, which may bring together an adventure among various fields in solving physics problems.

\begin{acknowledgements}
The initial stages of this work was done in part during the TCA Summer Student Program in 2022 by the National Center for Theoretical Sciences.
\end{acknowledgements}

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