# Quasars: standard candles up to z=7.5 with the precision of Supernovae Ia

M. G. Dainotti  First and second authors share the same contribution
Corresponding author. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>National Astronomical Observatory of Japan, 2
Chome-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  The Graduate University for
Advanced Studies, SOKENDAI, Shonankokusaimura, Hayama, Miura District,
Kanagawa 240-0193, Japan  Space Science Institute, 4765 Walnut St, Suite B,
80301 Boulder, CO, USA  G. Bargiacchi  Scuola Superiore Meridionale, Largo S.
Marcellino 10, 80138 Napoli, Italy  Istituto Nazionale di Fisica Nucleare
(INFN), Sez. di Napoli, Complesso Univ. Monte S. Angelo, Via Cinthia 9, 80126,
Napoli, Italy  A. Ł. Lenart  Astronomical Observatory, Jagiellonian
University, ul. Orla 171, 31-501 Kraków, Poland  S. Nagataki
Interdisciplinary Theoretical & Mathematical Science Program, RIKEN (iTHEMS),
2-1 Hirosawa, Wako, Saitama, Japan 351-0198  RIKEN Cluster for Pioneering
Research, Astrophysical Big Bang Laboratory (ABBL), 2-1 Hirosawa, Wako,
Saitama, Japan 351-0198  Astrophysical Big Bang Group (ABBG), Okinawa
Institute of Science and Technology Graduate University (OIST), 1919-1 Tancha,
Okinawa, Japan 904-0495  S. Capozziello  Scuola Superiore Meridionale, Largo
S. Marcellino 10, 80138 Napoli, Italy  Istituto Nazionale di Fisica Nucleare
(INFN), Sez. di Napoli, Complesso Univ. Monte S. Angelo, Via Cinthia 9, 80126,
Napoli, Italy  Dipartimento di Fisica ”E. Pancini”, Università degli Studi di
Napoli Federico II, Complesso Univ. Monte S. Angelo, Via Cinthia 9, 80126,
Napoli, Italy

###### Abstract

Currently, the $\Lambda$ Cold Dark Matter model, which relies on the existence
of cold dark matter and a cosmological constant $\Lambda$, best describes the
Universe. However, we lack information in the high-redshift ($z$) region
between Type Ia Supernovae (SNe Ia) (up to $z=2.26$) and the Cosmic Microwave
Background ($z=1100$), an interval crucial to test cosmological models and
their possible evolution. We have defined a sample of 983 Quasars up to
$z=7.54$ with reduced intrinsic dispersion $\delta=0.007$ which determines the
matter density parameter $\Omega_{M}$ with the same precision of SNe Ia.
Although previous analysis have been used Quasars as cosmological tools (e.g.
Risaliti and Lusso 2019), this is the first time that high-redshift sources,
in this case Quasars, as standalone cosmological probes yield such tight
constraints on $\Omega_{M}$. Our results show the importance of correcting
cosmological relationships for selection biases and redshift evolution and how
the choice of a golden sample reduces considerably the intrinsic scatter. This
proves the reliability of Quasars as standard cosmological candles.

Quasars

##  1 Introduction

Nowadays it is time for the scavenger hunt for cosmological parameters such as
the Hubble constant, $H_{0}$, and the matter density parameter, $\Omega_{M}$.
The main challenge is to further cast light on which is the most suitable
description of the Universe. Currently, the most accredited model is the
$\Lambda$CDM model, which includes a cold dark matter (CDM) component and a
dark energy associated with a cosmological constant $\Lambda$, as suggested by
the current accelerated expansion of the Universe (Riess et al., 1998;
Perlmutter et al., 1999). Predictions from this scenario are compatible with
observational probes such as the cosmic microwave background (CMB) (Planck
Collaboration et al., 2020) at redshift $z=1100$, the baryon acoustic
oscillations (Alam et al., 2021), and type Ia supernovae (SNe Ia) observed at
low $z$ up to $z=2.26$ (Rodney et al., 2015). However, we lack information on
intermediate redshifts between SNe Ia and CMB. Having reliable cosmological
candles at intermediate redshifts between the SNe Ia and CMB could be the
crucial turning point to understanding to what extent the current cosmological
$\Lambda$CDM model is the most suitable one. To this end, the community
strives to reduce as much as possible the uncertainties on cosmological
parameters and to find probes at intermediate redshifts which can be
considered standard candles (Yonetoku et al., 2004). Given the current
interest of the community in Quasars as reliable cosmological tools (Dainotti
et al., 2022a; Lenart et al., 2023; Risaliti & Lusso, 2019; Bargiacchi et al.,
2022; Khadka et al., 2022), we focus on the hunt for a gold sample for
Quasars, as already done for Gamma-ray Bursts (GRBs) (Dainotti et al., 2010,
2013, 2015, 2016, 2020, 2022d, 2021b, 2022c, 2022b, 2023a). We propose an
approach to the use of Quasars as standard candles similar to the one done for
SNe Ia 25 years ago, but Quasars are observed at much larger distances (up to
$z=7.54$). To tackle such an approach we need a standardizable candle and in
this regard, a relation exists between the X-ray and the Ultraviolet (UV)
luminosities of Quasars (known as the Risaliti-Lusso relation, hereafter
called RL). To use this, Quasar emission mechanisms need to be very well
understood and the relation at play should not be affected by redshift
evolution (a change in redshift) and selection biases. Namely, such a relation
should remain the same at all redshifts or if there is a redshift evolution
this should be properly accounted for before its use as a cosmological tool.
Dainotti et al. (2022a) have already demonstrated that the RL relation is not
induced by selection biases or redshift evolution, but it is intrinsic to the
physics of Quasars. Here we have presented two main gold samples of Quasars.
One is built from a flat $\Lambda$CDM model with $\Omega_{M}=0.3$ and
$H_{0}=70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ and is composed of
983 Quasars up to $z=7.54$ which constrains the parameter $\Omega_{M}$ in the
assumed cosmology with the same precision as SNe Ia in the Pantheon sample.
The other one is instead obtained through a circularity-free procedure and
consists of 975 sources with the same maximum redshift of $z=7.54$. The paper
is structured in data and methodology, Sect. 2, where we detail the sample and
all methods, results, Sect. 3 and 4, where the choice of the golden sample
assuming a given cosmology is explained and the solution for a golden sample
completely independent from the circularity problem including the Markov Chain
sampling uncertainty results is presented. We discuss our findings in Sec. 5.

##  2 Data set and Methodology

Our initial Quasar sample is the most recent one released for cosmological
studies (Lusso et al., 2020). It consists of 2421 sources in the redshift
range between $z=0.009$ and $z=7.54$ (Bañados et al., 2018) collected from
eight catalogues (Nardini et al., 2019; Salvestrini et al., 2019; Vito et al.,
2019) and archives (Menzel et al., 2016; Pâris et al., 2018; Webb et al.,
2020; Evans et al., 2010), with the addition of a subsample of low redshift
Quasars that present UV observations from the International Ultraviolet
Explorer and X-ray data in archives. To obtain this Quasar sample suitable for
cosmological analyses, as many as possible observational biases have been
meticulously inspected and removed (Risaliti & Lusso, 2015; Lusso & Risaliti,
2016; Risaliti & Lusso, 2019; Salvestrini et al., 2019; Lusso et al., 2020).
We note here some differences between our sample and the samples used in
previous works. Here, we study this final sample of 2421 sources without any
additional selection, such as the cut at redshift $z=0.7$ previously used in
some works (Lusso et al., 2020; Bargiacchi et al., 2022), to avoid any
possible induced bias due to the reduction on the redshift sample (Dainotti et
al., 2022a; Lenart et al., 2023).

###  2.1 Correction for the redshift evolution of the luminosities

We clarify that previous works have not considered selection biases and
redshift evolution with the exception of our works in the literatures
(Dainotti et al., 2022a; Lenart et al., 2023; Bargiacchi et al., 2023). Since
Quasars are high-redshift sources, we need to account for selection biases and
redshift evolution effects. These factors could induce artificial correlations
between intrinsic physical quantities of sources (Dainotti et al., 2013), such
as the relation between X-ray and UV luminosities for Quasars. To correct for
these effects, we have applied the statistical Efron & Petrosian method (Efron
& Petrosian, 1992) assuming that the luminosities evolve with redshift as
$(1+z)^{k}$. This method has already been employed for GRBs (Dainotti et al.,
2013, 2015, 2017, 2021a, 2023b) and Quasars (Dainotti et al., 2022a; Lenart et
al., 2023). The choice of a more complex function for the redshift evolution
would not affect the results (Singal et al., 2011; Dainotti et al., 2021c,
2022a). Thus, the de-evolved UV and X-ray luminosities are computed as
$L^{\prime}_{UV}=L_{UV}/(1+z)^{k_{UV}}$ and
$L^{\prime}_{X}=L_{X}/(1+z)^{k_{X}}$ by using $L_{UV}$ and $L_{X}$ obtained
from the measured flux densities $F_{UV}$ and $F_{X}$ (in units of
$\mathrm{erg\,s^{-1}\,cm^{-2}\,Hz^{-1}}$) according to
$L_{X,UV}=4\,\pi\,d_{l}^{2}\,F_{X,UV}$ where $d_{l}$ is the luminosity
distance in cm and the K-correction is assumed to be equal to 1 for Quasars
(Lusso et al., 2020). Here we consider a flat $\Lambda$CDM model. Our main
gold Quasar sample of 983 sources is obtained by fixing $\Omega_{M}=0.3$ and
$H_{0}=70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ in the distance
luminosity. The values of $k_{UV}$ and $k_{X}$ used to determine
$L^{\prime}_{UV}$ and $L^{\prime}_{X}$ are $k_{UV}=4.36\pm 0.08$ and
$k_{X}=3.36\pm 0.07$ (Dainotti et al., 2022a), which have been obtained within
a flat $\Lambda$CDM model with $\Omega_{M}=0.3$ and
$H_{0}=70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, consistently with
our assumption. The evolutionary parameter $k$ depends on $\Omega_{M}$, but
not on $H_{0}$ (Dainotti et al., 2022a, 2023b; Lenart et al., 2023). Thus,
when we change the value of $\Omega_{M}$ to $\Omega_{M}=0.1$ and
$\Omega_{M}=1$ to test the dependence of our results on the cosmological
assumptions, we accordingly use the values of $k_{UV}$ and $k_{X}$
corresponding to these cosmologies. More precisely, these values are obtained
from the functions $k_{UV}(\Omega_{M})$ and $k_{X}(\Omega_{M})$ reported in
Dainotti et al. (2022a) and shown in their Fig. 4. The resulting values are
$k_{UV}=4.79\pm 0.08$ and $k_{X}=3.81\pm 0.06$ for $\Omega_{M}=0.1$ and
$k_{UV}=3.89\pm 0.08$ and $k_{X}=2.88\pm 0.06$ for $\Omega_{M}=1$.

The choice of a specific $\Omega_{M}$ and thus, a value of $k$ for the
correction for evolution automatically induces the circularity problem. This
issue can be completely overcome in the cosmological fit if we do not fix $k$
a-priori to compute the luminosities, but we apply the functions
$k_{UV}(\Omega_{M})$ and $k_{X}(\Omega_{M})$ in the fitting procedure while
leaving also the cosmological parameters free to vary, as already done in
Lenart et al. (2023) for Quasars and in Dainotti et al. (2023b) for GRBs. This
allows us to avoid assuming a-priori an underlying cosmology and to leave the
evolutionary parameters free to vary together with the parameters of the fit.
In our work, we also employ this circularity-free methodology when we fit
$\Omega_{M}$ with the gold sample of 975 sources selected by applying the
$\sigma$-clipping procedure to the relation between X-ray and UV fluxes,
instead of luminosity. Indeed, the selection of the sample based on measured
fluxes does not require any assumption on the cosmological model, contrary to
the one based on luminosities, and, thus enables us to apply this circularity-
free procedure to fit $\Omega_{M}$ with a sample that is not biased by any
cosmological assumption.

###  2.2 Fitting procedure

We have performed all fits with the Bayesian D’Agostini method (D’Agostini,
2005). The likelihood function ($\mathcal{L}$) employed to constrain
parameters with Quasars is given by Khadka & Ratra (2020a, b); Lusso et al.
(2020); Khadka & Ratra (2021); Colgáin et al. (2022b); Khadka & Ratra (2022);
Colgáin et al. (2022a); Bargiacchi et al. (2022); Lenart et al. (2023) as:

$\text{ln}\mathcal{L}=-\frac{1}{2}\sum_{i=1}^{N}\left[\frac{(y_{i}-\phi_{i})^{2}}{s^{2}_{i}}+\text{ln}(s^{2}_{i})\right]$
(1)

where “ln” is the natural logarithm and $N$ is the number of sources. When we
fit the luminosities, $y_{i}=\mathrm{log_{10}}L^{\prime}_{X}$ of the Quasar at
redshift $z_{i}$, while $\phi_{i}$ is the logarithmic X-ray luminosity
predicted from the fitted model. The quantity
$s^{2}_{i}=(\Delta\mathrm{log_{10}}L^{\prime}_{X})^{2}_{i}+\gamma^{2}(\Delta\mathrm{log_{10}}L^{\prime}_{UV})^{2}_{i}+\delta^{2}$
includes the statistical 1 $\sigma$ uncertainties ($\Delta$) on both
luminosities and the intrinsic dispersion $\delta$ of the RL relation. The
free parameters of the fit are $\gamma$, $\beta$, $\delta$, and the ones of
the cosmological model studied. The same methodology is applied when we fit
fluxes instead of luminosities, just replacing
$\mathrm{log_{10}}L^{\prime}_{UV}$ and $\mathrm{log_{10}}L^{\prime}_{X}$ with
the measured $\mathrm{log_{10}}F_{UV}$ and $\mathrm{log_{10}}F_{X}$. In this
case, $\gamma_{\mathrm{F}}$ and $\delta_{\mathrm{F}}$ are the slope and the
intrinsic dispersion of the linear relation.

###  $\sigma$-clipping technique

Before applying the $\sigma$-clipping procedure, we have searched for possible
outliers. Thus, we have computed the maximum value of
$\Delta\mathrm{log_{10}}L^{\prime}_{UV}/\mathrm{log_{10}}L^{\prime}_{UV}$ and
$\Delta\mathrm{log_{10}}L^{\prime}_{X}/\mathrm{log_{10}}L^{\prime}_{X}$.
Points with large uncertainties on luminosities would not be removed by the
$\sigma$-clipping and would have a higher weight in the cosmological fits
compared to points with smaller uncertainties. This check has revealed that
our sample does not present any outliers of this kind. Independently of the
$\Omega_{M}$ value, the maximum values of
$\Delta\mathrm{log_{10}}L^{\prime}_{UV}/\mathrm{log_{10}}L^{\prime}_{UV}$ and
$\Delta\mathrm{log_{10}}L^{\prime}_{X}/\mathrm{log_{10}}L^{\prime}_{X}$ are
0.022 and 0.013, respectively, with only one source with
$\Delta\mathrm{log_{10}}L^{\prime}_{UV}/\mathrm{log_{10}}L^{\prime}_{UV}>0.02$
and four sources with
$\Delta\mathrm{log_{10}}L^{\prime}_{UV}/\mathrm{log_{10}}L^{\prime}_{UV}>0.01$.
Because these values are lower than $\sim 1-2\%$, we do not remove any Quasar
from the initial sample. Hence, we apply the $\sigma$-clipping selection to
all 2421 sources.

The $\sigma$-clipping allows us to reduce the intrinsic scatter of the RL
relation by removing the sources with a vertical distance from the best-fit
relation greater than a chosen threshold value by assuming a given
cosmological model. It is used when dealing with relations presenting an
intrinsic dispersion to remove possible outliers in the sample and assure a
better determination of the free parameters of the relation. This procedure
has already been successfully applied for Quasars to constrain cosmological
parameters (Lusso et al., 2020; Bargiacchi et al., 2021, 2022). We detail the
method for our case. First, we fit the RL relation with the whole Quasar
sample assuming a flat $\Lambda$CDM model with
$H_{0}=70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ and
$\Omega_{M}=0.3$ or $\Omega_{M}=0.1$ or $\Omega_{M}=1$, to compute the X-ray
and UV luminosities. The model used is the RL relation, thus $\phi_{i}$ of
Equation 1 is $\phi_{i}=\gamma\,\mathrm{log_{10}}L^{\prime}_{UV,i}+\beta$ for
the source at redshift $z_{i}$. From this fit, we obtain the best-fit values
of $\gamma$, $\beta$, and $\delta$ and their 1 $\sigma$ uncertainties. Then,
we evaluate for each Quasar at $z_{i}$ the quantity

$\Sigma=\frac{|\mathrm{log_{10}}L^{\prime}_{X,i}-(\gamma\,\mathrm{log_{10}}L^{\prime}_{UV,i}+\beta)|}{\sqrt{\Delta^{2}\mathrm{log_{10}}L^{\prime}_{X,i}+\gamma^{2}\,\Delta^{2}\mathrm{log_{10}}L^{\prime}_{UV,i}+\delta^{2}}}$
(2)

where we use the best-fit values previously obtained for $\gamma$, $\beta$,
and $\delta$. This quantity is exactly the one that is minimized in the
fitting algorithm (i.e. the first term in square bracket of Equation 1) to
determine the parameters of the RL relation, and thus it is the most
appropriate to estimate the discrepancy between the measured X-ray luminosity
and the one predicted from the RL relation. Once we have computed $\Sigma$ for
each source, we select only Quasars with
$\Sigma\leq\sigma_{\mathrm{clipping}}$ and we repeat the fit of the RL
relation with this reduced sample. Since the fit on this new sample yields
best-fit values of $\gamma$, $\beta$, and $\delta$ different from the ones of
the previous fit, and thus different $\Sigma$ values for each source, there
will be Quasars in the sample considered at this step that do no fulfil the
requirement $\Sigma\leq\sigma_{\mathrm{clipping}}$ anymore. Hence, we iterate
this procedure until all sources in the selected sample verify the requisite.
After this $\sigma$-clipping process, we obtain a final Quasar sample, with
the corresponding best-fit values and 1 $\sigma$ uncertainties of $\gamma$,
$\beta$, and $\delta$. We have chosen several $\sigma_{\mathrm{clipping}}$
values between 0.6 and 2 to investigate how the assumed
$\sigma_{\mathrm{clipping}}$, the best-fit value of $\delta$, and the number
of survived sources are related. This method selects the gold samples (983
Quasars) within a flat $\Lambda$CDM model shown in the left panel of Fig. 1
with its corresponding best-fit RL relation (purple line). We have also
applied the same method to select the Quasar samples using the observed fluxes
instead of the luminosities. We fit the relation between
$\mathrm{log_{10}}F_{X}$ and $\mathrm{log_{10}}F_{UV}$ for which
$\gamma_{\mathrm{F}}$, $\beta_{\mathrm{F}}$, and $\delta_{\mathrm{F}}$ are the
slope, intercept, and intrinsic dispersion, respectively. The gold sample of
975 Quasars obtained with the $\sigma$-clipping on fluxes is shown in the left
panel of Fig. 4 with the best-fit linear relation (purple line).

###  2.3 Cosmological fits

For each Quasar sample produced by the $\sigma$-clipping with a specific
threshold $\sigma_{\mathrm{clipping}}$ we have fitted a flat $\Lambda$CDM
model. Following the fitting procedure described above, if we use the Quasar
sample selected through the $\sigma$-clipping on the luminosities corrected
for a fixed redshift evolution, the quantities $y_{i}$ and $\phi_{i}$ of
Equation 1 for a source at $z_{i}$ are respectively
$y_{i}=\mathrm{log_{10}}F_{X,i}+\mathrm{log_{10}}(4\,\pi\,d_{l}^{2}(z_{i}))-k_{X}\mathrm{log_{10}}(1+z_{i})$
and
$\phi_{i}=\gamma\,[\mathrm{log_{10}}F_{UV,i}+\mathrm{log_{10}}(4\,\pi\,d_{l}^{2}(z_{i}))-k_{UV}\mathrm{log_{10}}(1+z_{i})]+\beta$,
where $k_{X}$ and $k_{UV}$ are the evolutions corresponding to the
cosmological model assumed. Instead, if we perform the fit on the sample
obtained from the $\sigma$-clipping on the measured fluxes, we require
$y_{i}=\mathrm{log_{10}}F_{X,i}+\mathrm{log_{10}}(4\,\pi\,d_{l}^{2}(z_{i}))-k_{X}(\Omega_{M})\,\mathrm{log_{10}}(1+z_{i})$
and
$\phi_{i}=\gamma\,[\mathrm{log_{10}}F_{UV,i}+\mathrm{log_{10}}(4\,\pi\,d_{l}^{2}(z_{i}))-k_{UV}(\Omega_{M})\,\mathrm{log_{10}}(1+z_{i})]+\beta$,
thus avoiding the circularity problem. The luminosity distance $d_{l}$ is
computed fixing $H_{0}=70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$
and considering $\Omega_{M}$ as a free parameter with a wide uniform prior
between 0 and 1. We also leave $\gamma$ and $\beta$ free to vary. Hence, we
obtain the best-fit values of $\Omega_{M}$, $\gamma$, $\beta$, and $\delta$
with their associated 1 $\sigma$ uncertainty.

##  3 Results

###  3.1 Golden samples with an assumed cosmology

Figure 1: Left panel: The golden sample of 983 Quasars obtained with the
$\sigma$-clipping on the $L_{\mathrm{X}}-L_{\mathrm{UV}}$ relation assuming a
flat $\Lambda$CDM model with $\Omega_{M}=0.3$ and
$H_{0}=70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$. The resulting
best-fit values of the parameters are: the slope $\gamma=0.576\pm 0.004$, the
normalization $\beta=8.7\pm 0.2$, and the dispersion $\delta=0.007\pm 0.004$.
Blue points are the sources with error bars representing the statistical 1
$\sigma$ uncertainties and the best-fit linear relation is drawn as a purple
line. The black dashed line is the best-fit line for the 2036 Quasars used in
Lusso et al. (2020), for which $\gamma=0.562\pm 0.011$, $\beta=9.2\pm 0.3$,
and $\delta=0.221\pm 0.004$ after correction for evolution. Right panel:
Cosmological results in a flat $\Lambda$CDM model from our golden sample shown
in the left panel. This shows the values of $\Omega_{M}$, $\gamma$, $\beta$,
and $\delta$. The contour levels at $68\%$ and $95\%$ are represented by the
inner dark and light blue regions, respectively.

Our initial Quasar sample is the most up-to-date for cosmological studies
(Lusso et al., 2020) and composed of 2421 sources between $z=0.009$ and
$z=7.54$. This sample is studied to define the gold Quasar sample to obtain
the tightest possible relation to be used as an efficient cosmological tool
with the same precision achieved with SNe Ia. Differently from recent works
(Lusso et al., 2020; Bargiacchi et al., 2021, 2022), we use the full sample of
Quasars at all redshifts and we correct for redshift evolution of the sample,
as detailed before. So, we retain also sources at small redshifts instead
excluded in most of previous analyses. Differently from Risaliti & Lusso
(2019), the RL relation is already corrected for selection biases and redshift
evolution as shown in Dainotti et al. (2022a) when the $\sigma$-clipping is
applied. This procedure allows us to obtain the smallest intrinsic scatter of
an already bias-free and evolution-free relation. Nevertheless, to constrain
cosmological parameters, such as $\Omega_{M}$, we are not only interested in
reaching the smallest dispersion, but also in relying on a statistically
sufficient number of sources. Thus, we need to find a compromise between these
two antagonistic factors. Indeed, increasing the data set results in a larger
intrinsic dispersion and vice versa. We divide our results in the search of
two main golden samples, one assuming a given cosmological model and the
second one without any cosmological assumption, thus completely overcoming the
so-called circularity problem.

If we assume a flat $\Lambda$CDM model with $\Omega_{M}=0.3$ and
$H_{0}=70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, the optimal
number of sources is 983, obtained by requiring a threshold for the
$\sigma$-clipping $\sigma_{\mathrm{clipping}}=1.788$, with a scatter
$\delta=0.007$ of the RL relation. The 983 Quasars created in such a manner
define the golden sample shown with the corresponding best-fit RL relation in
the left panel of Fig. 1.

Figure 2: Upper panel: The uncertainty of $\Omega_{M}$ as a function of the
number of sources and as function of the sigma-clipping showed with the colour
bar on the right. The solid black line shows the best-fit of the decreasing
trend of the uncertainty on $\Omega_{M}$ vs. the number of sources, while the
horizontal blue line denotes the uncertainty on $\Omega_{M}$ reached with the
Pantheon SNe Ia sample. This is obtained for our golden sample under the
assumption of a flat $\Lambda$CDM model with $\Omega_{M}=0.3$, and
$H_{0}=70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$. Lower panel: The
intrinsic dispersion of the RL relation as a function of the number of sources
and as a function of the sigma-clipping indicated on the right as a color bar.
Black lines mark the 1 $\sigma$ uncertainty on the $\delta$ values.

Figure 3: Upper panel: The values of $\Omega_{M}$ and their associated
uncertainties vs. the number of Quasars. The color bar on the right shows the
normalized probability density, indicating for each sample size the most
probable value of $\Omega_{M}$, thus the smallest uncertainty on $\Omega_{M}$.
This Fig. indicates that the smallest error bar on $\Omega_{M}$ (the red
contour) is achieved for $N\approx 2000$, which yields $\Omega_{M}=0.119\pm
0.019$. This is obtained for our golden sample assuming a flat $\Lambda$CDM
model. Bottom panel: Values of $\Omega_{M}$ with corresponding 1 $\sigma$
uncertianities as a function of the $\sigma$-clipping threshold and the
probability distribution function (PDF) showed with the colour bar on the
right side. The red line is the best-fit of $\Omega_{M}$ points.

Then, we use this sample to derive $\Omega_{M}$ with the Monte Carlo Markov
Chain (MCMC) computation and leaving contemporaneously free the values of
$\Omega_{M}$ in the range from 0 a 1 using a uniform distribution and the
parameters of the RL relation: $\gamma$ (the slope), $\beta$ (the
normalization) and $\delta$ (the dispersion). We fix
$H_{0}=70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ assuming a
standard flat $\Lambda$CDM model. Results of this analysis are shown in the
right panel of Fig. 1 where the corner plot of the RL relation parameters and
$\Omega_{M}$ are obtained when we perform the correction for the redshift
evolution. The best-fit value yields $\Omega_{M}=0.268\pm 0.022$ which carries
the same uncertainty on $\Omega_{M}$ found with the Pantheon sample (1048 SNe
Ia) assuming the same cosmology (Scolnic et al., 2018), as shown with the
horizontal blue line in Fig. 2. We also recover the $\Omega_{M}$ we have
assumed to build this sample within 0.68 $\sigma$.

To assess to what extent the number of sources impacts the values and the
uncertainty on $\Omega_{M}$, we show in the upper panel of Fig. 2 the values
of $\sigma_{\Omega_{M}}$ vs. the number of sources and the varying of the
sigma-clipping shown as a color bar. In the upper panel of Fig. 2, we start
from the total sample of sources and we apply a very restrictive criterion for
the $\sigma$-clipping=0.6 and then we continue with larger values of
$\sigma$-clipping until arriving at 2.0. When we change the values of the
$\sigma$-clipping, the smaller is the $\sigma$-clipping, the smaller the
sample size obtained if we consider sources from 300 to 1000 and from 1700 to
2000, but when the $\sigma$-clipping is too small (0.045 for example, see the
red point in the upper panel of Fig. 2), then the uncertainties on
$\Omega_{M}$ becomes larger. Since the $\sigma$-clipping and the number of
sources also determine the intrinsic dispersion of the RL, thus we show in the
bottom panel of Fig. 2 the intrinsic scatter as as a function of the number of
sources and of the $\sigma$-clipping shown as a color bar. It is clear from
this figure that the dispersion of the RL relation increases monotonically at
the increase of the sample sizes starting from around 1000 sources, while to
achieve smaller dispersion the trend is rather flat from 300 to 1000 sources.
This increasing trend of the dispersion of RL relation as a function of the
number of sources is reflected by the increasing values of the uncertainties
on $\Omega_{M}$ in the range between 1000 and 1300. However, the highly non-
linear process of obtaining cosmological parameters, in this case the value of
$\sigma_{\Omega_{M}}$, and the cut that the $\sigma$-clipping induces in the
initial sample does not allow a straightforward comparison between the upper
and lower panel of Fig. 2. To conclude, the number of 975 sources is the
optimal compromise to obtain the smallest uncertainty on $\Omega_{M}$ taking
into consideration the dispersion. Indeed, if we still would enlarge further
the $\sigma$-clipping the number of sources would become larger together with
the dispersion of the RL relation. This is the reason why we have mentioned
before the relevance to reach a compromise among the number of sources used
and the dispersion of the RL and consequently the uncertainty on $\Omega_{M}$
before the increase we observe between 1000 and 1700 sources. We stress that
the subsamples shown in Fig. 2 are not subsample of the Gold Sample, but they
are samples drawn independently, however with the same procedure of the
$\sigma$-clipping starting from the full sample assuming a flat $\Lambda$CDM
model. Continuing on the evaluation of how many number of sources are needed
to obtain the most probable value of $\Omega_{M}$ we show the colour map in
the upper panel of Fig. 3 where the probability density function (PDF) of
$\Omega_{M}$ is plotted as a function of the number of sources. We note from
this figure that 983 sources are the best sample as it provides closed
cosmological contours with the highest corresponding probability. To
complement this information, we also plotted in the bottom panel of Fig. 3 the
$\Omega_{M}$ values as a function of the $\sigma$-clipping and as a function
of the probability density for $\Omega_{M}$ shown in the color bar. From this
figure it is clear that the $\sigma$-clipping of 1.8 is the optimal value to
choose our Gold sample since the scatter on $\Omega_{M}$ becomes small enough
to reach the precision of SNe Ia Pantheon sample. To check this result against
the cosmological settings, we also test the assumptions of $\Omega_{M}=1$
(Universe filled by matter in which $\Lambda=0$), and $\Omega_{M}=0.10$ (very
close to the De Sitter Universe). We obtain that the best-fit values of
$\Omega_{M}$ are consistent within less than 3 $\sigma$ with the a-priori
assumption. In fact, when we choose the golden sample of 980 Quasars derived
with the $\sigma_{\mathrm{clipping}}=1.788$ assuming $\Omega_{M}=0.10$, we
obtain $\Omega_{M}=0.083\pm 0.009$, while when we select the sample of 968
Quasars assuming $\Omega_{M}=1$, we obtain $\Omega_{M}=0.910\pm 0.055$ for
$\sigma_{\mathrm{clipping}}=1.785$.

###  3.2 The Anderson Darling Test for different Gold samples and the parent
population

To check the similarity between the Gold sample and the parent population, we
have applied the Anderson-Darling test to compare both the fluxes-fluxes
distributions and the luminosity-luminosity distributions from the parent
population and the Gold sample. The result of the test shows that the Gold of
the luminosities and fluxes are not compatible both in X-rays and UV with the
parent population. In addition, the samples from the parent population used by
Lusso et al. (2020) also is not compatible with our Gold sample. It is not
surprising that we can have different results when other probes are added to
Quasars since the probes with smaller uncertainties ($s$ in Eq. 1) weigh more
than the ones with larger uncertainties, being the $s$ values at the
denominator of the likelihood function. In addition, since our Gold sample is
already a sample for which the selection biases have been removed, the
cosmological results from this sample should not necessarily be the same as
the enlarged sample (the parent population) if the parent population undergoes
selection biases and redshift evolution. Indeed, results of cosmological
parameters may change if evolutionary effects are not considered (Dainotti et
al., 2013, 2022b, 2023a; Lenart et al., 2023). In relation instead of our Gold
sample both in X-rays and UV derived from a given cosmology ($\Omega_{M}=0.3$
and $H_{0}=70\,kms^{-1}Mpc^{-1}$) these are compatible with the Gold sample
originated assuming other cosmologies ($\Omega_{M}=0.1$,
$H_{0}=70\,kms^{-1}Mpc^{-1}$, and $\Omega_{M}=1$ and
$H_{0}=70\,kms^{-1}Mpc^{-1}$). This ensures that the selection of our Gold
sample does not depend on cosmological models.

###  3.3 A circularity-free golden sample

To further guarantee that we completely avoid the circularity problem for the
choice of the golden sample of Quasars, we use the $F_{X}-F_{UV}$ relation,
the observer frame relation corresponding to the RL relation. As previously,
we apply the same $\sigma_{clipping}$ procedure to reduce the scatter of the
$F_{X}-F_{UV}$ relation. A $\sigma_{\mathrm{clipping}}=1.78$ identifies an
optimal sample of 975 sources, shown in the left panel of Fig. 4. We then use
this sample (free from any circularity problem) to derive $\Omega_{M}$ and the
RL parameters (see Methods), and we obtain $\Omega_{M}=0.107\pm 0.047$, as
reported in the right panel of Fig. 4.

Figure 4: Left panel: The gold sample of 975 Quasars generated with the
$\sigma$-clipping on the $F_{\mathrm{X}}-F_{\mathrm{UV}}$ relation with the
best-fit parameters being $\gamma_{\mathrm{F}}=0.625\pm 0.007$,
$\beta_{\mathrm{F}}=-14.3\pm 0.2$, and $\delta_{\mathrm{F}}=0.053\pm 0.002$.
Blue points are the sources with error bars representing the statistical 1
$\sigma$ uncertainties and the best-fit linear relation is drawn as a purple
line. Right panel: Cosmological results from the golden sample shown in the
left panel. This corner plot shows the values of $\Omega_{M}$, $\gamma$,
$\beta$, and $\delta$. The contour levels at $68\%$ and $95\%$ are represented
by the inner dark and light blue regions, respectively.

Additionally, to assure that our findings are not driven by the low-$z$
Quasars ($z<0.7$), which according to Lusso et al. (2020) could be affected by
host galaxy contamination and lower data quality, we have removed these
sources (47 Quasars) from the golden sample obtaining $\Omega_{M}=0.125\pm
0.040$.

##  4 Monte Carlo Markov Chain (MCMC) Sampling Uncertainty

To further guarantee that our results are not due to the use of a single run
in the MCMC calculation and that the sampling procedure is stable, we show
that the results of the computation when it is run 100 times. With this
procedure, we obtain $<\Omega_{M}>=0.112\pm 0.048$, where the symbol $<>$
denotes the average value. We have investigated the reliability of the best-
fit values and 1 $\sigma$ uncertainty on $\Omega_{M}$ obtained in each of the
cosmological fits. Our results are derived by fitting the free parameters of
the models studied with only one MCMC run. We test these results against the
sampling error on the parameters derived in the sampling procedure. To this
end, we have looped all the MCMC samplings 100 times for each model, then
computing the mean values of $\Omega_{M}$ and its uncertainty. The results
obtained with this method for both quantities and all the cosmological cases
investigated in our analysis are shown in Table 1. These results are
completely consistent with the ones obtained from only one run of the MCMC,
with a maximum discrepancy of 0.08 $\sigma$.

$\sigma_{\mathrm{clipping}}$ | $\Omega_{\mathrm{M,start}}$ | $<\Omega_{M}>$ | $S_{<\Omega_{M}>}$ | $<\sigma_{\Omega_{M}}>$ | $S_{<\sigma_{\Omega_{M}}>}$
---|---|---|---|---|---
Luminosities
1.788 | 0.3 | 0.2681 | 0.0010 | 0.0223 | 0.0007
1.788 | 0.1 | 0.0836 | 0.0004 | 0.0086 | 0.0004
1.785 | 1 | 0.9141 | 0.0024 | 0.0536 | 0.0013
Fluxes
1.78 | - | 0.1124 | 0.0021 | 0.0475 | 0.0021
Table 1: The results of the looped computations for 100 iterations. $S$
denotes the standard deviation of a given parameter,
$\Omega_{\mathrm{M,start}}$ is the value of $\Omega_{M}$ assumed to obtain the
corresponding golden sample with the threshold value
$\sigma_{\mathrm{clipping}}$. The cases in which the $\sigma$-clipping is
applied to the relation in luminosities or in fluxes are also shown
separately.

##  5 Discussion and Conclusions

In the choice of the golden samples with a fixed cosmology we have first
assumed a flat $\Lambda$CDM model with $\Omega_{M}=0.3$ and
$H_{0}=70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$. This is a
necessary starting cosmological model and value for $\Omega_{M}$, because our
aim is to compare the results of our uncertainties with the ones obtained
using the Pantheon sample for the same cosmological model. Regarding the
sample size, we note that the 1048 Pantheon SNe Ia have been slimmed down from
an original sample of 3473 events, with a cutting of the $70\%$ of the
starting data set (Scolnic et al., 2018). Instead, in our work, we reduce the
initial sample of 2421 Quasars of the $\sim 60\%$ to build all the golden
samples studied. Thus, the slimming of our sample is even less severe than the
one of SNe Ia. We here compare the SNe Ia and Quasars sample from a mere
statistical point of view and, not for the purpose to compare them from a
physical and an observational point of view, since they differ on both
aspects.

Our results highlight that the uncertainties on $\Omega_{M}$ depend on the
assumed cosmology, namely the smaller the uncertainties the more we are closer
to the most likely value of the cosmological parameters. The uncertainty on
$\Omega_{M}$ assuming $\Omega_{M}=0.10$ is 6.7 times smaller than the one in
the case of $\Omega_{M}=1$. With the evolutionary parameters included and with
no information about the underlying cosmology, we still obtain a value of
$\Omega_{M}$ compatible with the De-Sitter Universe at the 2.3 $\sigma$ level,
and is not compatible with the current value of $\Omega_{M}=0.338\pm 0.018$
found in Brout et al. (2022) at the level of 4.58 $\sigma$ level. We here
clarify that this estimate on $\Omega_{M}$ found by us is larger (more than 1
$\sigma$) than the observed baryonic matter today if we consider the value
found by Planck measurements of CMB (Planck Collaboration et al., 2020) which
is $0.0500\pm 0.0002$. Although the value of the uncertainties in the gold
sample identified by the $F_{X}-F_{UV}$ relation is 2.14 times larger than the
ones obtained by the Pantheon sample of SNe Ia (Scolnic et al., 2018), we can
still obtain uncertainties comparable with the 740 SNe Ia shown in Betoule et
al. (2014) where $\sigma_{\Omega_{M}}=0.042$.

The results on $\Omega_{M}$ obtained with the golden sample of 975 quasars
from the $F_{X}-F_{UV}$ relation (i.e. $\Omega_{M}=0.107\pm 0.047$) is
compatible with $\Omega_{M}=0.125\pm 0.040$ derived from the same golden
sample to which we have removed the low-$z$ sources due to the contamination
of the host galaxies (47 Quasars). However, when a larger sample is available
is necessary to check if the current results still hold within 1 $\sigma$.
Although there have been several studies that have measured $\Omega_{M}$ with
Quasars with lower precision (e.g. Khadka & Ratra 2020b, 2022; Colgáin et al.
2022a), we here for the first time obtain a value with a higher precision with
QSOs alone, which is not due to a circular argument, since it is based on a
flux-flux relation. In addition, the analysis performed by us using the
luminosities and assuming a given cosmological model is meant to show the
great potentiality of Quasars to be used as standardizable candles even
currently when an appropriate sample size and reduced uncertainties is used.
Indeed, the analysis we have shown here is similar to the analysis we
performed in Dainotti et al. (2023b) where we were not interested in knowing
the value of the cosmological parameters, but we were focused on how many
sources, in this case Quasars, are needed to reach the same precision of SNe
Ia Pantheon sample.

In conclusion, we have shown that Quasars alone with the RL relation can now
be upgraded to reliable standard candles to measure cosmological parameters
such as $\Omega_{M}$ with the same precision as SNe Ia, but at a large
redshift, up to 7.5, when a golden sample of Quasars is chosen.

##  6 Acknowledgement

We thank Beta Lusso and Risaliti Guido for the discussion on the role of
selection biases in the sample and to Biagio De Simone to help running a
couple of notebooks for the MCMC sampling.

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