# Updated observational constraints on spatially-flat and non-flat
$\Lambda$CDM and XCDM cosmological models

Javier de Cruz Pérez${}^{\href https://orcid.org/0000-0001-8603-5447}$
<EMAIL_ADDRESS>Departamento de Física Teórica, Universidad Complutense de
Madrid, 28040, Madrid, Spain  Chan-Gyung Park${}^{\href
https://orcid.org/0000-0002-3076-2781}$<EMAIL_ADDRESS>(corresponding author) Division of Science Education and Institute of Fusion
Science, Jeonbuk National University, Jeonju 54896, Republic of Korea  Bharat
Ratra${}^{\href https://orcid.org/0000-0002-7307-0726}$<EMAIL_ADDRESS>Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan,
KS 66506, USA

###### Abstract

We study the performance of six $\Lambda$CDM models, with four of them
allowing for non-flat spatial hypersurfaces (non-zero current value of the
spatial curvature density parameter $\Omega_{k}$) and three of them allowing
for a non-unity value of the lensing consistency parameter $A_{L}$. We also
study a set of six XCDM models where the non-evolving cosmological constant
$\Lambda$ dark energy density is replaced by a dynamical dark energy density
X-fluid parameterized by a non-evolving equation of state parameter $w$. For
the non-flat models we consider two different primordial power spectra, Planck
$P(q)$, used by the Planck collaboration, and new $P(q)$, resulting from
quantum fluctuations in a not-necessarily-very-slow-roll non-flat inflation
model. These models are constrained by and tested against: Planck 2018 CMB
temperature and polarization power spectra data (P18); Planck 2018 CMB lensing
potential power spectrum data (lensing); and, an updated compilation of baryon
acoustic oscillation, type Ia supernova, Hubble parameter [$H(z)$], and growth
factor [$f\sigma_{8}$] data points [collectively denoted by non-CMB (new)
data], individually and jointly. P18 data favor $\Omega_{k}<0$ (closed spatial
geometry) for the $\Lambda$CDM and XCDM models and $w<-1$ (phantom-like
dynamical dark energy) for the XCDM models while non-CMB (new) data favor
$\Omega_{k}>0$ (open geometry) in the case of the $\Lambda$CDM models and
$\Omega_{k}<0$ (closed geometry) and $w>-1$ (quintessence-like dynamical dark
energy) for the XCDM models. When P18 and non-CMB (new) data are jointly
analyzed there is weak evidence in favor of open spatial geometry and moderate
evidence in favor of quintessence-like dynamical dark energy. On the other
hand, regardless of data considered, $A_{L}>1$ is always favored, with
different degrees of evidence, even for P18+lensing+non-CMB (new) data.
According to Akaike and deviance information criterion results,
$A_{L}$-varying models are positively favored over the flat $\Lambda$CDM model
for P18+lensing+non-CMB (new) data. The XCDM model cosmological parameter
constraints obtained from P18 or P18+lensing data and from non-CMB (new) data
are incompatible at $>3\sigma$, ruling out the three $A_{L}=1$ XCDM models at
$>3\sigma$. In the nine models not ruled out by $>3\sigma$ incompatibilities
between parameter values determined from different data sets, for the
P18+lensing+non-CMB (new) data set we find little deviation from flat geometry
and moderate deviation from a cosmological constant. In all six non-flat
models that are not ruled out at $>3\sigma$, open geometry is mildly favored
(by at most $0.8\sigma$), and in all three XCDM+$A_{L}$ models (that are not
ruled out at $>3\sigma$) quintessence-like dynamical dark energy is moderately
favored (by at most $1.6\sigma)$. In the $A_{L}=1$ non-flat $\Lambda$CDM
cases, we find for P18+lensing+non-CMB (new) data $\Omega_{k}=0.0009\pm
0.0017$ [$0.0008\pm 0.0017$] for the Planck [new] $P(q)$ model, favoring open
geometry at 0.53$\sigma$ [0.47$\sigma$]. Given these results, the flat
$\Lambda$CDM model remains the simplest (largely) observationally-consistent
cosmological model. Our cosmological parameter constraints obtained for the
flat $\Lambda$CDM model (and other models), when P18+lensing+non-CMB (new)
data are considered, are the most restrictive results to date.

###### pacs:

98.80.-k, 95.36.+x

##  I Introduction

The six-parameter spatially-flat $\Lambda$CDM model, [1], built within the
framework of general relativity, is the simplest observationally-consistent
cosmological model, and is now commonly recognized as the standard model of
cosmology. As dominant low-redshift stress-energy building blocks, the model
uses a non-evolving cosmological constant $\Lambda$ dark energy density and a
pressure-less cold dark matter (CDM) component and assumes flat spatial
hypersurfaces. From the perspective of general relativity, the observed
currently accelerated cosmological expansion of the Universe is caused by
gravity sourced by the currently dominant cosmological constant.

The flat $\Lambda$CDM model is largely observationally consistent, however
some recent data hint at potential discrepancies, such as differences in the
values of the Hubble constant, $H_{0}$, and the amplitude of matter
fluctuations, $\sigma_{8}$, measured using different techniques [2, 3, 4, 5,
6, 7], or some anomalies that appear when we compare theoretical predictions
of the model, based on best-fit cosmological model parameter values, with
actual observations. These potential discrepancies motivate studying
extensions of the flat $\Lambda$CDM model.

Cosmic microwave background (CMB) radiation anisotropy measurements have so
far generally provided the most restrictive constraints on cosmological
parameters and these data have been used to test extensions of the
$\Lambda$CDM model with the aim of studying the issues mentioned above. In
particular, the Planck 2018 TT,TE,EE+lowE (hereafter denoted by P18) data set
[8] has been used to test three single parameter extensions of the standard
model: what we call the $\Lambda$CDM Planck $P(q)$ model which makes use of a
particular expression for $P(q)$, the primordial power spectrum of matter
inhomogeneities in non-flat models that allow for a non-zero $\Omega_{k}$
spatial curvature density parameter (see Sec. III below for details); the flat
XCDM parameterization which assumes a time-evolving dark energy density
$\rho_{\text{DE}}\sim a^{-3(1+w)}$ with $a$ being the scale factor and $w$ the
constant equation of state parameter of the dynamical dark energy fluid; and,
the flat $\Lambda$CDM+$A_{L}$ model where the amplitude of the gravitational
potential power spectrum is rescaled by the phenomenological lensing
consistency parameter $A_{L}$, [9], in such a way that $A_{L}=1$ corresponds
to recovering the theoretically predicted (using the best-fit cosmological
parameter values) amount of weak lensing of the CMB anisotropy. When analyzing
P18 data, the results for the new non-standard parameter in these three
extensions of the flat $\Lambda$CDM model are:
$\Omega_{k}=-0.044^{+0.018}_{-0.015}$ which represents 2.44$\sigma$ evidence
in favor of non-flat (closed) spatial hypersurfaces; $w=-1.58^{+0.16}_{-0.35}$
indicating a 3.63$\sigma$ preference for phantom-like behavior of the
dynamical dark energy component; and, $A_{L}=1.180\pm 0.065$ which says
$A_{L}>1$ is preferred over $A_{L}=1$ at 2.77$\sigma$. Below, using
statistical criteria, we will compare the performance of the standard flat
$\Lambda$CDM model to these three extensions when it comes to fitting P18
data. It turns out that, to varying degrees of significance, all three
extensions are favored over the standard model.

We showed in [10], that for P18 data the $\Omega_{k}$-varying and
$A_{L}$-varying one-parameter $\Lambda$CDM extension models can handle the so-
called lensing anomaly, which is related to the amount of weak gravitational
lensing in the CMB power spectra. The trajectory of the CMB photons on their
way to us are bent due to the gravitational effects produced by the
inhomogeneous mass distribution. This effect is commonly referred to as weak
gravitational lensing. When we compute the CMB temperature and polarization
spectra we must account for this weak lensing effect, getting as a result the
lensed CMB spectra. In order to provide a theoretical prediction for the
amount of lensing, given a cosmological model, one must assume values for the
cosmological parameters, [11]. Constraining the flat $\Lambda$CDM model by
using P18 CMB data and then using the obtained cosmological parameter values
to predict the amount of weak lensing expected in the CMB spectra, one finds a
mismatch with the observed CMB power spectra over a small range of multipoles,
[9, 8]. Due to the tight constraints provided by Planck CMB data, there seems
to be no room in the flat $\Lambda$CDM model to alleviate this anomaly, and
consequently alternate models that introduce one or more additional parameters
are considered.

We note that a recent analysis of the updated PR4 Planck data set, [12],
results in updated values, $\Omega_{k}=-0.012\pm 0.010$, $1.2\sigma$ in favor
of closed geometry, and $A_{L}=1.039\pm 0.052$, $0.75\sigma$ in favor of
$A_{L}>1$. These new measurements are more consistent with the flat
$\Lambda$CDM model values and show less evidence in favor of non-flat spatial
hypersurfaces and $A_{L}>1$ than do the P18 data set results, partly because
of updated PR4 Planck data and partly because of the different likelihoods
used in the new analysis.

In this work, in addition to considering $\Omega_{k}$ and $A_{L}$ as
additional fitting parameters (see [10] for a detailed study, some results of
which we update in the present paper where we use updated data), we also study
the possibility of having a time-evolving dark energy density where the
equation of state parameter $w$ is constant but allowed to vary from the
cosmological constant value of $w=-1$. Here $w$ is the ratio of the pressure
to the energy density of the dynamical dark energy X-fluid and we refer to
this as the XCDM parametrization or model. We emphasize that to remove the
instability, the speed of sound squared is arbitrarily set to $c^{2}_{s}=1$ in
the XCDM parameterization. The XCDM parameterization is not a physical model
but it has been widely used to analyze data and this is why we study it in
detail in this paper. For recent discussions of observational constraints on
the XCDM model see [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and
references therein. In this paper we present results from the most complete
analysis of the XCDM models to date.

In physical dynamical dark energy models the dark energy component can be
modelled as a dynamical scalar field $\phi$ with its associated potential
energy density $V(\phi)$, [26, 27]. For an appropriate $V(\phi)$ the scalar
field energy density $\rho_{\phi}$ evolves slowly over time until it overcomes
other contributions to the cosmological energy budget and becomes the dominant
component giving rise to the observed currently accelerated cosmological
expansion of the Universe.

In this paper we study twelve cosmological models (six of them are
$\Lambda$CDM models, which we previously studied in [10] and present updated
results for here, and the other six are XCDM models), namely: the flat
$\Lambda$CDM (+$A_{L}$), the non-flat $\Lambda$CDM Planck $P(q)$ (+$A_{L}$),
the non-flat $\Lambda$CDM new $P(q)$ (+$A_{L}$), the flat XCDM (+$A_{L}$), the
non-flat XCDM Planck $P(q)$ (+$A_{L}$), and the non-flat XCDM new $P(q)$
(+$A_{L}$) model. For the non-flat models we consider two different primordial
power spectra, Planck $P(q)$ and new $P(q)$, the details of which are given in
Sec. III. For recent discussions of observational constraints on spatial
curvature see [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] and references therein. In this
paper we present the most restrictive constraints on spatial curvature (in
$\Lambda$CDM and XCDM models) to date.

We use combinations of data to place constraints on the cosmological
parameters of each model and in particular we want to measure the values of
$\Omega_{k}$, $w$, and $A_{L}$. We also want to constrain the other six
primary parameters that all these models share (the conventional six
parameters of the flat $\Lambda$CDM model) as well as constrain the derived
parameters $H_{0}$, $\Omega_{m}$ (the current value of the non-relativistic
matter density parameter), and $\sigma_{8}$. We are also interested in
determining which parameters are measured in a cosmological model independent
manner from these data, amongst the twelve models we study.

The different data sets we use in this work are P18 data, Planck 2018 CMB weak
lensing data, and non-CMB data, consisting of baryon acoustic oscillation
(BAO) measurements, type Ia supernova (SNIa) data, Hubble parameter [$H(z)$]
data points, and a collection of $f\sigma_{8}$ growth factor measurements. We
consider two different non-CMB data compilations, the non-CMB (old) data set
we used in [10], and an updated non-CMB (new) data set, updated with respect
to the non-CMB (old) data set. See Sec. II for details. Adding P18 lensing
data, and especially adding non-CMB data to the mix, alter the conclusions
mentioned above that are based on just P18 data.

In the following we briefly summarize the main results obtained in this work.
Assuming that the data sets we use are correct and that there are no
unaccounted systematics, the three XCDM models with $A_{L}=1$ are ruled out at
$>3\sigma$ due to incompatibilities between P18 data and non-CMB (new) data
cosmological parameter constraints. Extending these three models by adding a
varying lensing consistency parameter, $A_{L}$, reduces the incompatibilities
between P18 data and non-CMB (new) data constraints, thus allowing for joint
analyzes of P18 and non-CMB (new) data in the context of these models. In
these models P18 data favor $\Omega_{k}<0$ (closed geometry), $w<-1$ (phantom-
like dynamical dark energy), and $A_{L}>1$, whereas when Planck CMB lensing
data are added to the mix the evidence in favor of $\Omega_{k}<0$ and
$A_{L}>1$ decreases but that in favor of $w<-1$ is barely affected. When non-
CMB (new) data are included in the analysis the conclusions obtained with P18
and P18+lensing data change and the evidence previously favoring
$\Omega_{k}<0$ and $w<-1$ subsides, giving rise to a preference for
$\Omega_{k}>0$ (open geometry) and $w>-1$ (quintessence-like dynamical dark
energy).

Considering only the nine models not ruled out by $>3\sigma$ incompatibilities
between parameter values determined from different data sets, for the
P18+lensing+non-CMB (new) data set we find little deviation from a flat
geometry and moderate deviation from a cosmological constant, with the biggest
deviations being $\Omega_{k}=0.0015\pm 0.0019$ in the XCDM Planck and new
$P(q)+A_{L}$ models, which favor open geometry and are 0.79$\sigma$ from flat
geometry, and $w=-0.958\pm 0.026$ in the XCDM Planck $P(q)+A_{L}$ model, which
favors quintessence-like dark energy and is 1.62$\sigma$ from a cosmological
constant. In all six non-flat models that are not ruled out at $>3\sigma$,
open geometry is mildly favored, and in all three XCDM+$A_{L}$ models (that
are not ruled out at $>3\sigma$), quintessence-like dark energy is moderately
favored. In the $A_{L}=1$ non-flat $\Lambda$CDM cases, we find for
P18+lensing+non-CMB (new) data $\Omega_{k}=0.0009\pm 0.0017$ [$0.0008\pm
0.0017$] for the Planck [new] $P(q)$ model, favoring open geometry at
0.53$\sigma$ [0.47$\sigma$]. Given these results, the flat $\Lambda$CDM model
remains the simplest (largely) observationally-consistent cosmological model.

Our cosmological parameter constraints, when P18+lensing+non-CMB (new) data
are considered, are the most restrictive results to date. In particular, for
the six primary parameters in the flat $\Lambda$CDM model we get for the
current value of the physical baryonic matter density parameter
$\Omega_{b}h^{2}=0.02249\pm 0.00013$, for the current value of the physical
cold dark matter density parameter $\Omega_{c}h^{2}=0.11849\pm 0.00084$, for
the angular size of the sound horizon at recombination
$100\theta_{\text{MC}}=1.04109\pm 0.00028$, for the reionization optical depth
$\tau=0.0569\pm 0.0071$, for the primordial scalar-type perturbation power
spectral index $n_{s}=0.9685\pm 0.0036$, and for the power spectrum amplitude
$\ln(10^{10}A_{s})=3.046\pm 0.014$, where $h$ is the Hubble constant in units
of 100 km s-1 Mpc-1. Additionally, for the derived parameters, we find
$H_{0}=68.05\pm 0.38$ km s-1 Mpc-1, $\Omega_{m}=0.3059\pm 0.0050$, and
$\sigma_{8}=0.8077\pm 0.0057$. Among models with $A_{L}=1$, these values show
almost model-independent consistency, with differences always below $1\sigma$.
However, when we compare these cosmological parameter values with those
obtained for the $A_{L}$-varying models, we observe larger differences. In
particular, for the six varying $A_{L}$ models relative to the flat
$\Lambda$CDM model we find the maximum differences for $\sigma_{8}$,
$1.08\sigma$ for models with $w=-1$ and $1.80\sigma$ when comparing to
varying-$w$ models, with all other parameters agreeing to better than
1$\sigma$. As in our previous work [10] we once again find that the
$A_{L}$-varying models are the most favored by the most complete data set
considered in this work, namely the P18+lensing+non-CMB (new) data set.

The outline of our paper is as follows. In Sec. II we provide details of
observational data we use to constrain cosmological parameters in, and to test
the performance of, the cosmological models we study. In Sec. III we briefly
describe the main features of the models studied, as well as the methods
employed for the analyses. Section IV, which represents the main part of the
article, is dedicated to presenting and commenting in detail on all the
results obtained in our analyses. In particular, we discuss the cosmological
parameter constraints obtained, compare the performance of the different
models under study, and analyze whether there are tensions among the
cosmological parameter constraints derived from different data. In Sec. V we
summarize the most significant results obtained in the previous section, and,
finally, in Sec. VI we present our conclusions.

##  II Data

The data we use in this work are the Planck cosmic microwave background
radiation temperature and polarization anisotropy power spectra and the
lensing potential power spectrum, the Pantheon+ type Ia supernovae
compilation, and baryon acoustic oscillation, Hubble parameter, and growth
rate measurements.

###  II.1 Planck 2018 CMB data

We use the Planck 2018 TT,TE,EE+lowE (P18) CMB temperature and polarization
power spectra as well as the Planck lensing potential power spectrum [8]. Here
TT, TE, and EE denote the temperature-only power spectra at low multipole
number $\ell$ ($2\leq\ell\leq 29$) and high $\ell$ ($30\leq\ell\leq 2508$), TE
cross-power spectrum, and $E$-mode polarization power spectrum at high $\ell$
($30\leq\ell\leq 1996$), respectively, while lowE denotes the $E$-mode
polarization power spectrum at low $l$ ($2\leq\ell\leq 29$). For high-$\ell$
P18 data we use the Planck 2018 baseline Plik likelihood (see Sec. 2.2.1 of
[8]). We can more restrictively constrain cosmological parameters by adding
the power spectrum of the lensing potential measured by Planck [55] to the P18
data. In the following we refer to the P18 plus lensing data combination as
P18+lensing data.

###  II.2 Non-CMB data

In addition to CMB data, we collect and use a number of non-CMB data sets to
constrain model parameters. We denote the non-CMB data compilation used in our
previous work [10] as non-CMB (old), and the new non-CMB data compilation
assembled and used here, with updates described below, as non-CMB (new).

Compared to our earlier work [10], we replace the 1048 Pantheon SNIa [56] and
the binned DES 3yr SNIa data points [57] with a subset (as discussed below) of
the new 1701 Pantheon+ compilation data points [58] that also include DES SNIa
measurements. We replace the BAO data point $D_{A}(z=0.81)/r_{d}=10.75\pm
0.43$ from [59] with $D_{M}(z=0.835)/r_{d}=18.92\pm 0.51$ from [60]; the
distances $D_{A}$ and $D_{M}$ are defined below. In the growth rate data we
now add the data point $f\sigma_{8}(z=0.013)=0.46\pm 0.06$ from [61]. In the
Hubble parameter data we now include the data point $H(z=0.75)=98.8\pm
33.6~{}\textrm{km}\textrm{s}^{-1}\textrm{Mpc}^{-1}$ from [62] and for some of
the $H(z)$ data points we now account for a non-diagonal covariance matrix as
explained below.

####  II.2.1 BAO data

We use a collection of the latest BAO data points measured at various
redshifts. Table 1 lists the data sets, effective redshifts, observables, and
measurement values for the 16 BAO data points we use. All BAO data we use
account for all known systematic errors. The six BOSS Galaxy, three eBOSS LRG,
three eBOSS Quasar, and two Ly$\alpha$-forest BAO measurements are correlated
and their covariance matrices are given below. We do not use the ELGs data
from [63, 64] because the corresponding posterior distribution is highly non-
Gaussian and the full likelihood (not a summary data point) must be used in
this case.

The BAO quantities listed in Table 1 correspond to several distances ($D_{V}$,
$D_{M}$, and $D_{H}$) and to the growth rate $f\sigma_{8}$. The angle-averaged
distance is $D_{V}(z)=\left[czD_{M}^{2}(z)/H(z)\right]^{1/3}$, where $H(z)$ is
the Hubble parameter at redshift $z$ and $D_{M}(z)$ is the transverse comoving
distance. Using the proper angular diameter distance

$D_{A}(z)=\frac{c}{H_{0}}\frac{f_{k}\left[H_{0}\sqrt{|\Omega_{k}|}\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}\right]}{(1+z)\sqrt{|\Omega_{k}|}},$
(1)

where $f_{k}[x]=\sin x$, $x$, and $\sinh x$ for closed ($k=1$;
$\Omega_{k}<0$), flat ($k=0$; $\Omega_{k}=0$), and open ($k=-1$,
$\Omega_{k}>0$) Universes, respectively, $D_{M}(z)$ is expressed as
$D_{M}(z)=(1+z)D_{A}(z)$. $D_{H}(z)=c/H(z)$ is the Hubble distance at redshift
$z$. In Table 1 BAO distances are divided by the radius of sound horizon
$r_{d}$ at the drag epoch $z_{d}$,

$r_{d}=\int_{z_{d}}^{\infty}\frac{c_{s}(z)}{H(z)}dz,$ (2)

where $c_{s}(z)$ is the sound speed of the photon-baryon fluid.

Table 1: BAO measurements.

Data Set | $z_{\textrm{eff}}$ | Observable | Measurement | Reference
---|---|---|---|---
6dFGS+SDSS MGS | $0.122$ |  $D_{V}(r_{d,\textrm{fid}}/r_{d})$ [Mpc] |  $539\pm 17$ [Mpc] | [65]
BOSS Galaxy | $0.38$ | $D_{M}/r_{d}$ | $10.274\pm 0.151$ | [66]
| $0.38$ | $D_{H}/r_{d}$ | $24.888\pm 0.582$ | [66]
| $0.38$ | $f\sigma_{8}$ | $0.49729\pm 0.04508$ | [66]
| $0.51$ | $D_{M}/r_{d}$ | $13.381\pm 0.179$ | [66]
| $0.51$ | $D_{H}/r_{d}$ | $22.429\pm 0.482$ | [66]
| $0.51$ | $f\sigma_{8}$ | $0.45902\pm 0.03784$ | [66]
eBOSS LRG | $0.698$ | $D_{M}/r_{d}$ | $17.646\pm 0.302$ | [66, 67]
| $0.698$ | $D_{H}/r_{d}$ | $19.770\pm 0.469$ | [66, 67]
| $0.698$ | $f\sigma_{8}$ | $0.47300\pm 0.04429$ | [66, 67]
DES Y3 | $0.835$ | $D_{M}/r_{d}$ | $18.92\pm 0.51$ | [60]
eBOSS Quasar | $1.48$ | $D_{M}/r_{d}$ | $30.21\pm 0.79$ | [68, 69]
| $1.48$ | $D_{H}/r_{d}$ | $13.23\pm 0.47$ | [68, 69]
| $1.48$ | $f\sigma_{8}$ | $0.462\pm 0.045$ | [68, 69]
Ly$\alpha$-forest | $2.334$ | $D_{M}/r_{d}$ | $37.5_{-1.1}^{+1.2}$ | [70]
| $2.334$ | $D_{H}/r_{d}$ | $8.99_{-0.19}^{+0.20}$ | [70]

Note: The sound horizon size of the fiducial model is
$r_{d,\textrm{fid}}=147.5~{}\textrm{Mpc}$ in [65].

The covariance matrix between measurement errors for BOSS Galaxy data is

$\mathbf{C}_{\textrm{BOSS-
Galaxy}}=\begin{pmatrix}0.022897&-0.02007&0.0026481&0.013487&-0.0081402&0.0010292\\\
-0.02007&0.33849&-0.0085213&-0.016024&0.13652&-0.0038002\\\
0.0026481&-0.0085213&0.0020319&0.001325&-0.0023012&0.000814158\\\
0.013487&-0.016024&0.001325&0.032158&-0.020091&0.0026409\\\
-0.0081402&0.13652&-0.0023012&-0.020091&0.23192&-0.0055377\\\
0.0010292&-0.0038002&0.000814158&0.0026409&-0.0055377&0.0014322\end{pmatrix}.$
(3)

The covariance matrix for eBOSS LRG data is

$\mathbf{C}_{\textrm{eBOSS-LRG}}=\begin{pmatrix}0.09114&-0.033789&0.0024686\\\
-0.033789&0.22009&-0.0036088\\\ 0.0024686&-0.0036088&0.0019616\end{pmatrix}.$
(4)

The covariance matrix for eBOSS Quasar data is

$\mathbf{C}_{\textrm{eBOSS-Quasar}}=\begin{pmatrix}0.6227&0.01424&0.02257\\\
0.01424&0.2195&-0.007315\\\ 0.02257&-0.007315&0.002020\end{pmatrix}.$ (5)

The covariance matrix for Ly-$\alpha$ forest data is

$\mathbf{C}_{\textrm{Ly}\alpha}=\begin{pmatrix}1.3225&-0.1009\\\
-0.1009&0.0380\end{pmatrix}.$ (6)

####  II.2.2 SNIa data

For SNIa data, we use a subset of the new 1701 data point Pantheon+
compilation [58], which is determined by removing all SNIa with $z<0.01$ since
these data points are model dependent due to the peculiar velocities that have
to be considered. This cut leaves us with 1590 data points, spanning the
redshift range $0.01016\leq z\leq 2.26137$. The covariance matrix includes not
only the statistical errors but also the systematic ones. The data set,
likelihoods, and all necessary information can be found at
https://github.com/PantheonPlusSH0ES. In this work we use the $\chi^{2}$ of
the SNIa data set where the absolute magnitude of the SN is marginalized over,
which differs from the $\chi^{2}$ value obtained from the analysis where the
SN absolute magnitude is not marginalized over.

####  II.2.3 $H(z)$ data

Table 2: $H(z)$ measurements. $z$ | $H(z)$ | Reference
---|---|---
| (km s-1 Mpc-1) |
$0.07$ | $69.0\pm 19.6$ | [71]
$0.09$ | $69.0\pm 12.0$ | [72]
$0.12$ | $68.6\pm 26.2$ | [71]
$0.17$ | $83.0\pm 8.0$ | [72]
$0.2$ | $72.9\pm 29.6$ | [71]
$0.27$ | $77.0\pm 14.0$ | [72]
$0.28$ | $88.8\pm 36.6$ | [71]
$0.4$ | $95.0\pm 17.0$ | [72]
$0.47$ | $89.0\pm 50.0$ | [73]
$0.48$ | $97.0\pm 62.0$ | [74]
$0.75$ | $98.8\pm 33.6$ | [62]
$0.88$ | $90.0\pm 40.0$ | [74]
$0.9$ | $117.0\pm 23.0$ | [72]
$1.3$ | $168.0\pm 17.0$ | [72]
$1.43$ | $177.0\pm 18.0$ | [72]
$1.53$ | $140.0\pm 14.0$ | [72]
$1.75$ | $202.0\pm 40.0$ | [72]
$0.1791$ | $74.91$ | [75]
$0.1993$ | $74.96$ | [75]
$0.3519$ | $82.78$ | [75]
$0.3802$ | $83.0$ | [75]
$0.4004$ | $76.97$ | [75]
$0.4247$ | $87.08$ | [75]
$0.4497$ | $92.78$ | [75]
$0.4783$ | $80.91$ | [75]
$0.5929$ | $103.8$ | [75]
$0.6797$ | $91.6$ | [75]
$0.7812$ | $104.5$ | [75]
$0.8754$ | $125.1$ | [75]
$1.037$ | $153.7$ | [75]
$1.363$ | $160.0$ | [75]
$1.965$ | $186.5$ | [75]

We use the 32 Hubble parameter measurements provided in Table 1 of [76], and
listed in Table 2 here, that cover the redshift range $0.070\leq z\leq 1.965$.
This new compilation includes an additional data point, $H(z=0.75)=98.8\pm
33.6~{}\textrm{km}\textrm{s}^{-1}\textrm{Mpc}^{-1}$, compared to the data set
used in our previous paper. We now also account for the correlation between
the 15 measurements provided in [77, 78, 79]. The corresponding covariance
matrix must be computed following the steps in the code at
https://gitlab.com/mmoresco/CCcovariance.

####  II.2.4 $f\sigma_{8}$ data

In addition to the growth rate data included in the BAO data compilation of
Table 1, we use other growth rate measurements. These $f\sigma_{8}$ measures
are obtained from peculiar velocity analysis [80, 81, 82] or redshift-space
distortion analysis [83, 84, 85, 86, 87], and are listed in Table 3, and
include an additional data point, $f\sigma_{8}(z=0.013)=0.46\pm 0.06$,
compared to the data set used in our previous paper.

Table 3: $f\sigma_{8}$ measurements. Survey | $z$ | Measurement | Reference
---|---|---|---
ALFALFA | $0.013$ | $0.46\pm 0.06$ | [61]
IRAS | $0.02$ | $0.398\pm 0.065$ | [80, 81]
6dFGS+SDSS | $0.035$ | $0.338\pm 0.027$ | [82]
SDSS DR7 | $0.1$ | $0.376\pm 0.038$ | [83]
eBOSS LRG | $0.18$ | $0.29\pm 0.10$ | [84, 85]
eBOSS LRG | $0.38$ | $0.44\pm 0.06$ | [84, 85]
VIPERS | $0.6$ | $0.49\pm 0.12$ | [86]
VIPERS | $0.86$ | $0.46\pm 0.09$ | [86]
FastSound | $1.36$ | $0.482\pm 0.116$ | [87]

##  III Methods

We use the Markov chain Monte Carlo (MCMC) method to determine the range of
$\Lambda$CDM and XCDM model parameters that are favored by the measurements we
consider. More precisely, we use the CAMB/COSMOMC program (October 2018
version) [88, 89, 90] for this purpose. CAMB computes the evolution of
spatially inhomogenous perturbations of radiation and matter densities and
predicts the power spectra of matter and CMB anisotropy perturbations as
functions of cosmological parameter values, while COSMOMC compares the Planck
CMB likelihood data and the non-CMB data sets with these model predictions to
find the likelihood distribution of cosmological model parameters.

When P18 data are used, the cosmological parameters that are measured, that
are common across all cosmological models we study, are the current baryon
density ($\Omega_{b}h^{2}$), the current cold dark matter density
($\Omega_{c}h^{2}$), the angular size of the sound horizon at recombination
defined in CAMB/COSMOMC ($\theta_{\textrm{MC}}$), the reionization optical
depth ($\tau$), and the amplitude ($A_{s}$) and the spectral index ($n_{s}$)
of the primordial scalar-type perturbation spectrum, with the addition of the
current curvature density parameter ($\Omega_{k}$) for non-flat cosmological
models and the dark energy equation of state parameter $w$ for XCDM models.
Here $\Omega_{b}$ and $\Omega_{c}$ are the current values of the baryon and
cold dark matter density parameters, respectively, and $h$ is the Hubble
constant in units of 100 km s-1 Mpc-1. The gravitational lensing consistency
parameter $A_{L}$ is set to unity in some cases, but in some cases $A_{L}$ is
a free parameter to be determined from data. For a detailed definition of the
parameters of cosmological models, see the Planck team’s paper on cosmological
parameter estimation [8]. In our paper we consider only the primary parameters
that define the cosmological model, and use the Planck team’s settings for
COSMOMC’s internal calibration or nuisance parameters (e.g., calPlanck). For
example, in the flat $\Lambda$CDM model, the primary cosmological parameters
are $(\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, $100\theta_{\textrm{MC}}$, $\tau$,
$\ln(10^{10}A_{s})$, $n_{s}$). In this paper, when we use P18 data to measure
cosmological parameters we assume flat priors for these parameters, non-zero
over: $0.005\leq\Omega_{b}h^{2}\leq 0.1$, $0.001\leq\Omega_{c}h^{2}\leq 0.99$,
$0.5\leq 100\theta_{\textrm{MC}}\leq 10$, $0.01\leq\tau\leq 0.8$, $0.8\leq
n_{s}\leq 1.2$ (for the Planck $P(q)$, see below), and
$1.61\leq\ln(10^{10}A_{s})\leq 3.91$. We also adopt flat priors non-zero over
$-0.5\leq\Omega_{k}\leq 0.5$ for non-flat models, $-3.0\leq w\leq 0.2$ for
XCDM models, $0.8\leq n_{s}<1$ for models with the new $P(q)$ (see below), and
$0\leq A_{L}\leq 10$ for $A_{L}$-varying models, except in the case of the P18
data constraints for the non-flat XCDM$+A_{L}$ models where $0.8<A_{L}\leq
10$. When estimating parameters from non-CMB data we fix the values of the
optical depth $\tau$ and the spectral index $n_{s}$ to those obtained from P18
data (since non-CMB data are unable to constrain these parameters) and
determine the remaining cosmological parameters. When showing the results of
the parameter constraints, we also list the three derived parameters $H_{0}$,
$\Omega_{m}$, and $\sigma_{8}$, which are obtained from the primary parameters
of the cosmological model.

The MCMC chains are considered to have converged when the Gelman and Rubin $R$
statistic, provided by COSMOMC, satisfies the condition $R-1<0.01$. For each
model and data set, we use the converged MCMC chains to compute the average
values, confidence intervals, and likelihood distributions of model
parameters. We utilize the GetDist code [91] for this purpose. As for the
confidence intervals, we typically use the standard deviation estimated from
the MCMC chains. However, for parameters of special interest to us ($H_{0}$,
$\Omega_{k}$, $w$, and $A_{L}$) with highly asymmetric likelihood
distributions or with bounds, we also compute and record (in parentheses in
the tables) a second set of limits (see $\S$2.4 of [92]). In the case of two
tail limits we compute the second 68% confidence interval as the one between
the two points with equal marginalized probability density such that 68% of
the samples lie within the two points with 16% of the sample in each tail. The
second one tail limit we compute and record is a 95% confidence limit with 5%
of the samples in the tail. We use these second limits in our discussions of
parameters with highly asymmetric likelihood distributions or with bounds.
When comparing the sizes of error bars of cosmological parameter values in
different models or derived using different data, we always compare the
standard deviation error bars. When we compute the increase or decrease in the
size of the error bars between two different estimates of a cosmological
parameter, we use the expression
$[]\%=100\left(1-\sigma_{X}/\sigma_{Y}\right)$ where $\sigma_{X}$
($\sigma_{Y}$) is the error bar of the result obtained with either the
smallest (largest) amount of data or with the smallest (largest) number of
free parameters.

In our study, we use $\Lambda$CDM and XCDM cosmological models with different
initial spectrum of scalar-type perturbations. For the flat tilted
$\Lambda$CDM and XCDM models, the primordial scalar-type energy density
perturbation power spectrum is given by

$P_{\delta}(k)=A_{s}\left(\frac{k}{k_{0}}\right)^{n_{s}},$ (7)

where $k$ is the wavenumber and $n_{s}$ and $A_{s}$ are the spectral index and
the amplitude of the spectrum at pivot scale $k_{0}=0.05~{}\textrm{Mpc}^{-1}$.
This power spectrum arises from quantum fluctuations during an early epoch of
power-law inflation with a scalar-field potential energy density of
exponential form in a spatially flat cosmological model [93, 94, 95].

For the tilted non-flat models, we use a primordial power spectrum of the form

$P_{\delta}(q)\propto\frac{(q^{2}-4K)^{2}}{q(q^{2}-K)}\left(\frac{k}{k_{0}}\right)^{n_{s}-1},$
(8)

where $q=\sqrt{k^{2}+K}$ is the wavenumber in a model with spatial curvature
$K=-(H^{2}_{0}/c^{2})\Omega_{k}$. This spectrum is a phenomenological version
that combines the untilted spectrum of primordial perturbations derived from
very-slow-roll inflation in non-flat Universes [96, 97] and a power-law
spectrum to account for tilt. Recently a numerical study of quantum
fluctuations in closed inflation models finds that it is possible to get a
primordial power spectrum very close to this one in closed models for some
initial conditions, [98]. We will denote this spectrum as Planck $P(q)$ as it
has been widely used in most studies of non-flat cosmological models,
including by the Planck team.

Finally we consider a third power spectrum from a not-necessarily-very-slow-
roll non-flat inflation model

$P_{\delta}(q)\propto(q^{2}-4K)^{2}\left|P_{\mathcal{R}}(A)\right|,$ (9)

where $P_{\mathcal{R}}(A)$ is given in Eqs. (14) and (17) of [10], depending
on whether the curvature of the Universe is positive or negative,
respectively, [99].

Equations (7), (8), and (9) correspond to the initial density power spectra
$P_{\delta}$’s that can be derived from the Poisson equation (see Eq. 3.43 of
Ref. [100]) but as the CAMB input for the primordial power spectrum we need
different forms of power spectra. The function ScalarPower(k,ix) of CAMB
power_tilt.f90 computes the input primordial power spectrum (Eq. 3.26 of
[100]), $\mathcal{P}_{\mathcal{R}}(k)=A_{s}(k/k_{0})^{n_{s}-1}$, which is the
appropriate form of the CAMB input primordial power spectrum for the tilted
flat and the tilted non-flat model with the Planck $P(q)$. However, for the
tilted non-flat model with the new $P(q)$, the primordial power spectrum
should be replaced with the one derived from the tilted non-spatially-flat
inflation model of [99]. There are slight differences in the notation between
[99] and [100]. First, the primordial power spectrum in $q$-space
$\tilde{\mathcal{P}}_{\mathcal{R}}(q)$, is related to
$\mathcal{P}_{\mathcal{R}}(k)$ by
$\tilde{\mathcal{P}}_{\mathcal{R}}(q)=\mathcal{P}_{\mathcal{R}}(k)q^{2}/(q^{2}-K)$
(see Eq. 3.29 of [100]), which is equivalent to $\mathcal{P}_{\mathcal{R}}(A)$
of [99]. Besides,
$\mathcal{P}_{\mathcal{R}}(A)=(A+1)^{3}P_{\mathcal{R}}(A)/(2\pi^{2})$ for the
closed model and
$\mathcal{P}_{\mathcal{R}}(A)=A^{3}P_{\mathcal{R}}(A)/(2\pi^{2})$ for the open
model (see Eqs. B10 and B14 of [99]), where $\nu=q/\sqrt{|K|}=A+1$ and $\nu=A$
for the closed and open models, respectively, and $P_{\mathcal{R}}(A)$ is
given in Eqs. (57) and (59) of [99] for closed and open models, respectively.
For example, for the closed model, the input primordial spectrum for CAMB
becomes
$\mathcal{P}_{\mathcal{R}}(k)=(q^{2}-K)/q^{2}\cdot(A+1)^{3}/(2\pi^{2})P_{\mathcal{R}}(A)$.
When applying the new primordial spectrum, the amplitude must be normalized so
that the amplitude at the pivot scale $k_{0}$ is $A_{s}$. For the open model,
we can also obtain a similar relation considering the negative sign of
curvature $K$ and $\nu=A$.

We constrain the various dark energy models described above by using various
combinations of observational data. We focus on how five combinations of data
constrain each model: P18 data, P18+lensing data, non-CMB data, P18+non-CMB
data, and P18+lensing+non-CMB data. We also investigated differences in
cosmological constraints resulting from old and new versions of non-CMB data.
We compare the performance of the models by examining how well each model fits
different combinations of observations using the AIC and DIC information
criteria.

The Akaike information criterion (AIC) is defined as

$\textrm{AIC}=\chi^{2}_{\textrm{min}}+2n,$ (10)

where $\chi^{2}_{\textrm{min}}$ is the minimum value of $\chi^{2}$ for the
best-fit cosmological parameters and $n$ is the number of independent
cosmological parameters. The term $2n$ corresponds to a penalty to the
goodness-of-fit for increasing the number of model parameters. The definition
of AIC above is valid only for data sets with a large number of data points.
All the data combinations we use here have a sufficiently large number of data
points to justify using this AIC definition.

To quantify how much a data set favors the model, we also use the deviance
information criterion (DIC) defined as

$\textrm{DIC}=\chi^{2}(\hat{\theta})+2p_{D},$ (11)

where $p_{D}=\overline{\chi^{2}}-\chi^{2}(\hat{\theta})$ and the term $2p_{D}$
penalizes the goodness-of-fit for increasing the number of model parameters.
Here $\overline{\chi^{2}}$ denotes the average of $\chi^{2}$’s estimated from
the MCMC chains and $\chi^{2}(\hat{\theta})$ is the value of $\chi^{2}$ at the
best-fit cosmological parameters $\hat{\theta}$.

We use the differences in the AIC and DIC values of the model under study,
computed relative to the tilted flat $\Lambda$CDM model constrained using the
same data set. (We also list similarly defined $\chi^{2}_{\textrm{min}}$
difference values in the Tables.) According to the usual scale, when
$-2\leq\Delta\textrm{AIC},\Delta\textrm{DIC}<0$ there is said to be weak
evidence in favor of the model under study, while when
$-6\leq\Delta\textrm{AIC},\Delta\textrm{DIC}<-2$ there is said to be positive
evidence, when $-10\leq\Delta\textrm{AIC},\Delta\textrm{DIC}<-6$ there is
strong evidence, and when $\Delta\textrm{AIC},\Delta\textrm{DIC}<-10$ there is
very strong evidence in favor of the model under study relative to the tilted
flat $\Lambda$CDM model. This scale also applies if $\Delta\textrm{AIC}$ and
$\Delta\textrm{DIC}$ are positive, but then favors the tilted flat
$\Lambda$CDM model over the model under study.

We want to quantitatively compare how consistent the data sets used to
constrain cosmological parameters in a cosmological model are with each other,
and how the level of consistency varies across models. We used two different
statistical estimators to check for consistency between data sets used in a
given model.

The first estimator we used to check for consistency is based on the DIC
values of the individual data sets, [101], and is defined as

$\mathcal{I}(D_{1},D_{2})=\textrm{exp}\left(-\frac{\mathcal{G}(D_{1},D_{2})}{2}\right),$
(12)

where
$\mathcal{G}(D_{1},D_{2})=\textrm{DIC}(D_{12})-\textrm{DIC}(D_{1})-\textrm{DIC}(D_{2})$
and $D_{1}$ and $D_{2}$ are the two data sets being compared. Here
$\textrm{DIC}(D_{1})$ and $\textrm{DIC}(D_{2})$ are the DIC values estimated
from the MCMC chains when $D_{1}$ and $D_{2}$ are used independently to
constrain the cosmological parameters of a given model, and
$\textrm{DIC}(D_{12})$ is the DIC value that results when $D_{1}$ and $D_{2}$
are used together to constrain cosmological parameters of the model. This
statistical estimator indicates $\log_{10}\mathcal{I}>0$ when the two data
sets used in the cosmological model are consistent with each other.
Conversely, $\log_{10}\mathcal{I}<0$ indicates that the two data sets are
inconsistent. Applying Jeffreys’ scale, we judge the degree of consistency or
inconsistency between two data sets as substantial if
$\lvert\log_{10}\mathcal{I}\rvert>0.5$, strong if
$\lvert\log_{10}\mathcal{I}\rvert>1$, and decisive if
$\lvert\log_{10}\mathcal{I}\rvert>2$, [101].

The second statistical estimator for determining the consistency between two
data sets is the tension probability. The details of this estimator are given
in [102, 103, 104] and in our previous work [10], and we briefly describe it
here. The tension probability is related to the suspiciousness
$S_{D}=R_{D}/I_{D}$ that is defined in terms of the Bayes ratio $R_{D}$ and
the information ratio $I_{D}$. The Bayes ratio
$R_{D}=\mathcal{Z}_{12}/(\mathcal{Z}_{1}\mathcal{Z}_{2})$ where
$\mathcal{Z}_{1}$, $\mathcal{Z}_{2}$, and $\mathcal{Z}_{12}$ are the Bayesian
evidences estimated from $D_{1}$, $D_{2}$, and $D_{12}$, respectively. The
Bayesian evidence is defined as

$\mathcal{Z}_{D}=\int\mathcal{L}_{D}(\theta)\pi(\theta)d\theta,$ (13)

where $\mathcal{L}_{D}(\theta)$ is the likelihood of a model $\theta$ given
the data and $\pi(\theta)$ is the prior of the model. We compute the Bayesian
evidence by using the method of [105]. The information ratio $I_{D}$ is
defined through $\ln(I_{D})=\mathcal{D}_{1}+\mathcal{D}_{2}-\mathcal{D}_{12}$.
Here $\mathcal{D}_{D}$ is the Kullback-Leibler divergence for data $D$, which
is the average over the posterior of the Shannon information

$\mathcal{I}_{S,D}(\theta)=\ln\frac{\mathcal{P}_{D}(\theta)}{\pi(\theta)},$
(14)

where $\mathcal{P}_{\mathcal{D}}(\theta)$ is the posterior distribution [106].
Finally, the tension probability, [102, 103, 104],

$p=\int^{\infty}_{d-2\ln(S_{D})}\\!\\!\frac{x^{d/2-1}e^{-x/2}}{2^{d/2}\Gamma(d/2)}dx,$
(15)

where $d$ is the Bayesian model dimensionality
$d=\tilde{d}_{1}+\tilde{d}_{2}-\tilde{d}_{12}$, where
$\tilde{d}/2=\langle\mathcal{I}_{S,D}^{2}\rangle_{\mathcal{P}_{D}}-\langle\mathcal{I}_{S,D}\rangle^{2}_{\mathcal{P}_{D}}$
and $\Gamma(z)$ is a Gamma function. If $p\lesssim 0.05$ the data sets are in
moderate tension whereas if $p\lesssim 0.003$ they are in strong tension.
Based on the Gaussian formula, we can convert $p$ into a “sigma value”,

$\sigma=\sqrt{2}\textrm{Erfc}^{-1}(1-p),$ (16)

where $\textrm{Erfc}^{-1}$ is the inverse complementary error function. The
value $p=0.05$ ($p=0.003$) corresponds to a 2$\sigma$ (3$\sigma$) Gaussian
standard deviation.

##  IV Results

###  IV.1 $\Lambda$CDM models

In this subsection we study how the inclusion of non-CMB (new) data, an
updated version of the non-CMB (old) data used in [10], in different data set
combinations we use here, affect the cosmological parameter constraints in the
$\Lambda$CDM models and goodness-of-fit results and consistencies between
different data subsets. In the last part of this subsection we summarize the
updated data constraints on cosmological parameters in the $\Lambda$CDM
models.

####  IV.1.1 Non-CMB (old) vs. non-CMB (new) cosmological parameter
constraints

Cosmological parameter constraints for the four-parameter flat $\Lambda$CDM
model from non-CMB (old) and non-CMB (new) data are in the lower half of Table
4 and Fig. 1. We observe some differences in the mean parameter values favored
by the two data sets, but all are below the 1$\sigma$ level. The differences
between the results provided by the two data sets for $\Omega_{c}{h^{2}}$,
$100\theta_{\textrm{MC}}$, and $\ln(10^{10}A_{s})$ are $-0.77\sigma$,
$-0.80\sigma$, and $+0.64\sigma$, respectively. Similar levels of differences
are observed for the derived parameters, with $\Omega_{m}$ showing the largest
at $-0.64\sigma$. On the other hand, a significant change is observed in the
size of the error bars. Compared to the results from the non-CMB (old) data,
the results from the non-CMB (new) data increase the error bars of
$\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, and $H_{0}$ by 24%, 23%, and 26%,
respectively. One major reason for this is the larger error bars associated
with updated $H(z)$ data (those for which there is now a non-diagonal
covariance matrix).

Non-CMB (old) and non-CMB (new) data results obtained for the five-parameter
non-flat $\Lambda$CDM Planck $P(q)$ cosmological model are in the lower half
of Table 5 and in Fig. 2. As in the flat $\Lambda$CDM model, the differences
in the mean values of the primary parameters remain below 1$\sigma$, with
$100\theta_{\textrm{MC}}$ (+0.57$\sigma$) being the largest. The value of the
primary spatial curvature parameter is $\Omega_{k}=-0.032^{+0.056}_{-0.046}$
for non-CMB (old) data which is 0.57$\sigma$ away from flat geometry and
$-0.66\sigma$ from the non-CMB (new) value
$\Omega_{k}=0.010^{+0.046}_{-0.030}$ which in turn is 0.33$\sigma$ in favor of
an open Universe. For mean derived parameters the largest difference is
$-0.39\sigma$ for $\sigma_{8}$. In contrast to the flat $\Lambda$CDM model,
where the error bars of all cosmological parameters obtained from non-CMB
(old) data are smaller than those obtained using non-CMB (new) data, we
observe a different pattern in the non-flat $\Lambda$CDM Planck $P(q)$ model.
This may be due to the degeneracy between parameters and differences in the
ability of the two data sets to constrain cosmological parameters. While for
$\Omega_{b}h^{2}$ (+18%) and $H_{0}$ (+32%) there are increases in the error
bars when we move from non-CMB (old) to non-CMB (new) data, for the rest of
the parameters it is the other way around, with the largest decreases
affecting $100\theta_{\textrm{MC}}$ ($-51$%), $\Omega_{k}$ ($-34$%), and
$\ln(10^{10}A_{s})$ ($-26$%).

When we compare the results obtained with non-CMB (old) and non-CMB (new)
data, displayed in Table 6 and in Fig. 3, for the five-parameter non-flat
$\Lambda$CDM new $P(q)$ model, we see very similar results to the case of the
non-flat $\Lambda$CDM Planck $P(q)$ model. We observe mild differences between
the mean values of the primary parameters, with the largest being
+0.57$\sigma$ for $100\theta_{\textrm{MC}}$. As for $\Omega_{k}$, the non-CMB
(old) result $\Omega_{k}=-0.036^{+0.056}_{-0.047}$ favors non-flat
hypersurfaces at 0.64$\sigma$ and differs by $-0.66\sigma$ with the non-CMB
(new) value $\Omega_{k}=0.006^{+0.051}_{-0.030}$, which is 0.20$\sigma$ away
from flat. In regard to changes in the size of the error bars, we observe an
increase for $\Omega_{b}h^{2}$ (+15%) and for $H_{0}$ (+28%), while reductions
affect $100\theta_{\textrm{MC}}$ ($-39$%), $\Omega_{k}$ ($-27$%), and
$\ln(10^{10}A_{s})$ ($-17$%).

The non-CMB (old) data and non-CMB (new) data $\Delta\text{DIC}$ and
$\Delta\text{AIC}$ values, between the flat $\Lambda$CDM model and the two
non-flat $\Lambda$CDM models, in Tables 5 and 6, do not show significant
differences.

Table 4: Mean and 68% confidence limits of flat $\Lambda\textrm{CDM}$ model parameters constrained by TT,TE,EE+lowE (P18) and P18+lensing, and non-CMB data sets. The Hubble constant $H_{0}$ has units of km s-1 Mpc-1.  | Flat $\Lambda$CDM models
---|---
Parameter | P18 | P18+lensing | P18+lensing+non-CMB (old) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.02236\pm 0.00015$ | $0.02237\pm 0.00014$ | $0.02250\pm 0.00013$ | $0.02249\pm 0.00013$
$\Omega_{c}h^{2}$ | $0.1202\pm 0.0014$ | $0.1200\pm 0.0012$ | $0.11838\pm 0.00083$ | $0.11849\pm 0.00084$
$100\theta_{\textrm{MC}}$ | $1.04090\pm 0.00031$ | $1.04091\pm 0.00031$ | $1.04110\pm 0.00029$ | $1.04109\pm 0.00028$
$\tau$ | $0.0542\pm 0.0079$ | $0.0543\pm 0.0073$ | $0.0569\pm 0.0071$ | $0.0569\pm 0.0071$
$n_{s}$ | $0.9649\pm 0.0043$ | $0.9649\pm 0.0041$ | $0.9688\pm 0.0036$ | $0.9685\pm 0.0036$
$\ln(10^{10}A_{s})$ | $3.044\pm 0.016$ | $3.044\pm 0.014$ | $3.046\pm 0.014$ | $3.046\pm 0.014$
$H_{0}$ | $67.28\pm 0.61$ | $67.34\pm 0.55$ | $68.09\pm 0.38$ | $68.05\pm 0.38$
$\Omega_{m}$ | $0.3165\pm 0.0084$ | $0.3155\pm 0.0075$ | $0.3053\pm 0.0050$ | $0.3059\pm 0.0050$
$\sigma_{8}$ | $0.8118\pm 0.0074$ | $0.8112\pm 0.0059$ | $0.8072\pm 0.0058$ | $0.8077\pm 0.0057$
$\chi_{\textrm{min}}^{2}$ | $2765.80$ | $2774.71$ | $3888.41$ | $4249.26$
DIC | $2817.93$ | $2826.45$ | $3940.70$ | $4301.20$
AIC | $2819.80$ | $2828.71$ | $3942.41$ | $4303.26$
| Flat $\Lambda$CDM models
Parameter | Non-CMB (old) | Non-CMB (new) | P18+non-CMB (old) | P18+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.0256\pm 0.0025$ | $0.0256\pm 0.0033$ | $0.02250\pm 0.00012$ | $0.02248\pm 0.00013$
$\Omega_{c}h^{2}$ | $0.1129\pm 0.0062$ | $0.1207\pm 0.0080$ | $0.11825\pm 0.00087$ | $0.11839\pm 0.00087$
$100\theta_{\textrm{MC}}$ | $1.0323\pm 0.0082$ | $1.0421\pm 0.0091$ | $1.04112\pm 0.00029$ | $1.04110\pm 0.00029$
$\tau$ | $0.0542$ | $0.0542$ | $0.0548\pm 0.0076$ | $0.0546\pm 0.0077$
$n_{s}$ | $0.9649$ | $0.9649$ | $0.9692\pm 0.0036$ | $0.9689\pm 0.0036$
$\ln(10^{10}A_{s})$ | $3.10\pm 0.11$ | $3.00\pm 0.11$ | $3.041\pm 0.015$ | $3.041\pm 0.016$
$H_{0}$ |  $69.8\pm 1.7$ ($69.8^{+1.8}_{-1.5}$) | $70.5\pm 2.3$ | $68.15\pm 0.39$ | $68.08\pm 0.39$
$\Omega_{m}$ | $0.286\pm 0.011$ | $0.296\pm 0.011$ | $0.3045\pm 0.0051$ | $0.3054\pm 0.0051$
$\sigma_{8}$ | $0.787\pm 0.027$ | $0.784\pm 0.026$ | $0.8048\pm 0.0068$ | $0.8052\pm 0.0067$
$\chi_{\textrm{min}}^{2}$ | $1106.54$ | $1469.93$ | $3879.35$ | $4240.24$
DIC | $1114.45$ | $1478.11$ | $3931.02$ | $4292.33$
AIC | $1114.54$ | $1477.93$ | $3933.35$ | $4294.24$

Figure 1: Likelihood distributions of flat $\Lambda$CDM model parameters favored by non-CMB data. Non-CMB (old) denotes the compilation of non-CMB data sets used in [10] while non-CMB (new) denotes the new compilation of non-CMB data sets of Sec. II.2.  Table 5: Mean and 68% confidence limits of non-flat $\Lambda$CDM model [Planck $P(q)$] parameters constrained by TT,TE,EE+lowE (P18) and P18+lensing, and non-CMB data sets. The Hubble constant $H_{0}$ has units of km s-1 Mpc-1.  | Non-flat $\Lambda$CDM models [Planck $P(q)$]
---|---
Parameter | P18 | P18+lensing | P18+lensing+non-CMB (old) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.02260\pm 0.00017$ | $0.02249\pm 0.00016$ | $0.02249\pm 0.00015$ | $0.02245\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.1181\pm 0.0015$ | $0.1186\pm 0.0015$ | $0.1187\pm 0.0013$ | $0.1190\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.04116\pm 0.00032$ | $1.04107\pm 0.00032$ | $1.04106\pm 0.00031$ | $1.04101\pm 0.00031$
$\tau$ | $0.0483\pm 0.0083$ | $0.0495\pm 0.0082$ | $0.0563\pm 0.0073$ | $0.0559\pm 0.0071$
$\Omega_{k}$ | $-0.043\pm 0.017$ | $-0.0103\pm 0.0066$ | $0.0004\pm 0.0017$ | $0.0009\pm 0.0017$
| ($-0.043^{+0.018}_{-0.015}$) | ($-0.0103^{+0.0071}_{-0.0060}$) |  |
$n_{s}$ | $0.9706\pm 0.0047$ | $0.9687\pm 0.0046$ | $0.9681\pm 0.0044$ | $0.9672\pm 0.0043$
$\ln(10^{10}A_{s})$ | $3.027\pm 0.017$ | $3.030\pm 0.017$ | $3.046\pm 0.014$ | $3.046\pm 0.014$
$H_{0}$ |  $54.5\pm 3.6$ ($54.5^{+3.1}_{-3.9}$) | $63.7\pm 2.3$ | $68.17\pm 0.55$ | $68.24\pm 0.54$
$\Omega_{m}$ | $0.481\pm 0.062$ | $0.351\pm 0.024$ | $0.3051\pm 0.0053$ | $0.3053\pm 0.0051$
$\sigma_{8}$ | $0.775\pm 0.015$ | $0.796\pm 0.011$ | $0.8080\pm 0.0066$ | $0.8094\pm 0.0066$
$\chi_{\textrm{min}}^{2}$ | $2754.73$ | $2771.53$ | $3887.99$ | $4248.74$
$\Delta\chi_{\textrm{min}}^{2}$ | $-11.07$ | $-3.18$ | $-0.42$ | $-0.52$
DIC | $2810.59$ | $2826.17$ | $3942.07$ | $4302.41$
$\Delta\textrm{DIC}$ | $-7.34$ | $-0.28$ | $+1.37$ | $+1.21$
AIC | $2810.73$ | $2827.53$ | $3943.99$ | $4304.74$
$\Delta\textrm{AIC}$ | $-9.07$ | $-1.18$ | $+1.58$ | $+1.48$
| Non-flat $\Lambda$CDM models [Planck $P(q)$]
Parameter | Non-CMB (old) | Non-CMB (new) | P18+non-CMB (old) | P18+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.0242\pm 0.0033$ | $0.0262\pm 0.0040$ | $0.02248\pm 0.00015$ | $0.02244\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.120\pm 0.012$ | $0.118\pm 0.011$ | $0.1185\pm 0.0013$ | $0.1189\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.10\pm 0.11$ | $1.025\pm 0.073$ | $1.04107\pm 0.00031$ | $1.04102\pm 0.00031$
$\tau$ | $0.0483$ | $0.0483$ | $0.0543\pm 0.0077$ | $0.0539\pm 0.0078$
$\Omega_{k}$ | $-0.032\pm 0.051$ | $0.010\pm 0.038$ | $0.0004\pm 0.0017$ | $0.0009\pm 0.0017$
| ($-0.032^{+0.056}_{-0.046}$) | ($0.010^{+0.046}_{-0.030}$) |  |
$n_{s}$ | $0.9706$ | $0.9706$ | $0.9687\pm 0.0043$ | $0.9677\pm 0.0044$
$\ln(10^{10}A_{s})$ | $2.90\pm 0.34$ | $3.07\pm 0.27$ | $3.040\pm 0.016$ | $3.040\pm 0.016$
$H_{0}$ | $70.1\pm 1.7$ | $70.5\pm 2.5$ | $68.25\pm 0.56$ | $68.31\pm 0.56$
$\Omega_{m}$ | $0.294\pm 0.018$ | $0.292\pm 0.015$ | $0.3040\pm 0.0055$ | $0.3043\pm 0.0054$
$\sigma_{8}$ | $0.771\pm 0.035$ | $0.790\pm 0.033$ | $0.8055\pm 0.0076$ | $0.8070\pm 0.0077$
$\chi_{\textrm{min}}^{2}$ | $1106.53$ | $1468.22$ | $3878.77$ | $4239.58$
$\Delta\chi_{\textrm{min}}^{2}$ | $-0.01$ | $-1.71$ | $-0.58$ | $-0.66$
DIC | $1116.92$ | $1479.52$ | $3933.33$ | $4293.78$
$\Delta\textrm{DIC}$ | $+2.47$ | $+1.41$ | $+2.31$ | $+1.45$
AIC | $1116.53$ | $1478.22$ | $3934.77$ | $4295.58$
$\Delta\textrm{AIC}$ | $+1.99$ | $+0.29$ | $+1.42$ | $+1.34$

Figure 2: Likelihood distributions of non-flat $\Lambda$CDM model [Planck $P(q)$] parameters favored by non-CMB data. Non-CMB (old) denotes the compilation of non-CMB data sets used in [10] while non-CMB (new) denotes the new compilation of non-CMB data sets of Sec. II.2.  Table 6: Mean and 68% confidence limits of non-flat $\Lambda$CDM model [new $P(q)$] parameters constrained by TT,TE,EE+lowE (P18) and P18+lensing, and non-CMB data sets. The Hubble constant $H_{0}$ has units of km s-1 Mpc-1.  | Non-flat $\Lambda$CDM models [new $P(q)$]
---|---
Parameter | P18 | P18+lensing | P18+lensing+non-CMB (old) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.02255\pm 0.00017$ | $0.02248\pm 0.00016$ | $0.02248\pm 0.00015$ | $0.02246\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.1188\pm 0.0015$ | $0.1188\pm 0.0014$ | $0.1186\pm 0.0013$ | $0.1190\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.04109\pm 0.00032$ | $1.04104\pm 0.00032$ | $1.04106\pm 0.00031$ | $1.04103\pm 0.00031$
$\tau$ | $0.0525\pm 0.0083$ | $0.0515\pm 0.0081$ | $0.0566\pm 0.0074$ | $0.0562\pm 0.0072$
$\Omega_{k}$ | $-0.033\pm 0.014$ | $-0.0086\pm 0.0057$ | $0.0003\pm 0.0017$ | $0.0008\pm 0.0017$
| ($-0.033^{+0.017}_{-0.011}$) | ($-0.0086^{+0.0063}_{-0.0050}$) |  |
$n_{s}$ | $0.9654\pm 0.0045$ | $0.9661\pm 0.0043$ | $0.9679\pm 0.0042$ | $0.9671\pm 0.0041$
$\ln(10^{10}A_{s})$ | $3.039\pm 0.017$ | $3.035\pm 0.016$ | $3.046\pm 0.014$ | $3.046\pm 0.014$
$H_{0}$ | $56.9\pm 3.6$ | $64.2\pm 2.0$ | $68.13\pm 0.54$ | $68.21\pm 0.54$
$\Omega_{m}$ | $0.444\pm 0.055$ | $0.345\pm 0.021$ | $0.3054\pm 0.0051$ | $0.3054\pm 0.0051$
$\sigma_{8}$ | $0.786\pm 0.014$ | $0.799\pm 0.010$ | $0.8079\pm 0.0067$ | $0.8094\pm 0.0065$
$\chi_{\textrm{min}}^{2}$ | $2757.38$ | $2771.75$ | $3887.55$ | $4248.50$
$\Delta\chi_{\textrm{min}}^{2}$ | $-8.42$ | $-2.96$ | $-0.86$ | $-0.76$
DIC | $2811.54$ | $2825.74$ | $3942.22$ | $4302.33$
$\Delta\textrm{DIC}$ | $-6.39$ | $-0.71$ | $+1.52$ | $+1.13$
AIC | $2813.38$ | $2827.75$ | $3943.55$ | $4304.50$
$\Delta\textrm{AIC}$ | $-6.42$ | $-0.96$ | $+1.14$ | $+1.24$
| Non-flat $\Lambda$CDM models [new $P(q)$]
Parameter | Non-CMB (old) | Non-CMB (new) | P18+non-CMB (old) | P18+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.0241\pm 0.0033$ | $0.0260\pm 0.0039$ | $0.02249\pm 0.00015$ | $0.02246\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.120\pm 0.013$ | $0.119\pm 0.012$ | $0.1184\pm 0.0013$ | $0.1188\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.11\pm 0.11$ | $1.033\pm 0.079$ | $1.04108\pm 0.00031$ | $1.04104\pm 0.00031$
$\tau$ | $0.0525$ | $0.0525$ | $0.0549\pm 0.0077$ | $0.0542\pm 0.0076$
$\Omega_{k}$ | $-0.036\pm 0.051$ | $0.006\pm 0.041$ | $0.0003\pm 0.0017$ | $0.0008\pm 0.0017$
| ($-0.036^{+0.056}_{-0.047}$) | ($0.006^{+0.051}_{-0.030}$) |  |
$n_{s}$ | $0.9654$ | $0.9654$ | $0.9684\pm 0.0041$ | $0.9675\pm 0.0042$
$\ln(10^{10}A_{s})$ | $2.88\pm 0.34$ | $3.05\pm 0.29$ | $3.042\pm 0.016$ | $3.041\pm 0.015$
$H_{0}$ | $70.1\pm 1.8$ | $70.4\pm 2.5$ | $68.21\pm 0.55$ | $68.29\pm 0.55$
$\Omega_{m}$ | $0.295\pm 0.018$ | $0.293\pm 0.016$ | $0.3043\pm 0.0054$ | $0.3045\pm 0.0053$
$\sigma_{8}$ | $0.770\pm 0.035$ | $0.787\pm 0.033$ | $0.8057\pm 0.0074$ | $0.8071\pm 0.0075$
$\chi_{\textrm{min}}^{2}$ | $1106.49$ | $1468.21$ | $3878.76$ | $4239.45$
$\Delta\chi_{\textrm{min}}^{2}$ | $-0.05$ | $-1.72$ | $-0.59$ | $-0.79$
DIC | $1117.31$ | $1480.16$ | $3932.56$ | $4293.50$
$\Delta\textrm{DIC}$ | $+2.86$ | $+2.05$ | $+1.54$ | $+1.17$
AIC | $1116.49$ | $1478.21$ | $3934.76$ | $4295.45$
$\Delta\textrm{AIC}$ | $+1.95$ | $+0.28$ | $+1.41$ | $+1.21$

Figure 3: Likelihood distributions of non-flat $\Lambda$CDM model [new $P(q)$]
parameters favored by non-CMB data. Non-CMB (old) denotes the compilation of
non-CMB data sets used in [10] while non-CMB (new) denotes the new compilation
of non-CMB data sets of Sec. II.2.

####  IV.1.2 P18+non-CMB (old) vs. P18+non-CMB (new) cosmological parameter
constraints

The aim of this subsubsection is to determine what happens to the changes
highlighted in the previous subsubsection when P18 data are added to the mix.

P18+non-CMB (old) and P18+non-CMB (new) cosmological parameter constraints for
the six-parameter flat $\Lambda$CDM model are in the lower half of Table 4 and
in Fig. 4. Moving from P18 + non-CMB (old) data to P18 + non-CMB (new) data
results in smaller changes in the mean parameter values than found in the non-
CMB data alone case of the previous subsubsection. For the primary parameters,
the most affected mean values are $\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$,
whose values differ by +0.11$\sigma$ and $-0.11\sigma$ when compared with the
P18+non-CMB (old) data results. Similar conclusions hold for derived parameter
means, with the differences in $H_{0}$ and $\Omega_{m}$ being +0.13$\sigma$
and $-0.12\sigma$, respectively. No significant changes are observed in the
size of the error bars, when we move from P18+non-CMB (old) data to P18+non-
CMB (new) data. The largest changes are increases of 7.7% for
$\Omega_{b}{h^{2}}$ and 6.3% for $\ln(10^{10}A_{s})$.

In the non-flat geometry cases there are slightly larger differences between
the mean values obtained with P18+non-CMB (old) and P18+non-CMB (new) data.
Table 5 [6] and Fig. 5 [6] show results for the seven-parameter non-flat
$\Lambda$CDM Planck [new] $P(q)$ cosmological model, obtained from P18+non-CMB
(old) and P18+non-CMB (new) data. The mean values of $\Omega_{b}{h^{2}}$,
$\Omega_{c}{h^{2}}$, and $n_{s}$ differ by +0.19$\sigma$ [+0.14$\sigma$],
$-0.22\sigma$ [$-0.22\sigma$], and +0.16$\sigma$ [+0.15$\sigma$] respectively.
For $\Omega_{k}$, when P18+non-CMB (old) data are analyzed we find
$\Omega_{k}=0.0004\pm 0.0017$ [$\Omega_{k}=0.0003\pm 0.0017$], a 0.24$\sigma$
[0.18$\sigma$] favoring of open geometry, and differing from the P18+non-CMB
(new) value $\Omega_{k}=0.0009\pm 0.0017$ [$\Omega_{k}=0.0008\pm 0.0017$] by
$-0.21\sigma$ [$-0.21\sigma$]. We do not find significant changes in the error
bars, the largest changes being those for $n_{s}$ with an increase of 2.3%
[2.4%], and for $\Omega_{m}$ with a decrease of $-1.9$% [$-1.9$%].

The lensing parameter $A_{L}$ does not play a significant role at low redshift
[10], but there are small differences in the results obtained with P18+non-CMB
(old) and P18+non-CMB (new) data when $A_{L}$ is allowed to vary. For the flat
$\Lambda$CDM+$A_{L}$ model, comparing results from P18+non-CMB (old) data and
from P18+non-CMB (new) data (see Table 7 and Fig. 7) we do not find
significant changes in primary parameter values (those of $\Omega_{c}h^{2}$
differ at $-0.12\sigma$) or in derived parameter values (with changes in
$H_{0}$ and $\Omega_{m}$ being $+0.12\sigma$ and $-0.12\sigma$ respectively).
In addition, the error bars are practically unchanged. The value of the
lensing parameter obtained from P18+non-CMB (old) data is $A_{L}=1.201\pm
0.061$, differing only +0.04$\sigma$ from $A_{L}=1.198\pm 0.060$ obtained from
P18-non-CMB (new) data.

When the curvature parameter is allowed to vary, the results also do not
change much when the non-CMB data set is updated. For the non-flat
$\Lambda$CDM+$A_{L}$ Planck [new] $P(q)$ model (see Table 8 [9] and Fig. 8
[9]), for the primary parameters obtained from P18+non-CMB (old) and P18+non-
CMB (new) data we find differences in $\Omega_{c}h^{2}$ and $n_{s}$ at
$-0.20\sigma$ [$-0.26\sigma$] and +0.18$\sigma$ [+0.11$\sigma$] with almost no
changes in the size of the error bars. For $\Omega_{k}$ and $A_{L}$ the
differences are at $-0.21\sigma$ [$-0.21\sigma$] and +0.080$\sigma$
[+0.12$\sigma$] respectively.

In summary, the combination of P18 CMB data with either non-CMB (old) or non-
CMB (new) data give almost identical cosmological parameter results. While it
is reassuring that updated non-CMB data do not significantly affect the
P18+non-CMB data constraints, this almost certainly is because P18 data have
much more weight than current non-CMB data.

The consistency [see Eqs. (12) and (16)] between the results obtained using
P18 data and using either non-CMB (old) data or non-CMB (new) data are quite
similar, as can be seen in Table 10, with a slight exception in the non-flat
Planck $P(q)$ model where the discordance between P18 and non-CMB data
decreases from $\sigma=3.005$ for non-CMB data (old) to $\sigma=2.704$ for
non-CMB (new) data and so there now is a lower level of tension than was found
using older data, [107].

No significant differences are observed between the P18+non-CMB (old) and
P18+non-CMB (new) data values of $\Delta$AIC and $\Delta$DIC obtained when
flat $\Lambda$CDM is compared with the non-flat models (see Tables 5 and 6)
and the $A_{L}$-varying models (Tables 7, 8 and 9).

Figure 4: Likelihood distributions of flat $\Lambda$CDM model parameters
favored by P18 and non-CMB data. Non-CMB (old) denotes the compilation of non-
CMB data sets used in [10] while non-CMB (new) denotes the new compilation of
non-CMB data sets of Sec. II.2. Likelihood results for P18 data are shown for
comparison.

Figure 5: Likelihood distributions of non-flat $\Lambda$CDM model [Planck
$P(q)$] parameters favored by P18 and non-CMB data. Non-CMB (old) denotes the
compilation of non-CMB data sets used in [10] while non-CMB (new) denotes the
new compilation of non-CMB data sets of Sec. II.2. Likelihood results for P18
data are shown for comparison.

Figure 6: Likelihood distributions of non-flat $\Lambda$CDM model [new $P(q)$]
parameters favored by P18 and non-CMB data. Non-CMB (old) denotes the
compilation of non-CMB data sets used in [10] while non-CMB (new) denotes the
new compilation of non-CMB data sets of Sec. II.2. Likelihood results for P18
data are shown for comparison.  Table 7: Mean and 68% confidence limits of
flat $\Lambda\textrm{CDM}$+$A_{L}$ model parameters constrained by P18,
P18+lensing, P18+non-CMB (old), P18+non-CMB (new), P18+lensing+non-CMB (old),
and P18+lensing+non-CMB (new) data sets. $H_{0}$ has units of km s-1 Mpc-1.

| Flat $\Lambda$CDM+$A_{L}$ models
---|---
Parameter | P18 | P18+lensing | P18+non-CMB (old) | P18+non-CMB (new) | P18+lensing+non-CMB (old) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.02259\pm 0.00017$ | $0.02251\pm 0.00017$ | $0.02265\pm 0.00014$ | $0.02264\pm 0.00014$ | $0.02258\pm 0.00014$ | $0.02257\pm 0.00014$
$\Omega_{c}h^{2}$ | $0.1180\pm 0.0015$ | $0.1183\pm 0.0015$ | $0.11736\pm 0.00092$ | $0.11752\pm 0.00091$ | $0.11747\pm 0.00091$ | $0.11759\pm 0.00090$
$100\theta_{\textrm{MC}}$ | $1.04114\pm 0.00032$ | $1.04109\pm 0.00032$ | $1.04120\pm 0.00029$ | $1.04120\pm 0.00029$ | $1.04118\pm 0.00029$ | $1.04116\pm 0.00029$
$\tau$ | $0.0496\pm 0.0082$ | $0.0487\pm 0.0087$ | $0.0484\pm 0.0083$ | $0.0480\pm 0.0084$ | $0.0476\pm 0.0085$ | $0.0477\pm 0.0086$
$n_{s}$ | $0.9710\pm 0.0050$ | $0.9695\pm 0.0048$ | $0.9726\pm 0.0038$ | $0.9722\pm 0.0038$ | $0.9715\pm 0.0038$ | $0.9711\pm 0.0038$
$\ln(10^{10}A_{s})$ | $3.030\pm 0.017$ | $3.028\pm 0.018$ | $3.026\pm 0.017$ | $3.025\pm 0.017$ | $3.023\pm 0.018$ | $3.024\pm 0.018$
$A_{L}$ | $1.181\pm 0.067$ | $1.073\pm 0.041$ | $1.201\pm 0.061$ | $1.198\pm 0.060$ | $1.089\pm 0.035$ | $1.087\pm 0.035$
$H_{0}$ | $68.31\pm 0.71$ | $68.14\pm 0.69$ | $68.62\pm 0.43$ | $68.55\pm 0.42$ | $68.52\pm 0.42$ | $68.45\pm 0.42$
$\Omega_{m}$ | $0.3029\pm 0.0093$ | $0.3048\pm 0.0091$ | $0.2988\pm 0.0054$ | $0.2997\pm 0.0054$ | $0.2998\pm 0.0053$ | $0.3005\pm 0.0053$
$\sigma_{8}$ | $0.7997\pm 0.0088$ | $0.7996\pm 0.0089$ | $0.7961\pm 0.0074$ | $0.7964\pm 0.0075$ | $0.7955\pm 0.0075$ | $0.7961\pm 0.0075$
$\chi_{\textrm{min}}^{2}$ (Total) | $2756.12$ | $2771.24$ | $3865.90$ | $4227.27$ | $3881.55$ | $4242.61$
$\Delta\chi_{\textrm{min}}^{2}$ | $-9.68$ | $-3.47$ | $-13.45$ | $-12.97$ | $-6.86$ | $-6.65$
DIC | $2812.41$ | $2825.53$ | $3922.11$ | $4283.86$ | $3935.15$ | $4297.19$
$\Delta\textrm{DIC}$ | $-5.52$ | $-0.92$ | $-8.91$ | $-8.47$ | $-5.55$ | $-4.01$
AIC | $2812.12$ | $2827.24$ | $3921.90$ | $4283.27$ | $3937.55$ | $4298.61$
$\Delta\textrm{AIC}$ | $-7.68$ | $-1.47$ | $-11.45$ | $-10.97$ | $-4.86$ | $-4.65$

!

Figure 7: Likelihood distributions of flat $\Lambda$CDM$+A_{L}$ model parameters favored by P18 and non-CMB data. Non-CMB (old) denotes the compilation of non-CMB data sets used in [10] while non-CMB (new) denotes the new compilation of non-CMB data sets of Sec. II.2. Likelihood results for P18 data are shown for comparison.  Table 8: Mean and 68% confidence limits of non-flat $\Lambda\textrm{CDM}$+$A_{L}$ [Planck $P(q)$] model parameters constrained by P18, P18+lensing, P18+non-CMB (old), P18+non-CMB (new), P18+lensing+non-CMB (old), and P18+lensing+non-CMB (new) data sets. $H_{0}$ has units of km s-1 Mpc-1.  | Non-flat $\Lambda$CDM+$A_{L}$ [Planck $P(q)$] models
---|---
Parameter | P18 | P18+lensing | P18+non-CMB (old) | P18+non-CMB (new) | P18+lensing+non-CMB (old) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.02258\pm 0.00017$ | $0.02251\pm 0.00017$ | $0.02268\pm 0.00017$ | $0.02265\pm 0.00016$ | $0.02259\pm 0.00016$ | $0.02255\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.1183\pm 0.0015$ | $0.1183\pm 0.0015$ | $0.1170\pm 0.0014$ | $0.1174\pm 0.0014$ | $0.1173\pm 0.0014$ | $0.1178\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.04116\pm 0.00033$ | $1.04110\pm 0.00032$ | $1.04125\pm 0.00032$ | $1.04120\pm 0.00032$ | $1.04118\pm 0.00032$ | $1.04113\pm 0.00032$
$\tau$ | $0.0478\pm 0.0081$ | $0.0489\pm 0.0085$ | $0.0485\pm 0.0087$ | $0.0482\pm 0.0085$ | $0.0479\pm 0.0085$ | $0.0480\pm 0.0085$
$\Omega_{k}$ | $-0.130\pm 0.095$ | $-0.005\pm 0.027$ | $-0.0006\pm 0.0017$ | $-0.0001\pm 0.0017$ | $-0.0002\pm 0.0017$ | $0.0004\pm 0.0017$
| ($-0.130^{+0.13}_{-0.055}$) | ($-0.0048^{+0.029}_{-0.0083}$) |  |  |  |
$n_{s}$ | $0.9704\pm 0.0048$ | $0.9696\pm 0.0049$ | $0.9735\pm 0.0046$ | $0.9723\pm 0.0046$ | $0.9718\pm 0.0045$ | $0.9705\pm 0.0044$
$\ln(10^{10}A_{s})$ | $3.027\pm 0.017$ | $3.028\pm 0.018$ | $3.025\pm 0.018$ | $3.025\pm 0.017$ | $3.024\pm 0.017$ | $3.025\pm 0.017$
$A_{L}$ | $0.88\pm 0.15$ | $1.09\pm 0.16$ | $1.203\pm 0.062$ | $1.196\pm 0.062$ | $1.090\pm 0.036$ | $1.084\pm 0.035$
| ($0.876^{+0.088}_{-0.19}$) | ($1.09^{+0.15}_{-0.17}$) |  |  |  |
$H_{0}$ |  $45\pm 11$ ($45^{+5}_{-10}$) | $69\pm 11$ | $68.48\pm 0.56$ | $68.52\pm 0.56$ | $68.49\pm 0.56$ | $68.53\pm 0.56$
$\Omega_{m}$ | $0.80\pm 0.35$ | $0.32\pm 0.11$ | $0.2994\pm 0.0055$ | $0.2998\pm 0.0055$ | $0.2998\pm 0.0055$ | $0.3003\pm 0.0054$
$\sigma_{8}$ | $0.733\pm 0.045$ | $0.796\pm 0.016$ | $0.7946\pm 0.0088$ | $0.7961\pm 0.0086$ | $0.7952\pm 0.0085$ | $0.7975\pm 0.0085$
$\chi_{\textrm{min}}^{2}$ (Total) | $2754.99$ | $2771.14$ | $3865.53$ | $4227.07$ | $3881.37$ | $4242.22$
$\Delta\chi_{\textrm{min}}^{2}$ | $-10.81$ | $-3.57$ | $-13.82$ | $-13.17$ | $-7.04$ | $-7.04$
DIC | $2811.63$ | $2827.14$ | $3924.07$ | $4285.58$ | $3936.85$ | $4298.73$
$\Delta\textrm{DIC}$ | $-6.30$ | $0.69$ | $-6.95$ | $-6.75$ | $-3.85$ | $-2.47$
AIC | $2812.99$ | $2829.14$ | $3923.53$ | $4285.07$ | $3939.37$ | $4300.22$
$\Delta\textrm{AIC}$ | $-6.81$ | $0.43$ | $-9.82$ | $-9.17$ | $-3.04$ | $-3.04$

Table 9: Mean and 68% confidence limits of non-flat $\Lambda\textrm{CDM}$+$A_{L}$ [new $P(q)$] model parameters constrained by P18, P18+lensing, P18+non-CMB (old), P18+non-CMB (new), P18+lensing+non-CMB (old), and P18+lensing+non-CMB (new) data sets. $H_{0}$ has units of km s-1 Mpc-1.  | Non-flat $\Lambda$CDM+$A_{L}$ [new $P(q)$] models
---|---
Parameter | P18 | P18+lensing | P18+non-CMB (old) | P18+non-CMB (new) | P18+lensing+non-CMB (old) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.02257\pm 0.00017$ | $0.02252\pm 0.00017$ | $0.02269\pm 0.00016$ | $0.02265\pm 0.00016$ | $0.02260\pm 0.00016$ | $0.02256\pm 0.00016$
$\Omega_{c}h^{2}$ | $0.1187\pm 0.0016$ | $0.1183\pm 0.0015$ | $0.1170\pm 0.0013$ | $0.1175\pm 0.0014$ | $0.1174\pm 0.0013$ | $0.1178\pm 0.0014$
$100\theta_{\textrm{MC}}$ | $1.04111\pm 0.00033$ | $1.04108\pm 0.00032$ | $1.04125\pm 0.00032$ | $1.04120\pm 0.00032$ | $1.04118\pm 0.00032$ | $1.04113\pm 0.00032$
$\tau$ | $0.0512\pm 0.0086$ | $0.0495\pm 0.0093$ | $0.0490\pm 0.0086$ | $0.0484\pm 0.0085$ | $0.0486\pm 0.0086$ | $0.0481\pm 0.0082$
$\Omega_{k}$ | $-0.10\pm 0.11$ | $0.003\pm 0.016$ | $-0.0006\pm 0.0017$ | $-0.0001\pm 0.0017$ | $-0.0002\pm 0.0017$ | $0.0004\pm 0.0017$
| ($-0.10^{+0.12}_{-0.23}$) | ($0.003^{+0.019}_{-0.010}$) |  |  |  |
$n_{s}$ | $0.9654\pm 0.0057$ | $0.9688\pm 0.0053$ | $0.9730\pm 0.0043$ | $0.9723\pm 0.0045$ | $0.9713\pm 0.0042$ | $0.9707\pm 0.0046$
$\ln(10^{10}A_{s})$ | $3.036\pm 0.018$ | $3.030\pm 0.019$ | $3.026\pm 0.018$ | $3.026\pm 0.017$ | $3.025\pm 0.017$ | $3.025\pm 0.017$
$A_{L}$ | $0.94\pm 0.20$ | $1.13\pm 0.15$ | $1.204\pm 0.061$ | $1.194\pm 0.061$ | $1.088\pm 0.035$ | $1.084\pm 0.034$
| ($0.94^{+0.22}_{-0.30}$) | ($1.13^{+0.12}_{-0.17}$) |  |  |  |
$H_{0}$ | $51\pm 14$ |  $72.0\pm 9.2$ ($72.0^{+7.7}_{-11}$) | $68.47\pm 0.56$ | $68.52\pm 0.55$ | $68.48\pm 0.56$ | $68.53\pm 0.55$
$\Omega_{m}$ | $0.70\pm 0.43$ | $0.287\pm 0.076$ | $0.2994\pm 0.0056$ | $0.2999\pm 0.0054$ | $0.2999\pm 0.0055$ | $0.3004\pm 0.0054$
$\sigma_{8}$ | $0.752\pm 0.052$ | $0.801\pm 0.011$ | $0.7948\pm 0.0083$ | $0.7964\pm 0.0084$ | $0.7956\pm 0.0082$ | $0.7974\pm 0.0084$
$\chi_{\textrm{min}}^{2}$ (Total) | $2756.33$ | $2770.45$ | $3865.41$ | $4227.11$ | $3880.69$ | $4242.01$
$\Delta\chi_{\textrm{min}}^{2}$ | $-9.47$ | $-4.26$ | $-13.94$ | $-13.13$ | $-7.72$ | $-7.25$
DIC | $2814.83$ | $2827.29$ | $3923.86$ | $4285.29$ | $3937.52$ | $4298.75$
$\Delta\textrm{DIC}$ | $-3.10$ | $0.84$ | $-7.16$ | $-7.04$ | $-3.18$ | $-2.45$
AIC | $2814.33$ | $2828.45$ | $3923.41$ | $4285.11$ | $3938.69$ | $4300.01$
$\Delta\textrm{AIC}$ | $-5.47$ | $-0.26$ | $-9.94$ | $-9.13$ | $-3.72$ | $-3.25$

Figure 8: Likelihood distributions of non-flat $\Lambda$CDM$+A_{L}$ model
[Planck $P(q)$] parameters favored by P18 and non-CMB data. Non-CMB (old)
denotes the compilation of non-CMB data sets used in [10] while non-CMB (new)
denotes the new compilation of non-CMB data sets of Sec. II.2. Likelihood
results for P18 data are shown for comparison.

Figure 9: Likelihood distributions of non-flat $\Lambda$CDM$+A_{L}$ model [new $P(q)$] parameters favored by P18 and non-CMB data. Non-CMB (old) denotes the compilation of non-CMB data sets used in [10] while non-CMB (new) denotes the new compilation of non-CMB data sets of Sec. II.2. Likelihood results for P18 data are shown for comparison.  Table 10: Consistency check parameter $\log_{10}\mathcal{I}$ and tension parameters $\sigma$ and $p$ for P18 vs. non-CMB data sets and P18+lensing vs. non-CMB data sets in the $\Lambda\textrm{CDM}$ models.  | Flat $\Lambda$CDM model
---|---
Data | P18 vs non-CMB (old) | P18 vs non-CMB (new) | P18+lensing vs non-CMB (old) | P18+lensing vs non-CMB (new)
$\log_{10}\mathcal{I}$ | $0.296$ | $0.805$ | $0.029$ | $0.730$
$\sigma$ | $1.749$ | $1.152$ | $1.747$ | $1.209$
$p$ (%) | $8.03$ | $24.9$ | $8.06$ | $22.7$
| Flat $\Lambda$CDM+$A_{L}$ model
Data | P18 vs non-CMB (old) | P18 vs non-CMB (new) | P18+lensing vs non-CMB (old) | P18+lensing vs non-CMB (new)
$\log_{10}\mathcal{I}$ | $1.033$ | $1.446$ | $1.033$ | $1.400$
$\sigma$ | $0.835$ | $0.164$ | $0.774$ | $0.0872$
$p$ (%) | $40.4$ | $87.0$ | $43.9$ | $93.1$
| Non-flat $\Lambda$CDM model [Planck $P(q)$]
Data | P18 vs non-CMB (old) | P18 vs non-CMB (new) | P18+lensing vs non-CMB (old) | P18+lensing vs non-CMB (new)
$\log_{10}\mathcal{I}$ | $-1.263$ | $-0.796$ | $0.297$ | $0.711$
$\sigma$ | $3.005$ | $2.704$ | $1.837$ | $1.555$
$p$ (%) | $0.265$ | $0.687$ | $6.62$ | $12.0$
| Non-flat $\Lambda$CDM+$A_{L}$ model [Planck $P(q)$]
Data | P18 vs non-CMB (old) | P18 vs non-CMB (new) | P18+lensing vs non-CMB (old) | P18+lensing vs non-CMB (new)
$\log_{10}\mathcal{I}$ | $0.972$ | $1.210$ | $1.641$ | $1.719$
$\sigma$ | $0.793$ | $0.595$ | $0.516$ | $0.241$
$p$ (%) | $42.8$ | $55.2$ | $60.6$ | $80.9$
| Non-flat $\Lambda$CDM model [new $P(q)$]
Data | P18 vs non-CMB (old) | P18 vs non-CMB (new) | P18+lensing vs non-CMB (old) | P18+lensing vs non-CMB (new)
$\log_{10}\mathcal{I}$ | $-0.806$ | $-0.391$ | $0.143$ | $0.775$
$\sigma$ | $2.577$ | $2.308$ | $1.886$ | $1.544$
$p$ (%) | $0.996$ | $2.10$ | $5.93$ | $12.3$
| Non-flat $\Lambda$CDM+$A_{L}$ model [new $P(q)$]
Data | P18 vs non-CMB (old) | P18 vs non-CMB (new) | P18+lensing vs non-CMB (old) | P18+lensing vs non-CMB (new)
$\log_{10}\mathcal{I}$ | $1.798$ | $2.107$ | $1.500$ | $1.887$
$\sigma$ | $0.402$ | $0.289$ | $0.573$ | $0.312$
$p$ (%) | $68.7$ | $77.2$ | $56.7$ | $75.5$

####  IV.1.3 P18+lensing+non-CMB (old) vs. P18+lensing+non-CMB (new)
cosmological parameter constraints

In this subsubsection we check the impact on the cosmological parameter
constraints when we move from P18+lensing+non-CMB (old) data to
P18+lensing+non-CMB (new) data. As we shall see, and as expected, the
differences presented in this subsubsection are even smaller than the ones
presented in the previous one for P18+non-CMB data.

P18+lensing+non-CMB (old) and P18+lensing+non-CMB (new) cosmological parameter
constraints for the six-parameter flat $\Lambda$CDM model are in the upper
half of Table 4 and Fig. 10. For primary parameters the largest differences
are for $\Omega_{c}h^{2}$ and $n_{s}$, at $-0.093\sigma$ and +0.059$\sigma$
respectively. Derived parameters $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$
differ at +0.074$\sigma$, $-0.085\sigma$, and $-0.061\sigma$. The error bars
obtained with P18+lensing+non-CMB (old) data and P18+lensing+non-CMB (new)
data are very similar, the largest differences affect primary parameter
$100\theta_{\textrm{MC}}$ ($-3.57$%) and derived parameter $\sigma_{8}$
($-1.75$%).

As for the non-flat spatial geometry models, the results obtained with
P18+lensing+non-CMB (old) and P18+lensing+non-CMB (new) data, for the seven-
parameter $\Lambda$CDM Planck [new] $P(q)$ model are in the upper half of
Table 5 [6] and Fig. 11 [12]. The primary parameters $\Omega_{b}{h^{2}}$,
$\Omega_{c}{h^{2}}$, and $n_{s}$, differ at +0.19$\sigma$ [+0.094$\sigma$],
$-0.16\sigma$ [$-0.22\sigma$], and +0.15$\sigma$ [+0.14$\sigma$] respectively.
For $\Omega_{k}$, when P18+lensing+non-CMB (old) data are analyzed we find
$\Omega_{k}=0.0004\pm 0.0017$ [$0.0003\pm 0.0017$], which is $+0.24\sigma$
[$+0.18\sigma$] away from zero and differing by $-0.21\sigma$ [$-0.21\sigma$]
with the P18+lensing+non-CMB (new) value, $\Omega_{k}=0.0009\pm 0.0017$
[$0.0008\pm 0.0017$] which is only $+0.53\sigma$ [$+0.47\sigma$] in favor of
an open Universe. The error bars do not change much. For the primary
parameters $\tau$ and $n_{s}$ we get a reduction of $-2.82$% [$-2.78$%] and
$-2.33$% [$-2.43$%] whereas for the derived parameters $H_{0}$ and
$\sigma_{8}$ the error bars are reduced by $-1.85$% [0.00%] and 0.00%
[$-3.08$%], respectively.

Results obtained with P18+lensing+non-CMB (old) and P18+lensing+non-CMB (new)
data, provided in Table 7 and Fig. 13, for the seven-parameter flat
$\Lambda$CDM+$A_{L}$ model, are very similar. The primary parameters
$\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ differ at +0.05$\sigma$ and
$-0.09\sigma$ respectively whereas the derived parameter $H_{0}$ differs at
+0.12$\sigma$. When P18+lensing+non-CMB (old) data are analyzed we obtain
$A_{L}=1.089\pm 0.035$ which shows a small difference of +0.04$\sigma$ with
the value $A_{L}=1.087\pm 0.035$ obtained after analyzing P18+lensing+non-CMB
(new) data.

Cosmological parameter constraints for the eight-parameter non-flat
$\Lambda$CDM Planck [new] $P(q)$ model from P18+lensing+non-CMB (old) and
P18+lensing+non-CMB (new) data are in Table 8 [9] and in Fig. 14 [15].
Although the differences are greater than in the case of the flat
$\Lambda$CDM+$A_{L}$ model, they are still small. Primary parameters
$\Omega_{c}h^{2}$, $n_{s}$, and $\Omega_{b}h^{2}$ differ at $-0.26\sigma$
[$-0.21\sigma$], +0.21$\sigma$ [$+0.10\sigma$], and +0.18$\sigma$
[$+0.18\sigma$], respectively, while derived parameter $\sigma_{8}$ differs at
$-0.19\sigma$ [$-0.15\sigma$]. For $\Omega_{k}$ and $A_{L}$ the differences
are $-0.25\sigma$ [$-0.25\sigma$] and +0.12$\sigma$ [+0.082$\sigma$].

In summary, the combination of P18+lensing data with either non-CMB (old) or
non-CMB (new) data gives almost identical cosmological parameter results,
similar to the P18 case of the previous subsubsection, but with even smaller
differences.

Again, similar to the previous subsubsection for P18+non-CMB data, there are
no significant differences between the P18+lensing+non-CMB (old) and
P18+lensing+non-CMB (new) data values of $\Delta$AIC and $\Delta$DIC obtained
when flat $\Lambda$CDM is compared with the non-flat models (see Tables 5 and
6) and the $A_{L}$-varying models (Tables 7, 8 and 9).

Contrary to the previous subsubsection for P18 data and non-CMB data, we do
not observe a qualitative change between the results obtained using
P18+lensing data and using either non-CMB (old) data or non-CMB (new) data
regarding the concordance/discordance (see Table 10) provided by the two
statistical estimators in Eqs. (12) and (16).

Figure 10: Likelihood distributions of flat $\Lambda$CDM model parameters
favored by P18+lensing and non-CMB data. Non-CMB (old) denotes the compilation
of non-CMB data sets used in [10] while non-CMB (new) denotes the new
compilation of non-CMB data sets of Sec. II.2. Likelihood results for
P18+lensing data are shown for comparison.

Figure 11: Likelihood distributions of non-flat $\Lambda$CDM model [Planck
$P(q)$] parameters favored by P18+lensing and non-CMB data. Non-CMB (old)
denotes the compilation of non-CMB data sets used in [10] while non-CMB (new)
denotes the new compilation of non-CMB data sets of Sec. II.2. Likelihood
results for P18+lensing data are shown for comparison.

Figure 12: Likelihood distributions of non-flat $\Lambda$CDM model [new
$P(q)$] parameters favored by P18+lensing and non-CMB data. Non-CMB (old)
denotes the compilation of non-CMB data sets used in [10] while non-CMB (new)
denotes the new compilation of non-CMB data sets of Sec. II.2. Likelihood
results for P18+lensing data are shown for comparison.

Figure 13: Likelihood distributions of flat $\Lambda$CDM$+A_{L}$ model
parameters favored by P18+lensing and non-CMB data. Non-CMB (old) denotes the
compilation of non-CMB data sets used in [10] while non-CMB (new) denotes the
new compilation of non-CMB data sets of Sec. II.2. Likelihood results for
P18+lensing data are shown for comparison.

Figure 14: Likelihood distributions of non-flat $\Lambda$CDM$+A_{L}$ model
[Planck $P(q)$] parameters favored by P18+lensing and non-CMB data. Non-CMB
(old) denotes the compilation of non-CMB data sets used in [10] while non-CMB
(new) denotes the new compilation of non-CMB data sets of Sec. II.2.
Likelihood results for P18+lensing data are shown for comparison.

Figure 15: Likelihood distributions of non-flat $\Lambda$CDM$+A_{L}$ model
[new $P(q)$] parameters favored by P18+lensing and non-CMB data. Non-CMB (old)
denotes the compilation of non-CMB data sets used in [10] while non-CMB (new)
denotes the new compilation of non-CMB data sets of Sec. II.2. Likelihood
results for P18+lensing data are shown for comparison.

####  IV.1.4 Summary of updated constraints in $\Lambda$CDM models

Here we summarize updated results for the $\Lambda$CDM models constraints that
follow from the updated non-CMB (new) data we use. The results of cosmological
parameter constraints from P18 data and P18+lensing data in the flat and non-
flat $\Lambda$CDM ($+A_{L}$ ) models are described in detail in Sec. IV A 1
and 2 of our previous paper [10].

For numerical values of cosmological parameters and to see how the
consideration of different data sets affect the two-dimensional contour plots,
see Tables 4—9 and Figs. 1—15.

Within the context of a given cosmological model, it is possible to assess
whether the constraints obtained from two different data sets are in tension
or not at some level of significance, and when they are in tension they cannot
be jointly used to constrain that cosmological model, which is ruled out at
that level of significance, if we assume the two data sets are correct. In
[10] we utilized the same statistical estimators employed in this work, namely
Eqs. (12) and (16). One of the most important results found there was that for
the non-flat $\Lambda$CDM Planck $P(q)$ model P18 data and non-CMB (old) data
could not be jointly analyzed since the level of discordance between the two
data sets was $\sigma=3.005$, larger than $3\sigma$, meaning that the non-flat
$\Lambda$CDM Planck $P(q)$ model was ruled out at more than 3$\sigma$. On the
other hand, the P18 data set and the non-CMB (new) data set have a reduced
discordance level of $\sigma=2.704$, indicating that although there is still a
significant level of tension, it is less than $3\sigma$ and so both data sets
can now be jointly analyzed. As can be seen from Table 10, the discordance
between the P18 and P18+lensing data sets and the non-CMB (new) data set is
less than that between the two CMB data sets and the non-CMB (old) data set. A
major reason for this is the larger error bars associated with updated $H(z)$
data (those for which there is now a non-diagonal covariance matrix). We
return to these points in Sec. IV.4.

Two other significant results obtained in [10] remain unchanged for the
updated non-CMB (new) data. The first is the compatibility of the measured
values of the curvature parameter $\Omega_{k}$, in the non-flat models, with
spatially-flat hypersurfaces, when P18 or P18+lensing data are jointly
analyzed with non-CMB (new) data. The second is the evidence that the lensing
parameter $A_{L}$ deviates from 1. When P18+lensing+non-CMB (new) data are
utilized with the $\Lambda$CDM Planck [new] $P(q)$ model we measure
$\Omega_{k}=0.0009\pm 0.0017$ [$0.0008\pm 0.0017$], whereas for the
$\Lambda$CDM Planck [new] $P(q)$+$A_{L}$ model we find $\Omega_{k}=0.0004\pm
0.0017$ [$0.0004\pm 0.0017$], therefore an open Universe is very mildly
favored in all the cases. In regard to the second result, for P18+lensing+non-
CMB (new) data, we still detect $\sim 2\sigma$ evidence in favor of $A_{L}>1$
for all models in which this phenomenological parameter is allowed to vary and
so can be measured. In particular, in the flat $\Lambda$CDM+$A_{L}$ model we
find $A_{L}=1.087\pm 0.035$, a deviation of 2.49$\sigma$ from the expected
value $A_{L}=1$. For the non-flat models, the $\Lambda$CDM Planck
$P(q)$+$A_{L}$ model yields $A_{L}=1.084\pm 0.035$, while the $\Lambda$CDM new
$P(q)$+$A_{L}$ model yields $A_{L}=1.084\pm 0.034$. Both results are
approximately 2.4$\sigma$ away from 1. When P18+non-CMB (new) data are used
with the $\Lambda$CDM Planck [new] $P(q)$ model, the resulting value for
$\Omega_{k}$ is $0.0009\pm 0.0017$ [$0.0008\pm 0.0017$]. For the $\Lambda$CDM
Planck [new] $P(q)$+$A_{L}$ model, $\Omega_{k}=-0.0001\pm 0.0017$ [$-0.0001\pm
0.0017$]. For P18+non-CMB (new) data we find $\sim 3\sigma$ evidence in favor
of $A_{L}>1$ for all the models. Specifically, in the case of the flat
$\Lambda$CDM+$A_{L}$ model, $A_{L}=1.198\pm 0.060$, indicating a deviation of
3.30$\sigma$ from the theoretically expected value of $A_{L}=1$. For the non-
flat models, the $\Lambda$CDM Planck $P(q)$+$A_{L}$ model gives
$A_{L}=1.196\pm 0.062$, while the $\Lambda$CDM new $P(q)$+$A_{L}$ model yields
$A_{L}=1.194\pm 0.061$ with both values deviating from 1 by 3.2$\sigma$.

The six-parameter tilted flat $\Lambda$CDM model, with $\Omega_{k}=0$ and
$A_{L}=1$, once again seems to pass all the consistency tests that we have
considered. The P18+lensing+non-CMB (new) data set is the largest one that we
have analyzed in this work. The primary cosmological parameter constraints
obtained from it for this standard model are $\Omega_{b}h^{2}=0.02249\pm
0.00013$, $\Omega_{c}h^{2}=0.11849\pm 0.00084$,
$100\theta_{\text{MC}}=1.04109\pm 0.00028$, $\tau=0.0569\pm 0.00071$,
$n_{s}=0.9685\pm 0.0036$, and $\ln(10^{10}A_{s})=3.046\pm 0.014$. Among the
different primary parameters the least well-determined is the reionization
optical depth at $8.0\sigma$ and the spectral index $n_{s}$ deviates from the
scale invariant value $n_{s}=1$ by 8.75$\sigma$. As we noted in [10] these
values are very similar to the corresponding ones in the other models under
study, therefore, they are almost model-independent results. As for the
derived parameters, for the Hubble constant we get $H_{0}=68.05\pm 0.38$ km
s-1 Mpc-1, which is in agreement with not only the result obtained from a
median statistics analysis $H_{0}=68\pm 2.8$ km km s-1 Mpc-1 [108, 109, 110],
but also with some other local measurements like for instance the flat
$\Lambda$CDM model value provided in [76] $H_{0}=69.5\pm 2.4$ km s-1 Mpc-1
from a joint analysis of $H(z)$, BAO, Pantheon+ SNIa, quasar angular size,
reverberation-measured Mg ii and C iv quasar, and 118 Amati correlation gamma-
ray burst data, or the local $H_{0}=69.8\pm 1.7$ km s-1 Mpc-1 from TRGB and
SNIa data [111]. On the other hand our measured $H_{0}$ value is still in
tension with the local $H_{0}=73.04\pm 1.04$ km s-1 Mpc-1 from Cepheids and
SNIa data [112], also see [113]. As for the other two derived parameters, we
get $\Omega_{m}=0.3059\pm 0.0050$ and $\sigma_{8}=0.8077\pm 0.0057$, the first
of which is in good agreement with the flat $\Lambda$CDM model value of
$\Omega_{m}=0.313\pm 0.012$ of [76] (for the same data used to measure $H_{0}$
listed above). These results have been obtained using the P18+lensing+non-CMB
(new) data set, which is possibly the currently largest combination of
mutually consistent data sets, and so these results are likely the currently
most-restrictive constraints on flat $\Lambda$CDM model parameters.

###  IV.2 XCDM models

In this subsection we study the XCDM models cosmological parameters
constraints, goodness of fit results, and the consistencies between different
data set combinations we use.

Before discussing these we compare our XCDM model results to prior results in
the literature for a few cases (where the same data set has previously been
used). The Planck team [114] measure, for the flat XCDM model and P18 data,
the dark energy equation of state parameter to be $w=-1.58^{+0.16}_{-0.35}$
(see $\S$17.5 of [114]), and for P18+lensing data they get
$w=-1.57^{+0.16}_{-0.33}$ (see $\S$17.6 of [114]), both of which are in good
agreement with our results shown in Table 11.

Interestingly, while Planck CMB data favor a large phantom-like region with
$w<-1$, non-CMB data favor a quintessence-like region with $w>-1$, see Figs.
16—27. This fact, as we shall see, goes hand in hand with the inconsistencies
between results, discussed below, obtained from Planck data and from non-CMB
(new) data in the XCDM models. When Planck jointly analyze such data (see
$\S$17.21 of [114] for P18+BAO+SNIa), they measure $w=-1.028\pm 0.033$
favoring phantom-like dynamical dark energy at 0.85$\sigma$, while using our
larger non-CMB (new) data compilation we find $w=-0.986\pm 0.024$ for P18+non-
CMB (new) data, disfavoring phantom-like dynamical dark energy at 0.58$\sigma$
in the flat XCDM model.

Another important point is that in all the XCDM cases, non-CMB data better
determine $w$ than do P18 or P18+lensing data. From Figs. 16—27 we see that
P18 or P18+lensing data alone are not sensitive to the dark energy equation of
state parameter $w$ because in these cases $w$ is strongly degenerate with all
other cosmological parameters. From these figures we can also see that this is
not the case for non-CMB (new) data. This is because SNIa as well as
$H(z)$+BAO data, both included in the non-CMB data compilation we use, have
the ability to reasonably restrictively constrain $w$, and the SNIa+$H(z)$+BAO
combination constrains $w$ very tightly. For example, the flat XCDM model
result from SNIa Pantheon+ data is $w=-0.90^{+0.17}_{-0.12}$, Table V of [76],
$w=-0.90\pm 0.14$ from Pantheon+SH0ES SNIa data [58], $w=-0.963\pm 0.070$ is
obtained from analyzing SN+BAO data [115], and the $H(z)$+BAO+SNP+ data
compilation result is $w=-0.886\pm 0.053$, Table V of [76], the last of which
is very consistent with our flat XCDM non-CMB (new) data result of
$w=-0.853\pm 0.039$.

Additionally, possibly as a consequence of the fact that non-CMB (new) data
more restrictively constrain $w$ than do P18 or P18+lensing data, non-CMB
(new) data also more restrictively constrain the derived parameters, $H_{0}$,
$\Omega_{m}$, and $\sigma_{8}$, than do P18 or P18+lensing data, see Tables
11—13.

Table 11: Mean and 68% (or 95%) confidence limits of flat XCDM ($+A_{L}$) model parameters from non-CMB (new), P18, P18+lensing, P18+non-CMB (new), and P18+lensing+non-CMB (new) data. $H_{0}$ has units of km s-1 Mpc-1.  | Flat XCDM models
---|---
Parameter | Non-CMB (new) | P18 | P18+lensing | P18+non-CMB (new) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.0316\pm 0.0043$ | $0.02240\pm 0.00015$ | $0.02243\pm 0.00015$ | $0.02251\pm 0.00014$ | $0.02250\pm 0.00014$
$\Omega_{c}h^{2}$ | $0.0994\pm 0.0087$ | $0.1200\pm 0.0014$ | $0.1193\pm 0.0012$ | $0.1181\pm 0.0010$ | $0.11830\pm 0.00095$
$100\theta_{\textrm{MC}}$ | $1.020\pm 0.010$ | $1.04094\pm 0.00032$ | $1.04100\pm 0.00031$ | $1.04114\pm 0.00030$ | $1.04112\pm 0.00029$
$\tau$ | $0.0537$ | $0.0537\pm 0.0078$ | $0.0524\pm 0.0074$ | $0.0558\pm 0.0078$ | $0.0577\pm 0.0075$
$n_{s}$ | $0.9654$ | $0.9654\pm 0.0044$ | $0.9667\pm 0.0041$ | $0.9696\pm 0.0039$ | $0.9690\pm 0.0038$
$\ln(10^{10}A_{s})$ | $3.57\pm 0.20$ | $3.043\pm 0.016$ | $3.038\pm 0.015$ | $3.043\pm 0.016$ | $3.048\pm 0.015$
$w$ | $-0.853\pm 0.039$ | $-1.59\pm 0.26$ | $-1.55\pm 0.26$ | $-0.986\pm 0.024$ | $-0.990\pm 0.023$
| ($-0.853^{+0.043}_{-0.033}$) | ($-1.59^{+0.15}_{-0.34}$) | ($-1.55^{+0.16}_{-0.35}$) |  |
$H_{0}$ | $69.8\pm 2.5$ |  $86.8\pm 8.9$ ($>70.2$) |  $86.0\pm 9.2$ ($>69.6$) | $67.78\pm 0.63$ | $67.81\pm 0.63$
$\Omega_{m}$ | $0.270\pm 0.012$ | $0.197\pm 0.046$ | $0.200\pm 0.048$ | $0.3075\pm 0.0063$ | $0.3077\pm 0.0062$
$\sigma_{8}$ | $0.824\pm 0.027$ | $0.974\pm 0.071$ | $0.960\pm 0.071$ | $0.801\pm 0.010$ | $0.8047\pm 0.0089$
$\chi_{\textrm{min}}^{2}$ | $1459.18$ | $2761.40$ | $2770.58$ | $4239.85$ | $4249.05$
$\Delta\chi_{\textrm{min}}^{2}$ | $-10.75$ | $-4.40$ | $-4.13$ | $-0.39$ | $-0.21$
DIC | $1468.74$ | $2815.67$ | $2824.21$ | $4294.20$ | $4303.30$
$\Delta\textrm{DIC}$ | $-9.37$ | $-2.26$ | $-2.24$ | $+1.87$ | $+2.10$
AIC | $1469.18$ | $2817.40$ | $2826.58$ | $4295.85$ | $4305.05$
$\Delta\textrm{AIC}$ | $-8.75$ | $-2.40$ | $-2.13$ | $+1.61$ | $+1.79$
| Flat XCDM$+A_{L}$ models
Parameter |  | P18 | P18+lensing | P18+non-CMB (new) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ |  | $0.02258\pm 0.00017$ | $0.02250\pm 0.00017$ | $0.02272\pm 0.00015$ | $0.02263\pm 0.00014$
$\Omega_{c}h^{2}$ |  | $0.1181\pm 0.0015$ | $0.1184\pm 0.0015$ | $0.1166\pm 0.0011$ | $0.1168\pm 0.0011$
$100\theta_{\textrm{MC}}$ |  | $1.04114\pm 0.00033$ | $1.04109\pm 0.00032$ | $1.04130\pm 0.00030$ | $1.04126\pm 0.00030$
$\tau$ |  | $0.0493\pm 0.0085$ | $0.04908\pm 0.0084$ | $0.0500\pm 0.0085$ | $0.0496\pm 0.0083$
$n_{s}$ |  | $0.9706\pm 0.0049$ | $0.9691\pm 0.0049$ | $0.9746\pm 0.0041$ | $0.9733\pm 0.0040$
$\ln(10^{10}A_{s})$ |  | $3.029\pm 0.018$ | $3.029\pm 0.018$ | $3.027\pm 0.017$ | $3.026\pm 0.017$
$w$ |  | $-1.23\pm 0.42$ | $-1.34\pm 0.37$ | $-0.964\pm 0.024$ | $-0.968\pm 0.024$
|  | ($-1.23^{+0.31}_{-0.59}$) | ($-1.34^{+0.26}_{-0.51}$) |  |
$A_{L}$ |  | $1.180\pm 0.097$ | $1.054\pm 0.055$ | $1.222\pm 0.063$ | $1.101\pm 0.037$
|  | ($1.180^{+0.062}_{-0.10}$) | ($1.054^{+0.039}_{-0.059}$) |  |
$H_{0}$ |  |  $77\pm 14$ ($77^{+20}_{-10}$) |  $80\pm 12$ ($>58.6$) | $67.83\pm 0.63$ | $67.79\pm 0.63$
$\Omega_{m}$ |  | $0.27\pm 0.11$ | $0.242\pm 0.083$ | $0.3043\pm 0.0062$ | $0.3050\pm 0.0062$
$\sigma_{8}$ |  | $0.86\pm 0.12$ | $0.89\pm 0.11$ | $0.784\pm 0.011$ | $0.785\pm 0.011$
$\chi_{\textrm{min}}^{2}$ |  | $2755.89$ | $2770.43$ | $4224.98$ | $4240.92$
$\Delta\chi_{\textrm{min}}^{2}$ |  | $-9.91$ | $-4.28$ | $-15.26$ | $-8.34$
DIC |  | $2813.08$ | $2825.81$ | $4283.50$ | $4296.89$
$\Delta\textrm{DIC}$ |  | $-4.85$ | $-0.64$ | $-8.83$ | $-4.31$
AIC |  | $2813.89$ | $2828.43$ | $4282.98$ | $4298.92$
$\Delta\textrm{AIC}$ |  | $-5.91$ | $-0.28$ | $-11.26$ | $-4.34$

Table 12: Mean and 68% (or 95%) confidence limits of non-flat XCDM ($+A_{L}$) model [Planck $P(q)$] parameters from non-CMB (new), P18, P18+lensing, P18+non-CMB (new), and P18+lensing+non-CMB (new) data. $H_{0}$ has units of km s-1 Mpc-1.  | Non-flat XCDM models [Planck $P(q)$]
---|---
Parameter | Non-CMB (new) | P18 | P18+lensing | P18+non-CMB (new) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.0285\pm 0.0043$ | $0.02260\pm 0.00017$ | $0.02249\pm 0.00016$ | $0.02245\pm 0.00015$ | $0.02246\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.118\pm 0.014$ | $0.1181\pm 0.0015$ | $0.1186\pm 0.0015$ | $0.1188\pm 0.0013$ | $0.1190\pm 0.0012$
$100\theta_{\textrm{MC}}$ | $1.44\pm 0.19$ | $1.04117\pm 0.00032$ | $1.04107\pm 0.00032$ | $1.04105\pm 0.00032$ | $1.04102\pm 0.00031$
$\tau$ | $0.0480$ | $0.0480\pm 0.0083$ | $0.0495\pm 0.0082$ | $0.0548\pm 0.0076$ | $0.0573\pm 0.0074$
$\Omega_{k}$ | $-0.177\pm 0.067$ | $-0.048\pm 0.035$ | $-0.011\pm 0.017$ | $0.0017\pm 0.0019$ | $0.0016\pm 0.0019$
| ($-0.177^{+0.064}_{-0.072}$) | ($-0.048^{+0.041}_{-0.012}$) | ($-0.0111^{+0.013}_{-0.00070}$) |  |
$n_{s}$ | $0.9706$ | $0.9706\pm 0.0047$ | $0.9687\pm 0.0046$ | $0.9678\pm 0.0044$ | $0.9674\pm 0.0042$
$\ln(10^{10}A_{s})$ | $2.89\pm 0.36$ | $3.027\pm 0.017$ | $3.030\pm 0.017$ | $3.042\pm 0.015$ | $3.048\pm 0.015$
$w$ | $-0.786\pm 0.041$ | $-1.27\pm 0.71$ | $-1.28\pm 0.45$ | $-0.975\pm 0.026$ | $-0.980\pm 0.026$
| ($-0.786^{+0.044}_{-0.037}$) | ($-1.27^{+0.97}_{-0.45}$) | ($-1.28^{+0.41}_{-0.54}$) |  |
$H_{0}$ | $70.6\pm 2.4$ |  $60\pm 16$ ($60^{+9}_{-20}$) |  $73\pm 15$ ($73^{+20}_{-10}$) | $67.95\pm 0.67$ | $67.95\pm 0.66$
$\Omega_{m}$ | $0.294\pm 0.018$ | $0.47\pm 0.23$ | $0.30\pm 0.15$ | $0.3074\pm 0.0062$ | $0.3078\pm 0.0062$
$\sigma_{8}$ | $0.774\pm 0.037$ | $0.83\pm 0.15$ | $0.87\pm 0.12$ | $0.801\pm 0.010$ | $0.8049\pm 0.0088$
$\chi_{\textrm{min}}^{2}$ | $1460.80$ | $2754.91$ | $2770.40$ | $4238.67$ | $4248.26$
$\Delta\chi_{\textrm{min}}$ | $-13.29$ | $-10.89$ | $-4.31$ | $-1.57$ | $-1.00$
DIC | $1468.14$ | $2810.86$ | $2827.00$ | $4294.75$ | $4303.54$
$\Delta\textrm{DIC}$ | $-9.97$ | $-7.07$ | $+0.55$ | $+2.42$ | $+2.34$
AIC | $1468.64$ | $2812.91$ | $2828.40$ | $4296.67$ | $4306.26$
$\Delta\textrm{AIC}$ | $-9.29$ | $-6.89$ | $-0.31$ | $+2.43$ | $+3.00$
| Non-flat XCDM$+A_{L}$ models [Planck $P(q)$]
Parameter |  | P18 ($A_{L}>0.8$) | P18+lensing | P18+non-CMB (new) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ |  | $0.02260\pm 0.00017$ | $0.02250\pm 0.00017$ | $0.02268\pm 0.00016$ | $0.02259\pm 0.00016$
$\Omega_{c}h^{2}$ |  | $0.1182\pm 0.0015$ | $0.1184\pm 0.0015$ | $0.1171\pm 0.0014$ | $0.1175\pm 0.0014$
$100\theta_{\textrm{MC}}$ |  | $1.04117\pm 0.00032$ | $1.04107\pm 0.00032$ | $1.04124\pm 0.00032$ | $1.04118\pm 0.00032$
$\tau$ |  | $0.0479\pm 0.0081$ | $0.0484\pm 0.0085$ | $0.0499\pm 0.0084$ | $0.04923\pm 0.0082$
$\Omega_{k}$ |  | $-0.073\pm 0.051$ | $-0.012\pm 0.027$ | $0.0011\pm 0.0019$ | $0.0015\pm 0.0019$
|  | ($-0.073^{+0.065}_{-0.029}$) | ($-0.012^{+0.027}_{-0.011}$) |  |
$n_{s}$ |  | $0.9706\pm 0.0048$ | $0.9689\pm 0.0049$ | $0.9734\pm 0.0047$ | $0.9717\pm 0.0046$
$\ln(10^{10}A_{s})$ |  | $3.027\pm 0.017$ | $3.027\pm 0.018$ | $3.028\pm 0.017$ | $3.027\pm 0.017$
$w$ |  | $-1.36\pm 0.77$ | $-1.32\pm 0.58$ | $-0.958\pm 0.027$ | $-0.958\pm 0.026$
|  | ($-1.36^{+1.1}_{-0.53}$) | ($-1.32^{+0.71}_{-0.38}$) |  |
$A_{L}$ |  |  $0.95\pm 0.12$ ($<1.20$) | $1.02\pm 0.16$ | $1.217\pm 0.064$ | $1.102\pm 0.037$
$H_{0}$ |  |  $54\pm 14$ ($54.3^{+5.9}_{-17}$) | $72\pm 15$ | $67.92\pm 0.67$ | $67.94\pm 0.68$
$\Omega_{m}$ |  | $0.57\pm 0.24$ | $0.31\pm 0.15$ | $0.3045\pm 0.0062$ | $0.3049\pm 0.0063$
$\sigma_{8}$ |  | $0.80\pm 0.13$ | $0.86\pm 0.13$ | $0.784\pm 0.011$ | $0.785\pm 0.011$
$\chi_{\textrm{min}}^{2}$ |  | $2754.46$ | $2770.28$ | $4224.83$ | $4239.70$
$\Delta\chi_{\textrm{min}}^{2}$ |  | $-11.34$ | $-4.43$ | $-15.41$ | $-9.56$
DIC |  | $2811.61$ | $2829.13$ | $4285.15$ | $4298.54$
$\Delta\textrm{DIC}$ |  | $-6.32$ | $+2.68$ | $-7.18$ | $-2.66$
AIC |  | $2814.46$ | $2830.28$ | $4284.83$ | $4299.70$
$\Delta\textrm{AIC}$ |  | $-5.34$ | $+1.57$ | $-9.41$ | $-3.56$

Table 13: Mean and 68% (or 95%) confidence limits of non-flat XCDM ($+A_{L}$) model [new $P(q)$] parameters from non-CMB (new), P18, P18+lensing, P18+non-CMB (new), and P18+lensing+non-CMB (new) data. $H_{0}$ has units of km s-1 Mpc-1.  | Non-flat XCDM models [new $P(q)$]
---|---
Parameter | Non-CMB (new) | P18 | P18+lensing | P18+non-CMB (new) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ | $0.0282\pm 0.0044$ | $0.02256\pm 0.00017$ | $0.02248\pm 0.00016$ | $0.02246\pm 0.00015$ | $0.02246\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.119\pm 0.015$ | $0.1188\pm 0.0015$ | $0.1188\pm 0.0014$ | $0.1188\pm 0.0013$ | $0.1190\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.47\pm 0.24$ | $1.04109\pm 0.00033$ | $1.04104\pm 0.00032$ | $1.04104\pm 0.00031$ | $1.04102\pm 0.00032$
$\tau$ | $0.0524$ | $0.0524\pm 0.0082$ | $0.0511\pm 0.0081$ | $0.0552\pm 0.0079$ | $0.0576\pm 0.0076$
$\Omega_{k}$ | $-0.186\pm 0.076$ | $-0.034\pm 0.025$ | $-0.008\pm 0.010$ | $0.0016\pm 0.0020$ | $0.0014\pm 0.0020$
| ($-0.186^{+0.083}_{-0.067}$) | ($-0.0338^{+0.029}_{-0.0086}$) | ($-0.0080^{+0.0098}_{-0.0023}$) |  |
$n_{s}$ | $0.9653$ | $0.9653\pm 0.0044$ | $0.9663\pm 0.0044$ | $0.9677\pm 0.0043$ | $0.9673\pm 0.0042$
$\ln(10^{10}A_{s})$ | $2.85\pm 0.38$ | $3.038\pm 0.017$ | $3.034\pm 0.016$ | $3.043\pm 0.016$ | $3.049\pm 0.015$
$w$ | $-0.785\pm 0.042$ | $-1.27\pm 0.61$ | $-1.27\pm 0.41$ | $-0.976\pm 0.026$ | $-0.982\pm 0.026$
| ($-0.785^{+0.045}_{-0.038}$) | ($-1.27^{+0.79}_{-0.44}$) | ($-1.27^{+0.40}_{-0.49}$) |  |
$H_{0}$ | $70.6\pm 2.5$ |  $63\pm 15$ ($63^{+10}_{-20}$) | $74\pm 14$ | $67.94\pm 0.66$ | $67.96\pm 0.66$
$\Omega_{m}$ | $0.296\pm 0.018$ | $0.41\pm 0.19$ | $0.29\pm 0.12$ | $0.3074\pm 0.0062$ | $0.3077\pm 0.0061$
$\sigma_{8}$ | $0.771\pm 0.036$ | $0.85\pm 0.13$ | $0.87\pm 0.11$ | $0.801\pm 0.010$ | $0.8056\pm 0.0089$
$\chi_{\textrm{min}}^{2}$ | $1459.51$ | $2757.86$ | $2770.57$ | $4238.57$ | $4247.96$
$\Delta\chi_{\textrm{min}}^{2}$ | $-13.27$ | $-7.94$ | $-4.14$ | $-1.67$ | $-1.30$
DIC | $1468.73$ | $2811.78$ | $2826.60$ | $4294.90$ | $4304.26$
$\Delta\textrm{DIC}$ | $-9.38$ | $-6.15$ | $+0.15$ | $+2.57$ | $+3.06$
AIC | $1468.66$ | $2815.86$ | $2828.57$ | $4296.57$ | $4305.96$
$\Delta\textrm{AIC}$ | $-9.27$ | $-3.94$ | $-0.14$ | $+2.33$ | $+2.70$
| Non-flat XCDM$+A_{L}$ models [new $P(q)$]
Parameter |  | P18 ($A_{L}>0.8$) | P18+lensing | P18+non-CMB (new) | P18+lensing+non-CMB (new)
$\Omega_{b}h^{2}$ |  | $0.02260\pm 0.00017$ | $0.02250\pm 0.00017$ | $0.02268\pm 0.00017$ | $0.02258\pm 0.00016$
$\Omega_{c}h^{2}$ |  | $0.1182\pm 0.0015$ | $0.1185\pm 0.0016$ | $0.1171\pm 0.0014$ | $0.1175\pm 0.0014$
$100\theta_{\textrm{MC}}$ |  | $1.04116\pm 0.00033$ | $1.04106\pm 0.00032$ | $1.04124\pm 0.00032$ | $1.04117\pm 0.00032$
$\tau$ |  | $0.0478\pm 0.0083$ | $0.0503\pm 0.0086$ | $0.0500\pm 0.0084$ | $0.04936\pm 0.0082$
$\Omega_{k}$ |  | $-0.072\pm 0.051$ | $-0.003\pm 0.018$ | $0.0011\pm 0.0019$ | $0.0015\pm 0.0019$
|  | ($-0.072^{+0.065}_{-0.030}$) | ($-0.003^{+0.018}_{-0.011}$) |  |
$n_{s}$ |  | $0.9706\pm 0.0048$ | $0.9676\pm 0.0055$ | $0.9731\pm 0.0046$ | $0.9717\pm 0.0046$
$\ln(10^{10}A_{s})$ |  | $3.027\pm 0.018$ | $3.032\pm 0.018$ | $3.028\pm 0.017$ | $3.027\pm 0.017$
$w$ |  | $-1.39\pm 0.77$ | $-1.18\pm 0.48$ | $-0.959\pm 0.026$ | $-0.959\pm 0.027$
|  | ($-1.39^{+1.1}_{-0.54}$) | ($-1.18^{+0.54}_{-0.37}$) |  |
$A_{L}$ |  | $0.95\pm 0.13$ | $1.07\pm 0.14$ | $1.213\pm 0.064$ | $1.101\pm 0.038$
|  | ($<1.19$) | ($1.07^{+0.12}_{-0.16}$) |  |
$H_{0}$ |  |  $55\pm 14$ ($54.5^{+5.7}_{-17}$) | $74\pm 15$ | $67.92\pm 0.65$ | $67.95\pm 0.66$
$\Omega_{m}$ |  | $0.56\pm 0.24$ | $0.30\pm 0.13$ | $0.3046\pm 0.0061$ | $0.3047\pm 0.0062$
$\sigma_{8}$ |  | $0.80\pm 0.12$ | $0.84\pm 0.12$ | $0.785\pm 0.011$ | $0.786\pm 0.011$
$\chi_{\textrm{min}}^{2}$ |  | $2754.40$ | $2770.27$ | $4224.52$ | $4239.76$
$\Delta\chi_{\textrm{min}}^{2}$ |  | $-11.40$ | $-4.44$ | $-15.72$ | $-9.50$
DIC |  | $2811.71$ | $2828.10$ | $4284.84$ | $4298.27$
$\Delta\textrm{DIC}$ |  | $-6.22$ | $+1.65$ | $-7.49$ | $-2.93$
AIC |  | $2814.40$ | $2830.27$ | $4284.52$ | $4299.76$
$\Delta\textrm{AIC}$ |  | $-5.40$ | $+1.56$ | $-9.72$ | $-3.50$

Figure 16: Likelihood distributions of flat XCDM model parameters favored by
non-CMB (new) data. Results for P18 and P18+non-CMB (new) data are shown for
comparison.

Figure 17: Likelihood distributions of flat XCDM$+A_{L}$ model parameters
favored by non-CMB (new) data. Results for P18 and P18+non-CMB (new) data are
shown for comparison.

Figure 18: Likelihood distributions of non-flat XCDM model [Planck $P(q)$]
parameters favored by non-CMB (new) data. Results for P18 and P18+non-CMB
(new) data are shown for comparison.

Figure 19: Likelihood distributions of non-flat XCDM$+A_{L}$ model [Planck
$P(q)$] parameters favored by non-CMB (new) data. Results for P18 and P18+non-
CMB (new) data are shown for comparison.

Figure 20: Likelihood distributions of non-flat XCDM model [new $P(q)$]
parameters favored by non-CMB data (new). Results for P18 and P18+non-CMB
(new) data are shown for comparison.

Figure 21: Likelihood distributions of non-flat XCDM$+A_{L}$ model [new
$P(q)$] parameters favored by non-CMB (new) data. Results for P18 and P18+non-
CMB (new) data are shown for comparison.

####  IV.2.1 Non-CMB (new) cosmological constraints

Here we summarize the XCDM models cosmological parameter constraints from the
non-CMB (new) data set. The non-CMB (new) data constraints on the XCDM$+A_{L}$
models are identical to those on the XCDM models so we do not distinguish
between these models in this subsubsection.

The results for the five-parameter flat XCDM model obtained with non-CMB (new)
data are in Table 11 and Figs. 16 and 17. For the equation of state parameter
we obtain $w=-0.853^{+0.043}_{-0.033}$ which is 4.45$\sigma$ away from the
cosmological constant $w=-1$ value and favors quintessence-like evolution. For
earlier indications that non-CMB data favor quintessence-like, $w>-1$,
evolution in flat and non-flat XCDM models, see [115, 116, 117, 76, 25, 118].
Regarding the derived parameters, we obtain $H_{0}=69.8\pm 2.5$ km s-1 Mpc-1
and $\Omega_{m}=0.270\pm 0.012$, in good agreement with the flat XCDM model
values of $H_{0}=69.05\pm 2.4$ km s-1 Mpc-1 and $\Omega_{m}=0.292\pm 0.016$
from a joint analysis of $H(z)$, BAO, Pantheon+ SNIa, quasar angular size,
reverberation-measured Mg ii and C iv quasar, and 118 Amati correlation gamma-
ray burst data, [76] Table VII.

Table 12 [13] and Figs. 18 and 19 [20 and 21] show results for the six-
parameter non-flat XCDM Planck [new] $P(q)$ cosmological model from non-CMB
(new) data. We obtain $\Omega_{k}=-0.177^{+0.064}_{-0.072}$
[$-0.186^{+0.083}_{-0.067}$] and $w=-0.786^{+0.044}_{-0.037}$
[$-0.785^{+0.045}_{-0.038}$]. The first result indicates that closed non-flat
hypersufaces are favored by 2.77$\sigma$ [2.24$\sigma$] and the second one
favors quintessence-like evolution over a cosmological constant by
5.78$\sigma$ [5.66$\sigma$]. As for the derived parameters, as in the flat
XCDM model, we find agreement with the $H_{0}=69.26\pm 2.45$ km s-1 Mpc-1 and
$\Omega_{m}=0.296\pm 0.020$ values of [76] from a joint analysis of the data
sets listed at the end of the previous paragraph. We obtain $H_{0}=70.6\pm
2.4$ km s-1 Mpc-1 [$70.6\pm 2.5$ km s-1 Mpc-1] and $\Omega_{m}=0.294\pm 0.018$
[$0.296\pm 0.018$].

####  IV.2.2 P18 data cosmological constraints

In the case of the flat XCDM cosmological model with seven primary parameters
(see Table 11 and Fig. 16), from P18 data alone we obtain $\Omega_{m}=0.197\pm
0.046$, which differs by $-1.94\sigma$ from the flat XCDM model value
$\Omega_{m}=0.292\pm 0.016$ from a joint analysis of $H(z)$, BAO, Pantheon+
SNIa, quasar angular size, reverberation-measured Mg ii and C iv quasar, and
118 Amati correlation gamma-ray burst data, [76] Table VII. The error bars
associated to the $H_{0}$ parameter cannot be determined and the best
estimation possible is $H_{0}>70.2$ km s-1 Mpc-1 (95% confidence limit).

The improvement in the fit with respect to the flat $\Lambda$CDM cosmological
model with $w=-1$ is positive according to the DIC and AIC statistical
criteria, see Table 11. This is reflected in the P18 data value for the
X-fluid equation of state parameter $w=-1.59^{+0.15}_{-0.34}$, a 3.93$\sigma$
deviation from $w=-1$. This difference in the equation of state parameter is
not accompanied by equally significant changes in the other primary parameters
when compared to the values obtained in the flat $\Lambda$CDM model, with the
largest being $-0.19\sigma$ for $\Omega_{b}{h^{2}}$. On the other hand, there
are significant differences in derived parameters when compared to the flat
$\Lambda$CDM values. In particular, for $\Omega_{m}$, and $\sigma_{8}$ we find
differences of $2.56\sigma$ and $-2.27\sigma$, respectively. As for the error
bars, those associated with the primary parameters barely change (the largest
change is an increase of 3.1% for 100$\theta_{\textrm{MC}}$). However, the
error bars of the derived parameters are significantly affected, with
increases of 82% and 90% for $\Omega_{m}$ and $\sigma_{8}$, respectively.

Using Table 11 we can compare P18 data results for the seven-parameter flat
XCDM model (upper half) and for the eight-parameter flat XCDM+$A_{L}$ model
(lower half and Fig. 17). There are changes in the values of the primary
parameters but none above 1$\sigma$. In particular, the values of
$\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, and $n_{s}$ differ by $-0.79\sigma$,
$+0.93\sigma$, and $-0.79\sigma$, respectively. For the equation of state
parameter in the flat XCDM model, we find $w=-1.59^{+0.15}_{-0.34}$ (deviating
by $3.93\sigma$ from $w=-1$). However, in the flat XCDM+$A_{L}$ model we find
$w=-1.23^{+0.31}_{-0.59}$ ($0.74\sigma$ away from $w=-1$), resulting in a
difference of $-0.59\sigma$ between the flat XCDM and flat XCDM+$A_{L}$ values
from P18 data. In both cases there is a preference for phantom-like behavior.
In the flat XCDM+$A_{L}$ model, we find $A_{L}=1.180^{+0.062}_{-0.10}$ which
is 1.8$\sigma$ away from $A_{L}=1$. As for the derived parameters, the matter
density parameters $\Omega_{m}$ differ by $-0.61\sigma$, while the values of
$\sigma_{8}$ differ by $0.82\sigma$. When comparing the flat XCDM and flat
XCDM+$A_{L}$ results, we observe an increase of 38% in the size of the error
bars of the dark energy equation of state parameters. For the rest of the
primary parameters the largest increase is 12% for $\Omega_{b}h^{2}$. As for
the derived parameters $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$, the
corresponding increases are 36%, 58%, and 41% respectively.

As expected, the simultaneous consideration of both $w$ and $\Omega_{k}$ (see
Tables 12 and 13 and Figs. 18 and 20) makes the already existing degeneracies
even bigger. In particular, in the non-flat XCDM cases the values of $w$ and
$\Omega_{m}$ increase and the value of $H_{0}$ decreases with respect to the
flat XCDM model values. Therefore, from the results obtained, it is clear that
P18 data alone cannot break the degeneracies between $H_{0}$, $\Omega_{m}$,
$\Omega_{k}$, and $w$. For the non-flat XCDM Planck $P(q)$ and the non-flat
XCDM new $P(q)$ models, we find $H_{0}=60^{+9}_{-20}$ km s-1 Mpc-1 and
$63^{+10}_{-20}$ km s-1 Mpc-1, respectively, whereas for the matter parameter
we get $\Omega_{m}=0.47\pm 0.23$ and $0.41\pm 0.19$, respectively, which are
in agreement within 1$\sigma$ with the values obtained in [76],
$H_{0}=69.26\pm 2.45$ km s-1 Mpc-1 and $\Omega_{m}=0.296\pm 0.020$ .

Comparing the results of the seven-parameter non-flat $\Lambda$CDM Planck
$P(q)$ model (see Table 5) and the results for the eight-parameter non-flat
XCDM Planck $P(q)$ model (see Table 12) we observe that there is almost no
difference in the values of the six primary cosmological parameters in common
with the flat $\Lambda$CDM model, with the largest difference being
+0.026$\sigma$ for $\tau$. Regarding the curvature parameter, for the
$\Lambda$CDM Planck $P(q)$ model we obtain
$\Omega_{k}=-0.043^{+0.018}_{-0.015}$ while for the XCDM Planck $P(q)$ case
the corresponding value is $\Omega_{k}=-0.048^{+0.041}_{-0.012}$, indicating a
difference of $+0.11\sigma$. Both models show a clear preference for closed
geometry, deviating from flat by 2.39$\sigma$ and 1.17$\sigma$, respectively.
The $\Omega_{k}$ error bars are 51% larger in the XCDM Planck $P(q)$ model
compared to the $\Lambda$CDM Planck $P(q)$ model value. For the equation of
state parameter in the XCDM Planck $P(q)$ case we obtain
$w=-1.27^{+0.97}_{-0.45}$ which indicates a preference for phantom-like
behavior at 0.28$\sigma$. As for the derived parameters $H_{0}$, $\Omega_{m}$,
and $\sigma_{8}$, the values obtained for both models differ at $-0.27\sigma$,
0.05$\sigma$, and $-0.36\sigma$ and they have error bars +75.9%, +73.0%, and
+90.0% larger in the XCDM Planck $P(q)$ case compared to the $\Lambda$CDM
Planck $P(q)$ model.

If we look at Tables 6 and 13 we can compare the results obtained with P18
data for the seven-parameter non-flat $\Lambda$CDM new $P(q)$ model and the
eight-parameter non-flat XCDM new $P(q)$ model. As we noted in the Planck
$P(q)$ case, there are also no significant differences in the values of the
primary cosmological parameters in common with the flat $\Lambda$CDM model in
the new $P(q)$ case, the largest being $-0.05\sigma$ for $\Omega_{b}h^{2}$. As
for the curvature parameter, the $\Lambda$CDM new $P(q)$ model yields
$\Omega_{k}=-0.033^{+0.017}_{-0.011}$ (1.94$\sigma$), while the XCDM new
$P(q)$ model gives $\Omega_{k}=-0.0338^{+0.029}_{-0.0086}$ (1.17$\sigma$).
Again, the results indicate a preference for closed geometry. The equation of
state parameter value in the XCDM new $P(q)$ model obtained from an analysis
of P18 data is $w=-1.27^{+0.79}_{-0.44}$ which differs from the cosmological
constant by 0.34$\sigma$. Regarding the derived parameters, the differences in
$H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ are $-0.30\sigma$, +0.17$\sigma$, and
$-0.49\sigma$, and the error bars for the $w\neq-1$ model are larger by 76.0%,
71.1%, and 89.2%, respectively.

Comparing P18 data results for the eight-parameter non-flat XCDM Planck [new]
$P(q)$ model and the nine-parameter non-flat XCDM Planck (new) $P(q)+A_{L}$
model (see Tables 12 and 13 and Figs. 18 — 21), we find similar values of the
primary parameters in common with the flat $\Lambda$CDM standard model, with
the largest difference affecting $\Omega_{c}h^{2}$ [$\tau$] by $-0.047\sigma$
[$0.39\sigma$]. While for the non-flat XCDM Planck [new] $P(q)$ model we get
$\Omega_{k}=-0.048^{+0.041}_{-0.012}$ [$-0.0338^{+0.029}_{-0.0086}$], which is
$1.17\sigma$ [$1.17\sigma$] away from flat hypersurfaces, for the non-flat
XCDM Planck [new] $P(q)+A_{L}$ model we obtain
$\Omega_{k}=-0.073^{+0.065}_{-0.029}$ [$-0.072^{+0.065}_{-0.030}$], favoring a
closed geometry at a significance level of 1.12$\sigma$ [1.11$\sigma$]. The
difference between the $P(q)$ and $P(q)+A_{L}$ model values is $0.38\sigma$
[$0.58\sigma$]. For the dark energy equation of state parameter, in the non-
flat XCDM Planck [new] $P(q)$ model we find $w=-1.27^{+0.97}_{-0.45}$
[$-1.27^{+0.79}_{-0.44}$] deviating from $w=-1$ by $0.28\sigma$ [$0.34\sigma$]
while for the non-flat XCDM Planck [new] $P(q)+A_{L}$ model we get
$w=-1.36^{+1.1}_{-0.53}$ (0.33$\sigma$) [$-1.39^{+1.1}_{-0.54}$
(0.35$\sigma$)] with $P(q)$ and $P(q)+A_{L}$ model values differing by
0.076$\sigma$ [0.10$\sigma$]. For the non-flat XCDM Planck [new] $P(q)+A_{L}$
model, P18 data are only able to provide a 95% upper bound of $<1.20$
[$<1.19$] on the lensing consistency parameter $A_{L}$. The differences
observed in the values of the derived parameters are also not significant, in
particular for $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ we find $0.27\sigma$
[$0.38\sigma$], $-0.30\sigma$ [$-0.49\sigma$], and $0.15\sigma$
[$0.28\sigma$]. Due to degeneracies, in some cases we find smaller error bars
when $A_{L}$ is allowed to vary (compared to the $A_{L}=1$ case). For
$\Omega_{k}$ and $w$, they increase by +31.37% [+50.98%] and +7.79% [20.78%],
respectively, while for the derived parameters, $H_{0}$, $\Omega_{m}$, and
$\sigma_{8}$, the corresponding error bar changes are $-14.29$% [$-7.14$%],
$+4.17$% [$+20.83$%] and $-15.38$% [$-8.33$%].

Figure 22: Likelihood distributions of flat XCDM model parameters favored by
non-CMB (new) data. Results for P18+lensing and P18+lensing+non-CMB (new) data
are shown for comparison.

Figure 23: Likelihood distributions of flat XCDM$+A_{L}$ model parameters
favored by non-CMB (new) data. Results for P18+lensing and P18+lensing+non-CMB
(new) data are shown for comparison.

Figure 24: Likelihood distributions of non-flat XCDM model [Planck $P(q)$]
parameters favored by non-CMB (new) data. Results for P18+lensing and
P18+lensing+non-CMB (new) data are shown for comparison.

Figure 25: Likelihood distributions of non-flat XCDM$+A_{L}$ model [Planck
$P(q)$] parameters favored by non-CMB (new) data. Results for P18+lensing and
P18+lensing+non-CMB (new) data are shown for comparison.

Figure 26: Likelihood distributions of non-flat XCDM model [new $P(q)$]
parameters favored by non-CMB (new) data. Results for P18+lensing and
P18+lensing+non-CMB (new) data are shown for comparison.

Figure 27: Likelihood distributions of non-flat XCDM$+A_{L}$ model [new
$P(q)$] parameters favored by non-CMB (new) data. Results for P18+lensing and
P18+lensing+non-CMB (new) data are shown for comparison.

####  IV.2.3 P18+lensing cosmological constraints

Comparing the seven-parameter flat XCDM model primary cosmological parameter
constraints for P18 and P18+lensing data, shown in Table 11 and in Figs. 16
and 22, we observe only minor differences, with the largest occurring in
$\Omega_{c}h^{2}$ (+0.38$\sigma$) and $n_{s}$ ($-0.22\sigma$). When
P18+lensing data are analyzed, the resulting cosmological parameter error bars
are similar to those for P18 data but slightly smaller, with the largest
decrease being $-16$% for $\Omega_{c}h^{2}$. For the equation of state
parameter, we obtain $w=-1.55^{+0.16}_{-0.35}$ using P18+lensing data, which
is 3.44$\sigma$ in favor of phantom-like behavior and differs by only
0.11$\sigma$ from the P18 data value ($w=-1.59^{+0.15}_{-0.34}$), which is
3.93$\sigma$ away from $w=-1$ of the cosmological constant. For the derived
parameters, $H_{0}$ cannot be properly constrained and the 95% limit is
$H_{0}>69.6$ km s-1 Mpc-1, whereas for the other two we have
$\Omega_{m}=0.200\pm 0.048$ and $\sigma_{8}=0.960\pm 0.071$, which differ by
$-0.045\sigma$ and 0.14$\sigma$ from the corresponding values obtained with
P18 data. Interestingly, for $\Omega_{m}$ we find that the error bars obtained
using P18 data are smaller than those obtained using P18+lensing data, by
4.17%. This could mean that adding lensing data to P18 data still does not
break the large degeneracies between cosmological parameters.

The results in Table 11 (and Figs. 17 and 23) allow us to compare P18 and
P18+lensing data constraints on the eight-parameter flat XCDM+$A_{L}$ model.
No significant differences in the values of the primary parameters are seen
when lensing data are added to the mix, with the largest changes appearing in
the $\Omega_{b}h^{2}$ (0.33$\sigma$) and $n_{s}$ (0.22$\sigma$) mean values.
When P18 data are considered we obtain $w=-1.23^{+0.31}_{-0.59}$ whereas when
we use P18+lensing data we get $w=-1.34^{+0.26}_{-0.51}$, both values favoring
phantom-like behavior at 0.74$\sigma$ and 1.31$\sigma$. The difference between
the two result is 0.17$\sigma$, and the size of the $w$ error bars decreases
by $-13.51$% when moving from P18 data to P18+lensing data. For the lensing
consistency parameter $A_{L}$, from P18 data we obtain
$A_{L}=1.180^{+0.062}_{-0.10}$ which deviates from the expected value
$A_{L}=1$ by 1.80$\sigma$, while from P18+lensing data we obtain
$A_{L}=1.054^{+0.039}_{-0.059}$ which prefers $A_{L}>1$ at 0.92$\sigma$. We
observe a shrinkage in the $A_{L}$ error bars of $-76.36$% when lensing data
are included, and the two mean $A_{L}$ values differ at 1.17$\sigma$. In
regard to the derived parameters, $\Omega_{m}$ and $\sigma_{8}$, the mean
value results differ by 0.20$\sigma$ and $-0.18\sigma$ and the error bars
decrease by $-32.53$% and $-9.09$% when lensing data are also included.

From the results shown in Table 11 we can compare P18+lensing data constraints
on the seven-parameter flat XCDM (upper half of the table and Fig. 22) and the
eight-parameter flat XCDM+$A_{L}$ (lower half of the table and Fig. 23)
cosmological models. For the primary parameters the differences between the
two sets of results are less than 1$\sigma$. The larger changes affect
$\Omega_{c}h^{2}$ and $n_{s}$ which differ at 0.47$\sigma$ and $-0.57\sigma$
respectively, whereas for the dark energy equation of state parameter $w$ the
difference between the two results is $-0.39\sigma$. As expected, in the case
of the flat XCDM+$A_{L}$ model we find larger error bars for the primary
parameters, than those found in the flat XCDM model, with the largest
increases of 20%, 17%, and 30% corresponding to the error bars of
$\Omega_{c}h^{2}$, $\ln(10^{10}A_{s})$, and $w$. For the derived parameters,
the $\Omega_{m}$ and $\sigma_{8}$ values differ by $-0.44\sigma$ and
0.53$\sigma$ with increases in the size of the error bars of 42% and 35% when
moving from the flat XCDM model to the flat XCDM+$A_{L}$ model.

Tables 12 and 13 and Figs. 24 and 26 list and show results for the non-flat
XCDM Planck $P(q)$ and the non-flat XCDM new $P(q)$ models when P18 and
P18+lensing data are used in the analyses. Regarding the primary parameters in
common with the flat $\Lambda$CDM model, we do not find significant changes
(the differences are less than 1$\sigma$) when we move from P18 data to
P18+lensing data, with the largest difference for the XCDM Planck [new] $P(q)$
case being 0.47$\sigma$ [0.34$\sigma$] for $\Omega_{b}h^{2}$. As for the error
bars, we find a general reduction in size when lensing data are added to the
mix. In the XCDM Planck $P(q)$ case we find a change of $-6.25$% in the
$\Omega_{b}h^{2}$ error bar value, while in the XCDM new $P(q)$ model we
obtain changes of $-6.25$% for both $\Omega_{b}h^{2}$ and $\ln(10^{10}A_{s})$.
On the other hand, there are non-negligible changes in $\Omega_{k}$ and $w$
when lensing data are included in the analysis. For the XCDM Planck [new]
$P(q)$ model the value $\Omega_{k}=-0.0111^{+0.013}_{-0.00070}$
[$-0.0080^{+0.0098}_{-0.0023}$] indicates a $0.85\sigma$ [$0.82\sigma$]
preference for a closed Universe and a mild difference with the P18 data
value, $\Omega_{k}=-0.048^{+0.041}_{-0.012}$ [$-0.0338^{+0.029}_{-0.0086}$],
of $-0.90\sigma$ [$-0.89\sigma$]. The $\Omega_{k}$ error bars obtained with
P18 data are approximately a factor of 2.1 (2.5) larger than those obtained
with P18+lensing data. Interestingly, for both of the non-flat XCDM models,
the central value of $w$ is barely affected by the addition of lensing data,
however there is a reduction in the size of the $w$ error bars (of 50% for the
XCDM Planck $P(q)$ and 38% for the XCDM new $P(q)$ cases) which increases the
evidence favoring phantom behavior. In particular, for the XCDM Planck [new]
$P(q)$ case, from P18+lensing data, we get $w=-1.28^{+0.41}_{-0.54}$
[$-1.27^{+0.40}_{-0.49}$] which deviates from $w=-1$ by 0.68$\sigma$
[0.68$\sigma$]. As for the derived parameters, in the case of the XCDM Planck
(new) $P(q)$ model, the differences between the P18+lensing data and P18 data
values, for $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ are $-0.97\sigma$
[$-0.64\sigma$], +0.62$\sigma$ [0.53$\sigma$], and $-0.21\sigma$
[$-0.12\sigma$], respectively. The largest reduction in the error bars affects
$\Omega_{m}$, with a decrease of 53% [58%].

Results in the lower half of Table 12 [13] (also see Figs. 19 and 25 [21 and
27]) allows us to compare P18 data and P18+lensing data constraints on the
nine-parameter non-flat XCDM Planck [new] $P(q)+A_{L}$ model. The differences
in the mean values of the primary parameters are less than 1$\sigma$, with the
largest differences occurring in $\Omega_{b}h^{2}$ (0.42$\sigma$
[0.42$\sigma$]), $100\theta_{\text{MC}}$ (0.22$\sigma$ [0.22$\sigma$]), and
$n_{s}$ (0.22$\sigma$ [0.41$\sigma$]). The primary parameter error bars also
do not significantly differ, with the largest differences being $+4.71$%
[$-6.25$%] for $\ln(10^{10}A_{s})$ [$\Omega_{c}h^{2}$]. For P18 data we obtain
$\Omega_{k}=-0.073^{+0.065}_{-0.029}$ [$-0.072^{+0.065}_{-0.030}$] (favoring
closed spatial hypersurfaces by 1.12$\sigma$ [1.11$\sigma$]). When P18+lensing
data are used we find $\Omega_{k}=-0.012^{+0.027}_{-0.011}$
[$-0.003^{+0.018}_{-0.011}$] (in favor of closed geometry by 0.44$\sigma$
[0.17$\sigma$]), with the P18 and P18+lensing values differing by
$-0.93\sigma$ [$-1.05\sigma$]. The $\Omega_{k}$ error bars obtained with P18
data are about 1.9 [2.8] times larger than those obtained with P18+lensing
data. In regard to the dark energy equation of state parameter, from P18 data
we find $w=-1.36^{+1.1}_{-0.53}$ [$-1.39^{+1.1}_{-0.54}$], whereas when
P18+lensing data are used we get $w=-1.32^{+0.71}_{-0.38}$
[$-1.18^{+0.54}_{-0.37}$], with the two results differing by $-0.034\sigma$
[$-0.17\sigma$]. Both values favor phantom-like behavior at 0.33$\sigma$
[0.35$\sigma$] and 0.45$\sigma$ [0.42$\sigma$], respectively, with a shrinkage
in the error bars of $-32.76$% [$-60.42$%]. P18 data cannot properly constrain
the lensing consistency parameter $A_{L}$ and only provide a 95% upper bound
of $A_{L}<1.20$ [$<1.19$]. On the other hand, when P18+lensing data are used
we obtain $A_{L}=1.02\pm 0.16$ [$1.07^{+0.12}_{-0.16}$], which favors
$A_{L}>1$ values by only 0.13$\sigma$ [0.44$\sigma$]. When we look at the
values of the derived parameters $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$, we
observe differences at $-1.10\sigma$ [$-1.29\sigma$], 0.92$\sigma$
[0.95$\sigma$], and $-0.33\sigma$ [0.24$\sigma$], respectively. The most
significant reduction in the size of the error bars is for $\Omega_{m}$ with a
decrease of $-60$% [$-85$%].

Comparing the P18+lensing data results for the eight-parameter non-flat XCDM
Planck $P(q)$ model and the nine-parameter non-flat XCDM Planck $P(q)+A_{L}$
model, see Table 12 and Figs. 24 and 25, we see that there are no significant
differences in the mean values of the primary cosmological parameters. For
$\Omega_{c}h^{2}$ and $\ln(10^{10}A_{s})$ we find values differing at
0.094$\sigma$ and 0.12$\sigma$, the dark energy equation of state parameter
$w$ values differ by 0.054$\sigma$, and for the curvature parameter
$\Omega_{k}$ the difference is 0.031$\sigma$. In regard to the increase in the
size of the error bars of the primary parameters for the non-flat XCDM Planck
$P(q)+A_{L}$ model, the largest is 37% for $\Omega_{k}$, followed by the error
bars of $w$ (22%) and $n_{s}$ (6.5%). As for the derived parameters $H_{0}$,
$\Omega_{m}$, and $\sigma_{8}$, we observe minimal differences in the mean
values, 0.047$\sigma$, $-0.047\sigma$, and 0.057$\sigma$, respectively, with
an increase in error bar size only for $\sigma_{8}$ (8.3%).

When we compare P18+lensing data results for the eight-parameter non-flat XCDM
new $P(q)$ (upper half of Table 13 and Fig. 26) and the nine-parameter non-
flat XCDM new $P(q)+A_{L}$ (lower half of Table 13 and Fig. 27) cosmological
models we are lead to very similar conclusions to the ones obtained from the
comparison of P18+lensing data results for the non-flat XCDM Planck $P(q)$ and
XCDM Planck $P(q)+A_{L}$ models. For the primary parameters in common with the
flat $\Lambda$CDM model, the largest mean values differences are
$-0.086\sigma$, $0.14\sigma$, and $-0.18\sigma$ for $\Omega_{b}h^{2}$,
$\Omega_{c}h^{2}$, and $n_{s}$, respectively. For the curvature parameter
$\Omega_{k}$ the values differ at $-0.34\sigma$ and for the equation of state
parameter of dark energy they differ at $-0.17\sigma$. As expected, some
increases in the size of the error bars are observed when the $A_{L}$
parameter is allowed to vary, in particular for $n_{s}$, $\ln(10^{10}A_{s})$,
and $w$ the enlargement of the error bars is 20%, 11%, and 15%, respectively.
As for the derived parameters $\Omega_{m}$ and $\sigma_{8}$, we find
differences of $-0.057\sigma$ and 0.18$\sigma$ and the error bars increase by
8.3% and 9.1%.

Figure 28: Likelihood distributions of flat XCDM model parameters favored by
P18, P18+lensing, and P18+lensing+non-CMB (new) data sets.

Figure 29: Likelihood distributions of flat XCDM$+A_{L}$ model parameters
favored by P18, P18+lensing, and P18+lensing+non-CMB (new) data sets.

Figure 30: Likelihood distributions of non-flat XCDM model [Planck $P(q)$]
parameters favored by P18, P18+lensing, and P18+lensing+non-CMB (new) data
sets.

Figure 31: Likelihood distributions of non-flat XCDM$+A_{L}$ [Planck $P(q)$]
model parameters favored by P18, P18+lensing, and P18+lensing+non-CMB (new)
data sets. Note that in the model for the P18 data set, a prior of $A_{L}>0.8$
is applied.

Figure 32: Likelihood distributions of non-flat XCDM model [new $P(q)$]
parameters favored by P18, P18+lensing, and P18+lensing+non-CMB (new) data
sets.