# Current data are consistent with flat spatial hypersurfaces in the
$\Lambda$CDM cosmological model but favor more lensing than the model predicts

Javier de Cruz Pérez1, Chan-Gyung Park2, and Bharat Ratra1 1Department of
Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66506, USA
2Division of Science Education and Institute of Science Education, Jeonbuk
National University, Jeonju 54896, Republic of Korea<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the performance of three pairs of tilted $\Lambda$CDM cosmological
models, two pairs allowing for non-flat spatial hypersurfaces, cosmic
microwave background (CMB) temperature and polarization power spectrum data
(P18), measurements of the Planck 2018 lensing potential power spectrum
(lensing), and a large compilation of non-CMB data (non-CMB). For the six
models, we measure cosmological parameters and study whether or not pairs of
the data sets (as well as subsets of them) are mutually consistent in these
models. Half of these models allow the lensing consistency parameter $A_{L}$,
which re-scales the gravitational potential power spectrum, to be an
additional free parameter to be determined from data, while the other three
have $A_{L}=1$ which is the theoretically expected value. The tilted
spatially-flat models assume the usual primordial spatial inhomogeneity power
spectrum that is a power law in wave number. The tilted non-flat models assume
either the primordial power spectrum used in the Planck group analyses [Planck
$P(q)$], that has recently been numerically shown to be a good approximation
to what is quantum-mechanically generated from a particular choice of closed
inflation model initial conditions, or a recently computed power spectrum [new
$P(q)$] that quantum-mechanically follows from a different set of non-flat
inflation model initial conditions. In the tilted non-flat models with
$A_{L}=1$ we find differences between P18 data and non-CMB data cosmological
parameter constraints, which are large enough to rule out the Planck $P(q)$
model at 3$\sigma$ but not the new $P(q)$ model. No significant differences
are found when cosmological parameter constraints obtained with two different
data sets are compared within the standard tilted flat $\Lambda$CDM model.
While both P18 data and non-CMB data separately favor a closed geometry, with
spatial curvature density parameter $\Omega_{k}<0$, when P18+non-CMB data are
jointly analyzed the evidence in favor of non-flat hypersurfaces subsides.
Differences between P18 data and non-CMB data cosmological constraints subside
when $A_{L}$ is allowed to vary. From the most restrictive P18+lensing+non-CMB
data combination we get almost model-independent constraints on the
cosmological parameters and find that the $A_{L}>1$ option is preferred over
the $\Omega_{k}<0$ one, with the $A_{L}$ parameter, for all models, being
larger than unity by $\sim 2.5\sigma$. According to the deviance information
criterion, in the P18+lensing+non-CMB analysis, the varying $A_{L}$ option is
on the verge of being strongly favored over the $A_{L}=1$ one, which could
indicate a problem for the standard tilted flat $\Lambda$CDM model. These data
are consistent with flat spatial hypersurfaces but more and better data could
improve the constraints on $\Omega_{k}$ and might alter this conclusion. Error
bars on some cosmological parameters are significantly reduced when non-CMB
data are used jointly with P18+lensing data. For example, in the tilted flat
$\Lambda$CDM model for P18+lensing+non-CMB data the Hubble constant
$H_{0}=68.09\pm 0.38$ km s-1 Mpc-1, which is consistent with that from a
median statistics analysis of a large compilation of $H_{0}$ measurements as
well as with a number of local measurements of the cosmological expansion
rate. This $H_{0}$ error bar is 31% smaller than that from P18+lensing data
alone.

###### pacs:

98.80.-k, 95.36.+x

##  I Introduction

General relativity is the current best description of gravity on cosmological
scales. In general relativity gravity is responsible for the observed
expansion of the Universe and can be sourced by non-relativistic (cold dark
and baryonic) matter, relativistic radiation/matter, a cosmological constant
(or a dynamical dark energy density), and the curvature of space. In an
influential 1932 paper Einstein and de Sitter, (Einstein and de Sitter, 1932),
noted that available data then were unable to measure spatial curvature and so
decided to study whether a spatially-flat cosmological model was
observationally consistent. They acknowledged that the cosmological model had
to be dynamical, and so Einstein’s original argument for a cosmological
constant — to make the Universe static — was no longer valid and so the
cosmological constant did not have to be included in this Einstein-de Sitter
model. They ignored relativistic radiation/matter in this model (which was not
under discussion then, and is known to be negligible at late times when the
model was meant to be applicable). This Einstein-de Sitter model only included
non-relativistic (and then only baryonic) matter.

A little over half a century later, motivated by observations indicating a
lower than critical non-relativistic matter energy density and the first
inflation model, an improved standard model, the spatially-flat $\Lambda$CDM
model, was proposed, (Peebles, 1984). In this model the cosmological constant
$\Lambda$, which has a time- and space-independent energy density, is the
dominant contributor to the current cosmological energy budget, followed by
non-relativistic non-baryonic cold dark matter (CDM), and then non-
relativistic baryonic matter. Like the Einstein-de Sitter model, the standard
spatially-flat $\Lambda$CDM model assumes vanishing spatial curvature,
motivated by early models of spatially-flat inflation, (Guth, 1981; Sato,
1981a, b; Kazanas, 1980). Soon after, spatially-non-flat, open and closed,
inflation models were developed, (Gott, 1982; Hawking, 1984; Ratra, 1985).

A decade and a half later, the observed currently accelerated cosmological
expansion, discovered from type Ia supernova (SNIa) measurements Riess et al.
(1998); Perlmutter et al. (1999), greatly strengthened support for a
cosmological constant or a dynamical dark energy density that slowly varied in
time and space (Peebles and Ratra, 1988; Ratra and Peebles, 1988) — if general
relativity is an accurate model for gravity on cosmological length scales —
and for the spatially-flat $\Lambda$CDM model or a model close to it. For
reviews of the current situation see Refs. (Di Valentino et al., 2021a;
Perivolaropoulos and Skara, 2022; Abdalla et al., 2022).

A half-decade prior to the first SNIa observations indicating currently
accelerated cosmological expansion, evidence for a lower than critical value
of non-relativistic matter density, along with the development of an open
inflation model, (Gott, 1982), led to some discussion of an open CDM model,
(Ratra and Peebles, 1994, 1995; Bucher et al., 1995; Yamamoto et al., 1995;
Kamionkowski et al., 1994; Gorski et al., 1998; Ratra et al., 1999), but with
cosmic microwave background (CMB) observations indicating that space curvature
had to be a subdominant contributor to the current cosmological energy budget,
(Hinshaw et al., 2013; Aghanim et al., 2020a), and with SNIa observations
favoring a significant contribution to the energy budget from a cosmological
constant, interest in open CDM models soon faded.

More recently, especially because of results from Planck CMB anisotropy data,
(Aghanim et al., 2020a), there has been renewed interest in non-flat models.
In these models the current cosmological energy budget is dominated by
$\Lambda$, to be consistent with the observed currently accelerated
cosmological expansion, but they now have very mildly closed spatial
hypersurfaces instead of open ones. This is because from an analysis of the
final Planck 2018 TT,TE,EE+lowE (hereafter P18) data, that makes use of a
specific primordial power spectrum (see below for a fuller discussion of these
data and the power spectrum they use in this analysis), they find a spatial
curvature energy density parameter value $\Omega_{k}=-0.044^{+0.018}_{-0.015}$
that is closed and 2.7$\sigma$ away from flat, (Aghanim et al., 2020a), when
$\Omega_{k}$ is included as an additional free parameter in the analysis. We
note that from a combination of Atacama Cosmology Telescope (ACT) and
Wilkinson Microwave Anisotropy Probe CMB anisotropy data Ref. (Aiola et al.,
2020) find $\Omega_{k}=-0.001^{+0.014}_{-0.010}$, which is 2.1$\sigma$ from
the P18 value and consistent with flat spatial hypersurfaces, while the South
Pole Telescope (SPT) CMB anisotropy data results in
$\Omega_{k}=0.001^{+0.018}_{-0.019}$, (Balkenhol et al., 2021), which is
1.7$\sigma$ from the P18 value and also consistent with flat spatial
hypersurfaces. Both these analyses use the primordial power spectrum used in
the P18 analysis.

The above result led to the study of the so-called lensing anomaly. The
trajectory of CMB photons are bent by the gravitational effect of
inhomogeneities present in the mass distribution along their way to us. This
statistical phenomenon, predicted by general relativity, is known as weak
gravitational lensing of the CMB. When computing the predicted CMB temperature
and polarization spectra in a cosmological model that are to be compared to
the observed spectra, it is important to account for this effect and compute
what are known as lensed CMB spectra. If we use the tilted flat $\Lambda$CDM
model to measure cosmological parameter values from Planck CMB data, we can
use this model, with these parameter values, to compute the expected amount of
CMB weak gravitational lensing, Lewis and Challinor (2006). Incorrectly
predicting the amount of weak lensing present in the CMB power spectra would
indicate an inconsistency in the standard model when it is used to fit Planck
CMB temperature and polarization anisotropy data. It turns out that this is
actually the case, since an excess of CMB weak lensing is observed in the CMB
power spectra, compared to what is expected in the standard model with
parameter values determined from CMB data Calabrese et al. (2008); Aghanim et
al. (2020a). This is known as the lensing anomaly, since the effect is not yet
thought to be statistically significant enough to reject the standard
spatially-flat $\Lambda$CDM model.

A number of solutions have been proposed, with two being more widely debated.
The first of these is related to the aforementioned non-zero value for
$\Omega_{k}$ in the P18 data analysis, which favors closed spatial
hypersurfaces, when $\Omega_{k}$ is taken to be an additional free parameter,
e.g. (Aghanim et al., 2020a; Handley, 2021; Di Valentino et al., 2019, 2022;
Yang et al., 2022). Due to the excess of CMB weak lensing found, it is
desirable to have a higher value of the non-relativistic matter energy density
parameter $\Omega_{m}$ in order to increase the amount of gravitational
lensing of CMB photons. Because of the tight constraints imposed by CMB data
on this parameter there is no room, within the tilted flat $\Lambda$CDM model,
to do this. By allowing non-flat spatial hypersurfaces, a closed model with
$\Omega_{k}<0$ can resolve this problem, since the CMB power spectra are
affected by the combination $(\Omega_{m}+\Omega_{k})h^{2}$, where $h$ is the
Hubble constant $H_{0}$ in units of
$100~{}\textrm{km}~{}\textrm{s}^{-1}~{}\textrm{Mpc}^{-1}$, which can be held
constant by making $\Omega_{k}$ slightly more negative while slightly
increasing $\Omega_{m}$ to give more CMB weak lensing, and also slightly
adjusting $h$. Cosmological distances also depend on spatial curvature,
therefore in a non-flat cosmological model the positions of the acoustic peaks
are shifted relative to the flat model case. This would not be a welcome
change, since the constraints from the observed CMB power spectra are tight.
This can be resolved by reducing the value of $h$ which shifts the acoustic
peaks in the opposite direction. The fact that almost the same temperature and
polarization power spectra can be produced with different combinations of the
cosmological parameter values points to a geometrical degeneracy between these
three parameters, $H_{0}$-$\Omega_{m}$-$\Omega_{k}$.

While the first of the more widely debated resolutions is based on a change of
more-conventional cosmological parameters, the second one is more
phenomenological, e.g. (Aghanim et al., 2020a; Henning et al., 2018a;
Bianchini et al., 2020; Di Valentino et al., 2019, 2022). Reference Calabrese
et al. (2008) introduces the lensing consistency parameter $A_{L}$ which re-
scales the gravitational potential power spectrum in such a way that when
$A_{L}=1$ we recover the theoretically predicted amount of weak lensing. If
$A_{L}$ is allowed to vary in the analysis, to be determined from data, when
$A_{L}>1$ the predicted amount of lensing is greater than the case when
$A_{L}=1$. In Ref. Aghanim et al. (2020a) when P18 data are used to analyze
the tilted flat $\Lambda$CDM+$A_{L}$ model, the result is $A_{L}=1.180\pm
0.065$ which represents a 2.8$\sigma$ deviation from the theoretically
expected value $A_{L}=1$. We emphasize however that the measured Planck
lensing likelihood is consistent with $A_{L}=1$, see Fig. 3 of Ref. (Aghanim
et al., 2020a) and Ref. (Aghanim et al., 2020b). We also note that from ACT
CMB anisotropy data $A_{L}=1.01\pm 0.11$, (Aiola et al., 2020), consistent
with $A_{L}=1$ and 1.3$\sigma$ smaller than the P18 value, while from SPT CMB
anisotropy data $A_{L}=0.81\pm 0.14$, (Henning et al., 2018b), 1.4$\sigma$
smaller than $A_{L}=1$ and 2.4$\sigma$ smaller than the P18 value.

To analyze CMB anisotropy data one must assume a form for the primordial power
spectrum of spatial inhomogeneities as a function of wavenumber. In the
inflation scenario zero-point quantum-mechanical fluctuations during inflation
generate the spatial inhomogeneities, (Hawking, 1982; Starobinsky, 1982; Guth
and Pi, 1982; Bardeen et al., 1983; Fischler et al., 1985). In spatially-flat
inflation models, if the inflaton field slowly rolls down an almost flat
potential energy density the scale factor increases exponentially with time
and the primordial power spectrum is almost scale-invariant with hardly any
tilt, (Harrison, 1970; Peebles and Yu, 1970; Zeldovich, 1972). A steeper
inflaton potential energy density makes the inflaton evolve more rapidly, can
cause the scale factor to grow only as a power of time, and will increase the
power spectral tilt (Lucchin and Matarrese, 1985; Ratra, 1989, 1992).

There has been much less study of the quantum-mechanical generation of spatial
inhomgeneities in non-flat inflation models. Power spectra have been derived
in spatially open and closed inflation models, (Gott, 1982; Hawking, 1984;
Ratra, 1985), with a slowly-rolling inflation potential energy density, (Ratra
and Peebles, 1995; Ratra, 2017), but these are untilted power spectra. The
power spectrum assumed in the non-flat analyses of Refs. (Aghanim et al.,
2020a; Handley, 2021; Di Valentino et al., 2019) is tilted but was not derived
from an inflation model computation. Very recently, a numerical study in
closed inflation models that computes primordial power spectra generated for a
few different, initially slow-roll, inflation initial conditions finds that it
is possible to generate, in the closed case, a tilted power spectrum very
close to that used in Refs. (Aghanim et al., 2020a; Handley, 2021; Di
Valentino et al., 2019), Guth et al. (2022). Also recently, a different set of
initial conditions in closed and open inflation models were used to compute a
different tilted power spectrum, (Ratra, 2022).

In this paper we consider cosmological models with four different power
spectra. In the tilted flat $\Lambda$CDM model we use the usual spatially-flat
inflation model tilted power spectrum. In the untilted non-flat $\Lambda$CDM
model, we use the untilted non-flat inflation model power spectrum, (Ratra and
Peebles, 1995; Ratra, 2017). In the two different tilted non-flat $\Lambda$CDM
models, we use the power spectrum assumed in Ref. (Aghanim et al., 2020a;
Handley, 2021; Di Valentino et al., 2019) — which we call the Planck $P(q)$ —
as well as the power spectrum computed in Ref. (Ratra, 2022), which we call
the new $P(q)$. See Sec. III below for a fuller description of the four power
spectra we use.

We emphasize that we use only non-flat inflation model power spectra that can
be derived using a straightforward extension of the spatially-flat inflation
model initial conditions to the non-flat inflation case. The issue of non-flat
inflation model initial conditions is more complex than the flat inflation
case, see discussion in Ref. (Ratra, 2022), so we focus on the simplest
physically-consistent options, which also makes the analysis tractable. We
note that a number of other power spectra have been considered in closed
cosmological models, see Refs. (Lasenby and Doran, 2005; Masso et al., 2008;
Asgari and Abbassi, 2015; Bonga et al., 2016; Handley and Lemos, 2019a;
Thavanesan et al., 2021; Kiefer and Vardanyan, 2022; Hergt et al., 2022).

A desire to measure the spatial curvature energy density parameter
$\Omega_{k}$ provides part of the motivation for our work. The CMB anisotropy
data are currently the most restrictive cosmological data, but to use these to
measure $\Omega_{k}$ requires assumption of a primordial power spectrum for
spatial inhomgeneities. Other, less-restrictive, data that do not require
assuming a power spectrum can also be used to measure $\Omega_{k}$. These
include better-established lower redshift data (that reach to $z\sim 2.3$),
such as SNIa, Hubble parameter as a function of redshift [$H(z)$], and (non-
growth-rate) baryon acoustic oscillation (BAO) measurements, (Scolnic et al.,
2018; Yu et al., 2018; Alam et al., 2021), as well as emerging probes that
reach to higher $z$, such as H ii starburst galaxy apparent magnitude
observations as a function of $z$ that reach to $z\sim 2.5$, (González-Morán
et al., 2019; Cao et al., 2020, 2021a; Johnson et al., 2022; Mehrabi et al.,
2022); quasar angular size measurements that reach to $z\sim 2.7$, (Cao et
al., 2017; Ryan et al., 2019; Lian et al., 2021; Cao et al., 2022a); Mg ii and
C iv reverberation measured quasar data that reach to $z\sim 3.4$, (Yu et al.,
2021; Khadka et al., 2021a, 2022a, 2022b; Cao et al., 2022b; Yu et al., 2022;
Czerny et al., 2022); possibly quasar flux measurements that reach to $z\sim
7.5$, (Risaliti and Lusso, 2019; Khadka and Ratra, 2020a, b; Lusso et al.,
2020; Khadka and Ratra, 2021, 2022; Rezaei et al., 2021; Dainotti et al.,
2022a; Petrosian et al., 2022); and gamma-ray burst data that reach to $z\sim
8.2$, (Dirirsa et al., 2019; Khadka and Ratra, 2020c; Khadka et al., 2021b;
Wang et al., 2022; Hu et al., 2021; Cao et al., 2022c; Luongo and Muccino,
2021; Cao et al., 2022d; Liu et al., 2022; Dainotti et al., 2022b; Cao et al.,
2022e). Individually these low- and intermediate-redshift data sets are only
able to provide relatively weaker constraints on cosmological parameters in
general, and specifically on $\Omega_{k}$, compared to those from CMB data.
However, when many (or all) low- and intermediate-redshift data are analyzed
jointly they provide useful constraints on $\Omega_{k}$ — currently still not
nearly as restrictive as the CMB ones — favoring flat spatial hypersurfaces
but still allowing a small amount of spatial curvature energy density, (Park
and Ratra, 2019a; Cao et al., 2021b; Cao and Ratra, 2022). For other recent
discussions of constraints on spatial curvature, see Refs. (Arjona and
Nesseris, 2021; Dhawan et al., 2021; Gonzalez et al., 2021; Geng et al., 2022;
Wei and Melia, 2022; Mukherjee and Banerjee, 2022; Wu et al., 2022) and
references therein, and see Refs. (Baumgartner and Yoo, 2022; Anselmi et al.,
2022; Jimenez et al., 2022) and references therein for recent, more general,
discussions of non-flat cosmological models.

While the standard spatially-flat $\Lambda$CDM cosmological model is
attractive because of its simplicity — the model only has 6 free cosmological
parameters — it is not straightforward to understand how to consistently
generalize the current quantum-mechanical standard model of particle physics
to one that accommodates the cosmological constant that is part of the
standard $\Lambda$CDM model. Nonetheless, the standard cosmological model is
consistent with a wide variety of measurements, including CMB anisotropy
measurements Aghanim et al. (2020a), SNIa apparent magnitude observations
(Scolnic et al., 2018), BAO data (Alam et al., 2021), $H(z)$ observations (Yu
et al., 2018), and measurements of the growth of structure as a function of
redshift ($f\sigma_{8}$). It is important to bear in mind that these data do
not rule out mild evolution of the dark energy density Gómez-Valent and Solà
Peracaula (2018); Ooba et al. (2019); Ryan et al. (2018); Sola Peracaula et
al. (2019); Singh et al. (2019); Park and Ratra (2020); Gómez-Valent et al.
(2020); Moreno-Pulido and Solà (2020); Sinha and Banerjee (2021); Cao et al.
(2020); Ureña López and Roy (2020); Cao et al. (2021b); Solà Peracaula et al.
(2021); Khadka et al. (2021b); Cao et al. (2022a); Xu et al. (2022); Cao et
al. (2022c); Jesus et al. (2021); Cao et al. (2022d); Moreno-Pulido and Sola
Peracaula (2022); Cao et al. (2022b); Adil et al. (2022) or, as discussed in
detail above, mildly curved spatial hypersurfaces. These extensions, among
others, might alleviate some of the issues affecting the standard spatially-
flat $\Lambda$CDM model, such as differences in $H_{0}$ and $\sigma_{8}$
values determined using different techniques, (Di Valentino et al., 2021a;
Perivolaropoulos and Skara, 2022; Abdalla et al., 2022). In this paper however
we focus our efforts on the study of the lensing anomaly and on the
measurement of $\Omega_{k}$.

In this paper we study eight cosmological models, namely, the tilted flat
$\Lambda$CDM (+$A_{L}$) models, the untilted non-flat $\Lambda$CDM (+$A_{L}$)
models, the tilted non-flat $\Lambda$CDM (+$A_{L}$) Planck $P(q)$ models, and
the tilted non-flat $\Lambda$CDM(+$A_{L}$) new $P(q)$ models. Six of these are
non-flat models, characterized by three different primordial power spectra
(see Sec. III for the form of the power spectra). By using a number of
cosmological models with compilations of observational data to test how well
the models fit these data, and to constrain the cosmological parameters of the
models, we can measure, among other things, $\Omega_{k}$ and also determine
whether the cosmological parameter constraints set by different data are
model-dependent or not. The data sets we employ in this work are P18 data,
Planck 2018 CMB weak lensing data, non-growth-factor BAO (BAO′) data, BAO
(including growth-factor) data, and non-CMB data [composed of BAO,
$f\sigma_{8}$, $H(z)$, and SNIa data]. These data are described in more detail
in Sec. II.

A brief summary of the more significant results we find follows. These assume
that the data sets we use are correct and do not have unaccounted for
systematic errors. The untilted non-flat $\Lambda$CDM model with and without a
varying $A_{L}$ parameter is not able to properly fit the P18 CMB anisotropy
power spectra, due to the lack of the tilt ($n_{s}$) degree of freedom.
Consequently its performance in comparison with the tilted models turns out to
be very poor. Significant evidence in favor of a closed Universe is found when
P18 data are considered alone and the tilted non-flat models better fit these
data than does the standard tilted flat $\Lambda$CDM model. There are
disagreements between P18 data cosmological constraints and non-CMB data
cosmological constraints in the context of the tilted non-flat models with
$A_{L}=1$, with the tilted non-flat $\Lambda$CDM Planck $P(q)$ model ruled out
at 3$\sigma$. These tensions completely fade when the $A_{L}$ parameter is
allowed to vary. On the other hand no significant tension is found when the
cosmological parameter constraints obtained with two different data sets are
compared within the standard tilted flat $\Lambda$CDM model. The most-
restrictive P18+lensing+non-CMB data set clearly favors the varying $A_{L}$
option (with $A_{L}>1$) over the $A_{L}=1$ one — which could be a problem for
the standard tilted flat $\Lambda$CDM model — and when this data set is
utilized we get almost model-independent cosmological parameter constraints.
These data are consistent with flat spatial hypersurfaces — so we conclude
that current data do not favor curved geometry — but more and better data
could improve the constraints on $\Omega_{k}$ and might alter this conclusion.
We note that even though both P18 data and non-CMB data favor closed geometry,
the larger $H_{0}$ and smaller $\Omega_{m}$ values favored by non-CMB data
(compared to those favored by P18 data) result in P18+lensing+non-CMB data
favoring flat spatial hypersurfaces. The Hubble constant value measured using
these data in the tilted flat $\Lambda$CDM model is $H_{0}=68.09\pm 0.38$ km
s-1 Mpc-1, which is consistent with that from a median statistics analysis of
a large compilation of Hubble constant measurements as well as with a number
of local measurements of the cosmological expansion rate. This $H_{0}$ error
bar is 31% smaller than that from P18+lensing data alone; similarly augmenting
the P18+lensing data with our non-CMB data compilation reduces the
$\Omega_{m}$ error bar by 33% and also reduces error bars on all the other
cosmological parameters by smaller amounts.

The layout of our paper is as follows. In Sec. II we detail the observational
data sets we employ to test the different cosmological models. In Sec. III we
describe the cosmological models and primordial power spectra we study and
summarize the methods we use in our analyses. We dedicate Sec. IV to discuss
in detail the results obtained by testing the different cosmological models
against the several data sets we consider. In this section we also utilize
different statistical estimators to compare the performance of the models in
fitting data and to study possible tensions between different data sets in a
given model. In Sec. V we summarize the more significant results of the
previous (long) section. Finally in Sec. VI we deliver our conclusions.

##  II Data

We use CMB anisotropy data and non-CMB data to constrain cosmological
parameters, to determine how well the cosmological models we study fit these
data, and to study how mutually consistent these data sets are in each of the
cosmological models. We now list the data sets we use.

P18. Planck 2018 CMB temperature anisotropy data together with polarization
data and their corresponding cross-spectra (TT,TE,EE+lowE), Aghanim et al.
(2020a), which contain: TT power spectra at low-$\ell$ ($2\leq\ell\leq 29$)
and high-$\ell$ ($30\leq\ell\leq 2508$) — where $\ell$ is multipole number, TE
data at high-$\ell$ ($30\leq\ell\leq 1996$), and EE data at low-$\ell$
($2\leq\ell\leq 29$) and high-$\ell$ ($30\leq\ell\leq 1996$). We use the
Planck 2018 baseline Plik $\ell\geq 30$ likelihood, which is described in Sec.
2.2.1 of Ref. Aghanim et al. (2020a).

(P18) lensing. Planck 2018 lensing potential power spectrum, see Sec. 2.3 of
Ref. Aghanim et al. (2020a) or Sec. 2 of Ref. Aghanim et al. (2020b) for more
details.

BAO′. Twelve BAO data points from both anisotropic and isotropic BAO
estimators that probe the redshift range $0.122\leq z\leq 2.334$ Gil-Marin et
al. (2020); Bautista et al. (2020); Hou et al. (2020); Neveux et al. (2020);
Carter et al. (2018); Abbott et al. (2019a); du Mas des Bourboux et al.
(2020). These are BAO data with growth rates excluded from the original
papers, and are listed, along with the appropriate covariance matrices, in
Sec. 3 of Ref. Cao and Ratra (2022).

Table 1: BAO measurements.

$z_{\textrm{eff}}$ | Measurement | Reference
---|---|---
$0.122$ |  $D_{V}\left(r_{d,\textrm{fid}}/r_{d}\right)$ [Mpc] $=539\pm 17$ [Mpc] | Carter et al. (2018)
$0.38$ |  $D_{M}/r_{d}$ $=10.274\pm 0.151$ | Gil-Marin et al. (2020)
$0.38$ |  $D_{H}/r_{d}$ $=24.888\pm 0.582$ | Gil-Marin et al. (2020)
$0.51$ |  $D_{M}/r_{d}$ $=13.381\pm 0.179$ | Gil-Marin et al. (2020)
$0.51$ |  $D_{H}/r_{d}$ $=22.429\pm 0.482$ | Gil-Marin et al. (2020)
$0.38$ | $f\sigma_{8}=0.49729\pm 0.04508$ | Gil-Marin et al. (2020)
$0.51$ | $f\sigma_{8}=0.45902\pm 0.03784$ | Gil-Marin et al. (2020)
$0.698$ |  $D_{M}/r_{d}$ $=17.646\pm 0.302$ | Gil-Marin et al. (2020); Bautista et al. (2020)
$0.698$ |  $D_{H}/r_{d}$ $=19.770\pm 0.469$ | Gil-Marin et al. (2020); Bautista et al. (2020)
$0.698$ |  $f\sigma_{8}$ $=0.47300\pm 0.04429$ | Gil-Marin et al. (2020); Bautista et al. (2020)
$0.81$ |  $D_{A}/r_{d}$ $=10.75\pm 0.43$ | Abbott et al. (2019a)
$1.48$ |  $D_{M}/r_{d}$ $=30.21\pm 0.79$ | Hou et al. (2020); Neveux et al. (2020)
$1.48$ |  $D_{H}/r_{d}$ $=13.23\pm 0.47$ | Hou et al. (2020); Neveux et al. (2020)
$1.48$ |  $f\sigma_{8}$ $=0.462\pm 0.045$ | Hou et al. (2020); Neveux et al. (2020)
$2.334$ |  $D_{M}/r_{d}$ $=37.5^{+1.2}_{-1.1}$ | du Mas des Bourboux et al. (2020)
$2.334$ |  $D_{H}/r_{d}$ $=8.99^{+0.20}_{-0.19}$ | du Mas des Bourboux et al. (2020)

Note: For the data point at $z=0.122$ the sound horizon size (at the drag
epoch) of the fiducial model is $r_{d,\textrm{fid}}=147.5~{}\textrm{Mpc}$
Carter et al. (2018).

BAO. An extension of the BAO′ data described above, that also probe the
redshift range $0.122\leq z\leq 2.334$, but now include the correlated growth
rate ($f\sigma_{8}$) data points provided in Refs. Gil-Marin et al. (2020);
Bautista et al. (2020); Hou et al. (2020); Neveux et al. (2020). Table 1 lists
these BAO data points.

The quantities listed in Table 1 include transverse comoving distance at
redshift $z$

$D_{M}(z)=(1+z)D_{A}(z),$ (1)

where $D_{A}(z)$ is the angular size distance at $z$,

$D_{H}(z)=\frac{c}{H(z)},$ (2)

where $H(z)$ is the Hubble parameter and $c$ the speed of light, and the
angle-averaged distance

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

The measurements are provided as relative distances with respect to the radius
of the sound horizon at the drag epoch redshift $z_{d}$

$r_{d}=\int^{\infty}_{z_{d}}\frac{c_{s}(z)dz}{H(z)},$ (4)

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

For BAO data from Ref. Gil-Marin et al. (2020) the appropriate covariance
matrix is now

$\begin{pmatrix}0.022897&-0.02007&0.0026481&0.013487&-0.0081402&0.0010292\\\
-0.02007&0.33849&-0.0085213&-0.016024&0.13652&-0.0038002\\\
0.0026481&-0.0085213&0.0020319&0.001325&-0.0023012&0.000814158\\\
0.013487&-0.016024&0.001325&0.032158&-0.020091&0.0026409\\\
-0.0081402&0.13652&-0.0023012&-0.020091&0.23192&-0.0055377\\\
0.0010292&-0.0038002&0.000814158&0.0026409&-0.0055377&0.0014322\end{pmatrix},$
(5)

while the covariance matrix for BAO data from Refs. Gil-Marin et al. (2020);
Bautista et al. (2020) is

$\begin{pmatrix}0.09114&-0.033789&0.0024686\\\ -0.033789&0.22009&-0.0036088\\\
0.0024686&-0.0036088&0.0019616\end{pmatrix},$ (6)

and that for BAO data from Refs. Hou et al. (2020); Neveux et al. (2020) is

$\begin{pmatrix}0.6227&0.01424&0.02257\\\ 0.01424&0.2195&-0.007315\\\
0.02257&-0.007315&0.002020\end{pmatrix}.$ (7)

${\bm{f}\bm{\sigma}_{8}}$. $f\sigma_{8}$ data points, in addition to those
correlated with BAO data that are listed in Table 1. These independent
$f\sigma_{8}$ measurements are obtained either from peculiar velocity data
Turnbull et al. (2012); Hudson and Turnbull (2013); Said et al. (2020) or from
redshift space distortion (RSD) analyses Shi et al. (2018); Simpson et al.
(2016); Blake et al. (2013); Mohammad et al. (2018); Okumura et al. (2016).
These are listed in Table 2.

The combination $f(z)\sigma_{8}(z)$ is used to quantify the growth rate of the
matter density perturbation. Here, the growth rate

$f(z)=-(1+z)\frac{d\ln{D(z)}}{dz}$ (8)

where $D(z)$ is the growth function. The other function involved,
$\sigma_{8}(z)$, is the root mean square of matter fluctuations smoothed over
spheres of radius $R_{8}=8h^{-1}\textrm{Mpc}$ at a given value of the
redshift. It is computed as

$\sigma^{2}_{8}(z)=\int\frac{d^{3}k}{(2\pi)^{3}}P_{m}(z,\vec{k})W^{2}(k{R_{8}}),$
(9)

where $P_{m}(z,\vec{k})$ is the matter power spectrum and $W(k{R_{8}})$ is the
window function.

Table 2: $f\sigma_{8}$ measurements. $z_{\textrm{eff}}$ | $f\sigma_{8}$ | Reference
---|---|---
$0.02$ | $0.398\pm 0.065$ | Turnbull et al. (2012); Hudson and Turnbull (2013)
$0.035$ | $0.338\pm 0.027$ | Said et al. (2020)
$0.1$ | $0.376\pm 0.038$ | Shi et al. (2018)
$0.18$ | $0.29\pm 0.10$ | Simpson et al. (2016)
$0.38$ | $0.44\pm 0.06$ | Blake et al. (2013)
$0.6$ | $0.49\pm 0.12$ | Mohammad et al. (2018)
$0.86$ | $0.46\pm 0.09$ | Mohammad et al. (2018)
$1.36$ | $0.482\pm 0.116$ | Okumura et al. (2016)

SNIa. Apparent magnitude as a function of redshift measurements for 1048
Pantheon SNIa Scolnic et al. (2018), probing the redshift range $0.01<z<2.3$,
and 20 compressed data points, spanning the redshift range $0.015\leq z\leq
0.7026$, representing 207 DES 3yr SNIa Abbott et al. (2019b). The Pantheon and
DES 3yr data are independent of each other, but the data points within each
sample are correlated and we account for the corresponding covariance matrices
in our analyses.

${\bm{H(z)}}$. Hubble parameter measurements over the redshift range
$0.070\leq z\leq 1.965$ obtained using the differential age technique. The 31
data points employed are listed in Table 2 of Ref. Park and Ratra (2019b).

Hereafter we denote the combination of BAO, $f\sigma_{8}$, SNIa, and $H(z)$
data sets as the non-CMB data set.

##  III Methods

We apply the Markov chain Monte Carlo (MCMC) method, implemented in the
CAMB/COSMOMC package (version of Oct. 2018), Challinor and Lasenby (1999);
Lewis et al. (2000); Lewis and Bridle (2002), to explore the parameter space
of the different models under study. The CAMB program computes the matter and
CMB power spectra based on the evolution of density perturbations of the
matter and radiation components and the COSMOMC program uses the MCMC method
to estimate the parameter constraints that are favored by the given
observational data sets. We have performed cross-checks using the
CLASS/MontePython package, Blas et al. (2011); Audren et al. (2013). In
general a good agreement between the results is obtained unless significant
degeneracies between some of the fitting parameters are present. When this
happens, differences in the central values are found, but the two sets of
results remain compatible at 1$\sigma$ due to large error bars. The inclusion
of more data breaks the aforementioned degeneracies and the two sets of
results then agree really well.

In this paper we consider four cosmological models: the tilted flat, (two)
tilted non-flat, and the untilted non-flat $\Lambda$CDM models, as well as
their extensions through the inclusion of the $A_{L}$ parameter, for a total
of eight cases. $A_{L}$ is a phenomenological parameter that scales the
theoretical prediction of the gravitational potential power spectrum, with its
theoretical expected value being $A_{L}=1$, see Ref. (Calabrese et al., 2008).
$A_{L}>1$ causes the smoothing of acoustic peaks in the CMB angular power
spectrum, and Planck CMB data tend to prefer $A_{L}>1$ Aghanim et al. (2020a).

The tilted flat $\Lambda$CDM model is characterized by six cosmological
parameters ($\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, $\theta_{\textrm{MC}}$,
$\tau$, $A_{s}$, $n_{s}$), where $\Omega_{b}$ and $\Omega_{c}$ are the current
values of non-relativistic baryonic and cold dark matter density parameters,
$\theta_{\textrm{MC}}$ is the angular size of the sound horizon at
recombination defined in the CAMB/COSMOMC program, $\tau$ is the reionization
optical depth, and $A_{s}$ and $n_{s}$ are the amplitude and the spectral
index of the primordial scalar-type energy density perturbation power spectrum

$P_{\delta}(k)=A_{s}\left(\frac{k}{k_{0}}\right)^{n_{s}},$ (10)

where $k$ is wavenumber and the pivot scale for $A_{s}$ is
$k_{0}=0.05~{}\textrm{Mpc}^{-1}$. This power spectrum is generated by quantum
mechanical fluctuations during an early epoch of power-law inflation in an
exponential potential energy density scalar field cosmological model with flat
spatial hypersurfaces, Lucchin and Matarrese (1985); Ratra (1989, 1992).

In the non-flat very-slow-roll (so untilted) inflation $\Lambda$CDM model, the
presence of non-zero spatial curvature determines a new length scale, and the
power-law part of the primordial power spectrum is not relevant. Thus, this
model still has six cosmological parameters, with the spectral index $n_{s}$
being replaced by the current value of the spatial curvature density parameter
$\Omega_{k}$. For very-slow-roll inflation in this non-flat inflation mode,
the primordial power spectrum is, Ratra and Peebles (1995); Ratra (2017),

$P_{\delta}(q)\propto\frac{(q^{2}-4K)^{2}}{q(q^{2}-K)}$ (11)

where $q=\sqrt{k^{2}+K}$ is the wavenumber in a model with non-zero spatial
curvature $K=-(H_{0}^{2}/c^{2})\Omega_{k}$, and $A_{s}$ is defined to be the
amplitude of the power spectrum at the pivot-scale $k_{0}$. This power
spectrum form holds in both the open ($\Omega_{k}>0$) and closed
($\Omega_{k}<0$) cases, with $q|K|^{-1/2}\geq 0$ and continuous in the open
case and $q|K|^{-1/2}=3,4,5\dots$ in the closed case. It is the power spectrum
used in the non-flat model analyses in Refs. Ooba et al. (2018a, b, c); Park
and Ratra (2019b, c, 2018, 2020).

For the tilted non-flat $\Lambda$CDM model, there are seven cosmological
parameters, with $\Omega_{k}$ added to the six of the tilted flat $\Lambda$CDM
model. In this model it has been usual to assume, e.g. Aghanim et al. (2020a),
a primordial power spectrum of the form

$P_{\delta}(q)\propto\frac{(q^{2}-4K)^{2}}{q(q^{2}-K)}\left(\frac{k}{k_{0}}\right)^{n_{s}-1},$
(12)

where $q$ (and $A_{s}$) is defined in the previous paragraph. The above
expression, which we refer to as the Planck $P(q)$, is a phenomenologically
modified version of the non-flat very-slow-roll untilted primordial density
perturbation, given in Eq. (11), to now also allow for tilt, Lesgourgues and
Tram (2014). It assumes that tilt in a non-flat space works in a way similar
to how it does in flat space. This expression was known to be physically
reasonable in the cases when $K=0$ or $n_{s}=1$, since Eqs. (10) and (11) are
recovered, respectively, and these two expressions hold in physically-
consistent inflation models. Very recently, a numerical study in closed
inflation models that computes primordial power spectra generated for a few
different, initially slow-roll, inflation initial conditions finds that it is
possible to generate, in the closed case, a power spectrum very close to that
given in Eq. (12), Guth et al. (2022).

In this paper we also study another not-necessarily very-slowly-rolling non-
flat (closed and open) inflation model, Ratra (2022). These tilted non-flat
inflation models result in a primordial power spectrum that differs from that
of eq. (12) and assumes a different inflation initial condition than those
studied in Ref. Guth et al. (2022). For the closed and open inflation models,
the resulting power spectrum

${P_{\delta}(q)\propto(q^{2}-4K)^{2}|P_{\zeta}(A)|,}$ (13)

where $P_{\zeta}(A)$ is different in the closed and open cases. For the closed
inflation model

$\sqrt{|P_{\zeta}(A)|}=\left(\frac{16\pi}{m^{2}_{p}}\right)^{1/2}\\!\\!\\!\\!Q^{1/p}\frac{(2+q_{s})p}{\sqrt{\pi
q_{s}}}\Bigg{|}-1+\frac{W(A)}{p}\Bigg{|}\,\,\frac{2^{-(6-4q_{s}+2A-W(A))/p}}{\sqrt{A}(A-1)(A+3)}\,\,\Bigg{|}\frac{\Gamma\left(1+W(A)/p\right)\Gamma\left((2+q_{s})/(2p)\right)}{\Gamma\left((2+W(A))/p\right)}\Bigg{|},$
(14)

with

$W(A)=\sqrt{-8-4q_{s}+q_{s}^{2}+4A(A+2)},$ (15)

and

$A=\frac{q}{\sqrt{|K|}}-1.$ (16)

While for the open inflation model

$\sqrt{|P_{\zeta}(A)|}=\left(\frac{16\pi}{m^{2}_{p}}\right)^{1/2}\\!\\!\\!\\!Q^{1/p}\frac{(2+q_{s})p}{\sqrt{\pi
q_{s}}}\Bigg{|}-1+\frac{W(A)}{p}\Bigg{|}\,\,\frac{2^{-(6-4q_{s})/p+\textrm{Re}(W(A)/p)}}{\sqrt{A}(A^{2}+4)}\,\,\Bigg{|}\frac{\Gamma\left(1+W(A)/p\right)\Gamma\left((2+q_{s})/(2p)\right)}{\Gamma\left((2+W(A))/p\right)}\Bigg{|},$
(17)

with

$W(A)=\sqrt{-12-4q_{s}+q_{s}^{2}-4A^{2}},$ (18)

and

$A=\frac{q}{\sqrt{|K|}}.$ (19)

In these equations, $\Gamma(x)$ is the Gamma function, $m_{p}$ is the Planck
mass, $Q$ is a normalization constant, $q_{s}=(2-2n_{s})/(3-n_{s})$, and
finally $p=2-q_{s}$. In both the closed and open inflation models $0<q_{s}<2$,
so $-\infty<n_{s}<1$. In this paper we refer to the power spectrum in this
tilted non-flat $\Lambda$CDM as the new $P(q)$, which is shown in Eq. (13),
and following the procedure applied to the other power spectra, $A_{s}$ gives
the amplitude of the new $P(q)$ at the pivot-scale $k_{0}$.

Figure 1 compares the initial scalar-type perturbation spectra of the tilted
flat, untilted non-flat, and two tilted non-flat models with the Planck $P(q)$
and the new $P(q)$. In this figure we set the values of the cosmological
parameters, for all the models, to the mean values of the tilted non-flat
$\Lambda$CDM model with Planck $P(q)$ constrained by the P18+lensing data (see
Table 6 for the parameters), except in panel (b) for the open models where we
change the sign of $\Omega_{k}$.

Figure 1: Initial scalar-type perturbation spectra of the tilted flat,
untilted non-flat, and two tilted non-flat $\Lambda$CDM models. For the tilted
non-flat closed models, the cosmological parameters of the tilted non-flat
$\Lambda$CDM model with Planck $P(q)$ constrained by using P18+lensing data
(including $\Omega_{k}=-0.0103$) are used (see Table 6). For closed models,
the same value of $A_{s}$ was assumed for all models and the same value of
$n_{s}$ was assumed for all tilted models. The powers at the first 11 large-
scale wavenumbers are indicated by the filled (open) circles for the tilted
closed model with the new (Planck) $P(q)$. For open non-flat models,
$\Omega_{k}=+0.0103$ was assumed. For the tilted flat model, the generalized
wavenumber $q$ is equivalent to $k$.

In the cases where we include the $A_{L}$ parameter in the analysis, this
increases by one the number of cosmological model parameters to be determined
from data, so depending on model we then have either seven or eight
cosmological model parameters in these cases.

At the background level, the evolution of the scale factor $a$ in all models
we study is described by the Hubble function

$\begin{split}H^{2}(a)=H^{2}_{0}[\Omega_{\gamma}{a^{-4}}&+(\Omega_{b}+\Omega_{c}){a^{-3}}\\\
&+\Omega_{k}{a^{-2}}+\Omega_{\nu}(a)+\Omega_{\Lambda}].\end{split}$ (20)

Here $a=1/(1+z)$ is the cosmic scale factor normalized to unity at present,
$\Omega_{\Lambda}$ represents the cosmological constant dark energy density
parameter, $\Omega_{\gamma}$ is the current value of the CMB photon energy
density parameter, and $\Omega_{\nu}(a)$ represents the contribution of the
massless and massive neutrinos, for which it is not possible to get an
analytical expression. In all cases we study, we determine the contribution of
photons, and massless and massive neutrinos by assuming a present CMB
temperature $T_{0}=2.7255$ K, the effective number of neutrino species
$N_{\textrm{eff}}=3.046$, and a single massive neutrino species with neutrino
mass $0.06$ eV. During parameter exploration using the MCMC method, we set
wide non-zero flat priors on parameters in order that they not affect the
parameter estimation; these priors are listed in Table 3.

Table 3: Flat priors of the fitting parameters.

Parameters | Our | Handley |  Handley+$\Omega_{k}$
---|---|---|---
$\Omega_{b}h^{2}$ | [0.005,0.1] | [0.019,0.025] | [0.019,0.025]
$\Omega_{c}h^{2}$ | [0.001,0.99] | [0.095,0.145] | [0.095,0.145]
100$\theta_{\textrm{MC}}$ | [0.5,10] | [1.03,1.05] | [1.03,1.05]
$\tau$ | [0.01,0.8] | [0.01,0.4] | [0.01,0.4]
$\Omega_{k}$ | [-0.5,0.5] | [-0.1,0.05] | [-0.3,0.15]
$n_{s}$ | [0.8,1.2] | [0.885,1.04] | [0.885,1.04]
$\ln\left(10^{10}A_{s}\right)$ | [1.61,3.91] | [2.5,3.7] | [2.5,3.7]
$A_{L}$ | [0,10] | - | -

Note: In almost all the computations reported in this paper we use the priors
listed in the Our column in this table. A general exception is that in almost
all the computations in the tilted non-flat $\Lambda$CDM model with the new
$P(q)$ we use a more restrictive prior range for the spectral index, $0.8\leq
n_{s}<1$. In addition to these choices, in all cases, for the derived
parameter $H_{0}$ we restrict its range of variation to $0.2\leq h\leq 1$. In
Table 21 when only lensing data is used, in order to test the impact of
different choices of priors, we also provide results for the narrower priors
employed in Ref. Handley (2021) (listed in the Handley column above). The
Handley+$\Omega_{k}$ column priors above differ from Handley priors by
allowing for a broader prior for the $\Omega_{k}$ parameter.

Due to the lack of constraining power of some of the data sets, when they are
considered alone, we have to fix the values of some of the cosmological
parameters in the analyses of these data sets. In BAO′, BAO, (P18) lensing, or
non-CMB data alone analyses we set the values of $\tau$ and $n_{s}$ to those
obtained in the P18 data alone analysis for each model. Additionally, in BAO′
data alone analyses we also fix the value of $\ln\left(10^{10}A_{s}\right)$,
again, to the corresponding P18 data analysis value. Finally, in Sec. IV.1.7,
when we compare the constraints obtained from P18+lensing data and non-CMB
data, in the non-CMB data analyses the values of $\tau$ and $n_{s}$ are fixed
to the ones we get in the P18+lensing data analysis for each model.

We use the converged MCMC chains to compute mean values, their confidence
limits, and the posterior distributions of the model parameters with the
GetDist code Lewis (2019). The MCMC chains are considered to converge when the
Gelman and Rubin $R$ statistic provided by COSMOMC becomes $R-1<0.01$.

In addition to using the various combinations of data sets (see Sec. II) for
constraining cosmological parameters in the models we study, we want to also
determine which of these models better fit the data sets we study. For a fair
comparison between competing cosmological models with different numbers of
free parameters it is necessary to be able to conveniently penalize for extra
degrees of freedom. In this work we employ two different statistical criteria,
that differently penalize for extra degrees of freedom, to compare the
performance of the models.

The first one we use is the Akaike information criterion (AIC) Akaike (1974)
which is defined as

$\textrm{AIC}=\chi^{2}_{\textrm{min}}+2n.$ (21)

Here $n$ is the number of independent cosmological parameters $\theta$ and
$\chi^{2}_{\textrm{min}}\equiv\chi^{2}(\hat{\theta})=-2\ln\mathcal{L}(\hat{\theta})$
is the minimum value of $\chi^{2}(\theta)=-2\ln\mathcal{L}(\theta)$ evaluated
at the best-fit cosmological parameter values $\hat{\theta}$ where
$\mathcal{L}(\theta)$ is the likelihood function.

The expression in eq. (21) is valid only for a large number of data points.
According to Ref. Burnham and Anderson (2002), when the number of data points
$N$ obeys $N/n<40$, the expression in eq. (21) should be replaced by

$\textrm{AIC}_{c}=\textrm{AIC}+\frac{2n(n+1)}{N-n-1}=\chi^{2}_{\textrm{min}}+\frac{2nN}{N-n-1}.$
(22)

Note that when $N$ is large compared to $n$ we have $N/(N-n-1)\simeq 1$ and
then $\textrm{AIC}_{c}\simeq\textrm{AIC}$. This is the case for P18 data and
non-CMB data but not for the BAO, BAO′, and lensing data sets. In particular
for BAO data $N=16$, for BAO′ data $N=12$, for the lensing data set $N=9$, and
in all three cases $N/n<40$ so $\textrm{AIC}_{c}\neq\textrm{AIC}$.

The second one we use is the deviance information criterion (DIC)
Spiegelhalter et al. (2002) which is defined as

$\textrm{DIC}=\chi^{2}(\hat{\theta})+2p_{D}$ (23)

where $p_{D}=\overline{\chi^{2}}-\chi^{2}(\hat{\theta})$ is the penalization
for those models with more degrees of freedom. Here an overbar represents the
mean value of the corresponding quantity. Unlike the AIC, the DIC is both
computed from Monte Carlo posterior samples and also uses the effective number
of constrained parameters by taking into account whether or not a parameter is
unconstrained by data, see Refs. Spiegelhalter et al. (2002); Liddle (2007).
Therefore, we may say that the DIC is more reliable than the AIC.

We mostly use the differences in the AICc and DIC values that are defined as

$\displaystyle\Delta\textrm{AIC}_{c}\equiv$
$\displaystyle\textrm{AIC}_{c,\textrm{X}}-\textrm{AIC}_{c,\textrm{Y}}$ (24)
$\displaystyle\Delta\textrm{DIC}\equiv$
$\displaystyle\textrm{DIC}_{\textrm{X}}-\textrm{DIC}_{\textrm{Y}}.$ (25)

Here Y represents the tilted flat $\Lambda$CDM model and X represents the
model under study. When $-2\leq\Delta\textrm{AIC}_{c},\Delta\textrm{DIC}<0$
there is weak evidence in favor of the model under study relative to the
tilted flat $\Lambda$CDM model. If
$-6\leq\Delta\textrm{AIC}_{c},\Delta\textrm{DIC}<-2$ there is positive
evidence, whereas if $-10\leq\Delta\textrm{AIC}_{c},\Delta\textrm{DIC}<-6$
there is strong evidence for the model under study. Finally if
$\Delta\textrm{AIC}_{c},\Delta\textrm{DIC}<-10$ there is very strong evidence
in favor of the model under study relative to the tilted flat $\Lambda$CDM
model. This scale also holds when $\Delta\textrm{AIC}_{c}$ and
$\Delta\textrm{DIC}$ are positive, and then favors the tilted flat
$\Lambda$CDM model over the model under study.

We also want to determine whether some of the data sets we consider are
mutually consistent (or inconsistent) in a specified cosmological model, and
also whether or not the data set consistency (inconsistency) is model
dependent. We utilize two different statistical estimators for this purpose.
The first one makes use of DIC values and is presented in Sec. 2.1.7 of Ref.
Joudaki et al. (2017)). This estimator is based on

$\mathcal{I}(D_{1},D_{2})\equiv\textrm{exp}\left(-\frac{\mathcal{G}(D_{1},D_{2})}{2}\right),$
(26)

where

$\mathcal{G}(D_{1},D_{2})=\textrm{DIC}(D_{1}\cup
D_{2})-\textrm{DIC}(D_{1})-\textrm{DIC}(D_{2}).$ (27)

Here $D_{1}$ and $D_{2}$ represent the two data sets under comparison,
$\textrm{DIC}(D_{1})$ and $\textrm{DIC}(D_{2})$ are the DIC values that result
when data set $D_{1}$ and $D_{2}$, respectively, are individually used to
constrain cosmological parameters of the specified cosmological model, and
$\textrm{DIC}(D_{1}\cup D_{2})$ is the DIC value that results when data sets
$D_{1}$ and $D_{2}$ are jointly used to constrain cosmological parameters of
the specified model. The intuitive idea behind this estimator is that if two
data sets are mutually consistent in a given cosmological model, which means
that the cosmological parameter best-fit values determined from each data set
are approximately similar, we would have $\chi^{2}_{\textrm{min}}(D_{1}\cup
D_{2})\simeq\chi^{2}_{\textrm{min}}(D_{1})+\chi^{2}_{\textrm{min}}(D_{2})$.
This would lead to negative values of $\mathcal{G}(D_{1},D_{2})$, see eq.
(23), which in turn would lead to $\mathcal{I}(D_{1},D_{2})>1$. However if
$\chi^{2}_{\textrm{min}}(D_{1}\cup
D_{2})>\chi^{2}_{\textrm{min}}(D_{1})+\chi^{2}_{\textrm{min}}(D_{2})$, and is
large enough, then we would find $\mathcal{I}(D_{1},D_{2})<1$. Therefore
$\log_{10}\mathcal{I}>0$ when the two data sets are mutually consistent and
when $\log_{10}\mathcal{I}<0$ the two data sets are mutually inconsistent, in
the cosmological model under study. Applying Jeffreys’ scale, the level of
consistency or inconsistency between the two data sets is substantial if
$\lvert\log_{10}\mathcal{I}\rvert>0.5$, is strong if
$\lvert\log_{10}\mathcal{I}\rvert>1$, and is decisive if
$\lvert\log_{10}\mathcal{I}\rvert>2$, Joudaki et al. (2017).

We now summarize the second statistical estimator we utilize to determine
whether two data sets are mutually consistent (or inconsistent) in a specified
cosmological model. This is described in Refs. Handley and Lemos (2019b, a);
Handley (2021), also see references therein. Given a data set $D$ and a given
model $M$, we can express the posterior distribution for the independent model
parameters $\theta$ through Bayes’ theorem

$\displaystyle p(\theta|D,M)=\frac{p(D|\theta,M)p(\theta|M)}{p(D|M)}\,.$ (28)

In the above expression $\mathcal{L}_{D}(\theta)\equiv p(D|\theta,M)$ is the
likelihood function, $\pi(\theta)\equiv p(\theta|M)$ are the priors for the
model parameters $\theta$, $\mathcal{Z}_{D}\equiv p(D|M)$ represents the
evidence, and $\mathcal{P}_{D}(\theta)\equiv p(\theta|D,M)$ is the posterior
distribution. Taking advantage of the fact that $\mathcal{P}_{D}(\theta)$ is a
probability distribution function in $\theta$, which means that
$\int\mathcal{P}_{D}(\theta)d\theta=1$, we can express the evidence as

$\mathcal{Z}_{D}=\int\mathcal{L}_{D}(\theta)\pi(\theta)d\theta.$ (29)

We are interested in quantifying the tension between two independent data sets
$D_{1}$ and $D_{2}$. The total likelihood from a joint analysis of both these
data sets is the product of the likelihoods for each data set,
$\mathcal{L}_{12}$ = $\mathcal{L}_{1}\mathcal{L}_{2}$. Consequently,
$\mathcal{Z}_{12}=\int\mathcal{L}_{1}(\theta)\mathcal{L}_{2}(\theta)\pi(\theta)d\theta$.
Here and in what follows we index quantities with “1” or “2” when they have
been computed using data set $D_{1}$ or $D_{2}$ respectively, and we use index
“12” when the two data sets are jointly used. We define the Bayes ratio as

$R_{D}\equiv\frac{\mathcal{Z}_{12}}{\mathcal{Z}_{1}\mathcal{Z}_{2}}.$ (30)

This statistic is constructed in such a way that when $R_{D}\gg 1$ we can say
that data sets $D_{1}$ and $D_{2}$ are consistent in the context of the
particular model, while if $R_{D}\ll 1$ the two data sets are inconsistent.
However, $R_{D}$ is strongly prior-dependent and to avoid this problem we
instead use the suspiciousness $S_{D}$, Handley and Lemos (2019b, a); Handley
(2021), which we define in the following.

To define $S_{D}$ we will need the Shannon information Shannon (1948)

$\mathcal{I}_{S,D}(\theta)=\ln\frac{\mathcal{P}_{D}(\theta)}{\pi(\theta)},$
(31)

which is a measure of the amount of information, about the parameters
$\theta$, that has been gained when moving from the priors to the posterior.
The average value over the posterior of the Shannon information

$\mathcal{D}_{D}=\int\mathcal{P}_{D}(\theta)\mathcal{I}_{S,D}(\theta)d\theta\equiv\langle\mathcal{I}_{S,D}\rangle_{\mathcal{P}_{D}},$
(32)

is known as the Kullback-Leibler divergence and measures how data compresses
from prior to posterior. The suspiciousness $S_{D}$ is defined in terms of the
Bayes ratio $R_{D}$ and the information ratio $I_{D}$

$S_{D}=\frac{R_{D}}{I_{D}},$ (33)

where

$\ln(I_{D})=\mathcal{D}_{1}+\mathcal{D}_{2}-\mathcal{D}_{12}.$ (34)

By considering a Gaussian analogy we can turn $\ln(S_{D})$ into the tension
probability $p$ of two data sets being inconsistent, Handley and Lemos (2019b,
a); Handley (2021),

$p=\int^{\infty}_{d-2\ln(S_{D})}\\!\\!\\!\\!\\!\\!\\!\\!\chi^{2}_{d}(x)dx=\int^{\infty}_{d-2\ln(S_{D})}\\!\\!\frac{x^{d/2-1}e^{-x/2}}{2^{d/2}\Gamma(d/2)}dx,$
(35)

where $d$ is the Bayesian model dimensionality

$d=\tilde{d}_{1}+\tilde{d}_{2}-\tilde{d}_{12},\qquad\tilde{d}/2=\langle\mathcal{I}_{S,D}^{2}\rangle_{\mathcal{P}_{D}}-\langle\mathcal{I}_{S,D}\rangle^{2}_{\mathcal{P}_{D}}.$
(36)

If $p\lesssim 0.05$ the data sets are in moderate tension whereas if
$p\lesssim 0.003$ they are in strong tension. The value of $p$ can be
converted into a “sigma value” using the Gaussian formula

$\sigma=\sqrt{2}\textrm{Erfc}^{-1}(p),$ (37)

where $\textrm{Erfc}^{-1}$ is the inverse complementary error function. In
particular $p\lesssim 0.05$ and $p\lesssim 0.003$ correspond to 2$\sigma$ and
3$\sigma$ Gaussian standard deviation, respectively.

In Sec. IV.3 we use both these statistical estimators to examine the
consistency of five pairs of data, namely: P18 and lensing, P18 and BAO, P18
and BAO′, P18 and non-CMB, and P18+lensing and non-CMB, in the context of
different cosmological models. We shall see in Sec. IV.1 that when $A_{L}$ is
allowed to vary error bars and two-dimensional cosmological constraint
contours determined from each data set broaden (compared to the $A_{L}=1$
case) and so are mutually consistent between different data sets (even if they
are not mutually consistent when $A_{L}=1$). We find, in Sec. IV.3, a similar
improvement in consistency when $A_{L}$ is allowed to vary (compared to the
$A_{L}=1$ case).

##  IV Results

###  IV.1 Cosmological parameters

The cosmological parameter mean values and error bars favored by the P18,
P18+lensing, and P18+lensing+non-CMB data sets are summarized in Tables 4-7
for the tilted flat $\Lambda$CDM (+$A_{L}$) models, the untilted non-flat
$\Lambda$CDM (+$A_{L}$) models, the tilted non-flat $\Lambda$CDM (+$A_{L}$)
models with the Planck $P(q)$, and the tilted non-flat $\Lambda$CDM ($+A_{L}$)
models with the new $P(q)$, respectively. Likelihood distributions of
cosmological parameters of the four models with $A_{L}=1$ are shown in Figs.
2, 3, and 4 for the P18, P18+lensing, and P18+lensing+non-CMB data sets,
respectively. The likelihood results for these four models, but now with
$A_{L}$ allowed to vary, are shown in Figs. 5, 6, and 7. Figures 8—15 show, in
each of the eight cosmological models we study, the cosmological parameter
constraints for P18, P18+lensing, and P18+lensing+non-CMB data, to illustrate
how the cosmological parameter constraints change as we include more data.
These results are discussed in Secs. IV.1.1-IV.1.4. In the third paragraph of
Sec. IV.3 we briefly discuss some cosmological parameter constraints from
(P18) lensing only data and in Sec. V we discuss whether P18, P18+lensing,
P18+non-CMB and P18+lensing+non-CMB data cosmological parameter constraints
are model-independent or not.

Our results may indicate tensions between some of the the CMB data sets and
some non-CMB low-redshift data in the context of the non-flat models. Tension
between P18 data and BAO′/BAO data in the tilted non-flat $\Lambda$CDM Planck
$P(q)$ model has been noted in Refs. Handley (2021); Di Valentino et al.
(2019); Di Valentino et al. (2021b) (our updated BAO′/BAO data differ from
those used in these references, see Sec. II). Here we want to check whether
this tension is observed for our updated BAO′/BAO data, whether it is observed
in the context of the other models we study, and how this tension is affected
when we allow the $A_{L}$ parameter to vary. In addition to the P18 vs.
BAO′/BAO comparison, we also compare P18 data and non-CMB data as well as
P18+lensing and non-CMB data. These comparisons are discussed in Secs.
IV.1.5-IV.1.7.

Table 4: Mean and 68.3% confidence limits of tilted flat $\Lambda\textrm{CDM}$ (+$A_{L}$) model parameters constrained by TT,TE,EE+lowE (P18), P18+lensing, and P18+lensing+non-CMB data sets. $H_{0}$ has units of km s-1 Mpc-1.  | Tilted flat $\Lambda$CDM model
---|---
Parameter | P18 | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.02236\pm 0.00015$ | $0.02237\pm 0.00014$ | $0.02250\pm 0.00013$
$\Omega_{c}h^{2}$ | $0.1202\pm 0.0014$ | $0.1200\pm 0.0012$ | $0.11838\pm 0.00083$
$100\theta_{\textrm{MC}}$ | $1.04090\pm 0.00031$ | $1.04091\pm 0.00031$ | $1.04110\pm 0.00029$
$\tau$ | $0.0542\pm 0.0079$ | $0.0543\pm 0.0073$ | $0.0569\pm 0.0071$
$n_{s}$ | $0.9649\pm 0.0043$ | $0.9649\pm 0.0041$ | $0.9688\pm 0.0036$
$\ln(10^{10}A_{s})$ | $3.044\pm 0.016$ | $3.044\pm 0.014$ | $3.046\pm 0.014$
$H_{0}$ | $67.28\pm 0.61$ | $67.34\pm 0.55$ | $68.09\pm 0.38$
$\Omega_{m}$ | $0.3165\pm 0.0084$ | $0.3155\pm 0.0075$ | $0.3053\pm 0.0050$
$\sigma_{8}$ | $0.8118\pm 0.0074$ | $0.8112\pm 0.0059$ | $0.8072\pm 0.0058$
| Tilted flat $\Lambda$CDM+$A_{L}$ model
Parameter | P18 | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.02259\pm 0.00017$ | $0.02251\pm 0.00017$ | $0.02258\pm 0.00014$
$\Omega_{c}h^{2}$ | $0.1180\pm 0.0015$ | $0.1183\pm 0.0015$ | $0.11747\pm 0.00091$
$100\theta_{\textrm{MC}}$ | $1.04114\pm 0.00032$ | $1.04109\pm 0.00032$ | $1.04118\pm 0.00029$
$\tau$ | $0.0496\pm 0.0082$ | $0.0487\pm 0.0087$ | $0.0476\pm 0.0085$
$n_{s}$ | $0.9710\pm 0.0050$ | $0.9695\pm 0.0048$ | $0.9715\pm 0.0038$
$\ln(10^{10}A_{s})$ | $3.030\pm 0.017$ | $3.028\pm 0.018$ | $3.023\pm 0.018$
$A_{L}$ | $1.181\pm 0.067$ | $1.073\pm 0.041$ | $1.089\pm 0.035$
$H_{0}$ | $68.31\pm 0.71$ | $68.14\pm 0.69$ | $68.52\pm 0.42$
$\Omega_{m}$ | $0.3029\pm 0.0093$ | $0.3048\pm 0.0091$ | $0.2998\pm 0.0053$
$\sigma_{8}$ | $0.7997\pm 0.0088$ | $0.7996\pm 0.0089$ | $0.7955\pm 0.0075$

Table 5: Mean and 68.3% confidence limits of untilted non-flat $\Lambda\textrm{CDM}$ (+$A_{L}$) model parameters constrained by TT,TE,EE+lowE (P18), P18+lensing, and P18+lensing+non-CMB data sets. $H_{0}$ has units of km s-1 Mpc-1.  | Untilted non-flat $\Lambda$CDM model
---|---
Parameter | P18 | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.02320\pm 0.00015$ | $0.02307\pm 0.00014$ | $0.02301\pm 0.00014$
$\Omega_{c}h^{2}$ | $0.11098\pm 0.00088$ | $0.11108\pm 0.00086$ | $0.11176\pm 0.00083$
$100\theta_{\textrm{MC}}$ | $1.04204\pm 0.00030$ | $1.04196\pm 0.00029$ | $1.04189\pm 0.00029$
$\tau$ | $0.0543\pm 0.0091$ | $0.0580\pm 0.0087$ | $0.0799\pm 0.0089$
$\Omega_{k}$ | $-0.095\pm 0.024$ | $-0.0322\pm 0.0075$ | $-0.0065\pm 0.0014$
$\ln(10^{10}A_{s})$ | $3.021\pm 0.019$ | $3.027\pm 0.018$ | $3.075\pm 0.018$
$H_{0}$ | $47.1\pm 3.2$ | $58.9\pm 2.1$ | $67.90\pm 0.56$
$\Omega_{m}$ | $0.617\pm 0.082$ | $0.390\pm 0.027$ | $0.2938\pm 0.0049$
$\sigma_{8}$ | $0.730\pm 0.017$ | $0.765\pm 0.011$ | $0.7997\pm 0.0076$
| Untilted non-flat $\Lambda$CDM+$A_{L}$ model
Parameter | P18 | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.02320\pm 0.00015$ | $0.02312\pm 0.00014$ | $0.02310\pm 0.00014$
$\Omega_{c}h^{2}$ | $0.11097\pm 0.00087$ | $0.11092\pm 0.00087$ | $0.11100\pm 0.00085$
$100\theta_{\textrm{MC}}$ | $1.04202\pm 0.00030$ | $1.04193\pm 0.00029$ | $1.04195\pm 0.00030$
$\tau$ | $0.0540\pm 0.0087$ | $0.0554\pm 0.0097$ | $0.0566\pm 0.0083$
$\Omega_{k}$ | $-0.12\pm 0.12$ | $0.0161\pm 0.0094$ | $-0.0060\pm 0.0014$
$\ln(10^{10}A_{s})$ | $3.020\pm 0.018$ | $3.021\pm 0.020$ | $3.024\pm 0.017$
$A_{L}$ | $1.08\pm 0.27$ | $1.44\pm 0.15$ | $1.162\pm 0.036$
$H_{0}$ | $52\pm 18$ | $85.7\pm 8.5$ | $68.48\pm 0.58$
$\Omega_{m}$ | $0.70\pm 0.42$ | $0.190\pm 0.043$ | $0.2874\pm 0.0050$
$\sigma_{8}$ | $0.721\pm 0.053$ | $0.7805\pm 0.0094$ | $0.7764\pm 0.0078$

Table 6: Mean and 68.3% confidence limits of Planck-$P(q)$-based tilted non-flat $\Lambda\textrm{CDM}$ ($+A_{L}$) model parameters constrained by TT,TE,EE+lowE (P18), P18+lensing, and P18+lensing+non-CMB data sets. $H_{0}$ has units of km s-1 Mpc-1.  | Tilted non-flat $\Lambda$CDM model [Planck $P(q)$]
---|---
Parameter | P18 | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.02260\pm 0.00017$ | $0.02249\pm 0.00016$ | $0.02249\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.1181\pm 0.0015$ | $0.1186\pm 0.0015$ | $0.1187\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.04116\pm 0.00032$ | $1.04107\pm 0.00032$ | $1.04106\pm 0.00031$
$\tau$ | $0.0483\pm 0.0083$ | $0.0495\pm 0.0082$ | $0.0563\pm 0.0073$
$\Omega_{k}$ | $-0.043\pm 0.017$ | $-0.0103\pm 0.0066$ | $0.0004\pm 0.0017$
$n_{s}$ | $0.9706\pm 0.0047$ | $0.9687\pm 0.0046$ | $0.9681\pm 0.0044$
$\ln(10^{10}A_{s})$ | $3.027\pm 0.017$ | $3.030\pm 0.017$ | $3.046\pm 0.014$
$H_{0}$ | $54.5\pm 3.6$ | $63.7\pm 2.3$ | $68.17\pm 0.55$
$\Omega_{m}$ | $0.481\pm 0.062$ | $0.351\pm 0.024$ | $0.3051\pm 0.0053$
$\sigma_{8}$ | $0.775\pm 0.015$ | $0.796\pm 0.011$ | $0.8080\pm 0.0066$
| Tilted non-flat $\Lambda$CDM+$A_{L}$ model [Planck $P(q)$]
Parameter | P18 | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.02258\pm 0.00017$ | $0.02251\pm 0.00017$ | $0.02259\pm 0.00016$
$\Omega_{c}h^{2}$ | $0.1183\pm 0.0015$ | $0.1183\pm 0.0015$ | $0.1173\pm 0.0014$
$100\theta_{\textrm{MC}}$ | $1.04116\pm 0.00033$ | $1.04110\pm 0.00032$ | $1.04118\pm 0.00032$
$\tau$ | $0.0478\pm 0.0081$ | $0.0489\pm 0.0085$ | $0.0479\pm 0.0085$
$\Omega_{k}$ | $-0.130\pm 0.095$ | $-0.005\pm 0.027$ | $-0.0002\pm 0.0017$
$n_{s}$ | $0.9704\pm 0.0048$ | $0.9696\pm 0.0049$ | $0.9718\pm 0.0045$
$\ln(10^{10}A_{s})$ | $3.027\pm 0.017$ | $3.028\pm 0.018$ | $3.024\pm 0.017$
$A_{L}$ | $0.88\pm 0.15$ | $1.09\pm 0.16$ | $1.090\pm 0.036$
$H_{0}$ | $45\pm 11$ | $69\pm 11$ | $68.49\pm 0.56$
$\Omega_{m}$ | $0.80\pm 0.35$ | $0.32\pm 0.11$ | $0.2998\pm 0.0055$
$\sigma_{8}$ | $0.733\pm 0.045$ | $0.796\pm 0.016$ | $0.7952\pm 0.0085$

Table 7: Mean and 68.3% confidence limits of new-$P(q)$-based tilted non-flat $\Lambda\textrm{CDM}$ ($+A_{L}$) model parameters constrained by TT,TE,EE+lowE (P18), P18+lensing, and P18+lensing+non-CMB data sets. $H_{0}$ has units of km s-1 Mpc-1.  | Tilted non-flat $\Lambda$CDM model [new $P(q)$]
---|---
Parameter | P18 | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.02255\pm 0.00017$ | $0.02248\pm 0.00016$ | $0.02248\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.1188\pm 0.0015$ | $0.1188\pm 0.0014$ | $0.1186\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.04109\pm 0.00032$ | $1.04104\pm 0.00032$ | $1.04106\pm 0.00031$
$\tau$ | $0.0525\pm 0.0083$ | $0.0515\pm 0.0081$ | $0.0566\pm 0.0074$
$\Omega_{k}$ | $-0.033\pm 0.014$ | $-0.0086\pm 0.0057$ | $0.0003\pm 0.0017$
$n_{s}$ | $0.9654\pm 0.0045$ | $0.9661\pm 0.0043$ | $0.9679\pm 0.0042$
$\ln(10^{10}A_{s})$ | $3.039\pm 0.017$ | $3.035\pm 0.016$ | $3.046\pm 0.014$
$H_{0}$ | $56.9\pm 3.6$ | $64.2\pm 2.0$ | $68.13\pm 0.54$
$\Omega_{m}$ | $0.444\pm 0.055$ | $0.345\pm 0.021$ | $0.3054\pm 0.0051$
$\sigma_{8}$ | $0.786\pm 0.014$ | $0.799\pm 0.010$ | $0.8079\pm 0.0067$
| Tilted non-flat $\Lambda$CDM+$A_{L}$ model [new $P(q)$]
Parameter | P18 | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.02257\pm 0.00017$ | $0.02252\pm 0.00017$ | $0.02260\pm 0.00016$
$\Omega_{c}h^{2}$ | $0.1187\pm 0.0016$ | $0.1183\pm 0.0015$ | $0.1174\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.04111\pm 0.00033$ | $1.04108\pm 0.00032$ | $1.04118\pm 0.00032$
$\tau$ | $0.0512\pm 0.0086$ | $0.0495\pm 0.0093$ | $0.0486\pm 0.0086$
$\Omega_{k}$ | $-0.10\pm 0.11$ | $0.003\pm 0.016$ | $-0.0002\pm 0.0017$
$n_{s}$ | $0.9654\pm 0.0057$ | $0.9688\pm 0.0053$ | $0.9713\pm 0.0042$
$\ln(10^{10}A_{s})$ | $3.036\pm 0.018$ | $3.030\pm 0.019$ | $3.025\pm 0.017$
$A_{L}$ | $0.94\pm 0.20$ | $1.13\pm 0.15$ | $1.088\pm 0.035$
$H_{0}$ | $51\pm 14$ | $72.0\pm 9.2$ | $68.48\pm 0.56$
$\Omega_{m}$ | $0.70\pm 0.43$ | $0.287\pm 0.076$ | $0.2999\pm 0.0055$
$\sigma_{8}$ | $0.752\pm 0.052$ | $0.801\pm 0.011$ | $0.7956\pm 0.0082$

Figure 2: Planck 2018 TT,TE,EE+lowE (P18) data likelihood distributions of
parameters of the tilted non-flat $\Lambda$CDM model with the new initial
power spectrum [new $P(q)$] (red contours), of the tilted non-flat
$\Lambda$CDM model with the Planck team’s initial spectrum [Planck $P(q)$]
(green), of the untitled non-flat $\Lambda$CDM model (grey), and of the tilted
flat $\Lambda$CDM model (blue contours).

Figure 3: P18+lensing data likelihood distributions of parameters of the
tilted non-flat $\Lambda$CDM model with the new initial power spectrum [new
$P(q)$] (red contours), of the tilted non-flat $\Lambda$CDM model with the
Planck team’s initial spectrum [Planck $P(q)$] (green), of the untitled non-
flat $\Lambda$CDM model (grey), and of the tilted flat $\Lambda$CDM model
(blue contours).

Figure 4: P18+lensing+non-CMB data likelihood distributions of parameters of
the tilted non-flat $\Lambda$CDM model with the new initial power spectrum
[new $P(q)$] (red contours), of the tilted non-flat $\Lambda$CDM model with
the Planck team’s initial spectrum [Planck $P(q)$] (green), of the untitled
non-flat $\Lambda$CDM model (grey), and of the tilted flat $\Lambda$CDM model
(blue contours).

Figure 5: Planck 2018 TT,TE,EE+lowE (P18) data likelihood distributions of
parameters of the tilted non-flat $\Lambda$CDM+$A_{L}$ model with the new
initial power spectrum [new $P(q)$] (red contours), of the tilted non-flat
$\Lambda$CDM+$A_{L}$ model with the Planck team’s initial spectrum [Planck
$P(q)$] (green), of the untitled non-flat $\Lambda$CDM+$A_{L}$ model (grey),
and of the tilted flat $\Lambda$CDM+$A_{L}$ model (blue contours).

Figure 6: P18+lensing data likelihood distributions of parameters of the
tilted non-flat $\Lambda$CDM+$A_{L}$ model with the new initial power spectrum
[new $P(q)$] (red contours), of the tilted non-flat $\Lambda$CDM+$A_{L}$ model
with the Planck team’s initial spectrum [Planck $P(q)$] (green), of the
untitled non-flat $\Lambda$CDM+$A_{L}$ model (grey), and of the tilted flat
$\Lambda$CDM+$A_{L}$ model (blue contours).

Figure 7: P18+lensing+non-CMB data likelihood distributions of parameters of
the tilted non-flat $\Lambda$CDM+$A_{L}$ model with the new initial power
spectrum [new $P(q)$] (red contours), of the tilted non-flat
$\Lambda$CDM+$A_{L}$ model with the Planck team’s initial spectrum [Planck
$P(q)$] (green), of the untitled non-flat $\Lambda$CDM+$A_{L}$ model (grey),
and of the tilted flat $\Lambda$CDM+$A_{L}$ model (blue contours).

Figure 8: Likelihood distributions of tilted flat $\Lambda$CDM model
parameters constrained by P18 (gray contours), P18+lensing (red contours),
P18+lensing+non-CMB (blue contours) data sets.

Figure 9: Likelihood distributions of untilted non-flat $\Lambda$CDM model
parameters constrained by P18 (gray contours), P18+lensing (red contours),
P18+lensing+non-CMB (blue contours) data sets.

Figure 10: Likelihood distributions of tilted non-flat $\Lambda$CDM model
parameters with the Planck team’s initial power spectrum [Planck $P(q)$]
constrained by P18 (gray contours), P18+lensing (red contours),
P18+lensing+non-CMB (blue contours) data sets.

Figure 11: Likelihood distributions of tilted non-flat $\Lambda$CDM model
parameters with the new initial power spectrum [new $P(q)$] constrained by P18
(gray contours), P18+lensing (red contours), P18+lensing+non-CMB (blue
contours) data sets.

Figure 12: Likelihood distributions of tilted flat $\Lambda$CDM+$A_{L}$ model
parameters constrained by P18 (gray contours), P18+lensing (red contours),
P18+lensing+non-CMB (blue contours) data sets.

Figure 13: Likelihood distributions of untilted non-flat $\Lambda$CDM+$A_{L}$
model parameters constrained by P18 (gray contours), P18+lensing (red
contours), P18+lensing+non-CMB (blue contours) data sets.

Figure 14: Likelihood distributions of tilted non-flat $\Lambda$CDM+$A_{L}$
model parameters with the Planck team’s initial power spectrum [Planck $P(q)$]
constrained by P18 (gray contours), P18+lensing (red contours),
P18+lensing+non-CMB (blue contours) data sets.

Figure 15: Likelihood distributions of tilted non-flat $\Lambda$CDM+$A_{L}$
model parameters with the new initial power spectrum [new $P(q)$] constrained
by P18 (gray contours), P18+lensing (red contours), P18+lensing+non-CMB (blue
contours) data sets.

We now discuss the results obtained from the different data sets we consider.

####  IV.1.1 P18 data cosmological constraints

In the case of the tilted flat $\Lambda$CDM model, with just six primary (not
derived) cosmological parameters, and with $\Omega_{k}=0$, from P18 data alone
(see Table 4 and Figs. 2 and 5) the derived parameters $\Omega_{m}=0.3165\pm
0.0084$ and $H_{0}=67.28\pm 0.61$ km s-1 Mpc-1, which are consistent with many
other measurements of these quantities and which differ from the low-redshift
data measurements of Ref. Cao and Ratra (2022), $\Omega_{m}=0.295\pm 0.017$
and $H_{0}=69.7\pm 1.2$ km s-1 Mpc-1, by $1.1\sigma$ and $1.8\sigma$.

The improvement in the fit to P18 data when the $A_{L}$ parameter is allowed
to vary, in the tilted flat $\Lambda$CDM$+A_{L}$ model, is positive, according
to the DIC statistical criterion described in Sec. III (see the results in
Sec. IV.2 ). This fact is reflected in the measured (from P18 data) value of
this phenomenological parameter, $A_{L}=1.181\pm 0.067$, which differs from
the theoretically expected $A_{L}=1$ by $2.7\sigma$. The inclusion of the
$A_{L}$ parameter, introduced to deal with the lensing anomaly, does not
significantly affect the values of the other six primary parameters, leaving
them close to the values found for the six parameter ($\Omega_{k}=0$) tilted
flat $\Lambda$CDM model with $A_{L}=1$ (the largest difference is for
$\Omega_{c}h^{2}$, where it is 1.1$\sigma$ of the quadrature sum of the two
error bars); it does however increase the error bars somewhat, with the
largest increase being 16% for $n_{s}$. In addition, in the case when $A_{L}$
is allowed to vary, the derived parameters $\Omega_{m}$ and $H_{0}$ (as well
as $\sigma_{8}$) errors bars mildly increase, by 11% and 16%, resulting in
(for P18 data) $\Omega_{m}=0.3029\pm 0.0093$ and $H_{0}=68.31\pm 0.71$ km s-1
Mpc-1, that are consistent with many other measurements of these quantities,
now differing only by $0.41\sigma$ and $1.0\sigma$, respectively, from those
of Ref. Cao and Ratra (2022). These derived $\Omega_{m}$ and $H_{0}$ parameter
values in the $A_{L}$-varying case also differ from those in the $A_{L}=1$
case by at most 1.1$\sigma$.

The addition of the $\Omega_{k}$ parameter to the six primary cosmological
parameters of the tilted flat $\Lambda$CDM model introduces a strong
degeneracy between increasing $\Omega_{m}$ and decreasing $H_{0}$. The non-
flat models also show some degeneracy between $\Omega_{m}$ and $\Omega_{k}$ as
well as between $H_{0}$ and $\Omega_{k}$. These degeneracies can be seen in
the corresponding panels in Fig. 2. In the tilted non-flat $\Lambda$CDM models
(see Tables 6 and 7 and Fig. 2) we see that P18 data alone is unable to break
the strong geometrical degeneracy between $\Omega_{m}$-$H_{0}$-$\Omega_{k}$.
For the Planck $P(q)$ and the new $P(q)$, the measured values (from P18 data)
$\Omega_{m}=0.481\pm 0.062$ and $0.444\pm 0.055$, as well as $H_{0}=54.5\pm
3.6$ and $56.9\pm 3.6$ km s-1 Mpc-1, respectively, are in conflict with most
other measurements of these parameters; for example, see the low-redshift data
measurements of Ref. Cao and Ratra (2022) in the paragraph before last. Note
that even though the values of, and error bars on, the six primary
cosmological parameters in common between the two tilted non-flat models and
the tilted flat model are very similar (the largest difference is 1.1$\sigma$
for $\Omega_{b}h^{2}$ between the tilted flat and the tilted non-flat Planck
$P(q)$ models, and the biggest increase, 13%, in the error bars is also for
$\Omega_{b}h^{2}$, in both tilted non-flat models relative to the tilted flat
model), the additional primary cosmological parameter $\Omega_{k}$ in the two
tilted non-flat models is relatively poorly constrained, and the derived
cosmological parameters $\Omega_{m}$ and $H_{0}$ error bars in the two tilted
non-flat $\Lambda$CDM models are approximately factors of 7 and 6 larger than
those in the tilted flat $\Lambda$CDM model (and $\Omega_{m}$ and $H_{0}$ in
these tilted non-flat models differ by between 2.3$\sigma$ and 3.5$\sigma$
from the tilted flat model values). The evidence in favor of $\Omega_{k}<0$ is
significant in both of the tilted non-flat $\Lambda$CDM models. For the Planck
$P(q)$ case we find $\Omega_{k}=-0.043\pm 0.017$ while for the new $P(q)$ case
$\Omega_{k}=-0.033\pm 0.014$, being 2.5$\sigma$ and 2.4$\sigma$ away from flat
respectively from flat spatial hypersurfaces. In both cases there is clear
preference for a closed over an open spatial geometry. And we shall see in
Sec. IV.2 that the P18 data DIC statistical criterion strongly favors both
tilted non-flat models over the tilted flat $\Lambda$CDM model.

Allowing the $A_{L}$ parameter to vary in the non-flat models introduces an
additional strong degeneracy between $\Omega_{k}$, $\Omega_{m}$, $H_{0}$, and
$A_{L}$; compare the corresponding panels in Figs. 2 and 5. In the tilted non-
flat $\Lambda$CDM+$A_{L}$ models with the Planck $P(q)$ and with the new
$P(q)$, where the $A_{L}$ parameter is allowed to vary (see Tables 6 and 7 and
Fig. 5) P18 data alone is unable to break the strong geometrical degeneracy
between $\Omega_{m}$-$H_{0}$-$\Omega_{k}$-$A_{L}$. (In the tilted non-flat
$\Lambda$CDM+$A_{L}$ new $P(q)$ model some parameters have a somewhat bimodal
distribution for P18 data, see the one-dimensional posterior distributions in
Fig. 5. This is not the case for the tilted non-flat $\Lambda$CDM+$A_{L}$
Planck $P(q)$ model.) Like in the tilted flat $\Lambda$CDM case discussed in
the paragraph before last, the extra $A_{L}$ parameter does not significantly
affect any of the (primary, not derived) cosmological parameter constraints,
compared to the $A_{L}=1$ case, except, because of the additional
$\Omega_{k}$-$A_{L}$ degeneracy, the $\Omega_{k}$ constraints which are now
$\Omega_{k}=-0.130\pm 0.095$ for the Planck $P(q)$ case and
$\Omega_{k}=-0.10\pm 0.11$ for the new $P(q)$, being only 1.4$\sigma$ and
0.91$\sigma$, respectively, away from flat spatial hypersurfaces, with the
$\Omega_{k}$ error bars now being factors of 6 and 8, respectively, larger
than those in the $A_{L}=1$ case. Also, unlike the tilted flat $\Lambda$CDM
case of the paragraph before last, we measure, from the P18 data,
$A_{L}=0.88\pm 0.15$ and $0.94\pm 0.20$, which differ from the theoretically
expected $A_{L}=1$ by only 0.80$\sigma$ and 0.30$\sigma$. We will see in Sec.
IV.2 that in both these models the fit to P18 data is weakly or positively
better when $A_{L}=1$ compared to the case when the $A_{L}$ parameter is
allowed to vary. However, when $A_{L}$ varies the DIC statistical criterion
weakly favors [positively disfavors] the tilted non-flat Planck $P(q)$ [new
$P(q)$] model over the tilted flat $\Lambda$CDM$+A_{L}$ model. In addition, in
both these cases when $A_{L}$ is allowed to vary, the $\Omega_{m}$ and $H_{0}$
(as well as $\sigma_{8}$) errors bars significantly increase, resulting in
(for P18 data) $\Omega_{m}=0.80\pm 0.35$ and $0.70\pm 0.43$, as well as
$H_{0}=45\pm 11$ and $51\pm 14$ km s-1 Mpc-1, that are consistent with many
other measurements of these quantities. Again, the error bars on
$\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, $\theta_{\rm MC}$, $\tau$, $n_{s}$, and
$A_{s}$ are similar in the two tilted non-flat $\Lambda$CDM$+A_{L}$ models and
the tilted flat $\Lambda$CDM$+A_{L}$ model, however the $A_{L}$ error bars are
approximately a factor of 2.5 larger in the tilted non-flat models, with the
introduction of the seventh primary cosmological parameter $\Omega_{k}$ (that
is poorly constrained) also resulting in the $\Omega_{m}$ error bars being a
factor $\sim$42 larger and the $H_{0}$ error bars being a factor $\sim$18
larger in the tilted non-flat $\Lambda$CDM$+A_{L}$ models compared to the
tilted flat $\Lambda$CDM$+A_{L}$ model.

The restriction that $n_{s}=1$ in the untilted non-flat $\Lambda$CDM
(+$A_{L}$) models is an unwelcome feature when fitting the P18 CMB anisotropy
spectra, according to the statistical criteria outlined in Sec. IV.2. Because
of this we will focus less attention on the untilted non-flat $\Lambda$CDM
model compared to the two tilted non-flat models. Despite the poor performance
of the untilted non-flat $\Lambda$CDM model in this case (which also affects
what happens when additional data are jointly analyzed with P18 data), the
model shares some features with the two tilted non-flat $\Lambda$CDM models
(see Table 5 and Fig. 2), namely, the evidence in favor of closed spatial
geometry, now with $\Omega_{k}=-0.095\pm 0.024$ (4.0$\sigma$), and the
presence of the aforementioned geometrical degeneracy between
$\Omega_{m}$-$H_{0}$-$\Omega_{k}$. Also, as in the two tilted non-flat models,
in the untilted non-flat case the measured values (from P18 data)
$\Omega_{m}=0.617\pm 0.082$ and $H_{0}=47.1\pm 3.2$ km s-1 Mpc-1 are in
conflict with most other measurements of these parameters.

In the untilted non-flat $\Lambda$CDM+$A_{L}$ model where the $A_{L}$
parameter is allowed to vary (see Table 5 and Fig. 5) P18 data alone is again
unable to break the larger geometrical degeneracy between
$\Omega_{m}$-$H_{0}$-$\Omega_{k}$-$A_{L}$. Like in the tilted flat and non-
flat $\Lambda$CDM cases discussed earlier, the extra $A_{L}$ parameter does
not significantly affect any of the (primary, not derived) cosmological
parameter constraints in the untilted non-flat model, compared to the
$A_{L}=1$ case, except, because of the additional $\Omega_{k}$-$A_{L}$
degeneracy, for the $\Omega_{k}$ constraint which is now $\Omega_{k}=-0.12\pm
0.12$ and only 1.0$\sigma$ away from flat spatial hypersurfaces. Also, unlike
the tilted flat $\Lambda$CDM case, but like the tilted non-flat cases of the
paragraph before last, we measure, from the P18 data, $A_{L}=1.08\pm 0.27$
which differs from the theoretically expected value of $A_{L}=1$ by only
0.30$\sigma$. We will see in Sec. IV.2 that in this model the fit to P18 data
is slightly better when $A_{L}=1$ compared to the case when the $A_{L}$
parameter is allowed to vary. Similar to the two tilted non-flat models of the
paragraph before last, when $A_{L}$ is allowed to vary in the untilted non-
flat model, the $\Omega_{m}$ and $H_{0}$ (as well as $\sigma_{8}$) errors bars
significantly increase, resulting in (for P18 data) $\Omega_{m}=0.70\pm 0.42$
and $H_{0}=52\pm 18$ km s-1 Mpc-1, that are consistent with most other
measurements of these quantities.

In Fig. 2 we provide the 2$\sigma$ contour plots for all four of the $A_{L}=1$
models. The contours for the untilted non-flat $\Lambda$CDM model overlap
minimally or even do not overlap at all with those corresponding to the other
models. This is likely due to the lack of the degree of freedom encapsulated
in $n_{s}$ in the untilted non-flat model, which greatly hinders the fit of
the CMB anisotropy power spectra and causes the other parameters to shift from
the ranges preferred in the context of the other three models. As for the
other three cosmological models, there is a significant overlap of contours,
except when $\Omega_{m}$ or $H_{0}$ (or less so $\sigma_{8}$) is involved,
which can even lead to the contours not overlapping at all. This is presumably
related with the geometrical degeneracy previously mentioned.

The corresponding plots for the four models now including allowing $A_{L}$ to
vary are in Fig. 5. Allowing $A_{L}$ to vary broadens the contours, and for
some parameters there are two disconnected 1$\sigma$ regions. While the
untilted non-flat model contours still do not overlap in many cases with those
of the other three models, in the other three models the contours overlap even
when $\Omega_{m}$ or $H_{0}$ or $\sigma_{8}$ is involved.

####  IV.1.2  P18+lensing data cosmological constraints

Constraints on primary parameters derived from joint analyses of P18 and
lensing data are quite similar to those derived from P18 data alone (see
Tables 4-7 and Figs. 3 and 6), except for $\Omega_{k}$ and $A_{L}$
constraints. On the other hand, constraints on derived parameters $\Omega_{m}$
and $H_{0}$ are, in most non-flat cases, greatly affected by lensing data. In
this subsubsection we discuss parameter constraints from jointly analyzed P18
and lensing data and compare these to the P18 data alone constraints.

Ideally one would like to establish that cosmological parameter constraints
derived from P18 data and from lensing data are mutually consistent, prior to
using P18+lensing data in joint analyses. While it is not straightforward to
derive (P18) lensing data alone cosmological parameter constraints for the
wide flat Our priors of Table 3, we shall see, in Sec. IV.3 (where we do
briefly discuss some of these cosmological constraints), that P18 data and
lensing data are not significantly mutually inconsistent. This is also
consistent with the results we discuss in this subsubsection.

Comparing the six-parameter tilted flat $\Lambda$CDM primary cosmological
parameter constraints for P18 data and P18+lensing data, shown in the upper
half of Table 4, we see that there are no significant changes in parameter
values (the largest change is that $\Omega_{c}h^{2}$ is 0.11$\sigma$ smaller
in the P18+lensing case) with all but the $\theta_{\rm MC}$ error bars being
smaller in the P18+lensing case (the $\theta_{\rm MC}$ error bar is unchanged
and the largest decrease is 14% for the $\Omega_{c}h^{2}$ error bar). For the
derived parameters, the largest change is the 0.089$\sigma$ decrease in
$\Omega_{m}$ relative to the P18 data value, and the 20% smaller $\sigma_{8}$
error bar. For P18+lensing data we find $\Omega_{m}=0.3155\pm 0.0075$ and
$H_{0}=67.34\pm 0.55$ km s-1 Mpc-1 which are consistent with many other
measurements of these quantities and 1.1$\sigma$ larger and 1.8$\sigma$ lower
than the low-redshift data measurements of Ref. Cao and Ratra (2022).

Comparing the seven-parameter tilted flat $\Lambda$CDM$+A_{L}$ primary
cosmological parameter constraints for P18 data and P18+lensing data, shown in
the lower half of Table 4, we see more significant changes in the parameter
values (the largest change is that $A_{L}$ is 1.4$\sigma$ smaller in the
P18+lensing case, with the next largest being $\Omega_{b}h^{2}$ which is
0.33$\sigma$ smaller) with some of the error bars being larger in the
P18+lensing case (the largest increase is 6% for the $\tau$ and
$\ln(10^{10}A_{s})$ error bars) and some of the error bars being smaller (the
largest decrease is 39% for $A_{L}$). The reason the error bars of $\tau$ and
$\ln(10^{10}A_{s})$ increase, contrary to the common expectation that the
error bars of the parameters will become smaller as more data is added,
appears to be that the degeneracy between parameters is only partially broken
by the lensing data. Interestingly, these characteristics are common to all
other $A_{L}$-varying models (see Tables 5-7). For the derived parameters, the
largest change is the 0.17$\sigma$ decrease in $H_{0}$ relative to the P18
data value, and the 3% smaller $H_{0}$ error bar. For P18+lensing data in the
varying $A_{L}$ case we measure $\Omega_{m}=0.3048\pm 0.0091$ and
$H_{0}=68.14\pm 0.69$ km s-1 Mpc-1 which are consistent with many other
measurements of these quantities and 0.51$\sigma$ larger and 1.1$\sigma$ lower
than the low-redshift data measurements of Ref. Cao and Ratra (2022).

The improvement in the fit to P18+lensing data when the $A_{L}$ parameter is
allowed to vary, in the tilted flat $\Lambda$CDM$+A_{L}$ model, is only weak,
as discussed in Sec. IV.2. We now find $A_{L}=1.073\pm 0.041$ which is
1.8$\sigma$ away from the theoretically expected $A_{L}=1$. While there is
still a deviation from the predicted value, the tendency of the lensing data
is to push $A_{L}$ closer to 1, resulting in a smaller deviation than the
2.7$\sigma$ one found for $A_{L}=1.181\pm 0.067$ from P18 data in the tilted
flat $\Lambda$CDM$+A_{L}$ model. The inclusion of the $A_{L}$ parameter does
not significantly affect the values of the other six primary parameters,
leaving them close to the values found for the six parameter tilted flat
$\Lambda$CDM model with $A_{L}=1$ (the largest difference is for
$\Omega_{c}h^{2}$, where it is 0.88$\sigma$ of the quadrature sum of the two
error bars); it does however increase the error bars, more than what happens
in the P18 alone data case discussed in Sec. IV.1.1, with largest increase
being 29% for $\ln(10^{10}A_{s})$. In addition, in the case when $A_{L}$ is
allowed to vary, the derived parameters change somewhat and their error bars
increase, with the largest changes associated with $\sigma_{8}$, where it is
now 1.1$\sigma$ smaller with a 51% larger error bar.

In the six-parameter untilted non-flat $\Lambda$CDM model, including lensing
data in the mix results in a reduction in the size of the cosmological
parameter error bars relative to those from P18 data (see Table 5 and Fig. 3).
The most affected parameters are the primary parameter $\Omega_{k}$, whose
error bars decrease by 69%, and the derived parameters $H_{0}$, $\Omega_{m}$
and $\sigma_{8}$, for which we observe a shrinkage of the error bars by 34%,
67%, and 35%, respectively. As happens in the tilted flat $\Lambda$CDM model,
here also there are no significant changes in the values of the primary
parameters, with the exception of the curvature parameter $\Omega_{k}$. This
is not true for two of the derived parameters, $H_{0}$ and $\Omega_{m}$, which
together with the curvature parameter are involved in the
$\Omega_{m}$-$H_{0}$-$\Omega_{k}$ geometrical degeneracy. From P18+lensing
data we find $\Omega_{k}=-0.095\pm 0.024$, $H_{0}=47.1\pm 3.2$ km s-1 Mpc-1,
and $\Omega_{m}=0.390\pm 0.027$. These values differ by 2.5$\sigma$,
3.1$\sigma$, and 2.6$\sigma$, respectively, from the corresponding values
obtained in the P18 data alone analysis.

From the results obtained for the untilted non-flat $\Lambda$CDM+$A_{L}$ model
(see Table 5 and Fig. 6), we observe significant changes in the values of the
primary parameters $\Omega_{k}$ and $A_{L}$, as well as in the derived
parameters $H_{0}$ and $\Omega_{m}$. For the P18+lensing data set we get
$\Omega_{k}=0.0161\pm 0.0094$ (1.7$\sigma$ away from flat) and $A_{L}=1.44\pm
0.15$ (2.9$\sigma$ away from $A_{L}=1$). These values differ by 1.1$\sigma$
and 1.2$\sigma$, respectively, from the corresponding values obtained in the
P18 data alone analysis. For the derived parameters, from P18+lensing data we
find $H_{0}=85.7\pm 8.5$ km s-1 Mpc-1 and $\Omega_{m}=0.190\pm 0.043$, which
differ by 1.7$\sigma$ and 1.2$\sigma$ from the corresponding P18 data alone
values.

Joint analyses of the P18 and lensing data in the tilted non-flat models
result in constraints that differ more from those derived using just P18 data
compared to what happens in the tilted flat model case (see Tables 6 and 7 and
Fig. 3). This is because lensing data partially breaks the
$\Omega_{m}$-$H_{0}$-$\Omega_{k}$ geometrical P18 alone data degeneracy of the
tilted non-flat models (compare the corresponding panels in Figs. 2 and 3).

Comparing the seven-parameter tilted non-flat $\Lambda$CDM Planck $P(q)$ and
new $P(q)$ primary cosmological parameter constraints for P18 data and
P18+lensing data, shown in the upper halves of Tables 6 and 7, we see that
aside from $\Omega_{k}$ (discussed next) there are no significant changes in
parameter values [the largest change is that $\Omega_{b}h^{2}$ is 0.47$\sigma$
(0.30$\sigma$) smaller in the P18+lensing case, for the Planck (new) $P(q)$]
with some of the error bars being smaller in the P18+lensing case [leaving
aside $\Omega_{k}$ (discussed next) the largest decrease is 6% (7%) for the
$\Omega_{b}h^{2}$ ($\Omega_{c}h^{2}$) error bar, for the Planck (new) $P(q)$].
On the other hand, $\Omega_{k}$ changes significantly when lensing data are
added to the mix, becoming 1.8$\sigma$ (1.6$\sigma$) larger, and closer to
flat for the Planck (new) $P(q)$, with 61% (59%) smaller error bars, still
favoring closed geometry over flat but only by 1.6$\sigma$ (1.5$\sigma$),
respectively. For the derived parameters, the largest change is the
2.2$\sigma$ (1.8$\sigma$) increase in $H_{0}$ relative to the P18 data value
for the Planck (new) $P(q)$, with 61% (62%) smaller error bars for
$\Omega_{m}$. For P18+lensing data we find $\Omega_{m}=0.351\pm 0.024$
($0.345\pm 0.021$) and $H_{0}=63.7\pm 2.3$ ($64.2\pm 2.0$) km s-1 Mpc-1 for
the Planck (new) $P(q)$, which are consistent with many other measurements of
these quantities and 1.9$\sigma$ (1.9$\sigma$) larger and 2.3$\sigma$
(2.4$\sigma$) lower, respectively, than the low-redshift data measurements of
Ref. Cao and Ratra (2022).

Comparing the eight-parameter tilted non-flat $\Lambda$CDM$+A_{L}$ Planck
(new) $P(q)$ primary cosmological parameter constraints for P18 data and
P18+lensing data, shown in the lower half of Table 6 (7), we see that there
are smaller differences compared to the tilted flat $\Lambda$CDM$+A_{L}$ case.
For the Planck $P(q)$ case we mostly find less significant changes (the
largest changes are that $\Omega_{k}$ and $A_{L}$ are 1.3$\sigma$ and
0.95$\sigma$ larger in the P18+lensing case, with the next largest being
$\Omega_{b}h^{2}$ which is 0.29$\sigma$ smaller) with some of the error bars
being larger in the P18+lensing case (the largest increase is 7% for the
$A_{L}$ error bar, and this is the only model where the $A_{L}$ error bar is
larger for P18+lensing data than for P18 data) and some of the error bars
being smaller (the largest decrease is 72% for $\Omega_{k}$). In the new
$P(q)$ case we find roughly half the parameters change more significantly (the
largest changes again are that $\Omega_{k}$ and $A_{L}$ are 0.93$\sigma$ and
0.76$\sigma$ larger in the P18+lensing case, with the next largest being
$n_{s}$ which is 0.44$\sigma$ larger) with some of the error bars being larger
in the P18+lensing case (the largest increase is 8% for the $\tau$ error bar)
and some of the error bars being smaller (the largest decrease is 85% for
$\Omega_{k}$). From the P18+lensing analyses, we measure $\Omega_{k}=-0.005\pm
0.027$ for the Planck $P(q)$ case and $\Omega_{k}=0.003\pm 0.016$ for the new
$P(q)$, both being only 0.19$\sigma$ away from flat spatial hypersurfaces,
very different from the P18 data alone results. For the derived parameters,
the largest change is the 1.5$\sigma$ (1.3$\sigma$) increase in $H_{0}$
relative to the P18 data value for the Planck (new) $P(q)$, with 69% (82%)
smaller error bars for $\Omega_{m}$. For P18+lensing data we find
$\Omega_{m}=0.32\pm 0.11$ ($0.287\pm 0.076$) and $H_{0}=69\pm 11$ ($72.0\pm
9.2$) km s-1 Mpc-1 for the Planck (new) $P(q)$, which are consistent with many
other measurements of these quantities and 0.22$\sigma$ larger (0.10$\sigma$
smaller) and 0.063$\sigma$ lower (0.25$\sigma$ higher), respectively, than the
low-redshift data measurements of Ref. Cao and Ratra (2022). (Note that the
P18+lensing data Planck $P(q)$ $H_{0}$ error bar is unchanged, $\pm 11$ km
s-1, from the P18 data value, and this is the only model where this happens.)

We will see in Sec. IV.2 that in both tilted non-flat $\Lambda$CDM$+A_{L}$
models the fit to P18+lensing data is weakly better when $A_{L}=1$ compared to
the case when the $A_{L}$ parameter is allowed to vary; this differs from what
happens in the tilted flat $\Lambda$CDM$+A_{L}$ model. Also, unlike the tilted
flat $\Lambda$CDM$+A_{L}$ P18+lensing case, we measure, from P18+lensing data,
$A_{L}=1.089\pm 0.16$ and $1.13\pm 0.15$, for the Planck $P(q)$ and the new
$P(q)$, respectively, which differ from the theoretically expected $A_{L}=1$
by only 0.56$\sigma$ and 0.87$\sigma$. The inclusion of the $A_{L}$ parameter
does not significantly affect the values of the other seven primary
parameters, leaving them close to the values found for the seven parameter
tilted non-flat $\Lambda$CDM models with $A_{L}=1$ [the largest difference is
for $\Omega_{k}$, where it is 0.19$\sigma$ (0.68$\sigma$) for the Planck (new)
$P(q)$]; it does however increase the error bars, but less than what happens
in the P18 alone data case discussed in Sec. IV.1.1, with largest factor being
4 (3) for $\Omega_{k}$ for the Planck (new) $P(q)$. In addition, in the case
when $A_{L}$ is allowed to vary, the derived parameters change somewhat and
their error bars increase, with the largest changes associated with $H_{0}$,
where it is now 0.47$\sigma$ (0.83$\sigma$) larger for the Planck (new) $P(q)$
with a factor of 5 (5) larger error bar.

From the discussion above in this subsubsection, the fact that the
cosmological constraint contours displayed in Fig. 3 for the three tilted
models overlap should not come as a surprise. Unlike in the previous P18 data
alone case, the P18+lensing data contours that involve $\Omega_{m}$, $H_{0}$,
or $\Omega_{k}$ now overlap for the tilted models, indicating that the
geometrical degeneracy is, at least, partially broken. Figure 6 shows the
results when the $A_{L}$ parameter is included in the analysis. While the
overlap already found in the P18 data alone analysis (see Fig. 5) remains, the
bimodal 1$\sigma$ regions of that plot have now disappeared.

####  IV.1.3 P18+lensing+non-CMB data cosmological constraints

In this subsubsection we comment on the results obtained from a joint analysis
of the P18+lensing+non-CMB data set and how the cosmological constraints
change when compared to those obtained using P18+lensing data. As outlined in
Sec. II non-CMB data we use here is comprised of BAO, $f\sigma_{8}$, SNIa, and
$H(z)$ data, all of which provide useful information on the late-time
Universe.

Ideally one would like to establish that cosmological parameter constraints
derived from P18+lensing data and from non-CMB data are mutually consistent,
prior to using P18+lensing+non-CMB data in joint analyses. Given that P18 data
dominate the P18+lensing data compilation, it is instructive to also study
whether P18 data cosmological constraints are consistent with those from non-
CMB data. We shall see in Sec. IV.1.5 that, in some of the models we study
here, cosmological constraints from BAO′ and BAO data, the dominant part of
the non-CMB data compilation, are somewhat inconsistent with those derived
using P18 data. This is also consistent with the results we discuss in this
subsubsection, as well as, with the results presented in Sec. IV.1.6, where we
compare the cosmological parameter values obtained using P18 data and using
non-CMB data. In Sec. IV.1.7 we compare P18+lensing data cosmological
constraints and non-CMB data cosmological constraints, and find similar
tensions. In addition, in Sec. IV.3, we study tensions between some of the CMB
data sets and some of the low-redshift data sets, including the case of
P18+lensing data vs. non-CMB data, by using the two statistical estimators
presented in Sec. III.

Comparing the six-parameter tilted flat $\Lambda$CDM primary cosmological
parameter constraints for P18+lensing data and P18+lensing+non-CMB data, shown
in the upper half of Table 4, we see that there are no significant changes in
parameter values (the largest change is that $\Omega_{c}h^{2}$ is 1.1$\sigma$
smaller in the P18+lensing+non-CMB case) with all but the ln($10^{10}A_{s})$
error bars being smaller in the P18+lensing+non-CMB case (the
ln($10^{10}A_{s})$ error bar is unchanged and the largest decrease is 31% for
the $\Omega_{c}h^{2}$ error bar). For the derived parameters, the largest
changes are the 1.1$\sigma$ decrease in $\Omega_{m}$ and the 1.1$\sigma$
increase in $H_{0}$ relative to the P18+lensing data values, and the 33% (31%)
smaller $\Omega_{m}$ ($H_{0}$) error bar. For P18+lensing+non-CMB data we find
$\Omega_{m}=0.3053\pm 0.0050$ and $H_{0}=68.09\pm 0.38$ km s-1 Mpc-1 which are
consistent with many other measurements of these quantities and 0.58$\sigma$
larger and 1.3$\sigma$ lower than the low-redshift data measurements of Ref.
Cao and Ratra (2022).

Comparing the seven-parameter tilted flat $\Lambda$CDM$+A_{L}$ primary
cosmological parameter constraints for P18+lensing data and P18+lensing+non-
CMB data, shown in the lower half of Table 4, we see smaller changes in the
parameter values (the largest change is that $\Omega_{c}h^{2}$ is 0.47$\sigma$
smaller in the P18+lensing+non-CMB case, with the next largest being $n_{s}$
which is 0.33$\sigma$ larger) with all but the ln($10^{10}A_{s})$ error bars
being smaller in the P18+lensing+non-CMB case (the ln$(10^{10}A_{s})$ error
bar is unchanged and the largest decrease is 39% for the $\Omega_{c}h^{2}$
error bars). For the derived parameters, the largest changes are the
0.47$\sigma$ increase in $H_{0}$ and the 0.47$\sigma$ decrease in $\Omega_{m}$
relative to the P18+lensing data values, and the 42% smaller $\Omega_{m}$
error bar. For P18+lensing+non-CMB data in the varying $A_{L}$ case we measure
$\Omega_{m}=0.2998\pm 0.0053$ and $H_{0}=68.52\pm 0.42$ km s-1 Mpc-1 which are
consistent with many other measurements of these quantities and 0.27$\sigma$
larger and 0.93$\sigma$ lower than the low-redshift data measurements of Ref.
Cao and Ratra (2022).

The improvement in the fit to P18+lensing+non-CMB data when the $A_{L}$
parameter is allowed to vary, in the tilted flat $\Lambda$CDM$+A_{L}$ model,
is positive, as discussed in Sec. IV.2. We now find $A_{L}=1.089\pm 0.035$
which is now 2.5$\sigma$ away from the theoretically expected $A_{L}=1$,
larger than the 1.8$\sigma$ deviation for the P18+lensing case of Sec. IV.1.2;
the tendency of the non-CMB data is to push $A_{L}$ farther away from 1. The
inclusion of the $A_{L}$ parameter does not significantly affect the values of
the other six primary parameters, leaving them close to the values found for
the six parameter tilted flat $\Lambda$CDM model with $A_{L}=1$ (the largest
difference is for ln$(10^{10}A_{s})$, where it is 1.0$\sigma$ lower); it does
however increase the error bars, comparable to what happens in the P18+lensing
data case discussed in Sec. IV.1.2, with largest increase being 29% for
$\ln(10^{10}A_{s})$. In addition, in the case when $A_{L}$ is allowed to vary,
the derived parameters change somewhat and their error bars increase, with the
largest changes associated with $\sigma_{8}$, where it is now 1.2$\sigma$
smaller with a 29% larger error bar.

Adding non-CMB data to P18+lensing data strongly suppresses P18+lensing data
support for non-zero spatial curvature, (see Tables 6 and 7), except in the
case of the untilted non-flat $\Lambda$CDM model, for which
$\Omega_{k}=-0.0065\pm 0.0014$ (4.6$\sigma$ away from flat) and also for the
untilted non-flat $\Lambda$CDM$+A_{L}$ model where $\Omega_{k}=-0.0060\pm
0.0014$ (4.3$\sigma$ away from flat) (see Table 5).

Comparing the seven-parameter tilted non-flat $\Lambda$CDM Planck $P(q)$ and
new $P(q)$ primary cosmological parameter constraints for P18+lensing data and
P18+lensing+non-CMB data, shown in the upper halves of Tables 6 and 7, we see
that aside from $\Omega_{k}$ (discussed next) there are no significant changes
in parameter values [the largest change is that ln($10^{10}A_{s}$) is
0.73$\sigma$ (0.52$\sigma$) larger in the P18+lensing+non-CMB case, for the
Planck (new) $P(q)$] with all of the error bars being smaller in the
P18+lensing+non-CMB case [leaving aside $\Omega_{k}$ (discussed next) the
largest decrease is 18% (13%) for the log($10^{10}A_{s}$) error bar, for the
Planck (new) $P(q)$]. On the other hand, $\Omega_{k}$ changes significantly
when non-CMB data are added to the mix, becoming 1.6$\sigma$ (1.5$\sigma$)
larger, and closer to flat for the Planck (new) $P(q)$, with 74% (70%) smaller
error bars, now favoring open geometry over flat but only by 0.24$\sigma$
(0.18$\sigma$), respectively. For the derived parameters, the largest changes
are the 1.9$\sigma$ (1.9$\sigma$) increase in $H_{0}$ and the 1.9$\sigma$
(1.8$\sigma$) decrease in $\Omega_{m}$ relative to the P18+lensing data values
for the Planck (new) $P(q)$, with 78% (76%) smaller error bars for
$\Omega_{m}$ and 76% (73%) smaller error bars for $H_{0}$. For
P18+lensing+non-CMB data we find $\Omega_{m}=0.3051\pm 0.0053$ ($0.3054\pm
0.0051$) and $H_{0}=68.17\pm 0.55$ ($68.13\pm 0.54$) km s-1 Mpc-1 for the
Planck (new) $P(q)$, which are consistent with many other measurements of
these quantities and 0.57$\sigma$ (0.59$\sigma$) larger and 1.2$\sigma$
(1.2$\sigma$) lower, respectively, than the low-redshift data measurements of
Ref. Cao and Ratra (2022).

Comparing the eight-parameter tilted non-flat $\Lambda$CDM$+A_{L}$ Planck
(new) $P(q)$ primary cosmological parameter constraints for P18+lensing data
and P18+lensing+non-CMB data, shown in the lower half of Table 6 (7), we see
that there are approximately comparable differences to the tilted flat
$\Lambda$CDM$+A_{L}$ case. For the Planck (new) $P(q)$ case the largest change
is that $\Omega_{c}h^{2}$ is 0.49$\sigma$ (0.45$\sigma$) smaller in the
P18+lensing+non-CMB case, with the next largest being $\Omega_{b}h^{2}$
($n_{s}$) which is 0.34$\sigma$ (0.37$\sigma$) smaller, with most of the error
bars being smaller in the P18+lensing+non-CMB case (the largest decreases are
94% (89%) for $\Omega_{k}$ and 77% (77%) for $A_{L}$). From the
P18+lensing+non-CMB analyses, we measure $\Omega_{k}=-0.0002\pm 0.0017$ for
both $P(q)$ cases, both being only 0.12$\sigma$ away from flat spatial
hypersurfaces, very different from the P18 data alone results. For the derived
parameters, the largest change is the 0.18$\sigma$ (0.39$\sigma$) decrease in
$\Omega_{m}$ ($\sigma_{8}$) relative to the P18+lensing data value for the
Planck (new) $P(q)$, with 95% (93%) smaller error bars for $\Omega_{m}$ and
95% (94%) smaller error bars for $H_{0}$. For P18+lensing+non-CMB data we find
$\Omega_{m}=0.2998\pm 0.0055$ ($0.2999\pm 0.0055$) and $H_{0}=68.49\pm 0.56$
($68.48\pm 0.56$) km s-1 Mpc-1 for the Planck (new) $P(q)$, which are
consistent with many other measurements of these quantities and 0.27$\sigma$
(0.27$\sigma$) larger and 0.91$\sigma$ (0.92$\sigma$) lower, respectively,
than the low-redshift data measurements of Ref. Cao and Ratra (2022).

We will see in Sec. IV.2 that in both tilted non-flat $\Lambda$CDM$+A_{L}$
models the fit to P18+lensing+non-CMB data is positively better when the
$A_{L}$ parameter is allowed to vary compared to the $A_{L}=1$ case; this is
similar to what happens in the tilted flat $\Lambda$CDM$+A_{L}$ model. Also,
like the tilted flat $\Lambda$CDM$+A_{L}$ P18+lensing+non-CMB case, we
measure, from P18+lensing+non-CMB data, $A_{L}=1.090\pm 0.036$ and $1.088\pm
0.035$, for the Planck $P(q)$ and the new $P(q)$, respectively, which both
differ from the theoretically expected $A_{L}=1$ by 2.5$\sigma$. The inclusion
of the $A_{L}$ parameter does not significantly affect the values of the other
seven primary parameters, leaving them close to the values found for the
seven-parameter tilted non-flat $\Lambda$CDM models with $A_{L}=1$ [the
largest difference is for ln($10^{10}A_{s}$), where it is 1.0$\sigma$
(0.95$\sigma$) smaller for the Planck (new) $P(q)$]; it does however increase
the error bars, but less than what happens in the P18 alone and P18+lensing
data cases discussed in Secs. IV.1.1 and IV.1.2, with largest increase being
21% for ln($10^{10}A_{s}$) for both $P(q)$ cases. In addition, in the case
when $A_{L}$ is allowed to vary, the derived parameters change somewhat and
their error bars increase, with the largest changes associated with
$\sigma_{8}$, where it is now 1.3$\sigma$ (1.2$\sigma$) smaller for the Planck
(new) $P(q)$ with a 21% (22%) larger error bar.

When non-CMB data (that include $f\sigma_{8}$ data) are added to the mix and
the $A_{L}$ parameter is allowed to vary, $A_{L}>1$ is favored and there is a
decrease in the value of $\sigma_{8}$ compared to the $A_{L}=1$ case, which
helps to alleviate the corresponding tension. Since $A_{L}>1$ helps to resolve
the lensing anomaly, there is less or no need to increase the value of
$\Omega_{m}$ to predict more lensing. A lower value of $\Omega_{m}$ means less
structure formation in the past, consequently slightly alleviating the
$\sigma_{8}$ tension. While $\Omega_{k}$ plays a role at both early and late
times, the $A_{L}$ parameter only has an impact on CMB data. Since, as we
shall see in Sec. IV.1.6, non-CMB data prefer a flatter geometry than do P18
data, it is possible to understand why the evidence in favor of
$\Omega_{k}\neq 0$ subsides, while the evidence for $A_{L}>1$ does not, when
non-CMB data is added to the mix. A fairly large negative value of
$\Omega_{k}$ is required to resolve the P18 data lensing anomaly, thus
improving upon the performance shown by the tilted flat $\Lambda$CDM model,
however, such a large value of the curvature parameter is not supported by
lensing data or by non-CMB data. This fact raises the issue of whether it is
consistent to jointly use P18, lensing, and non-CMB data sets in the context
of the non-flat models. We try to answer this question, through the use of
different statistical criteria, in Sec. IV.3. Note that Figs. 4 and 7 show
that when P18+lensing+non-CMB data is used it is not necessary to consider
$A_{L}\neq 1$ in order to make the three sets of tilted model contours
overlap.

####  IV.1.4 Comparing P18, P18+lensing, and P18+lensing+non-CMB data
cosmological constraints for each model

Cosmological parameter contour plots allow us to easily see the degree of
correlation between the two variables. If the two variables are more
correlated then the corresponding constraint contours are more line-like, on
the other hand, if they are less correlated the contours are broader and
enclose 2-dimensional areas. In this subsubsection we comment on how the
constraint contours, for each cosmological model, change depending on whether
we consider P18, P18+lensing, or P18+lensing+non-CMB data. Figures 8-15 show,
for each of the eight cosmological models we study, the cosmological parameter
constraints for P18, P18+lensing, and P18+lensing+non-CMB data. The constraint
contours shrink as more data is included in the analysis used to determine
them.

From Fig. 8 for the six-parameter tilted flat $\Lambda$CDM we see that there
are significant overlaps between the contours obtained in the three data sets
considered. Along with the results discussed in Secs. IV.1.2 and IV.1.3 this
is an indication that there is not significant tension between P18,
P18+lensing, and P18+lensing+non-CMB data when these data are analyzed in the
tilted flat $\Lambda$CDM model. The $\Omega_{m}$-$H_{0}$ panel contours
indicate that these two parameters are strongly correlated. The inclusion of
lensing data and/or non-CMB data, which provide information about the late-
time Universe, partially breaks this correlation and induces a shift in the
one-dimensional posterior distributions of not only these two parameters but
also other parameters. Non-CMB data cause a larger shift.

For the six-parameter untilted non-flat $\Lambda$CDM model (see Fig. 9)
constraint contours determined from the three different data sets overlap only
for some parameters. In particular, for constraint contours in panels that
involve $\Omega_{k}$, $\Omega_{m}$, or $H_{0}$ there is no overlap between
those determined using P18 data and those determined using P18+lensing+non-CMB
data (there are larger than 2$\sigma$ differences between these contours when
one of these three parameters are involved and the differences are larger when
two of these three parameters are involved), and there is only a slight amount
of overlap between the P18 data contours and the P18+lensing data contours.
The $\Omega_{m}$-$H_{0}$-$\Omega_{k}$ geometrical degeneracy is prominent for
P18 data and is clearly seen in the $\Omega_{k}$-$\Omega_{m}$,
$\Omega_{k}$-$H_{0}$, and $\Omega_{m}$-$H_{0}$ panels, as these three
parameters are strongly correlated. Including lensing data and/or non-CMB data
partially breaks this degeneracy causing significant shifts in the one-
dimensional posterior distributions of not only these three parameters but
also other parameters. The shifts are bigger here than in the tilted flat
$\Lambda$CDM model and indicate significant tension between the data sets,
especially between the P18 and P18+lensing+non-CMB data sets, when they are
analyzed in the untilted non-flat $\Lambda$CDM model. Non-CMB data again
appear to cause the larger shift. As discussed in more detail in Sec. IV.3,
these shifts mean that P18 and non-CMB data are mutually inconsistent in the
untilted non-flat $\Lambda$CDM model and so cannot be jointly used to derive
cosmological parameter constraints in this model.

Similar, but quantitatively less discrepant, results are obtained for the
seven-parameter tilted non-flat $\Lambda$CDM Planck $P(q)$ and the tilted non-
flat $\Lambda$CDM new $P(q)$ models (see Figs. 10 and 11). The differences
between the untilted non-flat and tilted non-flat results is likely a
consequence of the additional $n_{s}$ parameter in the tilted non-flat models.
In the tilted non-flat models, the more-discrepant P18 and P18+lensing+non-CMB
data constraint contours overlap in all panels for pairs of the six primary
cosmological parameters excluding the $\Omega_{k}$ parameter as well as the
derived $\Omega_{m}$ and $H_{0}$ parameters. The differences are larger in the
Planck $P(q)$ case than in the new $P(q)$ case, largest for $H_{0}$, smallest
for $\Omega_{m}$, with $\Omega_{k}$ being in-between. In the new $P(q)$ case,
the 2$\sigma$ contours overlap for $\Omega_{m}$ and almost overlap for
$\Omega_{k}$. These results may be an indication of the tension found, in the
context of the tilted non-flat models, between P18 data and the BAO data set.
We shall study this tension in more detail in Sec. IV.3. As in the untilted
non-flat $\Lambda$CDM model, the geometrical degeneracy between
$\Omega_{m}$-$H_{0}$-$\Omega_{k}$ affects the tilted non-flat models. Again,
including lensing data and/or non-CMB data partially breaks this degeneracy
causing significant shifts in the one-dimensional posterior distributions of
not only these three parameters but also other parameters. The shifts are
bigger here than in the tilted flat $\Lambda$CDM model and but smaller than in
the untilted non-flat $\Lambda$CDM model, but still indicate some tension
between the data sets, especially between the P18 and P18+lensing+non-CMB data
sets, especially when they are analyzed in the tilted non-flat $\Lambda$CDM
Planck $P(q)$ model. Non-CMB data again appear to cause the larger shift.

When the $A_{L}$ parameter is allowed to vary, the three different sets of
constraint contours overlap in all four models (see Figs. 12–15). In the non-
flat models there now is a bigger degeneracy between the cosmological
parameters $\Omega_{m}$-$H_{0}$-$\Omega_{k}$-$A_{L}$ which causes the
constraint contours to expand relative to the $A_{L}=1$ case, especially for
P18 data. For some parameters in the untilted non-flat $\Lambda$CDM model and
the tilted non-flat $\Lambda$CDM new $P(q)$ model we observe a bimodal
distribution when only P18 data is used, and the same parameters in the tilted
non-flat $\Lambda$CDM Planck $P(q)$ model have an almost bimodal distribution
for P18 data. These bimodalities are likely a consequences of the above-
mentioned geometrical degeneracy.

Table 8: Mean and 68.3% confidence limits of tilted flat $\Lambda\textrm{CDM}$
(+$A_{L}$) model parameters constrained by TT,TE,EE+lowE (P18), BAO, and BAO′
data. $H_{0}$ has units of km s-1 Mpc-1.

| Tilted flat $\Lambda$CDM model
---|---
Parameter | P18 | P18+BAO | BAO | P18+BAO′ | BAO′
$\Omega_{b}h^{2}$ | $0.02236\pm 0.00015$ | $0.02243\pm 0.00013$ | $0.043\pm 0.016$ | $0.02241\pm 0.00014$ | $0.043\pm 0.016$
$\Omega_{c}h^{2}$ | $0.1202\pm 0.0014$ | $0.11926\pm 0.00097$ | $0.163\pm 0.042$ | $0.11946\pm 0.00098$ | $0.168\pm 0.044$
$100\theta_{\textrm{MC}}$ | $1.04090\pm 0.00031$ | $1.04102\pm 0.00029$ | $1.054\pm 0.026$ | $1.04099\pm 0.00029$ | $1.059\pm 0.025$
$\tau$ | $0.0542\pm 0.0079$ | $0.0581\pm 0.0081$ | $0.0542$ | $0.0555\pm 0.0077$ | $0.0542$
$n_{s}$ | $0.9649\pm 0.0043$ | $0.9673\pm 0.0037$ | $0.9649$ | $0.9665\pm 0.0038$ | $0.9649$
$\ln(10^{10}A_{s})$ | $3.044\pm 0.016$ | $3.051\pm 0.017$ | $3.01\pm 0.27$ | $3.045\pm 0.016$ | $3.044$
$H_{0}$ | $67.28\pm 0.61$ | $67.70\pm 0.43$ | $83\pm 12$ | $67.60\pm 0.44$ | $83\pm 12$
$\Omega_{m}$ | $0.3165\pm 0.0084$ | $0.3106\pm 0.0058$ | $0.294\pm 0.015$ | $0.3119\pm 0.0059$ | $0.300\pm 0.016$
$\sigma_{8}$ | $0.8118\pm 0.0074$ | $0.8119\pm 0.0073$ | $0.874\pm 0.037$ | $0.8102\pm 0.0070$ | $0.92\pm 0.12$
$\chi_{\textrm{min}}^{2}$ (Total) | $2765.80$ | $2786.66$ | $15.92$ | $2777.75$ | $10.98$
$\chi_{\textrm{min}}^{2}$ (BAO/BAO′) | $\cdots$ | $20.22$ | $15.92$ | $11.61$ | $10.98$
$\chi_{\textrm{BAO/BAO}^{\prime}}^{2}$ (at P18 B-F) | $\cdots$ | $22.24$ | $22.24$ | $12.58$ | $12.58$
DIC | $2817.93$ | $2839.25$ | $21.93$ | $2829.61$ | $14.93$
$\textrm{AIC}_{c}$ | $2819.80$ | $2840.66$ | $27.56$ | $2831.75$ | $19.98$
| Tilted flat $\Lambda$CDM+$A_{L}$ model
Parameter | P18 | P18+BAO | BAO | P18+BAO′ | BAO′
$\Omega_{b}h^{2}$ | $0.02259\pm 0.00017$ | $0.02258\pm 0.00015$ | $0.043\pm 0.015$ | $0.02256\pm 0.00014$ | $0.045\pm 0.013$
$\Omega_{c}h^{2}$ | $0.1180\pm 0.0015$ | $0.1183\pm 0.0010$ | $0.163\pm 0.042$ | $0.1185\pm 0.0010$ | $0.177\pm 0.042$
$100\theta_{\textrm{MC}}$ | $1.04114\pm 0.00032$ | $1.04113\pm 0.00029$ | $1.055\pm 0.024$ | $1.04109\pm 0.00030$ | $1.065\pm 0.018$
$\tau$ | $0.0496\pm 0.0082$ | $0.0522\pm 0.0080$ | $0.0496$ | $0.0492\pm 0.0084$ | $0.0496$
$n_{s}$ | $0.9710\pm 0.0050$ | $0.9705\pm 0.0038$ | $0.9710$ | $0.9698\pm 0.0039$ | $0.9710$
$\ln(10^{10}A_{s})$ | $3.030\pm 0.017$ | $3.036\pm 0.017$ | $3.00\pm 0.27$ | $3.030\pm 0.018$ | $3.030$
$A_{L}$ | $1.181\pm 0.067$ | $1.170\pm 0.060$ | $1.181$ | $1.174\pm 0.061$ | $1.181$
$H_{0}$ | $68.31\pm 0.71$ | $68.21\pm 0.46$ | $83\pm 12$ | $68.11\pm 0.47$ | $85\pm 10$
$\Omega_{m}$ | $0.3029\pm 0.0093$ | $0.3042\pm 0.0060$ | $0.294\pm 0.015$ | $0.3055\pm 0.0061$ | $0.302\pm 0.017$
$\sigma_{8}$ | $0.7997\pm 0.0088$ | $0.8031\pm 0.0077$ | $0.875\pm 0.037$ | $0.8011\pm 0.0079$ | $0.93\pm 0.11$
$\chi_{\textrm{min}}^{2}$ (Total) | $2756.12$ | $2776.71$ | $15.91$ | $2767.77$ | $10.98$
$\chi_{\textrm{min}}^{2}$ (BAO/BAO′) | $\cdots$ | $20.47$ | $15.91$ | $11.37$ | $10.98$
$\chi_{\textrm{BAO/BAO}^{\prime}}^{2}$ (at P18 B-F) | $\cdots$ | $20.78$ | $20.78$ | $11.88$ | $11.88$
DIC | $2812.41$ | $2832.92$ | $21.83$ | $2823.77$ | $15.04$
$\Delta\textrm{DIC}$ | $-5.52$ | $-6.33$ | $-0.10$ | $-5.90$ | $0.11$
$\textrm{AIC}_{c}$ | $2812.12$ | $2832.71$ | $27.55$ | $2823.77$ | $19.98$
$\Delta\textrm{AIC}_{c}$ | $-7.68$ | $-7.95$ | $-0.01$ | $-7.98$ | $0.00$

Note: $\Delta\textrm{DIC}$ ($\Delta\textrm{AIC}_{c}$) indicates an excess
value relative to that of the tilted flat $\Lambda$CDM model constrained with
the same data. The number of free parameters of the tilted flat $\Lambda$CDM
model is 27 for P18, P18+BAO, and P18+BAO′ data sets (including 21 internal
calibration parameters), 4 for BAO data, and 3 for BAO′ data.

Table 9: Mean and 68.3% confidence limits of untilted non-flat
$\Lambda\textrm{CDM}$ (+$A_{L}$) model parameters constrained by TT,TE,EE+lowE
(P18), BAO, and BAO′ data. $H_{0}$ has units of km s-1 Mpc-1.

| Untilted non-flat $\Lambda$CDM model
---|---
Parameter | P18 | P18+BAO | BAO | P18+BAO′ | BAO′
$\Omega_{b}h^{2}$ | $0.02320\pm 0.00015$ | $0.02298\pm 0.00014$ | $0.040\pm 0.015$ | $0.02299\pm 0.00014$ | $0.040\pm 0.015$
$\Omega_{c}h^{2}$ | $0.11098\pm 0.00088$ | $0.11184\pm 0.00089$ | $0.175\pm 0.046$ | $0.11171\pm 0.00089$ | $0.175\pm 0.047$
$100\theta_{\textrm{MC}}$ | $1.04204\pm 0.00030$ | $1.04188\pm 0.00029$ | $1.16\pm 0.13$ | $1.04189\pm 0.00030$ | $1.13\pm 0.12$
$\tau$ | $0.0543\pm 0.0091$ | $0.077\pm 0.010$ | $0.0543$ | $0.073\pm 0.010$ | $0.0543$
$\Omega_{k}$ | $-0.095\pm 0.024$ | $-0.0066\pm 0.0015$ | $-0.047\pm 0.059$ | $-0.0074\pm 0.0016$ | $-0.034\pm 0.057$
$\ln(10^{10}A_{s})$ | $3.021\pm 0.019$ | $3.069\pm 0.021$ | $2.70\pm 0.43$ | $3.059\pm 0.021$ | $3.021$
$H_{0}$ | $47.1\pm 3.2$ | $67.77\pm 0.60$ | $84\pm 12$ | $67.46\pm 0.63$ | $83\pm 12$
$\Omega_{m}$ | $0.617\pm 0.082$ | $0.2950\pm 0.0055$ | $0.303\pm 0.019$ | $0.2975\pm 0.0057$ | $0.307\pm 0.019$
$\sigma_{8}$ | $0.730\pm 0.017$ | $0.7977\pm 0.0093$ | $0.850\pm 0.048$ | $0.7927\pm 0.0090$ | $1.00\pm 0.18$
$\chi_{\textrm{min}}^{2}$ (Total) | $2789.77$ | $2837.93$ | $15.91$ | $2828.81$ | $10.67$
$\chi_{\textrm{min}}^{2}$ (BAO/BAO′) | $\cdots$ | $20.34$ | $15.91$ | $11.68$ | $10.67$
$\chi_{\textrm{BAO/BAO}^{\prime}}^{2}$ (at P18 B-F) | $\cdots$ | $1987.47$ | $1987.47$ | $1765.08$ | $1765.08$
DIC | $2847.14$ | $2895.04$ | $24.31$ | $2884.90$ | $17.55$
$\Delta\textrm{DIC}$ | $29.21$ | $55.79$ | $2.38$ | $55.29$ | $2.62$
$\textrm{AIC}_{c}$ | $2843.77$ | $2891.93$ | $31.91$ | $2882.81$ | $24.39$
$\Delta\textrm{AIC}_{c}$ | $23.97$ | $51.27$ | $4.35$ | $51.06$ | $4.41$
| Untilted non-flat $\Lambda$CDM+$A_{L}$ model
Parameter | P18 | P18+BAO | BAO | P18+BAO′ | BAO′
$\Omega_{b}h^{2}$ | $0.02320\pm 0.00015$ | $0.02318\pm 0.00015$ | $0.041\pm 0.015$ | $0.02320\pm 0.00015$ | $0.042\pm 0.014$
$\Omega_{c}h^{2}$ | $0.11097\pm 0.00087$ | $0.11117\pm 0.00086$ | $0.176\pm 0.045$ | $0.11095\pm 0.00087$ | $0.180\pm 0.044$
$100\theta_{\textrm{MC}}$ | $1.04202\pm 0.00030$ | $1.04198\pm 0.00030$ | $1.16\pm 0.13$ | $1.04199\pm 0.00030$ | $1.14\pm 0.12$
$\tau$ | $0.0540\pm 0.0087$ | $0.0598\pm 0.0087$ | $0.0540$ | $0.0557\pm 0.0089$ | $0.0540$
$\Omega_{k}$ | $-0.12\pm 0.12$ | $-0.0064\pm 0.0015$ | $-0.050\pm 0.060$ | $-0.0073\pm 0.0015$ | $-0.035\pm 0.058$
$\ln(10^{10}A_{s})$ | $3.020\pm 0.018$ | $3.033\pm 0.018$ | $2.68\pm 0.41$ | $3.023\pm 0.019$ | $3.020$
$A_{L}$ | $1.08\pm 0.27$ | $1.310\pm 0.062$ | $1.08$ | $1.319\pm 0.063$ | $1.08$
$H_{0}$ | $52\pm 18$ | $68.27\pm 0.61$ | $84\pm 12$ | $67.93\pm 0.62$ | $84\pm 11$
$\Omega_{m}$ | $0.70\pm 0.42$ | $0.2897\pm 0.0054$ | $0.304\pm 0.018$ | $0.2921\pm 0.0055$ | $0.307\pm 0.020$
$\sigma_{8}$ | $0.721\pm 0.053$ | $0.7799\pm 0.0083$ | $0.848\pm 0.049$ | $0.7750\pm 0.0085$ | $1.01\pm 0.18$
$\chi_{\textrm{min}}^{2}$ (Total) | $2787.76$ | $2809.82$ | $15.89$ | $2799.18$ | $10.68$
$\chi_{\textrm{min}}^{2}$ (BAO/BAO′) | $\cdots$ | $21.96$ | $15.89$ | $11.38$ | $10.68$
$\chi_{\textrm{BAO/BAO}^{\prime}}^{2}$ (at P18 B-F) | $\cdots$ | $106.63$ | $106.63$ | $80.18$ | $80.18$
DIC | $2846.45$ | $2869.28$ | $24.63$ | $2857.90$ | $17.89$
$\Delta\textrm{DIC}$ | $28.52$ | $30.03$ | $2.70$ | $28.29$ | $2.96$
$\textrm{AIC}_{c}$ | $2843.76$ | $2865.82$ | $31.89$ | $2855.18$ | $24.39$
$\Delta\textrm{AIC}_{c}$ | $23.96$ | $25.16$ | $4.33$ | $23.43$ | $4.41$

Note: $\Delta\textrm{DIC}$ ($\Delta\textrm{AIC}_{c}$) indicates an excess
value relative to that of the tilted flat $\Lambda$CDM model constrained with
the same data.

Table 10: Mean and 68.3% confidence limits of Planck-$P(q)$-based tilted non-
flat $\Lambda\textrm{CDM}$ (+$A_{L}$) model parameters constrained by
TT,TE,EE+lowE (P18), BAO, and BAO′ data. $H_{0}$ has units of km s-1 Mpc-1.

| Tilted non-flat $\Lambda$CDM model [Planck $P(q)$]
---|---
Parameter | P18 | P18+BAO | BAO | P18+BAO′ | BAO′
$\Omega_{b}h^{2}$ | $0.02260\pm 0.00017$ | $0.02241\pm 0.00015$ | $0.040\pm 0.015$ | $0.02241\pm 0.00015$ | $0.040\pm 0.016$
$\Omega_{c}h^{2}$ | $0.1181\pm 0.0015$ | $0.1195\pm 0.0014$ | $0.174\pm 0.047$ | $0.1195\pm 0.0014$ | $0.172\pm 0.047$
$100\theta_{\textrm{MC}}$ | $1.04116\pm 0.00032$ | $1.04099\pm 0.00032$ | $1.15\pm 0.13$ | $1.04099\pm 0.00032$ | $1.13\pm 0.12$
$\tau$ | $0.0483\pm 0.0083$ | $0.0578\pm 0.0077$ | $0.0483$ | $0.0550\pm 0.0078$ | $0.0483$
$\Omega_{k}$ | $-0.043\pm 0.017$ | $0.0005\pm 0.0018$ | $-0.046\pm 0.060$ | $-0.0001\pm 0.0018$ | $-0.033\pm 0.055$
$n_{s}$ | $0.9706\pm 0.0047$ | $0.9667\pm 0.0045$ | $0.9706$ | $0.9666\pm 0.0044$ | $0.9706$
$\ln(10^{10}A_{s})$ | $3.027\pm 0.017$ | $3.051\pm 0.016$ | $2.74\pm 0.43$ | $3.044\pm 0.016$ | $3.027$
$H_{0}$ | $54.5\pm 3.6$ | $67.83\pm 0.58$ | $83\pm 12$ | $67.58\pm 0.62$ | $83\pm 12$
$\Omega_{m}$ | $0.481\pm 0.062$ | $0.3100\pm 0.0060$ | $0.303\pm 0.019$ | $0.3122\pm 0.0063$ | $0.306\pm 0.019$
$\sigma_{8}$ | $0.775\pm 0.015$ | $0.8130\pm 0.0079$ | $0.850\pm 0.049$ | $0.8099\pm 0.0081$ | $0.98\pm 0.17$
$\chi_{\textrm{min}}^{2}$ (Total) | $2754.73$ | $2786.20$ | $15.88$ | $2776.90$ | $10.68$
$\chi_{\textrm{min}}^{2}$ (BAO/BAO′) | $\cdots$ | $20.09$ | $15.88$ | $11.71$ | $10.68$
$\chi_{\textrm{BAO/BAO}^{\prime}}^{2}$ (at P18 B-F) | $\cdots$ | $665.90$ | $665.90$ | $582.59$ | $582.59$
DIC | $2810.59$ | $2840.62$ | $24.34$ | $2832.28$ | $17.58$
$\Delta\textrm{DIC}$ | $-7.34$ | $1.37$ | $2.41$ | $2.67$ | $2.65$
$\textrm{AIC}_{c}$ | $2810.73$ | $2842.20$ | $31.88$ | $2832.90$ | $24.39$
$\Delta\textrm{AIC}_{c}$ | $-9.07$ | $1.54$ | $4.32$ | $1.15$ | $4.41$
| Tilted non-flat $\Lambda$CDM+$A_{L}$ model [Planck $P(q)$]
Parameter | P18 | P18+BAO | BAO | P18+BAO′ | BAO′
$\Omega_{b}h^{2}$ | $0.02258\pm 0.00017$ | $0.02260\pm 0.00017$ | $0.041\pm 0.014$ | $0.02262\pm 0.00017$ | $0.044\pm 0.013$
$\Omega_{c}h^{2}$ | $0.1183\pm 0.0015$ | $0.1180\pm 0.0015$ | $0.174\pm 0.045$ | $0.1178\pm 0.0015$ | $0.182\pm 0.043$
$100\theta_{\textrm{MC}}$ | $1.04116\pm 0.00033$ | $1.04115\pm 0.00033$ | $1.16\pm 0.14$ | $1.04118\pm 0.00032$ | $1.12\pm 0.11$
$\tau$ | $0.0478\pm 0.0081$ | $0.0522\pm 0.0081$ | $0.0478$ | $0.0496\pm 0.0085$ | $0.0478$
$\Omega_{k}$ | $-0.130\pm 0.095$ | $-0.0004\pm 0.0018$ | $-0.045\pm 0.063$ | $-0.0012\pm 0.0018$ | $-0.026\pm 0.054$
$n_{s}$ | $0.9704\pm 0.0048$ | $0.9712\pm 0.0047$ | $0.9704$ | $0.9716\pm 0.0047$ | $0.9704$
$\ln(10^{10}A_{s})$ | $3.027\pm 0.017$ | $3.035\pm 0.017$ | $2.74\pm 0.45$ | $3.029\pm 0.018$ | $3.027$
$A_{L}$ | $0.88\pm 0.15$ | $1.170\pm 0.061$ | $0.88$ | $1.178\pm 0.061$ | $0.88$
$H_{0}$ | $45\pm 11$ | $68.13\pm 0.60$ | $84\pm 11$ | $67.85\pm 0.61$ | $85\pm 10$
$\Omega_{m}$ | $0.80\pm 0.35$ | $0.3044\pm 0.0062$ | $0.303\pm 0.019$ | $0.3064\pm 0.0063$ | $0.307\pm 0.019$
$\sigma_{8}$ | $0.733\pm 0.045$ | $0.8020\pm 0.0089$ | $0.851\pm 0.048$ | $0.7983\pm 0.0091$ | $0.99\pm 0.16$
$\chi_{\textrm{min}}^{2}$ (Total) | $2754.99$ | $2776.32$ | $15.91$ | $2767.04$ | $10.73$
$\chi_{\textrm{min}}^{2}$ (BAO/BAO′) | $\cdots$ | $20.38$ | $15.91$ | $11.22$ | $10.73$
$\chi_{\textrm{BAO/BAO}^{\prime}}^{2}$ (at P18 B-F) | $\cdots$ | $593.77$ | $593.77$ | $518.08$ | $518.08$
DIC | $2811.63$ | $2835.10$ | $24.31$ | $2825.27$ | $17.54$
$\Delta\textrm{DIC}$ | $-6.30$ | $-4.15$ | $2.38$ | $-4.34$ | $2.61$
$\textrm{AIC}_{c}$ | $2812.99$ | $2834.32$ | $31.91$ | $2825.04$ | $24.45$
$\Delta\textrm{AIC}_{c}$ | $-6.81$ | $-6.34$ | $4.35$ | $-6.71$ | $4.47$

Note: $\Delta\textrm{DIC}$ ($\Delta\textrm{AIC}_{c}$) indicates an excess
value relative to that of the tilted flat $\Lambda$CDM model constrained with
the same data.

Table 11: Mean and 68.3% confidence limits of tilted new-$P(q)$-based non-flat
$\Lambda\textrm{CDM}$ (+$A_{L}$) model parameters with the new $P(q)$
constrained by TT,TE,EE+lowE (P18), BAO, and BAO′ data. $H_{0}$ has units of
km s-1 Mpc-1.

| Tilted non-flat $\Lambda$CDM model [new $P(q)$]
---|---
Parameter | P18 | P18+BAO | BAO | P18+BAO′ | BAO′
$\Omega_{b}h^{2}$ | $0.02255\pm 0.00017$ | $0.02242\pm 0.00015$ | $0.039\pm 0.015$ | $0.02243\pm 0.00016$ | $0.041\pm 0.016$
$\Omega_{c}h^{2}$ | $0.1188\pm 0.0015$ | $0.1194\pm 0.0014$ | $0.173\pm 0.048$ | $0.1193\pm 0.0014$ | $0.177\pm 0.048$
$100\theta_{\textrm{MC}}$ | $1.04109\pm 0.00032$ | $1.04100\pm 0.00032$ | $1.16\pm 0.14$ | $1.04102\pm 0.00032$ | $1.13\pm 0.12$
$\tau$ | $0.0525\pm 0.0083$ | $0.0582\pm 0.0081$ | $0.0525$ | $0.0562\pm 0.0080$ | $0.0525$
$\Omega_{k}$ | $-0.033\pm 0.014$ | $0.0003\pm 0.0018$ | $-0.051\pm 0.061$ | $-0.0004\pm 0.0018$ | $-0.032\pm 0.059$
$n_{s}$ | $0.9654\pm 0.0045$ | $0.9665\pm 0.0043$ | $0.9654$ | $0.9665\pm 0.0043$ | $0.9654$
$\ln(10^{10}A_{s})$ | $3.039\pm 0.017$ | $3.051\pm 0.016$ | $2.72\pm 0.45$ | $3.046\pm 0.016$ | $3.039$
$H_{0}$ | $56.9\pm 3.6$ | $67.79\pm 0.59$ | $83\pm 12$ | $67.52\pm 0.61$ | $83\pm 12$
$\Omega_{m}$ | $0.444\pm 0.055$ | $0.3102\pm 0.0060$ | $0.304\pm 0.019$ | $0.3124\pm 0.0063$ | $0.307\pm 0.020$
$\sigma_{8}$ | $0.786\pm 0.014$ | $0.8128\pm 0.0079$ | $0.846\pm 0.048$ | $0.8098\pm 0.0080$ | $0.99\pm 0.18$