Figure 33: Likelihood distributions of non-flat XCDM$+A_{L}$ [new $P(q)$]
model parameters favored by P18, P18+lensing, and P18+lensing+non-CMB (new)
data sets. Note that in the model for the P18 data set, a prior of $A_{L}>0.8$
is applied.

####  IV.2.4 P18+lensing+non-CMB (new) cosmological constraints

Seven-parameter flat XCDM model cosmological parameter constraints obtained
from analyses of P18+lensing and P18+lensing+non-CMB (new) data are listed in
Table 11 and shown in Figs. 22 and 28. Both sets of primary parameter values
are similar, with largest differences for $\Omega_{c}h^{2}$ ($+0.65\sigma$),
$\tau$ ($-0.50\sigma$), and $\ln(10^{10}A_{s})$ ($-0.47\sigma$). The error
bars decrease when non-CMB (new) data are added to the mix, with the $-26$%
decrease for $\Omega_{c}h^{2}$ being the largest. The equation of state
parameter value from P18+lensing data, $w=-1.55\pm 0.26$, is $-2.15\sigma$
away from the P18+lensing+non-CMB (new) data value, $w=-0.990\pm 0.023$, with
the error bars of the P18+lensing+non-CMB (new) data value being 11 times
smaller than the P18+lensing data value. The P18+lensing data value favors
phantom-like behavior at 2.12$\sigma$ significance while the P18+lensing+non-
CMB (new) data value signifies 0.43$\sigma$ support for quintessence-like
dynamical dark energy. As for the derived parameters $H_{0}$, $\Omega_{m}$,
and $\sigma_{8}$, we find some non-negligible differences at 1.97$\sigma$,
$-2.23\sigma$, and 2.17$\sigma$ respectively. Major reductions in the size of
the error bars are observed for these parameters when we move from P18+lensing
data to P18+lensing+non-CMB (new) data, in particular the error bars are 14.6,
7.7, and 8.0 times smaller, respectively.

If we examine the lower half of Table 11 and Figs. 23 and 29 we can compare
the results obtained for the eight-parameter flat XCDM+$A_{L}$ cosmological
model when analyzing P18+lensing and P18+lensing+non-CMB (new) data. Again,
both sets of primary parameter values are similar. In particular, the
$\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, $100\theta_{\text{MC}}$, and $n_{s}$
values differ by $-0.59\sigma$, $+0.86\sigma$, $-0.39\sigma$, and
$-0.66\sigma$, respectively. The size of the error bars is also affected and,
as expected, including the non-CMB (new) data results in smaller error bars,
particularly for the parameters $\Omega_{b}h^{2}$ ($-21.43$%),
$\Omega_{c}h^{2}$ ($-36.36$%), and $n_{s}$ ($-22.50$%). When P18+lensing data
are used we obtain $w=-1.34^{+0.26}_{-0.51}$, which is 1.31$\sigma$ away from
$w=-1$, while from P18+lensing+non-CMB (new) data we obtain $w=-0.968\pm
0.024$ which deviates from $w=-1$ by 1.33$\sigma$. The two data sets favor
different behaviors of the dark energy equation of state parameter (phantom-
like vs. quintessence-like) and the two values differ at $-1.42\sigma$ with
the $w$ error bars becoming 15 times smaller when moving from P18+lensing to
P18+lensing+non-CMB (new) data. Regarding the lensing consistency parameter
$A_{L}$, with P18+lensing data we find $A_{L}=1.054^{+0.039}_{-0.059}$
(0.92$\sigma$ away from $A_{L}=1$), whereas with P18+lensing+non-CMB (new)
data we get $A_{L}=1.101\pm 0.037$ (2.73$\sigma$ in favor of $A_{L}>1$), where
the two values differ by $-0.87\sigma$. In this case, the reduction in the
size of the error bars is $-48.65$%. The values of the derived parameters
$\Omega_{m}$ and $\sigma_{8}$ differ by $-0.76\sigma$ and 0.95$\sigma$,
respectively, with the error bars obtained with P18+lensing+non-CMB (new) data
being 13 and 10 times smaller than the ones obtained with P18+lensing data.

Looking at Table 11 we can compare the results obtained from P18+lensing+non-
CMB (new) data for the seven-parameter flat XCDM model (upper half of the
table and Figs. 22 and 28) and the eight-parameter flat XCDM+$A_{L}$ model
(lower half of the table and Figs. 23 and 29). We observe some differences in
the values of the primary parameter, in particular for $\Omega_{c}h^{2}$,
$\tau$, $n_{s}$, and $\ln(10^{10}A_{s})$ the differences are at 1.03$\sigma$,
0.72$\sigma$, $-0.78\sigma$, and $-0.97\sigma$, respectively. As for the
equation of state parameter of dark energy the difference between the two
values is $-0.66\sigma$. There are no significant increases in the error bars
when moving from the flat XCDM model to the flat XCDM+$A_{L}$ model, the
largest being those of $\tau$ (10%) and $\ln(10^{10}A_{s})$ (12%), whereas for
$w$ the error bars increase is 4.2%. In regard to the derived parameters
$H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ the mean values differ at
0.022$\sigma$, $0.31\sigma$, and 1.39$\sigma$, while only the error bars of
$\sigma_{8}$ increase (by 19%).

Comparing the eight-parameter non-flat XCDM Planck [new] $P(q)$ primary
cosmological parameter constraints for P18+lensing data and for
P18+lensing+non-CMB (new) data, listed in Table 12 [13] and shown in Figs. 24
and 30 [26 and 32], we observe small differences, all less than 1$\sigma$,
with those for $\tau$ and $\ln(10^{10}A_{s})$ being the largest at
$-0.71\sigma$ [$-0.59\sigma$] and $-0.79\sigma$ [$-0.68\sigma$] respectively.
The curvature parameter value obtained using P18+lensing data is
$\Omega_{k}=-0.011\pm 0.017$ [$-0.008\pm 0.010$] a 0.65$\sigma$ [0.80$\sigma$]
evidence in favor of closed geometry, whereas for P18+lensing+non-CMB (new)
data we get $\Omega_{k}=0.0016\pm 0.0019$ [$0.0014\pm 0.0020$] a 0.84$\sigma$
[0.70$\sigma$] preference for open geometry. The P18+lensing data and the
P18+lensing+non-CMB (new) data $\Omega_{k}$ values differ by $-0.74\sigma$
[$-0.92\sigma$]. It is interesting that when non-CMB (new) data are added to
the mix, the preference changes from a closed Universe to an open one; we
follow up on this point in the next paragraph. Regarding the dark energy
equation of state parameter value, from P18+lensing data we get $w=-1.28\pm
0.45$ [$-1.27\pm 0.41$] which differs by $-0.66\sigma$ [$-0.70\sigma$] from
the result obtained with P18+lensing+non-CMB (new) data, $w=-0.980\pm 0.026$
[$-0.982\pm 0.026$]. While the P18+lensing data result is 0.62$\sigma$
[0.66$\sigma$] in favor of phantom-like behavior, the P18+lensing+non-CMB
(new) data result prefers quintessence-like behavior at 0.73$\sigma$
[0.69$\sigma$]. The P18+lensing and P18+lensing+non-CMB (new) data values of
the derived parameters $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ differ at
0.34$\sigma$ [0.43$\sigma$], $-0.052\sigma$ [$-0.15\sigma$], and 0.54$\sigma$
[0.58$\sigma$], respectively, with error bars 23 [21], 25 [20], and 14 [12]
times smaller when P18+lensing+non-CMB (new) data are employed.

From Table 12 [13], we see that for the non-flat XCDM Planck [new] $P(q)$
model non-CMB (new) data favor closed geometry at 2.77$\sigma$ [2.24$\sigma$],
P18 data favor closed geometry at 1.17$\sigma$ [1.17$\sigma$], P18+lensing
data favor closed geometry at 0.85$\sigma$ [0.82$\sigma$], while joint
P18+non-CMB (new) data favor open geometry at 0.89$\sigma$ [0.80$\sigma$], and
joint P18+lensing+non-CMB (new) data favor open geometry at 0.84$\sigma$
[0.70$\sigma$]. As in the non-flat $\Lambda$CDM models, [10], this is likely
due to the $\Omega_{m}-\Omega_{k}-H_{0}$ degeneracy and the fact that P18 data
favor a smaller $H_{0}$ and a larger $\Omega_{m}$ than do non-CMB (new) data,
see Table 12 [13]. This point is also relevant for the non-flat XCDM$+A_{L}$
models discussed below.

We can compare the cosmological constraints obtained for the nine-parameter
non-flat XCDM Planck [new] $P(q)+A_{L}$ model from P18+lensing data and
P18+lensing+non-CMB (new) data by looking at the lower half of Table 12 [Table
13] and Figs. 25 and 31 [27 and 33]. We do not observe significant differences
in the values of the primary parameters, with $\Omega_{b}h^{2}$,
$\Omega_{c}h^{2}$, and $n_{s}$ showing the largest differences at
$-0.39\sigma$ [$-0.34\sigma$], 0.44$\sigma$ [0.47$\sigma$], and $-0.42\sigma$
[$-0.57\sigma$], respectively. As for the error bars, the most affected
parameters are again $\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, and $n_{s}$ with
decreases of $-6.25$% [$-6.25$%], $-7.14$% [$-14.29$%], and $-6.52$%
[$-19.57$%]. For the curvature parameter when P18+lensing data are utilized we
obtain $\Omega_{k}=-0.012^{+0.027}_{-0.011}$ [$-0.003^{+0.018}_{-0.011}$]
whereas when P18+lensing+non-CMB (new) data are considered we get
$\Omega_{k}=0.0015\pm 0.0019$ [$0.0015\pm 0.0019$] with the two values
differing at $-0.50\sigma$ [$-0.25\sigma$]. While the first result shows a
0.44$\sigma$ [0.17$\sigma$] preference for a closed Universe, the second one
is 0.79$\sigma$ [0.79$\sigma$] in favor of an open Universe. Furthermore, the
error bars from P18+lensing+non-CMB (new) data are 14 [9] times smaller than
the ones obtained with P18+lensing data. In regard to the dark energy equation
of state parameter, we get $w=-1.32^{+0.71}_{-0.38}$ (a $0.45\sigma$ phantom-
like deviation from $w=-1$) [$w=-1.18^{+0.54}_{-0.37}$ $(0.33\sigma)$] for
P18+lensing data, and $w=-0.958\pm 0.026$ (a 1.62$\sigma$ quintessence-like
deviation from $w=-1$) [$w=-0.959\pm 0.027$ $(1.52\sigma)$] for
P18+lensing+non-CMB (new) data, with P18+lensing+non-CMB (new) data error bars
22 [18] times smaller than the P18+lensing error bars. The two values differ
at $-0.51\sigma$ [$-0.43\sigma$]. For the third non-standard-model parameter,
$A_{L}$, with P18+lensing data we get $A_{L}=1.02\pm 0.16$
[$1.07^{+0.12}_{-0.16}$] and when we analyze P18+lensing+non-CMB (new) data we
find $A_{L}=1.102\pm 0.037$ [$1.101\pm 0.038$]. The differences between the
two values is $-0.50\sigma$ [$-0.25\sigma$]. Both results show a preference
for $A_{L}>1$ with a significance of 0.13$\sigma$ [0.44$\sigma$] and
2.76$\sigma$ [2.66$\sigma$], respectively, with error bars 4 [4] times smaller
in the second case. The differences in the values of the derived parameters
are not significant, where for $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ we
obtain 0.27$\sigma$ [0.43$\sigma$], 0.034$\sigma$ [$-0.036\sigma$], and
0.57$\sigma$ [0.45$\sigma$], respectively, with error bars obtained from
P18+lensing+non-CMB (new) being 22 [21], 24 [21], and 12 [11] times smaller,
respectively.

Results obtained from P18+lensing+non-CMB (new) data for the eight-parameter
non-flat XCDM Planck $P(q)$ model and the nine-parameter non-flat XCDM Planck
$P(q)+A_{L}$ model are listed in Table 12 and shown in Figs. 24, 30, 25, and
31. Comparing the results, we find that the values of primary parameters
$\Omega_{c}h^{2}$, $\tau$, $n_{s}$, and $\ln(10^{10}A_{s})$ differ at
0.81$\sigma$, 0.73$\sigma$, $-0.69\sigma$, and 0.93$\sigma$, respectively.
Looking at the values of the curvature parameter $\Omega_{k}$ we observe a
difference of 0.037$\sigma$ whereas if we compare the two values of the dark
energy equation of state parameter $w$ the difference is $-0.60\sigma$. No
significant increase in the size of the error bars for the primary parameters
is observed, with the largest error bars increases being associated with
$\Omega_{c}h^{2}$ (14%), $\tau$ (10%), and $\ln(10^{10}A_{s})$ (12%). For the
derived parameters $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$, the differences
between the mean values are at 0.011$\sigma$, 0.33$\sigma$, and 1.41$\sigma$
and the increase in the size of the error bars of $\sigma_{8}$ is 20%.

Comparing the P18+lensing+non-CMB (new) data constraints obtained for the
eight-parameter non-flat XCDM new $P(q)$ (upper half of Table 13 and Figs. 26
and 32) and the nine-parameter non-flat new $P(q)+A_{L}$ model (lower half of
Table 13 and Figs. 27 and 33), we find very similar results to the ones
obtained for the non-flat XCDM Planck $P(q)$ ($+A_{L}$) models from these
data. The values of the primary parameters $\Omega_{c}h^{2}$, $\tau$, $n_{s}$,
and $\ln(10^{10}A_{s})$ differ by 0.79$\sigma$, 0.74$\sigma$, $-0.71\sigma$,
and 0.97$\sigma$, respectively. For the curvature parameter $\Omega_{k}$,
there is a difference of $-0.036\sigma$ between the two values, and for the
equation of state parameter of dark energy $w$ the results differ at
$-0.61\sigma$. We do not observe significant changes in the size of the error
bars of the primary parameters. In particular, for $n_{s}$ and
$\ln(10^{10}A_{s})$, the error bars increase by 8.7% and 12%, respectively,
while that for the curvature parameter $\Omega_{k}$ decrease by $-5.3$%. For
the derived parameters $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$, the
differences between the two values are 0.011$\sigma$, 0.34$\sigma$, and
1.39$\sigma$, respectively. The corresponding increases in the size of the
error bars are 0.0%, 1.61%, and 19%.

####  IV.2.5 Comparing P18, P18+lensing, and P18+lensing+non-CMB (new) data
cosmological constraints

Cosmological parameter contour plots are very useful for understanding the
level of correlation between the different variables considered in the
analysis and for detecting inconsistencies between cosmological parameter
constraints obtained either for different cosmological models or for the same
model but from different data sets. In this subsubsection we discuss the
changes observed in the contour plots as we include more data in the analysis,
namely when we compare P18, P18+lensing, and P18+lensing+non-CMB (new) data
contours. We first discuss the $A_{L}=1$ models and then comment on the
$A_{L}$-varying cases.

For the seven-parameter flat XCDM $A_{L}=1$ cosmological model, as seen in
Fig. 28, there are significant overlaps of contours. When we compare P18
(grey) and P18+lensing (red) data contours we see significant overlaps of the
1$\sigma$ contours for all parameters. On the other hand, when we compare grey
P18 or red P18+lensing data with blue P18+lensing+non-CMB (new) data contours,
we observe a number of panels in which the two sets of 1$\sigma$ contours do
not overlap: these are panels labeled with either one or both of the primary
parameter $w$ and the derived parameters $H_{0}$, $\Omega_{m}$, and
$\sigma_{8}$. This is consistent with comments in Sec. IV.2.4 that noted that
non-CMB (new) data favored quintessence-like dark energy evolution while P18
data favored phantom-like dark energy evolution.

For the eight-parameter non-flat XCDM Planck $P(q)$ and new $P(q)$ $A_{L}=1$
cosmological models (see Figs. 30 and 32), we find non-overlapping $1\sigma$
regions, where two separate $1\sigma$ regions either do not overlap or the
overlap is infinitesimally small, even when we compare grey P18 and red
P18+lensing contours. For the Planck $P(q)$ model in Fig. 30 these are in the
$w-\Omega_{k}$, $\sigma_{8}-\Omega_{k}$, $H_{0}-w$, $\Omega_{m}-w$,
$\sigma_{8}-H_{0}$, and $\sigma_{8}-\Omega_{m}$ panels, while in the new
$P(q)$ model in Fig. 32 these are in the $\Omega_{k}-H_{0}$, $\Omega_{m}-w$,
$\sigma_{8}-H_{0}$, and $\sigma_{8}-\Omega_{m}$ panels. When we compare the
grey P18 and the blue P18+lensing+non-CMB (new) data contours in the non-flat
XCDM models we find less overlap, now with some cases where even the 2$\sigma$
contours do not overlap. For the Planck and the new $P(q)$ models in Figs. 30
and 32 these are in the $w-\Omega_{k}$, $H_{0}-\Omega_{k}$,
$\Omega_{m}-\Omega_{k}$, $\sigma_{8}-\Omega_{k}$, $H_{0}-w$, $\Omega_{m}-w$,
$\sigma_{8}-H_{0}$, and $\sigma_{8}-\Omega_{m}$ panels. These differences are
likely caused by non-CMB (new) data favoring slower quintessence-like dark
energy evolution while P18 and P18+lensing data favor more rapid phantom-like
evolution.

When $A_{L}$ is allowed to vary and no longer fixed to unity we see, from
Figs. 29, 31, and 33 for the flat and non-flat XCDM models, the differences
between the grey P18 data contours and the blue P18+lensing+non-CMB (new) data
contours subside, with no cases with non-overlapping 2$\sigma$ contours, but
with some non-overlapping 1$\sigma$ contours, for the non-flat XCDM models,
including some new ones in $A_{L}$ panels. For the XCDM Planck and new
$P(q)+A_{L}$ models these non-overlapping contours are in all panels that
contain $\Omega_{k}$ and in the $A_{L}-\sigma_{8}$, $A_{L}-H_{0}$, and
$A_{L}-w$ panels. If we now compare grey P18 and red P18+lensing contour plots
for the three flat and non-flat XCDM$+A_{L}$ models there are no panels where
the contour plots do not overlap at 1$\sigma$.

The results commented on in this subsubsection, as we shall see, largely agree
with those we discuss in Sections IV.3 and IV.4. Namely, P18 and P18+lensing
data results seem to be in tension with non-CMB (new) data results in the
context of the XCDM model with $A_{L}=1$. As stated previously when $A_{L}$ is
allowed to vary we no longer see significantly non-overlapping contours and
this translates into a better performance when it comes to fitting those data
sets that include non-CMB (new) data (see results in Table 15 for the
$\Lambda$CDM models and Table 16 for the XCDM models and discussion in Section
IV.4 below) and also into reduced tension between pairs of data sets (see
Table 10 for $\Lambda$CDM models and Table 14 for XCDM models and discussion
in Section IV.4 below). This may be indicating that in order to jointly
analyze CMB and non-CMB data, in the context of the XCDM models, the lensing
parameter $A_{L}$ should be considered as a free parameter.

####  IV.2.6 Comparing P18 data and non-CMB (new) data cosmological
constraints

In this subsubsection we study the mutual consistency of cosmological
parameter constraints derived from P18 and non-CMB (new) data in XCDM models.
If these constraints are mutually consistent in an XCDM model then P18 and
non-CMB (new) data can be jointly used to constrain cosmological parameters in
that XCDM model. If they are not mutually consistent in an XCDM model this
implies the model is inconsistent with at least one of these data sets and so
can be rejected at some level of confidence (if one assumes that both sets of
data are correct). The results presented here are complemented by those
provided in Sec. IV.4 where we use the two statistical estimators to assess
the level of tension between P18 and non-CMB (new) data in the context of a
given XCDM model.

Since non-CMB (new) data are unable to constrain $\tau$ and $n_{s}$, in the
non-CMB (new) data analyses we set their values to those obtained in the
corresponding P18 data analysis. Also, non-CMB (new) data are practically
insensitive to variations in the lensing consistency parameter $A_{L}$,
therefore when we compare P18 and non-CMB (new) cosmological constraints for
the XCDM+$A_{L}$ models, the constraints from non-CMB (new) data are those for
the corresponding models with $A_{L}=1$.

P18 and non-CMB (new) data results obtained for the seven-parameter and the
five-parameter flat XCDM cosmological models are listed in Table 11 and shown
in Fig. 16. We observe significant differences between the two sets of
cosmological parameter constraints not only in the primary parameters but also
in the derived parameters. In regard to the primary parameters we find that
the two values for $\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$,
100$\theta_{\text{MC}}$, and $\ln(10^{10}A_{s})$ disagree by $-2.14\sigma$,
2.34$\sigma$, 2.09$\sigma$, and $-2.63\sigma$, respectively. With non-CMB
(new) data we obtain $w=-0.853^{+0.043}_{-0.033}$, which indicates a
4.45$\sigma$ preference for quintessence-like behavior whereas with P18 data
we get $w=-1.59^{+0.15}_{-0.34}$ which indicates preference for phantom
behavior 3.93$\sigma$ away from the cosmological constant. The values of $w$
estimated from P18 and non-CMB (new) data differ by $-4.80\sigma$. In the flat
XCDM model P18 data and non-CMB (new) data values of all five primary
parameters are mutually inconsistent at greater than 2$\sigma$ confidence. As
for the derived parameters, we find for $\Omega_{m}$ and $\sigma_{8}$
differences of $-1.54\sigma$ and 1.97$\sigma$, respectively, between the P18
data and non-CMB (new) data values. These results might mean that in the
context of the flat XCDM model P18 data and non-CMB (new) data should not be
analyzed together, and that the flat XCDM model might be inconsistent with at
least one of the two data sets (under the assumption that both data sets are
correct). We will return to this issue in Sec. IV.4.

We can compare the results obtained for the eight-parameter flat XCDM+$A_{L}$
model and the five-parameter flat XCDM model, when P18 and non-CMB (new) data
are employed, respectively, if we look at Table 11 and Fig. 17. We find
significant differences in the values of the primary parameters, which exceed
2$\sigma$ confidence for four of the five primary parameters. In particular,
for $\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, $100\theta_{\text{MC}}$, and
$\ln(10^{10}A_{s})$ we observe differences of $-2.10\sigma$, $2.12\sigma$,
$2.11\sigma$, and $-2.69\sigma$, respectively. As for the equation of state
parameter of dark energy, for the flat XCDM+$A_{L}$ model constrained with P18
data we obtain $w=-1.23^{+0.31}_{-0.59}$, indicating preference for phantom-
like behavior at 0.74$\sigma$. Compared to the $w$ value for non-CMB (new)
data in the flat XCDM model, the two values differ at $-1.21\sigma$
confidence, significantly smaller than the corresponding $-4.80\sigma$ in the
flat XCDM model where $A_{L}$ is not allowed to vary and is set to unity. When
we compare the values of the derived parameters we find less severe
discrepancies than in the flat XCDM case, with none of them even reaching the
1$\sigma$ level. The two values of $H_{0}$ and $\sigma_{8}$ disagree by
$0.70\sigma$ and $0.29\sigma$, respectively. To determine whether P18 and non-
CMB (new) data can be jointly analyzed, and whether the flat XCDM+$A_{L}$
model is inconsistent with at least one of these data sets, we will make use
of the statistical estimators in Sec. IV.4.

The primary cosmological parameters values of the eight-parameter and six-
parameter non-flat XCDM Planck [new] $P(q)$ models from P18 data and from non-
CMB (new) data are listed in Table 12 [13] and shown in Fig. 18 [20]. Here we
observe smaller differences than those in the flat XCDM model, with two [none]
of the six primary parameters disagreeing at 2$\sigma$ confidence. The largest
differences are for $\Omega_{b}h^{2}$ and 100$\theta_{\text{MC}}$ with
disagreements at $-1.37\sigma$ [$-1.28\sigma$] and $-2.10\sigma$
[$-1.79\sigma$], respectively, as well as for $\Omega_{k}$ as discussed next.
When analyzing P18 data we obtain $\Omega_{k}=-0.048^{+0.041}_{-0.012}$
[$-0.0338^{+0.029}_{-0.0086}$] for the curvature parameter, indicating a
1.17$\sigma$ [1.17$\sigma$] preference for closed geometry and a 1.98$\sigma$
[1.82$\sigma$] disagreement with the value obtained from non-CMB (new) data
$\Omega_{k}=-0.177^{+0.064}_{-0.072}$ [$-0.186^{+0.083}_{-0.067}$], which in
turn is 2.77$\sigma$ [2.24$\sigma$] in favor of closed hypersurfaces. As for
the equation of state parameter, using P18 data we obtain
$w=-1.27^{+0.97}_{-0.45}$ [$-1.27^{+0.79}_{-0.44}$], while using non-CMB (new)
data we get $w=-0.786^{+0.044}_{-0.037}$ [$-0.785^{+0.045}_{-0.038}$]. The P18
value indicates a 0.28$\sigma$ [0.34$\sigma$] preference for phantom-like
behavior while the non-CMB (new) value represents a 5.78$\sigma$
[5.66$\sigma$] preference for quintessence-like behavior, and the P18 and non-
CMB (new) values differ at $-0.50\sigma$ [$-0.61\sigma$]. Regarding the
derived parameters, for $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ we observe
differences of $-1.14\sigma$ [$-0.74\sigma$], 0.76$\sigma$ [0.60$\sigma$], and
0.36$\sigma$ [0.59$\sigma$], almost all below 1$\sigma$.

P18 and non-CMB (new) data results obtained for the nine-parameter non-flat
XCDM Planck [new] $P(q)+A_{L}$ model and the six-parameter non-flat XCDM
Planck [new] $P(q)$ model can be found in Table 12 [13] and seen in Fig. 19
[21]. Non-negligible differences are observed when we look at the values of
some primary parameters with one [none] of the six primary parameters
disagreeing at 2$\sigma$ significance. For $\Omega_{b}h^{2}$,
$100\theta_{\text{MC}}$, and $\ln(10^{10}A_{s})$, we see differences at
$-1.37\sigma$ [$-1.27\sigma$], $-2.10\sigma$ [$-1.79\sigma$], and $0.38\sigma$
[$0.47\sigma$], respectively. For the non-flat XCDM Planck [new] $P(q)+A_{L}$
model, when P18 data is considered, we obtain
$\Omega_{k}=-0.073^{+0.065}_{-0.029}$ [$-0.072^{+0.065}_{-0.030}$], which is
$1.12\sigma$ [$1.11\sigma$] away from flat. For the non-flat XCDM Planck [new]
$P(q)$ model, using non-CMB (new) data, we find
$\Omega_{k}=-0.177^{+0.064}_{-0.072}$ [$-0.186^{+0.083}_{-0.067}$], indicating
a preference for a closed Universe at $2.77\sigma$ [$2.24\sigma$]. The two
values disagree at $1.48\sigma$ [$1.29\sigma$]. In regard to the equation of
state parameter of dark energy for the XCDM Planck [new] $P(q)+A_{L}$ model
$w=-1.36^{+1.1}_{-0.53}$ [$-1.39^{+1.1}_{-0.54}$] from P18 data, with this
value favoring phantom-like behavior at $0.33\sigma$ [$0.35\sigma$], whereas
from non-CMB data in the XCDM Planck [new] $P(q)$ model we obtain
$w=-0.786^{+0.044}_{-0.037}$ [$-0.785^{+0.045}_{-0.038}$], indicating a
preference for quintessence at $5.78\sigma$ [$5.66\sigma$], with a discrepancy
of $-0.52\sigma$ [$-0.55\sigma$] between the two values. As for the derived
parameters $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$, we find that the two
values disagree at $-2.56\sigma$ [$-2.55\sigma$], $1.15\sigma$ [$1.10\sigma$],
and $0.19\sigma$ [$0.23\sigma$], respectively.

####  IV.2.7 Comparing P18+lensing data and non-CMB (new) data cosmological
constraints

In the previous subsubsection we showed that in the XCDM models the
differences between P18 data and non-CMB (new) data cosmological parameter
constraints are not negligible. We now determine whether the same is true when
we compare P18+lensing data and non-CMB (new) data XCDM models cosmological
constraints.

Cosmological parameter constraints for the seven-parameter and the five-
parameter flat XCDM model obtained from P18+lensing data and non-CMB (new)
data are listed in Table 11 and shown in Fig. 22. As in the previous
subsubsection for the P18 and non-CMB (new) data primary parameter values,
there are significant differences between the two sets of cosmological
constraints here, with all five primary parameter values differing at more
than 2$\sigma$ significance. In particular, $\Omega_{b}h^{2}$,
$\Omega_{c}h^{2}$, 100$\theta_{\text{MC}}$, and $\ln(10^{10}A_{s})$ values
differ at $-2.13\sigma$, 2.27$\sigma$, 2.10$\sigma$, and $-2.65\sigma$. From
P18+lensing data, for the dark energy equation of state parameter we get
$w=-1.55^{+0.16}_{-0.35}$ showing a 3.44$\sigma$ preference for phantom-like
behavior and a difference of $-4.27\sigma$ with the non-CMB (new) data result
$w=-0.853^{+0.043}_{-0.033}$ which favors quintessence-like behavior at
4.45$\sigma$. Differences in the derived parameters are not as severe as those
found for the primary cosmological parameter values but they are non-
negligible. For $\Omega_{m}$ and $\sigma_{8}$ we obtain differences of
$-1.41\sigma$ and $1.79\sigma$. These results probably mean that P18+lensing
and non-CMB (new) data should not be jointly analyzed in the context of the
flat XCDM model, and that the flat XCDM model probably is inconsistent with at
least one of the two data sets (under the assumption that both data sets are
correct), but this needs to be confirmed with the help of the statistical
estimators.

The primary and derived parameter values of the eight-parameter flat
XCDM+$A_{L}$ model and the five-parameter flat XCDM model obtained with
P18+lensing and non-CMB (new) data, respectively, are listed in Table 11 and
shown in Fig. 23. When we look at the values of the primary parameters, we
observe significant differences, with four of the five primary parameter
values differing at more than 2$\sigma$. For $\Omega_{b}h^{2}$,
$\Omega_{c}h^{2}$, $100\theta_{\text{MC}}$, and $\ln(10^{10}A_{s})$ the
differences between the two values are $-2.11\sigma$, $2.15\sigma$,
$2.11\sigma$, and $-2.69\sigma$, respectively. When the non-CMB (new) data are
analyzed in the flat XCDM model we obtain $w=-0.853^{+0.043}_{-0.033}$ which
indicates a 4.45$\sigma$ preference for quintessence-like behavior. In
contrast, when we consider P18+lensing data in the context of the flat
XCDM+$A_{L}$ model we get $w=-1.34^{+0.26}_{-0.51}$ which indicates a
$1.31\sigma$ deviation from $w=-1$. The two values differ by $-1.86\sigma$,
significantly smaller than the corresponding $-4.27\sigma$ value in the flat
XCDM model discussed in the previous paragraph. As in the previous case less
significant differences are found when we look at the values of the derived
parameters. In particular for $\Omega_{m}$ and $\sigma_{8}$ the two results
disagree at $-0.33\sigma$ and $0.58\sigma$. As in the previous case with
$A_{L}=1$ these results may indicate that P18+lensing and non-CMB (new) data
should not be jointly analyzed in the flat XCDM+$A_{L}$ model but further
tests are needed before we definitively conclude this.

The results presented in Table 12 [13] and shown in Fig. 24 [26] allow us to
compare the eight-parameter and the six-parameter non-flat XCDM Planck [new]
$P(q)$ models primary cosmological parameter constraints obtained from
P18+lensing data and non-CMB (new) data. In this case two [one] of the six
primary parameters differ at more than 2$\sigma$ significance. For
$\Omega_{b}h^{2}$ and 100$\theta_{\text{MC}}$ the differences are at
$-1.40\sigma$ [$-1.30\sigma$] and $-2.10\sigma$ [$-1.79\sigma$], respectively,
which are very similar to the results found for the case of P18 and non-CMB
(new) data. Using P18+lensing data we obtain for the curvature parameter
$\Omega_{k}=-0.0111^{+0.013}_{-0.00070}$ [$-0.0080^{+0.0098}_{-0.0023}$] which
is 0.85$\sigma$ [0.82$\sigma$] away from flat and differs at 2.59$\sigma$
[2.14$\sigma$] with the non-CMB (new) data value
$\Omega_{k}=-0.177^{+0.064}_{-0.072}$ [$-0.186^{+0.083}_{-0.067}$] that also
favors closed geometry but now at 2.77$\sigma$ [2.24$\sigma$]. For the dark
energy equation of state parameter, P18+lensing data give
$w=-1.28^{+0.41}_{-0.54}$ [$-1.27^{+0.40}_{-0.49}$] representing a phantom-
like deviation of 0.68$\sigma$ [0.68$\sigma$] from the cosmological constant
value of $w=-1$ and differing at $-1.20\sigma$ [$-1.21\sigma$] from the non-
CMB (new) data value $w=-0.786^{+0.044}_{-0.037}$ [$-0.785^{+0.045}_{-0.038}$]
that favors quintessence-like behavior at 5.78$\sigma$ [5.66$\sigma$]. As for
the derived parameters we find mild differences, in particular for $H_{0}$,
$\Omega_{m}$, and $\sigma_{8}$ these are 0.23$\sigma$ [0.24$\sigma$],
$0.040\sigma$ [$-0.049\sigma$], and 0.76$\sigma$ [0.86$\sigma$].

P18+lensing and non-CMB (new) data results for the nine-parameter non-flat
XCDM Planck [new] $P(q)+A_{L}$ and the six-parameter non-flat XCDM Planck
[new] $P(q)$ model can be seen in Table 12 [13] and in Fig. 25 [27]. We
observe some non-negligible differences between the values of the primary
parameters, with two [one] of the six differing at more than 2$\sigma$
significance. For those parameters common to the flat $\Lambda$CDM model the
ones that show larger differences are $\Omega_{b}h^{2}$ and
$100\theta_{\text{MC}}$ with disagreements at $-1.39\sigma$ [$-1.29\sigma$]
and $-2.10\sigma$ [$-1.79\sigma$] respectively. With non-CMB (new) data and
the non-flat XCDM Planck [new] $P(q)$ model, we find
$\Omega_{k}=-0.177^{+0.064}_{-0.072}$ [$-0.186^{+0.083}_{-0.067}$] which
favors closed spatial hypersurfaces at 2.77$\sigma$ [2.24$\sigma$] whereas
when we analyze P18+lensing data within the context of the non-flat XCDM
Planck [new] $P(q)+A_{L}$ model we get $\Omega_{k}=-0.012^{+0.027}_{-0.011}$
[$-0.003^{+0.018}_{-0.011}$] which shows a preference for a closed Universe at
0.44$\sigma$ [0.17$\sigma$]. The two values differ at 2.54$\sigma$
[2.19$\sigma$] significance. In regard to the equation of state parameter of
dark energy, when non-CMB (new) data are utilized we find
$w=-0.786^{+0.044}_{-0.037}$ [$-0.785^{+0.045}_{-0.038}$] and when P18+lensing
data are considered we find $w=-1.32^{+0.71}_{-0.38}$
[$-1.18^{+0.54}_{-0.37}$], with the differences between the two values being
$-0.75\sigma$ [$-0.73\sigma$]. In the first case we observe evidence of
5.78$\sigma$ [5.66$\sigma$] in favor of quintessence-like behavior while in
the second case the result shows a preference for phantom-like behavior at
0.45$\sigma$ [0.33$\sigma$] significance. As for the derived parameters we
find less significant differences than for the primary parameters, in
particular for $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ the values disagree at
$0.092\sigma$ [$0.24\sigma]$, $0.11\sigma$ [$0.03\sigma$], and $0.64\sigma$
[$0.55\sigma]$, respectively.

###  IV.3 Model Selection

In this subsection we discuss how (relatively) well each data set is fit by
each of the twelve models we study. In particular we consider the following
combinations of data: non-CMB (new), P18, P18+lensing, P18+non-CMB (new), and
P18+lensing+non-CMB (new). We also consider three pairs of $\Lambda$CDM models
and three pairs of XCDM models, with one of each pair having $A_{L}=1$ and the
other with varying $A_{L}$, with one of the three being spatially flat and the
other two allowing for non-zero spatial curvature. As we have mentioned
previously, the lensing consistency parameter $A_{L}$ does not impact the non-
CMB data analysis and so when we comment on the results obtained with non-CMB
(new) data we do not mention the $A_{L}$-varying models. We focus on the
results obtained for the DIC, Eq. (11), since it is considered to be more
reliable than the AIC, Eq. (10). The different levels of significance for the
$\Delta$DIC values are defined below Eq. (11) in Sec. III.

Non-CMB (new). When the equation of state parameter of dark energy is fixed to
$w=-1$, the flat $\Lambda$CDM model is weakly [positively] favored over the
non-flat Planck [new] $P(q)$ model. On the other hand, when $w$ is allowed to
vary we find that the flat XCDM model is strongly favored over the flat
$\Lambda$CDM model. Simultaneous variations of $w$ and the curvature parameter
$\Omega_{k}$ are favored by non-CMB (new) data since both the non-flat XCDM
Planck $P(q)$ and new $P(q)$ models are strongly favored compared to the flat
$\Lambda$CDM model, with the XCDM Planck $P(q)$ case very weakly favored over
the XCDM new $P(q)$ case, while both non-flat XCDM models are very weakly
favored over the flat XCDM model. Finally, the $\Lambda$CDM Planck [new]
$P(q)$ model is very strongly [very strongly] disfavored when compared to the
XCDM Planck [new] $P(q)$ model. These results indicate that allowing $w$ to be
a freely varying parameter is significantly more important than allowing
$\Omega_{k}$ to be a freely varying parameter in improving the fit to non-CMB
(new) data relative to the performance of the flat $\Lambda$CDM model.

P18. For $A_{L}=1$ and $w=-1$, the non-flat $\Lambda$CDM Planck $P(q)$ and new
$P(q)$ models are strongly favored over the flat $\Lambda$CDM model, with the
Planck $P(q)$ model weakly favored over the new $P(q)$ one. However, when
$A_{L}=1$ and $w$ is allowed to vary, the non-flat XCDM Planck $P(q)$ and new
$P(q)$ models are positively favored over the flat XCDM model, with the Planck
$P(q)$ model weakly favored over the new $P(q)$ one. When the lensing
consistency parameter $A_{L}$ is allowed to vary, while keeping $w=-1$, the
flat $\Lambda$CDM+$A_{L}$ model is positively favored over the flat
$\Lambda$CDM model while the $\Lambda$CDM Planck [new] $P(q)+A_{L}$ model is
weakly [positively] disfavored compared to the $\Lambda$CDM Planck [new]
$P(q)$ model. When $w$ is allowed to vary while keeping $A_{L}=1$, the flat
XCDM model is positively favored over the flat $\Lambda$CDM model whereas the
non-flat XCDM Planck [new] $P(q)$ model is weakly [weakly] disfavored compared
to the $\Lambda$CDM Planck [new] $P(q)$ model. We now consider the results
obtained when both $w$ and $A_{L}$ are allowed to vary. Compared to the flat
$\Lambda$CDM model, we find that the flat XCDM+$A_{L}$ model is positively
favored, while the non-flat XCDM Planck and new $P(q)+A_{L}$ models are
strongly favored. The flat XCDM+$A_{L}$ model is positively favored over the
flat XCDM model. The non-flat XCDM Planck [new] $P(q)+A_{L}$ model is weakly
disfavored [weakly favored] compared to the non-flat XCDM Planck [new] $P(q)$
model and very weakly [positively] favored compared to the $\Lambda$CDM Planck
[new] $P(q)+A_{L}$ model. Additionally, the non-flat $\Lambda$CDM Planck [new]
$P(q)+A_{L}$ model is weakly favored [positively disfavored] over the flat
$\Lambda$CDM$+A_{L}$ model and the non-flat XCDM Planck and new $P(q)+A_{L}$
models are weakly favored over the flat XCDM$+A_{L}$ model. As we found
previously in the $\Lambda$CDM case, [10], we find for XCDM here that allowing
$A_{L}$ to vary weakens the support for the non-flat models over the
corresponding varying-$A_{L}$ flat model, with the additional caveat that
unlike in the $A_{L}=1$ $\Lambda$CDM case where both non-flat models are
strongly favored over the flat model, in the XCDM case the non-flat models are
only positively favored over the flat model.

P18+lensing. When we analyze the results obtained from P18+lensing data we do
not find the level of evidence found when lensing data is not included in the
mix. If we set $w=-1$ and $A_{L}=1$ we see that the non-flat $\Lambda$CDM
Planck $P(q)$ and new $P(q)$ models are weakly favored relative to the flat
$\Lambda$CDM model. For $w=-1$ and varying $A_{L}$, the flat
$\Lambda$CDM+$A_{L}$ model is weakly favored over the flat $\Lambda$CDM model,
and the non-flat $\Lambda$CDM Planck [new] $P(q)+A_{L}$ model is weakly
[weakly] disfavored when compared with the $\Lambda$CDM Planck [new] $P(q)$
model. When $w$ is allowed to vary but $A_{L}=1$, the flat XCDM model is
positively favored over the flat $\Lambda$CDM model. However, on the other
hand, the non-flat XCDM Planck $P(q)$ and new $P(q)$ models are both weakly
disfavored when compared to the non-flat $\Lambda$CDM Planck $P(q)$ and new
$P(q)$ models, respectively. When allowing simultaneous variation of $w$ and
$A_{L}$ we find that the flat XCDM+$A_{L}$ model is weakly favored over the
flat $\Lambda$CDM model, while the XCDM Planck [new] $P(q)+A_{L}$ model is
positively [weakly] disfavored compared to the standard model. Finally, the
XCDM Planck [new] $P(q)+A_{L}$ model is positively [positively] disfavored
when compared with the $\Lambda$CDM Planck [new] $P(q)$ model and is
positively [weakly] disfavored over the XCDM Planck [new] $P(q)$ model.

P18+non-CMB (new). We now consider how the P18 data analysis results change
when we include non-CMB (new) data in the analysis. For $w=-1$ and $A_{L}=1$
we find that the non-flat $\Lambda$CDM Planck and new $P(q)$ models are weakly
disfavored when compared to the flat $\Lambda$CDM model. Varying $A_{L}$ while
holding $w=-1$, we observe that the flat $\Lambda$CDM+$A_{L}$ model is
strongly favored over the flat $\Lambda$CDM model, and both the non-flat
$\Lambda$CDM Planck and new $P(q)+A_{L}$ models are strongly favored over the
$\Lambda$CDM Planck and new $P(q)$ models, respectively. Considering the
complementary case, when $w$ varies and $A_{L}=1$ we see that the flat XCDM
model is weakly disfavored when compared to the flat $\Lambda$CDM model,
whereas the $\Lambda$CDM Planck [new] $P(q)$ model is weakly [weakly] favored
over the XCDM Planck [new] $P(q)$ model. When $w$ and $A_{L}$ are both varied,
we observe that the three XCDM+$A_{L}$ models, namely the flat XCDM+$A_{L}$
one and the non-flat XCDM Planck and new $P(q)+A_{L}$ ones, are strongly
favored over the corresponding three $\Lambda$CDM models, the flat one and the
non-flat Planck and new $P(q)$ ones, respectively. On the other hand the flat
$\Lambda$CDM+$A_{L}$ model is very strongly favored over the flat XCDM model,
and the two non-flat $\Lambda$CDM+$A_{L}$ models are strongly favored over the
corresponding non-flat XCDM models. The flat XCDM+$A_{L}$ model is very
strongly favored over the flat XCDM model, the XCDM Planck $P(q)+A_{L}$ model
is on the verge of being very strongly preferred over the XCDM Planck $P(q)$
model, and the XCDM new $P(q)+A_{L}$ model is very strongly favored over the
XCDM new $P(q)$ model. Finally we observe that the XCDM Planck and new
$P(q)+A_{L}$ models are weakly preferred over the $\Lambda$CDM Planck and new
$P(q)+A_{L}$ models, respectively. When $A_{L}$ is allowed to vary, P18+non-
CMB (new) data typically strongly favor $A_{L}>1$.

P18+lensing+non-CMB (new). Finally, we now analyze the results obtained for
the most complete data set employed in this work. We note again that when
lensing data are included in the analysis such data compilations tend to more
poorly distinguish between the models. The flat $\Lambda$CDM model is weakly
favored over the $\Lambda$CDM Planck and new $P(q)$ models. When considering
the case where $w=-1$ but $A_{L}$ is freely varied in the analysis, we find
that the flat $\Lambda$CDM+$A_{L}$ model is positively favored over the flat
$\Lambda$CDM model. A similar level of support is found for the $\Lambda$CDM
Planck and new $P(q)+A_{L}$ models compared to the $\Lambda$CDM Planck and new
$P(q)$ models, respectively. For $A_{L}=1$ but varying $w$, we observe that
the flat XCDM model is positively disfavored in comparison to the flat
$\Lambda$CDM model. Additionally, the XCDM Planck [new] $P(q)$ model is weakly
[weakly] disfavored compared to the $\Lambda$CDM Planck [new] $P(q)$ model.
When both $w$ and $A_{L}$ are free parameters, we find that the flat
XCDM+$A_{L}$ model is positively preferred over the flat $\Lambda$CDM model
and strongly favored over the flat XCDM model, while the XCDM Planck [new]
$P(q)+A_{L}$ model is positively [positively and almost strongly] favored over
the XCDM Planck [new] $P(q)$ model. We also note that when we compare the XCDM
Planck [new] $P(q)+A_{L}$ model with the $\Lambda$CDM Planck [new]
$P(q)+A_{L}$ model, the former is only weakly [weakly] preferred over the
latter. Compared to our previous analysis with the non-CMB (old) data, [10],
with the non-CMB (new) data here, in the flat and two non-flat $\Lambda$CDM
models the varying $A_{L}$ option is no longer as positively favored over the
$A_{L}=1$ option, however it is positively or strongly favored over the
$A_{L}=1$ option in the flat and two non-flat XCDM models.

Table 14: Consistency check parameter $\log_{10}\mathcal{I}$ and tension parameters $\sigma$ and $p$ for P18 vs. non-CMB (new) data sets and P18+lensing vs. non-CMB (new) data sets in the XCDM ($+A_{L}$) models.  | Flat XCDM model | Flat XCDM+$A_{L}$ model
---|---|---
Data | P18 vs non-CMB | P18+lensing vs non-CMB | P18 vs non-CMB | P18+lensing vs non-CMB
$\log_{10}\mathcal{I}$ | $-2.125$ | $-2.247$ | $-0.364$ | $-0.506$
$\sigma$ | $3.448$ | $3.555$ | $2.095$ | $2.378$
$p$ (%) | $0.056$ | $0.039$ | $3.619$ | $1.742$
| Non-flat XCDM model [Planck $P(q)$] | Non-flat XCDM+$A_{L}$ model [Planck $P(q)$]
Data | P18 vs non-CMB | P18+lensing vs non-CMB | P18 vs non-CMB | P18+lensing vs non-CMB
$\log_{10}\mathcal{I}$ | $-3.421$ | $-1.824$ | $-1.173$ | $-0.275$
$\sigma$ | $4.294$ | $3.396$ | $2.611$ | $2.167$
$p$ (%) | $0.003$ | $0.069$ | $0.902$ | $3.026$
| Non-flat XCDM model [new $P(q)$] | Non-flat XCDM+$A_{L}$ model [new $P(q)$]
Data | P18 vs non-CMB | P18+lensing vs non-CMB | P18 vs non-CMB | P18+lensing vs non-CMB
$\log_{10}\mathcal{I}$ | $-3.125$ | $-1.942$ | $-0.957$ | $-0.312$
$\sigma$ | $3.960$ | $3.164$ | $2.662$ | $2.256$
$p$ (%) | $0.007$ | $0.155$ | $0.778$ | $2.409$

###  IV.4 Data Set Tensions

After analyzing in detail the cosmological parameter constraints for each of
the twelve models under study, obtained for the different data sets considered
in this work, and after comparing how (relatively) well each data set is fit
by each model, in this subsection we test whether, in the context of a given
cosmological model, there is concordance (discordance) between the results
obtained for pairs of some of the data sets and also whether this data set
consistency (inconsistency) is model independent. To accomplish this we use
the two statistical estimators, presented in Sec. III, whose expressions are
provided in Eqs. (12), (15), and (16) respectively. See there for discussions
of the different levels of significance of the estimator values. The values
for the two statistical estimators, $\log_{10}\mathcal{I}$ and $p(\sigma)$,
are provided in Table 10 for the $\Lambda$CDM models and in Table 14 for the
XCDM models, for the P18 and non-CMB and for the P18+lensing and non-CMB data
sets comparisons. For the $\Lambda$CDM models in Table 10 we provide both non-
CMB (old) and non-CMB (new) data results, and as mentioned earlier non-CMB
(new) data are more consistent with P18 and P18+lensing data than non-CMB
(old) data are. For the XCDM models in Table 14 we provide only non-CMB (new)
data results. In the following discussion we focus only on the results
obtained with the non-CMB (new) data set.

P18 vs. non-CMB (new). In the flat $\Lambda$CDM model P18 data and non-CMB
(new) data are not inconsistent. For the first statistical estimator we get
$\log_{10}\mathcal{I}=0.805$ which indicates substantial consistency whereas
for the second one we find $p=24.9\%$ ($\sigma=1.152$) which indicates neither
significant concordance nor significant discordance. For the flat
$\Lambda$CDM+$A_{L}$ model we obtain similar results with both estimators,
$\log_{10}\mathcal{I}=1.446$ and $p=87\%$ ($\sigma=0.164$), pointing to a
strong consistency between the results obtained with the P18 and non-CMB (new)
data sets.

When we consider varying $\Omega_{k}$ but keep $A_{L}=1$ and $w=-1$ we observe
non-negligible tensions between the two sets of cosmological parameter
constraints. For the non-flat $\Lambda$CDM Planck $P(q)$ model we get
$\log_{10}\mathcal{I}=-0.796$ and $p=0.687\%$ ($\sigma=2.704$) indicating a
substantial inconsistency between P18 data and non-CMB (new) data results, but
less than $3\sigma$ and also less than the $\sigma=3.005$ found when comparing
the P18 and non-CMB (old) data results, [10]. On the other hand, for the non-
flat $\Lambda$CDM new $P(q)$ model we have $\log_{10}\mathcal{I}=-0.391$ and
$p=2.10\%$ ($\sigma=2.308$), with the first estimator indicating neither
consistency nor inconsistency while the second estimator points to a moderate
inconsistency, which means that the tension between the two data sets is
reduced in the new $P(q)$ case relative to the Planck $P(q)$ case. In view of
these results the degree of discordance is not significant enough to prevent
P18 data and non-CMB (new) data being used together in the $\Lambda$CDM Planck
and new $P(q)$ models.

When $w=-1$ but $\Omega_{k}$ and $A_{L}$ are allowed to vary, there is
agreement between the results obtained with P18 and non-CMB (new) data. While
for the $\Lambda$CDM Planck $P(q)+A_{L}$ model we find
$\log_{10}\mathcal{I}=1.210$ and $p=55.2\%$ ($\sigma=0.595$), for the
$\Lambda$CDM new $P(q)+A_{L}$ model we get $\log_{10}\mathcal{I}=2.107$ and
$p=77.2\%$ ($\sigma=0.289$). In the first case the estimators indicate
substantial consistency between the pair of data sets whereas in the second
case the level of concordance is closer to being decisive.

When $A_{L}=1$ but the dark energy equation of state parameter $w$ varies we
find some very significant tensions between P18 and non-CMB (new) data in the
context of the XCDM models. For the flat XCDM model we get
$\log_{10}\mathcal{I}=-2.125$ and $p=0.056\%$ ($\sigma=3.448$), indicating a
decisive degree of discordance for this model. The simultaneous variation of
$\Omega_{k}$ and $w$ increases the disagreement between the two data sets. For
the XCDM Planck $P(q)$ model we obtain $\log_{10}\mathcal{I}=-3.421$ and
$p=0.003\%$ ($\sigma=4.294$) and similarly for the XCDM new $P(q)$ model we
get $\log_{10}\mathcal{I}=-3.125$ and $p=0.007\%$ ($\sigma=3.960$). In both
cases the two statistical estimators point to a decisive degree of
inconsistency, which makes the joint consideration of P18 and non-CMB (new)
data inadvisable in the context of the non-flat XCDM models with $A_{L}=1$.
When $A_{L}=1$ all three XCDM models, flat and non-flat, are inconsistent at
$>3\sigma$ with either P18 data or non-CMB data or both (if these data sets
are correct).

Allowing for a variation in the lensing consistency parameter $A_{L}$ results
in more mutually consistent P18 and non-CMB (new) data results in the context
of the XCDM models. In the flat XCDM+$A_{L}$ model we obtain
$\log_{10}\mathcal{I}=-0.364$ and $p=3.619\%$ ($\sigma=2.095$) which indicates
that the degree of discordance is at most moderate. For the XCDM Planck
$P(q)+A_{L}$ model we find $\log_{10}\mathcal{I}=-1.173$ and $p=0.902\%$
($\sigma=2.611$) whereas for the XCDM new $P(q)+A_{L}$ model we obtain
$\log_{10}\mathcal{I}=-0.957$ and $p=0.778\%$ ($\sigma=2.662$). Both
statistical estimators still show an almost strong discordance between P18 and
non-CMB (new) data results, but the degree of inconsistency is probably not
high enough to prevent us from considering these two data sets jointly in our
analysis in the context of the non-flat XCDM+$A_{L}$ models.

In summary, P18 and non-CMB (new) data are largely consistent in the six flat
and non-flat $\Lambda$CDM($+A_{L}$) models and in the three flat and non-flat
XCDM$+A_{L}$ models, but are inconsistent at $>3\sigma$ in the three flat and
non-flat XCDM $A_{L}=1$ models, thus ruling out these three models at
$>3\sigma$, if the two data sets are correct. We see next that very similar
results hold in the P18+lensing and non-CMB (new) data sets comparison.

P18+lensing vs. non-CMB (new). When looking at the results obtained for the
flat $\Lambda$CDM model, including lensing data with P18 data in the analysis
results in conclusions similar to the case when just P18 data results are
compared to the non-CMB (new) data results. From the P18+lensing and non-CMB
(new) data comparison, for the first statistical estimator we find
$\log_{10}\mathcal{I}=0.730$ and for the second one we obtain $p=22.7\%$
($\sigma=1.209$). In the first case we find a substantial degree of
concordance between the P18+lensing and non-CMB (new) data results whereas in
the second case neither significant consistency nor significant inconsistency
can be claimed.

Compared to the previous case, where we compare P18 and non-CMB (new) data
results, for $A_{L}=1$ and $w=-1$ but the curvature parameter $\Omega_{k}$ is
allowed to vary, we find less significant disagreement between the P18+lensing
and non-CMB (new) data results. For the non-flat $\Lambda$CDM Planck $P(q)$
model we get $\log_{10}\mathcal{I}=0.711$ in addition to $p=12\%$
($\sigma=1.555$) and very similarly for the non-flat $\Lambda$CDM new $P(q)$
model we get $\log_{10}\mathcal{I}=0.755$ for the first estimator and
$p=12.3\%$ ($\sigma=1.544$) for the second one. So, the first statistical
estimator, in both cases, points to substantial consistency between the
results obtained with P18+lensing and non-CMB (new) data while the second one
indicates neither significant consistency nor significant inconsistency
between the two data sets. This means that P18+lensing and non-CMB (new) data
can be jointly analyzed in the context of the non-flat $\Lambda$CDM models.

As expected, in light of the results obtained in the previous P18 vs. non-CMB
(new) data case, the simultaneous variation of $\Omega_{k}$ and $A_{L}$ in the
analysis enhances the consistency between results obtained from P18+lensing
data and from non-CMB (new) data. In the case of the non-flat $\Lambda$CDM
Planck $P(q)+A_{L}$ model we get $\log_{10}\mathcal{I}=1.719$ and for the
second estimator $p=80.9\%$ ($\sigma=0.241$) while for the non-flat
$\Lambda$CDM new $P(q)+A_{L}$ model we have $\log_{10}\mathcal{I}=1.887$ as
well as $p=75.7\%$ ($\sigma=0.312$), so again we find very similar results for
the two cosmological models. In the context of the non-flat
$\Lambda$CDM+$A_{L}$ models the degree of concordance between P18+lensing and
non-CMB (new) data results is strong.

Like in the P18 vs. non-CMB (new) data comparison analysis, when we look at
the cases that allow the equation of state parameter of dark energy $w$ to
vary, we find some non-negligible disagreements between results obtained with
P18+lensing and non-CMB (new) data. For the flat XCDM model we obtain for the
two statistical estimators $\log_{10}\mathcal{I}=-2.247$ and $p=0.039\%$
($\sigma=3.555$) respectively. These results indicate a decisive degree of
discordance between the P18+lensing and non-CMB (new) data sets in the context
of the flat XCDM model and rule out the flat XCDM model at $>3\sigma$, if both
data sets are correct.

As for the cases where $w$ and $\Omega_{k}$ are allowed to simultaneously
vary, we observe a slight reduction in the disagreement between P18+lensing
and non-CMB (new) data results, compared to the $\Omega_{k}=0$ case, although
they are probably still too large to allow a joint analysis of the two data
sets in the context of the non-flat XCDM models. For the non-flat XCDM Planck
$P(q)$ model the first and second statistical estimators give
$\log_{10}\mathcal{I}=-1.824$ and $p=0.069\%$ ($\sigma=3.396$) whereas for the
non-flat XCDM new $P(q)$ model we get $\log_{10}\mathcal{I}=-1.942$ and
$p=0.155\%$ ($\sigma=3.164$). Therefore, for both models there is still an
almost decisive degree of inconsistency between the results obtained with
P18+lensing data and with non-CMB (new) data, with both non-flat XCDM models
ruled out at $>3\sigma$.

As expected, when we also allow $A_{L}$ to vary the disagreements between the
results obtained with the two data sets subside. For the flat XCDM+$A_{L}$
model we find $\log_{10}\mathcal{I}=-0.506$ which indicates a substantial
degree of inconsistency and for the second estimator we have $p=1.742\%$
($\sigma=2.378$) which goes a step further and seems to point to a strong
inconsistency, but $<3\sigma$, between the P18+lensing and non-CMB (new) data
results. In the context of the flat XCDM+$A_{L}$ model it is probably safe to
jointly analyze P18+lensing and non-CMB (new) data.

In the case of the non-flat XCDM Planck $P(q)+A_{L}$ model for the two
statistical estimators we have $\log_{10}\mathcal{I}=-0.275$ and $p=3.026\%$
($\sigma=2.167$) whereas for the non-flat XCDM new $P(q)+A_{L}$ model we
obtain $\log_{10}\mathcal{I}=-0.312$ and $p=2.409\%$ ($\sigma=2.256$). While
the first estimator for both models do not indicate a significant degree of
discordance, the second one points to a strong disagreement between the two
sets of results. However, the level of disagreement does not seem to be enough
to disallow the joint consideration of P18+lensing and non-CMB (new) data
within the context of the non-flat XCDM+$A_{L}$ cosmological models.

In our previous work [10], we statistically confirmed that in the tilted non-
flat $\Lambda$CDM models with $A_{L}=1$ there is a tension between P18 data
and non-CMB data results that rules out the non-flat models at more than
$2.5\sigma$ significance (just over $3.0\sigma$ for the Planck $P(q)$ model
and just under $2.6\sigma$ for the new $P(q)$ model). Here, from the results
of the non-flat models with $A_{L}=1$ or varying $A_{L}$ (in Tables 10 and
14), except for the non-flat $\Lambda$CDM$+A_{L}$ new $P(q)$ model, we find
larger values of $\log_{10}\mathcal{I}$ and $p$ for the P18+lensing vs. non-
CMB (new) data comparison than for the P18 vs. non-CMB (new) data comparison,
which suggests that the addition of lensing data to P18+non-CMB (new) data
reduces the tension between P18 and non-CMB (new) data. However, this seems to
contradict our finding that adding lensing data to P18+non-CMB (new) data does
not improve the model fit. For example, in the non-flat $\Lambda$CDM models
with $A_{L}=1$, adding lensing data to the P18+non-CMB (new) data only
slightly decreases the DIC values. Furthermore, all $A_{L}$-varying
$\Lambda$CDM models constrained with P18+lensing+non-CMB (new) data are
positively disfavored compared to the corresponding models constrained with
P18+non-CMB (new) data, and similar behavior is seen for all XCDM models
except the non-flat XCDM Planck $P(q)$ model with $A_{L}=1$. This is likely
due to the fact that lensing data slightly shifts the values of the
cosmological parameters towards the region of parameter space that is
preferred by non-CMB (new) data, and thus appears to be more consistent with
non-CMB (new) data. Consequently, when we compare P18+lensing with non-CMB
(new) data we observe less disagreement between the two results than when we
compare P18 data and non-CMB (new) data results.

In summary, P18+lensing and non-CMB (new) data are largely consistent in the
six flat and non-flat $\Lambda$CDM($+A_{L}$) models and in the three flat and
non-flat XCDM$+A_{L}$ models, but are inconsistent at $>3\sigma$ in the three
flat and non-flat XCDM $A_{L}=1$ models, thus ruling out these three models at
$>3\sigma$, if the two data sets are correct. From Figs. 16, 18, and 20, we
can compare P18 and non-CMB (new) data contours for the XCDM models with
$A_{L}=1$. We see that in some panels the 3$\sigma$ contours are the first to
overlap, in agreement with our discussion in this subsection about
inconsistencies between these two data sets within the context of these
models. The effect of considering a varying $A_{L}$ parameter can be seen in
Figs. 17, 19, and 21. The inclusion of the lensing consistency parameter
brings the contours closer together thus reducing the tension between P18 and
non-CMB (new) cosmological parameter constraints and this agrees with the
information in Table 14 where this tension reduction can be appreciated in a
more quantitative way. In Sec. IV.2.5 we compare P18, P18+lensing, and
P18+lensing+non-CMB (new) contours obtained for the XCDM models for fixed and
varying $A_{L}$. We have some at most 2$\sigma$ non-overlapping regions (for
the non-flat XCDM models with $A_{L}=1$) and this was not unexpected since
when we jointly analyze either P18 or P18+lensing data with non-CMB (new) data
we are forcing the cosmological parameters to have values between P18 (or
P18+lensing) data values and non-CMB (new) data values. In Sec. IV.3 we
compare the performance of the different models tested against the different
combinations of data sets. The results provided in Tables 15 and 16 clearly
show that when non-CMB (new) data are included in the analysis it is possible
to improve the performance of the flat $\Lambda$CDM and flat XCDM models by
allowing $A_{L}$ to vary and this goes hand-in-hand with the fact that the
inclusion of a varying lensing consistency parameter in the analysis can
increase the consistency between P18 and non-CMB (new) data sets in the
context of the XCDM models. This again highlights the importance of
considering varying $A_{L}$ in the analysis since in the $\Lambda$CDM and the
XCDM models it helps to reduce the tension between CMB and non-CMB data
results. These results are very similar to those found above for the P18 data
set and the non-CMB (new) data sets comparison.

Table 15: Individual and total $\chi^{2}$ values for the best-fit flat and
non-flat $\Lambda\textrm{CDM}$ inflation models. Deviance information
criterion (DIC) and Akaike information criterion (AIC) are also listed.

Data sets | $\chi_{\textrm{plik}}^{2}$ | $\chi_{\textrm{lowl}}^{2}$ | $\chi_{\textrm{simall}}^{2}$ | $\chi_{\textrm{lensing}}^{2}$ | $\chi_{\textrm{prior}}^{2}$ | $\chi_{\textrm{SN}}^{2}$ | $\chi_{\textrm{BAO}}^{2}$ | $\chi_{H(z)}^{2}$ | $\chi_{f\sigma_{8}}^{2}$ | $\chi^{2}_{\textrm{total}}$ | $\Delta\chi^{2}$ | DIC | $\Delta\textrm{DIC}$ | $\Delta\textrm{AIC}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Flat $\Lambda\textrm{CDM}$ model
Non-CMB (new) |  |  |  |  |  | 1416.49 | 26.43 | 14.57 | 12.44 | 1469.93 |  | 1478.11 |  |
P18 | 2344.71 | 23.39 | 396.05 |  | 1.66 |  |  |  |  | 2765.80 |  | 2817.93 |  |
P18+lensing | 2344.66 | 23.39 | 396.06 | 8.79 | 1.82 |  |  |  |  | 2774.71 |  | 2826.45 |  |
P18+non-CMB (new) | 2347.28 | 22.53 | 396.08 |  | 1.83 | 1414.42 | 25.11 | 14.97 | 18.03 | 4240.24 |  | 4292.33 |  |
P18+lensing+non-CMB (new) | 2346.87 | 22.63 | 396.33 | 8.89 | 1.65 | 1414.32 | 25.03 | 14.96 | 18.57 | 4249.26 |  | 4301.20 |  |
Flat $\Lambda$CDM$+A_{L}$ model
P18 | 2337.23 | 21.92 | 395.66 |  | 1.31 |  |  |  |  | 2756.12 | $-9.68$ | 2812.41 | $-5.52$ | $-7.68$
P18+lensing | 2341.62 | 22.29 | 395.68 | 9.94 | 1.71 |  |  |  |  | 2771.24 | $-3.47$ | 2825.53 | $-0.92$ | $-1.47$
P18+non-CMB (new) | 2337.37 | 21.80 | 395.70 |  | 1.36 | 1415.51 | 25.39 | 14.90 | 15.23 | 4227.27 | $-12.97$ | 4283.86 | $-8.47$ | $-10.97$
P18+lensing+non-CMB (new) | 2342.15 | 22.01 | 395.69 | 9.80 | 1.84 | 1415.26 | 25.47 | 14.93 | 15.46 | 4242.61 | $-6.65$ | 4297.19 | $-4.01$ | $-4.65$
Non-flat $\Lambda\textrm{CDM}$ model [Planck $P(q)$]
Non-CMB (new) |  |  |  |  |  | 1414.52 | 26.98 | 14.57 | 12.15 | 1468.22 | $-1.71$ | 1479.52 | +1.41 | +0.29
P18 | 2336.45 | 21.29 | 395.60 |  | 1.38 |  |  |  |  | 2754.73 | $-11.07$ | 2810.59 | $-7.34$ | $-9.07$
P18+lensing | 2342.29 | 21.86 | 395.66 | 10.09 | 1.63 |  |  |  |  | 2771.53 | $-3.18$ | 2826.17 | $-0.28$ | $-1.18$
P18+non-CMB (new) | 2345.93 | 23.07 | 396.21 |  | 1.89 | 1414.50 | 24.80 | 14.84 | 18.32 | 4239.58 | $-0.66$ | 4293.78 | +1.45 | +1.34
P18+lensing+non-CMB (new) | 2345.79 | 23.29 | 395.95 | 8.99 | 1.97 | 1414.27 | 24.90 | 14.84 | 18.75 | 4248.74 | $-0.52$ | 4302.41 | +1.21 | +1.48
Non-flat $\Lambda$CDM$+A_{L}$ model [Planck $P(q)$]
P18 | 2336.57 | 21.51 | 395.61 |  | 1.29 |  |  |  |  | 2754.99 | $-10.81$ | 2811.63 | $-6.30$ | $-6.81$
P18+lensing | 2341.32 | 22.55 | 395.71 | 9.44 | 2.12 |  |  |  |  | 2771.14 | $-3.57$ | 2827.14 | $+0.69$ | $+0.43$
P18+non-CMB (new) | 2337.36 | 21.67 | 395.77 |  | 1.34 | 1415.46 | 25.50 | 14.95 | 15.02 | 4227.07 | $-13.17$ | 4285.58 | $-6.75$ | $-9.17$
P18+lensing+non-CMB (new) | 2341.49 | 22.19 | 395.70 | 9.78 | 1.82 | 1415.38 | 25.47 | 14.88 | 15.51 | 4242.22 | $-7.04$ | 4298.73 | $-2.47$ | $-3.04$
Non-flat $\Lambda\textrm{CDM}$ model [new $P(q)$]
Non-CMB (new) |  |  |  |  |  | 1414.72 | 26.79 | 14.55 | 12.15 | 1468.21 | $-1.72$ | 1480.16 | +2.05 | +0.28
P18 | 2338.26 | 21.42 | 396.28 |  | 1.42 |  |  |  |  | 2757.38 | $-8.42$ | 2811.54 | $-6.39$ | $-6.42$
P18+lensing | 2342.99 | 21.18 | 395.90 | 9.92 | 1.76 |  |  |  |  | 2771.75 | $-2.96$ | 2825.74 | $-0.71$ | $-0.96$
P18+non-CMB (new) | 2346.77 | 22.78 | 395.77 |  | 1.81 | 1414.47 | 25.00 | 14.85 | 17.98 | 4239.45 | $-0.79$ | 4293.50 | +1.17 | +1.21
P18+lensing+non-CMB (new) | 2345.86 | 22.98 | 396.27 | 8.94 | 1.73 | 1414.35 | 24.79 | 14.86 | 18.73 | 4248.50 | $-0.76$ | 4302.33 | +1.13 | +1.24
Non-flat $\Lambda$CDM$+A_{L}$ model [new $P(q)$]
P18 | 2337.56 | 21.31 | 395.93 |  | 1.52 |  |  |  |  | 2756.33 | $-9.47$ | 2814.83 | $-3.10$ | $-5.47$
P18+lensing | 2341.21 | 22.62 | 395.75 | 9.49 | 1.37 |  |  |  |  | 2770.45 | $-4.26$ | 2827.29 | $+0.84$ | $-0.26$
P18+non-CMB (new) | 2336.94 | 21.93 | 395.70 |  | 1.43 | 1415.23 | 25.42 | 14.93 | 15.54 | 4227.11 | $-13.13$ | 4285.29 | $-7.04$ | $-9.13$
P18+lensing+non-CMB (new) | 2342.43 | 21.94 | 395.77 | 9.12 | 1.71 | 1415.41 | 25.52 | 14.93 | 15.17 | 4242.01 | $-7.25$ | 4298.75 | $-2.45$ | $-3.25$

Note: $\Delta\chi^{2}$, $\Delta\textrm{DIC}$, and $\Delta\textrm{AIC}$
indicate the values relative to those of the tilted flat $\Lambda\textrm{CDM}$
model for the same combination of data sets. For the tilted flat $\Lambda$CDM
model AIC$=2819.80$ (P18), $2828.71$ (P18+lensing), and $4303.26$
(P18+lensing+non-CMB (new)). All $\chi^{2}$ values are computed at the
corresponding model best-fit cosmological parameter values. See section IV B
of [10] for detailed descriptions of individual CMB $\chi^{2}$’s.

Table 16: Individual and total $\chi^{2}$ values for the best-fit flat and
non-flat XCDM inflation models. Deviance information criterion (DIC) and
Akaike information criterion (AIC) are also listed.

Data sets | $\chi_{\textrm{plik}}^{2}$ | $\chi_{\textrm{lowl}}^{2}$ | $\chi_{\textrm{simall}}^{2}$ | $\chi_{\textrm{lensing}}^{2}$ | $\chi_{\textrm{prior}}^{2}$ | $\chi_{\textrm{SN}}^{2}$ | $\chi_{\textrm{BAO}}^{2}$ | $\chi_{H(z)}^{2}$ | $\chi_{f\sigma_{8}}^{2}$ | $\chi^{2}_{\textrm{total}}$ | $\Delta\chi^{2}$ | DIC | $\Delta\textrm{DIC}$ | $\Delta\textrm{AIC}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Flat XCDM model
Non-CMB (new) |  |  |  |  |  | 1411.93 | 21.66 | 14.84 | 10.75 | 1459.18 | $-10.75$ | 1468.74 | $-9.37$ | $-8.75$
P18 | 2341.65 | 22.46 | 395.79 |  | 1.51 |  |  |  |  | 2761.40 | $-4.40$ | 2815.67 | $-2.26$ | $-2.40$
P18+lensing | 2342.37 | 22.14 | 395.65 | 8.61 | 1.81 |  |  |  |  | 2770.58 | $-4.13$ | 2824.21 | $-2.24$ | $-2.13$
P18+non-CMB (new) | 2348.35 | 22.33 | 396.21 |  | 1.77 | 1413.17 | 25.60 | 14.97 | 17.44 | 4239.85 | $-0.39$ | 4294.20 | +1.87 | +1.61
P18+lensing+non-CMB (new) | 2347.50 | 22.54 | 396.48 | 9.01 | 1.57 | 1413.38 | 25.41 | 14.98 | 18.19 | 4249.05 | $-0.21$ | 4303.30 | +2.10 | +1.79
Flat XCDM+$A_{L}$ model
P18 | 2337.36 | 21.64 | 395.60 |  | 1.29 |  |  |  |  | 2755.89 | $-9.91$ | 2813.08 | $-4.85$ | $-5.91$
P18+lensing | 2341.84 | 21.98 | 395.62 | 9.04 | 1.94 |  |  |  |  | 2770.43 | $-4.28$ | 2825.81 | $-0.64$ | $-0.28$
P18+non-CMB (new) | 2338.34 | 21.56 | 395.67 |  | 1.29 | 1412.51 | 26.83 | 14.90 | 13.88 | 4224.98 | $-15.26$ | 4283.50 | $-8.83$ | $-11.26$
P18+lensing+non-CMB (new) | 2343.23 | 21.75 | 395.69 | 9.77 | 2.03 | 1412.70 | 26.87 | 14.93 | 13.95 | 4240.92 | $-8.34$ | 4296.89 | $-4.31$ | $-4.34$
Non-flat XCDM model [Planck $P(q)$]
Non-CMB (new) |  |  |  |  |  | 1412.51 | 22.67 | 14.77 | 10.85 | 1460.80 | $-13.29$ | 1468.14 | $-9.97$ | $-9.29$
P18 | 2336.63 | 21.31 | 395.63 |  | 1.34 |  |  |  |  | 2754.91 | $-10.89$ | 2810.86 | $-7.07$ | $-6.89$
P18+lensing | 2341.99 | 21.93 | 395.68 | 9.29 | 1.51 |  |  |  |  | 2770.40 | $-4.31$ | 2827.00 | +0.55 | $-0.31$
P18+non-CMB (new) | 2346.82 | 22.99 | 396.59 |  | 1.81 | 1412.60 | 25.56 | 14.79 | 17.50 | 4238.67 | $-1.57$ | 4294.75 | +2.42 | +2.43
P18+lensing+non-CMB (new) | 2346.33 | 23.13 | 396.00 | 9.28 | 2.09 | 1412.98 | 25.43 | 14.85 | 18.18 | 4248.26 | $-1.00$ | 4303.54 | +2.34 | +3.00
Non-flat XCDM+$A_{L}$ model [Planck $P(q)$]
P18 ($A_{L}>0.8$) | 2336.44 | 21.20 | 395.52 |  | 1.30 |  |  |  |  | 2754.46 | $-11.34$ | 2811.61 | $-6.32$ | $-5.34$
P18+lensing | 2341.46 | 22.28 | 395.66 | 9.04 | 1.83 |  |  |  |  | 2770.28 | $-4.43$ | 2829.13 | +2.68 | +1.57
P18+non-CMB (new) | 2337.83 | 21.81 | 395.74 |  | 1.37 | 1412.23 | 27.55 | 14.81 | 13.50 | 4224.83 | $-15.41$ | 4285.15 | $-7.18$ | $-9.41$
P18+lensing+non-CMB (new) | 2341.91 | 21.97 | 395.81 | 10.06 | 1.55 | 1412.20 | 27.24 | 14.79 | 14.17 | 4239.70 | $-9.56$ | 4298.54 | $-2.66$ | $-3.56$
Non-flat XCDM model [new $P(q)$]
Non-CMB (new) |  |  |  |  |  | 1411.93 | 21.98 | 14.88 | 10.71 | 1459.51 | $-13.27$ | 1468.73 | $-9.38$ | $-9.27$
P18 | 2338.49 | 21.43 | 396.14 |  | 1.81 |  |  |  |  | 2757.86 | $-7.94$ | 2811.78 | $-6.15$ | $-3.94$
P18+lensing | 2341.99 | 21.73 | 395.82 | 8.96 | 2.07 |  |  |  |  | 2770.57 | $-4.14$ | 2826.60 | +0.15 | $-0.14$
P18+non-CMB (new) | 2347.31 | 22.91 | 396.21 |  | 1.83 | 1412.59 | 25.67 | 14.81 | 17.24 | 4238.57 | $-1.67$ | 4294.90 | +2.57 | +2.33
P18+lensing+non-CMB (new) | 2345.80 | 23.23 | 396.59 | 9.00 | 1.82 | 1412.99 | 25.12 | 14.81 | 18.60 | 4247.96 | $-1.30$ | 4304.26 | +3.06 | +2.70
Non-flat XCDM+$A_{L}$ model [new $P(q)$]
P18 ($A_{L}>0.8$) | 2336.38 | 21.16 | 395.52 |  | 1.34 |  |  |  |  | 2754.40 | $-11.40$ | 2811.71 | $-6.22$ | $-5.40$
P18+lensing | 2341.66 | 21.74 | 395.68 | 9.20 | 1.98 |  |  |  |  | 2770.27 | $-4.44$ | 2828.10 | +1.65 | +1.56
P18+non-CMB (new) | 2337.48 | 21.79 | 395.75 |  | 1.35 | 1412.30 | 27.39 | 14.80 | 13.66 | 4224.52 | $-15.72$ | 4284.84 | $-7.49$ | $-9.72$
P18+lensing+non-CMB (new) | 2341.80 | 22.10 | 395.78 | 9.71 | 1.85 | 1412.28 | 27.47 | 14.79 | 13.97 | 4239.76 | $-9.50$ | 4298.27 | $-2.93$ | $-3.50$

Note: $\Delta\chi^{2}$, $\Delta\textrm{DIC}$, and $\Delta\textrm{AIC}$
indicate the values relative to those of the tilted flat $\Lambda\textrm{CDM}$
model for the same combination of data sets. For the tilted flat $\Lambda$CDM
model AIC$=2819.80$ (P18), $2828.71$ (P18+lensing), and $4303.26$
(P18+lensing+non-CMB (new)). All $\chi^{2}$ values are computed at the
corresponding model best-fit cosmological parameter values. See section IV B
of [10] for detailed descriptions of individual CMB $\chi^{2}$’s.

##  V Discussion

In this work we have employed P18 data, (P18) CMB lensing data, and non-CMB
data to place constraints on the cosmological parameters of twelve
cosmological models: flat $\Lambda$CDM (+$A_{L}$), non-flat $\Lambda$CDM
Planck $P(q)$ (+$A_{L}$), non-flat $\Lambda$CDM new $P(q)$ (+$A_{L}$), flat
XCDM (+$A_{L}$), non-flat XCDM Planck $P(q)$ (+$A_{L}$), and non-flat XCDM new
$P(q)$ ($+A_{L}$). In order to compare how well each of these models fit
observational data, we use two statistical criteria: the AIC (Eq. (10)) and
the DIC (Eq. (11)). Furthermore, with the help of two estimators (Eqs. (12)
and (15)), we determine the mutual (in)consistency between results obtained
from two different data sets when analyzed in a given model.

According to the statistical tools utilized, three of the twelve cosmological
models studied are rejected at 3$\sigma$ significance due to incompatibilities
between results obtained from different data sets, provided that these data
are correct and free from unaccounted systematic errors. From the P18 vs. non-
CMB (new) and P18+lensing vs. non-CMB (new) data comparisons, the three
cosmological models ruled out at confidence levels exceeding $3\sigma$ are the
flat XCDM, the non-flat XCDM Planck $P(q)$, and the non-flat XCDM new $P(q)$
models. Interestingly, when non-CMB (old) data is replaced by non-CMB (new)
data, the non-flat $\Lambda$CDM Planck $P(q)$ model is no longer rejected at
more than 3$\sigma$, which is what we found in [10] where we used non-CMB
(old) data.

When we compare the cosmological parameter constraints obtained with P18 data
and non-CMB (new) data in the flat XCDM model, we observe that the values of
all primary parameters disagree at more than $2\sigma$. In particular,
$\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, 100$\theta_{\text{MC}}$, and
$\ln(10^{10}A_{s})$ differ at $-2.14\sigma$, 2.34$\sigma$, 2.09$\sigma$, and
$-2.63\sigma$ respectively, with the difference in the values of the dark
energy equation of state parameter $w$ ($-4.80\sigma$) being the largest one.
Including the lensing data in the analysis and considering the P18+lensing
data vs. non-CMB (new) data comparison barely changes the level of
disagreement. In this case, the difference between the two sets of
cosmological parameter values for $\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$,
100$\theta_{\text{MC}}$, and $\ln(10^{10}A_{s})$ are $-2.13\sigma$,
2.27$\sigma$, 2.10$\sigma$, and $-2.65\sigma$, respectively, while the $w$
difference is $-4.27\sigma$. We also use two statistical estimators to measure
the degree of concordance/discordance between the results obtained with two
data sets in a given model. When comparing the P18 and non-CMB (new) results,
we have $\log_{10}\mathcal{I}=-2.125$ and $p=0.056\%$ ($\sigma=3.448$) with
both statistical estimators indicating a decisive degree of disagreement. As
for the results obtained when we study the case of P18+lensing data vs. non-
CMB data we obtain $\log_{10}\mathcal{I}=-2.247$ and $p=0.039\%$
($\sigma=3.555$), which is also decisive. Therefore, in light of these results
we conclude that the flat XCDM model is ruled out at more than 3$\sigma$
significance.

The other two cosmological models rejected by these data at 3$\sigma$
significance are the non-flat XCDM Planck $P(q)$ and the non-flat XCDM new
$P(q)$ models. For both models, when we compare the results obtained with P18
and non-CMB (new) data we find smaller differences compared to those for the
flat XCDM model. In the Planck $P(q)$ model, for the primary parameters common
to the flat $\Lambda$CDM model the largest difference is found for
$100\theta_{\text{MC}}$ ($-2.10\sigma$) whereas for the curvature parameter
$\Omega_{k}$ and the dark energy equation of state parameter $w$ we find
differences between the two values at $1.98\sigma$ and $-0.50\sigma$,
respectively. In regard to the incompatibilities found between P18 data and
non-CMB (new) data results we obtain $\log_{10}\mathcal{I}=-3.421$ and
$p=0.003\%$ ($\sigma=4.294$) which means that the level of discordance exceeds
3$\sigma$ and thus argues against the joint analysis of these two data sets in
the context of the non-flat XCDM Planck $P(q)$ model. When the lensing data
are included in the mix the level of disagreement between the results from the
two data sets is reduced, but is still high enough to reject this model at
more than 3$\sigma$. The primary parameters $100\theta_{\text{MC}}$,
$\Omega_{k}$, and $w$ values disagree at $-2.10\sigma$, $2.59\sigma$, and
$-1.20\sigma$ respectively whereas for the tension estimators we obtain
$\log_{10}\mathcal{I}=-1.824$ for the first one and $p=0.069\%$
($\sigma=3.396$) for the second one. Similar results are obtained for the non-
flat XCDM new $P(q)$ model but at a less severe level of disagreement. When we
compare P18 and non-CMB (new) data results, the largest differences between
the two sets of results are for $100\theta_{\text{MC}}$, $\Omega_{k}$, and
$w$, which differ by $-1.79\sigma$, $1.82\sigma$, and $-0.61\sigma$
respectively. As expected, the statistical estimators reconfirm the tension
observed in the differences in parameter values, with the first one being
$\log_{10}\mathcal{I}=-3.125$ while for the second one we have $p=0.007\%$
($\sigma=3.960$), with both indicating a decisive level of discordance between
the results. No significant changes are observed when the P18+lensing data vs.
non-CMB (new) data case is studied. The primary parameter
$100\theta_{\text{MC}}$, $\Omega_{k}$, and $w$ values differ at $-1.79\sigma$,
$2.14\sigma$, and $-1.21\sigma$ and for the statistical estimators of the
level of tension we find $\log_{10}\mathcal{I}=-1.942$ and $p=0.155\%$
($\sigma=3.164$). Given these results, both non-flat XCDM models with
$A_{L}=1$ are ruled out at $3\sigma$ confidence level.

Allowing the lensing consistency parameter $A_{L}$ to vary makes P18 data and
P18+lensing data results compatible with non-CMB (new) data results in the
XCDM models. In the flat XCDM model when P18 data are considered we find
$A_{L}=1.180^{+0.062}_{-0.10}$ which favors $A_{L}>1$ over $A_{L}=1$ by
$1.80\sigma$, and $w=-1.23^{+0.31}_{-0.59}$ indicating a preference for
phantom-behavior at $0.74\sigma$ significance, while for P18+non-CMB (new)
data we get $A_{L}=1.222\pm 0.063$ ($3.52\sigma$) and $w=-0.964\pm 0.024$
($1.50\sigma$ preference for quintessence-like behavior), and finally when
P18+lensing+non-CMB (new) data are analyzed we obtain $A_{L}=1.101\pm 0.037$
($2.73\sigma$) and $w=-0.968\pm 0.024$ ($1.33\sigma$ preference for
quintessence-like behavior). When $A_{L}$ is allowed to vary, the tensions
between cosmological parameter constraints obtained from the two different
data sets are alleviated with respect to the case when $A_{L}=1$. When we look
at the P18 vs. non-CMB (new) data case we find $\log_{10}\mathcal{I}=-0.364$
and $p=3.619\%$ ($\sigma=2.095$) whereas for the P18+lensing vs. non-CMB (new)
data case we get $\log_{10}\mathcal{I}=-0.506$ and $p=1.742\%$
($\sigma=2.378$), therefore, according to the statistical estimators the flat
XCDM+$A_{L}$ model is not ruled out at 3$\sigma$ significance by these data.
Regarding the performance of the flat XCDM+$A_{L}$ model with respect to the
flat $\Lambda$CDM model, when the P18+lensing+non-CMB (new) data set is
considered we obtain $\Delta\text{DIC}=-4.31$ which means that the first model
is positively favored over the flat $\Lambda$CDM model.

Very similar results are found for the non-flat XCDM Planck $P(q)+A_{L}$ and
the non-flat XCDM new $P(q)+A_{L}$ cosmological models. In particular, when
P18 data are analyzed we get for the XCDM Planck [new] $P(q)+A_{L}$ model
$\Omega_{k}=-0.073^{+0.065}_{-0.029}$ [$-0.072^{+0.065}_{-0.030}$],
$w=-1.36^{+1.1}_{-0.53}$ [$-1.39^{+1.1}_{-0.54}$], and $A_{L}<1.20$ [$<1.19$]
which indicates a preference for a closed Universe of $1.12\sigma$
[$1.11\sigma$] and a preference of $0.33\sigma$ [$0.35\sigma$] in favor of
phantom-like behavior. Jointly analyzing P18 and non-CMB (new) data, we obtain
$\Omega_{k}=0.0011\pm 0.0019$ [$0.0011\pm 0.0019$] for the curvature
parameter, favoring an open Universe by $0.58\sigma$ [$0.58\sigma$], while for
the dark energy equation of state parameter we get $w=-0.958\pm 0.027$
[$-0.959\pm 0.026$] indicating a preference for the quintessence-like behavior
by $1.56\sigma$ [$1.58\sigma$]. As for the lensing parameter, we find
$A_{L}=1.217\pm 0.064$ [$1.213\pm 0.064$] showing a preference for $A_{L}>1$
at $3.39\sigma$ [$3.33\sigma$]. Finally when we analyze the largest data set
considered in this work, namely P18+lensing+non-CMB (new), we find for the
three non-standard parameters $\Omega_{k}=0.0015\pm 0.0019$ [$0.0015\pm
0.0019$], $w=-0.958\pm 0.026$ [$-0.959\pm 0.027$], and $A_{L}=1.102\pm 0.037$
[$1.101\pm 0.038$]. According to these results there is a $0.79\sigma$
[$0.79\sigma$] preference for an open Universe, a preference for quintessence-
like behavior of $1.62\sigma$ [$1.52\sigma$], and the option $A_{L}>1$ is
preferred at $2.76\sigma$ [$2.66\sigma$]. It is important to note that while
P18 data show a preference for a closed Universe ($\Omega_{k}<0$) and phantom-
like behaviour ($w<-1$), when non-CMB (new) data are included in the mix
either in the P18+non-CMB (new) case or in the P18+lensing+non-CMB (new) case,
the opposite is true: an open Universe ($\Omega_{k}>0$) and quintessence-like
behavior ($w>-1$) are favored. In regard to the statistical estimators, for
the non-flat XCDM Planck [new] $P(q)+A_{L}$ model, in the case of P18 vs. non-
CMB (new) data we get $\log_{10}\mathcal{I}=-1.173$ [$-0.957$] and $p=0.902\%$
[$0.778\%$] ($\sigma=2.611$ [$2.662$]) and for P18+lensing vs. non-CMB (new)
data we have $\log_{10}\mathcal{I}=-0.275$ [$-0.312$] and $p=3.026\%$
[$2.409\%$] ($\sigma=2.167$ [$2.256$]). As noted previously, it is the
inclusion of a varying $A_{L}$ parameter that brings concordance to the
cosmological parameter constraints obtained from the two different data sets.
As for the performance of the non-flat XCDM+$A_{L}$ models we obtain
$\Delta\text{DIC}=-2.66[-2.93]$ pointing to a positive preference for the non-
flat models, with a varying equation of state parameter and lensing parameter,
over the flat $\Lambda$CDM model with $A_{L}=1$.

It is interesting and important to note that for the largest data set we
study, the P18+lensing+non-CMB (new) one, all models with a varying $A_{L}$
parameter are positively favored over the flat $\Lambda$CDM one. This seems to
indicate that consideration of this phenomenological parameter might be
necessary to get a better fit to these cosmological data. On the other hand,
we should take into account that the flat $\Lambda$CDM model is the simplest
observationally-consistent cosmological model among all the models studied in
this paper. It passes all consistency tests we have subjected it to and it
does so with a constant $\Lambda$ (not an evolving dark energy component) and
with flat hypersurfaces ($\Omega_{k}=0$) and with the lensing consistency
parameter set to unity ($A_{L}=1$). It is also interesting and important to
note that analyses of updated PR4 Planck data in the $\Lambda$CDM+$A_{L}$
model [119, 12] find $A_{L}$ values less inconsistent with $A_{L}=1$, and that
this change is partly due to the data set update and partly due to the
different likelihoods used.

There appears to be a mismatch between the observed smoothing of some parts of
the P18 CMB power spectra and the predicted smoothing due to weak
gravitational lensing in the six-parameter flat $\Lambda$CDM model that best
fits these data. One possible alternative that could resolve this issue is
considering negative values of the curvature parameter $\Omega_{k}<0$ (closed
geometry), which in turn allows for a larger non-relativistic matter density
parameter $\Omega_{m}$ values if $\Omega_{\Lambda}$ is held constant, compared
to the case with $\Omega_{k}=0$, thus incrementing the amount of lensing.
Another alternative, which is phenomenological rather than physical, is to re-
scale the gravitational potential power spectrum with the phenomenological
lensing consistency parameter $A_{L}$, which automatically increases the
amount of lensing for $A_{L}>1$. The second option allows $\Omega_{m}$ to
remain in the low-value region (compared to the case either with
$\Omega_{k}=0$ or with $\Omega_{k}<0$), so that the amount of structure
formation is reduced, affecting the value of $\sigma_{8}$, which is lower for
all $A_{L}$-varying models than for models with $A_{L}=1$. Consequently, the
better performance of $A_{L}$-varying models highlighted above may be related
to the simultaneous alleviation of the lensing anomaly and the $\sigma_{8}$
tension, when CMB and non-CMB (new) data are jointly analyzed, that affect the
flat $\Lambda$CDM model.

Consistent with what we found earlier for the $\Lambda$CDM models [10], when
only the CMB data (either P18 or P18+lensing data) are used it is not possible
to obtain nearly model-independent cosmological parameter constraints. To
obtain model-independent constraints it is necessary to include non-CMB data
in the mix. Actually, even considering P18+lensing+non-CMB (new) data, which
is the most restrictive data set we use, it is only possible to get almost
model-independent constraints when comparing the models with $A_{L}=1$ or when
comparing the models that allow for the variation of $A_{L}$. Among models
with $A_{L}=1$ the differences between the primary and derived cosmological
parameter values always remain below $1\sigma$. No particular parameter has
more model-dependent values uniformly across models; parameter value
differences vary, depending on the two models that are under comparison.
Therefore, it is not unreasonable to claim that P18+lensing+non-CMB (new) data
are able to set nearly model-independent cosmological parameter constraints
among the models with $A_{L}=1$. On the other hand, when comparing the
constraints obtained for the models with $A_{L}=1$ and those obtained for the
$A_{L}$-varying models we observe differences of close to $2\sigma$ for
$\sigma_{8}$ and close to $1\sigma$ for the second most affected parameter,
$\ln(10^{10}A_{s})$.

##  VI Conclusions

Under the condition that the data sets we have employed in this work are
correct and free from unaccounted systematics, the statistical estimators we
have used show that three of the twelve cosmological models we have studied
are rejected at 3$\sigma$ or more because they cannot simultaneously
accommodate either P18 and non-CMB (new) data or P18+lensing and non-CMB (new)
data. These models are the flat XCDM model and the non-flat XCDM Planck and
new $P(q)$ models.

As we showed in [10], in the $\Lambda$CDM models neither P18 nor P18+lensing
data are able to completely break the geometrical degeneracy between the
$\Omega_{m}-H_{0}-\Omega_{k}-A_{L}$ parameters and, as expected, this remains
true here when we additionally consider a varying dark energy equation of
state parameter $w$. According to the results obtained in this work, there are
three options for the flat $\Lambda$CDM model to deal with the lensing
anomaly, if we consider only one additional parameter at a time:
$\Omega_{k}<0$, $w>-1$, or $A_{L}>1$. In the case of the non-flat $\Lambda$CDM
models with $A_{L}=1$, P18 data favor a more negative value of the curvature
parameter, which is compensated by a larger value of the matter density
parameter $\Omega_{m}$ and a lower value of the Hubble constant $H_{0}$. On
the other hand, the results obtained for the flat XCDM model show that a more
negative dark energy equation of state parameter $w$ value is favored,
compensated by a smaller value of $\Omega_{m}$ and a higher value of $H_{0}$,
contrary to the case of the non-flat models. Finally in the case of the flat
$\Lambda$CDM+$A_{L}$ model, due to the weak correlations of the $A_{L}$
parameter with the other free parameters, we do not observe significant
changes in the values of $\Omega_{m}$ and $H_{0}$ with respect to the flat
$\Lambda$CDM model. According to the DIC values, among these three options the
most favored one when it comes to fitting P18 data turns out to be
$\Omega_{k}<0$: for the two non-flat $\Lambda$CDM models we get
$\Delta\textrm{DIC}=-7.34$ [Planck $P(q)$] and $\Delta\textrm{DIC}=-6.39$ [new
$P(q)$], respectively, which indicates that both models are strongly favored
over the flat $\Lambda$CDM model. On the other hand, for the flat
$\Lambda$CDM+$A_{L}$ model we get $\Delta\textrm{DIC}=-5.52$ whereas for the
flat XCDM model $\Delta\textrm{DIC}=-2.26$, meaning that both the
$\Lambda$CDM+$A_{L}$ model and the flat XCDM model are positively favored over
the standard flat $\Lambda$CDM model, with the flat $\Lambda$CDM+$A_{L}$ model
favored over the flat XCDM model.

Although CMB lensing data are not as restrictive as P18 data, in some cases we
find non-negligible changes in the cosmological parameter constraints when
P18+lensing data are used instead of P18 data alone. For the non-flat
$\Lambda$CDM models, we find that $\Omega_{k}$ is still negative but its
absolute value decreases, moving closer to flat spatial geometry. In addition,
$\Omega_{m}$ decreases whereas $H_{0}$ increases with respect to the values
from the P18 data analysis. When we move from P18 data to P18+lensing data,
the evidence in favor of $A_{L}>1$ decreases for the flat $\Lambda$CDM+$A_{L}$
model, but the results are barely changed for the flat XCDM model, indicating
that the XCDM model is not very sensitive to CMB lensing data.

Non-CMB (new) data by themselves do not support the results that we have just
summarized. For the non-flat $\Lambda$CDM models, with non-CMB (new) data the
evidence in favor of $\Omega_{k}<0$ (closed geometry) subsides and instead
positive values of the curvature parameter ($\Omega_{k}>0$, open geometry) are
found, contrary to what happens when analyzing non-CMB (old) data, [10]. Also,
with non-CMB (new) data smaller values of $\Omega_{m}$ and higher values of
$H_{0}$ are found with respect to the results obtained with P18 data. As for
the flat XCDM model, non-CMB (new) data favor larger values of $\Omega_{m}$
and lower values of $H_{0}$ compared to those from P18 data. From the DIC
values, we see that with non-CMB (new) data the non-flat $\Lambda$CDM Planck
$P(q)$ and $\Lambda$CDM new $P(q)$ models have $\Delta\textrm{DIC}=+1.41$ and
$\Delta\textrm{DIC}=+2.05$, respectively. These positive values indicate that
the two models are not favored over the flat $\Lambda$CDM model. However, the
flat XCDM model, which has $\Delta\textrm{DIC}=-9.37$ for non-CMB (new) data,
is strongly preferred when compared to the standard flat $\Lambda$CDM model.

The significant discrepancies found between the results obtained with
P18/P18+lensing data and non-CMB (new) data led us to make use of statistical
estimators to more properly quantify the tensions. As reported above, these
tests reveal that the three XCDM models with $A_{L}=1$ are ruled out at
$3\sigma$ or more. For the non-flat $\Lambda$CDM models, we observe that the
level of tension in the cases of P18 vs. non-CMB (new) data and P18+lensing
vs. non-CMB (new) data are reduced compared to the cases where non-CMB (old)
data were used instead of non-CMB (new) data, with the main consequence being
that here these models are not ruled out at the $3\sigma$ threshold. Regarding
the XCDM models with a varying $A_{L}$ parameter we note that while the levels
of tension are still high, they are less than $3\sigma$ and not enough to
disallow the joint analysis of P18, lensing, and non-CMB (new) data.

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

Our cosmological parameter constraints obtained for the flat $\Lambda$CDM
model, when P18+lensing+non-CMB (new) data are considered, are the most
restrictive results to date. In particular, for the six primary parameters we
get $\Omega_{b}h^{2}=0.02249\pm 0.00013$, $\Omega_{c}h^{2}=0.11849\pm
0.00084$, $100\theta_{\text{MC}}=1.04109\pm 0.00028$, $\tau=0.0569\pm 0.0071$,
$n_{s}=0.9685\pm 0.0036$, and $\ln(10^{10}A_{s})=3.046\pm 0.014$.
Additionally, for the derived parameters, we find $H_{0}=68.05\pm 0.38$ km s-1
Mpc-1, $\Omega_{m}=0.3059\pm 0.0050$, and $\sigma_{8}=0.8077\pm 0.0057$. Among
models with $A_{L}=1$, these values show almost model-independent consistency,
with differences always below $1\sigma$. However, when we compare these
cosmological parameter values with those obtained for the $A_{L}$-varying
models, we observe larger differences. In particular, for the six varying
$A_{L}$ models relative to the flat $\Lambda$CDM model we find the following
maximum differences for each parameter. For models with $w=-1$, we find
$-0.35\sigma$ for $\Omega_{b}h^{2}$, $0.53\sigma$ for $\Omega_{c}h^{2}$,
$-0.12\sigma$ for $100\theta_{\text{MC}}$, $0.81\sigma$ for $\tau$,
$-0.41\sigma$ for $n_{s}$, $0.95\sigma$ for $\ln(10^{10}A_{s})$, $0.71\sigma$
for $H_{0}$, $0.75\sigma$ for $\Omega_{m}$, and $1.08\sigma$ for $\sigma_{8}$,
whereas comparing $w$-varying model results we get $-0.55\sigma$ for
$\Omega_{b}h^{2}$, $0.81\sigma$ for $\Omega_{c}h^{2}$, $-0.27\sigma$ for
$100\theta_{\text{MC}}$, $0.69\sigma$ for $\tau$, $-0.66\sigma$ for $n_{s}$,
$0.88\sigma$ for $\ln(10^{10}A_{s})$, $0.21\sigma$ for $H_{0}$, $0.13\sigma$
for $\Omega_{m}$, and $1.80\sigma$ for $\sigma_{8}$.

When P18+lensing+non-CMB (new) data are analyzed, interesting trends related
to the lensing consistency parameter $A_{L}$ are observed. For models with
fixed $A_{L}=1$, we have on average evidence for open spatial geometry with
positive curvature parameter $\Omega_{k}>0$ ($0.63\sigma$) and a quintessence-
like dark energy equation of state parameter $w>-1$ ($0.63\sigma$). For models
with varying $A_{L}$ parameter, we get $\Omega_{k}>0$ ($0.52\sigma$), $w>-1$
($1.49\sigma$), and $A_{L}>1$ ($2.59\sigma$). Therefore, among the various
non-standard parameters explored in this work, the lensing consistency
parameter is the one that most deviates from the standard flat $\Lambda$CDM
model $A_{L}=1$ value. This conclusion is supported by the findings from the
DIC values which indicate that, when P18+lensing+non-CMB (new) data are used,
all $A_{L}$-varying models are positively favored over the flat $\Lambda$CDM
model.

The conclusions we have drawn in this work depend on the data sets we have
employed. The recent analysis of the updated PR4 Planck data set [12] results
in updated values for the curvature parameter ($\Omega_{k}=-0.012\pm 0.010$)
and the lensing consistency parameter ($A_{L}=1.039\pm 0.052$). These new
measurements show less evidence in favor of non-flat hypersurfaces and
$A_{L}>1$ compared to the results obtained from P18 data, partly as a
consequence of the updated PR4 Planck data set and partly as a consequence of
the different likelihoods used in the analyses. (One might view the second
source of difference as a systematic and it might then be appropriate to
account for it as an additional systematic error on cosmological parameter
values.)

More and better cosmological data are required in order to clarify whether the
tensions highlighted in our work, affecting the standard flat $\Lambda$CDM
model of cosmology, can really be interpreted as hints of new physics or if
they arise from some unaccounted systematics in the cosmological data we have
used. Given the circumstances, the flat $\Lambda$CDM model remains the
simplest (largely) observationally consistent cosmological model.

###### Acknowledgements.

J.d.C.P. was supported by the Margarita Salas fellowship funded by the
European Union (NextGenerationEU). C.-G.P. was supported by a National
Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)
No. RS-2023-00246367.

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