$\chi_{\textrm{min}}^{2}$ (Total) | $2757.38$ | $2786.27$ | $15.90$ | $2777.01$ | $10.67$
$\chi_{\textrm{min}}^{2}$ (BAO/BAO′) | $\cdots$ | $20.66$ | $15.90$ | $11.82$ | $10.67$
$\chi_{\textrm{BAO/BAO}^{\prime}}^{2}$ (at P18 B-F) | $\cdots$ | $278.54$ | $278.54$ | $236.71$ | $236.71$
DIC | $2811.54$ | $2840.16$ | $24.57$ | $2831.65$ | $17.69$
$\Delta\textrm{DIC}$ | $-6.39$ | $0.91$ | $2.64$ | $2.04$ | $2.76$
$\textrm{AIC}_{c}$ | $2813.38$ | $2842.27$ | $31.90$ | $2833.01$ | $24.39$
$\Delta\textrm{AIC}_{c}$ | $-6.42$ | $1.61$ | $4.34$ | $1.26$ | $4.41$
| Tilted non-flat $\Lambda$CDM+$A_{L}$ model [new $P(q)$]
Parameter | P18 | P18+BAO | BAO | P18+BAO′ | BAO′
$\Omega_{b}h^{2}$ | $0.02257\pm 0.00017$ | $0.02260\pm 0.00017$ | $0.039\pm 0.015$ | $0.02261\pm 0.00017$ | $0.042\pm 0.015$
$\Omega_{c}h^{2}$ | $0.1187\pm 0.0016$ | $0.1180\pm 0.0014$ | $0.174\pm 0.047$ | $0.1178\pm 0.0015$ | $0.177\pm 0.046$
$100\theta_{\textrm{MC}}$ | $1.04111\pm 0.00033$ | $1.04117\pm 0.00033$ | $1.17\pm 0.14$ | $1.04117\pm 0.00032$ | $1.13\pm 0.13$
$\tau$ | $0.0512\pm 0.0086$ | $0.0532\pm 0.0081$ | $0.0512$ | $0.0495\pm 0.0084$ | $0.0512$
$\Omega_{k}$ | $-0.10\pm 0.11$ | $-0.0005\pm 0.0017$ | $-0.055\pm 0.060$ | $-0.0012\pm 0.0018$ | $-0.035\pm 0.059$
$n_{s}$ | $0.9654\pm 0.0057$ | $0.9707\pm 0.0044$ | $0.9654$ | $0.9715\pm 0.0047$ | $0.9654$
$\ln(10^{10}A_{s})$ | $3.036\pm 0.018$ | $3.038\pm 0.017$ | $2.69\pm 0.43$ | $3.029\pm 0.018$ | $3.036$
$A_{L}$ | $0.94\pm 0.20$ | $1.168\pm 0.061$ | $0.94$ | $1.176\pm 0.062$ | $0.94$
$H_{0}$ | $51\pm 14$ | $68.09\pm 0.60$ | $83\pm 12$ | $67.85\pm 0.63$ | $84\pm 11$
$\Omega_{m}$ | $0.70\pm 0.43$ | $0.3048\pm 0.0062$ | $0.304\pm 0.019$ | $0.3065\pm 0.0065$ | $0.306\pm 0.020$
$\sigma_{8}$ | $0.752\pm 0.052$ | $0.8026\pm 0.0086$ | $0.844\pm 0.048$ | $0.7982\pm 0.0092$ | $0.99\pm 0.18$
$\chi_{\textrm{min}}^{2}$ (Total) | $2756.33$ | $2776.32$ | $15.90$ | $2767.43$ | $10.68$
$\chi_{\textrm{min}}^{2}$ (BAO/BAO′) | $\cdots$ | $20.30$ | $15.90$ | $11.21$ | $10.68$
$\chi_{\textrm{BAO/BAO}^{\prime}}^{2}$ (at P18 B-F) | $\cdots$ | $194.81$ | $194.81$ | $160.72$ | $160.72$
DIC | $2814.83$ | $2834.67$ | $24.75$ | $2824.97$ | $17.76$
$\Delta\textrm{DIC}$ | $-3.10$ | $-4.58$ | $2.82$ | $-4.64$ | $2.83$
$\textrm{AIC}_{c}$ | $2814.33$ | $2834.32$ | $31.90$ | $2825.43$ | $24.39$
$\Delta\textrm{AIC}_{c}$ | $-5.47$ | $-6.34$ | $4.34$ | $-6.32$ | $4.41$

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

Figure 16: Likelihood distributions constrained by the Planck 2018
TT,TE,EE+lowE (P18), BAO, and BAO′ data sets in the tilted flat $\Lambda$CDM
model.

Figure 17: Likelihood distributions constrained by the Planck 2018
TT,TE,EE+lowE (P18), BAO, and BAO′ data sets in the tilted flat
$\Lambda$CDM$+A_{L}$ model.

Figure 18: Likelihood distributions constrained by the Planck 2018
TT,TE,EE+lowE (P18), BAO, and BAO′ data sets in the untilted non-flat
$\Lambda$CDM model.

Figure 19: Likelihood distributions constrained by the Planck 2018
TT,TE,EE+lowE (P18), BAO, and BAO′ data sets in the untilted non-flat
$\Lambda$CDM$+A_{L}$ model.

Figure 20: Likelihood distributions constrained by the Planck 2018
TT,TE,EE+lowE (P18), BAO, and BAO′ data sets in the tilted non-flat
$\Lambda$CDM model with the Planck $P(q)$.

Figure 21: Likelihood distributions constrained by the Planck 2018
TT,TE,EE+lowE (P18), BAO, and BAO′ data sets in the tilted non-flat
$\Lambda$CDM$+A_{L}$ model with the Planck $P(q)$.

Figure 22: Likelihood distributions constrained by the Planck 2018
TT,TE,EE+lowE (P18), BAO, and BAO′ data sets in the tilted non-flat
$\Lambda$CDM model with the new $P(q)$.

Figure 23: Likelihood distributions constrained by the Planck 2018
TT,TE,EE+lowE (P18), BAO, and BAO′ data sets in the tilted non-flat
$\Lambda$CDM$+A_{L}$ model with the new $P(q)$.

####  IV.1.5 Comparing P18 data and BAO/BAO′ data cosmological constraints

In this subsubsection we compare BAO and BAO′ data cosmological constraints to
those obtained from P18 data. Prior to jointly analyzing P18+BAO/BAO′ data, we
need to determine whether P18 and BAO/BAO′ data cosmological constraints are
mutually consistent. In Sec. IV.3 we use two other statistical estimators to
examine whether or not P18 and BAO/BAO′ data are in tension.

The cosmological parameter mean values and error bars favored by the P18, BAO,
BAO′, P18+BAO, and P18+BAO′ data sets are summarized in Tables 8-11 for the
tilted flat $\Lambda$CDM (+$A_{L}$) models, the untilted non-flat $\Lambda$CDM
(+$A_{L}$) models, the tilted non-flat $\Lambda$CDM (+$A_{L}$) models with the
Planck $P(q)$, and the tilted non-flat $\Lambda$CDM ($+A_{L}$) models with the
new $P(q)$, respectively. Likelihood distributions of cosmological parameters
of the four models with $A_{L}=1$ and $A_{L}$ varying are shown in Figs. 16-23
for the P18, BAO, BAO′, P18+BAO, and P18+BAO′ data sets.

Since neither BAO′ nor BAO data have the ability to constrain $\tau$ or
$n_{s}$ or $A_{L}$, we set their values to those found in the corresponding
P18 data analysis. In addition, for the same reason, in the BAO′ data
analyses, we also set the value of $\ln(10^{10}A_{s})$ to that found in the
corresponding P18 data analysis. We see from the upper and lower panels of
Tables 8-11 that the BAO and BAO′ data results for the $A_{L}=1$ and
$A_{L}$-varying cases are similar, even though the fixed $\tau$ and $n_{s}$
[and $\ln(10^{10}A_{s})$] values are slightly different for the $A_{L}=1$ and
$A_{L}$-varying cases.

From Tables 8-11 we see that, in the six non-flat $\Lambda$CDM (+$A_{L}$)
models the constraints set by BAO′/BAO data on $\Omega_{m}$ are tighter than
the ones imposed by P18 data and in the three non-flat $\Lambda$CDM+$A_{L}$
models the constraints set by BAO′/BAO data on $\Omega_{k}$ are tighter than
the ones imposed by P18 data. P18 data more restrictively constrains all other
parameters in all eight cosmological models.

As we discuss below, there is a significant level of disagreement in the non-
flat models between P18 data cosmological constraints and BAO′/BAO data
cosmological constraints, in most cases. From Tables 9-11 we see that all
three data sets, P18, BAO′, and BAO, favor negative values of the curvature
parameter, with BAO′ and BAO data favoring closed geometry only weakly, at
0.48$\sigma$ to 0.96$\sigma$. However, we should take into account the
geometrical degeneracy between $H_{0}$-$\Omega_{k}$-$\Omega_{m}$ and note that
both BAO′ and BAO data favor higher values of $H_{0}$ and lower values of
$\Omega_{m}$ than do P18 data and this is what causes the P18 and BAO/BAO′
cosmological constraint differences.

We first discuss BAO′ data results (BAO′ data do not include $f\sigma_{8}$
data points, see Sec. II) and then consider results from BAO data. This will
allow us to test the impact of some $f\sigma_{8}$ data on the cosmological
constraints.

Comparing the six-parameter and the three-parameter tilted flat $\Lambda$CDM
model primary cosmological parameter constraints for P18 and BAO′ data, shown
in the upper half of Table 8, we see that the values of $\Omega_{b}h^{2}$ and
$\Omega_{c}h^{2}$ are in mild disagreement, at 1.3$\sigma$ and 1.1$\sigma$
respectively. We also observe a similar 1.3$\sigma$ level of tension in the
derived $H_{0}$ values, whereas the other two derived parameters, $\Omega_{m}$
and $\sigma_{8}$, show a better agreement, disagreeing by only 0.91$\sigma$
and 0.90$\sigma$ respectively.

Comparing the seven-parameter and the three-parameter tilted flat
$\Lambda$CDM+$A_{L}$ model primary cosmological parameter constraints for P18
and BAO′ data, shown in the lower half of Table 8, we see that the values of
$\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, and $100\theta_{\textrm{MC}}$ are in
1.7$\sigma$, 1.4$\sigma$ and 1.3$\sigma$ tension respectively. As for the
derived parameters, we find $H_{0}$ and $\sigma_{8}$ values are in 1.7$\sigma$
and 1.2$\sigma$ disagreement while $\Omega_{m}$ values differ by only
0.046$\sigma$. This means that only for the $\Omega_{m}$ parameter does the
inclusion of a varying $A_{L}$ reduce the disagreement found in the $A_{L}=1$
case, while increasing the disagreement for a number of other parameters.

P18 and BAO′ data results obtained for the six-parameter and the three-
parameter untilted non-flat $\Lambda$CDM model, shown in the upper half of
Table 9, indicate more significant differences than found in the tilted flat
$\Lambda$CDM model. The primary cosmological parameters $\Omega_{b}h^{2}$ and
$\Omega_{c}h^{2}$ values disagree at 1.1$\sigma$ and 1.4$\sigma$, while the
primary spatial curvature parameter value is $\Omega_{k}=-0.034\pm 0.057$ for
BAO′ data, which is 0.60$\sigma$ away from flat and in 0.99$\sigma$ tension
with the P18 value $\Omega_{k}=-0.095\pm 0.024$, which is 4.0$\sigma$ away
from flat. Regarding the derived parameters, we find that $\Omega_{m}$,
$H_{0}$, and $\sigma_{8}$ values are in 3.7$\sigma$, 2.9$\sigma$, and
1.5$\sigma$ disagreement. According to these results, P18 and BAO′ data
probably should not be jointly analyzed in the context of the untilted non-
flat $\Lambda$CDM model.

The results for the seven-parameter and the three-parameter untilted non-flat
$\Lambda$CDM+$A_{L}$ model, obtained considering P18 and BAO′ data, are in the
lower half of Table 9. While there is a slight increase in the disagreement
between the values of the primary parameters $\Omega_{b}h^{2}$ (1.3$\sigma$)
and $\Omega_{c}h^{2}$ (1.6$\sigma$), there are significant decreases for the
derived parameters $\Omega_{m}$, and $H_{0}$, but not for $\sigma_{8}$, that
now disagree by 0.93$\sigma$, 1.5$\sigma$, and 1.5$\sigma$ respectively. This
is caused by the increase in the size of the error bars in the $A_{L}$-varying
P18 case with respect to the corresponding values obtained with $A_{L}=1$. For
the BAO′ data primary spatial curvature parameter we find
$\Omega_{k}=-0.035\pm 0.058$, which is 0.60$\sigma$ away from flat
hypersurfaces and only in 0.64$\sigma$ tension with the P18 value
$\Omega_{k}=-0.12\pm 0.12$, which is now only 1.0$\sigma$ away from flat.
According to these results, unlike in the $A_{L}=1$ case, in the
$A_{L}$-varying case P18 and BAO′ data can probably be jointly analyzed in the
context of the untilted non-flat $\Lambda$CDM model. Note that in this case a
joint analysis of P18+BAO′ data favors closed geometry at 4.9$\sigma$, with
$\Omega_{k}=-0.0073\pm 0.0015$, although because of the lack of the tilt
($n_{s}$) degree of freedom this untilted non-flat $\Lambda$CDM+$A_{L}$ model
does not provide a good fit to smaller-angular-scale P18 data, which is
reflected in the large $\Delta$DIC and $\Delta$AICc values for the P18+BAO′
case in the lower half of Table 9.

Comparing the seven-parameter and the four-parameter tilted non-flat
$\Lambda$CDM Planck $P(q)$ model primary cosmological parameter constraints
for P18 and BAO′ data, we see, in the upper half of Table 10, that the values
of $\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ are both in 1.1$\sigma$
disagreement. The BAO′ data primary spatial curvature parameter value
$\Omega_{k}=-0.033\pm 0.055$ is 0.6$\sigma$ away from flat and only in
0.17$\sigma$ tension with the P18 value $\Omega_{k}=-0.043\pm 0.017$, which is
2.5$\sigma$ in favor of closed geometry. The derived parameters $\Omega_{m}$,
$H_{0}$, and $\sigma_{8}$ are in 2.7$\sigma$, 2.3$\sigma$, and 1.2$\sigma$
tension. These results reveal that P18 and BAO′ data cosmological constraints
are somewhat inconsistent in the tilted non-flat $\Lambda$CDM Planck $P(q)$
model and these data probably should not be used jointly to constrain this
model.

Looking at the lower half of Table 10 we can compare results obtained for the
eight-parameter and the four-parameter tilted non-flat $\Lambda$CDM+$A_{L}$
Planck $P(q)$ model from P18 and BAO′ data respectively. We observe that the
primary parameters $\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ are in 1.6$\sigma$
and 1.5$\sigma$ tension. For the BAO′ data primary spatial curvature parameter
we find $\Omega_{k}=-0.026\pm 0.054$, which is only 0.48$\sigma$ away from
flat and in 0.95$\sigma$ tension with the P18 value $-0.130\pm 0.095$, which
is 1.4$\sigma$ away from flat. Regarding the derived parameters we find that
$\Omega_{m}$, $H_{0}$, and $\sigma_{8}$ are in 1.4$\sigma$, 2.7$\sigma$ and
1.6$\sigma$ disagreement. Compared to the $A_{L}=1$ case, in the
$A_{L}$-varying case we find a significant reduction only in the $\Omega_{m}$
tension, with most of the other parameter disagreements being more
significant, which again suggests that P18 and BAO′ data should not be jointly
analyzed within the tilted non-flat $\Lambda$CDM+$A_{L}$ Planck $P(q)$ model.

Comparing the seven-parameter and the four-parameter tilted non-flat
$\Lambda$CDM new $P(q)$ model primary cosmological parameter constraints for
P18 and BAO′ data, from the upper half of Table 11 we see that the values of
$\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ both disagree at 1.2$\sigma$. The BAO′
data primary spatial curvature parameter value is $\Omega_{k}=-0.032\pm
0.059$, which is only a 0.54$\sigma$ deviation from flat and, similar to the
Planck $P(q)$ model, is only in 0.016$\sigma$ disagreement with the P18 value
$-0.033\pm 0.014$, which is 2.4$\sigma$ away from flat. Regarding the derived
parameters $\Omega_{m}$, $H_{0}$, and $\sigma_{8}$, we find that their values
disagree at 2.3$\sigma$, 2.1$\sigma$, and 1.1$\sigma$ respectively. While
these disagreements are smaller than the ones found in the tilted non-flat
$\Lambda$CDM Planck $P(q)$ model, they still are large enough to require we
more carefully test whether P18 and BAO′ data can be jointly used to constrain
cosmological parameters in this cosmological model.

The results for the eight-parameter and the four-parameter tilted non-flat
$\Lambda$CDM+$A_{L}$ new $P(q)$ model are in the lower half of Table 11, for
P18 and BAO′ data, respectively. As happens in the Planck $P(q)$ model, when
the $A_{L}$ parameter is allowed to vary the tensions found for the primary
parameters $\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ do not decrease (in fact
they slightly increase) with respect to the $A_{L}=1$ case, both now being
1.3$\sigma$. For the BAO′ data primary spatial curvature parameter we find
$\Omega_{k}=-0.035\pm 0.059$, which is 0.59$\sigma$ away from flat
hypersurfaces and only in 0.52$\sigma$ tension with the P18 value
$\Omega_{k}=-0.10\pm 0.11$, which is 0.91$\sigma$ away from flat. As for the
value of the derived parameters $\Omega_{m}$, $H_{0}$ and $\sigma_{8}$ we find
disagreements at 0.92$\sigma$, 1.9$\sigma$, and 1.3$\sigma$ respectively. The
tensions are reduced with respect to the case with $A_{L}=1$, due to the
increase of the error bars, but possibly still are not small enough to allow
the joint use of P18+BAO′ data for constraining tilted non-flat
$\Lambda$CDM+$A_{L}$ new $P(q)$ model cosmological parameters.

We now comment on the consistency between the cosmological constraints
obtained using the BAO data set (which contain some $f\sigma_{8}$ data points)
and the P18 data cosmological constraints. Here we also have to deal with the
$\sigma_{8}$ tension, namely the discrepancy between the larger value for
$\sigma_{8}$ obtained when P18 data are considered and the typically smaller
values that one gets from low-redshift structure formation data (the
$f\sigma_{8}$ data points we consider) or from weak lensing measurements. Note
that since BAO data include some $f\sigma_{8}$ measurements we allow for
ln($10^{10}A_{s}$) to vary in the BAO data only analyses (unlike the BAO′ data
only analyses where we fix the value of this parameter). We shall see that the
tilted non-flat $\Lambda$CDM new $P(q)$ model is the model that best
reconciles these measurements.

Comparing the six-parameter and the four-parameter tilted flat $\Lambda$CDM
primary cosmological parameter constraints for P18 and BAO data, shown in the
upper half of Table 8, we see that the values of $\Omega_{b}h^{2}$ and
$\Omega_{c}h^{2}$ are in 1.3$\sigma$ and 1.0$\sigma$ tension, respectively. A
similar level of disagreement is found if we look at the values of the derived
parameters. In particular for $\Omega_{m}$, $H_{0}$, and $\sigma_{8}$ we find
1.3$\sigma$, 1.3$\sigma$, and 1.6$\sigma$ disagreement. Here the greatest
disagreement is that affecting $\sigma_{8}$, which has to do with the
$\sigma_{8}$ tension mentioned above.

Considering the results presented in the lower half of Table 8 for the seven-
parameter and the four-parameter tilted flat $\Lambda$CDM+$A_{L}$ model,
obtained for P18 and BAO data, respectively, we find that including a varying
$A_{L}$ parameter does not decrease the primary parameter tensions found when
$A_{L}=1$. For $\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ the disagreement is now
1.4$\sigma$ and 1.1$\sigma$. On the other hand for the derived $\Omega_{m}$,
$H_{0}$, and $\sigma_{8}$ we find that their corresponding values disagree at
0.50$\sigma$, 1.2$\sigma$, and 2.0$\sigma$. Once again, allowing $A_{L}$ to
vary reduces the $\Omega_{m}$ disagreement and the largest disagreement is
between the $\sigma_{8}$ values.

Comparing the six-parameter and the five-parameter untilted non-flat
$\Lambda$CDM model primary cosmological parameter constraints for P18 and BAO
data, provided in the upper half of Table 9, we observe that the values of
$\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ show a disagreement of 1.1$\sigma$ and
1.4$\sigma$, respectively. The BAO data value for the primary spatial
curvature parameter is $\Omega_{k}=-0.047\pm 0.059$, which is 0.80$\sigma$
away from flat hypersurfaces and in 0.75$\sigma$ tension with the P18 value
$-0.095\pm 0.024$, which represents a 4.0$\sigma$ deviation from flat. The
level of tension is worse for the derived parameters $\Omega_{m}$, $H_{0}$,
and $\sigma_{8}$, the disagreements now being 3.7$\sigma$, 3.0$\sigma$, and
2.4$\sigma$. We may say that P18 and BAO data should not be jointly used to
constrain cosmological parameters in the untilted non-flat $\Lambda$CDM model.

Results for the seven-parameter and the five-parameter untilted non-flat
$\Lambda$CDM+$A_{L}$ model for P18 and BAO data, respectively, can be seen in
the lower half of Table 9. Again we do not observe a reduction in the tension
for the primary parameters $\Omega_{b}h^{2}$ (1.2$\sigma$) and
$\Omega_{c}h^{2}$ (1.4$\sigma$) compared with the results found for the
$A_{L}=1$ case. On the other hand, there is an important decrease for the
derived parameters $\Omega_{m}$, $H_{0}$, and $\sigma_{8}$, the disagreement
now being 0.94$\sigma$, 1.5$\sigma$, and 1.8$\sigma$, respectively. This is
probably caused by the increase in the size of the error bars in the
$A_{L}$-varying P18 case, with respect to the corresponding values obtained
with $A_{L}=1$. For the BAO data primary spatial curvature parameter we find
$\Omega_{k}=-0.050\pm 0.060$, which is 0.83$\sigma$ away from flat and in
0.52$\sigma$ tension with the P18 value $\Omega_{k}=-0.12\pm 0.12$, which is
1.0$\sigma$ in favor of a closed geometry.

Comparing the seven-parameter and the five-parameter tilted non-flat
$\Lambda$CDM Planck $P(q)$ model primary cosmological parameter constraints
for P18 and BAO data, shown in the upper half of Table 10, we see that the
values of $\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ are both in 1.2$\sigma$
tension. The BAO data primary spatial curvature parameter
$\Omega_{k}=-0.046\pm 0.060$ is 0.77$\sigma$ away from flat hypersurfaces and,
as in the BAO′ case, in good agreement with, differing only by 0.048$\sigma$
from, the P18 result $-0.043\pm 0.017$, which is 2.5$\sigma$ away from flat).
As for the derived parameters $\Omega_{m}$, $H_{0}$, and $\sigma_{8}$ we
observe disagreements of 2.7$\sigma$, 2.3$\sigma$ and 1.5$\sigma$. These
results reveal an inconsistency between P18 and BAO cosmological constraints
that probably mean P18 and BAO data should not be used to jointly constrain
cosmological parameters in the tilted non-flat $\Lambda$CDM Planck $P(q)$
model.

We provide results for the eight-parameter and the five-parameter tilted non-
flat $\Lambda$CDM+$A_{L}$ Planck $P(q)$ model, from P18 and BAO data, in the
lower half of Table 10. For the primary parameters $\Omega_{b}h^{2}$ and
$\Omega_{c}h^{2}$ we find a tension between the P18 and BAO values of
1.3$\sigma$ and 1.2$\sigma$, respectively. For the BAO data primary spatial
curvature parameter we find $\Omega_{k}=-0.045\pm 0.063$, which represents a
0.71$\sigma$ evidence in favor of closed geometry and is only in 0.75$\sigma$
tension with respect to the P18 value $-0.130\pm 0.095$, which represents a
1.4$\sigma$ deviation from flat. Regarding the derived $\Omega_{m}$, $H_{0}$,
and $\sigma_{8}$ parameters, the observed disagreements are 1.4$\sigma$,
2.5$\sigma$, and 1.8$\sigma$. The tension for $\Omega_{m}$ has reduced
significantly with respect to the $A_{L}=1$ case, however overall the
disagreements are still large enough to not allow one to jointly analyze P18
and BAO data in this cosmological model.

Comparing the seven-parameter and the five-parameter tilted non-flat
$\Lambda$CDM new $P(q)$ model primary cosmological parameter constraints for
P18 and BAO data, shown in the upper half of Table 11, we see that the values
of $\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ are both in 1.1$\sigma$
disagreement. The BAO data value of the primary spatial curvature parameter is
$\Omega_{k}=-0.051\pm 0.061$, which represents a 0.84$\sigma$ deviation from a
flat geometry and is only in 0.29$\sigma$ disagreement with the P18 value
$\Omega_{k}=-0.033\pm 0.014$, which is 2.4$\sigma$ away from flat. Regarding
the derived parameters $\Omega_{m}$, $H_{0}$, and $\sigma_{8}$, we find
2.4$\sigma$, 2.1$\sigma$, and 1.2$\sigma$ disagreements between the
corresponding values. It is necessary to further study the possible tension
between P18 and BAO within this model.

Results for the eight-parameter and the five-parameter tilted non-flat
$\Lambda$CDM+$A_{L}$ new $P(q)$ model, obtained from P18 and BAO data, can be
seen in the lower half of Table 11. For the primary parameters
$\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ the disagreement is at 1.1$\sigma$ and
1.2$\sigma$ respectively. For the BAO data primary spatial curvature parameter
we find $\Omega_{k}=-0.055\pm 0.060$, which represents 0.92$\sigma$ evidence
in favor of closed geometry and is in only 0.36$\sigma$ disagreement with the
P18 value $-0.10\pm 0.11$, which represents a 0.91$\sigma$ deviation from
flat. Regarding the derived parameters, $\Omega_{m}$, $H_{0}$, and
$\sigma_{8}$ we find 0.92$\sigma$, 1.7$\sigma$, and 1.3$\sigma$ disagreements.
The tensions for $H_{0}$ and $\Omega_{m}$ have reduced with respect to the
case with $A_{L}=1$, however they are still large enough to wonder whether we
can jointly analyze P18 and BAO data in the context of this model.

In Tables 8-11 $\chi^{2}_{\textrm{min}}$ (BAO/BAO′) is the value of $\chi^{2}$
for BAO or BAO′ data respectively, at the best-fit position for BAO or BAO′
data, while $\chi^{2}_{\textrm{BAO/BAO}^{\prime}}$ (at P18 B-F) is the value
of $\chi^{2}$ for BAO or BAO′ data evaluated at the best-fit position for P18
data. The values of $\chi_{\textrm{min}}^{2}$ (BAO/BAO′) and
$\chi_{\textrm{BAO/BAO}^{\prime}}^{2}$ (at P18 B-F) gives a qualitative
indication of the agreement or disagreement in the values of the cosmological
parameters obtained by considering P18 data and by considering BAO/BAO′ data.
If the cosmological parameters agree one might expect that
$\chi_{\textrm{min}}^{2}$ (BAO/BAO′)$\simeq$
$\chi_{\textrm{BAO/BAO}^{\prime}}^{2}$ (at P18 B-F). We see that this is the
case only for the tilted flat $\Lambda$CDM(+$A_{L}$) models for the BAO′ data,
but again emphasize that this is only a qualitative probe.

Figures 16-23 show one-dimensional likelihoods and two-dimensional contours
for cosmological parameters obtained using P18, BAO′, BAO, P18+BAO′, and
P18+BAO data. As mentioned above, BAO′ data constraints (shown in green) and
BAO data constraints (shown in grey) are comparatively less restrictive than
P18 constraints (shown in dark blue), are unable to put tight constraints on
the primary cosmological parameters (except for $\Omega_{k}$ in the three non-
flat $\Lambda$CDM$+A_{L}$ models), in most cases overlap at 2$\sigma$ with
each other, and in many cases they also overlap with the P18 data constraints.
Since the BAO data set contains more measurements than the BAO′ data set, the
BAO constraints are typically more restrictive, and BAO data, which includes
$f\sigma_{8}$ measurements, are much more effective at constraining
$\sigma_{8}$ than are BAO′ data.

Figures 16 and 17 are for tilted flat $\Lambda$CDM (+$A_{L}$) models. The
$\sim 1\sigma$ disagreements between the BAO′/BAO constraints and those
obtained with P18 data, discussed above, can be clearly seen in the contour
plots. For the tilted flat $\Lambda$CDM model the larger disagreements are in
panels for derived cosmological parameters, with the largest for $\sigma_{8}$.
Some of these disagreements decrease when the $A_{L}$ parameter is allowed to
vary.

Looking at the contour plots for the untilted non-flat $\Lambda$CDM (+$A_{L}$)
models (see Figs. 18 and 19) we observe non-overlapping contours in those
panels that involve the derived parameters $\Omega_{m}$ and $H_{0}$. These
disagreements are smaller when $A_{L}$ is allowed to vary. This may indicate
that in the context of this cosmological model we may jointly analyze P18 data
with BAO′/BAO data only when $A_{L}$ is allowed to vary.

Figures 20 and 21 show cosmological parameter constraints for the tilted non-
flat $\Lambda$CDM (+$A_{L}$) Planck $P(q)$ models, while the ones for the
tilted non-flat $\Lambda$CDM(+$A_{L}$) new $P(q)$ models are displayed in
Figs. 22 and 23. As expected, considering the results discussed above in this
subsubsection, the contour plots for these tilted non-flat models are quite
similar. We see in the panels that involve the primary cosmological parameters
there is overlap at 1$\sigma$, not only when $A_{L}$ is allowed to vary but
also when $A_{L}=1$. When $A_{L}=1$, for the Planck $P(q)$ model some P18 and
BAO′/BAO data constraint contours that involve $\Omega_{m}$ and $H_{0}$ do not
overlap even at 2$\sigma$. This is not true for the new $P(q)$ model with
$A_{L}=1$, where overlap is reached at $<2\sigma$. This may indicate that the
new $P(q)$ model is better able to reconcile P18 and BAO′/BAO data.

In view of the results discussed in this subsubsection, further tests are
needed to properly quantify the level of disagreement, in the context of non-
flat models, between P18 data and BAO′/BAO data cosmological constraints. We
return to this issue in Sec. IV.3.

Table 12: Mean and 68.3% confidence limits of tilted flat
$\Lambda\textrm{CDM}$ (+$A_{L}$) model parameters constrained by non-CMB, P18,
and P18+non-CMB data sets. $H_{0}$ has units of km s-1 Mpc-1.

| Tilted flat $\Lambda$CDM | Tilted flat $\Lambda$CDM$+A_{L}$
---|---|---
Parameter | Non-CMB | P18 | P18+non-CMB | P18 | P18+non-CMB
$\Omega_{b}h^{2}$ | $0.0256\pm 0.0025$ | $0.02236\pm 0.00015$ | $0.02250\pm 0.00012$ | $0.02259\pm 0.00017$ | $0.02265\pm 0.00014$
$\Omega_{c}h^{2}$ | $0.1129\pm 0.0062$ | $0.1202\pm 0.0014$ | $0.11825\pm 0.00087$ | $0.1180\pm 0.0015$ | $0.11736\pm 0.00092$
$100\theta_{\textrm{MC}}$ | $1.0323\pm 0.0082$ | $1.04090\pm 0.00031$ | $1.04112\pm 0.00029$ | $1.04114\pm 0.00032$ | $1.04120\pm 0.00029$
$\tau$ | $0.0542$ | $0.0542\pm 0.0079$ | $0.0548\pm 0.0076$ | $0.0496\pm 0.0082$ | $0.0484\pm 0.0083$
$n_{s}$ | $0.9649$ | $0.9649\pm 0.0043$ | $0.9692\pm 0.0036$ | $0.9710\pm 0.0050$ | $0.9726\pm 0.0038$
$\ln(10^{10}A_{s})$ | $3.10\pm 0.11$ | $3.044\pm 0.016$ | $3.041\pm 0.015$ | $3.030\pm 0.017$ | $3.026\pm 0.017$
$A_{L}$ | $\cdots$ | $\cdots$ | $\cdots$ | $1.181\pm 0.067$ | $1.201\pm 0.061$
$H_{0}$ | $69.8\pm 1.7$ | $67.28\pm 0.61$ | $68.15\pm 0.39$ | $68.31\pm 0.71$ | $68.62\pm 0.43$
$\Omega_{m}$ | $0.286\pm 0.011$ | $0.3165\pm 0.0084$ | $0.3045\pm 0.0051$ | $0.3029\pm 0.0093$ | $0.2988\pm 0.0054$
$\sigma_{8}$ | $0.787\pm 0.027$ | $0.8118\pm 0.0074$ | $0.8048\pm 0.0068$ | $0.7997\pm 0.0088$ | $0.7961\pm 0.0074$
$\chi_{\textrm{min}}^{2}$ (Total) | $1106.54$ | $2765.80$ | $3879.35$ | $2756.12$ | $3865.90$
$\chi_{\textrm{min}}^{2}$ (Non-CMB) | $1106.54$ | $\cdots$ | $1111.57$ | $\cdots$ | $1109.54$
DIC | $1114.45$ | $2817.93$ | $3931.02$ | $2812.41$ | $3922.11$
$\Delta\textrm{DIC}$ | $\cdots$ | $\cdots$ | $\cdots$ | $-5.52$ | $-8.91$
$\textrm{AIC}_{c}$ | $1114.54$ | $2819.80$ | $3933.35$ | $2812.1$ | $3921.90$
$\Delta\textrm{AIC}_{c}$ | $\cdots$ | $\cdots$ | $\cdots$ | $-7.68$ | $-11.45$

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

Table 13: Mean and 68.3% confidence limits of untilted non-flat
$\Lambda\textrm{CDM}$ (+$A_{L}$) model parameters constrained by non-CMB, P18,
and P18+non-CMB data sets. $H_{0}$ has units of km s-1 Mpc-1.

| Untilted non-flat $\Lambda$CDM | Untilted non-flat $\Lambda$CDM$+A_{L}$
---|---|---
Parameter | Non-CMB | P18 | P18+non-CMB | P18 | P18+non-CMB
$\Omega_{b}h^{2}$ | $0.0243\pm 0.0033$ | $0.02320\pm 0.00015$ | $0.02300\pm 0.00014$ | $0.02320\pm 0.00015$ | $0.02320\pm 0.00015$
$\Omega_{c}h^{2}$ | $0.120\pm 0.013$ | $0.11098\pm 0.00088$ | $0.11161\pm 0.00086$ | $0.11097\pm 0.00087$ | $0.11097\pm 0.00085$
$100\theta_{\textrm{MC}}$ | $1.10\pm 0.10$ | $1.04204\pm 0.00030$ | $1.04189\pm 0.00029$ | $1.04202\pm 0.00030$ | $1.04199\pm 0.00030$
$\tau$ | $0.0543$ | $0.0543\pm 0.0091$ | $0.0717\pm 0.0095$ | $0.0540\pm 0.0087$ | $0.0562\pm 0.0086$
$\Omega_{k}$ | $-0.033\pm 0.050$ | $-0.095\pm 0.024$ | $-0.0062\pm 0.0014$ | $-0.12\pm 0.12$ | $-0.0062\pm 0.0014$
$\ln(10^{10}A_{s})$ | $2.87\pm 0.34$ | $3.021\pm 0.019$ | $3.057\pm 0.019$ | $3.020\pm 0.018$ | $3.024\pm 0.018$
$A_{L}$ | $\cdots$ | $\cdots$ | $\cdots$ | $1.08\pm 0.27$ | $1.324\pm 0.063$
$H_{0}$ | $70.2\pm 1.7$ | $47.1\pm 3.2$ | $68.07\pm 0.56$ | $52\pm 18$ | $68.45\pm 0.58$
$\Omega_{m}$ | $0.294\pm 0.018$ | $0.617\pm 0.082$ | $0.2920\pm 0.0050$ | $0.70\pm 0.42$ | $0.2878\pm 0.0050$
$\sigma_{8}$ | $0.771\pm 0.034$ | $0.730\pm 0.017$ | $0.7921\pm 0.0085$ | $0.721\pm 0.053$ | $0.7759\pm 0.0078$
$\chi_{\textrm{min}}^{2}$ (Total) | $1106.53$ | $2789.77$ | $3926.27$ | $2787.76$ | $3895.24$
$\chi_{\textrm{min}}^{2}$ (Non-CMB) | $1106.53$ | $\cdots$ | $1107.71$ | $\cdots$ | $1107.45$
DIC | $1116.95$ | $2847.14$ | $3982.38$ | $2846.45$ | $3954.21$
$\Delta\textrm{DIC}$ | $2.50$ | $29.21$ | $51.36$ | $28.52$ | $23.19$
$\textrm{AIC}_{c}$ | $1116.53$ | $2843.77$ | $3980.27$ | $2843.76$ | $3951.24$
$\Delta\textrm{AIC}_{c}$ | $1.99$ | $23.97$ | $46.92$ | $23.96$ | $17.89$

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

Table 14: Mean and 68.3% confidence limits of Planck-$P(q)$-based tilted non-
flat $\Lambda\textrm{CDM}$ ($+A_{L}$) model parameters constrained by non-CMB,
P18, and P18+non-CMB data sets. $H_{0}$ has units of km s-1 Mpc-1.

| Tilted non-flat $\Lambda$CDM Planck $P(q)$ | Tilted non-flat $\Lambda$CDM$+A_{L}$ Planck $P(q)$
---|---|---
Parameter | Non-CMB | P18 | P18+non-CMB | P18 | P18+non-CMB
$\Omega_{b}h^{2}$ | $0.0242\pm 0.0033$ | $0.02260\pm 0.00017$ | $0.02248\pm 0.00015$ | $0.02258\pm 0.00017$ | $0.02268\pm 0.00017$
$\Omega_{c}h^{2}$ | $0.120\pm 0.012$ | $0.1181\pm 0.0015$ | $0.1185\pm 0.0013$ | $0.1183\pm 0.0015$ | $0.1170\pm 0.0014$
$100\theta_{\textrm{MC}}$ | $1.10\pm 0.11$ | $1.04116\pm 0.00032$ | $1.04107\pm 0.00031$ | $1.04116\pm 0.00033$ | $1.04125\pm 0.00032$
$\tau$ | $0.0483$ | $0.0483\pm 0.0083$ | $0.0543\pm 0.0077$ | $0.0478\pm 0.0081$ | $0.0485\pm 0.0087$
$\Omega_{k}$ | $-0.032\pm 0.051$ | $-0.043\pm 0.017$ | $0.0004\pm 0.0017$ | $-0.130\pm 0.095$ | $-0.0006\pm 0.0017$
$n_{s}$ | $0.9706$ | $0.9706\pm 0.0047$ | $0.9687\pm 0.0043$ | $0.9704\pm 0.0048$ | $0.9735\pm 0.0046$
$\ln(10^{10}A_{s})$ | $2.90\pm 0.34$ | $3.027\pm 0.017$ | $3.040\pm 0.016$ | $3.027\pm 0.017$ | $3.025\pm 0.018$
$A_{L}$ | $\cdots$ | $\cdots$ | $\cdots$ | $0.88\pm 0.15$ | $1.203\pm 0.062$
$H_{0}$ | $70.1\pm 1.7$ | $54.5\pm 3.6$ | $68.25\pm 0.56$ | $45\pm 11$ | $68.48\pm 0.56$
$\Omega_{m}$ | $0.294\pm 0.018$ | $0.481\pm 0.062$ | $0.3040\pm 0.0055$ | $0.80\pm 0.35$ | $0.2994\pm 0.0055$
$\sigma_{8}$ | $0.771\pm 0.035$ | $0.775\pm 0.015$ | $0.8055\pm 0.0076$ | $0.733\pm 0.045$ | $0.7946\pm 0.0088$
$\chi_{\textrm{min}}^{2}$ (Total) | $1106.53$ | $2754.73$ | $3878.77$ | $2754.99$ | $3865.53$
$\chi_{\textrm{min}}^{2}$ (Non-CMB) | $1106.53$ | $\cdots$ | $1111.36$ | $\cdots$ | $1109.27$
DIC | $1116.92$ | $2810.59$ | $3933.33$ | $2811.63$ | $3924.07$
$\Delta\textrm{DIC}$ | $2.47$ | $-7.34$ | $2.31$ | $-6.30$ | $-6.95$
$\textrm{AIC}_{c}$ | $1116.53$ | $2810.73$ | $3934.77$ | $2812.99$ | $3923.53$
$\Delta\textrm{AIC}_{c}$ | $1.99$ | $-9.07$ | $1.42$ | $-6.81$ | $-9.82$

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

Table 15: Mean and 68.3% confidence limits of new-$P(q)$-based tilted non-flat
$\Lambda\textrm{CDM}$ ($+A_{L}$) model parameters constrained by non-CMB, P18,
and P18+non-CMB data sets. $H_{0}$ has units of km s-1 Mpc-1.

| Tilted non-flat $\Lambda$CDM new $P(q)$ | Tilted non-flat $\Lambda$CDM$+A_{L}$ new $P(q)$
---|---|---
Parameter | Non-CMB | P18 | P18+non-CMB | P18 | P18+non-CMB
$\Omega_{b}h^{2}$ | $0.0241\pm 0.0033$ | $0.02255\pm 0.00017$ | $0.02249\pm 0.00015$ | $0.02257\pm 0.00017$ | $0.02269\pm 0.00016$
$\Omega_{c}h^{2}$ | $0.120\pm 0.013$ | $0.1188\pm 0.0015$ | $0.1184\pm 0.0013$ | $0.1187\pm 0.0016$ | $0.1170\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.11\pm 0.11$ | $1.04109\pm 0.00032$ | $1.04108\pm 0.00031$ | $1.04111\pm 0.00033$ | $1.04125\pm 0.00032$
$\tau$ | $0.0525$ | $0.0525\pm 0.0083$ | $0.0549\pm 0.0077$ | $0.0512\pm 0.0086$ | $0.0490\pm 0.0086$
$\Omega_{k}$ | $-0.036\pm 0.051$ | $-0.033\pm 0.014$ | $0.0003\pm 0.0017$ | $-0.10\pm 0.11$ | $-0.0006\pm 0.0017$
$n_{s}$ | $0.9654$ | $0.9654\pm 0.0045$ | $0.9684\pm 0.0041$ | $0.9654\pm 0.0057$ | $0.9730\pm 0.0043$
$\ln(10^{10}A_{s})$ | $2.88\pm 0.34$ | $3.039\pm 0.017$ | $3.042\pm 0.016$ | $3.036\pm 0.018$ | $3.026\pm 0.018$
$A_{L}$ | $\cdots$ | $\cdots$ | $\cdots$ | $0.94\pm 0.20$ | $1.204\pm 0.061$
$H_{0}$ | $70.1\pm 1.8$ | $56.9\pm 3.6$ | $68.21\pm 0.55$ | $51\pm 14$ | $68.47\pm 0.56$
$\Omega_{m}$ | $0.295\pm 0.018$ | $0.444\pm 0.055$ | $0.3043\pm 0.0054$ | $0.70\pm 0.43$ | $0.2994\pm 0.0056$
$\sigma_{8}$ | $0.770\pm 0.035$ | $0.786\pm 0.014$ | $0.8057\pm 0.0074$ | $0.752\pm 0.052$ | $0.7948\pm 0.0083$
$\chi_{\textrm{min}}^{2}$ (Total) | $1106.49$ | $2757.38$ | $3878.76$ | $2756.33$ | $3865.41$
$\chi_{\textrm{min}}^{2}$ (Non-CMB) | $1106.49$ | $\cdots$ | $1111.36$ | $\cdots$ | $1109.32$
DIC | $1117.31$ | $2811.54$ | $3932.56$ | $2814.83$ | $3923.86$
$\Delta\textrm{DIC}$ | $2.86$ | $-6.39$ | $1.54$ | $-3.10$ | $-7.16$
$\textrm{AIC}_{c}$ | $1116.49$ | $2813.38$ | $3934.76$ | $2814.33$ | $3923.41$
$\Delta\textrm{AIC}_{c}$ | $1.95$ | $-6.42$ | $1.41$ | $-5.47$ | $-9.94$

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

Figure 24: Likelihoods of the tilted flat $\Lambda$CDM model parameters
constrained by P18, non-CMB, and P18+non-CMB data sets.

Figure 25: Likelihoods of the tilted flat $\Lambda$CDM$+A_{L}$ model
parameters constrained by P18, non-CMB, and P18+non-CMB data sets. The
likelihoods for the non-CMB data set, which do not depend on $A_{L}$, are the
same as in Fig. 24.

Figure 26: Likelihoods of the untilted non-flat $\Lambda$CDM model parameters
constrained by P18, non-CMB, and P18+non-CMB data sets.

Figure 27: Likelihoods of the untilted non-flat $\Lambda$CDM$+A_{L}$ model
parameters constrained by P18, non-CMB, and P18+non-CMB data sets. The
likelihoods for the non-CMB data set, which do not depend on $A_{L}$, are the
same as in Fig. 26.

Figure 28: Likelihoods of the tilted non-flat $\Lambda$CDM model [with Planck
$P(q)$] parameters constrained by P18, non-CMB, and P18+non-CMB data sets.

Figure 29: Likelihoods of the tilted non-flat $\Lambda$CDM$+A_{L}$ model [with
Planck $P(q)$] parameters constrained by P18, non-CMB, and P18+non-CMB data
sets. The likelihoods for the non-CMB data set, which do not depend on
$A_{L}$, are the same as in Fig. 28.

Figure 30: Likelihoods of the tilted non-flat $\Lambda$CDM model [with new
$P(q)$] parameters constrained by P18, non-CMB, and P18+non-CMB data sets.

Figure 31: Likelihoods of the tilted non-flat $\Lambda$CDM$+A_{L}$ model [with
new $P(q)$] parameters constrained by P18, non-CMB, and P18+non-CMB data sets.
The likelihoods for the non-CMB data set, which do not depend on $A_{L}$, are
the same as in Fig. 30.

####  IV.1.6 Comparing P18 data and non-CMB data cosmological constraints

In the previous subsubsection we compared BAO and BAO′ data cosmological
constraints to those obtained from P18 data. In the non-flat models with
$A_{L}=1$ there is somewhat significant disagreement between the values of the
cosmological parameters (especially the derived parameters $\Omega_{m}$,
$H_{0}$, and $\sigma_{8}$) determined using P18 data and those determined from
BAO or BAO′ data. This disagreement motivates additional tests to decide
whether P18 data and BAO′/BAO data can be used together to constrain
parameters of the non-flat models. While both P18 data and BAO′/BAO data favor
negative $\Omega_{k}$ values, BAO′/BAO data favor higher values of $H_{0}$ and
lower values of $\Omega_{m}$ relative to the values obtained in the P18
analysis. Allowing for a varying $A_{L}$ parameter resolves these tensions,
which may indicate that we can only jointly analyze P18 data and BAO′/BAO data
in the non-flat models when $A_{L}$ is allowed to vary.

To further examine these inconsistencies, in this subsubsection we compare
non-CMB data (which include BAO as well as BAO′ data) cosmological constraints
to those obtained from P18 data. (Prior to jointly analyzing P18+non-CMB data,
we need to determine whether P18 and non-CMB data cosmological constraints are
mutually consistent.) This allows us to examine how the inclusion of SNIa,
$H(z)$, and $f\sigma_{8}$ data affects the P18 data vs. BAO′/BAO data
conclusions of Sec. IV.1.5 and provides a different, perhaps more expansive,
test of the consistency of cosmological parameters obtained from high-redshift
data and from low-redshift data. In Sec. IV.3 we use two other statistical
estimators to examine whether or not P18 and non-CMB data are in tension.

The cosmological parameter mean values and error bars favored by the P18, non-
CMB, and P18+non-CMB data sets are summarized in Tables 12-15 for the tilted
flat $\Lambda$CDM (+$A_{L}$) models, the untilted non-flat $\Lambda$CDM
(+$A_{L}$) models, the tilted non-flat $\Lambda$CDM (+$A_{L}$) models with the
Planck $P(q)$, and the tilted non-flat $\Lambda$CDM ($+A_{L}$) models with the
new $P(q)$, respectively. Likelihood distributions of cosmological parameters
of the four models with $A_{L}=1$ and with $A_{L}$ varying are shown in Figs.
24-31 for the P18, non-CMB, and P18+non-CMB data sets.

Since non-CMB data do not have the ability to constrain $\tau$ or $n_{s}$, we
set their values to those found in the corresponding P18 data analysis.
$A_{L}$ does not affect predictions for the non-CMB measurements we study so
we do not include $A_{L}$ in the non-CMB data analyses. (We saw, in Sec.
IV.1.5, that BAO′/BAO data constraints for $A_{L}=1$ and for varying $A_{L}$
were very similar, see Tables 8-11.)

From Tables 13-15 we see, in the six non-flat $\Lambda$CDM (+$A_{L}$) models
that the constraints set by non-CMB data on $H_{0}$ and $\Omega_{m}$ are
tighter than the ones imposed by P18 data, and in the three non-flat
$\Lambda$CDM+$A_{L}$ models that the constraints set by non-CMB data on
$\Omega_{k}$ and $\sigma_{8}$ are tighter than the ones imposed by P18 data.
P18 data more restrictively constrain all other parameters in all eight
cosmological models.

As we discuss below, there is at least one parameter in the three non-flat
models with $A_{L}=1$ with a more than 3$\sigma$ level of disagreement between
P18 data cosmological constraints and non-CMB data cosmological constraints
and one parameter in the tilted flat $\Lambda$CDM model with $A_{L}=1$ and in
the tilted non-flat $\Lambda$CDM$+A_{L}$ model with the Planck $P(q)$ with a
more than 2$\sigma$ level of disagreement between P18 data cosmological
constraints and non-CMB data cosmological constraints. From Tables 13-15 we
see that both P18 data and non-CMB data favor negative values of the curvature
parameter, with non-CMB data only weakly favoring closed spatial
hypersurfaces, at 0.66$\sigma$ to 0.71$\sigma$. However, we should take into
account the geometrical degeneracy between $H_{0}$-$\Omega_{k}$-$\Omega_{m}$
and note that, like both BAO′ and BAO data, non-CMB data favor higher values
of $H_{0}$ and lower values of $\Omega_{m}$ than do P18 data and this is what
causes the P18 and non-CMB cosmological constraint differences.

The dominant component of non-CMB data is BAO′/BAO data. This is why the
cosmological constraints obtained from BAO′/BAO data are similar to the ones
obtained from the complete non-CMB low-redshift data set. However, there are
some differences between these sets of constraints that are worth mentioning.
As expected, the error bars obtained considering non-CMB data are smaller than
the ones from BAO′/BAO data. While similar values for $\Omega_{m}$ are found
in both cases, the values of $H_{0}$ favored by non-CMB data are $\sim
1\sigma$ smaller than those favored by BAO′/BAO data. BAO′ data favor closed
spatial hypersurfaces at 0.48$\sigma$ to 0.60$\sigma$ while BAO data favor
them by 0.71$\sigma$ to 0.96$\sigma$, which are on either side of the
0.66$\sigma$ to 0.71$\sigma$ favoring of closed spatial hypersurfaces from
non-CMB data. We also find smaller values for the $\sigma_{8}$ parameter when
non-CMB data are considered, with BAO′ data favoring 1.1$\sigma$-1.3$\sigma$
larger values while BAO data favor $\sim 1.3\sigma$ larger values in the non-
flat models and a 1.9$\sigma$ larger value in the tilted flat $\Lambda$CDM
case. This might be because the non-CMB data set contain additional
$f\sigma_{8}$ data points that favor lower values of $\sigma_{8}$ than those
in the BAO data set.

Comparing the six-parameter and the four-parameter tilted flat $\Lambda$CDM
model primary cosmological parameter constraints for P18 and non-CMB data,
shown in the left half of Table 12, we see that the values of
$\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, and $\theta_{\textrm{MC}}$ are in mild
disagreement, at 1.3$\sigma$, 1.1$\sigma$, and 1.0$\sigma$, respectively. We
also observe a more significant 2.2$\sigma$ level of tension in the derived
$\Omega_{m}$ values, the derived $H_{0}$ values differ by 1.4$\sigma$, and
$\sigma_{8}$ values show a better agreement, disagreeing by only 0.89$\sigma$.

Comparing the seven-parameter and the four-parameter tilted flat
$\Lambda$CDM+$A_{L}$ model primary cosmological parameter constraints for P18
and non-CMB data, shown in Table 12, we see that the values of
$\Omega_{b}h^{2}$ and $\theta_{\textrm{MC}}$ are in 1.2$\sigma$ and
1.1$\sigma$ tension respectively. As for the derived parameters, we find
$\Omega_{m}$ values differ by 1.2$\sigma$ while $H_{0}$ and $\sigma_{8}$
values are in only 0.81$\sigma$ and 0.45$\sigma$ disagreement. So unlike in
the BAO data and the BAO′ data comparisons with P18 data of Sec. IV.1.5, the
inclusion of a varying $A_{L}$ reduces the disagreement for all three derived
parameters, but less successfully for $\Omega_{m}$ in the non-CMB case here
compared to the BAO/BAO′ cases there.

P18 and non-CMB data results obtained for the six-parameter and the four-
parameter untilted non-flat $\Lambda$CDM model, shown in the left half of
Table 13, indicate mostly less significant differences in primary parameters
but more significant differences in derived parameters than found in the
tilted flat $\Lambda$CDM model. The primary spatial curvature parameter value
is $\Omega_{k}=-0.033\pm 0.050$ for non-CMB data, which is 0.66$\sigma$ away
from flat and in 1.1$\sigma$ tension with the P18 value $\Omega_{k}=-0.095\pm
0.024$, which is 4.0$\sigma$ away from flat. Regarding the derived parameters,
we find that $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ values are in
6.4$\sigma$, 3.8$\sigma$, and 1.1$\sigma$ disagreement. These results probably
mean that P18 and non-CMB data should not be jointly analyzed in the context
of the untilted non-flat $\Lambda$CDM model.

The results for the seven-parameter and the four-parameter untilted non-flat
$\Lambda$CDM+$A_{L}$ model, obtained considering P18 and non-CMB data, are in
Table 13. There is a slight increase in the disagreement between the values of
the primary spatial curvature parameter $\Omega_{k}$ (now 0.67$\sigma$) and
decreases for the derived parameters $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$,
that now disagree by 1.0$\sigma$, 0.97$\sigma$, and 0.79$\sigma$ respectively.
This is caused by the larger error bars in the $A_{L}$-varying P18 case
compared to the corresponding values obtained with $A_{L}=1$. According to
these results, unlike in the $A_{L}=1$ case, in the $A_{L}$-varying case P18
and non-CMB data can probably be jointly analyzed in the context of the
untilted non-flat $\Lambda$CDM model. Note that in this case a joint analysis
of P18+non-CMB data favors closed geometry at 4.4$\sigma$, with
$\Omega_{k}=-0.0062\pm 0.0014$, although because of the lack of the tilt
($n_{s}$) degree of freedom this untilted non-flat $\Lambda$CDM+$A_{L}$ model
does not provide a good fit to smaller-angular-scale P18 data, which is
reflected in the large $\Delta$DIC and $\Delta$AICc values for the P18+non-CMB
case in the lower half of Table 13.

Comparing the seven-parameter and the five-parameter tilted non-flat
$\Lambda$CDM Planck $P(q)$ model primary cosmological parameter constraints
for P18 and non-CMB data, we see, in the left half of Table 14, that the
primary parameter values do not much disagree. The non-CMB data primary
spatial curvature parameter value $\Omega_{k}=-0.032\pm 0.051$ is 0.63$\sigma$
away from flat and only in 0.20$\sigma$ tension with the P18 value
$\Omega_{k}=-0.043\pm 0.017$, which is 2.5$\sigma$ in favor of closed
geometry. The derived parameters $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ are
in 3.9$\sigma$, 2.9$\sigma$, and 0.11$\sigma$ tension. These results show that
P18 and non-CMB data cosmological constraints are inconsistent in the tilted
non-flat $\Lambda$CDM Planck $P(q)$ model and these data probably should not
be used jointly to constrain this model.

Looking at Table 14 we can compare results obtained for the eight-parameter
and the five-parameter tilted non-flat $\Lambda$CDM+$A_{L}$ Planck $P(q)$
model from P18 and non-CMB data respectively. Aside from $\Omega_{k}$, the
primary parameter disagreements do not change much compared to the $A_{L}=1$
case. For the non-CMB data primary spatial curvature parameter we have
$\Omega_{k}=-0.032\pm 0.051$, which is 0.63$\sigma$ away from flat and in
0.91$\sigma$ tension with the P18 value $-0.130\pm 0.095$, which is
1.4$\sigma$ away from flat. Regarding the derived parameters we find that
$H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ are in 2.3$\sigma$, 1.4$\sigma$ and
0.67$\sigma$ disagreement. Compared to the $A_{L}=1$ case, in the
$A_{L}$-varying case we find significant reductions in the $H_{0}$ and
$\Omega_{m}$ tensions, with both disagreements still being significant, which
suggest that P18 and non-CMB data should not be jointly analyzed within the
tilted non-flat $\Lambda$CDM+$A_{L}$ Planck $P(q)$ model.

Comparing the seven-parameter and the five-parameter tilted non-flat
$\Lambda$CDM new $P(q)$ model primary cosmological parameter constraints for
P18 and non-CMB data, from the left half of Table 15 we see that the primary
parameter values do not much disagree. The non-CMB data primary spatial
curvature parameter value is $\Omega_{k}=-0.036\pm 0.051$, which is only a
0.71$\sigma$ deviation from flat and, similar to the Planck $P(q)$ model, is
only in 0.057$\sigma$ disagreement with the P18 value $-0.033\pm 0.014$, which
is 2.4$\sigma$ away from flat. Regarding the derived parameters $H_{0}$,
$\Omega_{m}$, and $\sigma_{8}$, we find that their values disagree at
3.3$\sigma$, 2.6$\sigma$, and 0.42$\sigma$ respectively. While the $H_{0}$ and
$\Omega_{m}$ disagreements are a little smaller than the ones found in the
tilted non-flat $\Lambda$CDM Planck $P(q)$ model, they still are large enough
to require we more carefully test whether P18 and non-CMB data can be jointly
used to constrain cosmological parameters in this model.

The results for the eight-parameter and the five-parameter tilted non-flat
$\Lambda$CDM+$A_{L}$ new $P(q)$ model are in Table 15, for P18 and non-CMB
data, respectively. As happens in the Planck $P(q)$ model, when the $A_{L}$
parameter is allowed to vary the mild tensions found for the primary
parameters, except for $\Omega_{k}$, do not change much compared to the
$A_{L}=1$ case. For the non-CMB data primary spatial curvature parameter we
have $\Omega_{k}=-0.036\pm 0.051$, which is 0.71$\sigma$ away from flat
hypersurfaces and now in 0.53$\sigma$ tension with the P18 value
$\Omega_{k}=-0.10\pm 0.11$, which is 0.91$\sigma$ away from flat. As for the
value of the derived parameters $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$ we
find disagreements at 1.4$\sigma$, 0.94$\sigma$, and 0.29$\sigma$
respectively. The tensions are reduced with respect to the case with
$A_{L}=1$, due to the increase of the error bars, but possibly the $H_{0}$
tension is still not small enough to allow the joint use of P18+non-CMB data
for constraining tilted non-flat $\Lambda$CDM+$A_{L}$ new $P(q)$ model
cosmological parameters.

Figures 24-31 show one-dimensional likelihoods and two-dimensional contours
for cosmological parameters obtained using P18, non-CMB, and P18+non-CMB data.
As mentioned above, non-CMB data constraints (shown with unfilled black lines)
are comparatively less restrictive than P18 constraints (shown in grey), are
unable to put tight constraints on the primary cosmological parameters (except
on $\Omega_{k}$ in the three non-flat $\Lambda$CDM$+A_{L}$ models), and in
many cases they at least partially overlap with the P18 data constraints.

Figures 24 and 25 are for tilted flat $\Lambda$CDM (+$A_{L}$) models. The
$\sim 1\sigma$ disagreements between the non-CMB constraints and those
obtained with P18 data, discussed above, can be clearly seen in the contour
plots. For the tilted flat $\Lambda$CDM model the larger disagreements are in
panels for derived cosmological parameters, with the largest for $\Omega_{m}$
and the next largest for $H_{0}$. These disagreements decrease when the
$A_{L}$ parameter is allowed to vary.

Looking at the contour plots for the untilted non-flat $\Lambda$CDM (+$A_{L}$)
models (see Figs. 26 and 27) we observe non-overlapping contours in those
panels that involve the derived parameters $H_{0}$ and $\Omega_{m}$ or the
primary parameter $\Omega_{k}$, especially in the $\Omega_{k}$-$\theta_{\rm
MC}$ subpanel of Fig. 26. These disagreements largely disappear when $A_{L}$
is allowed to vary, except perhaps for the $H_{0}$ one. This may indicate that
in the context of this cosmological model we may jointly analyze P18 data with
non-CMB data only when $A_{L}$ is allowed to vary.

Figures 28 and 29 show cosmological parameter constraints for the tilted non-
flat $\Lambda$CDM (+$A_{L}$) Planck $P(q)$ models, while the ones for the
tilted non-flat $\Lambda$CDM(+$A_{L}$) new $P(q)$ models are displayed in
Figs. 30 and 31. As expected, considering the results discussed above in this
subsubsection, the contour plots for these tilted non-flat models are quite
similar. We see in the panels that involve the primary cosmological parameters
that there is overlap at 1$\sigma$, not only when $A_{L}$ is allowed to vary
but also when $A_{L}=1$. When $A_{L}=1$, for the Planck $P(q)$ model, P18 and
non-CMB data constraint contours that involve $H_{0}$ and $\Omega_{m}$ do not
overlap even at 2$\sigma$. These disagreements are less severe for the new
$P(q)$ model with $A_{L}=1$, where overlap is reached in most cases at a
little over $2\sigma$.

In view of the results discussed in this subsubsection, further tests are
needed to properly quantify the level of disagreement, in the context of non-
flat models, between P18 data and non-CMB data cosmological constraints. We
return to this issue in Sec. IV.3.

Table 16: Mean and 68.3% confidence limits of tilted flat
$\Lambda\textrm{CDM}$ (+$A_{L}$) model parameters constrained by non-CMB,
P18+lensing, and P18+lensing+non-CMB data sets. The Hubble constant $H_{0}$
has a unit of km s-1 Mpc-1.

| Tilted flat $\Lambda$CDM | Tilted flat $\Lambda$CDM$+A_{L}$
---|---|---
Parameter | Non-CMB | P18+lensing | P18+lensing+non-CMB | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.0257\pm 0.0026$ | $0.02237\pm 0.00014$ | $0.02250\pm 0.00013$ | $0.02251\pm 0.00017$ | $0.02258\pm 0.00014$
$\Omega_{c}h^{2}$ | $0.1128\pm 0.0061$ | $0.1200\pm 0.0012$ | $0.11838\pm 0.00083$ | $0.1183\pm 0.0015$ | $0.11747\pm 0.00091$
$100\theta_{\textrm{MC}}$ | $1.0321\pm 0.0080$ | $1.04091\pm 0.00031$ | $1.04110\pm 0.00029$ | $1.04109\pm 0.00032$ | $1.04118\pm 0.00029$
$\tau$ | $0.0543$ | $0.0543\pm 0.0073$ | $0.0569\pm 0.0071$ | $0.0487\pm 0.0087$ | $0.0476\pm 0.0085$
$n_{s}$ | $0.9649$ | $0.9649\pm 0.0041$ | $0.9688\pm 0.0036$ | $0.9695\pm 0.0048$ | $0.9715\pm 0.0038$
$\ln(10^{10}A_{s})$ | $3.10\pm 0.11$ | $3.044\pm 0.014$ | $3.046\pm 0.014$ | $3.028\pm 0.018$ | $3.023\pm 0.018$
$A_{L}$ | $\cdots$ | $\cdots$ | $\cdots$ | $1.073\pm 0.041$ | $1.089\pm 0.035$
$H_{0}$ | $69.9\pm 1.7$ | $67.34\pm 0.55$ | $68.09\pm 0.38$ | $68.14\pm 0.69$ | $68.52\pm 0.42$
$\Omega_{m}$ | $0.285\pm 0.011$ | $0.3155\pm 0.0075$ | $0.3053\pm 0.0050$ | $0.3048\pm 0.0091$ | $0.2998\pm 0.0053$
$\sigma_{8}$ | $0.787\pm 0.026$ | $0.8112\pm 0.0059$ | $0.8072\pm 0.0058$ | $0.7996\pm 0.0089$ | $0.7955\pm 0.0075$
$\chi_{\textrm{min}}^{2}$ (Total) | $1106.55$ | $2774.71$ | $3888.41$ | $2771.24$ | $3881.55$
$\chi_{\textrm{min}}^{2}$ (Non-CMB) | $1106.55$ | $\cdots$ | $1112.05$ | $\cdots$ | $1109.64$
DIC | $1114.38$ | $2826.45$ | $3940.70$ | $2825.53$ | $3935.15$
$\Delta\textrm{DIC}$ | $\cdots$ | $\cdots$ | $\cdots$ | $-0.92$ | $-5.55$
$\textrm{AIC}_{c}$ | $1114.55$ | $2828.71$ | $3942.41$ | $2827.24$ | $3937.55$
$\Delta\textrm{AIC}_{c}$ | $\cdots$ | $\cdots$ | $\cdots$ | $-1.47$ | $-4.86$

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

Table 17: Mean and 68.3% confidence limits of untilted non-flat
$\Lambda\textrm{CDM}$ (+$A_{L}$) model parameters constrained by non-CMB,
P18+lensing, and P18+lensing+non-CMB data sets. The Hubble constant $H_{0}$
has a unit of km s-1 Mpc-1.

| Untilted non-flat $\Lambda$CDM | Untilted non-flat $\Lambda$CDM$+A_{L}$
---|---|---
Parameter | Non-CMB | P18+lensing | P18+lensing+non-CMB | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.0241\pm 0.0033$ | $0.02307\pm 0.00014$ | $0.02301\pm 0.00014$ | $0.02312\pm 0.00014$ | $0.02310\pm 0.00014$
$\Omega_{c}h^{2}$ | $0.121\pm 0.013$ | $0.11108\pm 0.00086$ | $0.11176\pm 0.00083$ | $0.11092\pm 0.00087$ | $0.11100\pm 0.00085$
$100\theta_{\textrm{MC}}$ | $1.11\pm 0.11$ | $1.04196\pm 0.00029$ | $1.04189\pm 0.00029$ | $1.04193\pm 0.00029$ | $1.04195\pm 0.00030$
$\tau$ | $0.0580$ | $0.0580\pm 0.0087$ | $0.0799\pm 0.0089$ | $0.0554\pm 0.0097$ | $0.0566\pm 0.0083$
$\Omega_{k}$ | $-0.037\pm 0.050$ | $-0.0322\pm 0.0075$ | $-0.0065\pm 0.0014$ | $0.0161\pm 0.0094$ | $-0.0060\pm 0.0014$
$\ln(10^{10}A_{s})$ | $2.84\pm 0.34$ | $3.027\pm 0.018$ | $3.075\pm 0.018$ | $3.021\pm 0.020$ | $3.024\pm 0.017$
$A_{L}$ | $\cdots$ | $\cdots$ | $\cdots$ | $1.44\pm 0.15$ | $1.162\pm 0.036$
$H_{0}$ | $70.2\pm 1.7$ | $58.9\pm 2.1$ | $67.90\pm 0.56$ | $85.7\pm 8.5$ | $68.48\pm 0.58$
$\Omega_{m}$ | $0.295\pm 0.018$ | $0.390\pm 0.027$ | $0.2938\pm 0.0049$ | $0.190\pm 0.043$ | $0.2874\pm 0.0050$
$\sigma_{8}$ | $0.769\pm 0.035$ | $0.765\pm 0.011$ | $0.7997\pm 0.0076$ | $0.7805\pm 0.0094$ | $0.7764\pm 0.0078$
$\chi_{\textrm{min}}^{2}$ (Total) | $1106.51$ | $2813.13$ | $3938.22$ | $2807.91$ | $3915.05$
$\chi_{\textrm{min}}^{2}$ (Non-CMB) | $1106.51$ | $\cdots$ | $1108.60$ | $\cdots$ | $1107.39$
DIC | $1117.24$ | $2869.06$ | $3992.71$ | $2856.10$ | $3973.55$
$\Delta\textrm{DIC}$ | $2.86$ | $42.61$ | $52.01$ | $29.65$ | $32.85$
$\textrm{AIC}_{c}$ | $1116.51$ | $2867.13$ | $3992.22$ | $2863.91$ | $3971.05$
$\Delta\textrm{AIC}_{c}$ | $1.96$ | $38.42$ | $49.81$ | $35.20$ | $28.64$

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

Table 18: Mean and 68.3% confidence limits of Planck-$P(q)$-based tilted
nonflat $\Lambda\textrm{CDM}$ ($+A_{L}$) model parameters constrained by non-
CMB, P18+lensing, and P18+lensing+non-CMB data sets. The Hubble constant
$H_{0}$ has a unit of km s-1 Mpc-1.

| Tilted non-flat $\Lambda$CDM Planck $P(q)$ | Tilted non-flat $\Lambda$CDM$+A_{L}$ Planck $P(q)$
---|---|---
Parameter | Non-CMB | P18+lensing | P18+lensing+non-CMB | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.0242\pm 0.0033$ | $0.02249\pm 0.00016$ | $0.02249\pm 0.00015$ | $0.02251\pm 0.00017$ | $0.02259\pm 0.00016$
$\Omega_{c}h^{2}$ | $0.120\pm 0.013$ | $0.1186\pm 0.0015$ | $0.1187\pm 0.0013$ | $0.1183\pm 0.0015$ | $0.1173\pm 0.0014$
$100\theta_{\textrm{MC}}$ | $1.10\pm 0.10$ | $1.04107\pm 0.00032$ | $1.04106\pm 0.00031$ | $1.04110\pm 0.00032$ | $1.04118\pm 0.00032$
$\tau$ | $0.0495$ | $0.0495\pm 0.0082$ | $0.0563\pm 0.0073$ | $0.0489\pm 0.0085$ | $0.0479\pm 0.0085$
$\Omega_{k}$ | $-0.033\pm 0.050$ | $-0.0103\pm 0.0066$ | $0.0004\pm 0.0017$ | $-0.005\pm 0.027$ | $-0.0002\pm 0.0017$
$n_{s}$ | $0.9687$ | $0.9687\pm 0.0046$ | $0.9681\pm 0.0044$ | $0.9696\pm 0.0049$ | $0.9718\pm 0.0045$
$\ln(10^{10}A_{s})$ | $2.90\pm 0.34$ | $3.030\pm 0.017$ | $3.046\pm 0.014$ | $3.028\pm 0.018$ | $3.024\pm 0.017$
$A_{L}$ | $\cdots$ | $\cdots$ | $\cdots$ | $1.09\pm 0.16$ | $1.090\pm 0.036$
$H_{0}$ | $70.1\pm 1.8$ | $63.7\pm 2.3$ | $68.17\pm 0.55$ | $69\pm 11$ | $68.49\pm 0.56$
$\Omega_{m}$ | $0.294\pm 0.018$ | $0.351\pm 0.024$ | $0.3051\pm 0.0053$ | $0.32\pm 0.11$ | $0.2998\pm 0.0055$
$\sigma_{8}$ | $0.771\pm 0.036$ | $0.796\pm 0.011$ | $0.8080\pm 0.0066$ | $0.796\pm 0.016$ | $0.7952\pm 0.0085$
$\chi_{\textrm{min}}^{2}$ (Total) | $1106.51$ | $2771.53$ | $3887.99$ | $2771.14$ | $3881.37$
$\chi_{\textrm{min}}^{2}$ (Non-CMB) | $1106.51$ | $\cdots$ | $1111.94$ | $\cdots$ | $1110.31$
DIC | $1117.27$ | $2826.17$ | $3942.07$ | $2827.14$ | $3936.85$
$\Delta\textrm{DIC}$ | $2.89$ | $-0.28$ | $1.37$ | $0.69$ | $-3.85$
$\textrm{AIC}_{c}$ | $1116.51$ | $2827.53$ | $3943.99$ | $2829.14$ | $3939.37$
$\Delta\textrm{AIC}_{c}$ | $1.96$ | $-1.18$ | $1.58$ | $0.43$ | $-3.04$

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

Table 19: Mean and 68.3% confidence limits of new-$P(q)$-based tilted nonflat
$\Lambda\textrm{CDM}$ ($+A_{L}$) model parameters constrained by non-CMB,
P18+lensing, and P18+lensing+non-CMB data sets. The Hubble constant $H_{0}$
has a unit of km s-1 Mpc-1.

| Tilted non-flat $\Lambda$CDM new $P(q)$ | Tilted non-flat $\Lambda$CDM$+A_{L}$ new $P(q)$
---|---|---
Parameter | Non-CMB | P18+lensing | P18+lensing+non-CMB | P18+lensing | P18+lensing+non-CMB
$\Omega_{b}h^{2}$ | $0.0242\pm 0.0033$ | $0.02248\pm 0.00016$ | $0.02248\pm 0.00015$ | $0.02252\pm 0.00017$ | $0.02260\pm 0.00016$
$\Omega_{c}h^{2}$ | $0.120\pm 0.013$ | $0.1188\pm 0.0014$ | $0.1186\pm 0.0013$ | $0.1183\pm 0.0015$ | $0.1174\pm 0.0013$
$100\theta_{\textrm{MC}}$ | $1.10\pm 0.10$ | $1.04104\pm 0.00032$ | $1.04106\pm 0.00031$ | $1.04108\pm 0.00032$ | $1.04118\pm 0.00032$
$\tau$ | $0.0515$ | $0.0515\pm 0.0081$ | $0.0566\pm 0.0074$ | $0.0495\pm 0.0093$ | $0.0486\pm 0.0086$
$\Omega_{k}$ | $-0.033\pm 0.050$ | $-0.0086\pm 0.0057$ | $0.0003\pm 0.0017$ | $0.003\pm 0.016$ | $-0.0002\pm 0.0017$
$n_{s}$ | $0.9654$ | $0.9661\pm 0.0043$ | $0.9679\pm 0.0042$ | $0.9688\pm 0.0053$ | $0.9713\pm 0.0042$
$\ln(10^{10}A_{s})$ | $2.89\pm 0.34$ | $3.035\pm 0.016$ | $3.046\pm 0.014$ | $3.030\pm 0.019$ | $3.025\pm 0.017$
$A_{L}$ | $\cdots$ | $\cdots$ | $\cdots$ | $1.13\pm 0.15$ | $1.088\pm 0.035$
$H_{0}$ | $70.1\pm 1.8$ | $64.2\pm 2.0$ | $68.13\pm 0.54$ | $72.0\pm 9.2$ | $68.48\pm 0.56$
$\Omega_{m}$ | $0.295\pm 0.017$ | $0.345\pm 0.021$ | $0.3054\pm 0.0051$ | $0.287\pm 0.076$ | $0.2999\pm 0.0055$
$\sigma_{8}$ | $0.771\pm 0.036$ | $0.799\pm 0.010$ | $0.8079\pm 0.0067$ | $0.801\pm 0.011$ | $0.7956\pm 0.0082$
$\chi_{\textrm{min}}^{2}$ (Total) | $1106.49$ | $2771.75$ | $3887.55$ | $2770.45$ | $3880.69$
$\chi_{\textrm{min}}^{2}$ (Non-CMB) | $1106.49$ | $\cdots$ | $1111.65$ | $\cdots$ | $1109.43$
DIC | $1117.14$ | $2825.74$ | $3942.22$ | $2827.29$ | $3937.52$
$\Delta\textrm{DIC}$ | $2.76$ | $-0.71$ | $1.52$ | $0.84$ | $-3.18$
$\textrm{AIC}_{c}$ | $1116.49$ | $2827.75$ | $3943.55$ | $2828.45$ | $3938.69$
$\Delta\textrm{AIC}_{c}$ | $1.94$ | $-0.96$ | $1.14$ | $-0.26$ | $-3.72$

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

Figure 32: Likelihoods of the tilted flat $\Lambda$CDM model parameters
constrained by P18+lensing, non-CMB, and P18+lensing+non-CMB data sets.

Figure 33: Likelihoods of the tilted flat $\Lambda$CDM$+A_{L}$ model
parameters constrained by P18+lensing, non-CMB, and P18+lensing+non-CMB data
sets. The likelihoods for the non-CMB data set, which do not depend on
$A_{L}$, are the same as in Fig. 32.

Figure 34: Likelihoods of the untilted non-flat $\Lambda$CDM model parameters
constrained by P18+lensing, non-CMB, and P18+lensing+non-CMB data sets.

Figure 35: Likelihoods of the untilted non-flat $\Lambda$CDM$+A_{L}$ model
parameters constrained by P18+lensing, non-CMB, and P18+lensing+non-CMB data
sets. The likelihoods for the non-CMB data set, which do not depend on
$A_{L}$, are the same as in Fig. 34.

Figure 36: Likelihoods of the tilted non-flat $\Lambda$CDM model [with Planck
$P(q)$] parameters constrained by P18+lensing, non-CMB, and P18+lensing+non-
CMB data sets.

Figure 37: Likelihoods of the tilted non-flat $\Lambda$CDM$+A_{L}$ model [with
Planck $P(q)$] parameters constrained by P18+lensing, non-CMB, and
P18+lensing+non-CMB data sets. The likelihoods for the non-CMB data set, which
do not depend on $A_{L}$, are the same as in Fig. 36.

Figure 38: Likelihoods of the tilted non-flat $\Lambda$CDM model [with new
$P(q)$] parameters constrained by P18+lensing, non-CMB, and P18+lensing+non-
CMB data sets.

Figure 39: Likelihoods of the tilted non-flat $\Lambda$CDM$+A_{L}$ model [with
new $P(q)$] parameters constrained by P18+lensing, non-CMB, and
P18+lensing+non-CMB data sets. The likelihoods for the non-CMB data set, which
do not depend on $A_{L}$, are the same as in Fig. 38.

####  IV.1.7 Comparing P18+lensing data and non-CMB data cosmological
constraints

In the previous subsubsection we compared non-CMB data cosmological
constraints to those obtained from P18 data. We found significant tensions in
the non-flat models with $A_{L}=1$ for the derived parameters $H_{0}$ and
$\Omega_{m}$, and a 2.2$\sigma$ tension between the two $\Omega_{m}$ values in
the flat $\Lambda$CDM model with $A_{L}=1$. In view of these results
additional tests are needed if one wants to know whether P18 and non-CMB data
can be jointly analysed to determine cosmological constraints. We study this
in Sec. IV.3. Interestingly, when the $A_{L}$ parameter is allowed to vary
these tensions decrease significantly, with the largest tension being
2.3$\sigma$ between the two $H_{0}$ values in the tilted non-flat Planck
$P(q)$ model, and the remaining tensions not exceeding 1.4$\sigma$, perhaps an
indication that P18 and non-CMB data can be use jointly to constrain
cosmological parameters when $A_{L}$ is allowed to vary.

In Secs. IV.1.1 and IV.1.2 we discussed the cosmological constrained obtained
from P18 data and from P18+lensing data. We shall see, in Sec. IV.3, that, in
the non-flat models, P18 data and lensing data are less mutually inconsistent
than P18 data and non-CMB data are, however it is necessary to perform an
additional test to determine whether or not P18, lensing, and non-CMB data can
be jointly analyzed to derive cosmological constraints in the non-flat models
with $A_{L}=1$.

In this subsubsection we describe the results of this additional test that
compares non-CMB data cosmological constraints to the ones obtained from
P18+lensing data. P18+lensing+non-CMB data cannot be jointly used in the
context of a given model unless the cosmological constraints obtained with
P18+lensing data and with non-CMB data are consistent. While in the previous
subsubsection we labelled the study of P18 vs. non-CMB as a study of high-
redshift data cosmological constraints vs. low-redshift data cosmological
constraints, we cannot do that in this subsubsection since most of the
information in the lensing data is from low-redshift data.

The cosmological parameter mean values and error bars favored by the
P18+lensing, non-CMB, and P18+lensing+non-CMB data sets are summarized in
Tables 16-19 for the tilted flat $\Lambda$CDM (+$A_{L}$) model, the untilted
non-flat $\Lambda$CDM (+$A_{L}$) model, the tilted non-flat $\Lambda$CDM
(+$A_{L}$) Planck $P(q)$ model, and the tilted non-flat $\Lambda$CDM
(+$A_{L}$) new $P(q)$ model. Likelihood distributions of cosmological
parameters of the four models with $A_{L}=1$ and with varying $A_{L}$ are
shown in Figs. 32-39 for the P18+lensing, non-CMB and P18+lensing+non-CMB data
sets.

Since non-CMB data do not have the ability to constrain $\tau$ or $n_{s}$, in
this subsubsection we set their values in the non-CMB data only analyses to
those found in the corresponding P18+lensing data analyses. Note that in the
previous subsubsection, where the case of P18 data versus non-CMB data was
studied, the values of $\tau$ and $n_{s}$ in the non-CMB data only analyses
were set to those found in the corresponding P18 data analyses, nevertheless,
the cosmological parameter constraints from the two non-CMB data analyses are
practically identical. This indicates that non-CMB data are mostly insensitive
to changes in the values of $\tau$ and $n_{s}$, as we have assumed. Again, we
do not include the $A_{L}$ parameter in the analyses when only non-CMB data
are considered, since it does not play a role at low redshift.

Similar to what happens in the previous subsubsection, when we compare
cosmological constraints from P18 data and from non-CMB data, by looking at
Tables 17-19 we observe that for the six non-flat $\Lambda$CDM (+$A_{L}$)
models the constraints imposed by non-CMB data on the $H_{0}$ and $\Omega_{m}$
parameters are tighter than the ones from P18+lensing data. P18+lensing data
more restrictively constrain all other parameters in all eight cosmological
models.

Comparing the six-parameter and the four-parameter tilted flat $\Lambda$CDM
model primary cosmological parameter constraints for P18+lensing and non-CMB
data, shown in the left part of Table 16, we observe that the values of
$\Omega_{b}h^{2}$, $\Omega_{c}{h^{2}}$, and $\theta_{\textrm{MC}}$ are in mild
disagreement, at 1.3$\sigma$, 1.2$\sigma$, and 1.1$\sigma$, respectively. We
also see tensions in the derived parameters. In particular, for the non-
relativistic matter density parameter $\Omega_{m}$, the level of tension
reaches 2.3$\sigma$, whereas the values of $H_{0}$ disagree by 1.4$\sigma$.

From Table 16 we can compare the seven-parameter and the four-parameter tilted
flat $\Lambda$CDM+$A_{L}$ model cosmological parameter constraints for
P18+lensing data and for non-CMB data. Regarding the primary cosmological
parameters we see that the values of $\Omega_{b}{h^{2}}$ and
$\theta_{\textrm{MC}}$ disagree at 1.2$\sigma$ and 1.1$\sigma$ respectively.
The inclusion of the $A_{L}$-varying parameter reduces significantly the
tension found in the $A_{L}=1$ case for $\Omega_{m}$ and $H_{0}$, with them
now disagreeing by only 1.4$\sigma$ and 0.96$\sigma$. We do not find any clear
evidence that prevents us from jointly analyzing P18+lensing and non-CMB data,
in the context of the tilted flat $\Lambda$CDM model, with and without a
varying $A_{L}$ parameter.

The results for the six-parameter and the four-parameter untilted non-flat
$\Lambda$CDM model obtained from P18+lensing and non-CMB data are in Table 17.
While for the primary cosmological parameters we do not observe significant
tensions, we do for the derived parameters. The primary spatial curvature
parameter is $\Omega_{k}=-0.037\pm 0.050$ for non-CMB data, which is
0.74$\sigma$ away from flat hypersurfaces and in 0.095$\sigma$ tension with
the P18+lensing analysis value $\Omega_{k}=-0.0322\pm 0.0075$, which is
4.3$\sigma$ away from flat. As for the derived parameters, we find that
$H_{0}$, $\Omega_{m}$ and $\sigma_{8}$ values are in 4.2$\sigma$, 2.9$\sigma$,
and 0.11$\sigma$ disagreement. The high levels of tensions reached for some of
the parameters may indicate that P18+lensing and non-CMB data should not be
jointly analyzed in the context of the untilted non-flat $\Lambda$CDM model.

P18+lensing and non-CMB data results obtained for the seven-parameter and the
four-parameter untilted non-flat $\Lambda$CDM+$A_{L}$ model are shown in Table
17. Regarding the values of the primary cosmological parameters, except for
$\Omega_{k}$ (discussed next), as was observed in the $A_{L}=1$ case, there
are no significant tensions. The value of the curvature parameter is
$\Omega_{k}=-0.037\pm 0.050$ (0.74$\sigma$ away from flat) for the non-CMB
data and $\Omega_{k}=0.0161\pm 0.0094$ for the P18+lensing data, which
indicates 1.7$\sigma$ evidence in favor of an open spatial geometry. The two
$\Omega_{k}$ values disagree at 1.0$\sigma$. The disagreement in the values of
the derived parameters $H_{0}$, $\Omega_{m}$ and $\sigma_{8}$ values are
1.8$\sigma$, 2.3$\sigma$, and 0.32$\sigma$, respectively, which clearly
represents a reduction with respect to the $A_{L}=1$ case. This is due to the
elongation of the error bars in the varying $A_{L}$ case compared to the
$A_{L}=1$ case. Given these results, the P18+lensing and the non-CMB data
should perhaps not be used together in the context of the untilted non-flat
$\Lambda$CDM+$A_{L}$ model. It may be noticed however that when we do so,
namely in the P18+lensing+non-CMB analysis, the obtained value for the
curvature parameter is $\Omega_{k}=-0.0060\pm 0.0014$, which is 4.3$\sigma$
away from flat. Nonetheless, according to the AIC and DIC this model is
strongly disfavoured when it is compared with the tilted models, due to the
lack of the degree of freedom contained in the $n_{s}$ parameter.

The results that allow us to compare the seven-parameter and the five-
parameter tilted non-flat $\Lambda$CDM Planck $P(q)$ model primary
cosmological parameter constraints for P18+lensing data and non-CMB data can
be seen in Table 18. There are no significant tensions in the values of the
primary cosmological parameters. The non-CMB data value of the spatial
curvature parameter, $\Omega_{k}=-0.033\pm 0.050$, is 0.66$\sigma$ away from
flat and in 0.45$\sigma$ tension with the value found in the P18+lensing
analysis, namely $\Omega_{k}=-0.0103\pm 0.0066$ which represents a 1.6$\sigma$
deviation from flat hypersurfaces. As for the values of the derived parameters
$H_{0}$, $\Omega_{m}$, and $\sigma_{8}$, the tensions are 2.2$\sigma$,
1.9$\sigma$, and 0.66$\sigma$, respectively. Given these results, further
tests are probably necessary in order to decide whether P18+lensing and non-
CMB data can be jointly analyzed in the context of the tilted non-flat
$\Lambda$CDM Planck $P(q)$ model.

P18+lensing and non-CMB data results obtained for the eight-parameter and the
five-parameter tilted non-flat $\Lambda$CDM+$A_{L}$ Planck $P(q)$ model are
shown in Table 18. Similar to the $A_{L}=1$ case we do not find significant
disagreements in the values of the primary cosmological parameters. For the
non-CMB data the value of the curvature parameter is $\Omega_{k}=-0.033\pm
0.050$ which is 0.66$\sigma$ away from flat and in 0.49$\sigma$ tension with
the P18+lensing value, $\Omega_{k}=-0.005\pm 0.027$, which in turn is only
0.19$\sigma$ in favor of a closed geometry. An important reduction in the
disagreements found in the derived parameters, with respect to the $A_{L}=1$
case, is observed. In particular, for $H_{0}$, $\Omega_{m}$, and $\sigma_{8}$
the disagreement found is 0.099$\sigma$, 0.23$\sigma$, and 0.63$\sigma$. We
may say that in the context of the tilted non-flat $\Lambda$CDM+$A_{L}$ Planck
$P(q)$ we are allowed to analyze together P18+lensing data and non-CMB data.
By doing so, we get for P18+lensing+non-CMB data no evidence in favor of a
non-flat geometry, $\Omega_{k}=-0.0002\pm 0.0017$, but still a clear
2.5$\sigma$ preference for $A_{L}\neq 1$ since $A_{L}=1.090\pm 0.036$.

Comparing the seven-parameter and the five-parameter tilted non-flat
$\Lambda$CDM new $P(q)$ model primary cosmological parameter constraints for
P18+lensing data and non-CMB data, from the left part of Table 19 we see no
important differences in the values of the primary parameters. The value for
the spatial curvature parameter is $\Omega_{k}=-0.033\pm 0.050$ for non-CMB
data, which represents a 0.66$\sigma$ deviation from flat and it is in
0.48$\sigma$ tension with the value obtained in the P18+lensing analysis,
$\Omega_{k}=-0.0086\pm 0.0057$, which is 1.5$\sigma$ away from flat
hypersurfaces. Regarding the triad of derived parameters $H_{0}$,
$\Omega_{m}$, and $\sigma_{8}$, the disagreement found for each of them is
2.2$\sigma$, 1.9$\sigma$, and 0.75$\sigma$ respectively. In light of these
results we deem that more testing is required to decide whether the
P18+lensing and non-CMB data can be jointly analyzed in the context of the
tilted non-flat $\Lambda$CDM new $P(q)$ model.

In Table 19 we provide the results for the eight-parameter and the five-
parameter tilted non-flat $\Lambda$CDM+$A_{L}$ new $P(q)$ model when
P18+lensing data and non-CMB data are considered. The tensions found for the
values of primary cosmological parameters are not significant, as in the
$A_{L}=1$ case. When non-CMB data is considered we find $\Omega_{k}=-0.033\pm
0.050$, which shows a 0.66$\sigma$ evidence in favor of a closed geometry and
is in 0.69$\sigma$ tension with the P18+lensing data value,
$\Omega_{k}=0.003\pm 0.016$, which shows only a 0.19$\sigma$ preference for an
open geometry. As for the derived parameters $H_{0}$, $\Omega_{m}$, and
$\sigma_{8}$ the level of agreement is really good, with the corresponding
values only in 0.20$\sigma$, 0.10$\sigma$, and 0.80$\sigma$ tension,
respectively. These results seem to indicate that in the context of the tilted
non-flat $\Lambda$CDM+$A_{L}$ new $P(q)$ model P18+lensing data and non-CMB
data can be jointly analyzed. In the P18+lensing+non-CMB analysis we find
$\Omega_{k}=-0.0002\pm 0.0017$, so no clear preference for an open or a closed
geometry. On the other hand, we find $A_{L}=1.088\pm 0.035$ which is
2.5$\sigma$ away from the predicted value $A_{L}=1$.

In Figs. 32-39 we show the one-dimensional likelihoods and the two-dimensional
contours for cosmological parameters obtained using P18+lensing, non-CMB, and
P18+lensing+non-CMB data. The constraints coming from non-CMB data (shown with
unfilled black lines) are less restrictive than P18+lensing constraints (shown
in grey), except for the $H_{0}$ and $\Omega_{m}$ constraints in the six non-
flat models. Except for the untilted non-flat model with $A_{L}=1$ we observe
at least partial overlaps between the three sets of contours even when the
$A_{L}$ parameter is not allowed to vary.

The contour plots for the tilted flat $\Lambda$CDM (+$A_{L}$) models are in
Figs. 32 and 33. The aforementioned $\sim$1$\sigma$ disagreements (and the
$\sim 2\sigma$ $\Omega_{m}$ disagreement in the $A_{L}=1$ case) found when we
compared the one-dimensional likelihood P18+lensing and non-CMB results can
also be observed here. The largest tensions are seen in the panels containing
one of the derived parameters and the inclusion in the analysis of the varying
$A_{L}$ parameter clearly reduces them.

Looking at the contour plots for the untilted non-flat $\Lambda$CDM (+$A_{L}$)
models displayed in Figs. 34 and 35 we observe significantly non-overlapping
contours either when the primary parameter $\Omega_{k}$ is involved or when
the derived parameter $H_{0}$ or $\Omega_{m}$ is involved. This reinforces the
idea that when $A_{L}$ is not allowed to vary the P18+lensing and non-CMB data
sets cannot be analyzed together in the untilted non-flat $\Lambda$CDM model.
Quite different results are found when we do allow $A_{L}$ to vary. The
disagreements observed in the $A_{L}=1$ case largely disappear. Therefore we
may say that in the context of this varying $A_{L}$ cosmological model we can
jointly analyze P18+lensing and non-CMB data.

Figures 36 and 37 show cosmological parameter constraints for the tilted non-
flat $\Lambda$CDM (+$A_{L}$) models, while the ones for the tilted non-flat
$\Lambda$CDM (+$A_{L}$) new $P(q)$ models are displayed in Figs. 38 and 39.
The contour plots for these tilted non-flat models are very similar, something
that was not unexpected given the results discussed above in this
subsubsection. In both cases, when $A_{L}$ is not allowed to vary and when it
is allowed to vary, we observe overlaps between the primary parameter panels
contours at 1$\sigma$. When $A_{L}=1$ we observe an improvement in the
overlapping in the current P18+lensing data vs. non-CMB data case, compared to
the P18 data vs. non-CMB data case of the previous subsubsection, where now
for both the Planck $P(q)$ model and the new $P(q)$ model the contours do
overlap at 2$\sigma$. On the other hand, in the varying $A_{L}$ case we
observe overlaps, even in those panels that involve some of the derived
parameters, at 1$\sigma$.

As in the P18 data vs. non-CMB data cosmological constraints comparison
discussed in the previous subsubsection, further tests are needed to determine
whether or not P18+lensing data and non-CMB data can be jointly analyzed in
the context of the non-flat models under study. We discuss this issue in
detail in Sec. IV.3.

Table 20: Individual and total $\chi^{2}$ values for the best-fit flat and
non-flat $\Lambda\textrm{CDM}$ inflation models. The deviance information
criterion (DIC) and the Akaike information criterion (AICc) 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 | $\Delta\textrm{DIC}$ | $\Delta\textrm{AIC}_{c}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Tilted flat $\Lambda\textrm{CDM}$ model
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+lensing+non-CMB | 2346.61 | 22.64 | 396.34 | 8.94 | 1.84 | 1058.99 | 20.10 | 14.76 | 18.20 | 3888.41 |  | 3940.70 |  |
Tilted 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+lensing+non-CMB | 2342.43 | 21.99 | 395.68 | 9.74 | 2.06 | 1059.14 | 21.46 | 14.73 | 14.31 | 3881.55 | $-6.86$ | 3935.15 | $-5.55$ | $-4.86$
Untilted non-flat $\Lambda\textrm{CDM}$ model
P18 | 2369.95 | 22.22 | 395.69 |  | 1.92 |  |  |  |  | 2789.77 | $23.97$ | 2847.14 | $29.21$ | $23.97$
P18+lensing | 2383.06 | 20.88 | 396.13 | 10.63 | 2.43 |  |  |  |  | 2813.13 | $38.42$ | 2869.06 | $42.61$ | $38.42$
P18+lensing+non-CMB | 2396.21 | 19.89 | 399.59 | 11.65 | 2.28 | 1059.51 | 20.65 | 15.68 | 12.77 | 3938.22 | $49.81$ | 3992.71 | $52.01$ | $49.81$
Untilted non-flat $\Lambda$CDM$+A_{L}$ model
P18 | 2369.32 | 20.34 | 395.87 |  | 2.23 |  |  |  |  | 2787.76 | $21.96$ | 2846.45 | $28.52$ | $23.96$
P18+lensing | 2378.87 | 20.09 | 395.65 | 11.25 | 2.05 |  |  |  |  | 2807.91 | $33.20$ | 2856.10 | $29.65$ | $35.20$
P18+lensing+non-CMB | 2379.11 | 19.95 | 395.82 | 10.72 | 2.06 | 1060.16 | 22.50 | 15.47 | 9.26 | 3915.05 | $26.64$ | 3973.55 | $32.85$ | $28.64$
Tilted non-flat $\Lambda\textrm{CDM}$ model [Planck $P(q)$]
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+lensing+non-CMB | 2345.82 | 22.90 | 396.53 | 8.92 | 1.88 | 1059.00 | 20.09 | 14.70 | 18.15 | 3887.99 | $-0.42$ | 3942.07 | $1.37$ | $1.58$
Tilted 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+lensing+non-CMB | 2341.91 | 22.16 | 395.77 | 9.62 | 1.60 | 1059.06 | 20.61 | 14.74 | 15.90 | 3881.37 | $-7.04$ | 3936.85 | $-3.85$ | $-3.04$
Tilted non-flat $\Lambda\textrm{CDM}$ model [new $P(q)$]
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+lensing+non-CMB | 2346.63 | 22.53 | 396.30 | 8.91 | 1.53 | 1058.99 | 20.12 | 14.75 | 17.79 | 3887.55 | $-0.86$ | 3942.22 | $1.52$ | $1.14$
Tilted non-flat $\Lambda\textrm{CDM}$+$A_{L}$ 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+lensing+non-CMB | 2342.85 | 21.35 | 395.81 | 9.72 | 1.53 | 1059.13 | 21.27 | 14.77 | 14.27 | 3880.69 | $-7.72$ | 3937.52 | $-3.18$ | $-3.72$

Note: $\Delta\chi^{2}$, $\Delta\textrm{DIC}$, and $\Delta\textrm{AIC}_{c}$
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${}_{c}=2819.8$ (P18), $2828.7$ (P18+lensing), and $3942.4$
(P18+lensing+non-CMB). All $\chi^{2}$ values are computed at the corresponding
model best-fit cosmological parameter values.

###  IV.2 Model selection

In Sec. IV.1 we determined and discussed the cosmological parameter mean
values and error bars in four pairs of cosmological models (with $A_{L}=1$ and
with varying $A_{L}$) from P18, P18+lensing, and P18+lensing+non-CMB data, as
well as the differences in the values of the cosmological parameters obtained
from P18 data and BAO/BAO′ data, from P18 data and non-CMB data, and from
P18+lensing data and non-CMB data. In this subsection we utilize the DIC, eq.
(23), to determine which of these models best-fit some combinations of these
data sets.

For the P18, P18+lensing, and P18+lensing+non-CMB data sets, the values of
$\Delta\textrm{AIC}_{c}$, $\Delta\textrm{DIC}$, and the individual
contributions to the $\chi^{2}_{\textrm{total}}$ for each model are in Table
20. Here the Planck CMB data $\chi^{2}$s are: $\chi^{2}_{\textrm{plik}}$ from
the TT data power spectra $30\leq\ell\leq 2508$ multipoles, the TE data
$30\leq\ell\leq 1996$ multipoles, and the EE data $30\leq\ell\leq 1996$
multipoles; $\chi^{2}_{\textrm{lowl}}$ from the TT data power spectra
$2\leq\ell\leq 29$ multipoles; $\chi^{2}_{\textrm{simall}}$ from the EE data
power spectra $2\leq\ell\leq 29$ multipoles; $\chi^{2}_{\textrm{lensing}}$
from the lensing potential data power spectrum; and
$\chi^{2}_{\textrm{prior}}$ from the priors for the Planck calibration and
dust foreground emission. The P18+BAO/BAO′ data values of
$\Delta\textrm{AIC}_{c}$ and $\Delta\textrm{DIC}$ are provided in Tables 8-11,
whereas the corresponding P18+non-CMB data results can be found in Tables
12-15.

In this subsection we do not discuss the results obtained for the untilted
non-flat $\Lambda$CDM models, without and with a varying $A_{L}$, since as
seen in the results presented in Tables 20, 9, and 13 this model is not able
to fit CMB data as well as the other (tilted) models do. According to the
statistical criteria we use, the untilted non-flat $\Lambda$CDM model is very
strongly disfavoured when it is compared with the rest of the models that
allow for a tilt ($n_{s}$) degree of freedom.

We also do not discuss results obtained when only BAO′, BAO, (P18) lensing
(but see Table 22 and the brief related discussion in the third paragraph in
Sec. IV.3), or non-CMB data are considered, because these data sets do not
much discriminate between models. From Tables 8-11 and 12-15 one sees that for
these three data sets the DIC values for all models, including the untilted
non-flat $\Lambda$CDM model, are very similar. In order to find more
significant differences among the models under study we must include CMB data.

In what follows we summarize results we find in a number of different
combinations of data sets for the three tilted models. For clarity we focus on
DIC results, since this is a more reliable indicator, Spiegelhalter et al.
(2002); Liddle (2007). The tables also list the AICc values.

P18. The results for these data are listed in Table 20. When $A_{L}=1$, the
non-flat Planck $P(q)$ and the non-flat new $P(q)$ models are strongly favored
over the tilted flat model while the Planck $P(q)$ model is weakly favored
over the new $P(q)$ model. When $A_{L}$ is allowed to vary, the non-flat
Planck $P(q)$ model is weakly favored over the flat model, with both models
being positively favored over the non-flat new $P(q)$ model. The flat$+A_{L}$
model is positively favored over the flat one, the Planck $P(q)$ model is
weakly favored over the Planck $P(q)+A_{L}$ one, and the new $P(q)$ model is
positively favored over the new $P(q)+A_{L}$ one. It is interesting that
compared to the varying $A_{L}$ case, when $A_{L}=1$ both tilted non-flat
models are strongly favored over the tilted flat $\Lambda$CDM model.

P18+lensing. The results for these data are listed in Table 20. These data
provide only weak discrimination between models. When $A_{L}=1$, the non-flat
new $P(q)$ model is weakly favored over the non-flat Planck $P(q)$ model and
both are weakly favored over the flat model. When $A_{L}$ is allowed to vary,
the tilted flat model is weakly favored over both non-flat models while the
non-flat Planck $P(q)$ model is weakly favored over the non-flat new $P(q)$
model. The flat$+A_{L}$ model is weakly favored over the flat one, the Planck
$P(q)$ model is weakly favored over the Planck $P(q)+A_{L}$ one, and the new
$P(q)$ model is weakly favored over the new $P(q)+A_{L}$ one.

P18+BAO/P18+BAO′. The results for these data are listed in Tables 8, 10, and
11. We discuss the P18+BAO data and P18+BAO′ data results together since the
conclusions are very similar. When $A_{L}=1$, the tilted flat model is weakly
(positively) favored over the non-flat Planck and new $P(q)$ models with the
non-flat new $P(q)$ model weakly (weakly) favored over the non-flat Planck
$P(q)$ model for P18+BAO (P18+BAO′) data. When $A_{L}$ is allowed to vary, the
tilted flat model is positively (weakly) favored over the non-flat Planck
(new) $P(q)$ model, and the non-flat new $P(q)$ model weakly is favored over
the non-flat Planck $P(q)$ model, for P18+BAO data, while for P18+BAO′ data
the tilted flat model is weakly favored over both non-flat Planck and new
$P(q)$ models, and the non-flat new $P(q)$ model is weakly favored over the
non-flat Planck $P(q)$ model. The flat$+A_{L}$ model is strongly (positively)
favored over the flat one, the Planck $P(q)+A_{L}$ model is positively
(strongly) favored over the Planck $P(q)$ one, and the new $P(q)+A_{L}$ model
is positively (strongly) favored over the new $P(q)$ one for P18+BAO
(P18+BAO′) data.

P18+non-CMB. The results for these data are listed in Tables 12, 14, and 15.
Since the dominant component of non-CMB data is BAO/BAO′ data, in the P18+non-
CMB case here we find similar conclusions to the ones presented in the
P18+BAO/P18+BAO′ cases above. When $A_{L}=1$, the tilted flat model is
positively (weakly) favored over the non-flat Planck (new) $P(q)$ model with
the non-flat new $P(q)$ model weakly favored over the non-flat Planck $P(q)$
model. When $A_{L}$ is allowed to vary, the tilted flat model is weakly
favored over the non-flat Planck $P(q)$ and non-flat new $P(q)$ models, with
the non-flat new $P(q)$ model weakly favored over the non-flat Planck $P(q)$
model. The flat$+A_{L}$ model is strongly favored over the flat one, the
Planck $P(q)+A_{L}$ model is strongly favored over the Planck $P(q)$ one, and
the new $P(q)+A_{L}$ model is strongly favored over the new $P(q)$ one.

P18+lensing+non-CMB. The results for these data are listed in Table 20. When
$A_{L}=1$, the tilted flat model is weakly favored over the non-flat Planck
$P(q)$ and non-flat new $P(q)$ models with the non-flat Planck $P(q)$ model
weakly favored over the non-flat new $P(q)$ model. When $A_{L}$ is allowed to
vary, the tilted flat model is weakly (positively) favored over the non-flat
Planck (new) $P(q)$ model, with the non-flat Planck $P(q)$ model weakly
favored over the non-flat new $P(q)$ model. The flat$+A_{L}$ model is
positively favored over the flat one, the Planck $P(q)+A_{L}$ model is
positively favored over the Planck $P(q)$ one, and the new $P(q)+A_{L}$ model
is positively favored over the new $P(q)$ one.

In summary: P18 data and P18+non-CMB data both strongly disfavor the tilted
flat $\Lambda$CDM model with $A_{L}=1$ relative to some of the tilted
$\Omega_{k}<0$ or varying $A_{L}$ options; P18+lensing data are largely
agnostic; and P18+lensing+non-CMB data, P18+BAO data, and P18+BAO′ data all
positively favor the varying $A_{L}$ options over the $A_{L}=1$ cases.

Table 21: $\log_{10}\mathcal{I}$ and tension ($\sigma$ and $p$) parameters for
P18 data versus lensing data, P18 data versus BAO (BAO′) data, P18 data versus
non-CMB data, and P18+lensing data versus non-CMB data in the six tilted flat
and non-flat $\Lambda$CDM models. Table 3 lists the Our, Handley, and
Handley+$\Omega_{k}$ priors.

|  Tilted flat $\Lambda$CDM model
---|---
Data: | P18 vs. lensing | P18 vs. lensing | P18 vs. lensing | P18 vs. BAO |  P18 vs. BAO′ | P18 vs. non-CMB | P18+lensing vs. non-CMB
Prior: | Our | Handley |  Handley+$\Omega_{k}$ | Our | Our | Our | Our
$\log_{10}{\mathcal{I}}$ | $1.240$ | $1.166$ | $\ldots$ | $0.132$ | $0.707$ | $0.296$ | $0.029$
$\sigma$ | $0.718$ | $0.390$ | $\ldots$ | $1.533$ | $0.426$ | $1.749$ | $1.747$
$p$ (%) | $47.3$ | $69.7$ | $\ldots$ | $12.5$ | $67.0$ | $8.03$ | $8.06$
|  Tilted flat $\Lambda$CDM$+A_{L}$ model
Data: | P18 vs. lensing | P18 vs. lensing | P18 vs. lensing | P18 vs. BAO |  P18 vs. BAO′ | P18 vs. non-CMB | P18+lensing vs. non-CMB
Prior: | Our | Handley |  Handley+$\Omega_{k}$ | Our | Our | Our | Our
$\log_{10}{\mathcal{I}}$ | $\ldots$ | $\ldots$ | $\ldots$ | $0.286$ | $0.810$ | $1.033$ | $1.033$
$\sigma$ | $\ldots$ | $\ldots$ | $\ldots$ | $1.402$ | $0.371$ | $0.835$ | $0.774$
$p$ (%) | $\ldots$ | $\ldots$ | $\ldots$ | $16.1$ | $71.0$ | $40.4$ | $43.9$
|  Tilted non-flat $\Lambda$CDM model [Planck $P(q)$]
Data: | P18 vs. lensing | P18 vs. lensing | P18 vs. lensing | P18 vs. BAO |  P18 vs. BAO′ | P18 vs. non-CMB | P18+lensing vs. non-CMB
Prior: | Our | Handley |  Handley+$\Omega_{k}$ | Our | Our | Our | Our
$\log_{10}{\mathcal{I}}$ | $-0.486$ | $-0.316$ | $-0.360$ | $-1.236$ | $-0.891$ | $-1.263$ | $0.297$
$\sigma$ | $2.479$ | $2.411$ | $2.403$ | $3.000$ | $2.478$ | $3.005$ | $1.837$
$p$ (%) | $1.32$ | $1.59$ | $1.63$ | $0.270$ | $1.32$ | $0.265$ | $6.62$
|  Tilted non-flat $\Lambda$CDM$+A_{L}$ model [Planck $P(q)$]
Data: | P18 vs. lensing | P18 vs. lensing | P18 vs. lensing | P18 vs. BAO |  P18 vs. BAO′ | P18 vs. non-CMB | P18+lensing vs. non-CMB
Prior: | Our | Handley |  Handley+$\Omega_{k}$ | Our | Our | Our | Our
$\log_{10}{\mathcal{I}}$ | $\ldots$ | $\ldots$ | $\ldots$ | $0.182$ | $0.847$ | $0.972$ | $1.641$
$\sigma$ | $\ldots$ | $\ldots$ | $\ldots$ | $1.460$ | $0.465$ | $0.793$ | $0.516$
$p$ (%) | $\ldots$ | $\ldots$ | $\ldots$ | $14.4$ | $64.2$ | $42.8$ | $60.6$
|  Tilted non-flat $\Lambda$CDM model [new $P(q)$]
Data: | P18 vs. lensing | P18 vs. lensing | P18 vs. lensing | P18 vs. BAO |  P18 vs. BAO′ | P18 vs. non-CMB | P18+lensing vs. non-CMB
Prior: | Our | Handley |  Handley+$\Omega_{k}$ | Our | Our | Our | Our
$\log_{10}{\mathcal{I}}$ | $-0.062$ | $-0.089$ | $-0.057$ | $-0.880$ | $-0.526$ | $-0.806$ | $0.143$
$\sigma$ | $2.201$ | $1.887$ | $1.843$ | $2.604$ | $2.108$ | $2.577$ | $1.886$
$p$ (%) | $2.77$ | $5.91$ | $6.54$ | $0.922$ | $3.50$ | $0.996$ | $5.93$
|  Tilted non-flat $\Lambda$CDM$+A_{L}$ model [new $P(q)$]
Data: | P18 vs. lensing | P18 vs. lensing | P18 vs. lensing | P18 vs. BAO |  P18 vs. BAO′ | P18 vs. non-CMB | P18+lensing vs. non-CMB
Prior: | Our | Handley |  Handley+$\Omega_{k}$ | Our | Our | Our | Our
$\log_{10}{\mathcal{I}}$ | $\ldots$ | $\ldots$ | $\ldots$ | $1.066$ | $1.655$ | $1.798$ | $1.500$
$\sigma$ | $\ldots$ | $\ldots$ | $\ldots$ | $1.052$ | $0.145$ | $0.402$ | $0.573$
$p$ (%) | $\ldots$ | $\ldots$ | $\ldots$ | $29.3$ | $88.4$ | $68.7$ | $56.7$

Note: The statistical estimator values in the tilted flat $\Lambda$CDM model
for the Handley+$\Omega_{k}$ priors are the same as for the Handley priors
because $\Omega_{k}=0$ in the flat model.

###  IV.3 Data set tensions

In this subsection we check whether there is concordance (discordance) between
pairs of some of the data sets we study (in the context of a given
cosmological model), as well as whether or not this concordance (discordance)
is model independent. To do this, we use the two Sec. III statistical
estimators, in eq. (26) and in eqs. (35) and (37). The values of these
statistical estimators for the six tilted flat and non-flat $\Lambda$CDM
($+A_{L}$) models are listed in Table 21; we do not compute these estimators
in the untilted non-flat $\Lambda$CDM model which does not include the tilt
($n_{s}$) degree of freedom that is strongly favored by data. As in Sec. IV.2,
here we only study pairs of data sets in which one of the data sets is or
includes the P18 data set. Conclusions based on either of the two statistical
estimators qualitatively agree, for the five pairs of data sets we compare in
this subsection, as discussed next.

Table 22: Mean and 68.3% confidence limits of tilted flat and non-flat
$\Lambda\textrm{CDM}$ model parameters constrained by lensing data alone.
Table 3 lists the Our, Handley, and Handley+$\Omega_{k}$ priors. The Hubble
constant $H_{0}$ has units of km s-1 Mpc-1.

| Lensing data constraints with Our priors
---|---
Parameter |  Tilted flat $\Lambda$CDM |  Tilted non-flat $\Lambda$CDM [Planck $P(q)$] |  Tilted non-flat $\Lambda$CDM [new $P(q)$]
$\Omega_{b}h^{2}$ | $0.049\pm 0.023$ | $0.052\pm 0.027$ | $0.048\pm 0.026$
$\Omega_{c}h^{2}$ | $0.125\pm 0.032$ | $0.120\pm 0.023$ | $0.116\pm 0.022$
$100\theta_{\textrm{MC}}$ | $1.016\pm 0.022$ | $1.41\pm 0.33$ | $1.47\pm 0.27$
$\tau$ | $0.0542$ | $0.0483$ | $0.0525$
$\Omega_{k}$ | $\ldots$ | $-0.26\pm 0.11$ | $-0.279\pm 0.095$
$n_{s}$ | $0.9649$ | $0.9706$ | $0.9654$
$\ln(10^{10}A_{s})$ | $3.23\pm 0.11$ | $3.10\pm 0.19$ | $3.13\pm 0.16$
$H_{0}$ | $83\pm 10$ | $65\pm 17$ | $66\pm 16$
$\Omega_{m}$ | $0.255\pm 0.070$ | $0.54\pm 0.48$ | $0.48\pm 0.36$
$\sigma_{8}$ | $0.779\pm 0.082$ | $0.85\pm 0.16$ | $0.88\pm 0.15$
$\chi_{\textrm{min}}^{2}$ | $3.67$ | $3.12$ | $3.38$
DIC (lensing) | $14.2$ | $13.3$ | $13.9$
AIC (lensing) | $13.7$ | $15.1$ | $15.4$
DIC (P18) | $2817.9$ | $2810.6$ | $2811.5$
DIC (P18+lensing) | $2826.5$ | $2826.2$ | $2825.7$
$\log_{10}\mathcal{I}$ | $1.240$ | $-0.486$ | $-0.062$
| Lensing data constraints with Handley priors
Parameter |  Tilted flat $\Lambda$CDM |  Tilted non-flat $\Lambda$CDM [Planck $P(q)$] |  Tilted non-flat $\Lambda$CDM [new $P(q)$]
$\Omega_{b}h^{2}$ | $0.0220\pm 0.0018$ | $0.0221\pm 0.0017$ | $0.0220\pm 0.0017$
$\Omega_{c}h^{2}$ | $0.1121\pm 0.0093$ | $0.1117\pm 0.0099$ | $0.1134\pm 0.0097$
$100\theta_{\textrm{MC}}$ | $1.0397\pm 0.0058$ | $1.0395\pm 0.0058$ | $1.0395\pm 0.0059$
$\tau$ | $0.21\pm 0.11$ | $0.20\pm 0.11$ | $0.21\pm 0.11$
$\Omega_{k}$ | $\ldots$ | $-0.032\pm 0.040$ | $-0.029\pm 0.040$
$n_{s}$ | $0.957\pm 0.043$ | $0.954\pm 0.043$ | $0.939\pm 0.033$
$\ln(10^{10}A_{s})$ | $3.26\pm 0.15$ | $3.20\pm 0.16$ | $3.21\pm 0.16$
$H_{0}$ | $69.7\pm 3.9$ | $62\pm 14$ | $63\pm 14$
$\Omega_{m}$ | $0.281\pm 0.050$ | $0.40\pm 0.15$ | $0.39\pm 0.15$
$\sigma_{8}$ | $0.869\pm 0.064$ | $0.826\pm 0.083$ | $0.836\pm 0.084$
$\chi_{\textrm{min}}^{2}$ | $6.81$ | $6.89$ | $6.79$
DIC (lensing) | $13.9$ | $14.1$ | $13.8$
AIC (lensing) | $20.8$ | $22.9$ | $22.8$
DIC (P18) | $2817.9$ | $2810.6$ | $2811.5$
DIC (P18+lensing) | $2826.5$ | $2826.2$ | $2825.7$
$\log_{10}\mathcal{I}$ | $1.166$ | $-0.316$ | $-0.088$
|  Lensing data constraints with Handley$+\Omega_{k}$ priors
Parameter |  Tilted flat $\Lambda$CDM |  Tilted non-flat $\Lambda$CDM [Planck $P(q)$] |  Tilted non-flat $\Lambda$CDM [new $P(q)$]
$\Omega_{b}h^{2}$ | $\cdots$ | $0.0221\pm 0.0017$ | $0.0221\pm 0.0017$
$\Omega_{c}h^{2}$ | $\cdots$ | $0.1088\pm 0.0088$ | $0.1104\pm 0.0089$
$100\theta_{\textrm{MC}}$ | $\cdots$ | $1.0395\pm 0.0058$ | $1.0396\pm 0.0059$
$\tau$ | $\cdots$ | $0.20\pm 0.11$ | $0.20\pm 0.11$
$\Omega_{k}$ | $\cdots$ | $-0.123\pm 0.095$ | $-0.122\pm 0.096$
$n_{s}$ | $\cdots$ | $0.951\pm 0.041$ | $0.939\pm 0.032$
$\ln(10^{10}A_{s})$ | $\cdots$ | $3.11\pm 0.16$ | $3.11\pm 0.16$
$H_{0}$ | $\cdots$ | $48\pm 15$ | $48\pm 15$
$\Omega_{m}$ | $\cdots$ | $0.70\pm 0.33$ | $0.71\pm 0.33$
$\sigma_{8}$ | $\cdots$ | $0.745\pm 0.096$ | $0.75\pm 0.10$
$\chi_{\textrm{min}}^{2}$ | $\cdots$ | $6.79$ | $6.77$
DIC (lensing) | $\cdots$ | $13.9$ | $13.9$
AIC (lensing) | $\cdots$ | $22.8$ | $22.8$
DIC (P18) | $\cdots$ | $2810.6$ | $2811.5$
DIC (P18+lensing) | $\cdots$ | $2826.2$ | $2825.7$
$\log_{10}\mathcal{I}$ | $\cdots$ | $-0.360$ | $-0.057$

Note: $\mathcal{I}=\exp(-\mathcal{F}/2)$ where
$\mathcal{F}=\textrm{DIC(P18+lensing)}-\textrm{DIC(P18)}-\textrm{DIC(lensing)}$.
The cosmological parameter values in the tilted flat $\Lambda$CDM model for
the Handley+$\Omega_{k}$ priors are the same as for the Handley priors because
$\Omega_{k}=0$ in the flat model.

  * •

P18 vs. lensing. Since, as mentioned earlier, lensing data (see Sec. II) alone
do not place significant constraints on cosmological parameters (even if we
fix the values of some of them), the role played by the priors is more
important in lensing data alone analyses than in other cases. Therefore, in
this case, we use three different sets of priors (see Table 3) in order to
determine whether and how the lensing data alone cosmological parameter
constraints and statistical estimator values depend on the priors used. In all
three cases we report results obtained from converged chains. Due to the weak
constraining power of lensing data alone, it is not possible to reach
convergence when the $A_{L}$ parameter is allowed to vary. Consequently, we
provide results only for the $A_{L}=1$ cases.

Here we first briefly comment on the lensing data alone cosmological parameter
constraints, which do depend on the set of priors used, see Table 22. For
instance, if we look at the value of the curvature parameter $\Omega_{k}$
(which is most affected by the choice of prior) obtained by employing Our
priors, for the titled non-flat $\Lambda$CDM Planck (new) $P(q)$ model,
$\Omega_{k}=-0.26\pm 0.11$ ($\Omega_{k}=-0.279\pm 0.095$), we find a
1.9$\sigma$ (2.4$\sigma$) difference with the Handley priors analysis value
$\Omega_{k}=-0.032\pm 0.040$ ($\Omega_{k}=-0.029\pm 0.040$) and a 0.94$\sigma$
(1.2$\sigma$) difference with the Handley+$\Omega_{k}$ priors analysis value
$\Omega_{k}=-0.123\pm 0.095$ ($\Omega_{k}=-0.122\pm 0.096$). Reassuringly, we
find that when we broaden the prior for $\Omega_{k}$, as we do when we move
from Handley priors to Handley+$\Omega_{k}$ priors, the results get closer to
those obtained with Our priors, the broadest priors we use. Additionally, our
lensing data alone analysis (and cosmological parameter constraints) differ
from those of the Planck team (Sec. 3.2.1 of Ref. Aghanim et al. (2020b)) in
that we fix $n_{s}$ and vary $\Omega_{b}h^{2}$ freely, whereas the Planck team
use Gaussian priors for $n_{s}$ and $\Omega_{b}h^{2}$. Also, in our analysis
$0.2<h<1.0$ was chosen as the prior, while the Planck team used $0.4<h<1.0$.
One notable difference is that when Our priors are used the value we find for
$\Omega_{b}h^{2}$ is larger than the Gaussian prior value
($\Omega_{b}h^{2}=0.0222\pm 0.0005$) adopted by the Planck team. In the tilted
flat $\Lambda$CDM model we find $\Omega_{b}h^{2}=0.049\pm 0.023$, and similar
results are seen in the tilted non-flat models with the Planck and the new
$P(q)$. However, when the Handley priors and the Handley+$\Omega_{k}$ priors
are used, due to the very narrow range of $\Omega_{b}h^{2}$ (between $0.019$
and $0.025$) in these priors, such a deviation disappears, and
$\Omega_{b}h^{2}$ is constrained with very consistent values in the tilted
flat and the two tilted non-flat $\Lambda$CDM models and is also consistent
with the Gaussian prior value adopted by the Planck team. Given the
significant dependence on prior of the lensing data alone cosmological
constraints, it is not possible to compare lensing data alone cosmological
constraints to cosmological constraints we have derived from the other data
sets.

On the other hand, looking at Table 21, we do not see significant differences
in the statistical estimator values from lensing only data analyses for the
three different priors. This being the case, in the following, for the sake of
consistency with our other discussions, we discuss only the lensing data alone
results obtained using Our priors.

For the tilted flat $\Lambda$CDM model we do not find discordance between P18
data and lensing data. We find $\textrm{log}_{10}\mathcal{I}=1.240$ which
indicates a strong consistency between the two data sets. A similar conclusion
is indicated by the other statistical estimator, $\sigma=0.718$ and
$p=47.3\%$. We conclude that P18 and lensing data can be jointly analyzed in
the context of the tilted flat $\Lambda$CDM model.

Looking at the results for the tilted non-flat $\Lambda$CDM Planck $P(q)$
model, for the first statistical estimator
$\textrm{log}_{10}\mathcal{I}=-0.486$ which is on the verge of indicating a
substantial discordance while for the second one $\sigma=2.479$ and $p=1.32\%$
which indicate a moderate tension. These results however may not be
significant enough to conclude that P18 and lensing data cannot be used
together in an analysis of the tilted non-flat $\Lambda$CDM Planck $P(q)$
model.

In the tilted non-flat new $P(q)$ $\Lambda$CDM model, the two statistical
estimators considered here point to somewhat different conclusions. While for
the first one we get $\textrm{log}_{10}\mathcal{I}=-0.062$, which indicates
neither consistency nor inconsistency between the two data sets, the second
one, $\sigma=2.201$ and $p=2.77\%$, indicates a moderate tension between the
two data sets. Taken together these results indicate that there is at most
moderate inconsistency between P18 and lensing data within the tilted flat new
$P(q)$ $\Lambda$CDM model.

  * •

P18 vs. BAO′. In the context of the tilted flat $\Lambda$CDM model there is no
sign of discordance between these two data sets. We find
$\textrm{log}_{10}\mathcal{I}=0.707$, which indicates a substantial
consistency. The other statistical estimator points to a similar conclusion,
with $\sigma=0.426$ and $p=67\%$. Very similar results are found for the
tilted flat $\Lambda$CDM+$A_{L}$ model. The value
$\textrm{log}_{10}\mathcal{I}=0.810$, once again, indicates a substantial
consistency between P18 and BAO′ data, whereas for the second estimator we
find $\sigma=0.371$ and $p=71\%$. The P18 and BAO′ data sets are mutually
consistent and can be jointly analyzed in the tilted flat $\Lambda$CDM
(+$A_{L}$) models.

On the other hand, the opposite is true in the tilted non-flat $\Lambda$CDM
models (with $A_{L}=1$). The comparison of P18 and BAO′ data in the tilted
non-flat Planck $P(q)$ model results in $\textrm{log}_{10}\mathcal{I}=-0.891$
which indicates a substantial disagreement between these two data sets.
Reassuringly the second statistical estimator points to the same conclusion,
in particular, $\sigma=2.478$ and $p=1.32\%$. As expected (see Sec. IV.1.5)
inclusion of the varying $A_{L}$ parameter reduces the tensions with respect
to the $A_{L}=1$ case. For the Planck $P(q)$+$A_{L}$ model we find
$\textrm{log}_{10}\mathcal{I}=0.847$, which indicates a substantial degree of
consistency between the two data sets, and $\sigma=0.465$ and $p=64.2\%$,
therefore, there is no tension between P18 data and BAO′ data in this model.

We noted in Sec. IV.1.5 that the tilted non-flat $\Lambda$CDM new $P(q)$ model
better accommodates P18 and BAO′ data than does the tilted non-flat
$\Lambda$CDM Planck $P(q)$ model. In particular, in tilted non-flat
$\Lambda$CDM new $P(q)$ model when $A_{L}=1$ we find
$\textrm{log}_{10}\mathcal{I}=-0.526$, that is just in the range of
substantial inconsistency. According to the values obtained for the other
statistical estimator, $\sigma=2.108$ and $p=3.50\%$, there is a moderate
tension between the two data sets. The inclusion of a varying $A_{L}$
parameter in the analysis completely changes the conclusions with respect to
the $A_{L}=1$ case. For the new $P(q)$+$A_{L}$ model we find
$\textrm{log}_{10}\mathcal{I}=1.655$, indicating strong agreement. The values
$\sigma=0.145$ and $p=88.4\%$ support this conclusion.

  * •

P18 vs. BAO. We comment now on the results obtained when the tension between
P18 data and BAO data is studied in the context of the different cosmological
models. We note that the BAO data set includes some $f\sigma_{8}$ data points
which, as we shall see, induces some changes in the results with respect to
the P18 data and BAO′ data case.

Both statistical estimators do not indicate significant disagreement between
P18 data and BAO data for the tilted flat $\Lambda$CDM model with $A_{L}=1$.
For the first one we have $\textrm{log}_{10}\mathcal{I}=0.132$, which neither
indicates consistency nor inconsistency, and this is supported by the second
one for which we obtain $\sigma=1.533$ and $p=12.5\%$. It is important to note
that in this case the statistical estimators are closer to indicating a
moderate tension than they are in the P18 data vs. BAO′ data case. This is
related to the previously mentioned $\sigma_{8}$ tension. We get similar
results for the tilted flat $\Lambda$CDM+$A_{L}$ model, in which case we find
$\textrm{log}_{10}\mathcal{I}=0.286$, which again neither indicates an
agreement nor a disagreement, while for the second estimator $\sigma=1.402$
and $p=16.1\%$, and again no tension is revealed. In view of these results we
find no evidence that P18 and BAO data cannot be considered together in the
analysis of the tilted flat $\Lambda$CDM (+$A_{L}$) models.

Given the P18 data vs. BAO′ data comparison results in the tilted non-flat
$\Lambda$CDM models, it should not come as a surprise that we find tensions
when P18 data and BAO data are compared. In the tilted non-flat Planck $P(q)$
$\Lambda$CDM model with $A_{L}=1$ we find, for the first estimator
$\textrm{log}_{10}\mathcal{I}=-1.236$, and $\sigma=3.000$ and $p=0.27\%$ for
the second one. Both results indicate a strong inconsistency between the two
data sets. This level of tension fades when the $A_{L}$ parameter is allowed
to vary. For the Planck $P(q)$+$A_{L}$ model we obtain
$\textrm{log}_{10}\mathcal{I}=0.182$, which does not indicate consistency or
inconsistency, and $\sigma=1.460$ and $p=14.4\%$. The P18 and BAO data can be
jointly used in the Planck $P(q)$+$A_{L}$ model. As happens in the case of the
P18 data vs. BAO′ data comparison, the tilted non-flat new $P(q)$ $\Lambda$CDM
model performs better than the Planck $P(q)$ when it comes to accommodating
the P18 and BAO data sets. For the $A_{L}=1$ case we find
$\textrm{log}_{10}\mathcal{I}=-0.880$, revealing substantial disagreement,
while for the other estimator $\sigma=2.604$ and $p=0.922\%$, which indicates
a moderate tension. Once again, the tensions observed when $A_{L}=1$, in the
context of non-flat models, disappear when this parameter is allowed to vary.
For the tilted non-flat $\Lambda$CDM+$A_{L}$ new $P(q)$ model, we find
$\textrm{log}_{10}\mathcal{I}=1.066$, which points out to a strong consistency
between the two data sets, and for the other estimator we obtain
$\sigma=1.052$ and $p=29.3\%$. The P18 and BAO data can be jointly used in the
new $P(q)$+$A_{L}$ model.

In summary, in the tilted non-flat models, in the Planck $P(q)$ model P18 and
BAO data should not be jointly analyzed unless the $A_{L}$ parameter is
allowed to vary, while in the new $P(q)$ models these two data sets can be
considered together to put constraints on the cosmological parameters even
when $A_{L}=1$.

  * •

P18 vs. non-CMB. We now discuss whether or not there is tension between P18
data and non-CMB data in the context of the different cosmological models.
Similar results to the ones obtained in the P18 data and BAO′/BAO data
comparisons are expected, since BAO′ data and BAO data are dominant components
of non-CMB data.

For the tilted flat $\Lambda$CDM model with $A_{L}=1$ we find
$\textrm{log}_{10}\mathcal{I}=0.296$, which neither indicates agreement nor
disagreement, and $\sigma=1.749$ together with $p=8.03\%$, with neither of the
two estimators pointing to tension between P18 and non-CMB data in this model.
Including a varying $A_{L}$ in the model improves the agreement between the
two data sets. For the tilted flat $\Lambda$CDM+$A_{L}$ model we find
$\textrm{log}_{10}\mathcal{I}=1.033$ which points to strong consistency
between the two data sets, and for the other estimator we get $\sigma=0.835$
and $p=40.4\%$, a result consistent with the first. There is no tension that
prevents us from jointly analyzing P18 data and non-CMB data in the tilted
flat $\Lambda$CDM (+$A_{L}$) models.

In the case of the tilted non-flat Planck $P(q)$ $\Lambda$CDM model with
$A_{L}=1$, the value $\textrm{log}_{10}\mathcal{I}=-1.263$ indicates a strong
inconsistency between the P18 and non-CMB data sets. The second statistical
estimator provides similar results, $\sigma=3.005$ and $p=0.265\%$. In the
light of these results, we conclude that P18 data and non-CMB data should not
be jointly analyzed in the context of this tilted non-flat $A_{L}=1$ model.
For the Planck $P(q)$+$A_{L}$ model, we get
$\textrm{log}_{10}\mathcal{I}=0.972$, so substantial agreement is observed
between P18 data and non-CMB data in this case. In agreement with the result
obtained employing the first statistical estimator, for the second one we find
$\sigma=0.793$ and $p=42.8\%$, which again does not indicate any tension.

Once again the tilted non-flat $\Lambda$CDM new $P(q)$ model does better in
jointly accommodating P18 and non-CMB data than does the tilted non-flat
$\Lambda$CDM Planck $P(q)$ model. In the new $P(q)$ case with $A_{L}=1$, the
values obtained for both statistical estimators,
$\textrm{log}_{10}\mathcal{I}=-0.806$ and $\sigma=2.577$ and $p=0.996\%$,
indicate a substantial discordance between P18 data and non-CMB data in the
context of this model. Allowing $A_{L}$ to vary reduces the tension found in
the $A_{L}=1$ cases. For the new $P(q)$+$A_{L}$ model we get
$\textrm{log}_{10}\mathcal{I}=1.798$, which points to a strong agreement
between the two data sets, whereas for the second estimator we find
$\sigma=0.402$ and $p=68.7\%$ and no tension. Therefore, we may say that in
the context of the tilted non-flat $\Lambda$CDM (+$A_{L}$) new $P(q)$ models,
P18 and non-CMB data can be jointly analyzed.

  * •

P18+lensing vs. non-CMB. In the previous cases we have detected some tensions
in the context of the non-flat models. Here we study the possible disagreement
between P18+lensing data and non-CMB data.

For the tilted flat $\Lambda$CDM model with $A_{L}=1$, both statistical
estimators, with values $\textrm{log}_{10}\mathcal{I}=0.029$ and
$\sigma=1.747$ and $p=8.06\%$, shed no light on a possible consistency or
inconsistency between P18+lensing data and non-CMB data. For the tilted flat
$\Lambda$CDM+$A_{L}$ model, we find $\textrm{log}_{10}\mathcal{I}=1.033$ which
indicates a strong consistency between the two data sets. On the other hand,
the second statistical estimator provides $\sigma=0.774$ and $p=43.9\%$, which
do not indicate consistency or inconsistency. As we noted at the beginning of
this subsection we do not always expect a perfect match in the conclusions
from the two estimators.

In the tilted non-flat $\Lambda$CDM Planck $P(q)$ model with $A_{L}=1$ we find
$\textrm{log}_{10}\mathcal{I}=0.297$ which neither indicates consistency nor
inconsistency between P18+lensing data and non-CMB data, whereas for the
second estimator we find $\sigma=1.837$ and $p=6.62\%$, which does not reveal
inconsistency. The consistency between P18+lensing and non-CMB data improves
considerably in the context of the Planck $P(q)$+$A_{L}$ model. We get
$\textrm{log}_{10}\mathcal{I}=1.641$ indicating a strong consistency between
the two data sets, while the second one gives $\sigma=0.516$ and $p=60.6\%$,
in agreement with the conclusion provided by the first estimator.

Very similar conclusions are found for the new $P(q)$ (+$A_{L}$) and the
Planck $P(q)$ (+$A_{L}$) models. When the $A_{L}$ parameter is not allowed to
vary for the new $P(q)$ $A_{L}=1$ model we find
$\textrm{log}_{10}\mathcal{I}=0.143$ which does not reveal either an
inconsistency or a consistency, with second estimator giving $\sigma=1.886$
and $p=5.927\%$ and again no tension is revealed. On the other hand, in the
context of the new $P(q)$+$A_{L}$ model, we get
$\textrm{log}_{10}\mathcal{I}=1.50$ indicating a strong consistency between
the two data sets and reassuringly we find similar conclusions from the second
statistical estimator, $\sigma=0.573$ and $p=56.7\%$.

Unlike in the comparisons of P18 data and BAO′/BAO data and the comparisons of
P18 data and non-CMB data, we do not find tensions in the context of the non-
flat models between P18+lensing data and non-CMB data, even when the $A_{L}$
parameter is not allowed to vary. This may be suggesting that if we want to
jointly analyze P18 data and a low-redshift data set, such as BAO′/BAO data or
non-CMB data, we should either consider a varying $A_{L}$ parameter or include
(P18) lensing data in the mix.

We have studied the tensions between pairs of data sets, in the context of a
given cosmological model, in three different ways based on Bayesian
statistics. In Secs. IV.1.5-IV.1.7 we quantified the level of tension by
comparing the (one- and two-dimensional) cosmological parameter constraints
favored by each of the pair of data sets. In the one-dimensional cases we
estimated the tension by considering the quadrature sum of the two error bars
for each parameter, while in the two-dimensional cases we looked at whether or
not the two sets of contours shared a common parameter space area. In this
subsection we study tensions between data set pairs by using the two more
precise statistical estimators of Sec. III, see eq. (26) and eqs. (35) and
(37). Reassuringly, all three techniques employed result in similar
conclusions in most cases.

Among all the data set comparisons we study there are two with significant
enough discordances to be ruled out: we find in the tilted non-flat
$\Lambda$CDM Planck $P(q)$ $A_{L}=1$ model that P18 data and BAO data, as well
P18 data and non-CMB data, are not mutually consistent. In the first case,
when P18 and BAO data are compared, we observe a 2.7$\sigma$ tension between
the derived cosmological parameter values of $\Omega_{m}$ and of $H_{0}$,
obtained with P18 data and with BAO data. Additionally, in Fig. 20, contour
plot panels that contain one of these derived parameters show non-overlapping
regions at more than 2$\sigma$. As for the P18 vs. non-CMB case, the tensions
are even greater than for P18 vs. BAO. Comparing the derived cosmological
parameter values of $\Omega_{m}$ and of $H_{0}$, obtained with P18 data and
non-CMB data, we observe a disagreement at 2.9$\sigma$ and 3.9$\sigma$
respectively. Again the contour plot panels in Fig. 28 containing $\Omega_{m}$
and/or $H_{0}$ show a non-overlapping region at more than 2$\sigma$.

For the two statistical estimators of Sec. III, if we choose to say two data
sets are mutually inconsistent (in a given model) when
$\textrm{log}_{10}\mathcal{I}\leq-1$ or $\sigma\geq 3$, then this is true only
in the two cases discussed in the previous paragraph. For the the tilted non-
flat $\Lambda$CDM Planck $P(q)$ $A_{L}=1$ model, in the P18 vs. BAO case we
find $\textrm{log}_{10}\mathcal{I}=-1.236$ (meaning a strong disagreement
between the two data sets) and $\sigma$ = 3.000 ($p=0.270\%$), while in the
P18 vs. non-CMB analysis we find $\textrm{log}_{10}\mathcal{I}=-1.263$ (again
a strong disagreement between the two data sets) and $\sigma$ = 3.005
($p=0.265\%$). These results are qualitatively consistent with those of the
previous paragraph. They mean that P18 data and BAO data, as well as P18 data
and non-CMB data, cannot be jointly analyzed in this model, alternatively it
means that the tilted non-flat $\Lambda$CDM Planck $P(q)$ $A_{L}=1$ model is
inconsistent with these data and ruled out at approximately 3$\sigma$, by
them. We note that the level of tensions seen in the P18 vs. BAO and P18 vs.
non-CMB comparisons are less severe in the context of the new $P(q)$ model,
which does not strongly rule out the joint analyses of P18 data and BAO data,
as well as P18 data and non-CMB data, in the tilted non-flat $\Lambda$CDM new
$P(q)$ $A_{L}=1$ model. What is more, none of the other combinations studied,
namely, P18 data vs. lensing data, P18 data vs. BAO′ data, and P18+lensing
data vs. non-CMB data, are strongly mutually inconsistent in the tilted non-
flat $\Lambda$CDM new $P(q)$ model, even when the $A_{L}$ parameter is not
allowed to vary.

We now turn to a comparison between some of our results in Table 21 and
results presented in Refs. Handley (2021) and Di Valentino et al. (2019). We
emphasize that these are only semi-quantitative comparisons, since the data
sets used are not identical and the priors used also differ.

Reference Handley (2021) compares P18 data and lensing data, as well P18 data
and BAO data (note that while we refer to both data sets as BAO there are some
significant differences between the BAO data points used in Ref. Handley
(2021) and the updated BAO data we use here), in the tilted flat $\Lambda$CDM
model and in the tilted non-flat $\Lambda$CDM Planck $P(q)$ model. As
described in Sec. III we use the same ($p,\sigma$) statistical estimator as
Ref. Handley (2021) does and so these are the results we compare. For the
tilted flat $\Lambda$CDM model, from the P18 vs. lensing analysis, Ref.
Handley (2021) Fig. 2 reports $\sigma\simeq 0.19$ and $p\simeq 85\%$, while we
get $\sigma$=0.72 and $p=47\%$ (for Our priors) and $\sigma=0.39$ and $p=70\%$
(for Handley priors). Some differences are expected due to the different set
of data and priors used and this is reflected in these results. Reassuringly,
when we employ the same priors for the lensing data (but not for P18 data) as
used in Ref. Handley (2021) the results get closer. From the P18 vs. BAO
analysis in the tilted flat $\Lambda$CDM model Ref. Handley (2021) finds
$\sigma\simeq 0.95$ and $p\simeq 65\%$ while we get $\sigma=1.5$ and $p=13\%$,
consequently the qualitative conclusions are the same, indicating that no
tension is found. As for the tilted non-flat $\Lambda$CDM Planck $P(q)$ model,
from the P18 vs. lensing analysis, Ref. Handley (2021) reports $\sigma\simeq
2.5$ and $p\simeq 1.2\%$ and we find $\sigma=2.5$ and $p=1.3\%$ (for Our
priors) and $\sigma=2.4$ and $p=1.6\%$ (for Handley priors) so there is very
good agreement between the results. Finally, in the tilted non-flat
$\Lambda$CDM Planck $P(q)$ model, from a comparison of P18 data and BAO data,
Ref. Handley (2021) finds $\sigma\simeq 3.0$ and $p\simeq 0.3\%$ whereas we
get $\sigma=3.0$ and $p=0.3\%$. Considering all results, and the fact that
somewhat different BAO data and priors are used in the two analyses, there is
good agreement between the results and conclusions of Ref. Handley (2021) and
our results and conclusions.

Reference Di Valentino et al. (2019) uses $\textrm{log}_{10}\mathcal{I}$ to
quantify tensions so here we compare our and their results for this
statistical estimator. Reference Di Valentino et al. (2019) compares P18 data
and lensing data, as well as P18 and BAO′ data (note that while we refer to
both data sets as BAO′ there are significant differences between the BAO′ data
used in Ref. Di Valentino et al. (2019) and the updated BAO′ data we use
here), in the tilted flat $\Lambda$CDM model and in the tilted non-flat
$\Lambda$CDM Planck $P(q)$ model. For the tilted flat $\Lambda$CDM model and
the P18 data vs. lensing data analysis, Ref. Di Valentino et al. (2019) find
$\textrm{log}_{10}\mathcal{I}=0.6$ (substantial concordance) while we get
$\textrm{log}_{10}\mathcal{I}=1.24$ (strong concordance). For the P18 data vs.
BAO′ data analysis in the tilted flat $\Lambda$CDM model, Ref. Di Valentino et
al. (2019) report $\textrm{log}_{10}\mathcal{I}=0.2$ (neither a concordance
nor a discordance) and we find $\textrm{log}_{10}\mathcal{I}=0.7$ (substantial
concordance). On the other hand, in the tilted non-flat $\Lambda$CDM Planck
$P(q)$ model, for the P18 vs. lensing data analysis, Ref. Di Valentino et al.
(2019) provide $\textrm{log}_{10}\mathcal{I}=-0.84$ (substantial discordance)
while we obtain $\textrm{log}_{10}\mathcal{I}=-0.49$ which is on the verge of
also indicating a substantial discordance between the two data sets. Finally,
from the P18 vs. BAO′ analysis in the tilted non-flat $\Lambda$CDM Planck
$P(q)$ model, Ref. Di Valentino et al. (2019) report
$\textrm{log}_{10}\mathcal{I}=-1.8$ (strong discordance) whereas we get
$\textrm{log}_{10}\mathcal{I}=-0.89$ (substantial discordance).

As can be appreciated from the preceding discussion, the agreement between our
results and the results presented in Ref. Di Valentino et al. (2019) is not as
good as the one obtained from a comparison of our results and those of Ref.
Handley (2021). It is important to note that the ($p,\sigma$) statistical
estimator of eqs. (35) and (37) is not as dependent on the priors as is the
$\textrm{log}_{10}\mathcal{I}$ statistical estimator of eq. (26). This may
explain the differences found in the comparisons of our results to those of
Refs. Handley (2021) and Di Valentino et al. (2019). All in all, we consider
that there is reasonable, and so reassuring, agreement between our results and
results available in the literature.

##  V Discussion

We have used P18 data, (P18) lensing data, BAO′ data, BAO data, and non-CMB
data to constrain cosmological parameters in eight cosmological models, the
tilted flat $\Lambda$CDM (+$A_{L}$) model, the untilted non-flat $\Lambda$CDM
(+$A_{L}$) model, the tilted non-flat $\Lambda$CDM (+$A_{L}$) Planck $P(q)$
model, and the tilted non-flat $\Lambda$CDM (+$A_{L}$) new $P(q)$ model, and
to determine the goodness-of-fit of these models to the data sets. We have
also used the models to examine whether or not pairs of data sets are mutually
consistent, studying five cases: P18 data vs. lensing data, P18 data vs.
BAO′/BAO data, P18 data vs. non-CMB data, and P18+lensing data vs. non-CMB
data.

Assuming these data are correct and that there are no unaccounted systematic
errors, three of the eight models we consider may be rejected because they are
incompatible with some of these data at levels of significance discussed in
Sec. IV and summarized next. These rejected models are the two untilted non-
flat $\Lambda$CDM (+$A_{L}$) models and the tilted non-flat $\Lambda$CDM
Planck $P(q)$ model.

When P18 data are included in the analyses the untilted non-flat $\Lambda$CDM
(+$A_{L}$) models are, according to the DIC, very strongly disfavoured when
compared with the tilted models. This is because the untilted models lack the
degree of freedom encapsulated in the power spectrum tilt ($n_{s}$) parameter
that is strongly favored by P18 data and so the untilted models are
incompatible with P18 data.

When we use the tilted non-flat $\Lambda$CDM Planck $P(q)$ model to compare
cosmological parameter values from P18 data and BAO′/BAO data, as well as from
P18 data and non-CMB data, we find disagreements in the one-dimensional values
of the $H_{0}$ and $\Omega_{m}$ derived parameters of 2.3$\sigma$ and
2.7$\sigma$ (BAO′), 2.3$\sigma$ and 2.7$\sigma$ (BAO), and 2.9$\sigma$ and
2.9$\sigma$ (non-CMB). In Figs. 20 and 28, in those panels containing $H_{0}$
and $\Omega_{m}$, the two-dimensional contours do not overlap even at more
than 2$\sigma$ significance. Additionally, in the P18 data vs. BAO data case
we find $\textrm{log}_{10}\mathcal{I}=-1.236$ (meaning a strong disagreement
between the two data sets) and $\sigma$ = 3.000 ($p=0.27\%$), while in the P18
data vs. non-CMB data analysis we get $\textrm{log}_{10}\mathcal{I}=-1.263$
(again a strong disagreement between the two data sets) and $\sigma$ = 3.005
($p=0.265\%$). At their levels of significance, these results mean that the
tilted non-flat $\Lambda$CDM Planck $P(q)$ model is unable to simultaneously
accommodate P18 data and non-CMB data and so is ruled out at 3$\sigma$. Note
that non-CMB data include BAO′/BAO data and Refs. (Handley, 2021; Di Valentino
et al., 2019) have previously noted the incompatibility of P18 data and older
BAO′/BAO data in the tilted non-flat $\Lambda$CDM Planck $P(q)$ model. We
return to this point below.

The six-parameter tilted flat $\Lambda$CDM model is the simplest, (largely,