# On the consistency of with CMB measurements in light of the latest Planck,
ACT and SPT data

Rodrigo Calderón*  |,11footnotetext: Corresponding author. Arman Shafieloo*  | Dhiraj Kumar Hazra*  | and Wuhyun Sohn*  |

###### Abstract

Using Gaussian Processes we perform a thorough, non-parametric consistency
test of the model when confronted with state-of-the-art TT, TE, and EE
measurements of the anisotropies in the Cosmic Microwave Background by the ,
ACT, and SPT collaborations. Using ’s best-fit predictions to the TTTEEE data
from , we find no statistically significant deviations when looking for
signatures in the residuals across the different datasets. The results of SPT
are in good agreement with the best-fit predictions to the data, while the
results of ACT are only marginally consistent. However, when using the best-
fit predictions to CamSpec—a recent reanalysis of the data—as the mean
function, we find larger discrepancies between the datasets. Our analysis also
reveals an interesting feature in the polarisation (EE) measurements from the
CamSpec analysis, which could be explained by a slight underestimation of the
covariance matrix. Interestingly, the disagreement between CamSpec and /ACT is
mainly visible in the residuals of the TT spectrum, the latter favoring a
scale-invariant tilt $n_{s}\simeq 1$, which is consistent with previous
findings from parametric analyses. We also report some features in the EE
measurements captured both by ACT and SPT which are independent of the chosen
mean function and could be hinting towards a common physical origin. For
completeness, we repeat our analysis using the best-fit spectra to ACT+WMAP as
the mean function. Finally, we test the internal consistency of the data alone
by studying the high and low-$\ell$ ranges separately, finding no discrepancy
between small and large angular scales.

##  1 Introduction

Observations stemming from the anisotropies in the Cosmic Microwave Background
(CMB) have played a major role in establishing the standard model of Cosmology
— the paradigm with power law primordial spectrum. Despite suffering from
certain theoretical issues, this paradigm has been extremely successful in
accounting for a wide variety of observations across many scales and epochs in
the cosmic history. The satellite provided the most precise estimation of its
6 main cosmological parameters to date [1, 2]. In addition, other ground-based
CMB experiments from the Atacama Cosmology Telescope (ACT) [3, 4] and South
Pole Telescope (SPT) [5, 6] collaborations have recently provided
complementary measurements of temperature and polarisation of the CMB
anisotropies. These focus on smaller, sub-degree angular scales (larger
multipoles $\ell\gtrsim 650$) and offer a new way of testing the robustness of
with higher-resolution CMB maps and independently of .

Despite the success of the standard model, increasingly precise (low-redshift)
measurements have reported a few statistically significant discrepancies [7,
8, 9]. The most notable example is the $\gtrsim 5\sigma$ discrepancy in the
value of the Hubble constant $H_{0}$, as measured by low-$z$ probes using the
distance ladder [10] and high-$z$ estimations, assuming . A milder ($\sim
2\sigma$) but longstanding discrepancy has also been reported between high and
low-redshift estimations of the amplitude of matter fluctuations—characterized
by $S_{8}\equiv\sigma_{8,0}\sqrt{\Omega_{\text{m},0}/0.3}$—the latter
preferring lower values compared to the early universe predictions [11, 12,
13, 14, 15, 16, 17]. The model is also facing other (less-relevant)
observational challenges, see _e.g._ [18, 19, 20, 21].

Moreover, CMB measurements are known to have mild inconsistencies between the
different angular scales (high vs low multipoles [22, 23]), and as measured by
the different collaborations [24, 25]. Indeed, even within data alone, the TT
spectrum seems to favor a lensing amplitude $A_{\rm L}>1$ [2, 26] and provides
“evidence” for a non-vanishing (positive) spatial curvature [27, 2, 28],
although it has been argued that these are purely stemming from statistical
fluctuations [29]; see also [30, 31, 32, 33] and references therein for
further discussions on this. Furthermore, the results from the ACT
collaboration seem to prefer a scale-invariant spectrum of primordial
fluctuations, with $n_{s}\simeq 1$, while data excludes such a value at more
than $3\sigma$ [34, 35, 36, 24, 37, 38]; see e.g. [39, 40, 41] for other CMB
anomalies. If these inconsistencies are not coming from systematics, they may
hint towards new physics beyond the standard model.

In recent years, a lot of effort has gone into investigating extensions of to
try and provide physical explanations for some of the aforementioned
discrepancies; see _e.g._ [42, 43, 44, 45] for a review. These often change
the Universe’s growth and/or expansion history at late times [46, 47, 48], or
introduce new physics at early times such that (i) the physical size of the
sound horizon $r_{d}\equiv r_{s}(z_{d})$ decreases with respect to [49, 50,
51, 52, 53], (ii) the redshift of recombination is shifted [54, 55, 56, 57] or
(iii) new features are introduced in the primordial spectrum of fluctuations
[58, 59]. While appealing from the theoretical standpoint, very few of the
proposed solutions are actually able to simultaneously address these tensions.
For example, it has been argued that no late-time modification to $H(z)$ is
able to raise the value of $H_{0}$ [60, 61], while modifications to the early
universe might create or exacerbate the tensions with low-$z$ observations;
see _e.g._ [62, 63, 64, 65, 66, 67, 68, 69] for discussions on this topic.

Given the fundamental role of the CMB in cosmological analyses, it is crucial
to understand whether the differences between the latest observations are
coming from either statistical fluctuations, unaccounted systematics, or new
physics beyond . In this work, we test their statistical consistency using
Gaussian Processes (GP) – a non-parametric method that can effectively
represent smooth deformations away from the model under consideration. If the
differences between the datasets are entirely consistent with random
fluctuations, then the GP regression should yield confidence levels consistent
with zero. If not, then the GP can provide insights into what shape of
deformation, either from systematics or theoretical inconsistency, is
preferred by the data.

The structure of this paper is as follows. In Section 2, we describe in detail
the method and the data used in the analysis. We then proceed to perform the
consistency tests using the best fit predictions, by looking for structures in
the residuals with respect to the mean function. We start by confronting to
space-based observations by the Planck satellite. We use both the Plik and
CamSpec likelihoods, which correspond to the ’s official likelihood and the
most recent reanalysis of the data with the highest sky fraction,
respectively, in Section 3.1. We then repeat the analysis using ground-based
measurements by the Atacama Cosmology Telescope (ACT) and the South Pole
Telescope (SPT) to look for discrepancies between the experiments in Section
3.2. Finally, we test the robustness of our conclusions by using a different
mean function in the analysis222Namely, we use ’s best fit to the Planck 2018
official plik data release and ACT+WMAP data as alternative mean functions.
and investigate the consistency of the data alone by studying the low-$\ell$
($\ell<650$) and high-$\ell$ ranges ($\ell>650$) separately in Appendix A.1,
A.2 and B.1. In Appendix B.2, we assess whether an absolute scaling of the
spectra (difference in calibration) can account for the mild inconsistencies
between the different collaborations. The main findings of this analysis are
summarized in 3.3. We then conclude and discuss future prospects in Section 4.

##  2 Method and Data

###  2.1 Data

Figure 1: TT, TE and EE residuals with respect to best fit spectra to CamSpec
data. Solid red and green lines correspond to the residuals in the best fit
predictions from ACT+WMAP and Plik with respect to the best fit to CamSpec
data, respectively.

In this work, we confront the predictions of with different space and ground-
based CMB experiments as a way of testing the consistency of the model and the
robustness of the measurements. Namely, we consider the following data

  * •

2018 \- Temperature (TT), Polarisation (EE) and their cross-correlation (TE)
from the final 2018 data release [1, 2]. More specifically, we use the data
from the base-plikHM-TTTEEE-lowl-lowE-lensing likelihood, which is publicly
available at the Planck legacy archive333https://pla.esac.esa.int/#cosmology.
These cover the range444We omit the $\ell<30$ measurements due to their non-
Gaussian nature. $\ell\in[30,2500]$ in TT and $\ell\in[30,2000]$ in TE and EE.
We refer to this data simply as Planck 2018 (or plik).

  * •

CamSpec PR4 \- We also consider TT, TE and EE measurements from the latest
CamSpec NPIPE PR4_v12.6 likelihood [70]; see [29] for more details on CamSpec.
These cover the range $\ell\in[30,2500]$ in TT and $\ell\in[30,2000]$ in TE
and EE. We refer to this data simply as CamSpec.

  * •

ACT DR4 \- We use the Atacama Cosmology Telescope (ACT) Temperature (TT),
$E$-mode Polarisation (EE) and their cross-correlation (TE) from the
ACTPolliteDR4 likelihood in the latest ACT data release [3, 4]. We refer to
these measurements simply as ACT.

  * •

SPT 3G \- Finally, we also include the latest results from the South Pole
Telescope Collaboration (SPT-3G) [6]. These are updated measurements of both
E-mode Polarisation (EE) and Temperature-Polarisation cross-correlation (TE)
from [5] but with the inclusion of TT measurements. These cover angular scales
$\ell\in[750,3000]$ for TT and $\ell\in[350,3000]$ in TE and EE.

We should note that we use the minimum-variance-combined band powers for ACT
and SPT data, which might not accurately reflect the full information
contained in these datasets. We believe however that this can be seen as a
zeroth-order approximation. A more rigorous analysis using the full (multi-
frequency) likelihood might be needed for a more robust interpretation of the
results.

Table 1: Differences in the best-fit parameters for the different datasets. Best-fit | $H_{0}$ | $100\,\omega_{b}$ | $100\,\omega_{c}$ | $n_{s}$ | $\ln{(10^{10}A_{s})}$ | $\tau$
---|---|---|---|---|---|---
Planck 2018 | $67.32$ | $2.238$ | $12.01$ | $0.9660$ | $3.044$ | $0.054$
CamSpec | $67.18$ | $2.218$ | $11.96$ | $0.9624$ | $3.035$ | $0.052$
ACT+WMAP | $67.10$ | $2.223$ | $12.12$ | $0.9714$ | $3.068$ | $0.061$

###  2.2 Gaussian Process Regression

Gaussian Processes (GP) [71] have been extensively used in the literature to
fit a smooth curve from noisy and/or sparse data without the need to write
down an explicit parametric model [see e.g. 72, 73, 74, 75, 76, 77]. GP excels
when the noise in the data is well approximated by a (multivariate) Gaussian
distribution. It provides a posterior distribution of smooth functions given
the data based on two assumptions on the functional form: the mean function
($\mu(x)$) and the kernel ($k(x,x^{\prime})$). In-depth analyses of GP’s
dependence on these assumptions are given in [e.g. 72, 73, 78]. The mean of
the GP posterior distribution evaluated at a set of ‘test’ points
$\boldsymbol{x}_{\star}$ can be easily calculated through

$\boldsymbol{\mu}=\boldsymbol{m}_{\star}+\mathbf{K_{\star}}\mathbf{K}^{-1}(\boldsymbol{y}-\boldsymbol{m})$
(2.1)

where $\boldsymbol{m}_{\star}\equiv\mu(\boldsymbol{x}_{\star})$,
$\boldsymbol{m}\equiv\mu(\boldsymbol{x})$, $\mathbf{K}_{\star}\equiv
k(\boldsymbol{x}_{\star},\boldsymbol{x})$, $\mathbf{K}\equiv
k(\boldsymbol{x},\boldsymbol{x})+\Sigma$, where the observations
$\boldsymbol{y}$ are made at data points $\boldsymbol{x}$ with the data
covariance matrix $\Sigma$. Similarly, the posterior of the covariance is
obtained using

$\boldsymbol{C}=\mathbf{K}_{\star\star}-\mathbf{K}_{\star}\mathbf{K}^{-1}\mathbf{K}^{T}_{\star}$
(2.2)

In practice, the calculation of such quantities amounts to a matrix inversion
of $\mathbf{K}$. Computationally, a Cholesky decomposition is often preferred
as it is a faster and numerically more stable procedure. Finally, the log-
marginal likelihood (LML) under a GP is given by

$\ln\mathcal{L}=-\frac{1}{2}\big{[}\boldsymbol{r}^{T}\,\mathbf{K}^{-1}\,\boldsymbol{r}+\ln{|\mathbf{K}|}+N\ln{(2\pi)}\big{]}$
(2.3)

where $\boldsymbol{r}=\boldsymbol{y-m}$ is the residual vector, $N$ is the
number of (observed) datapoints and $|\mathbf{K}|$ denotes the determinant of
the full covariance matrix. The GP predictions depend on the choice of kernel
describing the correlations between the data points. In this work, we use a
_squared exponential_ (SE) kernel given by

$k(x,x^{\prime};\sigma_{f},\ell_{f})=\sigma_{f}^{2}\,e^{-(x-x^{\prime})^{2}/2\ell_{f}^{2}},$
(2.4)

where $\sigma_{f}$ and $\ell_{f}$ determine the amplitude and typical length-
scale of the correlations, respectively. These hyperparameters are optimized
by maximizing the log-marginal likelihood in (2.3); we refer to [71] for a
more detailed discussion.

In this work, we focus on testing the consistency of the model, and thus
decide to work in residual space where the best-fit () predictions have been
subtracted from the data—effectively choosing as a GP mean function. More
specifically, we decide to work in the space of
$\mathcal{D}_{\ell}=\ell(\ell+1)\mathcal{C}_{\ell}/2\pi$, where the physical
(oscillatory) features would be more pronounced. Having a closer look at Eq.
(2.3), it is seen that if the mean function is a good (enough) fit to the
data, the first and second (penalty) terms in (2.3) will tend to prefer _no
extra-correlations_ (_i.e._ $\sigma_{f}\simeq 0$) or _diverging correlation-
lengths_ ($\ell_{f}\to\infty$), as encoded in the GP kernel (2.4). In the
presence of hidden systematics or in the need for a modification of the mean
function, however, a finite value for $(\sigma_{f},\ell_{f})$ might be
statistically preferred. Therefore, inspecting the two-dimensional likelihood
profile $\mathcal{L}(\sigma_{f},\ell_{f})$ can yield valuable information on
the model and the dataset under consideration [73, 79, 80, 75, 77]. Thus, if
the likelihood is maximized for $\sigma_{f}\to 0$ (or $\ell_{f}\to\infty$),
the mean function is consistent with the data. On the other hand, any
significant detection of $\sigma_{f}\neq 0$ can be interpreted as hints of
underlying structures or systematics in the data that cannot be properly
accounted for by the model, given by a smooth deformation with a typical
amplitude and correlation length given by the preferred values of
$(\sigma_{f},\ell_{f})$.

Table 2: Uniform prior ranges in the hyperparameters for TT,TE and EE. Parameter | $\log_{10}\sigma_{f}$ | $\log_{10}\ell_{f}$
---|---|---
TT | $[-3,2]$ | $[0,4]$
TE | $[-3,0.5]$ | $[0,4]$
EE | $[-3,0.5]$ | $[0,4]$

##  3 Results and Discussions

In this section, we confront the best-fit predictions with the various CMB
observations. Extending upon [79], our goal is to test the consistency of the
model, update the analysis to include the most recent CMB measurements
described before, namely the 2018 from Plik, CamSpec (NPIPE) PR4, ACT DR4 and
SPT-3G data releases, and test the consistency between these datasets.

Before we delve into the details of the analysis, we start by giving a brief
overview of the results. Fig. 2 nicely captures the main takeaways of our
analysis. The radial distance to the origin translates the level of
“tension”555The exact metric for quantifying the level of (dis)agreement is
not relevant here, but it is related to the number quoted in Fig. 11; see
Section 3.3 for a more detailed discussion. between a given set of
measurements and the ’s best-fit predictions. It is clear that the CamSpec and
ACT directions shows overall the largest discrepancies, but it is also seen
that this is statement is somewhat dependent on the choice of mean function.

Figure 2: Visual overview of the results of this analysis. The labels in the
angular direction correspond to the different measurements used in the GP
analysis. The radial direction indicates the level of “disagreement” between a
given set of measurements and the best-fit predictions that are used as mean
functions, shown as green, orange and blue lines. The closer a point lies to
the origin, the more in agreement the measurements are with the set of
$\mathcal{D}_{\ell}$’s from the mean function. Conversely, points far away
from the origin admit large improvements in fit, given by smooth deformations
of the corresponding () mean function.

To further investigate this, we will thoroughly discuss the results using a
single mean function in the main body of the paper. More specifically, we
choose the best-fit to CamSpec PR4 data as a mean function in our Gaussian
Process (i.e. in Eq. (2.1)) since it is obtained from the latest (re)analysis
of the Planck data [29, 70]. The CamSpec (NPIPE) PR4 analysis includes
differences in the treatment of polarisation, calibration, and systematic
corrections. It also contains significantly more sky fraction (more data than
the official Planck data release) and yields the tightest constraints in terms
of cosmological parameters. In Section 3.1, we will extensively discuss the
results using measurements, both from the CamSpec and Plik analyses. The
results of our analysis using ground-based observations, namely ACT and SPT,
will be discussed in Section 3.2. Finally, we comment on the role of the mean
function and summarize our results in Section 3.3. Further details are also
given in the Appendix.

###  3.1 Consistency of with

####  3.1.1 CamSpec PR4

Figure 3: 2D posterior distributions as a function of $(\sigma_{f},\ell_{f})$
for the _CamSpec PR4_ data, and best-fit to the same data as mean function.
The color bar shows the improvement in fit, where
$\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$
and $\ln\mathcal{L}^{\rm GP}(\sigma_{f},\ell_{f})$ is the log marginal
likelihood in Eq. (2.3).

We start by considering the ’s best fit to the latest CamSpec data [70] as the
mean function in our analysis. In Fig. 3, we show the two-dimensional
likelihood profiles for the CamSpec residuals with respect to ’s best-fit
$\mathcal{D}_{\ell}$’s, as a function of $(\sigma_{f},\ell_{f})$. The color
bar shows the goodness of fit, where $\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm
GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$ and $\ln{\mathcal{L}^{\rm GP}}$ is the
log-marginal likelihood (LML), defined in Eq. (2.3). Negative values of
$\Delta\chi^{2}$ (in blue) reflect regions in parameter space yielding an
improvement in the fit with respect to . Conversely, red-colored regions
correspond to deviations from the mean function leading to a degraded fit to
the data ($\Delta\chi^{2}>0$), whereas white-shaded regions represent no
improvement at all. The black dot represents the set of hyperparameters
$(\sigma_{f},\ell_{f})$ yielding the highest likelihood and we report the
corresponding improvement in fit666Strictly speaking, this quantity is
analogous to a $\chi^{2}$ difference, but the exact $\Delta\chi^{2}$ values
should not be interpreted in the usual sense, as the LML receives additional
(possibly large) contributions from the second and third terms in Eq.(2.3)
which depend on the values of the kernel hyperparameters
$(\sigma_{f},\ell_{f})$. ($\Delta\chi^{2}$) in Table 3. As mentioned before,
if the mean function is a good description of the data, the LML in (2.3)
should peak at $\sigma_{f}\to 0$ and/or $\ell_{f}\to\infty$. In other words,
no smooth deviations away from the best-fit are needed to explain the data.
However, if the LML peaks at finite (possibly large) values of
$(\sigma_{f},\ell_{f})$, it might point towards the need for a different mean
function or indicate the existence of hidden structures/systematics in the
data.

In this case, as can be seen from Fig. 3, the model provides a relatively good
fit to TT, TE, and EE data. The LML seems to prefer small deviations from the
best fit spectra: $\sigma_{f}\lesssim 1$ for TT and TE and $\sigma_{f}\lesssim
0.1$ for the case of EE. Any larger deviation from the mean function is highly
penalized by the data, as can be seen by the color bar on the right
($\Delta\chi^{2}>0$ meaning a degraded fit), and the improvement in fit by the
GP is negligible in both TT and TE; see Table 3. Meanwhile, there is a
noticeable improvement in fit in EE for $\ell_{f}\lesssim 1$. Such a GP
realisation is essentially a white noise where the values at each $\ell$ are
uncorrelated with each other. Preferences toward the inclusion of white noise
indicate that the fluctuations in the data around the mean are more than what
is expected from the data covariance matrix. In other words, the covariance
matrix may have been slightly underestimated for this EE data. This may alter
the weights and hence the optimality of the cosmological parameter estimation,
but is less likely to have a significant effect on the estimated parameter
values.

In fact, the LML is expected to have a local minimum at some $\sigma_{f}$ with
$\ell_{f}\ll 1$ about half of the time. Taking the limit $\ell_{f}\rightarrow
0$ in (2.3), we found that the necessary and sufficient condition for such a
minimum is given by
${\|\Sigma^{-1}\bf{r}\|^{2}>\rm{tr}\left(\Sigma^{-1}\right)}$, where $\bf{r}$
is the residual vector. Assuming that the mean and the covariance matrix are
exact, the expected value of the left-hand side is equal to the right-hand
side. When the residual vector is large enough either by statistical
fluctuations and/or underestimation of the errors, then we expect to see a
local minimum at some $\sigma_{f}$ satisfying
${\|(\sigma_{f}^{2}I+\Sigma)^{-1}{\bf r}\|^{2}={\rm
tr}\left[(\sigma_{f}^{2}I+\Sigma)^{-1}\right]}.$

Figure 4: GP reconstructions of
$\Delta\mathcal{D}_{\ell}\equiv\mathcal{D}_{\ell}^{\rm
GP}-\mathcal{D}_{\ell}^{\Lambda\rm CDM}$ for the set of hyperparameters
maximizing the log-marginal likelihood in Eq. (2.3) when using CamSpec PR4
data and the corresponding best-fit spectra as mean functions. The solid line
and shaded regions correspond to the mean and $2\sigma$ confidence intervals
around it, respectively.

Despite this fact, it is still intriguing that we find $\Delta\chi^{2}$ as low
as $-10.21$ in EE for CamSpec PR4. As we discuss in Appendix B.1, most of
these improvements in fit come from low-$\ell$ data, and notably is not
present in the Official Release (PR3). Our non-parametric approach using GP
indicates that the updated analysis pipeline of CamSpec PR4 has caused the
residuals in the EE coadded spectrum to vary more than the expected amount
given by the covariance matrix; the errors seem to have been slightly
underestimated. Indeed, we confirm that the usual chi-squared statistic for
the EE data in the range $30\leq\ell\leq 650$ is $\chi^{2}=709$, larger than
the expected value of $620$ by $2.51\sigma$.777For the full data range of
$30\leq\ell\leq 2000$, $\chi^{2}=2023$, which is $0.83\sigma$ above the
expected value of 1971. Note that here we look at the TT+TE+EE best-fit
values. This excess in residual error has been investigated in [70, see e.g.
Table 1] and is reported to be statistically consistent with random
fluctuations. Moving one step further than these simple chi-squared checks, we
investigate the scales where these excess residuals are correlated through GP
analyses. The fact that the marginal likelihood is maximised at small
$\ell_{f}$ implies that these excess variances appear to be uncorrelated
random fluctuations rather than smooth deformations in the mean, unlike the
cases of TT and TE. Nonetheless, the deviations from zero are
$\mathcal{O}(10^{-2})$, with relatively minor improvements in fit, suggesting
is a good description of the data.

We then use the set of $(\sigma_{f},\ell_{f})$ maximizing the likelihood in
Eq. (2.3) (shown as a black dot in Fig. 3) to obtain the mean (in orange),
$68\%$ and $95\%$ C.L. (gray-shaded bands) shown in Fig. 4, using Eqs. (2.1)
and (2.2), respectively. Again, we see that the reconstructions are perfectly
consistent with zero across the entire multipole range covered by the data.

These results suggest that despite the aforementioned issues in EE, the best-
fit model is overall consistent with the CamSpec data. This should not come as
a surprise, since we have chosen the best-fit predictions to CamSpec data as
the mean function in our analysis. However, as explained before, this serves
as a consistency test for the different CMB measurements. In the presence of
systematics, or physics beyond , some inconsistencies might appear when using
a (possibly incorrect) mean function.

Table 3: $\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$ improvements in fit for the ground and space-based experiments obtained with the GP using ’s best-fit to CamSpec PR4 as mean function. $\Delta\chi^{2}$ |  2018 | CamSpec PR4 | ACT DR4 | SPT-3G
---|---|---|---|---
TT | $-19.94$ | $-0.35$ | $-15.11$ | $-0.124$
TE | $-0.001$ | $-0.75$ | $-2.62$ | $-0.002$
EE | $-0.446$ | $-10.21$ | $-7.89$ | $-8.916$

####  3.1.2 Planck 2018 - Baseline/Official Release

Figure 5: 2D posterior distributions as a function of $(\sigma_{f},\ell_{f})$
for the _Planck 2018_ data, and best-fit to the _CamSpec PR4_ data as mean
function. The color bar shows the improvement in fit, where
$\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$
and $\ln\mathcal{L}^{\rm GP}(\sigma_{f},\ell_{f})$ is the log marginal
likelihood in Eq. (2.3).

Similarly, we take the ’s best-fit predictions to CamSpec data and confront it
to the official likelihood (Plik) as a way of testing the internal consistency
of the measurements and the differences between the CamSpec and Plik
likelihoods. In Fig. 5, we show the 2D LML profiles for TT, TE, and EE
measurements from Plik (which we simply refer to as Planck). As discussed
before, finding a significantly low $\Delta\chi^{2}$ by our GP means that a
smooth deformation away from the mean function is preferred, which would hint
towards inconsistencies between CamSpec and Plik in this case. While we find
TE and EE to be consistent, we find notable inconsistencies in TT for the two
likelihoods. There is a large improvement in fit from GP for $\sigma_{f}\simeq
5$ and $\ell_{f}\simeq 10^{3}$; corresponding to the blue regions in Fig. 5.

Figure 6: GP reconstructions of
$\Delta\mathcal{D}_{\ell}\equiv\mathcal{D}_{\ell}^{\rm
GP}-\mathcal{D}_{\ell}^{\Lambda\rm CDM}$ for the set of hyperparameters
maximizing the log-marginal likelihood in Eq. (2.3) when using data and the
best-fit spectra to CamSpec as mean functions. The solid line and shaded
regions correspond to the mean and $2\sigma$ confidence intervals around it,
respectively.

As seen in the upper panel of Fig. 6, the corresponding GP reconstruction also
shows large deviations ($>3\sigma$) from zero. The corresponding improvements
in fit, $\Delta\chi^{2}$, are reported in the first column of Table 3. See
also Table 3 in [70] and discussions therein for a more detailed comparison
between the official Planck release (PR3) and CamSpec (PR4) within the model.

Interestingly, our GP analysis shows that these smooth deformations away from
the CamSpec mean function not only fit the Plik likelihood significantly
better but also have a similar correlation length $\ell_{f}$ to those that
improve the fit to the ACT data (Figs. 7 and 8), albeit with a smaller
amplitude ($\sigma_{f}$); see Section 3.2.1 below. These results imply that
the best-fit predictions from CamSpec and Plik differ mainly in the TT power
spectrum.

The fact that the GP captures differences in temperature between CamSpec and
Plik is not so surprising. The temperature measurements from the official data
release are known to prefer deviations away from the standard model in some
simple (one-parameter) extensions to . Two well-known examples are the
tendencies of the TT spectra to favour an excess of lensing amplitude
$A_{L}>1$ – which should be equal to unity in – and the preference for a
closed Universe, with positive spatial curvature $\Omega_{k,0}<0$. As already
noted in [27], these tendencies in TT to favour higher lensing amplitudes have
similar effects in the $\mathcal{C}_{\ell}$’s as other parameters, such as
curvature or the dark energy equation of state in extended models.
Interestingly, these “anomalies” disappear when going from Plik to CamSpec
(i.e. $A_{L}\to 1$ and $\Omega_{k,0}\to 0$) and are attributed to statistical
fluctuations in the multipole range $800<\ell<1600$ [29]. Our GP
reconstruction for the TT power spectrum also shows significant deviations
from zero in the same multipole range; see the upper panel of Fig. 6. However,
we note that our GP analysis prefers larger correlation scales
($\ell_{f}\simeq 10^{3}$) than what one would expect from random fluctuations,
even considering the off-diagonal correlations from lensing.

Finally, we should stress that in this comparison, we use the best-fit power
spectra to the CamSpec likelihood. An alternative analysis is to study
deviations away from the TTTEEE best-fit to the official data release, shown
in green in Fig. 1. We refer the reader to Appendix A.1, where we use the
best-fit $\mathcal{D}_{\ell}$’s to Plik instead, and test the consistency with
both CamSpec measurements and other ground-based experiments.

###  3.2 Consistency of with ground-based experiments (within )

Next, we use the same mean function as before and look for potential
structures in the residuals of ACT and SPT data. If is the correct model
describing the CMB anisotropies up to $\ell\simeq 4000$, and their parameters
are accurately estimated by the CamSpec analysis, then the two-dimensional
distributions of the hyperparameters $(\sigma_{f},\ell_{f})$ should not prefer
any finite, non-vanishing values. The presence of unaccounted systematics, or
discrepancies between the experiments, would however be reflected in the two-
dimensional likelihood profiles if there are any.

####  3.2.1 ACT DR4

Figure 7: 2D posterior distributions as a function of $(\sigma_{f},\ell_{f})$
for the ACT DR4 data, and best-fit to CamSpecPR4 data as mean function. The
color bar shows the improvement in fit, where
$\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$
and $\ln\mathcal{L}^{\rm GP}(\sigma_{f},\ell_{f})$ is the log marginal
likelihood in Eq. (2.3).

In Fig. 7 we show the posteriors for $(\sigma_{f},\ell_{f})$ when using ACT
DR4 data and ’s best-fit to CamSpec as mean function. In this case, an
interesting feature appears in the TT data. The LML peaks at
$(\sigma_{f},\ell_{f})\simeq(9,2\times 10^{3})$ where the GP finds an
improvement in fit with respect to the mean function (), corresponding to a
$\Delta\chi^{2}=-15.11$; see Table 3. Interestingly, the TE and EE posteriors
show similar (bimodal) distributions, with a preference for non-vanishing
values of $(\sigma_{f},\ell_{f})$, although the statistical significance of
these deviations from is milder than in the TT case. The improvements in fit
are reported in the third column of Table 3.

Figure 8: GP reconstructions of
$\Delta\mathcal{D}_{\ell}\equiv\mathcal{D}_{\ell}^{\rm
GP}-\mathcal{D}_{\ell}^{\Lambda\rm CDM}$ for the set of hyperparameters
maximizing the log-marginal likelihood in Eq. (2.3) when using ACT DR4 data
and best-fit to CamSpec PR4 as mean function. The solid line and shaded
regions correspond to the mean and $2\sigma$ confidence intervals around it,
respectively. Note that the multipole range is extended with respect to Fig.
6.

In Fig. 8, we show the mean and $2\sigma$ reconstructions from the GP when
using the set of hyperparameters maximizing the LML in Eq. (2.3), shown as
black dots in Fig. 7. Note that the ACT reconstructions of the TT spectra seem
to prefer lower amplitudes at $\ell\lesssim 3000$ (at more than $2\sigma$) and
a slightly larger amplitude at $\ell\gtrsim 3000$ with respect to what is
predicted by ’s best-fit to the CamSpec data. This is yet another (non-
parametric) indication that the ACT data seem to favor a scale-invariant,
Harrison-Zel’dovich ($n_{s}\simeq 1$) spectrum of fluctuations [24, 37, 35,
36]. At the same time, a larger value for $n_{s}$ might also imply an
increased value of $H_{0}$888However, we should note that such a shift in the
cosmological parameters, $H_{0}\to 73\;\rm km/s/Mpc$ and $n_{s}\to 1$, would
typically lead to larger values of $S_{8}$, worsening the fit to low-redshift
(weak-lensing) measurements of the clustering amplitude., through a reduction
of the size of the sound horizon [34, 38].

While the LML improvements at $\ell_{f}\sim 2000$ relate to the overall
scaling through $n_{\mathrm{s}}$, the other mode in LML found at $\ell_{f}\sim
90$ in TE and EE spectra may be closely related to the cosmological parameters
affecting the width and height of the acoustic peaks. Roughly speaking, a
realisation of a GP with $\ell_{f}\sim 90$ has a typical full width at half
maximum (FWHM) of $\sim 210$ and is likely to have oscillations that mimic the
acoustic peaks in the CMB ($\Delta\ell\sim 300$). Indeed, the GP
reconstruction of Fig. 8 for EE spectra has oscillation scales similar to
those of the differences in the best-fit predictions from ACT+WMAP and CamSpec
data (red line in Fig. 1). The features present in our non-parametric
reconstructions can therefore be a manifestation of the mild discordance in
the ($\omega_{b},\omega_{c}$)-plane between Planck and ACT (seen for instance
in Fig. 7 in [6]), or vice versa. An interesting feature is also captured in
the EE reconstructions around $\ell\sim 400-700$, with milder oscillations
extending up to $\ell\sim 3000$. These might be a slight hint of new features,
a manifestation of the mild disagreement in the estimated cosmological
parameters, or some unaccounted systematics affecting the low/high-$\ell$ part
of the ACT and/or data. Together with the issue in TT, it might explain why
recent analyses reach slightly different conclusions when considering ACT data
alone, or performing cuts at a given $\ell_{\rm max}$ in the data [e.g. 81,
51, 82]. Our results seem to support previous findings in the context of and
simple extensions [e.g. 24, 55, 35, 83], suggesting that these mild
discrepancies are mainly driven by the ACT data and in particular by the TT
measurements. Whether such discrepancies arise from physical, systematic, or
statistical origin, however, remains to be determined by upcoming (more
precise) CMB observations. In particular, the ACT collaboration is soon
expected to update its results with the ACT DR6 data release.

####  3.2.2 SPT-3G

Figure 9: 2D posterior distributions as a function of $(\sigma_{f},\ell_{f})$
for the SPT-3G data, and best-fit to CamSpec data as mean function. The color
bar shows the improvement in fit, where $\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm
GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$ and $\ln\mathcal{L}^{\rm
GP}(\sigma_{f},\ell_{f})$ is the log marginal likelihood in Eq. (2.3). Figure
10: GP reconstructions of
$\Delta\mathcal{D}_{\ell}\equiv\mathcal{D}_{\ell}^{\rm
GP}-\mathcal{D}_{\ell}^{\Lambda\rm CDM}$ using the set of hyperparameters
maximizing the log-marginal likelihood in Eq. (2.3) when using SPT-3G data and
best-fit to CamSpec PR4 as mean function. Solid line and shaded regions
correspond to the mean and $2\sigma$ confidence intervals around it,
respectively. Note that the multipole range is extended with respect to Fig. 6

Similarly, we inspect for structures in the residuals of SPT-3G data, when
subtracting ’s best fit to CamSpecPR4 data. The results are shown in Fig. 9.
The posteriors of the temperature auto-correlation (TT) and
temperature/polarisation cross-correlation (TE) are in good agreement with the
predictions, and the GP finds negligible improvements with respect to ; see
also Fig. 10. However, this could also be explained by the larger
uncertainties in the temperature measurements with respect to (see Fig. 1).
The situation is slightly different for EE, suggesting again a bimodal
distribution in the $(\sigma_{f},\ell_{f})$-plane, with typical deviations
from a zero mean-function of the order $\sigma_{f}\lesssim 1$ and with a
preferred correlation length of $\ell_{f}\simeq 30$, indicating that the
improvement in fit is likely due to subtle oscillations around the best-fit
predictions. The improvements in fit with respect to are again reported in the
last column of Table 3, with a maximum $\Delta\chi^{2}=-8.196$ for EE, which
is of the same order of magnitude as the improvement in fit as for the case of
ACT data. Curiously, the SPT reconstructions also show a prominent feature at
intermediate scales ($\ell\sim 500-700$). As discussed before, these
oscillations might be linked to the mild differences in the cosmological
parameters (such as $\omega_{b}$ or $\omega_{c}$) affecting the width, height
and position of the acoustic peaks with respect to the ones inferred by . We
should mention that the SPT-3G results are overall consistent with those of at
the parameter level; see Table IV and Fig. 7 in [6]. However, our results
suggest that the very mild differences between the two are mostly driven by
the EE measurements; in agreement with the conclusions from Fig. 3 in [6].

###  3.3 Summary of the results

Figure 11: Summary plot assessing the consistency/discrepancy between the
different choices of mean function and the different datasets. The row
indicates the choice of mean function $\boldsymbol{m}$, and the column
corresponds to the choice of data with serves as “training points”
$\boldsymbol{y}$ for the GP, Eq. (2.3). We remind the reader that the quantity
$\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$
is analogous to a $\chi^{2}$, but the exact values should be interpreted with
caution. It is only meant to represent a statistical preference for a
particular GP-deformation of the considered mean function.

In the previous sections, we tested the consistency of the CMB measurements
assuming ’s best-fit predictions ($\mathcal{D}_{\ell}^{XY}$) to CamSpec data.
However, as understood from Eqs. (2.1) and (2.3), the assumed choice of mean
function $\boldsymbol{m}$ plays a crucial role in our analysis. In the
Appendix, we repeat our analysis using different best-fit () models as mean
functions; see also Table 1 for the differences at the parameter level. In
this section, we aim to summarize some of the key results of our analyses
using different mean functions, which correspond to the different rows in Fig.
11. We refer the reader to the Appendix for more figures and detailed
discussions on these results.

  * •

We find mild inconsistencies in the temperature auto-spectrum between CamSpec
and Plik. As discussed in Section 3.1.2, the temperature spectra from the
official 2018 (PR3) release are known to show preferences for a closed
Universe or an excess of lensing. In the base model, these parameters are
expected to be $\Omega_{k,0}=0$ and $A_{L}=1$, respectively. These issues were
reported and discussed already in the first 2013 results [22]; see also the
follow-up discussions in [27, 2]. However, these tendencies in TT disappear in
the CamSpec (re)analysis, and are attributed to random statistical
fluctuations in the multipole range $800<\ell<1600$ [29]. Our non-parametric
analysis captures those differences between Plik and CamSpec in the same
$\ell$-range (see Fig. 6) and is reflected as improvements in fit
($\Delta\chi^{2}$) by our GP, shown in Fig. 11.

  * •

The $\mathcal{D}_{\ell}^{\rm TT}$ reconstructed using both Planck data from
Plik and ACT data shows noticeable discrepancies (at more than $2\sigma$
level) with ’s best-fit predictions to CamSpec (see Figs. 6 and 8), with
preferences for large correlation lengths and less power at small/intermediate
angular scales with respect to the best-fit predictions, thus mimicking a
change in the spectral index $n_{s}\to 1$. However, the TT data from SPT is
consistent with both the best-fit predictions to CamSpec and Plik; see the
leftmost panel in Fig. 11 (see also Fig. 2).

  * •

The two-dimensional likelihood profiles in EE and posterior distribution of
$\Delta\mathcal{D}_{\ell}^{\rm EE}$ for the CamSpec data seems to be the same,
irrespective of the chosen mean function: the GP always seem to find an
improvement in fit ($\Delta\chi^{2}\sim 10.2$) by adding some uncorrelated
noise ($\ell_{f}\to 1$), which could potentially be reflecting a slight
underestimation of the covariance in EE; see the CamSpec column for EE in Fig.
11.

  * •

The marginal-likelihood profiles $\mathcal{L}(\sigma_{f},\ell_{f})$ for both
ACT and SPT are also stable under changes in the mean function. In particular,
the reconstructed (oscillatory) features in EE are always present (see Figs.
8, 10, 12 and 13), which might be pointing towards a common physical (possibly
primordial) origin, new physics beyond the model, or differences in the
inferred cosmological parameters related the width and height of the acoustic
peaks; see also the discussion in Section 3.2.1.

  * •

The ACT+WMAP best-fit predictions within seem to be in tension ($\gg 2\sigma$)
with both CamSpec and Plik TT measurements; see Fig. 13. A similar behaviour
is observed for SPT-3G TT measurements, although the statistical significance
is milder. We stress that such a test is performed merely for the sake of
completeness, to effectively replace Planck’s large-scale constraints by
WMAP’s. Statistically, it is not correct to compare the best-fit obtained from
ACT+WMAP and compare it to a dataset with much more constraining power, such
as /CamSpec.

From all these considerations, as is nicely illustrated in Fig. 2, we conclude
that the mean function that better describes the different data sets seems to
be the best-fit from the Plik analysis, being perfectly consistent with
itself, and only mildly discrepant with ACT and SPT (with the largest
discrepancies seen in EE and relatively minor differences in TT). However, as
mentioned before, the CamSpec TT measurements seem to be in tension with ’s
best-fit to Plik, which might be non-parametric manifestations of the well-
known $A_{L}$ and $\Omega_{k,0}$ anomalies present in the firsts (official)
data releases [22, 27, 2], but which are no longer present in the CamSpec re-
analysis [29, 70]. We should also note that the CamSpec EE data seem to
require the addition of uncorrelated noise, irrespective of the chosen mean
function.

At this stage, the mild disagreement between the experiments is not
statistically significant ($\lesssim 2\sigma$) to confidently claim a
discrepancy, although it is interesting to see that both ACT and SPT seem to
prefer additional features in EE with respect to the best-fit predictions,
which could be pointing towards a common physical origin. Note that with the
arrival of upcoming CMB surveys such as Simons Observatory, CMB-S4, LiteBird
and others, the situation might soon improve and might even shed light on the
origin of these mild differences.

##  4 Conclusion

In this work, we used Gaussian Processes (GP) to test the consistency of with
the most-recent CMB observations. In particular, we tested the robustness of
the predictions against space-based observations from the Planck satellite –
both using the latest CamSpec and the final data release (official) from the
Plik likelihood [1, 70, 29] – as well as with other ground-based temperature
and polarisation measurements by the Atacama Cosmology Telescope (ACT) [3, 4]
and South Pole Telescope (SPT-3G) [5, 6] collaborations. We find a mild but
noticeable inconsistency between the best fit predictions from the CamSpec
analysis with and ACT data, mainly seen in the TT spectra, where the GP finds
a non-negligible improvement in fit, with indications for a Harrison-
Zel’dovich spectrum with $n_{s}\simeq 1$. This is a non-parametric
confirmation of previous results, which supports the idea that the ACT data
seem to favor a scale-invariant primordial power spectrum. Our non-parametric
analysis also suggests that the covariance in the polarisation (EE)
measurements in the CamSpec likelihood might have been slightly
underestimated. Additionally, the EE measurements from both ACT and SPT seem
to require additional features at intermediate scales ($\ell\sim 400-700$),
extending up to $\ell\sim 2500$, which might be pointing towards a common
physical (possibly primordial) origin, differences in the inferred
cosmological parameters or (less-likely) minor unaccounted-for systematics in
the data.
Overall, our analysis again confirms the robustness of the predictions when
confronted with state-of-the-art CMB measurements; although our analysis shows
some interesting features, which could be hints of new physics. The arrival of
upcoming CMB experiments such as the Simons Observatory [84], CMB-S4 [85] and
others, will allow for further, more careful exploration of these issues. The
method discussed in this work can be readily applied to the upcoming data;
hopefully determining whether the mild discrepancies reported here are
actually coming from physical, systematic or statistical origin.

## Acknowledgements

The authors would like to thank Erik Rosenberg for providing us with the
latest CamSpec likelihood used in this analysis. RC would like to thank Adrien
La Posta for useful discussions. We also thank the anonymous referee for
suggesting a comparison between the official and CamSpec results. This work
was supported by the high-performance computing cluster Seondeok at the Korea
Astronomy and Space Science Institute. DKH would like to acknowledge the
support from CEFIPRA grant no. 6704-1. AS would like to acknowledge the
support by National Research Foundation of Korea NRF2021M3F7A1082053, and the
support of the Korea Institute for Advanced Study (KIAS) grant funded by the
government of Korea.

##  Appendix A Different mean functions

###  A.1 Best fit to Planck 2018

Figure 12: GP reconstructions of
$\Delta\mathcal{D}_{\ell}\equiv\mathcal{D}_{\ell}^{\rm
GP}-\mathcal{D}_{\ell}^{\Lambda\rm CDM}$ using the set of hyperparameters
maximizing the log-marginal likelihood in Eq. (2.3) when using the different
datasets, specified in the legend. These results assume the best-fit spectra
to Planck18 (Plik) as the mean function. The solid line and shaded regions
correspond to the mean and $2\sigma$ confidence intervals around it,
respectively.

We now consider the ’s best fit $\mathcal{D}_{\ell}$’s to the official Planck
2018 data release [2] as the mean function in our analysis. The reconstructed
residuals for the different datasets are shown in Fig. 12. These are obtained
using the set of hyperparameters $(\sigma_{f},\ell_{f})$ maximizing the LML in
Eq. (2.3), shown as black dots in Fig. 16. As noted before, the TT posteriors
for CamSpec data (under the assumption of the best-fit to Plik) reflect the
differences in the temperature measurements from these two analyses, which
might be non-parametric manifestations of the known $A_{L}$ and $\Omega_{k,0}$
anomalies [27, 2, 39] which are no longer present in CamSpec [29]; see also
the discussion in Section 3.1.2. The trend for the ACT reconstructed residuals
$\Delta\mathcal{D}_{\ell}^{\rm TT}$ to favour large correlation lengths and
less power at large scales (mimicking $n_{s}\to 1$), observed in Section 3.2.1
remains the same, even when using the best fit to Plik as mean function.
However, as shown in Fig 12, the reconstructed TT residuals show milder
(within $2\sigma$) deviations from zero, suggesting a slightly better
agreement between ACT DR4 and Plik than with CamSpec. Although the TT
measurements from SPT seem to be consistent with both CamSpec and Plik (see
the TT panel in Fig. 11), the reconstructed residuals remain consistent with
zero in the lower-right panel of Fig. 12. An interesting aspect of this
analysis is that of the reconstructed features in EE for both ACT and SPT,
which seem to be stable under changes in the mean function. In other words,
the EE posterior distributions for ACT and SPT are very similar, across Figs.
7, 9, 16 and 17, yielding similar reconstructed $\Delta\mathcal{D}_{\ell}$’s
in Figs. 8, 10, 12 and 13.

Table 4: $\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm GP}-\ln\mathcal{L}^{\Lambda\rm
CDM})$ improvements in fit for the ground and space-based experiments obtained
with the GP using ’s best-fit to Planck 2018 as mean function.

(Plik-Planck 2018)
$\Delta\chi^{2}$ 2018 CamSpec PR4 ACT DR4 SPT-3G TT $0$ $-36.08$ $-3.79$
$-0.65$ TE $0$ $-0.93$ $-3.08$ $-0.005$ EE $-0.30$ $-10.22$ $-6.94$ $-9.30$

Finally, we would like to mention that the analysis has been reproduced to
$0.1\sigma$ accuracy using the new pipeline for the Simons Observatory [86],
providing an independent cross-check of the results from Plik.

###  A.2 Best-fit to ACT+WMAP

Figure 13: GP reconstructions of
$\Delta\mathcal{D}_{\ell}\equiv\mathcal{D}_{\ell}^{\rm
GP}-\mathcal{D}_{\ell}^{\Lambda\rm CDM}$ using the set of hyperparameters
maximizing the log-marginal likelihood in Eq. (2.3) when using the different
datasets, specified in the legend. These results assume the best-fit spectra
to ACT+WMAP as the mean function. The solid line and shaded regions correspond
to the mean and $2\sigma$ confidence intervals around it, respectively.

In Fig. 17, we show the posteriors for $(\sigma_{f},\ell_{f})$ when using yet
another (different) mean function, namely ’s best-fit to ACT+WMAP
data999https://github.com/ACTCollaboration/pyactlike/blob/master/pyactlike/data/bf_ACTPol_WMAP_lcdm.minimum.theory_cl
as given by Table 4 (fourth column) in [3]. The top row shows the posteriors
of the CamSpec and Plik residuals with respect to ACT+WMAP best-fit. The
likelihood profiles for TE and EE are similar to the ones obtained using the
best fit to ; see Fig. 3. However, the posteriors for TT show a preference for
even larger deviations from the mean function $\sigma_{f}\simeq 10$, with
significant improvements in fit found by the GP. This reflects the discordance
between TT measurements (both from Plik and CamSpec) and the best fit to
ACT+WMAP, as shown in Figs. 13 and 17. More specifically, the TT
reconstructions in Fig. 13 for both Plik and CamSpec show large ($\gg
3\sigma$) deviations from zero (dashed line), while the TE and EE measurements
are consistent with the best-fit predictions from ACT+WMAP across the entire
multipole range.

Table 5: $\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm GP}-\ln\mathcal{L}^{\Lambda\rm
CDM})$ improvements in fit for the various datasets obtained with the GP using
’s best-fit to ACT+WMAP as mean function.

(ACT+WMAP)
$\Delta\chi^{2}$ 2018 CamSpecPR4 ACT DR4 SPT-3G TT $-40.75$ $-225.64$ $-2.69$
$-7.469$ TE $-0.025$ $-1.15$ $-1.31$ $-0.0001$ EE $0$ $-10.20$ $-4.28$
$-13.735$

In the case of ACT DR4 data (third row in Fig. 17), despite the large
correlation lengths ($\ell_{f}\simeq 10^{3}$ for TT & TE and $\ell_{f}\simeq
10^{2}$ for EE), the deviations from the mean function are mild
$\sigma_{f}\simeq 1$ for all cases. Curiously, the posteriors suggest (again)
a mild bimodal distribution for $(\sigma_{f},\ell_{f})$ in TT, TE, and EE with
one mode located at $(\sigma_{f},\ell_{f})\simeq(1,10^{3})$ and the second one
at $(1,10^{2})$. This seems to be the case, irrespective of the chosen mean
function (best-fit to or ACT+WMAP), and might suggest new features in the data
that cannot be properly taken into account by a model; see also the discussion
on this in Section 3.2.1. For comparison, we show the GP reconstructions of
the ACT residuals with respect to ACT+WMAP best-fit predictions in Fig. 13, to
be compared with Fig. 8 and 12 where the best-fit to CamSpec and Plik were
used, respectively. The EE reconstructions show similar features irrespective
of the chosen mean function (best-fit to Camspec, or ACT+WMAP); see also the
discussion in Section 3.2.1.

Finally, we find that the posteriors for SPT-3G data in TT are consistent with
the posteriors of the data, (both from Plik and CamSpec) suggesting that the
difference in TT is mainly coming from ACT. However, both ACT and SPT seem to
agree on the bimodal distributions captured in EE. As discussed before, these
particular values for $(\sigma_{f},\ell_{f})$ lead to oscillations (features)
around the best-fit predictions, and ultimately to an improvement in fit with
respect to . The improvements in fit in terms of $\Delta\chi^{2}$ values with
respect to the mean-function ( best-fit to ACT+WMAP) are reported in Table 5.

##  Appendix B Supplementary Material:

###  B.1 : High vs low-$\ell$

Motivated by previous works [23, 51, 81], we study a low-high split of
multipoles in the data and see if there are indications of systematic errors
specific to either of these parts. Fig 14 show the individual contributions
from different $\ell$ ranges from the log marginal likelihoods for the CamSpec
and Plik likelihoods, depicted in Figs 3 and 5, respectively. We see that for
the CamSpec best-fit mean function, the corresponding $\Delta\chi^{2}$ is
small for TT measurements, and we find no significant dependence on the
high/low-$\ell$ either. Little tendencies to prefer $\ell_{f}\lesssim 1$ for
low-$\ell$ is due to our Gaussian noise approximation slightly underestimating
the non-Gaussian noise in low $\ell$s, for which the GP compensates by adding
some uncorrelated noise. We conclude that no smooth deformation of the best-
fit prediction is preferred to explain the original CamSpec data, and no
low/high-$\ell$-specific systematics are detected in TT. For the EE spectrum,
we find that most of the improvements to the LML come from low-$\ell$ and
around $\ell_{f}\lesssim 1$. This is consistent with our discussion in Section
3.1.1; the CamSpec PR4 might have slightly underestimated the low-$\ell$
error, and the inclusion of uncorrelated random noise through GP
($\ell_{f}\lesssim 1$) noticeably improves the LML. Finally, as seen in Fig.
14, the disagreement between Plik and CamSpec is mainly driven by the TT
($\ell>650$) part of the spectrum. The improvement in fit seen at smaller
multipoles is likely due to the non-gaussian nature of the errors.

Figure 14: 2D posterior distributions as a function of $(\sigma_{f},\ell_{f})$
for the various datasets, using best-fit to CamSpec data as mean function.

###  B.2 Possible Calibration Issue

To account for possible differences in the absolute magnitude calibration of
the CMB maps, we introduce temperature and polarisation scaling parameters
$y_{t}$ and $y_{p}$, which are close to unity. This is analogous to the
nuisance parameters introduced by the and ACT collaborations, in terms of
$A_{\rm Planck}$ and $y_{p}$, respectively. These parameters are multiplying
the theoretical spectrum such that:

$\displaystyle\mathcal{D}_{\ell}^{\rm TT,obs}$
$\displaystyle=y_{t}^{2}\cdot\mathcal{D}_{\ell}^{\rm TT,th},$
$\displaystyle\mathcal{D}_{\ell}^{\rm TE,obs}$ $\displaystyle=y_{t}\cdot
y_{p}\cdot\mathcal{D}_{\ell}^{\rm TE,th},$ (B.1)
$\displaystyle\mathcal{D}_{\ell}^{\rm EE,obs}$
$\displaystyle=y_{p}^{2}\cdot\mathcal{D}_{\ell}^{\rm EE,th}.$

The values of these parameters are chosen such that the combined
$\chi^{2}_{\rm tot}(y_{t},y_{p})=\chi^{2}_{\rm TT}(y_{t})+\chi^{2}_{\rm
TE}(y_{t},y_{p})+\chi^{2}_{\rm EE}(y_{p})$ is minimized. In the ideal case,
one would expect to have $y_{t}=y_{p}=1$. As an example, let us note that the
best-fit spectra provided by the collaboration is to be scaled by an overall
calibration parameter $A_{\rm Planck}\equiv 1/\texttt{calPlanck}^{2}$ for the
theoretical predictions to be properly compared with the coadded spectra.
Thus, for a given theoretical $\mathcal{D}_{\ell}$, we find the value of
$y_{t,p}$ minimizing the $\chi^{2}$ to the observed data and proceed to
subtract the _(scaled)_ theoretical spectrum from the _actual observed_ data.
In practice, we use scipy’s “differential evolution” (stochastic) minimizer
[87] to find the optimal values for $y_{t,p}$. We then apply our GP formalism
and inspect for changes in the posteriors under such scaling of the spectra.
The results are shown in Fig. 15, where we report the values of the scaling
parameters ($y_{t}$ and $y_{p}$, minimizing the combined $\chi^{2}$) in
parenthesis on top of each figure. For other consistency checks and discussion
on possible scaling issues in the data, we refer the reader to [e.g. 88, 89,
90].

Figure 15: 2D posterior distributions as a function of $(\sigma_{f},\ell_{f})$
for the various datasets, using the (scaled) best-fit to CamSpec data as mean
function. The value of the scaling parameters $(y_{t},y_{p})$ is reported in
parenthesis on the top of each figure. The color bar shows the improvement in
fit, where $\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm
GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$.

Figure 16: 2D posterior distributions as a function of $(\sigma_{f},\ell_{f})$
for the various datasets, using the best-fit to the (Plik) data as mean
function. The color bar shows the improvement in fit, where
$\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$
and $\ln\mathcal{L}^{\rm GP}(\sigma_{f},\ell_{f})$ is the log marginal
likelihood in Eq. (2.3).

Figure 17: 2D posterior distributions as a function of $(\sigma_{f},\ell_{f})$
for the various datasets, using best-fit to ACT+WMAP data as mean function.
The color bar shows the improvement in fit, where
$\Delta\chi^{2}=-2(\ln\mathcal{L}^{\rm GP}-\ln\mathcal{L}^{\Lambda\rm CDM})$.

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