# Excess of lensing amplitude in the Planck CMB power spectrum

Rahima Mokeddem,11footnotetext: Corresponding author. Wiliam S. Hipólito-
Ricaldi  Armando Bernui

###### Abstract

Precise measurements of the Planck cosmic microwave background (CMB) angular
power spectrum (APS) at small angles have stimulated accurate statistical
analyses of the lensing amplitude parameter $A_{L}$. To confirm if it
satisfies the value expected by the flat-$\Lambda$CDM concordance model, i.e.
$A_{L}=1$, we investigate the spectrum difference obtained as the difference
of the measured Planck CMB APS and the Planck best-fit $\Lambda$CDM APS model.
To know if this residual spectrum corresponds to statistical noise or if it
has a hidden signature that can be accounted for with a larger lensing
amplitude $A_{L}>1$, we apply the Ljung-Box statistical test and find, with
high statistical significance, that the spectrum difference is not statistical
noise. This spectrum difference is then analysed in detail using simulated
APS, based on the Planck $\Lambda$CDM best-fit model, where the lensing
amplitude is a free parameter. We explore different binnations of the
multipole order $\ell$ and look for the best-fit lensing amplitude parameter
that accounts for the spectrum difference in a $\chi^{2}$ procedure. We find
that there is an excess of signal that is well explained by a $\Lambda$CDM APS
with a non-null lensing amplitude parameter $A_{lens}$, with values in the
interval $[0.10,0.29]$ at 68% confidence level. Furthermore, the lensing
parameter in the Planck APS should be $1+A_{lens}>1$ at $\sim 3\sigma$ of
statistical confidence. Additionally, we perform statistical tests that
confirm the robustness of this result. Important to say that this excess of
lensing amplitude, not accounted in the Planck’s flat-$\Lambda$CDM model,
could have an impact on the theoretical expectation of large-scale structures
formation once the scales where it was detected correspond to these matter
clustering processes.

##  1 Introduction

The current standard model of cosmology was recently confirmed analysing the
precise cosmic microwave background (CMB) radiation data measured by the
Planck satellite [48]. The 6-parameters cosmological model that best fits the
observational data, from CMB radiation and other cosmological observables, is
the flat-$\Lambda$CDM, where the 3-dimensional space is Euclidean (i.e., zero
spatial curvature or $k=0$), and the universe is filled in with cold dark
matter (CDM) and dark energy –in the form of a cosmological constant
$\Lambda$– in addition to the standard baryonic and electromagnetic
ingredients [48, 36].

One distinctive feature of the Planck measurements is the high angular
resolution that has allowed to measure the CMB temperature fluctuations at the
scales $\sim 4^{\prime}$ (i.e., $\ell\sim 2500$). This made it possible to
reconstruct, by inverse procedure, the lensing potential map that causes the
weak lensing action of matter on the CMB photons path from the last scattering
surface to us [3, 5, 4, 10]. In fact, the CMB photons detected by Planck are
already weakly lensed by the matter they encounter in their paths, and the
prediction of this phenomenon in the standard model is a tiny effect of
smoothing the CMB acoustic peaks at very small angular scales (i.e.,
$\ell\stackrel{{\scriptstyle>}}{{{}_{\sim}}}1000$). Since the observed CMB
photons can not be delensed, the solution adopted by the Planck collaboration
for the CMB data analyses is ma the weak lensing effect in the model that best
fits the CMB angular power spectrum (APS). In practice, this solution
contemplates that the amplitude of the weak lensing effect on CMB photons is,
in the flat-$\Lambda$CDM model, $A_{L}=1$.

The precise measurements of the Planck CMB temperature-temperature (TT) APS at
small angles have stimulated accurate statistical analyses about the
suitability of the lensing amplitude parameter $A_{L}=1$, as a way to validate
the $\Lambda$CDM concordance model. As a result, some works have reported
discordance with this amplitude. In fact, previous analyses indicate a clear
preference for a higher lensing amplitude, i.e., $A_{L}>1$ [8, 10, 7, 9, 11,
6, 36]222see the table 9 in page 32, in ref.
https://arxiv.org/pdf/1605.02985.pdf. More recently, the Planck collaboration
reported: $A_{L}=1.180\,\pm\,0.065$ (68%, Planck TT,TE,EE+lowE) and
$A_{L}=1.243\,\pm\,0.096$ (68 %, Planck TT+lowE), where the lensing amplitude
$A_{L}$ is varied in the parameter analysis (see section 6.2 in [48]). These
results motivate us to perform a detailed examination of the CMB acoustic
peaks sensitive to the lens phenomenon, i.e. $\ell>1000$, due to a possible
excess of lensing power in the TT APS data, a signal that can be accounted for
with a lensing amplitude $A_{L}>1$, that is, higher than the value adopted by
the flat-$\Lambda$CDM Planck’s best-fit model $A_{L}=1$.

The outline of this work is the following: In section 2 we describe the Planck
data employed in this work. In section 3 we perform the statistical analyses
of the spectra difference (Planck CMB APS minus Planck best-fit $\Lambda$CDM
APS) to investigate: (i) if it corresponds to white noise or not; in negative
case, (ii) if it has signature and amplitude that can be explained by an extra
lensing amplitude $A_{lens}>0$ interpreted as a possible excess of lensing
signature not accounted by the Planck best-fit $\Lambda$CDM APS; (iii) if this
spectrum difference can be explained by modifying some other cosmological
parameters in the APS of the $\Lambda$CDM model. In section 4 we discuss our
results, and also present our conclusions and final remarks.

##  2 Data

The most accurate current measurements of the CMB temperature fluctuations are
part of the third public data release of the Planck collaboration [48],
precise data that allow to re-examine many interesting features already
reported with diverse CMB data sets [30, 2, 26, 20, 27, 21, 25, 28, 29, 18,
19, 22, 24, 23, 51]. Further, CMB products combined with data sets from other
cosmological tracers are being used to study models alternative to the
flat-$\Lambda$CDM model (see, e.g. [31, 32, 33, 34]).

In particular, the large set of data products released by the Planck team
offers the possibility to perform exhaustive analyses to learn more about
features of the matter clustering [14, 12, 13], investigating the small
angular scales where the weak lensing phenomenon left its signature imprinted.
In fact, the Planck CMB data are specially accurate at small angular scales
due to their high angular resolution [15, 17, 16]. Thus, we shall analyse in
details the small angular scales of the CMB TT APS which is part of the third
data release of the PLANCK collaboration [48]. Our main goal is to study if
the difference between the measured Planck APS and the best-fit $\Lambda$CDM
APS (based on the 6-parameters flat-$\Lambda$CDM model) from the data fit
analyses done by the Planck team, is just statistical noise or it has indeed
some indicative signature of an incorrect modeling of the weak lensing
amplitude (i.e., $A_{L}\neq 1$), or of other cosmological parameters. Clearly,
the null hypothesis that deserves investigation establishes that the
difference between the observed Planck APS and the $\Lambda$CDM APS is just
white noise.

For our analyses some features are important to be considered. Firstly, the
possible signal would be significantly detected only at the angular scales
with the smallest error measurements, that is, for the interval $\ell\sim
1000-2000$; at the same time, one knows that these scales correspond to the
regime where the lensing phenomenon has a large impact on the CMB APS.
Secondly, for these analyses we use the original unbinned data from Planck
repository pages, then our analyses shall consider binnation inspections
beside the $\Delta\ell=30$ case considered by the Planck team.

In our scrutiny, the released Planck CMB TT APS, the Planck APS 333COM-
PowerSpect-CMB-TT-full-R3.01.txt, covers the range of multipoles
$\ell=2-2508$, where the multipoles of interest here, $\ell\geq 30$, were
derived from the cross-half-mission likelihood Planck, Plik (for details see
ref. [35]). The Planck APS was derived from the Commander component-separation
algorithm applied to the combination of Planck 2018 temperature data between
30 and 857 GHz, including 86% of the sky (Planck-2018 results VI) (associated
$1\sigma$ errors include beam uncertainties).

The analysis was done by optimally combining the spectra in the frequency
range $100-217$ GHz, and correcting them for unresolved foregrounds using the
best-fit foreground solution (regions were chosen as to avoid the areas
dominated by noise). For the best-fit $\Lambda$CDM CMB APS, to be subtracted
to the Planck APS for the aim of our analyses, we use the 6-parameter
flat-$\Lambda$CDM minimal model, also released by the Planck team and
hereafter termed the $\Lambda$CDM APS 444COM-PowerSpect-CMB-base-plikHM-
TTTEEE-lowl-lowE-lensing-minimum-theory-R3.01.txt. This $\Lambda$CDM APS uses
the Planck TT, TE, EE+lowE+lensing data. Both power spectra, the Planck APS
and $\Lambda$CDM APS, were publicly released by the Planck collaboration555
https://archives.esac.esa.int/doi/html/data/astronomy/planck/Cosmology.html .

##  3 Methodology and data analyses

This section describes the methodological approach we shall follow to study
the Planck data APS, specifically concerning the possibility that the lensing
amplitude parameter that best-fits the APS data is larger than 1, $A_{L}>1$,
where the value used by the Planck collaboration to obtain the
flat-$\Lambda$CDM model is $A_{L}=1$.

###  3.1 Methodology to investigate the spectrum difference

We want to investigate if there is an excess of lensing power in the TT APS
measured by the Planck Collaboration $C^{\mbox{\footnotesize Planck}}_{\ell}$,
with 1$\sigma$ standard deviation $\sigma^{\mbox{\footnotesize
Planck}}_{\ell}$, with respect to the flat-$\Lambda$CDM APS,
$C^{\Lambda\mbox{\footnotesize CDM}}_{\ell}$, obtained through a best-fit
procedure of the 6 parameter flat-$\Lambda$CDM cosmological model to the TT
Planck APS data [48]. For this, we first define the spectrum difference :

$\delta^{obs}_{\ell}\equiv\frac{C^{\mbox{\footnotesize
Planck}}_{\ell}-C^{\Lambda\mbox{\footnotesize
CDM}}_{\ell}}{C^{\Lambda\mbox{\footnotesize CDM}}_{\ell}}\,.$ (3.1)

Notice that the $\Lambda$CDM APS, $C^{\Lambda\mbox{\footnotesize
CDM}}_{\ell}$, takes into account the CMB lensing phenomenon with $A_{L}=1$,
which is the value assumed by the Planck Collaboration in the
flat-$\Lambda$CDM model.

We define the symbol $A_{lens}\in[0,1]$ as the parameter that quantifies the
excess of lensing amplitude to explain the residual $\delta^{obs}_{\ell}$. We
construct synthetic APS, $C^{syn,L}_{\ell}(A_{lens})$, with the Planck 2018
flat-$\Lambda$CDM cosmological parameters but for arbitrary lensing amplitude
$A_{lens}\geq 0$. We also define

$\delta^{\mbox{\,\footnotesize
exc}}_{\ell}(A_{lens})\,\equiv\,\frac{C^{\mbox{\,\footnotesize
syn},L}_{\ell}(A_{lens})-C^{\mbox{\,\footnotesize
syn},uL}_{\ell}}{C^{\mbox{\,\footnotesize syn},uL}_{\ell}}\,,$ (3.2)

where

$\displaystyle C^{\Lambda\mbox{\footnotesize CDM}}_{\ell}$ $\displaystyle=$
$\displaystyle C^{\mbox{\,\footnotesize syn},L}_{\ell}(A_{lens}=1)\,,$ (3.3)
$\displaystyle C^{\mbox{\,\footnotesize syn},uL}_{\ell}$ $\displaystyle=$
$\displaystyle C^{\mbox{\,\footnotesize syn},L}_{\ell}(\mbox{A}_{lens}=0)\,,$
(3.4)

where the upper letters $L$ and $uL$ mean lensed and unlensed APS,
respectively.

The quantity $\delta^{\mbox{\,\footnotesize exc}}_{\ell}(A_{lens})$ measures
the possible excess of lensing power represented by the relative difference of
synthetic lensed and unlensed APS (the difference that depends on the
parameter $A_{lens}$; clearly, the case $A_{lens}=0$ implies that
$\delta^{\mbox{\,\footnotesize exc}}_{\ell}=0$). Then, a good statistical
agreement between the excess quantity $\delta^{\mbox{\,\footnotesize
exc}}_{\ell}$ and the observed difference $\delta^{\mbox{\,\footnotesize
obs}}_{\ell}$, measured with the reduced $\chi^{2}$ best-fit for some
$A_{lens}>0$ value, will provide an explanation for
$\delta^{\mbox{\,\footnotesize obs}}_{\ell}$ as being an excess of lensing
power in the Planck APS not accounted for by the $\Lambda$CDM lensing
amplitude $A_{L}=1$. Of course, other interpretations for
$\delta^{\mbox{\,\footnotesize obs}}_{\ell}$ would be possible, as for
instance that it is just statistical noise, and for this reason it is
interesting to examine them too. In order to evaluate if the spectrum
difference $\delta^{\mbox{\,\footnotesize obs}}_{\ell}$ corresponds to white
noise, we shall apply the Ljung-Box test [44].

Our analyses include the simulation of APS, from now on termed synthetic APS,
which considers the modeling of the weak lensing phenomenon on CMB photons for
cases with $A_{lens}=0$ (unlensed) and $A_{lens}\neq 0$ (lensed). The details
for the production of these synthetic APS are the following:

  * •

The Boltzmann code $CLASS$ [50] was modified to allow the code to consider the
lensing amplitude $A_{lens}$ as a parameter.

  * •

We use this modified code with the cosmological parameters corresponding to
the observed Planck TT APS, as detailed in the table 1, to produce synthetic
lensed and unlensed $\Lambda$CDM TT APS, $C^{\mbox{\,\footnotesize
syn},L}_{\ell}$ and $C^{\mbox{\,\footnotesize syn},uL}_{\ell}$, respectively.

Table 1: The cosmological parameters used to generate the synthetic angular power spectra. parameters | $\Omega_{b}h^{2}$ | $\Omega_{c}h^{2}$ | $100\theta_{MC}$ | $n_{s}$ | $\ln(10^{10}A_{s})$ | $\tau$
---|---|---|---|---|---|---
$\Lambda$CDM | 0.022383 | 0.12011 | 1.040909 | 0.96605 | 3.0448 | 0.0543
  * •

We then vary the value of the lensing amplitude $A_{lens}$ and construct the
quantity $\delta^{\mbox{\,\footnotesize exc}}_{\ell}(A_{lens})$.

Moreover the relative error for $\delta^{obs}_{\ell}$, $error(\delta)_{\ell}$
, is calculated by considering the error from the observed Planck TT APS,
$\sigma^{\mbox{\footnotesize Planck}}_{\ell}$, and using the standard approach
for propagation of uncertainties from equation (3.1).

Our analysis shall consider binnation inspections beside that one adopted by
the Planck team. Thus, after calculating the spectrum difference for each
$A_{lens}$ considered, and the corresponding errors, we have to choose the bin
length $\Delta\ell$, which is the size of the bin that we are dividing our
$\delta^{obs}_{\ell}$ array into. Each bin will be represented by its mean
value with error $\sigma_{\ell}\equiv
error(\delta)_{\ell}/\sqrt{\,\Delta\ell}$. In the last step, we compute the
$\chi^{2}$, after binning the spectrum difference and computing the error

$\chi^{2}(A_{lens})=\sum\frac{[\delta^{\mbox{\,\footnotesize
obs}}_{\ell}-\delta^{\mbox{\,\footnotesize
exc}}_{\ell}(A_{lens})]^{2}}{\sigma^{2}_{\ell}}\,,$ (3.5)

where $\delta^{obs}_{\ell}$ and $\delta^{exc}_{\ell}(A_{lens})$ are defined in
equations (3.1) and (3.2), respectively. The sum above is performed over the
binned spectra difference data $\\{\delta_{\ell}^{\mbox{\,\footnotesize obs /
exc}}\\}$.

Finally, the space of parameters is extended to include the neutrino mass
$\sum m_{\nu}$ and the spatial curvature $\Omega_{k}$ to explore the
possibility that any lensing excess signal can be mimicked by the effects of
some of these parameters.

###  3.2 Null hypothesis analyses

In this subsection we shall test if the set of values
$\\{\delta^{obs}_{\ell}\\}$ is a statistical noise or not, i.e. we test the
randomness of the spectrum difference. To perform this test we consider the
null hypothesis:

$H_{0}$: The spectrum difference, $\\{\delta^{obs}_{\ell}\\}$, corresponds to
a residual statistical (or white) noise.

That $H_{0}$ be true means that $A_{L}=1$ completely accounts for with the
lensing signal in the observed Planck APS. In order to examine this hypothesis
we shall use the Ljung-Box (LB) test [44], which is a modification of the Box-
Pierce Portmanteau Q statistic [43]. The LB test is used to look for
correlation in a data series, determining whether there is or not a remaining
signature in the residuals after a forecast model has been fitted to the data.
Basically, the LB test is a useful tool to evaluate the autocorrelation
between the data in analysis, and to quantify its statistical significance.

As a first step, it is necessary to compute the autocorrelation in a given
data set $\\{\delta^{obs}_{i}\\}$ in an interval $M=[\ell_{min},\ell_{max}]$
with $N$ data points

$\displaystyle\rho_{k}=\frac{\sum^{N-k}_{i=1}\left(\delta^{obs}_{i}-\overline{\delta^{obs}_{i}}\right)\left(\delta^{obs}_{i+k}-\overline{\delta^{obs}_{i}}\right)}{\sum^{N}_{i=1}\left(\delta^{obs}_{i}-\overline{\delta^{obs}_{i}}\right)^{2}}\,,$
(3.6)

where $\overline{\delta^{obs}_{i}}$ is the average of all $N$ points in the
$M$ interval, $k$ is commonly called the lag and $\rho_{k}$ is called the lag
$k$ autocorrelation.

Since $\rho_{k}$ measures the correlation between multipoles separated by $k$,
the autocorrelation $\rho_{k}$ can be used, in principle, to detect non-
randomness in the data. However, it is more recommended to use tests
considering multiple (sometimes called global or total) correlations across
all the data interval and for several lags jointly, like the LB test [44]. The
null hypothesis $H_{0}$ for this test establishes that the first $h$ lags
autocorrelations are jointly zero, i.e.

$\displaystyle H_{0}:\rho_{1}=\rho_{2}=\cdots=\rho_{k}=\cdots=\rho_{h}=0\,,$
(3.7)

where $h$ is the maximum lag considered in the test. In other words, $H_{0}$
being true implies that all the analyzed data are uncorrelated and correspond
to a white noise signal. The LB statistic is defined by

$\displaystyle Q_{h}=N(N+2)\sum^{h}_{k=1}\frac{\rho^{2}_{k}}{N-k}\,,$ (3.8)

where $\rho_{k}$ is the estimated correlation using equation (3.6). Thus , LB
test does not consider just a particular lag $k$ but a set of $h$ estimated
correlations.

Since $Q$ asymptotically follows a $\chi^{2}$ distribution, to determine the
statistical significance of the test it is compared to a $\chi^{2}$
distribution with $h^{\prime}=h-q$ degrees of freedom under the condition [44]

$\displaystyle Q_{h}>\chi^{2}_{1-\alpha,h^{\prime}}\,,$ (3.9)

where $q$ is the number of parameters used to fit the observed Planck APS and
$\alpha$ is the significance level. Then, small $p$-values ($p<\alpha$) will
imply that significant correlation exists between the data in the set
$\\{\delta^{obs}_{i}\\}$ and thus, $H_{0}$ is rejected in the $M$ interval.

A choice of the $h$ parameter in equation (3.8) requires a more detailed
discussion. Several studies has been performed to define which is the optimal
value. For instance, empirically [37] suggests $h=5$, [38] suggests $h\sim\ln
N$, [39] $h=min(10,N/5)$, [40] $h=20$. Furthermore, [41] employed a study with
simulations to show that for very large values of $h$, LB test could lead to
not so reliable results. Recently, [42] also used simulations to evaluate the
optimal value for the number of lags $h$ involved in the LB test. Their
results have shown that for the order of thousands data, the optimal values
are $h=50$ for $\alpha=0.05$ and $h=25$ for $\alpha=0.01$.

To perform the LB test we first compute $\\{\delta^{obs}_{l}\\}$ in different
$M$ intervals and apply it for each interval. Our results are summarized in
table 2 where $q=7$, $\alpha=0.01$ and, as recommended by [42], $h=25$ was
used. We rerun the test for different values of $h$ mentioned in the previous
paragraph and confirm that the results in the table 2 are robust. The first
interval analyzed is $2\leq\ell\leq 2500$, i.e. the complete range of
multipoles measured by Planck. In such interval we found a rejection of null
hypothesis $H_{0}$ with $p=0.0$. Then, we applied the test for three different
intervals: $2\leq\ell\leq 100$, $2\leq\ell\leq 800$ and $2\leq\ell\leq 1200$,
where it was found a no rejection of $H_{0}$. These results are indicating
that the CMB power spectrum is well fitted by the cosmological parameters
found by Planck team, and thus residual $\\{\delta^{obs}_{\ell}\\}$ is white
noise until, roughly, $\ell\sim 1200$. We also considered three more
intervals, $1100\leq\ell\leq 2500$, $1600\leq\ell\leq 2500$ and
$2000\leq\ell\leq 2500$. Results in such intervals are indicating a rejection
of the $H_{0}$ hypothesis at more than $99\%$ CL, i.e.
$\\{\delta^{obs}_{\ell}\\}$ is not white noise and some signal could be hidden
in such multipole intervals. One possibility is that rejection of $H_{0}$
would be due to the not so accurate measurements, represented by the large
error bars, observed in the last multipoles (especially for $\ell\gtrsim
2200$). In order to evaluate this possibility we consider one more interval.

Table 2: Statistical LB analyses to investigate the rejection or acceptance of the null hypothesis $H_{0}$ Multipole intervals |  $p$-value | Reject $H_{0}$ |  $\%$repetition
---|---|---|---
$\ell=[2,2500]$ | $0$ | Yes | $100\%$
$\ell=[2,100]$ | $6.0346\times 10^{-1}$ | No | $95\%$
$\ell=[2,800]$ | $1.1739\times 10^{-1}$ | No | $56\%$
$\ell=[2,1200]$ | $1.7097\times 10^{-2}$ | No | $20\%$
$\ell=[1100,2500]$ | $1.0436\times 10^{-14}$ | Yes | $100\%$
$\ell=[1600,2500]$ | $3.7059\times 10^{-8}$ | Yes | $94\%$
$\ell=[2000,2500]$ | $1.5677\times 10^{-3}$ | Yes | $59\%$
$\ell=[1100,2200]$ | $2.6334\times 10^{-13}$ | Yes | $100\%$

In fact, we also consider the statistical LB analyses in the interval:
$1100\leq\ell\leq 2200$, and according to the very small $p$-values obtained,
we also confirm the rejection of the null hypothesis $H_{0}$ in this interval
at $>99\%$ confidence level (CL) as we can see in the table 2. These results
confirm a significant correlation among the $\\{\delta^{obs}_{\ell}\\}$ and
support the hypothesis of the presence of some structure left in the spectrum
difference, even if one does not consider the data with largest errors for
$\ell>2200$. Roughly, the null hypothesis is rejected for $\ell\gtrsim
1000-1200$, precisely at the angular scales where theoretical estimates
indicate that lensing signal starts to be more relevant [1].

So far, we have applied the LB test to the main values of the spectrum
difference, i.e. without considering the errors $error(\delta)_{\ell}$.

Now, let us investigate the robustness of our results by introducing error
measurements in the LB test. In order to continue the analyses we construct
artificial sets $\\{{\widetilde{\delta}}^{obs}_{\ell}\\}$ defined as
$\\{{\widetilde{\delta}}^{obs}_{\ell}\\}\equiv\\{\delta^{obs}_{\ell}+R_{\ell}\\}$
and generated as follows: first, for each $\ell$ we generate a $R_{\ell}$
which is a random number obtained from a Gaussian distribution –with zero mean
and standard deviation: $error(\delta)_{\ell}$– that is added to
$\\{\delta^{obs}_{\ell}\\}$ to get $\\{{\widetilde{\delta}}^{obs}_{\ell}\\}$.
These sets $\\{{\widetilde{\delta}}^{obs}_{\ell}\\}$ are hypothetical points
representing simulated spectrum differences inside 1$\sigma$ error. We then
apply the LB test, in different M intervals, for $100$ simulated spectrum
differences $\\{{\widetilde{\delta}}^{obs}_{\ell}\\}$ and compute the
percentage of repetition of rejection, or not rejection, of the null
hypothesis. The results are shown in the last column in table 2.

For example, in the first interval $2\leq\ell\leq 2500$, the $100\%$ of
repetitions reject $H_{0}$, while in the second interval the $95\%$ of
repetitions do not reject $H_{0}$. The percentage of repetitions for $H_{0}$
no rejection is decreasing for the next two intervals: $2\leq\ell\leq 800$ and
$2\leq\ell\leq 1200$, which can be an indicative that around $\ell\sim 1000$
there exists the probability of some correlation between the data
$\\{\delta^{obs}_{\ell}\\}$. Nevertheless in all the other cases, percentage
of repetitions for rejection, or no rejection, of $H_{0}$ is sufficiently high
to reinforce the results shown in the third column of table 2.

As a conclusion of this subsection we must point out that there exist a strong
evidence to reject the null hypothesis in the interval $1100\leq\ell\leq
2500$, where lensing effects on CMB photons are more relevant [1].
Nevertheless, to be conservative, we can leave out the last multipoles due to
their large error bars, i.e., $\ell>2200$, and will focus our next analyses in
the interval $1100\leq\ell\leq 2200$.

Clearly, the CMB temperature data have noise, that is not isotropic due to the
observational strategy of the Planck probe. However, the conclusion of this
subsection remains roubust because in Appendix A.2 we show the results from
complementary analyses considering the APS of Planck noise maps (from two
frequencies: 545 and 857 GHz). Moreover, in the current LB tests we consider
the best-fit values from table (1), but not their corresponding errors.
Analyses taking into account the error bars, done in Appendix A.1, show that
our original LB test results remain the same.

###  3.3 Measuring the excess of lensing power

The first aim of this work was to investigate if the spectra difference
$\delta^{obs}_{\ell}$ defined in equation (3.1), corresponds to statistical
noise or not. Results of the previous subsection showed that for several
intervals with $\ell\gtrsim 1000-1200$ there seems to be a rejection of the
null hypothesis $H_{0}$. Now, our second aim is to explore the possibility
that the $\delta^{obs}_{\ell}$ data could be well reproduced through the
$\chi^{2}$ best-fit analyses, described in the subsection 3.1, by a synthetic
APS according to the equation (3.2), $\delta^{\mbox{\,\footnotesize
exc}}_{\ell}(A_{lens})$, for some value $A_{lens}>0$. As illustrative examples
for several $A_{lens}$ cases and diverse binnation schemes $\Delta\ell$, we
show some plots of these best-fit analyses in figures 1, 2, and 3. The fact
that one can find a good fit, i.e. $\chi^{2}\simeq 1$ as seen in table 3, for
the spectrum difference $\delta^{obs}_{\ell}$ for a synthetic APS with
$A_{lens}>0$ suggests that the lensing amplitude parameter in the Planck APS
should be indeed larger than 1, i.e. $1+A_{lens}>1$, meaning that this
parameter was underestimated in the analyses of the APS,
$C^{\mbox{\footnotesize Planck}}_{\ell}$, done by the Planck collaboration.

Assuming that the spectrum difference $\delta^{obs}_{\ell}$ has the signature
of the lensing phenomenon, as suggested by the plots shown in figures 1–3, one
can find the parameter value $A_{lens}$ of the synthetic APS that best-fits
these data. In the table 3 we display the $\chi^{2}$ values obtained for
different $A_{lens}$ values and diverse $\Delta\ell$ bin lengths. One notice
that in several cases, $\chi^{2}\simeq 1$ is fully possible, including the
$A_{lens}=0$ case which is also considered in our analyses. This suggests that
a more detailed likelihood analyses are in due.

We start considering the $\chi^{2}$ procedure, of the synthetic APS that best-
fits the spectrum difference, $\delta^{obs}_{\ell}$, for a continuous set of
$A_{lens}$ values. As a result, one obtains a $\chi^{2}$ value for each
$A_{lens}$ considered, that is, $\chi^{2}$ becomes a function of the
$A_{lens}$ parameter: $\chi^{2}=\chi^{2}(A_{lens})$, information that can be
plotted as curves $\chi^{2}$ versus $A_{lens}$, as observed in the figure 4,
considering moreover different bin lengths $\Delta\ell$. In the table 4 we
show the numerical values of the intersections between the curves
$\chi^{2}(A_{lens})$ appearing in figure 4, obtained for different binnations,
and the straight line $\chi^{2}=1$. According to these results, we conclude
that values $A_{lens}\neq 0$ exist, which imply that the spectrum difference
$\delta^{obs}_{\ell}$ can be well fitted by $\delta^{exc}_{\ell}(A_{lens})$
for some $A_{lens}\neq 0$ values and several $\Delta\ell$ choices.
Furthermore, we perform statistical likelihood analyses to find the best-fit
$A_{lens}$ values, as shown in table 5, also for several bin lengths.
Observing the figure 4 one can argue that several binnations provide a best-
fit with reduced $\chi^{2}=1$ for some $A_{lens}=A_{lens}(\Delta\ell)$ values,
meaning that a solution with $A_{lens}\neq 0$ exists but is not unique.
Complementing this $\chi^{2}$ studies, the maximum likelihood analyses shown
in figure 5 let us to conclude that $A_{lens}\in[0.10,0.29]$ at 68% CL. See
also the figure 6 for a detailed study of the illustrative case
$\Delta\ell=17$, where it can be seen that $A_{lens}>0$ at 2.6 $\\!\sigma$.
The same results are found for other binnation schemes.

As a result of the analyses in this subsection we found that the signal hidden
in $\\{\delta^{obs}_{\ell}\\}$ is related to an excess of lensing amplitude.
However, we reached this conclusion by considering only the best-fit values of
table (1). It is then necessary to test for a possible bias with this choice,
and for this we shall perform complementary analyses considering the measured
error bars of the best-fit cosmological parameters. Results of these tests are
discussed in the Appendix A.1, where we have proven that not only the hidden
signal in $\\{\delta^{obs}_{\ell}\\}$ is very weakly correlated with the
cosmological parameters from table (1), but also the fact that a $\sim 20\%$
lensing amplitude excess in $\\{\delta^{obs}_{\ell}\\}$ is robust.

Figure 1: Illustrative examples: plots of the spectrum difference
$\delta^{obs}_{\ell}$ together with $\delta^{exc}_{\ell}(A_{lens})$, for the
cases $A_{lens}=0.20$ (first and second panels) and $A_{lens}=0.22$ (third and
fourth panels), for $\ell\in[1100,2000]$ and $\ell\in[1100,2200]$ as indicated
in the plots. In all these plots we consider $\Delta\ell=63$. Although the
best-fit curve in the first and third panels seem equals, they are indeed
slightly different.

Figure 2: Illustrative examples: plots of the spectrum difference
$\delta^{obs}_{\ell}$ together with $\delta^{exc}_{\ell}(A_{lens})$ for
$\ell\in[1100,2000]$ and $\ell\in[1100,2200]$ as indicated in the plots. In
all these plots we consider $A_{lens}=0.20$. They were obtained for the bin
lengths $\Delta\ell=32$, for the first and the second panels and
$\Delta\ell=40$ for the third and the fourth panels.

Figure 3: Illustrative analyses comparing the spectrum difference $\delta^{obs}_{\ell}$ and $\delta^{exc}_{\ell}(A_{lens})$ for $\ell=[1100,2200]$ and $\ell=[1100,2000]$ with $\Delta\ell=63$ for $A_{lens}=0.20$ (upper 2 plots) and $A_{lens}=0.22$ (lower 2 plots) Figure 4: $\chi^{2}$ as a function of $A_{lens}$ and for different number of bins. This plot shows that, independent of the bin size $\Delta_{\ell}$, the $\chi^{2}$ function exhibits solutions for the case $\chi^{2}=1$ for $A_{lens}\neq 0$, which justify our search for the value $A_{lens}$ that best-fits the observed data $\delta^{obs}_{\ell}$.  Table 3: Illustrative examples of $\chi^{2}$ calculations for different values of $A_{lens}$ and several binnation choices $\Delta\ell$. Some of these cases can be seen in figures 1, 2, and 3. $A_{lens}$ | $\Delta\ell=17$ | $\Delta\ell=32$ | $\Delta\ell=41$ | $\Delta\ell=51$ | $\Delta\ell=63$
---|---|---|---|---|---
0.00 | 1.0363 | 1.2809 | 1.02590 | 0.95802 | 1.63446
0.10 | 0.9414 | 1.0747 | 0.7610 | 0.6416 | 1.0798
0.20 | 0.9329 | 1.0305 | 0.7094 | 0.5851 | 0.8395
0.24 | 0.9525 | 1.0561 | 0.7459 | 0.6319 | 0.8274
0.34 | 1.0565 | 1.2238 | 0.9734 | 0.9150 | 0.9980

###  3.4 Cosmological parameters dependency

In the previous section we explore the possibility that the residual structure
in CMB lensing signal can be explained by an excess in lensing amplitude
$A_{lens}>0$. However, there is also the possibility that such residual
structure could be mimicked by varying in other cosmological parameters. The
scrutiny of this dependency is important to confirm if the signal found in the
previous section is indeed due to the lensing effect or it also has
contribution of other cosmological parameters. This analysis is the main
objective in this subsection.

As it is well known, the APS, $C^{\Lambda\mbox{\footnotesize CDM}}_{\ell}$, is
sensitive to the cosmological parameters [52, 45, 53]. Such sensitivity is
inherited by the quantity $\delta^{obs}_{\ell}$ defined in equation (3.1).
Despite that, although the cosmological parameters in table 1 have a high
impact in the CMB APS, this is not the case when the spectrum difference is
considered, where their impact is weaker. In some sense, the analysis of
$\delta^{obs}_{\ell}$ will diminish possible degeneracies between those
cosmological parameters and $A_{lens}$, $\sum m_{\nu}$, and $\Omega_{k}$.

So far, the model considered was the simplest $\Lambda$CDM with the six
parameters listed in table 1 plus lensing amplitude $A_{L}=1$. Now, the space
of parameters is extended to include the neutrino mass $\sum m_{\nu}$ and the
spatial curvature $\Omega_{k}$. Massive neutrinos slow down the growth of
matter perturbations and prevent clustering process [46, 47], resulting,
therefore, in an anticorrelation with gravitational lensing amplitude.
Moreover, spatial curvature also has correlation with lensing and thus, CMB
lensing signal is sensitive to the neutrino mass and the spatial curvature of
the universe.

In order to investigate the space of parameters ($A_{lens}$, $\sum m_{\nu}$,
$\Omega_{k}$) in the light of the spectrum difference $\delta^{obs}_{\ell}$,
the other cosmological parameters were fixed by using best-fit values found by
Planck 2018 data [48] (see table 1) and the parameters $A_{lens}$, $\sum
m_{\nu}$, and $\Omega_{k}$ were allowed to vary. Then we use MCMC techniques
to find constraints on these three free parameters by considering flat priors
and marginalizing to compute the one dimensional posterior distributions.
Results are presented in figure 7 and table 6.

In figure 7 the one dimensional posterior distributions for $\Delta\ell=17$
and $\Delta\ell=63$ are shown. We can say, in general, that our results are
consistent at $68\%$ CL with an almost flat spatial curvature, with $\sum
m_{\nu}<0.47$ eV, and with a $\sim 20\%$ lensing amplitude excess quantified
by the $A_{lens}$ parameter. This $A_{lens}$ excess is still present at around
three standard deviations (see table 6), even though these analyses considered
a larger number of parameters. All these results are robust when changing
$\Delta\ell$ indicating that the neutrino mass or the spatial curvature weakly
impact in the $\delta^{obs}_{\ell}$ intensity signal. Therefore their effects
are not sufficient to explain the signature in the spectrum difference
$\delta^{obs}_{\ell}$, which is more likely explained by a lensing amplitude
$A_{lens}\simeq 0.20$, which implies that the lensing amplitude in the
Planck’s CMB APS should be $\sim 20\%$ larger than the value expected in the
flat-$\Lambda$CDM model.

Table 4: Table with the numerical values obtained due to the intersection between the horizontal line, representing the reduced-$\chi^{2}=1$, with the curves produced calculating the reduced-$\chi^{2}$ for the parameter $A_{L}$, and considering various bin-length cases $\Delta\ell$. The first and second points in the table below represent the values of $A_{L}$ where $\chi^{2}=1$. As observed in figure 4 there is no intersection for the case $\Delta\ell=32$.  bin length | first point | second point
---|---|---
$\Delta\ell=17$ | $0.028675$ | $0.29479$
$\Delta\ell=32$ |  – |  –
$\Delta\ell=41$ | $0.02495$ | $0.3305$
$\Delta\ell=51$ | $0.0023$ | $0.3477$
$\Delta\ell=63$ | $0.16053$ | $0.29425$
Table 5: Values of $A_{L}$ from the likelihood plot. The last column indicates the confidence level (CL) for having $A_{lens}>0$ in each binnation case. bin length |  $\\!\\!A_{lens}$ at max. likelihood | 1$\sigma$ interval | CL for $A_{lens}\\!>\\!0$
---|---|---|---
$\Delta\ell=17$ | $0.16036$ | $[0.0993,0.2213]$ | 2.6$\sigma$
$\Delta\ell=32$ | $0.177477$ | $[0.1164,0.2384]$ | 2.8$\sigma$
$\Delta\ell=41$ | $0.17567$ | $[0.1136,0.2376]$ | 2.8$\sigma$
$\Delta\ell=51$ | $0.172072$ | $[0.1090,0.2350]$ | 2.7$\sigma$
$\Delta\ell=63$ | $0.22702$ | $[0.1630,0.2910]$ | 3.5$\sigma$
Figure 5: Maximum likelihood analyses, complementing the analyses done in
figure 4, showing the 68% CL of the $A_{lens}$ values that perform the reduced
$\chi^{2}$ best-fit of the spectrum difference $\delta^{obs}_{\ell}$, that is
$A_{lens}\in[0.10,0.29]$, for several bin sizes. Figure 6: Illustrative
example of $\sigma$ intervals for the $\Delta\ell=17$ case. As observed, for
this binnation case the value $A_{lens}=0$ (equivalently $A_{L}=1$) is
excluded with $2.6\sigma$ (i.e., at 99% CL).

##  4 Discussions and Final Remarks

We now discuss the exhaustive analyses done with the precise Planck APS data
and the results obtained investigating the statistical features of the
spectrum difference $\delta^{obs}_{\ell}$. Recent analyses have reported the
preference of the lensing amplitude parameter $A_{L}$ for a value larger than
1, with high statistical significance [36, 48, 10, 7], and we want to study if
this result could be sourced by a statistical artifact or if it has a physical
origin. We emphasize that our analyses must be intended as complementary (due
to the methodology employed) and independent (with respect to previous works
and those done by the Planck collaboration [48]).

In this respect, it is interesting that the underestimated lensing amplitude
has also been observed in the lensing-potential power spectrum,
$C_{\ell}^{\phi\phi}$, as shown in [48], where it was measured $A_{L}=1.19$
for the best-fit model (see the discussion in section 2.3 in ref. [48]), a
value in perfect agreement with the result of our analyses.

On the other hand, we discuss some interesting issues regarding the study of
the lensing amplitudeproblem that are not usually discussed in the literature.
Firstly, in the usual statistical analyses of the CMB APS, the signature left
in the spectrum difference is not discussed, for this we consider that
previous to any best-fitting model procedure considering the lensing amplitude
as a parameter, one needs to discuss the nature of the spectrum difference
leftover: : is it a residual white noise, or not? Second, because we find that
the spectrum difference has a definite signature (i.e., it is not white
noise), the binnation procedure may have an impact in the measurement
of$A_{lens}$, and therefore deserves to be analyzed . Third, since the
spectrum difference has a definite signature, not any parameter –or
combination of parameters– can suitably best-fit it, a fact that when is
considered contributes to eliminate, or at least diminish, possible
degeneracies (because each cosmological parameter has a different effect in
the fitting of$\delta^{obs}_{\ell}$).

In the previous section we have performed a detailed examination of the
unbinned CMB APS. We have proved that the spectrum difference
$\delta^{obs}_{\ell}$ is not statistical noise; additionally we found several
multipole intervals, starting at $\ell\gtrsim 1000$, where this result is true
(see table 2), supporting the crucial point that the spectrum difference is
not statistical noise at small scales. This result is suggesting that some
signal is present in the spectra difference, and the signature shown in these
data must be an indication of the phenomenon that caused it. However,
degeneracy can also happen, that is, the signature present in the data could
be reproduced by more than one source and for this reason we also explore
diverse hypotheses to explain the signature and the amplitude of the spectrum
difference data.

Bearing in mind that, at the scales $\ell\gtrsim 1000$, the acoustic peaks are
extremely sensitive to the lensing phenomenon, the first hypothesis examined
was that the signature in the spectrum difference corresponds to an
underestimated lensing amplitude in the Planck best-fitting procedure. As
shown in the analyses of section 3.3 this hypothesis is indeed verified, where
we find a lack of lensing amplitude of around $20\%$ with respect to the
Planck APS fitted assuming the flat-$\Lambda$CDM model with $A_{L}=1$. As a
matter of fact, we find that there is an excess of signal in
$\delta^{obs}_{\ell}$ (see equation (3.1)) that is well explained by a
$\Lambda$CDM APS with a non-null lensing amplitude parameter $A_{lens}>0$,
with values in the interval $[0.10,0.29]$ at 68% CL. Moreover, we found
several scheme binnations that best-fits the spectrum difference
$\delta^{obs}_{\ell}$, with $\chi^{2}=1$, with lensing amplitudes
$A_{lens}\simeq 0.2$ (see tables 4 and 5). According to our likelihood
analyses, the synthetic APS produced for these best-fit procedures of
$\delta^{obs}_{\ell}$, with lensing amplitude $A_{lens}$ as a parameter, show
that $A_{lens}>0$, or equivalently $A_{L}>1$, with statistical significance of
$\sim 3\sigma$.

Additionally, we have also investigated a possible dependency of the spectrum
difference $\delta^{obs}_{\ell}$, in signature and intensity, on some
cosmological parameters associated to lensing amplitude, neutrino mass and
spatial curvature. Despite neutrino mass and spatial curvature can have impact
in lensing signal, the impact in $\delta^{obs}_{\ell}$ residual is weak and
cannot be enough to reproduce the excess in $A_{lens}$, in signature and
intensity. Our likelihood analyses for the examination of the cosmological
parameters $A_{lens}$, $\sum m_{\nu}$, and $\Omega_{k}$ are displayed in
figure 7.

Figure 7: One dimensional posterior distributions for $A_{lens}$, $\sum m_{\nu}$ and $\Omega_{k}$ for $\Delta\ell=17$ and $\Delta\ell=63$, as illustrative examples. The $\delta^{obs}_{\ell}$ signal is consistent with: $\sum m_{\nu}<0.47$eV, an almost flat spatial curvature, and $\sim 20\%$ of excess in $A_{lens}$. These results show the consistency of the different values that these parameters can assume to explain the spectrum difference $\delta^{obs}_{\ell}$. Table 6: Values $A_{lens}$, $\sum m_{\nu}$ and $\Omega_{k}$ at 1$\sigma$ for $\Delta\ell=17$ and $\Delta\ell=63$. The last column indicates the confidence level (CL) for having $A_{lens}>0$ in each binnation case. bin length |  $\\!A_{lens}$ | $\sum m_{\nu}$ | $\\!\Omega_{K}\\!$ | CL for $A_{lens}\\!>\\!0$
---|---|---|---|---
$\Delta\ell=17$ | $0.18^{+0.063}_{-0.063}$ |  $<0.47$ eV | $0.0029^{+0.009}_{-0.009}$ | 3$\sigma$
$\Delta\ell=63$ | $0.22^{+0.065}_{-0.065}$ |  $<0.45$ eV | $0.0013^{+0.0071}_{-0.0071}$ | 2.9$\sigma$

##  Appendix A Appendix: Robustness Tests

We perform a variety of robustness tests to support the validity of our
results. The first set of tests study the hypothesis of a possible bias in our
results coming from the fact that we used the best-fit Planck cosmological
parameters, displayed in table 1, ignoring their measured uncertainty. Because
this is a possible source of bias in the production of the synthetic CMB APS,
we repeat our analyses constructing the synthetic maps with cosmological
parameters randomly chosen from the $1\sigma$ confidence interval (table 2 of
[48]). The second set of tests investigate the possibility that a residual
noise in the CMB temperature maps could have some impact in the Planck APS
data possibly contributing to the excess of lensing amplitude found in our
analyses.

###  A.1 Variation of cosmological parameters

In this section we present the results of the analyses that include the
measured errors of the Planck cosmological parameters in the production of the
synthetic APS. To include the errors measured by the Planck in our tests, we
first generate 100 different points in the six-dimensional $\Lambda$CDM space
of parameters. Then, each point consists of a set of the six cosmological
parameters given in table 1, which were randomly generated following a normal
distribution within 1$\sigma$ interval (using the table 2 of [48]). For each
of the 100 sets of six parameters each, we construt 100 synthetic APS to
replace the set of $C^{\Lambda CDM}_{l}$ in equation (3.1), generating thus
their correspondig 100 $\\{\delta^{obs}_{\ell}\\}$ sets, then the same
analyses done in sec. 3.1 and sec. 3.2 are performed again.

The first set of analyses is meant to test the possible existence of a bias in
the acceptance/rejection of the null hypothesis $H_{0}$ in the relevant
multipole interval $[1100,2200]$ caused by the inclusion of the 1$\sigma$
errors measured by the Planck Collaboration. To do this, we apply the LB test
for each of 100 sets of $\\{\delta^{obs}_{\ell}\\}$ constructed as described
in the previous paragraph. The results show that, considering all the 100
times where $H_{0}$ was tested it is rejected. The distribution of the
$p-$values obtained has a mean $\simeq 9.9\times\sim 10^{-14}$ and a median
$\simeq 9.6\times 10^{-14}$, which means that even considering the 1$\sigma$
errors of the best-fit values of the table 1, the rejection of the null
hypothesis is robust, supporting our original result that the residual
$\\{\delta^{obs}_{\ell}\\}$ signal in the multipole interval $[1100,2200]$ is
not noise.

The second analyses in this subsection is to test whether our conclusion of a
lensing excess hidden in $\\{\delta^{obs}_{\ell}\\}$ signal is biased by the
inclusion of the best-fit error bars. To do that, we use equation (3.5) to
compute the likelihood of the 100 $\\{\delta^{obs}_{\ell}\\}$ sets. All
likelihoods present the same excess of lensing around $\sim 20\%$. In figure
(8) we present the outcoming curves from five illustratives cases: likelihood
of the best-fit value; likelihoods for the maximum and minimum $p-$value from
the LB test and three likelihoods randomly choosen from the set of 100
computed likelihoods. In all cases the maximum values are in agreement at
1$\sigma$. This result means that the lensing amplitude excess found in sec.
3.3 is robust, and the signal hidden in $\\{\delta^{obs}_{\ell}\\}$ is not
correlated, or at least, it is not strongly correlated with the parameters
given in table (1).

Figure 8: Illustrative example of the $\Delta\ell=17$, likelihoods for
different set of cosmological parameters. Shadowed region represents the 1
$\sigma$ interval for the best-fit case. The maximum values of the likelihoods
for the other sets of parameters are inside that region.

###  A.2 Noise maps

In this section we investigate the possibility that a residual noise in the
CMB temperature maps could have some impact in the Planck APS data possibly
modifiying the conclusion of lensing amplitude found in our analyses. To test
this we consider 20 simulated Planck noise maps for each of the two
frequencies: 857 GHz and 545 GHz to construct new simulated APS with noise,
$C^{Planck+noise}_{l}$, which replace the $C^{Planck}_{l}$ data in equation
(3.1) [3] 666https://pla.esac.esa.int. Thus the resulting
$\\{\delta^{obs+noise}_{\ell}\\}$ include the noise information and we apply
the LB test in several multipole intervals as shown in table 2. As described
in sec. 3.2, we now simulate 100 $\\{\delta^{obs+noise}_{\ell}\\}$ for each
noise map. Results are summarised in the table 7, where we show the good
agreement with the percentage values of rejection/acceptance previously found
in sec. (3.2) and displayed in table 2. The $p-$values in all analyzed
intervals were very small, in particular, for the interval $[1100,2200]$, the
$p-$value distribution obtained has a mean $\simeq 1.39\times 10^{-11}$ and a
median $\simeq 1.96\times 10^{-13}$ for both frequencies considered here. This
confirms that the porcentage values for rejection/acceptance of the $H_{0}$
hypothesis originally obtained are robust when the CMB maps noise are
considered.

Table 7: Statistical LB analyses to investigate the rejection or acceptance of the null hypothesis $H_{0}$ in the case that the CMB APS in study include the contribution of simulated noise from two Planck CMB frequency maps: $545$ GHz and $857$ GHz. As observed in this table, the results are the same as those obtained in table 2, showing that our original results are robust.  Multipole intervals | Reject $H_{0}$ | % for $545$ GHz | % for $857$ GHz
---|---|---|---
$\ell=[2,2500]$ | Yes | $100\%$ | $100\%$
$\ell=[2,800]$ | No | $65.8\%$ | $66\%$
$\ell=[2,1200]$ | No | $41.4\%$ | $42.2\%$
$\ell=[1100,2500]$ | Yes | $100\%$ | $100\%$
$\ell=[1100,2200]$ | Yes | $100\%$ | $100\%$

## Acknowledgements

We acknowledge the use of data from the Planck/ESA mission, downloaded from
the Planck Legacy Archive. RM and WHR thank CNPq, TWAS, and FAPES (PRONEM No
503/2020) for the fellowships and finantial support under which this work was
carried out. AB acknowledges a CNPq fellowship. We acknowledge that our work
made use of the CHE cluster, managed and funded by the COSMO/CBPF/MCTI, with
financial support from FINEP and FAPERJ, and operating at Javier Magnin
Computing Center/CBPF.

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[1]Thomas A. Witten

These authors contributed equally to this work.

1]James Franck Institute, University of Chicago, 929 E. 57th Street, Chicago,
60637, Illinois, USA

# Granular convergence as an iterated local map

Anna Movsheva<EMAIL_ADDRESS><EMAIL_ADDRESS>[

###### Abstract

Granular convergence is a property of a granular pack as it is repeatedly
sheared in a cyclic, quasistatic fashion, as the packing configuration changes
via discrete events. Under suitable conditions the set of microscopic
configurations encountered converges to a periodic sequence after sufficient
shear cycles. Prior work modeled this evolution as the iteration of a pre-
determined, random map from a set of discrete configurations into itself.
Iterating such a map from a random starting point leads to similar periodic
repetition. This work explores the effect of restricting the randomness of
such maps in order to account for the local nature of the discrete events.
$\bullet$The number of cycles needed for convergence shows similar statistical
behavior to that of numerical granular experiments. The number of cycles in a
repeating period also behaves qualitatively as seen in these granular studies.

##  1 Introduction

In the 1980’s soft matter science was largely confined to molecular structures
such as polymers. Polymers showed that qualitative rules of construction of a
many-body system could yield powerful predictions about their behavior.
Already in 1982 in the Pincus-Edwards workshop in Santa
Barbara[Edwards:1985dq] this same power began to be applied to assemblies of
macroscopic objects, such as granular packs[Edwards:1982vf].

$\bullet$_[AM: In this paragraph ”configuration” and ”cyclic shearing” is
first mentioned but without a good definition. The definition comes at a later
point. Maybe include a definition right here? Configuration is defined by the
positions of all the particles. Since the material is static there is no
momentum. And that is due to jamming.]_ We focus here on a remarkable form of
self-organization in a granular pack that we call “convergence,” revealed in
atomistic computer simulations[Fiocco:2014uq, Mungan:2019bc, Mungan:2021we,
Sastry:2021vk, Regev:2013fk, Regev:2015fk, Lavrentovich:2017rz]. One subjects
a dense, disordered _[AM: should we include that its also jammed?]_ packing of
spheres to repeated cycles of shear _[AM: first mention of cyclic shearing,
should we include a definition?]_ deformation while tracking the sequence of
microscopic configurations that occur. Initially, the sequence shows no
particular organization. But after a number of shear cycles, the sequence of
configurations _[AM: what sort of configuration do we have in mind?]_ repeats
exactly with every cycle, as evidenced in Fig 1b. A cycle may consist of
dozens of configurational shifts, selected from the vast number of possible
configurations. Any variations in the shearing cycles yield convergence to a
different set of configurations. Though the sequence may repeat after a single
shear cycle, it may also reach a state requiring two or more shear cycles in
order to return to its initial state.

Figure 1: Final slips and cycles in a sheared bead pack. a) Configuration of
simulated pack, after Ref. [Regev:2015fk][permission pending] showing a slip
event. b) Graphs of potential energy vs cycle number showing initial secular
motion followed by a periodic motion for three different shear amplitudes
$\Gamma$. The lowest amplitude in the top graph shows periodic behavior after
many cycles; a graph with a slightly larger amplitude requires a longer time
to converge. Small additional amplitude yields variability that never becomes
periodic. _[AM: Suggestion: instead of using a) and b), use left and right,
since the figures do not have a) and b) labels.]_

In this paper we aim for an improved understanding of this convergence process
using the concept of discrete maps defined in Ref. [Mungan:2019fk]. First, we
review the rationale for the discrete map picture.

$\bullet$Under cyclic annealing, one imposes shear strains e.g. by translating
the upper confining boundary of the pack alternately left and right, thus
cycling the shear strain $\gamma$ between bounds $-\Gamma$ and $\Gamma$. This
changing shear strain produces abrupt slip events such as that pictured in
Fig. 1a in which particle contacts form or break. Each slip event amounts to a
discrete change in the positional configuration $c$ of the pack. Given a
configuration $c_{i}$ just after a slip event, there is a continuous _[AM:
should we mention that its reversible?]_ change of $c$ as the shear gradually
increases, leading to an unstable state and a slip event to a new
configuration denoted $c_{j}$. The resulting $c_{j}$ depends on the initial
$c_{i}$. _[AM: The following sentence is very confusing to me. Should be
reworded.]_ Following Ref. [Mungan:2019fk], we postulate that this $c_{j}$ is
a fixed function $\mu(c_{i})$, defined for any encountered $c_{i}$ and fixed
for every point in the shear cycle, under increasing shear. A separate
function $\nu(c_{i})$ governs changes under decreasing shear. (In Ref.
[Mungan:2019fk] the functions $\mu$ and $\nu$ are denoted $U$ and $D$,
respectively.) The result of an entire cycle of shear than thus _[AM: ”than
thus”?]_ be determined by iterating the functions $\mu$ and $\nu$, as pictured
in Fig. 2. We denote the configuration obtained after an entire shear cycle by
$M(c_{i})$.

Each slip event produces a displacement field around the reconstructed
contacts _[AM: we haven’t talked about contact networks up until this point in
the paper, so this is a bit abrupt mention of contacts. and why are they
reconstructed?]_ and extends over the rest of the pack, as shown in Fig. 1a.
We shall assume that these displacements fall off rapidly with distance so
that they have negligible impact on subsequent slip events beyond a local
distance independent of system size.

_[AM: In this paragraph, the contents of the paper are discussed. Should we
perhaps be a bit more specific and refer to different sections and the summary
of their content?]_ In this paper, we extend the deterministic map picture
sketched above. We ask how it should be modified to take into account two
features of granular annealing: extensivity with system size and the locality
of the slip events. The next section addresses how local maps can be
compatible with the extensive behavior of the pack as a whole. Following this
section we define a crude implementation of local maps. The degree of locality
is controllable with a parameter. We report the scaling behavior of two
characteristic features: the number of cycles needed for convergence, and the
number of cycles making up the final multicycle repetition.

##  2 Extensivity and Locality

As noted above, granular packs in the regime of interest are extensive, so the
number of elementary slip events in a given interval of shear must be
proportional to the number of particles in the pack. _[AM: Why must they be
proportional? Is this a known result? Maybe we should reference a paper since
this doesn’t seem to be trivial to me.]_ Thus the map for two successive
elementary slip events is not local in the sense outlined above. To recover a
sense of locality one must introduce a notion of a correlation volume for a
given range of shear amplitudes. We are interested in full cycle maps
$M(c_{i})$ that repeat in one or more shear cycles, i.e. $M(c_{i})=c_{i}$ or
$M(M(c_{i}))=c_{i}$, etc.

_[AM: Here the idea of contacts is raised in more detail. I think we should
either talk more about this during the first mention of contacts or refer the
reader at the first mention of contacts to this section for further detail.]_
Above it was noted that an elementary slip event generally involves the making
or breaking of a small number of contacts and the strain field arising from it
falls off rapidly with distance. In both of these senses, the slip event may
be regarded as local. The volume $V_{o}$ affected is comparable to the volume
of a few particles. We shall neglect any causal effect between this slip
region and all others located outside of its volume during a shear cycle.
$\bullet$Still, the probability that another event will overlap the given one
during a shear cycle is nonzero.

This probability depends on the shear amplitude $\Gamma$. It can be expressed
in terms of the (intensive) density $\rho(\Gamma)$ of elementary events in a
shear cycle. This density becomes larger as $\Gamma$ $\bullet$increases. We
shall consider the regime where $\rho(\Gamma)\ll 1/V_{o}$, so that the $V_{o}$
region evolves independently of other regions; it is assumed to be autonomous.
This region’s subsequent change of state is thus dictated by elementary slip-
event maps $\mu(c)$ and $\nu(c)$. By iterating this map _[AM: instead of ”this
map”, which is ambiguous, say ”either of these maps”?]_ , one may infer the
effect of a given finite change of the imposed shear, including an entire
shear cycle, as pictured in Fig. 2. The result of $k$ shear cycles is then
given by the $k$-th iterate of $M$ from the initial state $c$, i.e.
$M(M(M(...M(c)....)))$. _[AM: is a slip really discontinuous? I thought it is
just not reversible.]_

Figure 2: Comparison of incremental maps $\mu(c)$ and $\nu(c)$ vs cycle maps
$M(c)$, as used in Ref. [Mungan:2019fk]. The lines show a possible motion of
some particle within the region as the shear increases from $-\Gamma$ to
$+\Gamma$ and back to $-\Gamma$. The motion is continuous _[AM: and
reversible?]_ within a configuration and discontinuous between configurations;
these are labeled $c_{1},\dots,c_{8}$ _[AM: denoted by thin lines]_. The first
elementary slip event from $c_{1}$ to $c_{2}$ is dictated by the elementary
map for increasing shear, i.e. $\mu(c_{1})$ shown by light-colored (red)
arrow, and similarly to configuration $c_{4}$ _[AM: iterated? ”similarly to
configuration $c_{4}$” is confusing phrasing]_. The elementary maps under
decreasing shear are controlled by dark-colored (blue) arrows. The cycle map
$M(c_{1})\equiv c_{8}$ is shown by a heavy black arrow.

In the numerical studies of Refs. [Lavrentovich:2017rz, Mungan:2019bc,
Regev:2013fk, Regev:2015fk], all slip events occurring throughout the
simulation are recorded. In our interpretation, this includes many autonomous
$V_{o}$ regions. Each cycle of shear alters the state $c$ of a region via its
own map $M(c)$. The system as a whole only reaches its final repeating state
when all of its autonomous regions have reached theirs. The number of shear
cycles needed for convergence is thus governed by the longest cycle in a
sample of the $M(c)$’s in the system. This feature will be important in the
discussion below.

##  3 Maps and graphs

The scheme above can be used for any map from a set of configurations to
itself. _[AM: The previous sentence is confusing to me.]_ A number of
varieties were explored in Ref. [Mungan:2019fk]. It is convenient to represent
a given map by its directed graph, in which the configurations $c$ are the
nodes and the image of a given configuration and the map from $c$ to its image
$c^{\prime}\equiv M(c)$ is represented by an arrow from $c$ to $c^{\prime}$.
Thus the graph has exactly one directed edge from each of its nodes.

Figure 3: Graph of a random map on 32 nodes. The node positions are arranged
to show connectivity clearly. Small arrows indicate the map from each node to
its image. This map has three components. The path from the leaf node at left
to its successive images is shown in color (red). This path terminates in a
repetition of two nodes. The middle component has a final repetition of four
nodes. The right-hand component has a fixed point, shown as an arrow from a
node to itself.

The properties of generic graphs are described in Ref. [Flajolet:1990fk].
Several of these general properties will be useful below. First, a general
graph consists of disjoint, unconnected subgraphs called components, as shown
in Fig. 3. Within each component there are generally nodes that are not images
of any others; their only connection is from the node to its image. These
nodes are the leaves of the component. All the nodes in a component can be
traversed by iterating from each of its leaves. During iteration from a leaf,
each new image may be a node not previously encountered. However, at some
stage, a node is reached that was encountered. Further iterations repeat the
sequence of nodes already encountered, thus forming a loop. The sequence of
nodes from a given node to the loop is called the tail of that node. Any two
leaves with different loops are necessarily disjoint and must belong to
separate components. Thus every component has exactly one loop, which may be a
single node.

Figure 4: Example graphs of local maps. The black dots are nodes. The 1024
nodes are placed randomly in a square with periodic boundaries. Colored arrows
indicate the image node for every node. A different color is chosen randomly
for every component. Black arrows indicate the loop for each component. Single
node loops are not shown. Gray disk indicates the maximum image distance
$d_{\text{max}}$. a) Small $d_{\text{max}}$. b) Larger $d_{\text{max}}$

##  4 Minimal local map

Figure 5: Sampled iterates of randomly-selected nodes with local maps cf. Sec.
4 . (Top left) Graph for a full map with $m=32.169$. A blue circle around a
point means the point has a directed edge to itself. Each point only has one
edge leaving it. (Two top rows) Sample graph iterates of random initial nodes
in maps with $m=32.169$ _[AM: Maybe “sampled map iterates of random initial
nodes with $m=32.169$”]_. Light-colored (red) arrows indicate the terminating
loop. (Bottom row) Sample graphs for $m=804.248$.

_[AM: I feel like the minimal local map needs a little motivation. Why did we
choose it? -¿ We want to see if a simple way of building graphs can capture
the main statistics of cyclic shearing maps $M(M(M\dots(c)\dots))$\- the
convergence time and length of converged repeated sequences.]_ Here we define
a further type of map that aims to implement the property of locality defined
in the Introduction. That is, for each configuration $c$, the range of
possible images $c^{\prime}$ is strongly limited. The notion of locality
requires a notion of distance; accordingly, we view our nodes, i.e. the
configurations $c$, as occupying points in a space. For most of our work, we
have used a two-dimensional space. $\bullet$This hypothetical dimension should
not be confused with the dimensionality of the pack, cf. Sec. 6.1.

We define our maps such that the distance from a node to its image must be
less than a limiting distance denoted $d_{\text{max}}$. A simple
implementation of such a map is to place the node points at random in the
space and then choose the image of each point by selecting the available
points within distance $d_{\text{max}}$ at random. By making this selection
for every node point in the space, we have defined a local map, as shown in
Fig. 4. Given an explicit map graph, one may easily determine successive
images by following the arrows of the graph. As with the generic graphs
described above, these graphs have components, tails, loops, and leaves.

The structure of these graphs depends on the limiting distance
$d_{\text{max}}$. The essential effect of $d_{\text{max}}$ is to alter the
number $m$ of potential images. The average $m$ is evidently the density of
points $\rho$ times the area within $d_{\text{max}}$:

$m\equiv\pi\rho~{}d_{\text{max}}^{2}.$ (1)

Henceforth we will use this “range parameter” $m$ as our measure of decreasing
locality.

###  4.1 Implementation

####  4.1.1 Tail and loop statistics

As noted in Sec. 3, the properties of greatest relevance for granular packs
are the lengths of the tails and loops as a function of our range parameter
$m$. _[AM: Sec. 3 the relevance is not fully explained. Somewhere need to
explain in plain text: tail = convergence time, loop = repeated sequence of
shearing cycles]_ These features are especially easy to sample in for the
local maps defined above. We wish to know the distribution of tail and loop
lengths for graphs containing randomly-chosen nodes.

We construct the maps using the object constructs in the Python language. Our
typical simulations use a density of $2^{10}$ i.e. 1024\. Each node is an
object containing its x-y position and a pointer to its eventual image node.

Once the node positions have been assigned, we may measure tail and loop
lengths without constructing the entire map. We first make a list of the node
objects with randomly-chosen x-y point coordinates. Next, for each node, we
compile a neighbor list of the nodes within distance $d_{\text{max}}$ of that
node. This list includes the node itself. Then for each node, we choose an
image node from its neighbor list at random. We set the pointer in the node to
its image node. To construct a sample of tail lengths, we choose a node at
random and we follow its pointer to the next node again and again, marking
each node as “visited”. As we follow the pointers to the next node and the
next node, we count the number of nodes we have visited. This counting and
traversing of nodes end once we come across a node that has already been
marked as “visited”. Thus the prior node was the beginning of a loop, and all
further iterations repeatedly traverse this loop. The nodes we have traversed
so far are the tail-plus-loop for the given starting node. The tail is the
sequence from the starting node to the node that began the loop. The remaining
nodes in the tail-plus-loop form the loop. We then store the lengths of the
tail, $\tau$, and the loop, $T$, as one sample of the desired distribution. We
repeat this process to obtain the entire distribution. Fig. 5 shows typical
samples.

####  4.1.2 Component statistics

To find a component of a graph, we must identify all the nodes of each
component. For our needs, it suffices to determine the average number of nodes
in a component weighted by per-node sampling. Given an arbitrary node $c_{i}$,
we can determine the number of nodes $C_{i}$ in its component by generalizing
the procedure above for identifying the tail and loop for a given node.

Before connecting the nodes, all the nodes start off being leaves.
Accordingly, we may set a logical “is_leaf” variable in each node indicating
that it is a leaf. As each node is being connected to another node, the
“is_leaf” variable of its image node is cleared. Once the full map is
constructed, its full set of leaves is established.

Unique components are fully disjoint so they do not share any nodes; in other
words, each node will belong to only one component. To determine the nodes of
a component, we begin by assigning a unique index to each leaf. We then
iterate the map from each leaf, labeling each node encountered with the leaf
index. When such a node already has a leaf index (because it is the descendent
of a previously-iterated leaf), the current leaf index is changed to match the
previously assigned one. Its descendants are also relabeled. Any two leaves
that belong to the same component will eventually acquire a common index, and
so will the descendants of these leaves.

When all the leaves of the map have been processed in this way, all nodes in a
given component will have the same index, and each component will have a
unique index111Any component consisting entirely of a loop will not be counted
in this way, since such a component has no leaves. We treat these in Appendix
A To find the size $C_{i}$ of a particular component, we find and count all
the nodes with its index. The average of all the $C_{i}$ sampled in this way
is denoted $C$.

In a map with many configurations, it is not practical or necessary to
determine all the components. Instead, we create many maps and study a small
class of components with nodes near the center of the sample box. For small
$m$ these components thus found are concentrated near this initial node, with
no nodes near the boundaries.

For larger $m$ such that the identified nodes extend beyond the middle quarter
of the box, it is possible that the component would change if the box were
larger. When such components are encountered, we create new nodes outside of
the box, adding a new box adjacent to the component being traced. We then
construct the map images for these new nodes. _[AM: We extend the box in all
directions filling it with the same density of randomly placed points.]_ and
determine the leaves and component indices for these nodes as above. We
continue this process until the new nodes have no effect on the component
under study. We provide further details in Appendix B.

##  5 Scaling of convergence properties

In this section we report the effects of our range parameter $m$; this is our
proxy for the shear amplitude $\Gamma$. This proxy is plausible, since
increasing the shear amplitude increases the number of ways an instability can
occur in an autonomous region, and thence the number of nodes needed to
represent it. Average tail and loop lengths increase with $m$ corresponding to
the increased number of shear cycles for annealing. For fixed $m$ these
lengths remain finite for arbitrarily large map sizes. One readily obtains
tail lengths that are numerically similar to the number of cycles to
convergence observed in particle-scale simulations[Regev:2015fk,
Lavrentovich:2017rz].

Another aspect to be compared to simulations is the ratio of loop to tail
lengths. For random maps, these are equal up to a factor of order
unity[Flajolet:1990fk]. By contrast, the granular simulations have loop
lengths many times larger than their cycle lengths. _[AM: this is vague. by
how much? is the relation constant or more]_

_[AM: The next two paragraphs seem a bit too brief. I think they should either
be integrated into next subsection or got rid of. ]_

Considering the distributions of tail and loop lengths, these have a minimal
structure; the lengths _[AM: the count of lengths?]_ fall off exponentially at
a rate consistent with their averages. This agrees well with the simulation
results of Ref. [Regev:2015fk]._[AM: reference fig 9. describe more.]_

We also considered a less model-dependent measure of locality, namely the
average component size $C(m)$ produced by the map. This parameter allows one
to make a correspondence between random maps with finite numbers of nodes and
local maps with an infinite number. _[AM: what is the conclusion?]_

Figure 6: Average loop and tail size of graphs, for a given range parameter
$m$ (the average number of candidate image points for each point). The
underlying parameter is $d_{\text{max}}$.  Figure 7: The orange line is the
average tail length given the average component size for graphs made using
local maps. The blue line is a square root best fit. The green line is for the
random map graphs [Flajolet:1990fk]. The behavior of tails for the local map
and random map seems to be very similar. Figure 8: The orange line is the
average loop length given the average component size for graphs made using
local maps. The blue line is a square root fit. The green line is the average
loop length for a given average component size in random maps. The local map
greatly diminishes the loop size compared to the random map, though the
general shape of the graph stays the same.

###  5.1 Average tail and loop lengths

Fig. 6 compares average tail and loop lengths, indicating a sublinear scaling
with $m$. When the average component size $C$ is used as a parameter, there is
a close correspondence between tail lengths of our local maps and that of
random maps [Flajolet:1990fk], as shown in Fig. 7. Remarkably, our measured
average for the local maps is equal within our uncertainty to the
corresponding average for random maps. For the random maps, this average
scales as $\sqrt{C}$ [Flajolet:1990fk]; this suggests that local maps have the
same scaling. As for the loops, they also show a close correspondence as shown
in Fig. 8. However, here the loop length is 2-3 times smaller than that of the
random maps. This implies that the asymptotic ratio of loop to tail length is
substantially smaller in local vs random maps, but only by a finite factor.

Figure 9: Graphs of energy differences $R(n)$ as defined in Ref.
[Regev:2015fk][permission pending] versus cycle number $n$. Curves correspond
to strain amplitudes $\Gamma=$ 0.06, 0.07, 0.075, 0.085, 0.088, 0.09, 0.093,
0.095 from left to right. Inset shows graphs from strain amplitudes $\Gamma=$
0.12 (blue) and $\Gamma=$ 0.15 (green), for which there was never a limiting
cycle achieved. See also Ref. [Lavrentovich:2017rz] for similar findings.
Figure 10: Tail length frequency count for 5000 generated local maps. The
graph is well-fitted with an exponential decay function.

###  5.2 Tail length distribution

In numerical annealing experiments, the activity during a cycle decreases to
zero as the cycles proceed. Regev et al._[AM: ref number?]_ monitored this
decrease by measuring the net change of potential energy in a shear cycle.
This $R(n)$ decreases approximately exponentially with increasing cycle number
$n$, with a slower decay at larger shear amplitudes, as shown in Fig. 9.

In our map picture, a large sample contains many autonomous regions each
traversing its tail to arrive at a loop. The number of active regions
producing displacements at the $n$th cycle is thus the number of tails of
length greater than $n$. Thus the activity $R(n)$ _[AM: activity R(n)? I
understand the idea but I think this could be expressed a bit more clearly.]_
corresponds to the number of tails with a length greater than $n$. Fig. 10
shows that this tail length distribution is consistent with exponential decay.
The width of the exponential is determined by the average tail length,
determined above. This indicates a good consistency between the Regev et
al.granular behavior and the present map picture.

##  6 Discussion

The preliminary exploration described above suggests that the convergent
annealing of granular packs may be usefully treated as an instance of dynamics
on discrete sets. The best-known examples of this dynamics are cellular
automatons [wolfram1984cellular], wherein the states are spatial arrays of
binary degrees of freedom. In this work, we explore the predictive
consequences of this type of dynamics for granular annealing. Specifically, we
aim to understand the collapse of the large set of initial states into a much
smaller set of final states, steerable by the annealing protocol.

The primitive constraint of locality studied above allows an improved
correspondence between the discrete map dynamics and the observed granular
annealing dynamics of Refs. [Regev:2015fk, Lavrentovich:2017rz]. It creates
convergence in a number of cycles that is independent of system size, unlike
unrestricted maps. It also resembles the progression toward the final cycle
seen in these annealing simulations. However it fails to account for a
striking feature of annealing: the dominance of final states with the same
period as the shear cycle, i.e. loops with a single node. Further, our model
shows no counterpart of the observed transition from convergence to chaos,
$\bullet$i.e. the behavior shown in Fig. 1. These indicate that our primitive
implementation of locality does not capture the essential elements of the
annealing process.

Below we consider further qualitative aspects of map properties that might
account for these unexplained features.

###  6.1 Implementing locality

Our rationale for considering locality was the observed locality of slip
events near specific positions in the pack. Yet in our model, the objects with
definite positions were not the events. Rather, they were the nodes, i.e. the
configurations of the system. This notion of locality has no clear
justification from the granular behavior. The direct counterpart of the
granular locality would be the links of the map. Thus it would be appropriate
to assign the edges of the map graph to definite positions, and to restrict
the maps based on this locality. We did not see a way to do this.

Even granting the notion of localized nodes, there is no obvious reason that
the distance measure should be that of a two-dimensional space, as we have
assumed. We explored the effect of placing our nodes in three or four
dimensions. The resulting maps could not be made large enough to show any
useful scaling behavior.

###  6.2 Relation to hysteron-based models

The minimalist approach used here complements hysteron models that restrict
the nature of the elementary slip events. Though we do not specify the events,
hysteron events are consistent with our locality picture. The physical slip
events certainly influence each other as explored e.g. in [Edwards:1985dq,
Keim:2021aa, Lindeman:2021aa]. We have given no account of this influence in
our model. This mutual influence could well be the key to explaining the
transition to chaos and to characterizing the final states. This would imply
that our model misses the point.

## Conclusion

The rich and controllable self-organization seen in cycling annealing arises
in part from the qualitative features of discrete, deterministic dynamics. The
principles underlying such self organization surely apply much more broadly.
The present exploratory study suggests a pathway towards identifying these
principles.

Acknowledgments TW is grateful to Sidney Nagel and the participants at the
2022 Annual Meeting of the Simons Collaboration on Cracking the Glass
transition [Nagel:2022aa] for helpful discussions early in this work. AM
acknowledges partial support from the University of Chicago Materials Research
Science and Engineering Center, which is funded by the National Science
Foundation under award number DMR-2011854.

Declarations

  * •

The authors have no relevant financial or non-financial interests to disclose.

  * •

The authors contributed equally to the work. AM co-developed the research
plan, devised and executed all the numerical work, identified relevant
references, prepared the figures and co-wrote the article. TW formulated the
model to be studied, suggested variations and quantities to measure, and co-
wrote the article.

  * •

Data availability statement

The data supporting the findings of this study are available within the paper,
and in the archive at [Movsheva2023Paper-archive]

## Appendix

Here we describe further details used in generating and measuring our maps.

##  Appendix A Finding components with no leaves

To complete the component-finding procedure described in Sec. 4.1.2 we
describe our procedure for identifying components with no leaves, consisting
entirely of a loop. To find these leafless components, we have to check if any
nodes in the full list of nodes haven’t been identified. Any found
unidentified node in this list would be a part of a loop full of other
unidentified nodes. As before, we give this node a unique identifier, and
using its pointer we iterate through the loop, giving each node the same
identifier as the first node in the loop. To make sure not to get stuck in
this loop when following the pointers, we must check that the next node hasn’t
been already identified. If it has, it means we have made a full circle in the
loop. After this, we continue going down the list of all the nodes checking
for unidentified nodes. Since the nodes are objects, the list gets dynamically
updated as we follow a node’s pointer and update any nodes in its path; thus
we don’t have to update the list itself.

##  Appendix B Finding nodes of large components

If a map is confined to a nonperiodic box, a graph that extends to the edge,
in general, has missing nodes that would have been included if the initial box
had been enlarged. Here we describe our procedure for eliminating this
distortion in our local maps.

We generated nodes with coordinates within a limited area for practicality, so
we have to be careful of what happens on the edges of this area. When we build
an iterative directed map (starting from a random node and composed only of
tail and loop) its nodes could happen to be next to the boundary of the area
populated by nodes. Without any alterations, the directed graph’s node
proximate to the boundary would be prohibited from sampling potential image
nodes on the other side of the boundary and be forced to select its image
confined to the area where the nodes have been pre-generated. We are
interested in the loop and tail size statistics for such graphs so this effect
from the boundary could bias these statistics by prematurely forcing the
directed graph to go back to the area where it has already been and have a
higher chance of closing and forming a loop. To fix this, whenever during the
building of a graph a node comes close enough to the boundary, we generate
nodes with coordinates in that direction at the fixed density to match the
previously generated node coordinates.

In order to find components unhindered by the boundaries of this space, we
generate nodes with coordinates filling a large enough area (a square) at a
fixed average density. Once all the image nodes for all the nodes are found,
the component that is at the center of the area should not go close enough to
the boundary, lest the nodes could sample outside the boundary (the nodes are
within $d_{\text{max}}$ of the boundary). If the component’s nodes do come too
close to the boundary, to remedy this we just generated nodes for an even
larger area and rebuild the maps. This precaution is so that during the
building of the maps, the boundary does not deprive the component of growing
beyond the set area of generated coordinates of the nodes.