# Conformal Cyclic Cosmology Signatures and Anomalies of the CMB Sky

Eve Bodnia  Vlad Isenbaev  Kellan Colburn  Joe Swearngin  Dirk Bouwmeester

###### Abstract

Circles of low-variance and Hawking points in the Cosmic Microwave Background
(CMB), resulting from black hole mergers and black hole evaporation,
respectively, in a previous cycle of the universe, have been predicted as
possible evidence for the Conformal Cyclic Cosmology model (CCC) introduced by
R. Penrose. We present a high-resolution search for such low-variance circles
in the Planck and WMAP CMB data, and introduce HawkingNet, our machine
learning open-source software based on a ResNet18 algorithm, to search for
Hawking points in the CMB. We find that CMB anomalies, consisting of a single
or a few bright pixels, erroneously lead to regions with many low-variance
circles when applying the search criteria used in previous works [1]. After
removing the anomalies from the data no statistically significant low-variance
circles can be found. Concerning Hawking points, also no statistically
significant evidence is found when using a Gaussian temperature amplitude
model over $\sim 1^{\circ}$ opening angle and after accounting for CMB
anomalies. That CMB anomalies themselves might be remnants of Hawking points
is not supported by low-variance and/or high-temperature circles around them.
The absence of such distinct features in the CMB does not disprove CCC because
if the density of such circles and Hawking points is large an interference
speckle pattern in the CMB might arise instead. This would lead to non-
Gaussian fluctuation in the CMB, a feature not unique to CCC. We do observe
significant local deviation of the real CMB sky from Gaussian noise on a local
scale when using best-fit $\Lambda CDM$ and COM PowerSpect CBM TT-full power
spectra.

.

##  1 Introduction

The Conformal Cyclic Cosmology model (CCC) introduced by R. Penrose [2, 3],
and inspired by work of P. Tod [4, 5], led to the qualitative prediction of
concentric low-variance circular rings in the cosmic microwave background
(CMB) resulting from super-massive black-hole mergers in the previous aeon -
former version of our current Universe [1]. Furthermore, Hawking evaporation
of black holes, the expected final fate of isolated super-massive black holes
in the previous aeon, are predicted to give rise to bright spots in the CMB,
referred to as Hawking points. Qualitative arguments predict that Hawing
points are surrounded by concentric low-variance circular rings that are
somewhat higher in temperature than the average CMB temperature, and that
concentric low-variance rings without a central Hawking point should be
somewhat lower in temperature [1].

V. Gurzadyan and R. Penrose (GP) reported that they found numerous concentric
families of the low-variance rings in the Wilkinson Microwave Anisotropy Probe
(WMAP)- the release of 7 years CMB data [1, 23]. They concluded that these
circles are statistically relevant structures in the CMB data, rather than
resulting from Gaussian random noise, thus providing evidence of conformal
cyclic cosmology [6]. They compared the numbers of the ring families with the
corresponding numbers of the elliptically distorted rings in the CMB, which is
referred in their text as the "sky-twist" procedure. In the CMB data, GP
examined 10,885 circular patches in the sky of equal number of pixels,
systematically covering the sky but excluding the region of our galactic disk.
The temperature variance, i.e. the standard deviation $\sigma$ in the
temperature, was calculated over circular rings of annular width 0.5 degree
and of angular radii in the range 2.5-16 degrees in steps of 0.5 degree. If
among those rings around a given centre, the temperature variance was at least
15 $\mu K$ below the average variance for that specific centre, the ring was
considered "low variance". Many instances of 3 and also of 4 concentric
circles where identified in Ref. [1], seemingly providing support to the CCC
hypothesis.

At this point we emphasize the decisive role of the average of the variance of
the rings with annular width 0.5 degree and of angular radii in the range
2.5-16 degrees in steps of 0.5 degree, which we will refer to as the baseline
variance, in determining which circles are considered to be of low variance.
We note that if for some reason this baseline variance is unexpectedly high,
which can happen when there is an unusually high or low temperature pixel in
the data in the region from 2.5 to 16 degrees, there will be low-variance
circles identified that would not be there without this one bright or dark
pixel. Those erroneously identified circles will come in clusters, leading to
the possibility of multiple concentric low-variance circles. Such unusually
bright or dark pixels, or perhaps a few such pixels together, are known to
reside within the CMB and are referred to as CMB anomalies. Such anomalies
might be the result of the extensive data processing, or of inaccuracies in
the measuring system, or perhaps represent real, but yet unknown, astronomical
phenomena.

The occurrence of low-varience circles in the same WMAP data was reexamined in
Refs. [7] and [8]. The authors did confirmed the concentric circles in the CMB
data, and found that the WMAP signal is indeed dominated by a cosmological
signal rather than by instrumental noise. However, they claimed that such
variations are the result of usual CMB anisotropies and therefore
statistically expected. We would like to note that the technical approach
towards the problem by these groups is drastically different from the one
taken by GP. GP’s way of simulating data is constructing the map of the
Gaussian sky injected with the WMAP’s noise, in contrast to the groups of [7,
8] who simulate the data using the CMB power spectrum plus the WMAP noise for
Lambda cold dark matter model ($\Lambda CDM$). Penrose and Gurzadyan argued
that the treatment in Ref. [7] of the CMB signal as random Gaussian noise
seems to be invalid, due to Kolmogorov’s theorem and von Neumann’s method [9].

The search for low variance circles was also repeated in Ref.[10] on the same
WMAP data set and on the newer Planck data. The authors also confirmed the
existence of the special directions claimed by [1]; however, they consider
those consistent with the predictions of the standard cosmological model. For
their analysis, the authors used the same parameter set from [1] on the WMAP
W-band smoothed with a 20 arcminute full-width at half-maximum (FWHM) Gaussian
beam data set and the newer Planck COMMANDER data. The COMMANDER data were
downgraded from 2048 pixels to 512 and also smoothed with a 20 arcminute FWHM
Gaussian beam to save computation time. The simulations were created by the
Planck best-fit $\Lambda CDM$ angular power spectra generated for 2048 pixels
and also downgraded to 512 pixels and smoothed out with the same Gaussian beam
[18]. The authors discovered that the distribution of low variance rings on
the hemispheres is prevalent in the COMMANDER data, but concluded the
asymmetry to be statistically insignificant, given that nearly $25\%$ of their
simulations produce a distribution of low variance directions with a higher
dipole asymmetry than seen in the data. The origin of the CMB dipole asymmetry
evidence based on WMAP and Planck data has been questioned [11], and possible
explanations are ranging from inhomogeneous noise distributions at the
considered scales to the exotic dark matter theories and cosmic strings [12,
11, 13, 14].

The predicted Hawking points, representing the Hawking radiation resulting
from massive black hole evaporation in the previous aeon, have been studied by
[16, 15, 17]. In a preceding aeon, all of the mass in that former Universe
eventually falls into supermassive black holes, which then evaporate via
Hawking radiation. At its late life stage, the eon undergoes enormous
conformal compressing, forcing the remnant Hawking radiation to become Hawking
points at the crossover surface. The Hawking points are predicted to show up
in the CMB sky in the form of a spatially Gaussian signal spread over a small
data patch with an amplitude of the order of 100 mK [15].

In summary, there is significant controversy about the claims in support of
the CCC model, and a lot of relevant CMB data has been smoothed out in order
to facilitate the computational intensive tasks of searching for signatures of
CCC in the CMB. We noted above a potential pitfall, which seems to have been
overlooked so far, for statistical analyses set by CMB anomalies in the form
of a few unusual bright or dark pixels.

In this article we take advantage of a supercomputer and machine learning
approaches to firstly try to reproduce the analysis by [1] on WMAP data and
the analysis of [10], but with 500,000 centers in the sky, instead of 10,885
centers. We find that CMB anomalies play a decisive role, and we repeat the
search for low-variance circles after removing the small collection of CMB
anomalies. Secondly, we search for the Hawking points on higher resolution
Planck data sets using different power spectra and a novel ResNet-based
machine learning (ML) tool on over 50 million centers in the sky [19, 24].

##  2 The GP method with high resolution

We repeated the search method of Gurzadyan and Penrose [1], and for this it is
essential to establish an appropriate "baseline" temperature variance for a
certain region in the sky around which the variance of ring-like structures is
to be determined. We performed the baseline calculations and searched for the
low variance circles across both hemispheres using 500,000 centers in WMAP and
Planck COMMANDER data.

Figure 1 presents the baseline variance underlying the GP analysis, of the
WMAP data [1]. Figure 2, shows the baseline variance in Planck COMMANDER data.
Both figures show pronounced higher baseline variance regions with sharp
circular boundaries. Note that the scale bars indicate changes in the baseline
variance by up to $\sim 30\mu K$.

Figure 1: The baseline calculation for the Gurzadyan and Penrose search method
on the WMAP data [1] Figure 2: The baseline in Planck COMMANDER data

The origin of those regions turns out to be the presence of CMB anomalies,
tiny regions of 1-2 pixels diameter with a significantly higher temperatures,
combined with the method of determining the baseline. Figure 3 shows the
anomaly that gives rise to the most pronounced ring feature of relatively high
baseline variance.

Figure 3: One of the anomalously bright spot in the WMAP data that
significantly raises the baseline temperature in the surrounding region.
Comparing to the mean CMB temperature of approximately 2.725K, the temperature
of the anomalous spot is 2.7294K, about 400 to 500 $\mu$K above the CMB mean!

The consequence of this is that one will find many low-variance ($15\mu K$ or
lower, compared to the baseline) circles that are not real low-variance
circles. Those rings will all be clustered around the anomalies, and therefore
can also lead to the appearance of families of nearly concentric circle. The
smoothing of date in previous works has masked this important feature.

The orange curve in Fig. 4 shows our data on finding low-variance circles of
different radii with the anomalies still in the CMB data.

Figure 4: Search for low variance circles in Planck COMMANDER data using the
GP method. The orange trend correspond to the data including the anomalies as
done by [1]. The green line is the same analysis as in [1] but with removed
anomalies. Finally, the blue line are 500 simulations with the error bars
based on histograms. The statistically significant excess disappears as soon
as the data set is corrected for CMB anomalies.

The blue curve shows our results from simulations based on random (Gaussian)
fluctuations with the appropriate power spectrum of the CMB. One is inclined
to conclude that there is a statistically significant excess of low variance
rings which will be found in clusters in the sky, consistent with [1]. However
upon excluding the CMB anomalies, identified by pixels with high temperature
$\geq 2.7294K$, leading to the green curve in Fig. 4, the statistical
significance of low-variance rings vanishes. In other words, the number of
low-variance circles in the real data is approximately consistent with the
number of circles to be found in random (Gaussian) noise. There is also no
longer any statistically significant evidence for a preference for concentric
low-variance circles.

##  3 A Machine Learning Search of Hawking Points

We now turn to the investigation of possible Hawking points in the CMB, as
predicted by CCC. This is a type of problem, searching for a well-defined
feature in a large and noisy data set, that is ideally suited for a machine
learning approach. We build upon experience in machine learning approaches to
reinforcement learning [21]. Using a GPU 40 TFlops supercomputer and our
adopted machine learning software HawkingNet (see Appendix A), we first
trained a neural network in distinguishing between the presence and absence of
artificially introduced Hawking points at the center of small patches in the
sky, and subsequently performed this task on 50 million such centers in the
real CMB sky.

We implement two kinds of training simulations for the CMB sky: one based on a
Gaussian noise model and another on a Pseudo-synthetic model that uses real
CMB data before adding artificial Hawking points. The common feature of the
two models is that every considered center is enclosed in a $4\times 4$
degrees patch in the sky - a matrix of the CMB temperature of $128\times 128$
elements, which approximately corresponds to the Planck data resolutions. For
every patch, we subtract the mean temperature in order to avoid temperature
bias between all the patches used in our analysis. Here we do have to stay
alert to the pitfall of CMB anomalies in a certain patch! Then the Hawking
point simulation signature suggested in Ref. [15] is added and the neural
network is trained to optimally identify such points. The signature is an
added spatially Gaussian-shaped signal with an amplitudes varying from 100 to
400$\mu$K and a width ranging from 0.7 to 1.2 degree. After the training, the
neural network searched for Hawking points in the real data as well as in CMB
simulation data without added Hawking points for comparison.

The Gaussian simulation method: In our Gaussian simulation method, 50 million
centers with surrounding patches are assumed to follow a Gaussian global
spatial noise distribution and were generated by the HEALPix library which
ensures the spatial uniformity for all the points across the sphere [20].
HEALPix allows to input the CMB power spectra and generates the (Gaussian)
random temperature fluctuations data according to these spectra. We used the
best-fit $\Lambda$CDM CMB power spectra from the baseline Planck
TT,TE,EE+lowE+lensing and "COM PowerSpect CMB-TT-full R30" - empirically
constructed spectra from the real data without any assumption about the CMB
model [22]. Note that we use Gaussian profiles for both the spatial shape of
the Hawking points and for the noise model. After the training of the neural
network both the real CMB and simulated CMB data (without added artificial
Hawking points) are subject to a Hawking point search and the results are
shown in Fig. 5

(a)

(b)

Figure 5: The Hawking points hypothesis tested on the real data from SMICA
Planck and the Gaussian simulations using ResNet18 ML architecture for 200
$\mu$K added Gaussian amplitude. (a) Blue represents the real data, and orange
the Gaussian simulations without Hawking points generated using HEALPix and
the best-fit $\Lambda$CDM power spectrum. (b) Blue represents the real data,
and orange the Gaussian simulations without Hawking points generated using
HEALPix and "COM PowerSpect CMB-TT-full R301" power spectrum. The
classification score is a measure of how likely it is that Hawking points are
present in patches.

Figure 5 (a) shows two probability density distributions, the real data from
SMICA Planck (blue) and the simulations (orange) using the Gaussian method
with the best-fit $\Lambda$CDM CMB power spectrum. Figure 5 (b) shows the
density distributions for the real data from SMICA Planck (blue) and the
simulations (orange) with the "COM PowerSpect CMB-TT-full R30" power spectrum.
The classification score is a number assigned as the result of the MLE
regression by the ML network indicating presence of the Hawking points
signatures in a scanned patch (see Appendix A); negative/positive values
correspond to the likelihood of the absence/presence of Hawking-point-like
features.

What should be expected if this method finds statistically significant Hawking
points in a reliable way in the real data? First, one should expect a
contribution from statistically insignificant Hawking points ("fake" Hawking
points resulting from noise) to overlap with the simulation data (without
added Hawking points), and second one should expect the real data to contain
significant contributions at classification scores extending on the positive
side of the simulation data. The higher the classification score the higher
the confidence of the network in identifying Hawking points in specific
patches in the sky.

From Fig. 5 (a) we see that the SMICA Planck CMB data is locally quite
different from Gaussian noise with the best-fit $\Lambda$CDM CMB power
spectrum. The blue histogram represents the real data and it falls almost
completely in the range of negative classification scores, implying that
hardly any Hawking-point-like features have been identified, not even
statistically insignificant Hawking points (resulting from noise as
represented by the orange histogram). This observation suggest that the SMICA
Planck CMB data is locally smoother than what follows from the $\Lambda$CDM
model. The absence of proper overlap between the simulations and the bulk of
the real data implies that the network was trained on simulation data that was
not faithful to the real CMB noise spectrum in the 4 $\times$ 4 degree
patches. In such cases one should be suspicious about the validity of the
findings of the machine learning search algorithm and better training data is
required.

In Fig. 5 (b), we see a better, but still far from ideal, overlap of the real
data and simulation data, based on training of the neural network with the
model-free "COM PowerSpect CMB-TT-full R30" power spectrum. Having gained some
confidence in this neural network in identifying real Hawking points, which
should have classification scores with positive values above the values in the
simulations, we zoom in on the region of high classification scores, as shown
in Fig. 6. We do indeed observe contributions with high classification scores
in the real SMICA Planck data. We also used the same neural network to analyse
the Planck COMMANDER data as show in Fig.7. Figure 7 (a) shows a reasonable
overlap between the simulations and real data and Fig.7 (b) shows significant
contributions, much more pronounced than in the Planck SMICA data, in the real
data at high classification scores.

Figure 6: The zoomed-in outliers from Fig. 5 (b) which were classified as
Hawking points on SMICA Planck data with the ML network trained on Gaussian
(COM PowerSpect CMB-TT-full R30) simulation data for identifying Hawking
points with a 200 $\mu$K amplitude Gaussian signal.

(a) The classification histogram and outlier tail.

(b) The zoomed in tail of the distribution from (a).

Figure 7: The Hawking points hypothesis tested on Planck COMMANDER data using
the Gaussian simulation method in the HawkingNet for 200 $\mu$K amplitude.

Having learned to be extremely cautious in drawing scientific conclusions from
searchers for features in the CMB we subsequently investigated the specific
patches in the sky that correspond to the high classification scores. In the
case of the SMICA Planck data it turned out that there is a single high
temperature anomaly in the sky, shown in Fig. 8, that misled the neural
network in identifying Hawking points in patches of the sky containing this
point. Analyzing the patches in the sky in the Planck COMMANDER data that
correspond to the outliers in Fig. 7(b) we find 11 anomalies one of which is
shown in Fig. 9 and the remaining 10 listed in Appendix B, Figs. 17(j) (a)–(j)
with temperature scales with respect to the overall CMB mean temperature of
2.725 K.

Figure 8: The anomaly in the SMICA Planck data set that is the reason for the
outliers with high classification scores in Fig. LABEL:fig:рp_outliers.

We observe that all these anomalies are single to few-pixel low temperatures,
with the coldest anomaly being at 2.6763 K (Fig. 9) and the warmest anomaly
(among these cold anomalies) at 2.7203 K. The true nature of these cold points
in the sky is a mystery; however, given that we haven’t observed such cold
points in the SMICA data set, they could be artifacts of the original data
processing for the COMMANDER data, or they are real and have been removed from
the SMICA data by data processing.

The high temperature anomaly in the Planck SMICA data was apparently picked up
by the neural network to have a large likelihood of representing a Hawking
point (with the anomaly near the center of the patches). In the case of the
Planck COMMANDER data the low temperature anomalies must have reduced the
average background temperature of a patch and thereby favoring those patches
(with an anomaly near the boundary of the patches) to detect a raised
temperature at the center, suggesting the presence of a Hawking point.

We conclude that, just as in the case of the search for rings of low variance,
anomalies deceive the search algorithms in identifying Hawking points, and no
statistically significant Hawking points have been identified.

Figure 9: One of the 11 anomalies found by the HawkingNet in the Planck
COMMANDER data.

Can we improve our machine learning approach for finding Hawking points? As
pointed out above the neural network training starting from Gaussian noise
with the best-fit $\Lambda$CDM CMB power spectrum was not very successful, and
the training starting from the "COM PowerSpect CMB-TT-full R30" power spectrum
was better, but not optimal. Given the fact that the later approach did
identify patches with potential Hawking points, which turned out to be indeed
regions that likely will be identified as potential Hawking points by any
other searching algorithms, unless the single or few pixel anomalies are
filtered out from the data, indicates that the ML algorithm served it’s
purpose fairly well. Therefor we can at this point already make the claim that
with the current CMB data resolution, Hawking points with a magnitude in the
range from 100 to 400 $\mu$K amplitude and spatial Gaussian extend between 0.7
and 1.2 degrees, and small distortions thereof, can be ruled out. We did
however explore one more neural network training method, the pseudo-synthetic
approach, which could support the above claim with higher confidence.

The pseudo-synthetic approach: In this method the classifier was trained on
real CMB SMICA Planck data with and without added Hawking points, hence the
name pseudo-synthetic. The choice of the Planck SMICA set over COMMANDER was
motivated by the smaller amount of CMB anomalies in the data. We considered
500,000 patches of $4\times 4$ degree size from both the real data Northern
and Southern hemispheres.

A concern that one could raise for this approach is that the patches of the
real data might already contain real Hawking points and if we train the
network on such patches with and without additional artificial Hawking points
this might lead to a corrupt classification score. This scenario has been
studied in the machine learning literature with the conclusion that, as long
as the set of training data (before adding or not adding the feature of
interest to individual element of the training data) do not already contain
the feature of interest in more than 50% of the cases of the individual
elements of the training data, the training will still converge to a reliable
classifier. For our case this means that we work under the assumption that in
the 500,000 patches of the Northern (or Southern) Hemisphere use in the
training procedure less than half of them contains a real Hawking point. This
is indeed a very reasonable assumption.

After training of the classifier on the Northern (Southern) Hemisphere, we
applied the classifier to the Southern (Northern) hemisphere date and to the
same Gaussian simulation data without Hawking points as used in Fig. 5 (b).
The resulting classification scores on the real data is shown in blue in Fig.
10 and the classifier score on the simulation data is shown in orange.

(a)

(b)

Figure 10: The Hawking points statistical test on the Pseudo-synthetic data
trained on the (a) northern CMB hemisphere compared to the real data from the
southern hemisphere (blue distribution), (b) southern CMB hemisphere compared
to the real data from the northern hemisphere (blue distribution), and
compared to the simulations based on "COM PowerSpect CMB-TT-full R301" power
spectrum (orange distribution).

Next we repeated the statistical test on pseudo-synthetic data set similarly
to Fig. 10, but excluded the very few anomalies in the Planck SMICA data and
plotted the zoomed in classification score in the positive tail of the
distribution, shown in Fig. 11. The real data and the simulation data now
completely overlap, confirming and strengthening our conclusion that Hawking
points with a magnitude in the range from 100 to 400 $\mu$K amplitude and
spatial Gaussian extend between 0.7 and 1.2 degrees, and small distortions
thereof, cannot be found in the currently available CMB data.

Figure 11: The Zoomed-in tail of the Hawking points statistical test on SMICA
for the Pseudo-synthetic data trained on the southern CMB hemisphere compared
to the real data from the northern hemisphere with anomalies removed. Figure
12: The location of the found outlier in Fig. 11 Figure 13: The zoomed-in red-
boarded patch of Fig. 12

Searching for Concentric Circles Around Anomalies: The CCC predictions of low
variance circles are very qualitative because details would depend on the very
complex dynamics of interfering and dissipating gravitational shock waves in
the plasma at the moment of decoupling of light (now the CMB) from matter,
378,000 years after the beginning of our universe. Also the Hawking point
predictions are very qualitative. This led us to a final attempt to find a
signature for the CCC hypothesis, namely by consider the identified anomalies
as possible Hawking points and search for concentric circles of slightly
higher temperatures, or circles of low-variance, around them.

We repeated the low variation circles analysis around the identified anomalies
with small steps of 0.2 degrees, assuming that the circles might be thinner
than suggested by Penrose. We did not find any circles around these anomalies
with the currently available CMB data resolution.

##  4 Discussion and Conclusion

We presented a high-resolution search for low-variance circles in the Planck
and WMAP Cosmic Microwave Background (CMB) data, and introduced HawkingNet,
our machine learning open-source software based on a ResNet18 algorithm, to
search for Hawking points in the CMB. We found that CMB anomalies, consisting
of a single or a few bright or dim pixels, erroneously lead to regions with
many low-variance circles as well as Hawking points when applying the search
criteria used in previous works. Given the fact that our analysis is based on
the use of a state-of-the-art supercomputer and machine learning approach, and
unlike previous studies, uses the maximum available resolution of CMB data, we
conclude that no statistically relevant sets of concentric low-variance
circles and Hawking points can be found in the currently available CMB data.
Perhaps highly deformed circles and distorted Hawking points can still be
extracted but only if there are clear predictions for this.

There might also be the possibility that the density of gravitational shock
waves from merging black holes in the previous eon has been so high that we
should expect to observe the result of many interfering spherical waves,
leading to a complex interference pattern. Such a complex interference pattern
at the moment of decoupling would require a very different characterization
method than a search for clearly distinguishable ring patterns and Hawking
points in the CMB.

Our work shows that all previous attempts on identifying signatures of
conformal cyclic cosmology are inconclusive, and suffer from the dramatic
impact of CMB anomalies on the data analyses. Unlike our study, the previous
studies were significant constrained by limited computational power, and as
such had largely overlooked the decisive role of anomalies at the level of
single pixels in the CMB date. The Hawking point analysis has shown the
significant deviation of the CMB real data from being locally Gaussian and
from following the best-fit $\Lambda$CDM model. This analyses can be expended
upon in order to test various cosmological models. We provide the open-source
code for our machine learning method and hope that this will inspire more
research on the CMB anomalies and on testing cosmological models:
https://github.com/winger/hawking_net. Higher resolution CMB data and/or new
predictions from CCC seem to be needed to continue the search for signatures
of the CCC model.

## Acknowledgments

We would like to thank an Jan Willem Dalhuisen for fruitful discussions and
express our admiration and appreciation to Sir Roger Penrose for proposing the
intriguing CCC model, and to scientists around the world who have contribute
to the discussion on CCC.

##  Appendix A The HawkingNet

Our HawkingNet software https://github.com/winger/hawking_net is designed to
search for Hawking points in the noisy large CMB data sets. Its architecture
is fully based on the deep residual network (ResNet18) - a subclass of the
convolution neural networks (CNN) family. ResNet is very popular in various
scientific fields that demand visual grid-based image analysis. The HawkingNet
consists of 18 neural layers - topological arrangements of convolutional
(locally connected) neurons. The connectivity pattern is very much inspired by
biological systems resembling the structure of the animal visual cortex, where
the individual cortical neurons are stimulated only in a restricted region of
the visual (receptive) field. The algorithm inputs the CMB data, processes the
data through its internal network trained for data classification, and outputs
the result in a form of a classification score, depending on how confident it
is the image patch contains a Hawking point. The most important building
blocks of the neural net are convolution layers and pooling layers (Figure
14).

Figure 14: The ResNet18 architecture used in HawkingNet. "Conv" and "Pool"
labels refer to convolution and pooling layers. The numbers 3x3, 7x7, 64-512
represent the convolution kernel size, and the number of available filters. Fc
1 stands for "fully connected layer with one filter". The bold lines
illustrate skipping connections discussed in the text, and dashed lines -
skipping connection with the maximized pooling power.

First, the input CMB data goes through the convolution layers whose
organization mimics the response of a neuron in the visual cortex to a
specific stimulus in biological systems. Specifically, it performs the
Frobenius inner product - a binary operation that takes the convolution kernel
in a matrix form and the specific layer’s input (also a matrix) and returns an
output matrix. By convolving the input image in such a way, the number of
initial free parameters is drastically reduced bypassing the input image
deeper in the neural network a form of a abstract feature map. Next, the
feature map is processed by the pooling layers which even more reduce the
dimensions of data. These layers combine the outputs of neuron clusters in the
feature map at one layer into a single neuron and pass it to the next layer.
The uniqueness of the ResNet basis is the ability of the network to skip
connections to jump over some layers, similarly to pyramidal cells in the
cerebral cortex, where cortical layer VI neurons skipping intermediate layers
on their way. Skipping the connection is beneficial both for the brain and for
artificial neural net because it speeds up the learning process overall and
mitigates larger training error for larger number of participating layers
(known as Degradation problem). The final outcome from the HawkingNet
performance is the classification metric which analyses the feature map
generated by the network. Classification in ML is generally a statistical
model that uses a logistic function to model a binary dependent variable (the
larger subgroup is referred to as logistic regression methods). So, in our
case, the binary logistic function inputs the feature map from the residual
network and defines the probability - a numerical value between 0 and 1 for
every image patch whether the Hawking point is found or not in a form of
positive and negative samples respectively (Figure 15). The network calculates
the probability density distribution to account for the probability of finding
the Hawking point in a considered data patch as a function of assigned
classified scores. We see in Figure 15 that most of the positive samples are
more likely contain the Hawking point for classification score 2, while for
the negative samples it is more likely for the classification score -2. These
values are a function of the overall neural network sensitivity with a binary
classifier which is typically measured by comparing its logistic function with
the theoretical model.

Figure 16 illustrates the measure of our sensitivity towards classifying
Hawking points. On the y-axis, we show the theoretical fraction of positive
samples with a given score, and the same fraction observed in practice at
Figure 15. Additionally, the aligned trends of empirical and theoretical
curves indicate that our network sensitivity is maximized, which was done by
11 hours of neural network training.

Figure 15: The HawkingNet empirical probability distributions for the positive
and negative samples. The positive samples indicate the data with classified
Hawking points in contrast to the negative samples.  Figure 16: The
theoretical fraction of positive samples found by the HawkingNet- data with
detected Hawking points as a function of the classification score. The aligned
trend of both theoretical and empirical curves indicates maximized sensitivity
of the neural network to the Hawking points.

##  Appendix B Figure 17: Planck COMMANDER Anomalies

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

## References

  * [1] V.G. Gurzadyan, R. Penrose, _On CCC-predicted concentric low-variance circles in the CMB sky_ _Eur. Phys. J. Plus_ , 128 p.22 (2013) [doi:10.1140/epjp/i2013-13022-4]
  * [2] R. Penrose, _Cycles of Time: An Extraordinary New View of the Universe_ _Bodley Head, London_ , (2010)
  * [3] R. Penrose, _The basic ideas of conformal cyclic cosmology_ _AIP Conference Proceedings_ , 1446 233 (2012) [doi:10.1063/1.4727997]
  * [4] K. P. Tod _Isotropic cosmological singularities: other matter models_ _Class. Quantum Grav._ , 521 20 (2003) [doi:10.1088/0264-9381/20/3/309]
  * [5] K. P. Tod, _The equations of CCC_ , _arXiv: General Relativity and Quantum Cosmology_ , (2013) [arXiv:1309.7248]
  * [6] V.G. Gurzadyan, R. Penrose, _More on the low variance circles in CMB sky_ , (2010) [arXiv:1012.1486]
  * [7] A. Moss, Douglas Scott, J.P. Zibin, _No evidence for anomalously low variance circles on the sky_ , _Journal of Cosmology and Astroparticle Physics_ , 4, 4, (2010) [arXiv:1012.1305]
  * [8] I.K. Wehus, H.K. Eriksen , _A search for concentric circles in the 7-year WMAP temperature sky maps_ , _The Astrophysical Journal Letters_ , 733, 2, article id. L29, 6 pp. (2011) [arXiv:1012.1268]
  * [9] V.G. Gurzadyan, A.E. Allahverdyan, T. Ghahramanyan, A.L. Kashin, H.G. Khachatryan, A.A. Kocharyan, S. Mirzoyan, E. Poghosian, D. Vetrugno, G.Yegorian, _A weakly random Universe?_ , _Astronomy and Astrophysics_ pg. 525. (2011) [arXiv:1011.3711v2]
  * [10] A. DeAbreu, C. Dagoberto, and Douglas Scott, _Searching for concentric low variance circles in the cosmic microwave background_ , _Journal of Cosmology and Astroparticle Physics_ pg. 12. (2015) [arXiv:1508.05158]
  * [11] P. Naselsky, W. Zhao, J. Kim, S. Chen, _Is the Cosmic Microwave Background Asymmetry due to the Kinematic Dipole_ _The Astrophysical Journal_ , 749 (1) (2012) [arXiv:1108.4376]
  * [12] Q. Yang, H. Yu, H. Di, _Cosmic microwave background dipole asymmetry could be explained by axion monodromy cosmic strings._ _Physics of the Dark Universe_ , 26 (2) (2019) [arXiv:1712.09917]
  * [13] L. Santos, P. Cabella, T. Villela, A. Balbi, N. Vittorio, and C. A. Wuensche _Quadrant asymmetry in the angular distribution of cosmic microwave background in the Planck satellite data_ _Astronomy and Astrophysics_ , 569 (A75) (2014) [doi:10.1051/0004-6361/201322943]
  * [14] J. Creswell, P. Naselsky _Asymmetry of the CMB map: local and global anomalies_ _Journal of Cosmology and Astroparticle Physics_ , 103 03 (2021) [arXiv:2102.13442]
  * [15] D. An, K. A. Meissner, P. Nurowski, R. Penrose, _Apparent evidence for Hawking points in the CMB Sky_ _Monthly Notices of the Royal Astronomical Society_ , (2020) [arXiv:1808.01740]
  * [16] M. Lopez, P. Bonizzi, K. Driessens, G. Koekoek, J. de Vries, R. Westra, _Searching for ring-like structures in the Cosmic Microwave Background_ , _Education, Physics_ (2021) [Corpus ID: 234343106]
  * [17] D. L. Jow, Douglas Scott _Re-evaluating evidence for Hawking points in the CMB_ , _Journal of Cosmology and Astroparticle Physics_ 021 (2020) [arXiv:1909.09672]
  * [18] Planck 2018 results, _V. CMB power spectra and likelihoods_ _Cosmology (including clusters of galaxies)_ , 641 (A5) p.92 (2020) [arXiv:1907.12875]
  * [19] K. He, X. Zhang, S. Ren, J.Sun, _Deep residual learning for image recognition_ _Deep residual learning for image recognition_ , (2016) [arXiv:1512.03385]
  * [20] K. M. Gorski, B. D. Wandelt, E. Hivon, F. K. Hansen, and A. J. Banday, _The HEALPix primer_. (1999) _arXiv preprint astro-ph/9905275_ [arXiv:9905275]
  * [21] Z. Liu, Z. Cen, V. Isenbaev, W. Liu, Z.S. Wu, B. Li, D. Zhao _Constrained Variational Policy Optimization for Safe Reinforcement Learning_ , _In: DeepAI, RLDM22]_ The Multi-disciplinary Conference on Reinforcement Learning and Decision Making (2022) [DeepAI22]
  * [22] _Planck Legacy Archive_ , [http://pla.esac.esa.int/pla/cosmology]
  * [23] _Wilkilson Microwave Anisotropy Probe_ , [Data]
  * [24] H. Kaiming, X. Zhang, S. Ren, J. Sun, _Springer International Publishing. Computer Vision – ECCV 2016_ , (2016) [ECCV16]