# First High Resolution Interferometric Observation of a Solar Prominence With
ALMA

Nicolas Labrosse,1 Andrew S. Rodger,1 Krzysztof Radziszewski,2 Paweł Rudawy,2
Patrick Antolin,3 Lyndsay Fletcher,1,4 Peter J. Levens,1 Aaron W. Peat,1
Brigitte Schmieder1,5 and Paulo J. A. Simões6,1
1SUPA, School of Physics & Astronomy, University of Glasgow, Glasgow G12 8QQ,
UK
2 Astronomical Institute, University of Wrocław, PL
3Department of Mathematics, Physics and Electrical Engineering, Northumbria
University, Newcastle Upon Tyne, NE1 8ST, UK
4Rosseland Centre for Solar Physics, University of Oslo, P.O.Box 1029
Blindern, 0315 Oslo, NO
5Centre for mathematical Plasma-Astrophysics, KU Leuven, Leuven, BE
6Centro de Rádio Astronomia e Astrofísica Mackenzie, Escola de Engenharia,
Universidade Presbiteriana Mackenzie, São Paulo, BR E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the first observation of a solar prominence at $84-116$ GHz using
the high resolution interferometric imaging of ALMA. Simultaneous observations
in H$\alpha$ from Białkaw Observatory and with SDO/AIA reveal similar
prominence morphology to the ALMA observation. The contribution functions of 3
mm and H$\alpha$ emission are shown to have significant overlap across a range
of gas pressures. We estimate the maximum millimetre-continuum optical
thickness to be $\tau_{\mathrm{3mm}}\approx 2$, and the brightness temperature
from the observed H$\alpha$ intensity. The brightness temperature measured by
ALMA is $\sim 6000-7000$ K in the prominence spine, which correlates well with
the estimated brightness temperature for a gas temperature of 8000 K.

###### keywords:

Sun: filaments, prominences

††pubyear: 2022††pagerange: First High Resolution Interferometric Observation
of a Solar Prominence With ALMA–First High Resolution Interferometric
Observation of a Solar Prominence With ALMA

##  1 Introduction

Solar prominences are one of the most striking illustrations of how the
magnetic field shapes the atmosphere of our Sun (Labrosse et al., 2010;
Schmieder et al., 2013; Vial & Engvold, 2015), and constitute the best example
in the solar system of natural plasma confinement: the magnetic field somehow
supports a cool ($T\sim 10^{4}$ K), dense ($n_{e}\sim 10^{10}$ cm-3) plasma in
a hot ($T\sim 10^{6}$ K), tenuous ($n_{e}\sim 10^{8}$ cm-3) environment. If
the magnetic field loses its equilibrium configuration, the prominence may
become unstable and erupt, throwing vast amounts of magnetized plasma into the
heliosphere. These prominence eruptions are often associated with the most
energetic events in the solar system: flares and coronal mass ejections.
Understanding their formation, and their temporal evolution, is relevant for
our understanding of the interplay between plasma and magnetic field in
stellar coronae. Attempts at inferring the temperature structure of solar
prominences have been limited due to the techniques used and available
spectral windows (Parenti, 2014). Yet, the gas temperature is a fundamental
quantity to measure to understand the lifecycle of solar prominences, from
their formation, to their evolution and their disappearance (Wedemeyer et al.,
2016).

The Atacama Large Millimeter/submillimeter Array (ALMA) offers a new window on
the solar prominence plasma. The formation mechanism of the sub-mm radiation
means that brightness temperature measurements are in principle a good proxy
for the electron temperature (Dulk, 1985). We carried out the first
observations of a solar prominence in April 2018 (ALMA Cycle 5). Section 2
describes the observations and data reduction procedures for ALMA, ground-
based H$\alpha$, and SDO/AIA. Section 3 presents a millimetre-continuum
optical thickness diagnostic using the H$\alpha$ integrated intensity, the
brightness temperature results, and a discussion of the prominence morphology
as seen by ALMA compared to H$\alpha$ and AIA. Finally, Section 4 presents our
conclusions.

##  2 Observations and Data Reduction

The small prominence appearing above the south-west solar limb on 19th April
2018 was well observed by ALMA together with other ground-based and space-
based observatories. Peat et al. (2021) carried out a detailed analysis of its
emission in the Mg II h and k lines observed by the IRIS spacecraft, while
Barczynski et al. (2021) investigated its dynamics in relation to its magnetic
structure.

ALMA observed off the solar limb near the heliocentric coordinates
($+650\arcsec,-750\arcsec$) between 15:20 and 17:45 UTC on the
$\mathrm{19^{th}}$ of April $2018$. The mode of operation planned for this
observation was a mosaic with a proposed total field of view (FOV) of $40$″ by
$80$″. This FOV was produced by cycling through a total of $5$ pointings, each
with a smaller FOV determined by the observing wavelength and array
configuration. In this case the observing band was Band $3$ ($84$–$116$ GHz)
and the array configuration was C$43$-$3$, the largest array configuration
available for solar observations at the time, giving the highest possible
spatial resolution. This prominence was also observed two days earlier in ALMA
Band 6. Here we focus our analysis on ALMA Band 3 data as they offer a better
diagnostic potential for solar prominences (Rodger & Labrosse, 2017).

###  2.1 ALMA Band $3$

The ALMA observation was split into two main observing blocks: from 15:38 –
16:32 (Block $1$) and from 16:50 – 17:44 (Block $2$). Within each block ALMA
conducted $7$ scans of the total solar FOV, with off-target calibration
occurring between each scan. The average time taken for each scan was $\sim 6$
minutes.

Issues with the system temperature were found in several of the 43 12m array
antennas where their XX and YY polarizations were too large, resulting in very
large estimates for the noise in the synthesised images. To combat this, the
problem antennas were identified, which included $8$ antennas for Block $1$
and $10$ for Block $2$, which were subsequently removed from the image
synthesis. Whilst the reliability of the data is improved by removing the
antennas, the resolution suffers somewhat due to the reduction in overall
baseline number.

To produce an image from the imperfectly sampled visibility data, we used the
deconvolution algorithm of the _Common Astronomical Software Applications_
(CASA) program tclean with the multiscale extension to the CLEAN algorithm
(Högbom, 1974) presented in Cornwell (2008). The resultant cleaned
interferometric images for the whole time range are shown in Figure 1.

Figure 1: Synthesised ALMA interferometric images for Blocks 1 and 2 of the
prominence on 19 April $2018$. Each image is produced by averaging over 7
scans ($\sim 42$ min), with the start time of each block given in the image
titles. The colourbars show the relative brightness temperature in Kelvin.

Interferometric images do not yield absolute brightness temperatures, but
rather relative brightness temperature values around some background value
determined by the largest spatial scales. To produce absolute brightness
temperature images the interferometric images must be combined with single
dish images to provide measurements of the total flux in the image (Koda et
al., 2011). The process used here is to combine the interferometric and the
single dish measurements from the Total Power (TP) array – this is called
“feathering”. Feathering involves combining two images together through the
weighted sum of the Fourier transform components in the _uv_ -plane. This will
also provide the low spatial resolution component of the image due to the
limitations in the shortest possible baseline in the interferometer array.
Across the entire observation ALMA conducted a total of $14$ observing scans
of the entire solar disc, each taking $\sim 10$ minutes. The method for TP
full-disc observations is described in White et al. (2017). For each
interferometric synthesised image a corresponding TP image was chosen to
feather with it to produce an absolute brightness temperature image. The
choice of corresponding TP image was by either having the longest period of
co-temporal observation, or if that was not possible, the shortest difference
in time between it and the synthesised interferometric image.

###  2.2 H$\alpha$

The prominence was observed in the H$\alpha$ line using the Large Coronagraph
and the Multi-Chanel Subtractive Double Pass Spectrograph (MSDP) located at
Białkow Observatory (University of Wrocław in Poland). The MSDP spectrograph
measured the intensity at $23$ different wavelengths ranging from $-1.1$ Å to
$+1.1$ Å from the line centre. The MSDP quasi-monochromatic images used in the
analysis were obtained from a combination of 13 sub-images during observing in
slow-mode using a scanner. A total of $739$ scans were taken between 10:02 and
15:54 UTC. In this study we use a singular scan from this observation to
estimate the optical thickness of the Band $3$ emission which is largely
dominated by the free-free thermal continuum. This scan was recorded between
15:35:08 and 15:35:24 UTC, roughly $3$ minutes before the start of the first
ALMA observation. The data has been calibrated in absolute units and the
scattered light was subtracted (Radziszewski et al., 2007).

###  2.3 AIA

In Section 3.3 we compare the image of the prominence synthesised with ALMA in
Band $3$ across observing Block $2$ to simultaneously observed imaging from
the _Atmospheric Imaging Assembly_ (AIA) on board the _Solar Dynamics
Observatory_ (SDO) (Lemen et al., 2012). As the ALMA image for Block $2$ is
time-averaged, small scale motions will be undetectable. To allow for direct
comparison with this ALMA image and the prepped AIA images, the AIA images are
also averaged over the corresponding time range. In AIA bands where the
prominence signal is faint, or there was a low signal-to-noise ratio, this
time averaging produced a more distinct view of the prominence to compare to
the ALMA image. AIA data where the signal was strong, e.g. at $304$ Å, were
also averaged with respect to time so that any time-dependent fine structure
will be unresolved in a similar manner to the lower temporal resolution ALMA
image.

##  3 Results and Analysis

###  3.1 H$\alpha$ Intensity and the Millimetre-Continuum Optical Thickness

The first step to estimate the optical thickness of the prominence in ALMA
Band $3$ is to calculate the integrated intensity across the H$\alpha$ line
profile. The result is shown in Figure 2.

Figure 2: Prominence observed in H$\alpha$ integrated intensity by the MSDP
spectrograph. The cyan cross marks the centre of the prominence, where
$E({\mathrm{H_{\alpha}}})\approx 2.2\times 10^{5}$ erg s-1 cm-2 sr-1.

The optical thickness of the mm-continuum is related to the H$\alpha$
integrated intensity $E({\mathrm{H_{\alpha}}})$ for an isothermal line of
sight (LOS) through

$\tau_{\nu}\approx 4.55\times
10^{17}g_{\mathrm{ff}}e^{-T_{0}/T}E({\mathrm{H_{\alpha}}})/(\nu^{2}b_{3}(T))\propto
g_{\mathrm{ff}}(\nu,T)f(T)\ ,$ (1)

where $g_{\mathrm{ff}}$ is the Gaunt factor, $T$ the temperature,
$T_{0}=17534$ K, $\nu$ the frequency, $b_{3}(T)$ the LTE (Local Thermodynamic
Equilibrium) departure coefficient of the third energy state of hydrogen
(varying from 0.75 at 6000 K to 2.97 at 10000 K according to Jejčič & Heinzel,
2009), and $f(T)$ is defined by (Heinzel et al., 2015; Rodger, 2019)

$f(T)={e^{-T_{0}/T}}/{b_{3}(T)}.$ (2)

In Heinzel et al. (2015) the authors use $3$ values for the $f(T)$ factor
calculated using $1$D prominence models at constant temperatures of $6000$,
$8000$ and $10000$ K. In fact, the variation with temperature of the optical
thickness estimation using H$\alpha$ integrated intensity is fairly small for
temperatures up to $\sim 20000$ K. We use the same values for $f(T)$ as quoted
in Heinzel et al. (2015) and the resulting optical thickness and brightness
temperature estimations are shown for each of $6000$, $8000$ and $10000$ K
constant temperature plasmas in Figure 3.

Figure 3: Optical thickness and brightness temperature estimations for ALMA
Band $3$ emission as calculated using the H$\alpha$ integrated intensities
from Figure 2 and an isothermal assumption. The estimation was calculated at
$3$ constant temperatures, where the values for the $f$-factor (Equation 2)
were the same as quoted in Heinzel et al. (2015).

From Figure 3, it can be seen that if the observed prominence were to have a
constant temperature between $6000$ and $10000$ K, the optical thickness of
the mm-continuum should lie in the densest parts between $\approx 1$ and $3$,
with the value at the centre of the prominence spine (cyan cross) at $\approx
2$, and going below 1 at the edges. An optical thickness of $2$ in Band 3 has
been shown to be too low to make a direct electron temperature diagnostic from
the mm-continuum brightness temperature for isothermal prominence models
(Rodger & Labrosse, 2017).

If the temperature of the plasma is higher than the $10000$ K considered here,
which is perhaps unlikely due to the observed H$\alpha$ intensities, the
optical thickness of the mm-continuum could be higher due to an increased
magnitude of the $f$ factor. Our calculations (see next section) show that,
even for the unlikely case where the prominence plasma is up to $20000$ K in
temperature, the optical thickness of the mm-continuum would not exceed
$\approx 4-5$.

###  3.2 Time Averaged ALMA Images

Figure 3 suggests that the brightness temperature appears to be uniformly
distributed across the width of the prominence spine. The prominence models of
Rodger & Labrosse (2017) indicate that the mm-continuum emission peaks in
lines of sight passing through optically thick material from the prominence-
to-corona transition region (PCTR). This suggests that the PCTR is either
optically thin in this observation or that the PCTR is not resolved, or both.
The resolution of the ALMA maps is defined by the semi-major and semi-minor
axes of the synthesised beam, which is $1.98$″ x $1.52$″ for Block $1$ and
$2.17$″ x $1.56$″ for Block $2$. The visibility of fine structures is also
affected by the long time range used in the production of this image as well
as the frequency bandwidth, as emission at different frequencies will be
produced from different layers in the fine structures.

Our estimated maximum optical thickness of the prominence spine at 3 mm ($\sim
2$) is consistent with the hypothesis that LOSs going through the sparser PCTR
material are optically thin in this observation. As an optical thickness of
$2$ is below that which is required for a direct measurement of the local
electron temperature of the emitting plasma, an absolute brightness
temperature measurement presents a lower boundary to the electron temperature
measurement.

Figure 4 shows maps of absolute brightness temperature from the feathered ALMA
data. The brightness temperatures for the prominence spine are between $6000$
and $7000$ K.

Figure 4: Absolute brightness temperature images of the prominence observed
with ALMA Band 3 on $19$ April $2018$.

Further from the limb there is some brightness temperature enhancement in the
background corona. These are likely to be caused by imaging artifacts caused
by the step-function-like brightness temperature variation of the solar limb
and the finite sampling of the _u-v_ plane. Artifacts like these have been
previously observed in other off-limb ALMA observations (Shimojo et al., 2017;
Yokoyama et al., 2018).

We coaligned the MSDP and ALMA observations closest in time to study the
relation between the observed (Fig. 4) and estimated (Fig. 3, right panels)
brightness temperatures. Figure 5 shows a very good correlation between the
two for a gas temperature of $T=8000$ K.

Figure 5: Observed brightness temperature from ALMA (15:38) vs that modelled
from H$\alpha$ MSDP data (15:35) in the prominence. The modelled brightness
temperatures are estimated for three gas temperarures (6, 8, and 10 kK) as in
Fig. 3. The dotted line shows when the observed and estimated brightness
temperatures are equal.

The scatter in the plots could be explained by a non-uniform gas temperature
throughout the prominence (e.g. Rodger & Labrosse, 2017) and/or by a filling
factor less than 1 (Irimajiri et al., 1995).

Through comparison between the Band $3$ interferometric image (Figure 1) and
the H$\alpha$ integrated intensity image (Figure 2) it is clear that the
prominence has a very similar morphology in both observations. The prominence
spine is clearly visible, with what looks to be some barb-like structures at
the sides, although the ALMA FOV only catches the southward side of the full
prominence observed in H$\alpha$. This similarity suggests that the emission
in ALMA Band $3$ is formed either by the same cool, dense material which forms
the H$\alpha$ emission; or in a closely fitting hot plasma sheath around the
cool, dense core; or it is formed by the contribution of multiple, unresolved
hot and cool fine-structures.

To address this question we computed the contribution functions at 3 mm and in
H$\alpha$ using the 2D non-LTE cylindrical radiative transfer models used in
Rodger & Labrosse (2017). In figure 6 we see a cross-section of a horizontal
cylinder (radius 1000 km) located above the solar disc and observed from the
left side. Optically thick emission will be seen on the left-most parts of the
cylinder cross-section (right panel, higher pressure). An enhanced
contribution function at the bottom of the cross-section (left panel, lower
pressure) reveals the important role played by the incident radiation coming
from the solar disc underneath. Figure 6 shows that the contribution functions
at 3 mm and in H$\alpha$ have clear and consistent overlap across a wide range
of prominence pressures. The larger width of the mm-continuum contribution
function compared to that of H$\alpha$ suggests a contribution from hotter
plasma at the base of the PCTR.

Figure 6: Contour maps for contribution functions for H$\alpha$ compared to
the continuum at 3 mm, for multi-thermal prominence models at different
pressures. The LOS is directed such that the observer is to the left of each
cylinder. The blue contours show H$\alpha$ while the black contours describe
the mm-continuum. The contour levels from light to dark correspond to 20, 40,
60,80, 100% of the maximum of the contribution function in the model
prominence.

###  3.3 ALMA–AIA

Our aim here is to compare the morphology of the prominence in the mm-
continuum to the various AIA bands which are each produced at different layers
within the solar atmosphere. For this analysis we use time-averaged AIA images
for each band, as the $3$ mm images in Figure 1 were produced across the whole
time range of both observations. The image chosen to co-align with AIA
corresponds to ALMA Block $2$ data. To co-align the images from the two
instruments the FOV of the data sets were set to be equal. An example of the
co-aligned images can be seen in Figure 7, where the ALMA observation is
displayed as a contour map on top of the various time-averaged AIA band
images.

Figure 7: Co-aligned time-averaged images of the $19$ April $2018$ prominence
observed with ALMA and AIA during the second ALMA observing block. The ALMA
image is displayed as the white contours overlaid on the AIA images. The
contours show the bright parts of the synthesised image with relative
brightness temperatures of 1000, 2000, and 3000 K.

Figure 7 shows that the contours of the Band $3$ prominence spine follow
clearly the dark structures observed in AIA $171$, $193$ and $211$ Å. This
finding agrees well with previous knowledge of the optical thicknesses of ALMA
Band $3$ and in the Lyman continuum at $195$ Å. The optical thickness of Band
$3$ emission is similar to that of the H$\alpha$ at line centre, and it has
been previously shown by Heinzel et al. (2008) that the opacity of the Lyman
continuum at $195$ Å is comparable to that of the line centre of H$\alpha$.
The bright prominence structure in $304$ Å is however significantly wider than
that observed in either the Band $3$ or H$\alpha$ images. This is not
surprising as most of the emission at $304$ Å is expected to be formed under
optically thick conditions in prominences, thus allowing fine-structures away
from the main body of the prominence to be more easily observed. The $94$ Å,
$131$ Å, $335$ Å, $1600$ Å and $1700$ Å data show similar results to $171$,
$193$ and $211$ Å, however the prominence signal is significantly fainter in
these channels. Emission at 1600 and 1700 Å above the limb should be formed by
cool lines ($10^{4}-10^{5}$ K), since the metallic continua from the
photosphere cannot contribute here (Simões et al., 2019). The dark structures
observed in AIA $193$ and $211$ Å follow the bright Band $3$ prominence closer
than the other AIA bands.

##  4 Discussion and Conclusions

This paper presents the first high resolution ALMA interferometric observation
of a solar prominence. Comparison between the ALMA Band $3$ and H$\alpha$
images reveals that both forms of emission present similar morphologies, with
both showing a clear prominence spine and barb-like structures. This suggests
that the mm-continuum at Band $3$ must be formed from the same cold, dense
plasma which is known to emit in H$\alpha$, or from hotter material contained
within a similarly shaped structure. This could either be explained by a
structure with a cool, dense core surrounded by a tight hot plasma sheath; or
a structure consisting of multiple unresolved cold and hot plasma fine
structures (Peat et al., 2021). This is consistent with the contribution
function maps computed for H$\alpha$ and $3$ mm, where both types of emission
have largely overlapping formation regions.

The H$\alpha$ integrated intensities are used to estimate the optical
thickness in the mm-continuum. We find that the maximum Band $3$ optical
thickness is $\approx 2$, which is too low for the brightness temperature to
be used as a direct proxy for the electron temperature without further
modelling. This optical thickness and the view of the prominence in Band $3$
are in good agreement with what would be expected for a solar prominence from
the models of Rodger & Labrosse (2017). Fully optically thick prominences
would be expected to show brightness temperature enhancement through
predominantly PCTR material. As no such brightening is observed the prominence
must either be optically thin for Band $3$ in these LOSs crossing high
temperature material, and/or this material is unresolved. The latter is
strongly supported by observations and simulations, which estimate the PCTR
widths to be on the order of $0.1\arcsec-0.3\arcsec$ (Antolin et al., 2015).
The brightness temperature in ALMA Band 3 (100 GHz) lies within $\approx
6000-7000$ K. Our measured and estimated brightness temperatures correlate
well for a gas temperature of $T=8000$ K. The range of measured brightness
temperatures is in agreement with previous prominence modelling (e.g. Heinzel
et al., 2015; Rodger & Labrosse, 2017). These results are also consistent with
earlier observations by e.g. Harrison et al. (1993); Bastian et al. (1993) at
similar wavelengths.

The Band $3$ prominence observation was also compared to co-aligned images
from each of the nine AIA bands. The AIA bands which appear to be most highly
correlated with the mm-continuum emission are the $193$ and $211$ Å bands,
where the bright ALMA prominence spine appeared as clear dark structures
against the bright surrounding corona. The $304$ Å prominence is also clearly
visible, however, the extent of this structure is considerably larger than in
the corresponding mm-continuum image. This indicates that there is a
significant amount of unresolved material which is optically thin in the mm-
continuum beyond the core of the prominence spine/barbs observed in the ALMA
Band $3$ image.

Future observations of prominences with ALMA will be improved when Bands $1$
and $2$ are available for solar observations. The prominence plasma is
optically thicker at these wavelengths, allowing more regions to be probed.
Knowledge of the temperature structure at different layers within the
prominence will help improve understanding into the prominence energy balance
and how they are sustained within the solar corona.

## Acknowledgements

We thank the referee for their comments. Support from STFC grants ST/T000422/1
(NL, LF), ST/N504075/1 (ASR), ST/K502005/1 (PJL) and ST/S505390/1 (AWP) is
gratefully acknowledged. P.A. acknowledges STFC support from Ernest Rutherford
Fellowship grant number ST/R004285/2. PJAS acknowledges support from CNPq
(contract 307612/2019-8) and FAPESP (2013/24155-3).

## Data Availability

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2017:1:01138.S.
ALMA is a partnership of ESO (representing its member states), NSF (USA) and
NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ.

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# PredictChain: Empowering Collaboration and Data Accessibility for AI in a
Decentralized Blockchain-based Marketplace

Matthew T. Pisano
Department of Computer Science
Rensselaer Polytechnic Institute
Troy, NY, USA
&Connor J. Patterson
Department of Computer Science
Rensselaer Polytechnic Institute
Troy, NY, USA
&Oshani Seneviratne
Department of Computer Science
Rensselaer Polytechnic Institute
Troy, NY, USA

###### Abstract

Limited access to computing resources and training data poses significant
challenges for individuals and groups aiming to train and utilize predictive
machine learning models. Although numerous publicly available machine learning
models exist, they are often unhosted, necessitating end-users to establish
their computational infrastructure. Alternatively, these models may only be
accessible through paid cloud-based mechanisms, which can prove costly for
general public utilization. Moreover, model and data providers require a more
streamlined approach to track resource usage and capitalize on subsequent
usage by others, both financially and otherwise. An effective mechanism is
also lacking to contribute high-quality data for improving model performance.
We propose a blockchain-based marketplace called _“PredictChain”_ for
predictive machine-learning models to address these issues. This marketplace
enables users to upload datasets for training predictive machine learning
models, request model training on previously uploaded datasets, or submit
queries to trained models. Nodes within the blockchain network, equipped with
available computing resources, will operate these models, offering a range of
archetype machine learning models with varying characteristics, such as cost,
speed, simplicity, power, and cost-effectiveness. This decentralized approach
empowers users to develop improved models accessible to the public, promotes
data sharing, and reduces reliance on centralized cloud providers.

_Keywords_ Blockchain $\cdot$ Decentralized AI and Data Marketplace $\cdot$
Oracle $\cdot$ Algorand

##  1 Introduction

AI is becoming pervasive in our lives, from voice assistants to self-driving
cars [1]. However, developing and training many machine learning models
requires enormous data and computing resources. For instance, state-of-the-art
image classification models like ResNet-50 [2] are designed to be trained on
high-end GPUs on the ImageNet dataset [3], which has over 14 million hand-
annotated images. Such resource requirements have led to the concentration of
AI development, which creates an obstacle to innovation by the masses.

Decentralized AI marketplaces (such as SingularityNET [4] and Ocean Protocol
[5]) have emerged as promising solutions to these challenges, which allow
individuals and organizations to collectively share their data and
computational resources to develop and train machine learning models. By
leveraging blockchain technology, decentralized AI marketplaces can facilitate
secure, transparent, incentivized, and decentralized access to machine
learning models and data, promoting innovation and collaboration.

In this paper, we propose a blockchain-based marketplace for AI called
“PredictChain” that aims to democratize machine learning model training and
inference using the Algorand blockchain on several machine learning models.
When users upload their datasets to PredictChain, they allow a model to be
trained on the datasets uploaded. Higher-quality datasets will produce higher-
quality models. The model owners and the dataset contributors are rewarded for
their contributions when a model is queried. The amount of this reward is
based on the correctness of the prediction, which encourages users to
participate in contributing the resources needed for good predictions while
leaving a public record for other users to view, which is in contrast to large
cloud organizations, such as AWS or Google Cloud. These cloud providers allow
users to train models on custom datasets, and come at a price, with no chance
of reward if a model does well.

##  2 Related Work

The development of decentralized AI marketplaces using blockchain technology
is an evolving field that has gained significant attention recently. While the
concept of AI resource marketplaces is not new, utilizing blockchain
technology enables the creation of decentralized platforms between untrusted
parties, potentially transforming AI model training and development. Various
studies have investigated the application of blockchain technology in
establishing decentralized marketplaces for AI resources, as we outline below.

Harris and Waggoner outline a blockchain-based predictive marketplace that
utilizes AI for predictive analysis. In their solution, an initial model is
trained on the data collaboratively provided by multiple users [6, 7, 8]. To
encourage the submission of high-quality data, the authors propose an
incentive mechanism that rewards honest users and punishes dishonest users.
Their model is trained off-chain on the client’s machine. All the data
submission and incentives logic is done within an Ethereum smart contract.
This system leaves the exact definition of the model flexible between
different clients running the system. They suggest that the specific class of
model would be some predictive model, giving examples of Conditional Random
Fields as Recurrent Neural Networks (CRF-RNNs) and Support Vector Machines
(SVMs). Depending on the model used, the authors mention that the
corresponding incentive mechanism would change from model to model. The
incentives proposed in this work fall into gamification and monetary
incentives. In the gamification strategy, the authors suggest that there
should be no monetary incentive for contributing data, instead using tokens or
badges in a game-theoretic manner. For the monetary route, they suggest that
users who upload data may also have to submit some stake into the contract as
collateral. Our work is most similar to this but with some notable
differences. We have deployed our solution on the Algorand blockchain, which
provides several performance advantages over an Ethereum-based solution, and
we have leveraged some advanced machine learning models in our solution
offering.

Sarpatwar et al. take some of the decentralization ideas of the previous work
and expand upon them by structuring their project with security as a main
focus [9]. They develop a marketplace for health insurance entities needing
models trained on confidential data, and the primary features of their work
are trust, fairness, and auditability. The datasets they used are unique since
the bulk of the data is not tied directly to any one training request. For
instance, a user uploads only a small chunk of data. This data will then be
used as the validation set for the training. Most of the training data for the
model inference process is pulled from private sources, requiring users to
upload only a small portion of their data. However, this has the disadvantage
of requiring external data to train models. To accommodate this, the authors
implemented several layers of security to protect private data sources. The
models they target need to be more integrated and trained in a decentralized
manner. This training strategy reflects federated learning [10], where each
model is given only some data in bulk and only sections with confidential
information stripped out. Additionally, the models are eventually shared with
the requesting user along with the model results. An off-chain component
communicates with this system’s private, permissioned blockchain (Hyperledger
Fabric [11]), which allows for more complexities in training the models, as
on-chain training has significant speed and memory constraints.

Montes and Goertzel have also noticed the problem of accessibility in the
realm of AI and have proposed a solution using blockchain technologies [12].
Their marketplace implementation, SingularityNET, was designed to make AI
development and training more accessible to everyone. They argue that this
democratization is useful for allowing more people to access AI technologies,
and it significantly impacts the development of artificial general
intelligence (AGI). Furthermore, they also note that the current development
of AI systems is controlled primarily by only a handful of large tech giants
and that this degree of centralization would hurt any eventual AGI due to the
bias that it may be given when trained only by a few thousand individuals,
missing out on important data that can be gathered from outside these
organizations. Their solution to this centralization problem is the
SingularityNET marketplace. There, users can request or provide AI services.
These actions are organized using simple, smart contracts that take in payment
from the requesting users and distribute rewards to service-providing users.
Utilizing blockchain and smart contract technologies is designed to avoid the
closed and centralized qualities that many contemporary AI marketplaces
embody.

Leveraging blockchain to enhance the security and privacy of machine learning
models in specific domains has also been explored.

Shamsi and Cuffe demonstrate a market similar to ours but using a more
decentralized architecture and a more specific scope for a use case of
blockchain-empowered prediction markets for the renewable energy market [13].
A key problem with many renewable energy sources is their unpredictable
supply. Wind turbines need wind. Solar panels need sunshine. Forecasting the
output of these energy production methods at various times or places helps
alleviate the pitfalls that prevent them from growing in market share. On
their platform, anyone can propose an event and request predictions for that
event with the promise of providing a reward. Reporters then propose several
predictions throughout the seven-day window that the market is active. In this
implementation, the hosts of this predictive market do not take responsibility
for the training and execution of the predictive machine learning models;
instead, it is all done on the side of the reporters. Using their framework,
reporters who provide correct data are rewarded, and those who provide
inconsistent data are punished.

Shen and Shafiq focus on using big data to train predictive machine learning
models to forecast the short-term future behavior of stocks within the Chinese
stock market [14]. For their dataset, they gathered the past performance of
3558 stocks from the Chinese stock market for two years. To gather this data,
they utilized the Tushare API***https://pypi.org/project/tushare, along with
web-scraping from Sina Finance†††https://finance.sina.cn and the SWS
Research‡‡‡https://www.swsresearch.com/institute_sw/home websites. The authors
utilized several techniques to reduce the noise inherent to the stock market
to allow for better predictions. Their primary method for doing this was
through feature engineering, for which they utilized three primary techniques.
First, they applied feature extension to their dataset. Using this technique,
they added additional meta-attributes to the entries, such as polarizing or
calculating the fluctuation percentage. Adding this metadata to the dataset
helps to give the model a more complete picture of the stock and how it
performs over time. Next, they eliminated some features, based on their
influence, by using the Recursive Feature Elimination (RFE) algorithm.
Eliminating unnecessary features allows their model to pay more attention to
the most influential features, improving its predictive capabilities. For
testing, the authors used a two-layered Long Short Term Memory (LSTM) model
with only an input layer and an output layer, and as for their output, they
kept it as simple as possible to ensure the model was focused only on
predicting the movement of the stock. The model either outputted a $1$ for the
stock going up at a given time step or a $0$ if the model believed the stock
would go down. This work helped us better understand some of the future
improvements we could make to our dataset preprocessing and model training.

##  3 Methodology

We have expanded upon the ideas mentioned above and addressed some of their
shortcomings in PredictChain, which presents a more scalable architecture
using off-chain oracles and the Algorand blockchain that supports the instant
finality of transactions. Our aim with this strategy is to focus on simplicity
and ease of use. Our implementation allows anyone to quickly set up an oracle
instance and begin connecting to clients without requiring a large network of
oracles. Another design choice that we made was in our suite of built-in
archetype machine learning models. Each client has a preset list of models to
choose from. It is no longer the client’s responsibility to train and
implement the models, as in Harris and Waggoner (2019) [6], which helps to
make setup and usage much cleaner and more accessible. Thus, PredictChain aims
to be more generalized and accessible to non-technical users.

Our solution consists of several key components to enable the sharing and
utilization of AI resources. These components include a blockchain, an oracle,
tokenomics, and the client. Figure 2 illustrates the system’s core components,
along with their methods of communication.

Figure 1: PredictChain Overall Architecture

Figure 2: PredictChain Oracle

###  3.1 Blockchain

Blockchain is useful in its role due to its immutability and its transparency.
Using the Algorand blockchain means that all requests are permanently stored
and public, so other users can see what type of models are useful for specific
datasets and what results (predictions, classifications) those models have
produced. In PredictChain, blockchain serves as a records keeper and a
messenger between the client and the oracle, accomplished by using
transactions as a form of direct communication. With every transaction sent,
there is a note. This note is a JSON-encoded string (encoded in base64) that
communicates information about the operation that the transaction is
requesting and arguments for that operation. A series of opcodes represent
these operations. These codes are abbreviations of the operation names
enclosed in angle brackets, for example, $\langle$QUERY_MODEL$\rangle$. The
arguments to these operations are represented as a _named dictionary_ , with
each key being the name of the argument and each value being the argument
itself. This named strategy allows the program to be very flexible and
agnostic about the exact ordering of the arguments.

We utilize the Algorand Blockchain [15] to implement PredictChain. The
Algorand blockchain is a decentralized, permissionless, and high-performing
blockchain well-suited for creating a decentralized AI marketplace. First,
Algorand’s unique Pure Proof of Stake (PPoS) consensus algorithm [16] and
architecture provides several advantages that make it an ideal choice for this
type of application. The PPoS consensus algorithm ensures fast and secure
transaction processing, enabling efficient transactions within the
marketplace, which means that participants can quickly and easily exchange
resources without needing a centralized intermediary, reducing transaction
costs and promoting decentralization. Second, Algorand’s smart contract
capabilities allow for the creation of smart contracts that can be used to
govern transactions within the marketplace. Smart contracts can be used to
enforce rules and regulations around the use of data and computing resources,
ensuring that participants’ privacy and security are protected. Third,
Algorand’s Layer-1 architecture enables scalable and high-performing
applications. The platform can support many transactions per second, enabling
participants to quickly and efficiently exchange resources within the
marketplace. Finally, Algorand’s commitment to sustainability and energy
efficiency aligns with the goals of creating more environmentally friendly and
sustainable machine learning models. The platform’s low energy consumption and
carbon footprint make it an attractive choice for those seeking to reduce the
environmental impact of AI development and training. Algorand’s unique
features and architecture make it an ideal choice for this application,
enabling greater collaboration and innovation. Furthermore, using ALGOs as a
payment method demonstrates that cryptocurrencies can effectively be used as a
direct form of payment for useful services.

###  3.2 Oracle

The oracle (Figure 2) acts as the bridge between the off-chain world and the
underlying blockchain, facilitating the retrieval and verification of external
data required for training and prediction. It constantly polls for updates
coming from the client through the blockchain by using its monitor. Upon
receiving these updates, it begins the execution of one of its three main
operations:  1. Downloading a user-specified dataset and saving it.  2.
Training one of the raw models based on user-inputted parameters.  3. Querying
one of the trained models on user-inputted data and comparing it to the real-
world result.  After each of these operations, the oracle sends out several
blockchain transactions. These can be either rewards to contributors of a
model or confirmations/results of the operation that has been performed.

When working with user-submitted datasets, the oracle uses a handler to manage
the operations performed on that dataset. The handler can save datasets to a
specified environment, load datasets from a specified environment, parse them
as a data frame, and split the dataset by the values of one of its attributes.
The environments that the handler recognizes are local and remote via the
InterPlanetary File System (IPFS) [17]. When using either of these
environments, the handler abstracts away the complexities of working with them
into a unified interface.

The oracle uses a similar, common interface when working with user-trained
models. This interface can create the model architecture, train the model on a
selected dataset, query the trained model, evaluate its performance, save the
model, and load it back from a specified environment. When creating and
training a model, there are affordances in the interface to choose among a
group of pre-existing archetype or template models. These models in the
initial iteration of PredictChain are:  1. Multi-layered perceptron (MLP)
neural network [18],  2. Recurrent neural network (RNN) [19],  3. Long short-
term memory neural network (LSTM) [20], and  4. Gated recurrent unit (GRU)
neural network [21].

Each of these models has a model_complexity attribute, which is a simple float
value designed to give users a general idea of how performant a model can be
once trained and serves as a method of calculating the cost of using or
training that given model. The attribute itself is calculated using the size
of the network and a linear multiplier to account for more complex model
architectures. The complexity is higher for models like GRUs or LSTMs as they
are more complex and often better-performing models [22]. Our RNN models can
predict several time steps, but their quality often degrades when a lag
between evidence and prediction is introduced [23]. For these networks and our
MLPs, the complexity is lower, which gives the desired effect of faster,
simpler models being cheaper than the slower, more complex models without
heavy calculations. The interface abstracts most of the complexities of
training, querying, and evaluating these models. The only difference between
them is the inclusion of several optional parameters.

###  3.3 Tokenomics for Resource Sharing

As the scale and power of machine learning models grow, the resources required
to train them also grow. To get useful results from these models, users often
have to pay large, centralized organizations, and they need to be incentivized
to provide good datasets or model parameters. PredictChain changes this
paradigm by incentivizing the thoughtful creation of useful and accessible
models and datasets with a transparent pricing scheme. The underlying
blockchain acts as a secure and transparent ledger that facilitates
transactions between market participants, enabling them to exchange resources
in a decentralized manner. The PredictChain marketplace allows individuals and
organizations to share their data and computational resources, which can then
be used to train machine learning models collaboratively. Tokenomics is
crucial in incentivizing participation, rewarding contributors, and
facilitating efficient transactions.

In PredictChain, the medium of exchange is ALGO, which helps to simplify the
transaction process as users will not be required to pay into yet another
token hoping that its value does not fluctuate. We expect that users will feel
more secure, as the lack of a separate token would make “rug-pull” deceptions
much more difficult to execute. Our usage of ALGO also simplifies paying into
or out of PredictChain. In this model, participants can contribute their
unused computing resources to the marketplace by hosting a node or submitting
datasets that would increase the model performance, and in exchange, they
receive ALGOs that represent their contribution. These tokens can then be used
to access computing resources from other participants in the marketplace,
purchase machine learning models trained on the data contributed by other
participants, or be used as a medium of exchange anywhere in the Algorand
ecosystem.

In PredictChain, the models and the datasets are publicly accessible to other
users via IPFS. The rewards, given as $R$ microALGOs, for uploading datasets
or training models are calculated using a public pre-determined equations.
$ds\\_size$ is the size of the dataset in bytes, $accuracy$ is the model’s
accuracy given the validation dataset or a real-world event, and the
$*\\_mult$ values are arbitrary values that the oracle sets to adjust the
price as needed due to prevailing market conditions. Each calculation has its
own value to balance out the calculations. A complete log of these adjustments
is stored on the blockchain.

To incentivize the good-faith hosting and maintenance of a PredictChain, those
who host the hardware for the network will receive a small portion of each
transaction. The oracle can determine this rate, similar to the $mult$ values
for the rewards. The results of every transaction would again be stored on the
blockchain. Clients could then examine the history of any given oracle and
determine if they offer fair transaction fees. Given enough oracles hosting
the network, this would allow clients great freedom in choosing the most
competitive oracle.

$\text{Dataset Usage and Querying Reward}=\lfloor
ds\\_size*mult*accuracy\rfloor$ (1)

$\text{Model Training Reward}=\lfloor mult*accuracy\rfloor.$ (2)

The exact value of $mult$ would vary across each implementation of
PredictChain. The following are several example values and rewards across
hypothetical models and datasets. All values are in MicroALGO.

In the following two tables, the values in the first row represent the values
of $mult$, the constant value used in reward calculations. Their variance
allows flexibility when adjusting to fluctuating ALGO prices of other market
pressures. The values in the first column represent the accuracy of the
rewarded models or those trained on a given dataset. The values in the body of
the graph are the reward values for model trainers or dataset uploaders in
MicroALGO.

| mult
---|---
accuracy |  | $\mathbf{10^{7}}$ | $\mathbf{3\times 10^{7}}$
0.3 | $3\times 10^{6}$ | $9\times 10^{6}$
0.5 | $5\times 10^{6}$ | $1.5\times 10^{7}$
0.7 | $7\times 10^{6}$ | $2.1\times 10^{7}$
0.9 | $9\times 10^{6}$ | $2.7\times 10^{7}$
0.99 | $9.9\times 10^{6}$ | $2.97\times 10^{7}$
Table 1: Training Reward Example

$ds\\_size=5MB$ | mult
---|---
accuracy |  | $\mathbf{2}$ | $\mathbf{6}$
0.3 | $3\times 10^{6}$ | $9\times 10^{6}$
0.5 | $5\times 10^{6}$ | $1.5\times 10^{7}$
0.7 | $7\times 10^{6}$ | $2.1\times 10^{7}$
0.9 | $9\times 10^{6}$ | $2.7\times 10^{7}$
0.99 | $9.9\times 10^{6}$ | $2.97\times 10^{7}$
Table 2: Database Upload Reward Example

Note: The lower $mult$ values in the dataset reward calculations 2 is due to
the size of the dataset. When the $5\times 10^{6}$ bytes of the 5MB dataset
are factored into the calculation 1, the rewards for the two calculations 2 1
are the same.

###  3.4 Client

The client is a middleman between the user and the blockchain. It is run as a
server, serving user interface content to the user, taking in requests from
the user, and parsing those requests into a form suitable for both the
blockchain and the oracle. It is separate from the oracle, being independent
of any oracle implementation. Additionally, the client constantly polls for
updates from oracle through the blockchain and its monitor. These updates are
queued and sent to the user interface upon request, allowing the user to
interact with the blockchain and see the important updates that come from it.
Furthermore, the client sets the user’s impression of PredictChain. For
example, the homepage discusses our mission, how we differ from our
competitors (such as the Shareable Updateable Sum framework by Harris and
Wagonner [7, 8, 6], and example model sets we provide (LSTM, GRU, MLP, etc.).
Users can create an account or log in to a pre-existing account via the
client, and then use PredictChain to its fullest functionality, such as adding
datasets, checking prices for model inferencing, or querying models with
provided datasets.

###  3.5 Software and Libraries Used

This work described in this paper is primarily built in Python, using the
Algorand Software Development Kit (SDK)§§§https://developer.algorand.org. The
SDK makes interacting with the Algorand blockchain very straightforward, i.e.,
through the tools provided by this library, we can easily read and write
transactions from the Algorand blockchain.

We also used data science libraries such as Pandas¶¶¶https://pandas.pydata.org
and pyTorch∥∥∥https://pytorch.org. These libraries encapsulate many of the
complexities of data preparation and model training.
Flask******https://flask.palletsprojects.com is also an important part of
PredictChain’s implementation. We used Flask to allow the client and oracle
nodes to function as servers. The client would take in requests from the user
and send out requests to the oracle. The oracle would then take in those
requests and issue responses. We chose to use Flask in client-oracle
communication to reduce the number of trivial transactions that would
otherwise be made. For example, it would not benefit the accessibility or
transparency of PredictChain greatly if the exchanged transactions were
dominated by simple ’what is the price of XXX?’ requests. Additionally, the
front-end client utilizes the React framework††††††https://react.dev and
Firebase‡‡‡‡‡‡https://firebase.google.com. React was useful to us as it
streamlined the process of making dynamic, modular code for the web interface.
Firebase was invaluable for handling administrative tasks such as keeping
track of registered users and their associated metadata within a given node
instance. For the recommended configuration of PredictChain, we use
Redis******https://redis.io as well. Redis helps to provide a reliable store
for our metadata about models and datasets in a simple, persistent manner.

##  4 Evaluation

###  4.1 Transactions

The usage of transactions is central to the communications protocol of
PredictChain. Algorand SDK allowed us to encode notes when sending
transactions, which enabled us to ensure the protocol functions correctly.
Specifically, the monitors are classes inside the client and oracle that
listen for any transactions with their node address as the recipient. This
listening uses the Algorand indexer class and constant polling for new
transactions. However, we noticed that this monitor would sometimes skip or
duplicate incoming transactions. Therefore, we constantly evaluated the
monitor’s performance, which was done by programmatically sending one or more
transactions to the client or oracle address and checking to see how the
monitor handled them. We evaluated the monitor by comparing the unique
transaction ids of the sent transactions to those processed. For example, we
constantly update the minimum timestamp the indexer can look for transactions
after, and we also keep a registry of the ids of all past transactions. This
process helps to eliminate the duplicate transactions that were getting
through.

###  4.2 Models

Another critical component of PredictChain is its usage of a variety of
models. At present, all of these models are different types of neural
networks. However, these networks are not created equally, each having
different qualities, such as model parameters, architectures, and outputs. To
evaluate the performance of these models, we ran several experiments on the
models where their hyper-parameters and architectures were kept constant as
they were tested. For an exemplary model evaluation, we measured the
performance on our sample dataset, The University of California, Irvine’s Dow
Jones Index data set [24]. This dataset has 16 different parameters detailing
the attributes of a range of stocks from the first half of 2011. As for the
models, they were all initialized with the parameters outlined in Table 3.

Table 3: Model Evaluation Parameters Parameter Name |  Value |  Description
---|---|---
epochs |  70 |  The number of epochs that the model would train for
target_attrib |  close |  The attribute that the model was trying to predict, this is the daily closing price of the stock
hidden_dim |  5 |  The number of neurons in each hidden layer
num_hidden_layers |  1 |  The number of hidden layers
time_lag |  0 |  The number of time steps that pass between the input window and the prediction
training_lookback |  10 |  The number of time steps that recurrent models receive as input
sub_split_value |  0 |  The integer id of the stock to predict, in this case, it is $AA (Alcoa Corp)

We performed this test on all of our basic model structures, specifically our
GRU, LSTM, RNN, and MLP models. In figures 6, 6, 6, and 6, we show the input
data and predictions for each model. We also show the final loss and the final
accuracy.

Figure 3: The results from our GRU model

Figure 4: The results from our LSTM model

Figure 5: The results from our RNN model

Figure 6: The results from our MLP model

Each of these results reflects the individual strengths and weaknesses of each
model. As is reflected in our model_complexity calculations, the GRU model is
the most performant, with a loss of only 0.57, which is closely followed by
the LSTM model, which still offers a good prediction, but is slightly less
consistent overall. In our evaluation of our RNNs, we noticed a good example
of the exploding gradients problem. Without the limits provided by the GRU and
LSTM models, the predictions of the RNN become increasingly erratic. Finally,
our MLP model does not suffer from an exploding gradient, but it does lack the
recurrence of the other three models, only being able to output one day at a
time.

###  4.3 End-To-End Testing

The final component of our testing was the end-to-end series of tests we
performed. These tests gave us a holistic picture of how well PredictChain
functioned and how well that functioning met our stated goals. We started with
front-end user interaction for all these tests and ended with the response to
those actions.

#### Price Queries

The three price query operations were the first and most straightforward
operations we tested. We performed this test by submitting query requests on
the user interface, then tracing the functions called by that operation from
the client to the oracle, then to the oracle getting the price, then finally
back to the client. We performed this test for all three queries, testing
different sets of valid and invalid inputs to see if they were handled
properly.

#### Major Transactions

The next set of tests we performed focused on our major transactions: Upload
Dataset, Train Model, and Query Model. These tests were similar to those for
the price queries. As we performed these tests, we noted that, while the
operations did complete successfully, the user feedback and results were
ambiguous. Additionally, we realized that it might be confusing for the user
to remember the exact names of the models and datasets to use them. To address
these issues, we made several additions. To address the issue of users having
to input the exact names, we added a dropdown feature instead of a test input.
We now store a list of the system’s current datasets and models, updated upon
reloading the page or when a user submits a transaction. This process ensures
that the list is always updated for better ease of use. To address the
feedback issue, we added a feedback section that periodically pinged the
client to see if any new response transactions came in from the oracle. If
these transactions did come in, we would display the operation, the model or
dataset name, and any extra data (like the result of a query) to the user.
This way, the user would not manually check the transaction note on the
Algorand block explorer.

By performing these tests, we evaluated the quality of PredictChain as a whole
and made important additions to improve the user experience. Our transaction
tests helped us to identify the issues that had previously existed and to
verify that the current communication protocol was working properly. Our model
tests helped us to confirm our previous assumptions about the nature of our
various models and which scenarios they are most useful. Finally, our end-to-
end testing confirmed the overall functioning of PredictChain and inspired us
to make some valuable improvements to the user experience.

##  5 Conclusion

Through PredictChain, users can upload datasets for training predictive
machine learning models, request that basic models be trained on any
previously uploaded datasets, or submit queries to those trained models.
Anybody participating in the PredictChain ecosystem can host a node with the
necessary computing resources to make machine learning models available as a
public good and benefit financially by naming a market-driven price for
training these models on user-supplied datasets. PredictChain can enable
various models, ranging from cheap, fast, and simple to more expensive,
slower, and powerful. This will allow for various predictive abilities for
both simple and complex needs. Furthermore, these models’ past predictions are
stored on the blockchain for public viewing and faster and free access by
subsequent users.

As our work on PredictChain progresses, we recognize several opportunities for
further enhancement. One key improvement involves providing users who upload
datasets with greater flexibility in data preprocessing, potentially
incorporating feature engineering techniques to optimize model performance.
Additionally, expanding the repertoire of models offered on PredictChain
beyond neural networks to include decision trees and statistical models based
on Bayesian inference would enhance the marketplace’s versatility.
Furthermore, allowing users to customize the training process by pruning
irrelevant attributes from datasets could lead to more tailored and efficient
model training. PredictChain can evolve into a more comprehensive and refined
platform by implementing these future improvements while adhering to its core
principles. Empowering users with greater control over their data and
expanding the variety of models and datasets available will foster
collaboration, innovation, and democratization of AI, propelling us towards
new frontiers of knowledge and applications.

In conclusion, we have successfully developed PredictChain, a fully functional
blockchain-based marketplace for predictive machine learning models. Users can
leverage PredictChain to upload datasets for model training, request training
on existing datasets, and submit queries to trained models. All past
predictions are securely stored on the blockchain, promoting transparency and
public accessibility. With ongoing development and community involvement,
PredictChain has the potential to revolutionize the AI landscape, shaping a
future where accessibility, transparency, and shared intelligence are
paramount.

Resource Availability: Our source code, demo, and comprehensive documentation
are available in our GitHub repo at https://github.com/AI-and-
Blockchain/S23_PredictChain.

## Acknowledgements

This work was supported by the Algorand Centres of Excellence program managed
by the Algorand Foundation. Any opinions, findings, conclusions, or
recommendations expressed in this material are those of the author(s) and do
not necessarily reflect the views of the Algorand Foundation.

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Konstantinos Dimopoulos

# Explaining the Hubble tension and dark energy from $\alpha$-attractors

Lucy Brissenden  Samuel Sánchez López

###### Abstract

A compelling unified model of dark energy and early dark energy (EDE) is
presented, using a scalar field with an exponential runaway potential, in the
context of alpha-attractors. The field is originally trapped at an enhanced
symmetry point, subsequently thaws to become successful EDE and eventually
slow-rolls to become dark energy. EDE ameliorates the observed Hubble tension.

##  1 Introduction

Since the arrival of high precision cosmological observations, a standard
model of cosmology, called the concordance model, has been constructed. In
short called $\Lambda$CDM, this concordance model recently has starting to
suffer from a number of challenges, the most important of which is arguably
the Hubble tension.

In a nutshell, the Hubble tension amounts to a disagreement in the estimate of
the Hubble constant $H_{0}$ as inferred by early (mainly CMB) and late (mainly
SNe) observations. Indeed, CMB observations from the Planck satellite [1]
suggest

$H_{0}=67.44\pm 0.58\ \textnormal{km}\
\textnormal{s}^{-1}\textnormal{Mpc}^{-1},$ (1)

while a distance scale measurement using Cepheid-SN-Ia data from the SH0ES
collaboration [2] results in

$H_{0}=73.04\pm 1.04\ \textnormal{km}\
\textnormal{s}^{-1}\textnormal{Mpc}^{-1}.$ (2)

This is a $5\sigma$ tension (4.56-6.36$\sigma$). The most compelling proposal
to overcome this problem is early dark energy (EDE).

##  2 Early Dark Energy

EDE amounts to momentary importance of dark energy near matter-radiation
equality. Investigated first by Refs. [3, 4, 5, 6], this proposal does not
necessarily consider that EDE is the same dark energy substance which is
responsible for the accelerated expansion at present (for a recent review see
Ref. [7]).

How does EDE manage to increase the value of $H_{0}$ as inferred from CMB
observations? Even though CMB and BAO observations tightly constrain the
cosmological parameters, they constrain the combination $H(z)r_{s}$, where
$H(z)$ is the Hubble parameter as a function of redshift $z$, and $r_{s}$ is
the comoving sound horizon at decoupling, given by

$r_{s}=\int^{\infty}_{z_{\rm dec}}\frac{c_{s}(z)}{H(z)}dz,$ (3)

where $c_{s}(z)$ is the sound speed. An additional amount of dark energy in
the Universe increases the total density, which in turn increases the Hubble
parameter because of the Friedmann equation $H^{2}=(\rho_{B}+\rho_{{}_{\rm
EDE}})/3m_{m_{\text{P}}}^{2}$, where $\rho_{B}$ is the density of the
radiation and matter background. EDE amounts to a brief increase of $H(z)$
before decoupling, which lowers the value of the sound horizon in Eq. (3).
Thus, EDE manages to simultaneously lower the value of $r_{s}$ and increase
$H_{0}$ without violating CMB observations.

The fractional energy density required for EDE to work is about 10%,
$f_{{}_{\rm EDE}}=0.10\pm 0.02$ at redshift $z_{c}=4070^{+400}_{-840}$.
Therefore, the EDE proposal amounts to as injection of energy at around the
time of matter-radiation equality ($z_{c}\simeq z_{\rm eq}\simeq 3600$), which
then decays away faster than the background radiation, such that it becomes
negligible at the time of last scattering, before it can be detected in the
CMB [4].

The original proposal in Ref. [3] suggested that the EDE was an axion scalar
field $\phi=\theta f$ with potential $V(\theta)=m^{2}f^{2}(1-\cos\theta)^{n}$
with $n>2$. The authors of Ref. [3] found that the fractional energy density
must be $f_{{}_{\rm EDE}}=0.08\pm 0.04$, which results in $H_{0}=70.0\pm
1.5\,$km$\,{\rm s}^{-1}{\rm Mpc}^{-1}$. After thawing, the EDE field
oscillates around its vacuum expectation value (VEV) with average barotropic
parameter $w=\frac{n-1}{n+1}$. To redshift faster than radiation, it is needed
that $w>\frac{1}{3}$, which implies that the minimum is of order higher than
quartic. Note that the density of the oscillating EDE redshifts as
$a^{-6n/(n+1)}$, which reduces to $a^{-6}$ (free-fall) in the limit $n\gg 1$.
The situation is similar to many other EDE models, where typically the EDE
scalar field oscillates around its VEV in a high order potential (see however,
Ref. [8]). In contrast, in our model presented below, the EDE scalar field
experiences a period of kinetic domination, where the field is in non-
oscillatory free-fall and its density decreases as $\propto a^{-6}$.

##  3 $\alpha$-attractors

Our model unifies EDE with late dark energy in the context of
$\alpha$-attractors. Ref. [9] is an earlier attempt for such unification in
the same theoretical context. However, that proposal is also of oscillatory
EDE.

$\alpha$-attractors appear naturally in conformal field theory or supergravity
theories [10, 11, 12, 13]. The scalar field has a non-canonical kinetic term,
featuring two poles, which the field cannot cross. The field can be
canonically normalised via a field redefinition. Then, the finite poles for
the non-canonical field are transposed to infinity for the canonical one. As a
result, the scalar potential is “stretched” near the poles, featuring two
plateau regions, which have been used for modelling inflation, see e.g. Refs.
[14, 15, 16, 17, 18, 19, 20] or quintessence [21], or both, in the context of
quintessential inflation [22, 23, 21].

The Lagrangian density features two poles at
$\varphi=\pm\sqrt{6\alpha}\,m_{\text{P}}$ and has the form

$\mathcal{L}=\frac{-\frac{1}{2}(\partial\varphi)^{2}}{\left(1-\frac{\varphi^{2}}{6\alpha\,m_{\text{P}}^{2}}\right)^{2}}-V(\varphi)\,,$
(4)

where $\varphi$ is the non-canonical scalar field and
$(\partial\varphi)^{2}\equiv
g^{\mu\nu}\partial_{\mu}\varphi\,\partial_{\nu}\varphi$. Redefining $\varphi$
in terms of the canonical scalar field $\phi$, we have

$\text{d}\phi=\frac{\text{d}\varphi}{1-\frac{\varphi^{2}}{6\alpha
m_{\text{P}}^{2}}}\quad\Rightarrow\quad\varphi=m_{\text{P}}\sqrt{6\alpha}\,\tanh{\left(\frac{\phi}{\sqrt{6\alpha}\,m_{\text{P}}}\right)}\,.$
(5)

The poles $\varphi=\pm\sqrt{6\alpha}\,m_{\text{P}}$ are transposed to infinity
and the Lagrangian density now reads

$\mathcal{L}=-\frac{1}{2}(\partial\phi)^{2}-V(\phi).$ (6)

##  4 The Model

In contrast to most EDE literature, we investigate non-oscillating EDE. Thus,
we require the scalar potential to be steep enough, such that, after equality
of matter and radiation, the EDE scalar field becomes dominated by its kinetic
energy density and engages in "free-fall" roll. Therefore, we study the
following toy-model.

Consider a potential of the form

$V(\varphi)=V_{X}\exp(-\lambda e^{\kappa\varphi/m_{\text{P}}}),$ (7)

where $\alpha,\kappa,\lambda$ are dimensionless model parameters, $V_{X}$ is a
constant energy density scale and $\varphi$ is the non-canonical scalar field
with kinetic poles given by the typical $\alpha$-attractors form with the
Lagrangian density in Eq. (4).

To assist our intuition, we switch to the canonically normalised (canonical)
scalar field $\phi$, using the transformation in Eq. (5). The Lagrangian
density is then given by Eq. (6), where the scalar potential is

$V(\phi)=\exp(\lambda e^{\kappa\sqrt{6\alpha}})V_{\Lambda}\exp[-\lambda
e^{\kappa\sqrt{6\alpha}\tanh(\phi/\sqrt{6\alpha}\,m_{\text{P}})}]\,,$ (8)

where $V_{\Lambda}$ is the vacuum density at present related to the model
parameters as

$V_{\Lambda}\equiv\exp(-\lambda e^{\kappa\sqrt{6\alpha}})V_{X}\,.$ (9)

Note that the model parameter is $V_{X}$ and not $V_{\Lambda}$, the latter
being generated by $V_{X}$ and the remaining model parameters as shown above.

##  5 Analytic study

We are interested in two limits for the potential: matter-radiation equality
and the present time. At matter-radiation equality, we consider
$\phi\rightarrow 0$ ($\varphi\rightarrow 0$). In this limit, we have

$V_{\textnormal{eq}}\simeq\exp[\lambda(e^{\kappa\sqrt{6\alpha}}-1)]V_{\Lambda}\exp(-\kappa\lambda\,\phi_{\textnormal{eq}}/m_{\text{P}})\,,$
(10)

where the subscript ‘eq’ denotes the time of matter-radiation equality. It is
assumed that the field was originally frozen there and at the time of equality
in unfreezes (thaws). We discuss and justify this assumption in Sec. 7.

After thawing the field soon rolls towards large values. Today, we consider
$\phi\rightarrow+\infty$ ($\varphi\rightarrow+\sqrt{6\alpha}\,m_{\text{P}}$).
The potential in this limit is

$V_{0}\simeq V_{\Lambda}\left[1+2\kappa\lambda
e^{\kappa\sqrt{6\alpha}}\sqrt{6\alpha}\,\exp\left(-\frac{2\phi_{0}}{\sqrt{6\alpha}\,m_{\text{P}}}\right)\right],$
(11)

where the subscript ‘0’ denotes the present time. Note that, in this limit,
the potential approaches $V_{\Lambda}$, which corresponds to positive vacuum
density with $w=-1$, as in $\Lambda$CDM.

The above approximations describe well the scalar potential near equality and
the present time. As explained below, between these regions, the scalar field
free-falls and becomes oblivious of the scalar potential.

Let us investigate the evolution of the EDE field. Originally the field is
frozen at zero (see Sec. 7). Its energy density is such that it remains frozen
there until equality, when it thaws following the appropriate exponential
attractor, since $V_{\rm eq}$ in Eq. (10) is approximately exponential [24].

For convenience, we assume this is the subdominant attractor, which requires
that the strength of the exponential is [25, 26]

$\kappa\lambda>\sqrt{3}\,.$ (12)

The subdominant exponential attractor is called the scaling attractor. In the
scaling attractor the energy density of the rolling scalar field mimics the
dominant background energy density. Thus, the fractional energy density of the
field is constant, given by the value [24, 25, 26]

$f_{{}_{\rm EDE}}\simeq\frac{3}{(\kappa\lambda)^{2}}<1$ (13)

This provides an estimate of the moment when the originally frozen scalar
field, unfreezes and begins rolling down its potential. Before unfreezing
$f_{{}_{\rm EDE}}$ is growing, because the background density decreases with
the expansion of the Universe, until $f_{{}_{\rm EDE}}$ obtains the above
value.

However, after unfreezing, the field soon experiences the full exp(exp)
steeper than exponential potential so, it does not follow the subdominant
attractor any more but it is dominated by its kinetic energy density only (it
free-falls). Then, its density scales as
$\rho_{\phi}\simeq\frac{1}{2}\dot{\phi}^{2}\propto a^{-6}$, until it refreezes
at a larger value $\phi_{0}$. This value is estimated as follows.

In free-fall, the equation of motion is reduced to
$\ddot{\phi}+3H\dot{\phi}\simeq 0$, where $H=2/3t$ after equality. The
solution is

$\phi(t)=\phi_{\rm eq}+\frac{C}{t_{\rm eq}}\left(1-\frac{t_{\rm
eq}}{t}\right)\,,$ (14)

where $C$ is an integration constant. From the above, it is straightforward to
find that $\dot{\phi}=Ct^{-2}$. Thus, at equality we have

$\displaystyle f_{{}_{\rm EDE}}=\left.\frac{\rho_{\phi}}{\rho}\right|_{\rm
eq}=\frac{\frac{1}{2}C^{2}t_{\rm
eq}^{-4}}{\frac{4}{3}(\frac{m_{\text{P}}}{t_{\rm
eq}})^{2}}=\frac{3}{8}\frac{C^{2}}{(m_{\text{P}}t_{\rm eq})^{2}}$
$\displaystyle\Rightarrow$ $\displaystyle
C=\sqrt{\mbox{$\frac{8}{3}$}f_{{}_{\rm EDE}}}\,m_{\text{P}}\,t_{\rm
eq}=\frac{\sqrt{8}}{\kappa\lambda}\,m_{\text{P}}\,t_{\rm eq}\;,$ (15)

where we used Eq. (13), $\rho_{\phi}\simeq\frac{1}{2}\dot{\phi}^{2}$ and that
$\rho=1/6\pi Gt^{2}=\frac{4}{3}(m_{P}/t)^{2}$. Therefore, the field freezes at
the value

$\phi_{0}=\phi_{\rm eq}+C/t_{\rm eq}=\phi_{\rm
eq}+\frac{\sqrt{8}}{\kappa\lambda}\,m_{\text{P}}\;,$ (16)

where we considered that $t_{\rm eq}\ll t_{\rm freeze}<t_{0}$ .

Using that $t_{\rm eq}\sim 10^{4}\,$y and $t_{0}\sim 10^{10}\,$y, we can
estimate

$\frac{V_{\rm eq}}{V_{0}}\simeq\frac{f_{{}_{\rm EDE}}\rho_{\rm
eq}}{0.7\,\rho_{0}}\simeq\frac{30}{7(\kappa\lambda)^{2}}\left(\frac{t_{0}}{t_{\rm
eq}}\right)^{2}\simeq\frac{3}{7(\kappa\lambda)^{2}}\times 10^{13}\,.$ (17)

Now, from Eqs. (10) and (11) we find

$\frac{V_{\rm
eq}}{V_{0}}\simeq\frac{e^{\lambda(e^{\kappa\sqrt{6\alpha}}-1)}\exp(-\kappa\lambda\,\phi_{\rm
eq}/m_{\text{P}})}{1+2\kappa\lambda\,e^{\kappa\sqrt{6\alpha}}\sqrt{6\alpha}\exp(-2\phi_{0}/\sqrt{6\alpha}\,m_{\text{P}})}\,.$
(18)

Considering that $\phi_{\rm eq}\simeq 0$ and Eq. (16), the above can be
written as

$\frac{V_{\rm
eq}}{V_{0}}\simeq\frac{e^{\lambda(e^{\kappa\sqrt{6\alpha}}-1)}}{1+2\kappa\lambda\,e^{\kappa\sqrt{6\alpha}}\sqrt{6\alpha}\,e^{-2\sqrt{8}/\kappa\lambda\sqrt{6\alpha}}}\,.$
(19)

Taking $f_{{}_{\rm EDE}}\simeq 0.1$ as required by EDE, Eq. (13) suggests

$\kappa\lambda\simeq\sqrt{30}\,.$ (20)

Combining this with Eq. (17) we obtain

$e^{\frac{\sqrt{30}}{\kappa}(e^{\kappa\sqrt{6\alpha}}-1)}\sim 10^{12}/7\,,$
(21)

where we have ignored the 2nd term in the denominator of the right-hand-side
of Eq. (19).

From the above we see that, $\kappa$ is large when $\alpha$ is small. Taking,
as an example, $\alpha=0.01$ we obtain $\kappa\simeq 18$ and $\lambda\simeq
0.30$ (from Eq. (20)). With these values, the second term in the denominator
of the right-hand-side of Eq. (19) is of order unity and not expected to
significantly influence our results.

For the selected values, Eq. (16) suggests that the total excursion of the
field is

$\Delta\phi=\phi_{0}-\phi_{\rm eq}=\frac{\sqrt{8}}{\kappa\lambda}\,m_{P}\simeq
0.5\,m_{P}\;,$ (22)

i.e. it is sub-Planckian. A sub-Planckian excursion of the field implies that
5th force considerations are suppressed.

##  6 Numerical investigation

We have thoroughly analysed this model in Ref. [27]. Here, we will present our
main results.

We have aimed to obtain a value of $H_{0}$ in the window

$72\leq\frac{H_{0}}{\rm km\;sec^{-1}\,Mpc^{-1}}\leq 74\,.$ (23)

With this requirement, the parameter space arrived at for our model parameters
is

$\displaystyle 0<\alpha<0.00071$ $\displaystyle 0<\kappa<700$ $\displaystyle
0<\lambda<0.027\;,$ (24)

with $V_{\Lambda}=10^{-120.068}\,m_{\text{P}}^{4}$. We see that the above
numbers are reasonable. In particular, the value of $\kappa\sim 10^{2}$
implies that the mass scale suppressing the exponent in our model in Eq. (7)
is near the scale of grand unification $m_{\text{P}}/\kappa\sim 10^{16}\,$GeV,
which is a rather natural scale.

In the above ranges, we find that $0.015<f_{{}_{\rm EDE}}<0.107$ at equality,
while it becomes lower than $10^{-3}$ by decoupling; the time the CMB
radiation is emitted. The barotropic parameter of dark energy at present is
$w_{\phi}=-1.000$ with negligible running (less than $10^{-11}$), which is
indistinguishable from $\Lambda$CDM.

One important finding is that the condition in Eq. (12),
$\kappa\lambda>\sqrt{3}$ assumed in the previous section, is not valid.
However, this was chosen only for convenience, as explained before Eq. (12).
If the condition is violated then the thawing EDE does not follow the scaling
exponential attractor but the dominant exponential attractor instead. In both
cases however, once the EDE field rolls away from zero, it starts experiencing
the full exp(exp) potential and goes into free-fall, as discussed. Thus, the
qualitative behaviour is the same, as also demonstrated by our numerical
results shown below.

As a concrete example, we choose the following values for the model parameters

$\displaystyle\alpha=0.0005$ $\displaystyle\kappa=145$
$\displaystyle\lambda=0.008125\,.$ (25)

The above suggest that $\kappa\lambda=1.178<\sqrt{3}$. The value of the Hubble
constant obtained in this case is

$H_{0}=73.27\;{\rm km}\;{\rm sec}^{-1}\,{\rm Mpc}^{-1},$ (26)

which evidently is well in agreement with the SH0ES observations. Comparison
of the Hubble parameter in this scenario with the one in $\Lambda$CDM is shown
in Fig. 1.

Figure 1: The Hubble parameter (in units of $\textnormal{km}\
\textnormal{s}^{-1}\textnormal{Mpc}^{-1}$) of the Universe in our model
(green), a classical $\Lambda$CDM simulation (black), and one with only matter
and radiation (blue), as a function of the redshift (top) and the e-folds
(bottom) elapsed since the beginning of the simulation. It is evident that our
model corresponds to a larger value of $H(z)$ than $\Lambda$CDM, as desired.

The behaviour of the fractional energy density $f_{{}_{\rm EDE}}$, which is
identified with the EDE density parameter $\Omega_{\phi}(z)$ is shown in Fig.
2. It is evident that, for this example, $f_{{}_{\rm
EDE}}=\Omega_{\phi}(z_{\rm eq})\simeq 0.08$.

In view of Eq. (9), we find $\log(V_{X}/V_{\Lambda})=9.926$. Thus, our model
parameter $V_{X}=10^{-110.142}\,m_{\text{P}}^{4}$ is fine-tuned at the same
level (slightly less) than $V_{\Lambda}$ in $\Lambda$CDM. However, it has to
be stressed that, in contrast to $\Lambda$CDM, our proposal addresses
simultaneously two cosmological problems; not only late dark energy but also
the Hubble tension.

Figure 2: The density parameter of the scalar field $\Omega_{\phi}$ as a
function of the redshift (top) and e-folds (bottom) elapsed since the
beginning of the simulation. As shown, at equality, there is a bump with
$f_{{}_{\rm EDE}}=\Omega_{\phi}(z_{\rm eq})$ with $f_{{}_{\rm EDE}}\simeq
0.08$.

The barotropic parameters, of EDE and the background, are shown in Fig. 3. It
can be seen clearly that, after thawing, the barotropic parameter of EDE is
$w_{\phi}=1$ and the field is in free-fall as discussed. Its density decreases
as $a^{-6}$ as clearly shown in Fig. 4, which corresponds to the
$n\rightarrow\infty$ limit of the oscillating EDE in Ref. [3] and it is never
attained by any oscillating EDE model. Thus, our model disturbs the emission
of the CMB at decoupling in the least amount possible.

Figure 3: Barotropic parameter of the scalar field (dotted green), of the
background perfect fluid (full blue) and of the sum of both components (full
black). It is evident that, after unfreezing, the EDE scalar field is in free-
fall, with $w_{\phi}=1$, until it refreezes again. Figure 4: The logarithmic
densities of matter (dot-dashed red), radiation (dotted orange), the sum of
both (solid blue) and the EDE scalar field (dashed green), as a function of
the redshift (top) and the e-folds (bottom) elapsed since the beginning of the
simulation. The horizontal full line represents the (S$H_{0}$ES) energy
density of the Universe at present. As shown, after it briefly becomes
important near equality, the density of EDE scalar field is reducing
drastically and it is orders of magnitude smaller than that of the matter
background by the time of decoupling ($z_{\rm dec}=1090$), as required.

Finally, for our example we obtain that the total excursion of the EDE field
from thawing to refreezing is sub-Planckian: $\Delta\phi/m_{\text{P}}=0.4274$,
in agreement with Eq. (22). This implies both that our model does not suffer
from fifth force problems and our potential is stable against radiative
corrections.

##  7 Trapping at the origin

A compelling explanation why the EDE scalar field finds itself frozen at the
origin in the first place is the following. If the origin is an enhanced
symmetry point (ESP), then at very early times, an interaction of $\varphi$
with some other scalar field $\sigma$ can trap the rolling $\varphi$ at zero
[28]. The scalar potential includes the interaction

$\Delta V=\frac{1}{2}g^{2}\varphi^{2}\sigma^{2}\,,$ (27)

where the coupling $g<1$ parametrises the strength of the interaction.

We assume that initially $\varphi$ is rolling down its steep potential, which
away from the origin, does not have to be of the form in Eq. (7). In fact, it
is conceivable that $\varphi$ might play the role of the inflaton field too
[27]. The original kinetic energy density of $\varphi$ is depleted due to
particle production of $\sigma$-particles, because their mass $\sim g\varphi$
changes non-adiabatically near the origin [28]. Note that, near the origin,
the $\varphi$-field is approximately canonically normalised.

As the field moves past the ESP, the produced $\sigma$ particles give rise to
an effective linear potential $\sim gn_{\sigma}|\varphi|$ [28], where
$n_{\sigma}$ is the number density of the produced $\sigma$-particles. This
linear potential halts the roll of $\varphi$ and reverses its variation. More
$\sigma$-particles are created when $\varphi$ crosses the origin again,
resulting in a steeper linear potential, which reverses the variation of
$\varphi$ again, closer to the origin this time. The process continues until
the $\varphi$-field is trapped at the origin [26, 28].

The trapping of a rolling scalar field at an ESP can take place only if the
$\sigma$-particles do not decay at maximum displacement. The end result of
this process is that all the kinetic energy density of the rolling $\varphi$
has been given to the $\sigma$-particles. Since $\varphi$ is trapped at zero,
the $\sigma$-particles are relativistic, which means that their density scales
as radiation, being a subdominant part of the thermal bath. As far as
$\varphi$ is concerned, it is trapped at the origin and its density is
$\rho_{\varphi}=V(\varphi=0)=e^{-\lambda}V_{X}=\,$constant.

After some time, the $\sigma$-particles may decay into the standard model
particles, which comprise the thermal bath of the hot Big Bang. Because the
confining potential is proportional to $n_{\sigma}$, it disappears. However,
the EDE $\varphi$-field remains frozen at the origin because the scalar
potential $V(\varphi)$ in Eq. (7) is flat enough there. The EDE
$\varphi$-field unfreezes again in matter-radiation equality.

The above scenario is one possible explanation of the initial condition
considered. Numerical simulations simply assume that the field begins frozen
at the origin. Other possibilities to explain our initial condition exist, for
example considering a thermal correction of the form $\delta V\propto
T^{2}\varphi^{2}$, which would drive the $\varphi$-field towards the origin at
high temperatures.

##  8 Conclusions

The concordance model $\Lambda$CDM suffers from the Hubble tension at
5$\sigma$. A prominent resolution of this tension is early dark energy (EDE).
EDE amounts to a dark energy substance, which momentarily becomes about 10% of
the total energy density near matter-radiation equality, but decays faster
than radiation afterwards.

EDE in the context of $\alpha$-attractors can unify EDE with late dark energy
without more fine-tuning than $\Lambda$CDM. We studied such a model of EDE,
characterised by the exp(exp) potential in Eq. (7). Our EDE is originally
frozen at the origin. Near equality it thaws, then it free-falls down its
runaway potential until it refreezes before today, when it becomes late dark
energy.

We have investigated numerically our model and demonstrated that it works for
natural values of the parameters. We also showed that the field excursion
between the initial and final frozen values is sub-Planckian, which means that
our model does not suffer from a fifth force problem and it is not unstable
against radiative corrections.

Acknowledgements: LB is supported by STFC. KD is supported (in part) by the
Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC
grant: ST/T001038/1. SSL is supported by the FST of Lancaster University. For
the purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.

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