# Deep Learning for Enhanced Scratch Input

Aman Bhargava<EMAIL_ADDRESS>University of Toronto27 King’s
College CircleTorontoOntarioCanadaM5S 1A1 ,  Alice X. Zhou
<EMAIL_ADDRESS>University of Toronto27 King’s College
CircleTorontoOntarioCanadaM5S 1A1 ,  Adam Carnaffan
<EMAIL_ADDRESS>University of Toronto27 King’s College
CircleTorontoOntarioCanadaM5S 1A1 and  Steve Mann<EMAIL_ADDRESS>University
of Toronto & MannLab Canada27 King’s College Circle330 Dundas St.
WTorontoOntarioCanada

###### Abstract.

The vibrations generated from scratching and tapping on surfaces can be highly
expressive and recognizable, and have therefore been proposed as a method of
natural user interface (NUI). Previous systems require custom sensor hardware
such as contact microphones and have struggled with gesture classification
accuracy.

We propose a deep learning approach to scratch input. Using smartphones and
tablets laid on tabletops or other similar surfaces, our system achieved a
gesture classification accuracy of 95.8%, substantially reducing gesture
misclassification from previous works. Further, our system achieved this
performance when tested on a wide variety of surfaces, mobile devices, and in
high noise environments.

The results indicate high potential for the application of deep learning
techniques to natural user interface (NUI) systems that can readily convert
large unpowered surfaces into a user interface using just a smartphone with no
special-purpose sensors or hardware.

scratch input, natural user interface, reality user interface, machine
learning, deep learning, gestural interface, audio classification

††conference: ; ; ††ccs: Human-centered computing Gestural input††ccs: Human-
centered computing Sound-based input / output††ccs: Human-centered computing
Accessibility technologies††ccs: Human-centered computing HCI theory, concepts
and models††ccs: Human-centered computing User interface toolkits

Figure 1. Generalized flow diagram for a scratch input system. Note that
multiple vibrational receivers and control systems may be used, assuming that
the gesture classifier is correctly configured.

Generalized scratch input system flow diagram.

##  1\. Introduction

There is substantial information, both frequency and time-based, contained in
scratches, taps, and swipes made on a textured surface. One can easily observe
that the type of gesture, speed, intensity, and shape significantly alter the
sound produced. While other factors such as different surface materials and
background noise alter the sounds, most gestures remain recognizable.

Scratch input offers the possibility of rapidly deployable, low cost, large
scale touch input surfaces. With a sufficiently high-performance system, a
wide range of environments could be configured on an ad hoc basis to become an
immersive, expressive user interface. Nearly any “scratchable” surface could
potentially become an interface, producing a form of natural user interface
augmented reality (Figure 1). Potential use cases include home automation,
accessibility technology, and user interface in hazardous environments
(discussed in Section 1.2).

###  1.1. Prior Work on Scratch Input

This expressiveness of scratching, tapping, and swiping surfaces is likely
what inspired the first documented instance of scratch input (Mann, 2007).
Mann et al initially proposed scratch input as a source signal and modulation
trigger for a novel musical instrument known as a “physiphone” or a
“hyperacoustic instrument” (Mann et al., 2008). Shortly thereafter, scratch
input was proposed in a human-computer interaction system by Chris Harrison in
2008 (Harrison and Hudson, 2008).

In this work, a custom sensing apparatus made of a microphone and a modified
stethoscope was used in low-noise environments to classify a set of 6
gestures. Based on energy peak counting and amplitude variation, a hand-
written shallow decision tree was used for classification. The gestures
investigated all differed clearly in terms of number of energy peaks (e.g.,
single tap, double tap) or amplitude variation (i.e., scratch vs. tap). To
evaluate the effectiveness of the system, users were given five minutes to
practice the gestures as they received real-time feedback on the quality of
their gestures from the system. Experimenters also provided advice during this
time on how participants should adjust their gestures for maximum accuracy.
Finally, the 15 participants were requested to perform each of the 6 gestures
5 times each in random order as gesture classification accuracy was recorded.

The study yielded an overall accuracy of 89.5%, with lower accuracy for
complex gestures (triple swipe, quadruple swipe). The investigation offered an
overview of potential applications of scratch input including device
enclosures and clothing fabric, though the noise in such environments was
noted as a challenge for the practical application of scratch input
technology.

Further work has been done by Harrison and others on similar topics since,
including investigations on the human body as a transmission medium and input
surface in 2010 called Skinput (Harrison et al., 2010). In this work, Harrison
et al successfully localize finger taps on the arm and hand into 20 areas. 186
features were computed based on signal amplitude, standard deviation, Fourier
transform, and total energy from each of the 5 receivers in the arm band.
Using a support vector machine, a mean classification accuracy of 87.6% was
achieved. In 2011, Harrison introduced TapSense, a system to enable the
identification of the object being used to tap a surface (Harrison et al.,
2011). A modified stethoscope similar to the apparatus of (Harrison and
Hudson, 2008) was used for sensing. Using a support vector machine with 559
features extracted from a Fourier transform computed on a sliding 43ms window,
the system achieved roughly 95% accuracy for differentiating fingertip,
fingernail, knuckle, and finger pad taps. Similar methods were used by Ono et
al in 2013 to recognize different types of touches and grasps on objects using
a paired vibration speaker and piezo-electric microphone (Ono et al., 2013). A
support vector machine with 400 features extracted from a windowed Fourier
transform was used to recognize a variety of object grasp positions.

The general progression in user input and gesture recognition based on
vibrations has been toward making use of progressively higher dimensionality
features and more complex classifier algorithms. Naturally, questions arise on
the applicability of deep learning techniques in such systems to enhance
classification accuracy.

###  1.2. Potential Use Cases

In addition to presenting a highly accurate scratch input gesture classifier,
we highlight the utility of the proposed system for “internet of things” (IoT)
implementations in home automation, for human-computer interaction
accessibility, and for use in harsh environments where conventional computer
interaction methods are infeasible (e.g., front-line workers in biohazardous
and other extreme environments).

####  1.2.1. Home Automation

The proposed system could readily be used in a variety of home automation use
cases. High-accuracy user input on a relatively small set of gestures is
readily applied to control tasks in home automation (Villanueva and Drögehorn,
2018). Advantages of the proposed scratch input system include the rapid
deployment of extremely large, low-cost interface surfaces and environments.
With several low-cost microphones or networked mobile devices, any space could
be converted into an interactive, controllable, and responsive environment.

Since any scratching or tapping gesture can be used for the system that
consistently differs from the rest acoustically, gestures could also be made
intuitive and language-independent. For example, the vertical scratch and
circular scratch gestures, resembling a binary ‘1’ (on) and ‘0’ (off)
respectively, may be used to turn on and off a lamp. Similarly, the W-scratch,
resembling iconographic depictions of water (Siegeltuch, 2017), may be used
for faucets, flushing, or other related functions.

When compared with existing commonly sold voice assistants for home
automation, the proposed system also stands out in terms of privacy. All
computation for gesture classification and noise discrimination could easily
be implemented onboard with a low-cost microprocessor or system on a chip.
Meanwhile, voice assistants today are nearly universally “cloud based”,
entailing a host of security concerns (Lei et al., 2017). To execute commands,
voice recordings are recorded and sent to a remote system (“cloud”) for
processing before execution. In production, the proposed deep learning-based
scratch user interface system could easily separate the gesture classification
system from the networked components, drastically increasing user privacy and
security.

####  1.2.2. Accessibility

As addressed in (Harrison and Hudson, 2008), the majority of computer systems
in common use today are relatively small for ease of transportation (e.g.,
mobile phones, laptops). Even in the home, where it may be possible to deploy
a larger scale user interface system, it is often inconvenient and
prohibitively expensive. This problem is especially important for those people
affected by disabilities relating to pain, flexibility, and dexterity.
According to a 2012 Canadian Survey on Disability, over 25% of those reported
to have some type of disability were identified as having dexterity disability
that limited their daily activity (Christine Bizier, 2012).

A natural user interface (NUI) system that could easily expand to the size of
a table could help to ameliorate these limitations on daily activity as they
relate to the use of ubiquitous computing systems. As will be discussed later,
the proposed system could also be configured to continuously learn and improve
on a user’s data, enabling adaptation to each individual’s accessibility
needs. Since the proposed system was validated and tested with a wide variety
of consumer smartphone and tablet microphones, it would be rapidly deployable
and very likely low-cost for the end user given the widespread use of
smartphones.

####  1.2.3. Hazardous Environments

Given the proposed system’s ease of use for system control tasks like home
automation in with little to no reliance on dexterity, it is logical to
propose its use in extreme environments. In areas of extreme cold, for
instance, thick protective gear must be worn that severely inhibits manual
dexterity. While conventional computer system input devices may be difficult
to use, scratches and taps remain easy to replicate. A set of gestures
optimized for use with protective equipment could be easily designed and
implemented with the following system, aiding in the maintenance of safety and
efficiency as safety equipment would need not be removed to use the system.

###  1.3. Design Goals, Scope, and Principles

Deep learning and machine learning are a natural step forward for scratch
input technology. With enough data on a well-defined problems, these learning
algorithms are often able to perform extremely well on hitherto unsolved
automated classification and regression problems such as object recognition,
speech recognition, and natural language processing. This investigation seeks
to understand the extent to which deep learning techniques can be used to
enhance scratch input technology. We seek improvement in terms of the
following metrics:

  1. (1)

Increased gesture classification accuracy.

  2. (2)

Decreased reliance on custom apparatus.

  3. (3)

Increased performance with a diversity of users and noisy environments on
similar-sounding gestures (Fig. 2).

Figure 2. Examples waveforms from a variety of users, devices, and
environments from the circular scratch gesture class. Note how background
noise, amplitude, speed, and shape differ substantially in real-world use
cases.

Examples waveforms for each input class.

To investigate this, a dataset of over 8,000 audio gesture clips were
collected using generic mobile devices by a range of users in diverse
environments. The proposed deep convolutional neural network achieved 95.8%
testing accuracy despite increased noise, gesture similarity, and surface
diversity. Several “baseline” machine learning models utilizing waveform
representation techniques similar to (Hershey et al., 2017) were also trained
to validate performance gains made by the deep learning approach.

##  2\. Proposed System

We propose a high-performance audio gesture processing system (Figure 3) that
uses a convolutional neural network (CNN) for classification in conjunction
with a Mel-spectrogram. In order to test the applicability of the system, a
dataset of 8,640 audio samples was collected, processed, and used to train and
test the proposed system and two baseline models. Full code for data
processing, feature extraction, model architecture, training, and validation
can be found in the project GitHub repository (link).

Figure 3. Flowchart of the proposed system utilizing Mel-spectrograms for
feature extraction and a convolutional neural network for classification.

Generalized scratch input system flow diagram.

###  2.1. Data Collection

The dataset for this experiment was evenly split between 6 gesture classes:

  1. (1)

Fingertip Tap

  2. (2)

Fingernail Tap

  3. (3)

Vertical Scratch

  4. (4)

Circle Scratch

  5. (5)

W-Scratch

  6. (6)

Silence/null class

This gesture set was chosen so that results obtained would be comparable to
those in (Harrison and Hudson, 2008) where 6 gesture classes were also tested.
Video instructions on gathering the data were sent to 24 participants, each
with different surfaces, environments, and mobile devices for recording. The
video provided instructions and examples for how to execute each gesture along
with a metronome recording to help users achieve consistent timing between
gestures. Each participant recorded 70 audio gesture segments of each input
classes. During the silence/null class recording, participants were encouraged
to do other work and activities, resulting in realistic background noise
(e.g., typing, page rustling, talking). To clean mis-timed and improper audio
gesture segments, the first 10 gestures from each recording were removed from
the dataset, resulting in a final dataset of 8,640 audio gesture segments.

Recordings contained both steady-state and transient background noise. Gesture
recordings varied substantially in amplitude and speed. Different mobile
device models also varied in terms of frequency filtering, microphone
frequency response, and recording bitrate. 12 of the 24 participants used
iPhones or iPads from 2017 or later while the rest used devices produced by
Huawei, Xiaomi, Samsung, Pocophone, Google, Motorola, LG, or One Plus in a
similar time period. The full dataset with device specifications is available
from the authors upon request.

###  2.2. Data Processing $\&$ Feature Extraction

The raw audio files were manually cropped and aligned such that the gestures
all occurred at the same times. Audio was then re-sampled at 44.1 kHz. The
clips were segmented into individual gesture recordings, resulting in a
dataset of 8,640 individual gesture audio files, each between 0.75 and 1.5
seconds long. For training and testing purposes, each gesture was
symmetrically cropped to the minimum length of 0.75 seconds (see Fig. 4).

Figure 4. Examples waveforms for each input class.

Examples waveforms for each input class.

To represent the waveforms to the machine learning algorithm, additional
feature extraction was required. If the audio recordings were directly
inputted, each audio vector would have dimensionality on the order of $10^{4}$
due to the 44.1 kHz re-sampling rate. Past studies have confirmed that deep
learning on such feature vectors is incredibly computationally expensive and
tends to lead to poor results (overfitting) (Wyse, 2017) (Palanisamy et al.,
2020).

The feature extraction method chosen for this study was to convert the audio
gesture recordings into Mel-frequency spectrograms. Mel-frequency spectrograms
are a type of spectrogram with a logarithmic frequency and amplitude scale
(Zwicker, 1961). They are produced by generating Fourier transforms of a
signal on short time windows and calculating the energy contained in each
logarithmically-spaced frequency interval (known as a “Mel”).

This technique has yielded excellent results in state-of-the-art deep learning
audio classification studies (Hershey et al., 2017) (Thornton, [n.d.])
(Tanveer et al., 2021) due to its favourable representation of the time-
frequency attributes of a signal. Because of the logarithmic frequency scale,
an increased or decreased fundamental frequency simply results in vertical
translation on the plot without the dilation or contraction associated with
typical spectrograms. Thus, differently pitched scratches and taps tend to
differ only in location on the spectrogram, improving the CNN’s ability to
generalize a rule for recognizing each gesture class. Mel-spectrograms of size
100 by 65 pixels were generated, corresponding to a 0.75 second long rolling
“processing window”.

###  2.3. System Design and Training

The CNN consists of two convolutional layers each composed of a number of 3x3
kernels followed by two fully connected layers with 100 hidden units. 2x2 max
pooling was used after each of the convolutional layers, and ReLU activation
functions were used throughout. For training, cross entropy loss was minimized
via stochastic gradient descent (SGD). Batch normalization was employed (Ioffe
and Szegedy, 2015) to aid in training. A softmax function was applied to the
output of the CNN to obtain class probabilities. A graphical depiction of the
CNN can be seen in Fig. 3.

Other hyperparameters including SGD batch size, learning rate, and the number
of 3x3 kernels per convolutional layer were selected via grid search (see
Table 4). To train the classifiers, the dataset was split into a training
dataset (60%), a validation dataset (15%), and a testing dataset (25%). The
training set was used in each grid search iteration to train the model for 50
epochs while the validation dataset was used to select the best model from the
grid search procedure. The final accuracies are reported in terms of
performance on the separate testing set (per (Goodfellow et al., 2016), (Yaser
et al., 2012)).

##  3\. Results

###  3.1. Baseline Classifiers

To validate the utility of computationally expensive deep learning approache
over simpler machine learning approaches, two baseline classifiers were
trained on the dataset.

The first baseline classifier was trained using feature vectors representing
signal energy as a function of time similar to (Harrison and Hudson, 2008).
Features for this classifier were extracted by summing along the columns of
the Mel-frequency spectrograms (Fig 5), resulting in 65-dimensional feature
vectors. A Multi-Layer Perceptron (MLP) model was used with 1 hidden layer
with ReLU activation functions. 100 neurons were used in the hidden layer, and
with the Adam optimizer, the model achieved a test set accuracy of 71.2%. This
accuracy is considerably lower than that in (Harrison and Hudson, 2008)
despite the substantially more complicated MLP model, indicating that time-
energy features may not be sufficient in higher noise and more more diverse
environments for this gesture set.

The second baseline classifier used the same MLP model as the first. The
feature vectors in this case represented energy as a function of frequency,
similar to (Harrison et al., 2010), (Ono et al., 2013). Feature vectors were
calculated by summing along the rows of the Mel-frequency spectrograms. The
100-dimensional feature vectors yielded a test accuracy of 81.2%. This result
indicates that frequency-energy information may be more helpful for gesture
classification in higher noise/variance environments.

Figure 5. Example spectrograms from the fingernail tap and the circular
scratch classes.

###  3.2. CNN Model Results

The overall test set accuracy of the final CNN classifier selected using grid
search 95.8%. Several methods were used to validate the performance of the
classifier in addition to overall accuracy on the testing dataset. First, the
receiver-operating characteristic (ROC) curves were calculated. ROC curves
plot the rate of false positives against true positive rates for different
“certainty cutoffs” on the model output. In the best case, the ROC curve would
pass through the top left corner (Fig. 6), indicating that the model could
achieve 100% true positive rate and 0% false positive rate. The area under the
ROC curve is the de facto measurement for classifier success in machine
learning, and was calculated for each gesture’s classification. An area of 1
represents perfect classification and an area of 0.5 represents random
guessing (Fawcett, 2006).

Figure 6. ROC curves for final CNN model on testing dataset.

Plot demonstrating ROC for cropped model

The ROC curves show extremely high performance on all classes, with the lowest
class being Fingernail Tap and the highest being W-scratch, each with area-
under-curve value of 0.9902 and 0.9975 respectively (Fig. 6).

A confusion matrix (Table. 1) was also calculated for the model’s
classification performance on the test set. The horizontal labels at the top
represent the algorithm’s classification while the vertical labels to the left
represent the ground truth class value for each sample in the testing set. The
confusion matrix shows that the algorithm most commonly output the silence
class when “uncertain”. The fingernail tap class was most commonly confused
for another class.

Table 1. Confusion Matrix for Cropped Test Data | Circle Scratch | Fingernail Tap | Fingertip Tap | Silence | Vertical Scratch | W-Scratch
---|---|---|---|---|---|---
Circle Scratch | 302 | 0 | 0 | 4 | 2 | 2
Fingernail Tap | 2 | 292 | 4 | 5 | 1 | 1
Fingertip Tap | 3 | 5 | 269 | 3 | 0 | 0
Silence | 6 | 3 | 4 | 300 | 4 | 3
Vertical Scratch | 2 | 0 | 0 | 3 | 266 | 1
W-Scratch | 2 | 0 | 0 | 2 | 1 | 265

During model testing, certain practical qualitative patterns were observed. In
particular, it was observed that the circle scratch class was frequently
misclassified as silence.

Figure 7. Live Demo Perception of Circle Scratch Input - Confused

A chart of probabilities for different output classes coupled with the image
being analyzed

Confounding examples of this and other classes showed substantial visual
similarity to the improper class (Fig. 7). In the case of the circular
scratch, misclassified gestures often lasted longer than the total length of
the spectrogram frame and therefore appeared similar to steady-state
background noise instead of a transient scratch.

##  4\. Discussion

The high and consistent accuracy from the CNN model shown in Table 1 and
Figure 6 show that deep learning can be used to enhance scratch input to a
great extent. Despite environmental noise resulting from the use of non-
contact microphones and variation in environment and surface, the proposed
system for scratch input yielded 95.8% test set accuracy.

The experiments also raised important questions relating to the practical
implementation of such a deep learning scratch input system. Namely, the
active time window duration of each segment and the generalizability of the
model to a new surface/environment are of interest to characterize the
applicability of the findings. Advantages and disadvantages of the deep
learning approach as compared to previous works are also of interest.

###  4.1. Comparison to Prior Work

The use case for the proposed system most closely resembles that of Harrison
and Hudson 2008 paper on scratch input (Harrison and Hudson, 2008) with key
differences in apparatus, feature extraction, and algorithmic complexity.

In large part due to the increased algorithmic complexity and lengthy training
procedure, the proposed deep learning system was able to achieve a 95.8%
classification accuracy on the 6 gesture types while Harrison and Hudson’s
system achieved an 89.5% classification accuracy. The proposed system also
achieved this accuracy in spite of additional background noise and variation
resulting from the use of miscellaneous phone and tablet microphones rather
than the stethoscope apparatus used by Harrison and Hudson. The gestures in
the present investigation did not differ in terms of energy profile as cleanly
as those of (Harrison and Hudson, 2008), and the users did not have the
opportunity to practice with the system before recording their data.

This increase in classification accuracy comes at a cost, however. The CNN
algorithm and the Mel-spectrogram feature extraction in the proposed system is
substantially more computationally expensive. It also lacks the
interpretability of Harrison and Hudson’s shallow decision tree algorithm
where one could theoretically trace the genesis of a classification error to
either a miscount of the energy peaks or an incongruent amplitude variation
measurement. As well, the proposed system requires a “silence” class to
differentiate null input from other gestures while prior work does not.

It is clear that a deep learning approach to scratch input has the potential
to drastically decrease gesture misclassification for scratch input and
related interaction modalities, even in noisy and high-variance environments
with diverse apparatus. The system increases classification by a roughly 5%
from past work, corresponding to a decrease in gesture misclassification from
10.5% ((Harrison and Hudson, 2008)) to 4.2%, or a 60% decrease in gesture
misclassification.

###  4.2. Window Sizing Tradeoff

One of the important trade-offs that must be made when used with a
spectrogram/convolutional neural network for gesture classification relates to
the duration of each gesture recording. In practice, the algorithm has an
“active window” of audio that it processes at a given time. This corresponds
to the width (i.e. duration) of the spectrogram images used as input for the
model. For convolutional neural networks, this size must be constant.

For these tests, the duration of each gesture segment was cropped to the exact
same length (0.75 seconds). This places a lower limit on the spacing between
two subsequent gestures as well as an upper limit on the length (and amount of
information available) on each gesture. It was observed that longer gestures
such as W-scratch and circular scratch would at times exceed the duration of
the frame, resulting in the problem discussed in subsection Section 3.2.

To investigate the amount of information that is lost by cropping longer
gestures, multiple alternatives for gesture spectrogram resizing were tested.
The original gesture recordings ranged from 0.75 to 1.5 seconds long, there
was an opportunity to experiment with scaling up the shorter segments, scaling
down the larger ones, or simply padding all of the gestures to a uniform size
with black space, as seen in Figures 8, 9, 10. While these approaches are not
feasible in practice, the results aid in understanding information lost or
gained because of the “active window” size.

Table. 2 lists the highest accuracies achieved the model using different image
transformation methods, with the highest accuracy achieved from padding,
lowest accuracy from cropping, and scaling (squeeze/ stretch) in the middle.
This result likely stems from information being lost from cropping the images,
distorted from scaling, and mostly retained in padding.

Table 2. Best model accuracies for different image transformations) | Cropped | Padded | Scaled Small | Scaled Large | Scaled Medium
---|---|---|---|---|---
Seed | 6 | 6 | 6 | 6 | 6
Learning Rate | 0.3 | 0.1 | 0.1 | 0.1 | 0.1
Batch Size | 126 | 48 | 48 | 64 | 48
Kernels | 26 | 4 | 12 | 12 | 8
Test Accuracy | 95.80% | 97.15% | 96.41% | 96.36% | 96.07%

Figure 8. Spectrogram image after cropping transformation (vertical scratch)

Figure 9. Spectrogram image after padding transformation (circle scratch)

Figure 10. Spectrogram image after scaling transformation (silence/null)

In the context of user interface and human-computer interaction, small
differences in classification accuracy can substantially change the user’s
experience. Therefore, meaningful engineering design and research is warranted
to determine the optimal window size and gesture set for use in a production
system. Investigation on the use of non-windowed machine learning approaches
such as recurrent neural networks may also be of value to address these
limitations.

###  4.3. Calibration Requirements

Per Section 2.2, the full 8,640-element dataset was randomly split into
testing, training, and validation subsets. Thus, each model was trained and
tested on every participants data. The same background noise and environments
were represented in the training, validation, and testing subsets. This likely
made classification easier since the model had already “seen” each environment
and noise distribution. A production system would not necessarily have this
benefit, unless some calibration were used.

To understand the model’s performance in new environments without calibration,
the training and testing were re-executed, now split on a subject-by-subject
basis. The hyperparameters selected via grid search were re-used for the
experiment. The model therefore was not exposed to the particular environment
or noise distribution of the subjects in the testing set, emulating a
production environment. This resulted in a 76% overall accuracy and the
following results (Fig. 11, Table. 3):

Figure 11. Test set ROC curves for participant-based train/test split model.

Plot demonstrating ROC for cropped model

Table 3. Confusion Matrix for Cropped Test Data (From an Unseen Surface) | Circle Scratch | Fingernail Tap | Fingertip Tap | Silence | Vertical Scratch | W-Scratch
---|---|---|---|---|---|---
Circle Scratch | 261 | 15 | 10 | 24 | 19 | 35
Fingernail Tap | 2 | 291 | 31 | 1 | 6 | 5
Fingertip Tap | 4 | 10 | 325 | 32 | 1 | 0
Silence | 61 | 8 | 16 | 299 | 18 | 38
Vertical Scratch | 7 | 2 | 1 | 6 | 291 | 10
W-Scratch | 25 | 34 | 37 | 34 | 25 | 272

Since performance is clearly inferior to the initial results, it would be
necessary to include this calibration step in a production system to expose
the model to the noise distribution and surface characteristics of a new
environment. To emulate the training of the initial 95.8% accuracy model, only
a few minutes would need to be spent to recording the 6 gestures to augment
the training dataset. This is a challenge to consider for this deep learning
approach when comparing its overall applicability with previous works.

###  4.4. Future Work

One of the primary areas for further exploration relates to the user
experience of a deep learning-based scratch input system. Optimization of
“active window” size and gesture duration would have an outsized effect on
user experience. Similarly, closer examination of problematic noises (e.g.,
pen scratches or typing) and associated mitigation methods are of interest.
Power efficient implementation of the processing pipeline would also be
important for production implementations of deep learning scratch input.

Another topic of interest is the incorporation of top-down approaches similar
to those presented in previous work (e.g., the shallow decision tree used in
(Harrison and Hudson, 2008)) with the proposed deep learning methods. As
mentioned in Section 4.1, the deep learning approach is not without its
disadvantages despite its increased classification accuracy. Interpretability,
reliability, and energy concerns may be addressed via a hybrid system that
incorporates alternative approaches to scratch input.

The calibration process required to use the deep learning scratch input system
with high accuracy is worthy of investigation. In particular, the use of
transfer learning and data augmentation could aid in further reducing the time
taken to calibrate the system. A natural next step for the deep learning
system could also involve continuous learning and improvement as the system is
used.

As proposed, the system is a discrete input system with a relatively small set
of classes. Incorporating one or more spatial dimension into the input signal
could enhance the usability. This could potentially be accomplished via
multiple microphones or by using a computer vision system to locate the user’s
gesture. As again, user experience testing would be paramount in understanding
the applicability of this augmentation.

Expanding the types of surfaces upon which the system works would also aid in
the creation of more immersive interaction modalities with deep learning
scratch input. Initial interest from Mann et al (Mann et al., 2008) originated
in part from work in water-based musical instrumentation. Perhaps deep
learning scratch input could be applied similarly to other states of matter.
Similarly, a deep learning approach to scratch input for musical expression
may yield interesting results.

##  5\. Conclusion

The extent to which deep learning can be used to enhance scratch input was
investigated. A convolutional neural network-based system was proposed and
tested, achieving a final testing accuracy of 95.8% on a dataset collected
from a variety of environments and surfaces using generic smartphones and
tablets. These results reduced the gesture misclassification rate by 60% as
compared to the previous state-of-the-art (Harrison and Hudson, 2008) despite
the additional challenge associated with multiple environments and non-contact
microphones.

We concluded a high potential for deep learning in the creation of natural
user interface (NUI) systems and other rapidly deployable user interfaces that
make use of vibrational signals arising from user action (i.e., scratching,
tapping, swiping). Home automation, accessibility, and hazardous environment
use cases were highlighted, and challenges associated with the methodology in
production were addressed. Next steps include further user testing, transfer
learning for calibration, and experimentation with alternative deep learning
models.

##  6\. Appendices

Table 4. Fixed and variable hyperparameters used in the CNN model Fixed Hyperparameters | Variable Hyperparameters (Name + Range)
---|---
2 Convolutional Layers |  Input Image Transformation |  Cropping/ padding/ squeezing/ stretching
2 Fully Connected Layers |  Batch Size |  [24, 48, 64, 126]
Batch Normalization |  Number of Kernels in the Convolutional Layers |  [4, 8, 12, 26]
Cross Entropy Loss function |  Learning Rate |  [0.3, 0.1, 0.01]
3x3 Kernel Size |  Seed |  [6, 42]
100 Neurons in the First Fully Connected Layer |  |

###### Acknowledgements.

We acknowledge Prof. Jonathan Rose of the University of Toronto for his
feedback and advice on this project.

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# Jumping to male-dominated occupations: A novel way to reduce gender wage gap
for Chinese women

Wei Bai  Zhongtao Yue  Tao Zhou***Corresponding author at: CompleX Lab,
University of Electronic Science and Technology of China, Chengdu 611731,
China.
E-mail address<EMAIL_ADDRESS>(T. Zhou).

###### Abstract

Occupational segregation is widely considered as one major reason leading to
the gender discrimination in labor market. Using large-scale Chinese resume
data of online job seekers, we uncover an interesting phenomenon that
occupations with higher proportion of men have smaller gender wage gap
measured by the female-male ratio on wage. We further show that the severity
of occupational segregation in China is low both overall and regionally, and
the inter-occupational discrimination is much smaller than the intra-
occupational discrimination. That is to say, Chinese women do not face large
barriers when changing their occupations. Accordingly, we suggest Chineses
women a new way to narrow the gender wage gap: to join male-dominated
occupations. Meanwhile, it is worth noticing that although the gender wage gap
is smaller in male-dominated occupations, it does not mean that the gender
discrimination is smaller there.

###### keywords:

Gender wage gap , occupational segregation , Gender inequality

organization=CompleX Lab, University of Electronic Science and Technology of
China, addressline=
Chengdu 611731, China,

##  1 Introduction

As one of the 17 Sustainable Development Goals, reducing gender inequality is
a major policy concern around the world [1]. The gender wage gap is a
prominent part of gender inequality. We use the ratio of females’ wage to
males’ (female-male ratio, $r_{fm}$) as the primary measure of the gender wage
gap. Globally, $r_{fm}\approx 0.76$ [2]. Zhang et al. [3] showed that from
1988 to 2004, $r_{fm}$ in China decreased from 0.863 to 0.762. Although there
are fluctuations in some years, it is very clear that the gender wage gap
keeps widening over time.

Gender wage gap is often closely related to occupational gender segregation
(occupational segregation for short) [4], which refers to the fact that
workers in the labor market are assigned to different occupational categories
due to gender differences, observed as the concentration of most female labor
force in some “feminine” occupations with low wage and low prestige.
Occupational segregation is usually considered as a main way of gender
discrimination and a major cause of gender wage gap. In terms of professions,
male and female students have different educational processes and outcomes
even in the same fields. Although STEM occupations are often high-paying,
women in them tend to be concentrated in lower-wageing ones [5]. Even when
women get the highest-wageing jobs in computer science and engineering, they
still earn less than their male counterparts. In different regions of the
United States, the wage of female doctors is generally lower than that of male
[6]. In Brazilian tourism industry, women are valued less than men even when
they have the same occupational characteristics [7]. In finance, although
American male and female MBA graduates earn almost the same at the start of
their careers, the ratio of females’ logarithmic annual wage to males’ has
been as low as 0.625 after 10 to 16 years [8]. Gender differentiation has also
appeared in different sectors in China: the gender wage gap within the
political sector is gradually disappearing, while outside is widening [9].
Yang et al. [10] notice that Chinese male job seekers have remarkably higher
salary expectation than females. In the American public sector, occupational
segregation is still the main cause of wage disparity [11].

The impact of occupational segregation on gender wage gap also varies by
country and generation. Todd et al. [12] found that occupational segregation
increased in Australia from 1995 to 2011. However, Busch [13] found that
women’s earnings in male-dominated occupations increased in Germany between
1992 and 2015. By using data from the European Structure of Earnings Survey
(2010), Boll et al. [14] showed that sectoral isolation has 0%-15% explanatory
power in salary decomposition. Laine [15] analyzed the impact of different
types of occupational segregation in Finland and found that the explanatory
power of corporate and job segregation dropped from 23.6% to 19.1% between
1995 and 2004. Using data from the 1988-1992 Chinese Urban Household Survey,
Ng [16] found that the total explanatory power of occupational segregation was
20%-40%, after controlling the influence of education, work experience,
province, industry, occupation, and company ownership.

Up to now, most related studies rely on questionnaire survey [17] or
experimental data [18], both of which are based on small samples. This work
explores the relationship between occupational segregation and gender wage gap
at occupational granularity by using large-scale samples. Our aim is to make
practical suggestions to women about how to reduce the gender wage gap. The
remainder of this paper is organized as follows. Section 2 presents the
relationship between gender wage gaps and levels of occupational segregation
in different occupations, as well as the overall severity of Chinese
occupational segregation. Section 3 analyzes the severities of discrimination
in different occupations by decomposing wages into those determined by human
endowments and those caused by discrimination. Section 4 further decomposes
gender wage gaps into intra- and inter-occupational differences. Finally,
Section 5 briefs the conclusions along with some discussions.

##  2 Gender Wage Gap and Occupational Segregation

We use resume data of 10,318,484 job seekers crawled from various online
recruitment websites from 2014 to 2015. The data includes basic personal
information, educational experience, work experience and expected occupation
and position. Among them, gender, age, previous wage and occupation from work
experience are mainly utilized. A job seeker is accepted for further analysis
if the information about gender, age, last year salary and occupation are all
presented, and the age is at least 16 years old. After data screening,
3,266,272 job seekers are accepted, including 1,957,747 for males and
1,308,525 for females.

In the following analysis, we mainly focus on the top-20 occupations with the
largest number of employees. The total number of employees in those 20
occupations is 2,607,503, accounting for 79.83% of the entire samples, as
detailed in Fig. 1.

Figure 1: The number of employees of the top-20 occupations.

In Fig. 2, we present the relationship between gender wage gaps and severities
of occupational segregation (measured by proportions of male employees) in the
top-20 occupations. Though gender wage gaps exist (i.e., $r_{fm}<1$) in almost
all occupations, it is observed that male-dominated occupations (such as
engineering and professional technicians) have relatively smaller $r_{fm}$,
while accountants, office clerks and other occupations that are generally
perceived as more feminine have relatively larger $r_{fm}$. The result
suggests that whether the occupation is dominated by men is significantly
related to the gender wage gap (the Pearson correlation coefficient reaches
0.6558, with $p$-value$<$0.01 according to the Student’s $t$-test). To some
extent, the observed correlation indicates that although male-dominated
occupations may have higher entry barriers for women, they exhibit narrower
gender wage gap than female-dominated ones.

Figure 2: The relationship between proportions of male employees and female-
male ratios in the top-20 occupations. The line is the linear fit of the
scatter plot, and the blue shaded area is the 95% confidence interval.

We calculate one of the most classic occupational segregation indices, the
Ducan index [19], which is a measure of what percentage of workers of another
gender would have to change jobs in order to achieve an equal distribution of
genders across occupations, when workers of one gender stayed in their current
jobs. The Ducan index ranges from 0 to 1: if all occupations are completely
dominated by one gender, it equals 1, while if proportions of male employees
in all occupations are the same, it equals 0. The mathematical formula of the
Ducan index is

$D={\frac{1}{2}}\sum_{j=1}^{n}|(F_{j}/F)-(M_{j}/M)|,$ (1)

where $F_{j}$ and $M_{j}$ are the number of female and male employees in
occupation $j$ respectively, $F$ and $M$ are the number of all female and male
employees, and $n$ is the number of considered occupations.

The Ducan index for the current data set is 0.398, which is remarkably lower
than those of developed countries in the world (for example, the Ducan index
in the United States has decreased from about 0.60 to 0.47 in the recent 60
years [20], and in Finland and Sweden, the Ducan index is still more than 0.40
in 2020 after a long decline [21]). At the same time, the Ducan index based on
the 2010 Population Census of the People’s Republic of China is 0.243, even
lower than the resume data. Fig. 3 shows the Ducan indices for different
provinces with lighter color corresponding to less segregation. One can
observe that, except for Tibet, severities of occupational segregation for
different provinces are similar and low. As Tibet is an outlier, it is removed
from the later statistics.

Figure 3: The Ducan indices for Chinese provinces.

Fig. 4 shows the relationships between the Ducan index and per capita GDP
($GDP_{pc}$) and the number of college students per $10^{5}$ people ($S_{h}$).
As the majority of updatings in the resume data happened in 2015, the relevant
statistical data at provincial level are taken from the China Statistical
Yearbook 2016, which presents statistics of social and economic status in
2015. It can be seen that the Ducan index is significantly correlated with the
level of economy and education ($p$-values are all less than 0.01 according to
the Student’s $t$-test). The above result suggests that the improvement of
education and economy is helpful in reducing occupational segregation,
however, the change of $D$ is not remarkable.

Figure 4: The relationship between $D$ and $GDP_{pc}$ (A), and the
relationship between $D$ and $S_{h}$ (B) at the provincial level. The lines
are the linear fits of the scatter plots, and the blue shaded areas are the
95% confidence intervals.

##  3 Inter-occupational Gender Discrimination

In order to better understand the impact of occupation-related discrimination
on the gender wage gap, we implement Brown decomposition on the resume data
[22]. Specifically, the gender wage gap is decomposed into differences within
and between occupations, and gender occupational distribution is estimated
from the perspective of job acquisition, thus separating out the inequality
caused by gender-specific occupational entry barriers. In this paper,
considering the distribution of individual occupations as an endogenous
variable, we use the two-stage process and multiple choice model to estimate
the entry probability of female (or male) in “barrier-free” career choice.
Finally, the Mincer wage equation is used to estimate the wage function of men
and women in each occupation. Brown decomposition emphasizes the impact of
occupational segregation on wage gaps and constructs a counterfactual
framework where women face the same occupational structure as men. The
mathematical formula of Brown decomposition is

$\displaystyle\bar{S}_{M}-\bar{S}_{F}=$
$\displaystyle\underbrace{\sum\nolimits_{j}p_{j}^{F}(\bar{X}_{j}^{M}-\bar{X}_{j}^{F})\beta_{j}^{M}}_{PD}+\underbrace{\sum\nolimits_{j}p_{j}^{F}\bar{X}_{j}^{F}(\bar{\beta}_{j}^{M}-\bar{\beta}_{j}^{F})}_{WD}$
(2)
$\displaystyle+\underbrace{\sum\nolimits_{j}\bar{S}_{j}^{M}(p_{j}^{M}-\tilde{p}_{j}^{F})}_{QD}+\underbrace{\sum\nolimits_{j}\bar{S}_{j}^{M}(\tilde{p}_{j}^{F}-p_{j}^{F})}_{OD},$

where $S$ represents the average annual wage, $X$ is the influencing factors
of wage (i.e. the characteristic matrix), $p_{j}$ is the probability of an
employee working in occupation $j$, $M$ and $F$ are short for male and female,
and $\tilde{p}_{j}^{F}$ is the expected probability of $p_{j}^{F}$ if both
occupational distributions of two genders are the same. PD and QD are the
intra-occupational and inter-occupational gender differences of endowments,
respectively. WD is intra-occupational gender wage discrimination, and OD
accounts for the occupational segregation.

The result of Brown decomposition is shown in Fig. 5, where WD accounts for
the largest proportion (52.06%), and QD accounts for the smallest proportion
(2.18%). It can be seen that the inter-occupational discrimination is low,
which is in line with the fact that the Ducan index is also low. This
indicates that occupations are not much segregated by gender, and thus women
are relatively easy to jump to male-dominated occupations to avoid being
trapped in occupations with very small $r_{fm}$.

Figure 5: The result of Brown decomposition. PD and WD are intra-occupational
differences, and the latter two are inter-occupational differences.

##  4 Intra-occupational Gender Discrimination

According to the result of Brown decomposition, the intra-occupational gender
discrimination is the main cause of the gender wage gap (79.79%). We then
apply the Blinder-Oaxaca (BO) decomposition for each occupation [23, 24],
which decomposes wage gaps between gender groups into the explainable part
caused by differences in individual characteristics, and the unexplainable
part attributed to discrimination. Regression of the wage yields the following
formula

$\displaystyle S_{M}=\beta_{M}X_{M}+\epsilon_{M},$ (3) $\displaystyle
S_{F}=\beta_{F}X_{F}+\epsilon_{F},$

where $\beta$ is the regression coefficient that captures the effects of human
endowment on wage for male (or female), and $\epsilon$ is the error term.
Then, the gap $1-r_{fm}$ can be expressed as

$1-r_{fm}\approx(\bar{X}_{M}-\bar{X}_{F})\beta_{M}+(\bar{\beta}_{M}-\bar{\beta}_{F})\bar{X}_{F},$
(4)

where $(\bar{X}_{M}-\bar{X}_{F})\beta_{M}$ represents the gap coming from the
differences in individual characteristics assuming that there is no gender
discrimination (explainable part), and
$(\bar{\beta}_{M}-\bar{\beta}_{F})\bar{X}_{F}$ is the gap resulted from
discrimination (unexplainable part).

In the following analysis, we concentrate on job seekers whose gender, age,
seniority, degree, school, last year salary, marital status, profession,
industry, occupation, expected city, living city and home city are all known
(in total there are 753,616 such job seekers). In addition, the wage is taken
logarithm, and variables except salary and dummy variables are treated
centrally before decomposition.

The proportions of the explainable and unexplainable parts are

$\displaystyle
P_{e}=\frac{(\bar{X}_{M}-\bar{X}_{F})\beta_{M}}{(\bar{X}_{M}-\bar{X}_{F})\beta_{M}+(\bar{\beta}_{M}-\bar{\beta}_{F})\bar{X}_{F}},$
(5) $\displaystyle
P_{u}=\frac{(\bar{\beta}_{M}-\bar{\beta}_{F})\bar{X}_{F}}{(\bar{X}_{M}-\bar{X}_{F})\beta_{M}+(\bar{\beta}_{M}-\bar{\beta}_{F})\bar{X}_{F}},$

Obviously, $P_{e}+P_{u}=1$ and $P_{u}$ is usually considered as
discrimination. Overall speaking, according to the BO decomposition, $P_{e}$
is only 18.53%, while $P_{u}=81.47\%$. Notice that, if $P_{e}<0$, for example
$P_{e}=-0.1$, it means female employees should earn $0.1G$ more than male
employees according to their individual characteristics, where $G$ denotes the
gender wage gap. That is to say, the discrimination equals 1.1G, even larger
than the observed gap. Fig. 6A shows the results of BO decomposition for the
top-20 occupations, showing that the discrimination ($P_{u}$) of the gender
wage gap is all relatively high, and even exceed 1 for a few occupations. Fig.
6B presents the relationship between the discrimination $P_{u}$ and the
severities of occupational segregation, showing a strong and significant
correlation ($p$-value$<$0.01 according to the Student’s $t$-test). In
addition, as shown in Fig. 6C, $P_{u}$ and $r_{fm}$ are also strongly
correlated. In other words, although $r_{fm}$ in a male-dominated occupation
may be smaller, the corresponding discrimination $P_{u}$ is very probably
larger than a female-dominated occupation.

Figure 6: (A) The proportions of explainable ($P_{e}$) and unexplainable
($P_{u}$) parts in different occupations. (B) The relationship between the
severities of occupational segregation and the proportions of the
discrimination parts according to the BO decomposition. (C) The relationship
between the gender wage gaps and the proportions of the discrimination parts
according to the BO decomposition. The lines are the linear fits of the
scatter plots, and the blue shaded areas are the 95% confidence intervals.

In the above analysis, the occupation “other transportation and general
machinery operators” is an outlier. In this occupation, $r_{fm}=1.19$, the
proportion of men is 96%, while $P_{u}>200\%$. Looking into the education
experiences of employees in this occupation (see Fig. 7), we find that more
than 75% of men are junior college students, while more than half of women
have a bachelor’s degree or even above. This means that women’s high wage is
achieved through a better educational background. While it is a good strategy
to narrow the gender wage gap by jumping to male-dominated occupations, women
also need to pay more efforts than men, and in those less-gap occupations, the
discrimination may be even larger (see Fig. 6C).

Figure 7: Distribution of the education experiences of male and female
employees in “other transportation and general machinery operators”.

##  5 Conclusions and Discussion

In this paper, using Chinese resume data from $\sim 3.3\times 10^{6}$ online
job seekers, we study the severity of occupational segregation in China and
its impact on the gender wage gap. Our results show that the gender wage gap
in male-dominated occupations is relatively small, while the occupational
segregation is not serious (indicated by a smaller Ducan index than many other
countries) and the inter-occupational discrimination is low (according to the
Brown decomposition). Therefore, to join male-dominated occupations is a
feasible way to narrow the gender wage gap. However, we also show that the
occupations with smaller gender wage gaps usually suffer even larger gender
discrimination. That is to say, in those occupations, female employees ought
to earn much more than male employees according to their individual
characteristics. As all results come from large-scale natural data, we believe
the reported phenomena are statistically solid [25, 26].

Occupational segregation mainly comes from the innate physiological
characteristics and physiques and the resulting differences in acquired skills
of men and women. Men are physically superior to women, while women have
innate advantages in communication and language. In terms of industry
distribution, men are concentrated in manufacturing, construction and
telecommunication sectors, while the proportion of women in services such as
education and medical care is relatively high. Whether in male-dominated or
female-dominated occupations, women’s wage and time spent are generally lower
than men’s [27, 28]. Studies have also shown that the occupational segregation
may be due to women’s reluctance to choose skilled occupations [29].

To reduce the occupational segregation, known studies mainly consider the
demand side. For example, in terms of political management of enterprises, a
gender quota system can be adopted to ensure a minimum proportion of female
managers [30]. However, Shaikh et al. [31] show that social policies don’t
seem to affect intra-group gender distributions and thus address inequality in
wage distribution. Our results provide a solution from the supply side of the
labor market. In contemporary China, if women want to narrow the gender wage
gap, they can actively cultivate male characteristic skills, change the
difference of individual endowment with men and jump to male-dominated
occupations to obtain the same remuneration as men. But at the same time,
women should also be prepared for more severe discrimination. Indeed, a lower
gender wage gap is statistically associated with higher gender discrimination.
Before the arrival to true general equality, the improvement of personal
endowment seems to be the only credible and critical way to break the wage
limit.

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# Dark age consistency in the 21cm global signal

Fumiya Okamatsu  Graduate School of Science and Engineering, Saga University,
Saga 840-8502, Japan  Teppie Minoda  Tsinghua University, Department of
Astronomy, Beijing 100084, China  Tomo Takahashi  Department of Physics, Saga
University, Saga 840-8502, Japan  Daisuke Yamauchi  Department of Physics,
Faculty of Science, Okayama University of Science, 1-1 Ridaicho, Okayama,
700-0005, Japan  Shintaro Yoshiura  Mizusawa VLBI Observatory, National
Astronomical Observatory Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

###### Abstract

We propose a new observable for the 21cm global signal during the dark ages,
the dark-age consistency ratio, which is motivated from the fact that the
shape of the functional form of the brightness temperature against the
frequency is cosmological-parameter independent in the standard $\Lambda$CDM
model. The dark-age consistency ratio takes a certain definite value in the
$\Lambda$CDM case, which can serve as a critical test of the model and probe
those beyond the standard one. The new observable just needs measurements of
the brightness temperature at a few frequency bands during the dark ages, and
thus it allows us to test cosmological scenarios even with limited information
on the global signal.

###### pacs:

98.80.-k

Introduction— The 21cm line of neutral hydrogen can probe the evolution of the
Universe over a wide range of redshift, especially at its higher end. Current
observations of the global signal (sky-averaged signal) of the 21cm line such
as EDGES Bowman et al. (2018), SARAS Singh et al. (2022) and its fluctuations
such as LOFAR Gehlot et al. (2019, 2020), MWA Ewall-Wice et al. (2016);
Yoshiura et al. (2021), OVRO-LWA Eastwood et al. (2019) and so on have already
reached high redshifts at the cosmic dawn. In the near future, SKAO SKA  will
also be operative on such high redshift with much better sensitivities.
Actually, the 21cm signal can also appear even during the so-called dark ages
($30\lesssim z\lesssim 150$), however, the observation of which is really
challenging due to the Earth’s ionosphere, the radio frequency interference
(RFI), and so on.

Indeed the dark ages have never been observed by any means, however this era
lies prior to the formation of the first objects in the Universe, and hence
the pristine information of cosmology can be derived without astrophysical
uncertainties. The 21cm line of neutral hydrogen would be a unique probe of
this era and there have been theoretical works to study what aspects of
cosmology can be probed by using its signal during the dark ages such as
primordial power spectrum Loeb and Zaldarriaga (2004); Balaji et al. (2022),
primordial non-Gaussianities Cooray (2006); Muñoz et al. (2015a); Flöss et al.
(2022); Yamauchi (2022); Orlando et al. (2023), isocurvature fluctuations
Gordon and Pritchard (2009); Kawasaki et al. (2011), gravitational waves Book
et al. (2012); Hirata et al. (2018), primordial black holes Cole and Silk
(2021); Yang (2021, 2022), dark matter-baryon interaction Tashiro et al.
(2014); Muñoz et al. (2015b), dark matter annihilation/decay Furlanetto et al.
(2006a); Valdes et al. (2007); Yang (2016), a test of statistical isotropy and
homogeneity Shiraishi et al. (2016), neutrino masses Loeb and Zaldarriaga
(2004) and so on, in most of which fluctuations of the 21cm line are mainly
studied although the global signal can also be useful.

Currently, various observation plans are being discussed to probe the 21cm
signal during the dark ages using a telescope on the moon or a satellite
orbiting around the moon, which can avoid the Earth’s ionosphere and the RFI
and make its detection possible, such as FARSIDE FAR ; Burns et al. (2021a),
DAPPER DAP ; Burns et al. (2021b), NCLE NCL ; Chen et al. (2020), LCRT LCR ;
Goel et al. (2022), DSL Chen et al. (2019), and so on (see also Koopmans et
al. (2021)). On the theoretical side, a precision cosmology using the dark age
21cm signal has been put forward by Mondal and Barkana (2023) (for an early
work on the parameter estimation using the global signal during the dark ages,
see Pritchard and Loeb (2010)). Although the 21cm fluctuations would bring a
lot of information on various cosmological aspects, the first target of these
lunar missions would be the global signal. It is therefore a timely issue
indeed to consider what we can learn from the 21cm global signal during the
dark ages, having planned these missions in mind.

21cm global signal— Here we give some basic formulas to calculate the 21cm
signal, particularly during the dark ages. For reviews of the 21cm physics, we
refer the readers to e.g., Furlanetto et al. (2006b); Pritchard and Loeb
(2012). The 21cm signal is characterized by the so-called differential
brightness temperature $T_{b}$:

$T_{b}=\frac{T_{s}-T_{\gamma}}{1+z}\left(1-e^{-\tau_{\nu}}\right)\,,$ (1)

where $T_{s}$ and $T_{\gamma}$ are the spin and radiation temperatures. Since
we consider photons of cosmic microwave background (CMB) as a backlight,
$T_{\gamma}$ coincides with the CMB temperature so that $T_{\gamma}=T_{\rm
CMB}$. $\tau_{\nu}$ is the optical depth, which is given by

$\tau_{\nu}=\frac{3ch_{p}\lambda_{21}^{2}A_{10}x_{\rm HI}(1-Y_{p})n_{b}}{32\pi
k_{B}T_{s}H}\,,$ (2)

where $\lambda_{21}$ is the wave length for the 21cm line, $A_{10}$ is the
Einstein A coefficient for the spontaneous decay, $x_{\rm HI}$ is the neutral
fraction of hydrogen, $Y_{p}$ is the primordial mass fraction of helium,
$n_{b}$ is the number density of baryon and $H$ is the Hubble parameter.
$c,h_{p}$ and $k_{B}$ are the speed of light, the Planck constant and the
Boltzmann constant. By inserting Eq. (2) to Eq. (1), and assuming that the
Universe is matter dominated and $\tau_{\nu}\ll 1$, one obtains

$\displaystyle T_{b}\simeq 85\,{\rm
mK}~{}\left(\frac{T_{s}-T_{\gamma}}{T_{s}}\right)\left(\frac{\omega_{b}}{0.02237}\right)$
$\displaystyle~{}~{}~{}~{}\times\left(\frac{0.144}{\omega_{m}}\right)^{1/2}\left(\frac{1-Y_{p}}{1-0.24}\right)\left(\frac{1+z}{100}\right)^{1/2}x_{\rm
HI}\,.$ (3)

From this expression one can see that we just need to specify the following
cosmological parameters to calculate $T_{b}$ in the standard $\Lambda$-Cold-
Dark-Matter ($\Lambda$CDM) model): baryon density
$\omega_{b}\equiv\Omega_{b}h^{2}$, cold dark matter density
$\omega_{m}\equiv\Omega_{c}h^{2}$, with $\Omega_{i}$ the energy density of
$i$-th component normalized by the critical energy density and $h$ the Hubble
constant in units of $100\,{\rm km}/{\rm s}/{\rm Mpc}$.

The evolution of the spin temperature can be given by Field (1958)

$T_{s}^{-1}=\frac{T_{\gamma}^{-1}+x_{c}T_{k}^{-1}+x_{\alpha}T_{k}^{-1}}{1+x_{c}+x_{\alpha}}\,,$
(4)

where $T_{k}$ is the matter temperature and $x_{c}$ is the coefficient for
atomic interactions, which is mainly determined by HH collisions during the
dark ages and depends on the cosmological parameters as
$x_{c}\propto\omega_{b}(1-Y_{p})$ since it depends on the number density of
the hydrogen. $x_{\alpha}$ is the coefficient for the Wouthuysen–Field effect
Wouthuysen (1952); Field (1958), in which Lyman-$\alpha$ photons effectively
induce the transition between the hyperfine states. Actually it can be
neglected during the dark ages in the standard $\Lambda$CDM case, although one
needs to take it into account in some cosmological scenarios beyond the
$\Lambda$CDM model. Notice that Eq. (4) in the $\Lambda$CDM model during the
dark ages gives,

$\frac{T_{s}-T_{\gamma}}{T_{s}}=\frac{x_{c}}{1+x_{c}}\left(1-\frac{T_{\gamma}}{T_{k}}\right)\,,$
(5)

and in the later stage of the dark ages ($30\lesssim z\lesssim 80$), $x_{c}\ll
1$ is realized, and then one can find the the scaling of the brightness
temperature against the cosmological parameters as

$T_{b}\propto\frac{\omega_{b}^{2}(1-Y_{p})^{2}}{\omega_{m}^{1/2}}\,.$ (6)

The scaling of $\omega_{b}$ and $\omega_{m}$ has also been noticed in Mondal
and Barkana (2023). Actually, by defining the following quantity

$C(\omega_{b},\omega_{m},Y_{p})\equiv\frac{\omega_{b}^{2}(1-Y_{p})^{2}}{\omega_{m}^{1/2}}\,,$
(7)

and rescaling $T_{b}$ as

$T_{b}^{\rm
sc}(\nu;\widetilde{\bm{\theta}},{\bm{\theta}})=T_{b}(\nu;{\bm{\theta}})\frac{C(\widetilde{\bm{\theta}})}{C({\bm{\theta}})}\,,$
(8)

where $\bm{\theta}=(\omega_{b},\omega_{m},Y_{p})$, one can obtain an almost
identical brightness temperature. In Fig. 1, we show the $T_{b}$ with and
without the rescaling according to Eq. (8), where we varied the cosmological
parameters in the range of 5$\sigma$ bounds from Planck data Aghanim et al.
(2020) for $\omega_{b}$ and $\omega_{m}$, and that from Hsyu et al. Hsyu et
al. (2020) for $Y_{p}$ yp_ . For calculations of the brightness temperature,
we used a modified version of recfast Seager et al. (1999, 2000); Wong et al.
(2008); Scott and Moss (2009). As seen from the figure, the rescaled $T_{b}$
(red) have almost the identical shape, due to the fact that $T_{b}$ scales as
Eq. (6) and the position of the absorption trough remains unchanged even when
the cosmological parameters are varied. On the other hand, $T_{b}$ without the
rescaling (blue) are widely distributed.

Figure 1: $T_{b}$ with (without) the rescaling of Eq. (8) is shown in red
(blue). Here we vary $\omega_{b}$, $\omega_{m}$ and $Y_{p}$ within 5$\sigma$
ranges from Planck $\omega_{b}=0.02237\pm 0.00015,\omega_{c}=0.12\pm 0.0012$
Aghanim et al. (2020) and Hsyu et al. $Y_{p}=0.2436\pm 0.00395$ Hsyu et al.
(2020).

To see more clearly that the shape of $T_{b}$ as a function of the frequency
is cosmological-parameter independent in the $\Lambda$CDM model, in Fig. 2, we
plot $T_{b}(\nu)$ divided by that at some reference frequency $\nu_{\ast}$
(red), which we take $\nu_{\ast}=30\,{\rm MHz}$ for illustration purposes,
with $\omega_{b}$, $\omega_{m}$ and $Y_{p}$ being varied within 5$\sigma$
ranges as done in Fig. 1. For comparison, we also show $T_{b}(\nu)$ just
divided by $-20.3~{}{\rm mK}$ (blue), which corresponds to the value of
$T_{b}(\nu=30\,{\rm MHz})$ for the case assuming the mean values for the
cosmological parameters. As seen from the figure, different cosmological
parameters give almost identical shapes for $T_{b}(\nu)$, particularly for the
frequency range of $20\,{\rm MHz}<\nu<50\,{\rm MHz}$, which corresponds to the
later stage of the dark ages $30\lesssim z\lesssim 80$.

Figure 2: Plots of $T_{b}(\nu)/T_{b}(\nu=30\,{\rm MHz})$ with $\omega_{b}$,
$\omega_{m}$ and $Y_{p}$ being varied within 5$\sigma$ ranges (red). For
comparison, the case where $T_{b}(\nu)$ is just normalized by a constant as
$T_{b}(\nu)/(-20.3\,{\rm mK})$ is also depicted (blue).

Consistency ratio as a new observable— The above arguments motivate us to
consider the ratio of $T_{b}$ at two different frequencies, which should take
an almost certain definite value regardless of the cosmological parameters and
can be used as a consistency check of the model.

We define the ratio as

$\displaystyle
R_{\nu_{i}/\nu_{j}}\equiv\frac{T_{b}(\nu=\nu_{i}\,{\rm[MHz]})}{T_{b}(\nu=\nu_{j}\,{\rm[MHz]})}\,,$
(9)

where $\nu_{i}$ and $\nu_{j}$ are two different frequencies, which we call the
“the dark-age consistency ratio” since this ratio would remain to take the
same value in the $\Lambda$CDM model to a high accuracy (although it is not
exact) even when we vary the cosmological parameters, particularly for
$\nu_{i}$ and $\nu_{j}$ being taken in the range of $20\,{\rm
MHz}<\nu<50\,{\rm MHz}$. In Table 1, we show the ratios for several values of
$\nu_{i}$ with the reference frequency $\nu_{j}=30\,{\rm MHz}$. As one can see
from the table, the dark-age consistency ratios in the $\Lambda$CDM model are
determined better than one percent accuracy regardless of the values of the
cosmological parameters. Therefore if some observation indicates a deviation
of $R_{\nu_{i}/\nu_{j}}$ from the prediction of the $\Lambda$CDM model, it
suggests a model beyond the standard one. In particular, the consistency ratio
proposed here would be very useful, even in the early stage of lunar missions
mentioned in the introduction where the data of some limited frequency bands
may be available. Even in such a case, the consistency ratio just needs
measurements of $T_{b}$ at just two separate frequency bands. Detailed
discussion on expected constraints on the consistency ratio in future missions
will be given in a separate work Okamatsu et al. .

$\nu_{i}$ | $R_{\nu_{i}/30}$ | $T_{b}(\nu_{i})\,[{\rm mK}]$
---|---|---
$~{}~{}40~{}~{}$ |  $0.3873\pm 0.0029$ ($0.76\%$) |  $-7.923\pm 1.0107$ ($12.76\%$)
$35$ |  $0.6401\pm 0.0016$ $(0.24\%)$ |  $-13.10\pm 1.7301$ $(13.20\%)$
$25$ |  $1.4454\pm 0.0023$ $(0.16\%)$ |  $-29.59\pm 3.9133$ $(13.23\%)$
$20$ |  $1.8487\pm 0.0126$ $(0.68\%)$ |  $-37.82\pm 4.8074$ $(12.71\%)$
Table 1: Ratio $R_{\nu_{i}/30}$ for several cases of $\nu_{i}$ in the
$\Lambda$CDM model. The uncertainty refers to the variation when the
cosmological parameters ($\Omega_{b}h^{2},\Omega_{m}h^{2},Y_{p})$ are varied
within the $5\sigma$ range. For comparison, the range of $T_{b}$ for the
5$\sigma$ variation of the cosmological parameters are also tabulated.

Testing cosmology with the consistency ratio — The consistency ratio defined
in Eq. (9) should be useful to probe cosmological models since, as discussed
above, it takes a definite constant value to a high accuracy during the dark
ages in the $\Lambda$CDM model as shown in Table 1. A possible deviation from
the $\Lambda$CDM value can arise by violating (one or more) following
assumptions during the dark ages:

  1. (i)

The Universe is matter-dominated.

  2. (ii)

Lyman-$\alpha$ sources are negligible.

  3. (iii)

Matter and photons are coupled via the Compton scattering.

  4. (iv)

Radiation field is determined by CMB.

An example of the violation of (i) is the so-called early dark energy (EDE)
scenario where a component behaving like dark energy exists in some early
times much before the current accelerating Universe. EDE has been attracting
attention in several occasions such as a possible solution to the Hubble
tension Kamionkowski and Riess (2022); Poulin et al. (2023) (for the current
status of the tension, see, e.g., Di Valentino et al. (2021); Perivolaropoulos
and Skara (2022)). EDE may also be able to address the so-called Helium
anomaly Takahashi and Yamashita (2023) where the primordial Helium abundance
measured by EMPRESS Matsumoto et al. (2022) may suggest a non-standard
cosmological scenario. Actually EDE has also been considered to explain the
EDGES signal Hill and Baxter (2018), in which EDE can become a non-negligible
component, or even a dominant one during the dark ages. To describe the energy
density of EDE, we can consider the following functional form adopted in Hill
and Baxter (2018):

$\rho_{\rm EDE}=C_{\rm EDE}\frac{1+a_{c}^{p}}{a^{p}+a_{c}^{p}}\,,$ (10)

where $a_{c}$ is the scale factor at which the behavior of the EDE energy
density changes from $\rho_{\rm EDE}={\rm const.}$ to $\rho_{\rm EDE}\propto
a^{-p}$. $C_{\rm EDE}$ can be fixed by giving the fraction of EDE at $a_{c}$
which is defined as

$f_{\rm EDE}=\left.\frac{\rho_{\rm EDE}(z)}{\rho_{\rm
tot}(z)}\right|_{z=z_{c}}=\frac{\rho_{\rm
EDE}(z_{c})}{\rho_{r,m,\Lambda}(z_{c})+\rho_{\rm EDE}(z_{c})}\,,$ (11)

where $\rho_{r,m,\Lambda}(z)$ is the sum of energy densities of radiation,
matter and the cosmological constant. In Fig. 3, we show $T_{b}$ in the EDE
model with $(f_{\rm EDE},z_{c},p)=(0.8,150,6)$ and $(0.8,300,4)$ as examples.

Figure 3: Plots of $T_{b}$ for several models: EDE with $(f_{\rm
EDE},z_{c},p)=(0.8,150,6)$ and $(0.8,300,4)$ (blue solid and dashed), dark
matter decay with $m_{\rm DM}=3\,{\rm MeV}$ and $10\,{\rm MeV}$ (orange solid
and dashed), excess radio background with $A_{R}=0.05$ and $0.005$ (purple
solid and dashed). The cosmological parameters are fixed as
$\Omega_{b}h^{2}=0.02237,\Omega_{c}h^{2}=0.12$ and $Y_{p}=0.2436$. For
reference, $T_{b}$ for the $\Lambda$CDM case is shown with the cosmological
parameters varied within 5$\sigma$ ranges.

The assumptions (ii) and/or (iii) can be violated, for instance, in models
where dark matter (DM) annihilates or decays since DM annihilation/decay can
produce photons in the energy range of Lyman-$\alpha$ and give an extra
heating source for the gas temperature $T_{k}$. Indeed there have been many
works regarding the effects of DM annihilation/decay on the 21cm signal, in
particular see Furlanetto et al. (2006a); Valdes et al. (2007); Yang (2016)
for its implications for the 21cm signal during the dark ages. In Fig. 3, we
show $T_{b}$ in models with light DM decay for the mass of $3\,{\rm MeV}$ and
$10\,{\rm MeV}$ for illustration which are calculated in the same manner as in
Valdes et al. (2007). The details of the calculations and cases with some
other scenarios will be given in a separate paper Okamatsu et al. . Other
examples of the violation of the assumption (iii) include models with baryon-
dark matter interaction Tashiro et al. (2014); Dvorkin et al. (2014); Barkana
(2018), primordial magnetic field Schleicher et al. (2009); Sethi and
Subramanian (2009); Minoda et al. (2019); Bera et al. (2020), and so on, which
have also been discussed in the context of the EDGES signal.

The assumption (iv) can be affected by extra radio background, and it has been
discussed as a possible explanation of the EDGES signal Feng and Holder
(2018); Yang (2018); Pospelov et al. (2018); Fialkov and Barkana (2019); Moroi
et al. (2018). Such an extra radio source is also suggested by ARCADE2 Fixsen
et al. (2011) and LWA1 Dowell and Taylor (2018), whose results motivate the
following parametrization for the radiation temperature Fialkov and Barkana
(2019):

$\displaystyle T_{\gamma}=T_{\rm
CMB,0}(1+z)\left[1+A_{R}\left(\frac{\nu}{\nu_{\rm
ref}}\right)^{\beta}\right]\,.$ (12)

Here $A_{R}$ is the relative size of the extra source to the CMB temperature
at the reference frequency $\nu_{\rm ref}$ and $\beta$ describes the frequency
dependence of the radiation. We should note that the functional form should
depend on the generation mechanism, and it may have some cut-off at some
frequency. However, we adopt the form (12) for illustration purpose. In
Fialkov and Barkana (2019), it has been suggested that the case with
$A_{R}=5.7$, $\nu_{\rm ref}=78\,{\rm MHz}$ and $\beta=-2.6$ could explain the
EDGES signal at $z=17$, which however would significantly distort $T_{b}$ at
the dark ages. In Fig. 3, $T_{b}$ for the cases with $A_{R}=0.05$ and $0.005$
for $\nu_{\rm ref}=78\,{\rm MHz}$ and $\beta=-2.6$ are shown.

Most models mentioned above can be tested by using the consistency ratio
introduced in Eq. (9). In Fig. 4, the predictions of $R_{40/30}$ and
$R_{20/30}$ for the $\Lambda$CDM model, and some other example models, such as
EDE, excess radio background and DM decay, are shown. As discussed above, the
$\Lambda$CDM model predicts certain definite values for the ratios with a very
small uncertainty regardless of the values of the cosmological parameters, and
its prediction is represented just by a point in the $R_{40/30}$–$R_{20/30}$
plane. In other models, the ratios are deviated from those of $\Lambda$CDM,
from which one can clearly see that the new observable $R_{40/30}$–$R_{20/30}$
should be useful to probe models beyond the standard model. Notice that, when
model parameters are varied, its predictions for $R_{\nu_{i}/\nu_{j}}$ also
change.

Figure 4: Predictions for $R_{40/30}$ and $R_{20/30}$ of several example
models are shown. Red point corresponds to the prediction of the $\Lambda$CDM
model. Blue, orange and purple points are those for models with EDE
($p=4,z_{c}=150$), excess radio background ($\nu_{\rm ref}=78\,{\rm
MHz},\beta=-2.6$), and DM decay, respectively. Model parameters are depicted
in the figure.

Conclusion — We proposed a new observable for the 21cm global signal during
the dark ages, the dark-age consistency ratio, which is motivated from the
fact that the shape of $T_{b}$ as a function of the frequency is almost
independent of the cosmological parameters in the $\Lambda$CDM model. Since it
takes a certain definite value in the $\Lambda$CDM, it can be used as a
consistency check of the model. If the deviation from the $\Lambda$CDM value
is observed, it would signal a model beyond the standard scenario. The new
observable only needs measurements at a few separate frequency bands, and
hence fruitful information on cosmology can be derived from the dark age 21cm
global signal even at the early stage of lunar or satellite missions in the
foreseeable future.

###### Acknowledgements.

This work was supported by JSPS KAKENHI 19K03874 (TT), 23K17691 (TT), 19H01891
(DY), 22K03627 (DY), 21J00416 (SY), 22KJ3092 (SY) and MEXT KAKENHI 23H04515
(TT). SY is supported by JSPS Research Fellowships for Young Scientists. This
research was also supported by the grant of OML Project by the National
Institutes of Natural Sciences (NINS program No, OML022303).

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