# Use of Transfer Learning and Wavelet Transform for Breast Cancer Detection

Ahmed Rasheed, Muhammad Shahzad Younis, Junaid Qadir and Muhammad Bilal Ahmed
Rasheed and Muhammad Shahzad Younis are with School of Electrical Engineering
and Computer Science, National University of Sciences and Technology,
Islamabad, Pakistan.
E-mail<EMAIL_ADDRESS>E-mail:
<EMAIL_ADDRESS>Junaid Qadir is with Information Technology
University (ITU)-Punjab, Lahore, Pakistan.
E-mail<EMAIL_ADDRESS>Muhammad Bilal is with University of the West
of England (UWE), Bristol, England.
E-mail<EMAIL_ADDRESS>
###### Abstract

Breast cancer is one of the most common cause of deaths among women.
Mammography is a widely used imaging modality that can be used for cancer
detection in its early stages. Deep learning is widely used for the detection
of cancerous masses in the images obtained via mammography. The need to
improve accuracy remains constant due to the sensitive nature of the datasets
so we introduce segmentation and wavelet transform to enhance the important
features in the image scans. Our proposed system aids the radiologist in the
screening phase of cancer detection by using a combination of segmentation and
wavelet transforms as pre-processing augmentation that leads to transfer
learning in neural networks. The proposed system with these pre-processing
techniques significantly increases the accuracy of detection on Mini-MIAS.

###### Index Terms:

Neural nets, Segmentation, Transfer Learning, Breast Cancer Detection, Deep
Learning, Medical Imagery.

##  1 Introduction

Based on the research of World Health Organization (WHO), 22.9% of the total
cancer cases diagnosed are of breast cancer and accounts for 13.7% cancer-
related deaths all over the world [1], which places breast cancer at second
place in the leading causes of cancer-deaths among women behind heart diseases
[16]. Early detection of tumors or irregular masses can save the lives of
many. Breast cancer has over 90% chance of being cured at its early stage but
it does not get much attention until it gets severe as it is not painful in
its early stage.

Computer-aided systems are now of great interest to reduce the number of
cancer casualties by detecting it in the early stages and helping out the
radiologists in making important decisions [2]. There are two types of
mammography examinations, screening and diagnostic. Diagnostic examination is
a follow-up method for patients who are already classified with abnormal
tissues by a proper channel in early screening process [3]. Screening of
mammography generally has four views—the Craniocaudal (CC) view and
Mediolateral (MLO) view for each breast. For further diagnostic more views are
also referred for in-depth analysis by radiologists. In our system we worked
on the classification of screening problem which will help the radiologists in
correct diagnostics and determining the next steps.

We propose a system that utilizes the knowledge and techniques of image
processing and deep learning resulting in the classification of masses found
in the x-ray scans of breasts known as mammograms. The abnormalities are
classified in the different classes based on the structure, shape, and
composition of tissues present. Furthermore, the system can also tell the
severity of the abnormality in two more classes (i.e., is the abnormality
benign and malignant). In particular, Deep Learning and Image Processing
techniques are used in our proposed system to classify the abnormalities in
the mammographic scans obtained from “Mammographic Image Analysis Society”
(MIAS) database.

In this work, we use a combination of Wavelet transforms, Segmentation and
Convolutional Neural Networks (CNN) for breast cancer abnormality detection,
which provides promising results with increased efficiency of detection. CNN
is best for learning different features from each class under observation. It
uses filters of different sizes which helps the network in learning and
recognizing the patterns in the images [24]. Wavelet transform and
segmentation are widely used image processing techniques that enhance the
features present in the image scan. These features are then learned by the CNN
to classify the images. Wavelet transform takes the image in frequency domain
and then decompose it into sub-band images from which different features can
be extracted easily [21] [22]. Segmentation on the other hand enhances the
boundaries and corners in the image, it splits the the image into several
portions. Each portion signifies a useful data in the form of different gray-
levels. Segmentation of medical images highlight the important parts under
study like the tumors in the problem under consideration [23]. The combination
of wavelet transform and segmented image used as the inputs of the CNN helps
the network in learning the significant features and improves the overall
efficiency of the classification done by the network [25]. This study can
further be used to develop Computer-aided Design (CAD) which would be useful
for the radiologists to detect cancer in its early stages and take the
necessary actions needed.

Training deep CNNs from scratch is not easy as a very large dataset along with
high-end computing resources are needed. One way to resolve this constraint is
to use Transfer learning in which already-trained deep ML models are used in a
related but different setting and tuned according to one’s needs through the
removal or addition of a small number of new layers at the end of the network
and then training of only the newly added layers [15]. This method is
especially effective when we have smaller datasets and wish to augment the
learning on these datasets with more general readily-available information
captured by the existing models [20]. Pre-trained Deep Neural Networks (DNNs)
are tested on the database using transfer learning techniques and pre-
processed images are passed to them.

The proposed system presents a state of the art architecture which combines
the knowledge of image processing and neural networks to form a single network
in order to detect the breast cancer from the mammograms. Moreover, the system
proposed does two step image classification. The first one, a seven class
classification detects the abnormality in the image mammogram and the latter
one, a three-class classification, tells the severity of the case using the
mammogram and information obtained from the first network combined. The
arrangement forms a hybrid system which in turns improves the results of the
classification by a significant factor and provides state of the art results
for a three-class classification of mammograms.Breast cancer classification
has been majorly accomplished via binary classification. [42][43][44] The
proposed methodology goes a step further and uses three class-classification
for breast cancer detection where, in addition to detecting the tumor, it also
identifies whether the detected tumor is benign or malignant.

The rest of the paper is organized in the following way. Background and
related works are presented in Section II. The details of the dataset are
presented in Section III. The details of our methodology are provided in
Section IV. Our results are provided in Section V. Finally, the paper is
concluded in Section VI.

##  2 Background and Related Work

Deep learning models are trained to achieve human-like intelligence and
decision making. Lots of work already has been done in medical field to
achieve models with state-of-the-art accuracy on different medical datasets
and different problems [4][5][11]. Use of deep learning models can be traced
back to late 90s. CNN architecture is widely used for the classification of
image-based data. Already trained networks can be found on databases to
classify the images into unique classes. Significant amount of work has
already been done in classifying the mammograms based on their abnormalities.
For classification various techniques are used for image pre-processing and
augmentation to aid the neural network and improve its performance.

The background and related work done in medical imaging and most importantly
in breast cancer detection has utilized a lot of techniques however
separately. Most breast cancer detection techniques have utilized wavelets for
feature enhancement[27][29], segmentation for diagnosis[4][46] and transfer
learning CNN networks for classification.[47][48][49] in their individuality.
The proposed system differs as it combines all three of these techniques and
consequently gives a much improved results than achieved before with the
individual techniques. The resultant AUC [Area Under the Receiver Operating
Characteristic (ROC) curve] gives a much improved classification.

###  2.1 AUC-ROC

Area Under the Curve - Receiver Operating Characteristic (AUC-ROC) Curve is a
metric for establishing the extent to which a model can distinguish classes.
AUC-ROC curves have been used as an apt measurement standard for medical
imagery [41].Higher the AUC-ROC, better the performance of the model. Our
system generates an AUC-ROC curve, on the mini-MIAS dataset for breast cancer
detection, that shows more promising results than previous works done.

###  2.2 Segmentation

The first technique used by our proposed system is image segmentation based on
the intensities of the mammogram images. The segmentation helps the system to
find the Region of Interests (ROI).
For this purpose, noise is removed first by adaptive mean filtering as this
filter removes the noise effectively and is better among noise removal
filters. Adaptive filter works by comparing each pixel of the image to its
neighboring pixel. If the pixel does not match with the majority of its
neighboring pixels then it is considered as noise and is then replaced by the
mean value.
After the noise removal K-mean clustering algorithm is applied to the image.
This algorithm selects the k number of centroids for the regions and then
allocates the pixels to the regions based on the clustering of neighboring
pixels. It is widely used in the segmentation of medical images as it is
important to extract only the desired regions from an image which may not be
easily distinguishable [34] [36]. For the classification purposes segmentation
is used as an aid to improve the performance. BichenZheng [35] used a hybrid
network of K-means segmentation and Support Vector Machine (SVM) to show the
improved accuracies in the detection of breast cancer.

###  2.3 Wavelet Tranform

Wavelet transform is a well-known image processing technique to extract the
features from an image. This technique is used to augment the available data
by providing the wavelets of original and segmented mammographic scans. The
main advantage of using wavelets is that they do a simultaneous localization
in frequency and time domain and is faster than other methods like FFTs [13]
[17] [18].

Wavelet transform has been used in several researches to aid radiologists in
understanding the image scans [26][27][40]. It enhances the features in the
image scan which are also beneficial for neural networks to learn and classify
the images more effectively. In medical imaging wavelets are mostly used as
pre-processing technique for image feature extraction. Liu et al. [28],
Ferreira and Borges [29] and Rashed et al. [30] showed that using wavelet
transform in classification improves its results in medical imaging.

Wavelet transforms helps to enhance the important features like the edges of
tumor present in the breast scan. The general equation of the wavelet
transform is provided next.

$W_{k}^{j}=\int f(x)\Psi(\dfrac{x}{2^{j}}-k)dx$ (1)

Here $\Psi$ is transforming function, f(x) represents the original signal
while k and j are translation and scale parameters respectively [14]. Figure 1
shows the visual representation of 2D wavelet transforms. Here $W_{D}^{j}$,
$W_{H}^{j}$ and $W_{V}^{j}$ presents detailed components in diagonal,
horizontal and vertical direction at the desired level j.

Figure 1: 2D Wavelet transform layer representation up to 3 levels.

###  2.4 Augmentation

Dataset augmentation refers to the task of generating new images from the
existing dataset, which improves the data scarcity and prevents the network
from over-fitting [8]. Image augmentation is used as the data enhancement
technique when you do not have a big database of images. It is considered an
important aspect when using CNN for classification as it improves the
generality of the network [31] [32]. Holger R. Roth [33] proved that data
augmentation significantly improves the performance of the network in
classification problems.

###  2.5 Deep Neural Network (DNN)

DNN is used at the end for the classification among selected classes. DNN is
now widely used in medical classification. A lot of work has also been done in
classifying the breast cancer using DNN [37] [38] [39].
These layers are then trained on the training dataset while the weights of all
other layers in the pre-trained models are kept frozen. This technique is
known as “Transfer Learning”. The purpose of using transfer learning is to
effectively utilize the complex structure of premade CNNs and to efficiently
train the network on a smaller dataset.

  1. 1.

Pre-trained CNN: An already trained network is more useful instead of building
a whole new CNN architecture from the scratch and then training it on our
smaller dataset. This would not give good results on the output of the
network. For this purpose, the number of already trained models are used by
replacing the final three layers of these models with the new fully connected
and classification layers according to our desired number of classes.

For the purpose of transfer learning, our proposed system uses the previously
trained state-of-the-art CNNs.

     * •

AlexNet: Krizhevsky in 2012 [6] achieved 15.3% top-5 test error rate in the
ImageNet Large Scale Visual Recognition Challenge (ILSVRC) with their unique
CNN architecture called AlexNet that consisted of convolutional layers,
dropout layers, pooling layers, and at last fully connected layers [12]. The
success of AlexNet ushered in the era of deep learning and CNNs.

     * •

VGGNet: Simonyan and Zisserman in 2014 introduced a 19-layer deeper CNN
achieving top results in ImageNet ILSVRC [7]. Their proposed network had small
convolutional filters which in the end showed a notable increase in accuracy.
This work led CNNs architectures to have deep network of layers for better
feature representations.

     * •

GoogleNet: Szegedy et al. in 2014 entered ImageNet ILSVRC with a deeper
network. This network was trained on high end resources and achieved a 6.7%
error rate in top-5 test [8].This network used a parallel inception network
instead of sequentially stacking the layers one after the other in the
architecture.

     * •

ResNet: He et al. in 2015 introduced a network architecture consisting of 152
layers [9].This set a new record in ILSVRC by achieving a 3.57% top-5 error
rate in the classification of the ImageNet database. They introduced a huge
number of smaller layers which made the model a very deep network but doing so
reduced the complexity of it.

All the aforementioned deep network architectures have been trained on the
ImageNet database and have been made designed to classify the input image into
one of 1000 classes. To use them for the proposed problem, the concept of
transfer learning is used. In transfer learning, the last three or more
network layers are removed and new layers are appended with the desired number
of neurons in the last fully-connected layer based on the number of classes
for the desired task. Learning rate is set high for the newly added fully-
connected layers. The network above remains unchanged and the new layers are
trained on the newer dataset. A validation check is performed after every few
iterations to check the performance of the network training.

  2. 2.

Augmentation: The above-mentioned techniques, segmentation, and wavelet
transforms, are also used as data augmentation and fed to the network for the
training purpose. Apart from this, the basic augmentation is also done which
includes the rotation, translation, scaling, and sheer of the images with
random values.

  3. 3.

Transfer Learning: Transfer learning the transfer of knowledge obtained from
one domain to another domain of different application. Transfer learning is a
part of deep learninng and comes in handy when the data on which the system is
to be trained is a small database [15] [19]. In the proposed system the it is
used by transferring the knowledge of a pre-trained network to our setting
which is abnormality detection in mammogram scans.

Fig. 2 shows the visual representation of transfer learning process.

Figure 2: Transfer Learning Process

Our system proposes a novel structure using the combination of above mentioned
techniques. A lot of work has been done on the given problem of classification
using these techniques individually. Our proposed system made with the
combinations shows a significant improve in the efficiency. The segmentation
and wavelet transforms act as the pre-processed augmentation of image data
which helps out the DNN in learning features in a more effective way.
Furthermore, the proposed system uses two DNNs in a cascaded way, reported in
the section below, which also helps the system learn the classes in a better
way.

##  3 Dataset

Neural networks require a large number of images as input for the training
purposes. Because of limited resources of large dataset availability, the
dataset used for this research is mini MIAS dataset [10]. It comprises of
total 322 images which are further divided into a number of classes. Original
size of each image in the dataset is 1024$\times$1024\. Sample images are
shown in Figure 1.

  * •

CALC—Calcification: deposition of calcium carbonates around the tissues (34
images).

  * •

CIRC—Well-defined/Circumscribed masses: irregular shaped masses present in
breast (24 images).

  * •

SPIC—Spiculated masses: poorly defined masses in the form of abnormality (24
images).

  * •

MISC—Other, ill-defined masses: miscellaneous abnormalities (18 images).

  * •

ARCH—Architectural Distortion: abnormal tissue arrangement causes
architectural distortions in the breast tissues (12 images).

  * •

ASYM—Asymmetry: increased mass density of the breast tissue (21 images).

  * •

NORM represents a normal image devoid of any abnormalities. (189)

TABLE I: 322 cases of Mini-MIAS dataset classified into 7 abnormalities. Classes | Number of Images
---|---
CALC | 34
CIRC | 24
SPIC | 24
MISC | 18
ARCH | 12
ASYM | 21
NORM | 189
Total | 322

These classes are further divided into the severity of the abnormality present
among the tissues based on the mammogram scans. Severity is divided into two
classes.

  * •

Benign

  * •

Malignant

TABLE II: 322 cases of Mini-MIAS dataset classified into 3 classes Class | Number of Images
---|---
Benign | 67
Malignant | 54
Normal | 201
Total | 322

Table I and II shows the exact number of image distribution among 7 classes of
abnormality and 3 classes of severity respectively.

For training and testing purposes the dataset is further divided into two
subsets: training data and validation data. 75% images from each class are
allocated for training of the network while the remaining 25% are allocated
for the validation of the trained network. This makes 90 images for validation
and 232 for training.

From 322 images, 133 images are of cases with abnormalities present and 189
images are of normal cases. Classes are divided in the following way as per
the abnormality present. This splitting of dataset is totally random to ensure
unbiased network training. The classes are also balanced at through random
selection in order to get unbiased classification for both the networks while
training.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Sample images from the mini MIAS database. (a) ARCH (architectural
distortion), (b) ASYM (asymmetry), (c) CALC (calcification), (d) CIRC
(circumscribed masses), (e) MISC (miscellaneous/ill-defined masses), and (f)
NORM (normal image devoid of masses).

##  4 Methodology

The flow diagram in Figure 4 illustrates our overall methodology.

Figure 4: Proposed system workflow diagram.

###  4.1 Segmentation

The first step done is image segmentation based on the intensities of the
mammogram images. K-means algorithm is used with adaptive noise removal to
obtain the best results of segmentation. Image scan of mamogram is divided
into portions and each portion is represented with a different gray level
intensity. These different intensities extracts the features of ROI in the
image. The abnormality or tumor can easily be detected in the segmented image.

Figure 5 shows a few sample images obtained after segmentation.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Sample images after segmentation (a) ARCH, (b) ASYM, (c) CALC, (d)
CIRC, (e) MISC, (f) NORM.

###  4.2 Wavelet Transform

Wavelet transforms of the image scans are done in the next step of our
purposed network. It works as the feature extraction for our CNN working at
the end to classify the breast cancer classes. Three image outputs are
obtained from wavelet transform, vertical, horizontal and diagonal. Each
containing the information of its respective component.

Three level wavelets transform as shown in Figure 6 is performed on the
original image as well as on the segmented image taken from the section above.
These images are then re-sized in the size of original image to be fed into
the CNN.

Figure 6: Sample images Level 1 Wavelet transform. (a) Diagonal, (b) Vertical,
(c) Horizontal.

###  4.3 Deep Neural Network (DNN)

DNN is used at the end of our proposed sytem for the classification among
selected classes. A hybrid system of two DNNs working together is proposed.
The first DNN is trained on 7 classes of abnormalities and gives its output to
the second DNN which then trains on the output of first DNN along with all the
images. This is a novel approach towards this kind of problem which uses the
knowledge of two DNNs for classification. Figure 7 shows the workflow of both
the networks working together.
Pre-trained networks are used with the transfer learning technique to train
our networks. The weights of first few layers of the pre-trained network are
kept frozen and last three layers are replaced by new modified layers which
are learnable. AlexNet, VGG, GoogleNet, ResNet50 networks are further trained
on the dataset discussed.

Figure 7: Proposed hybrid system of two DNNs

In the training phase, hyper-parameters are to be set in order to achieve best
results from the training on our dataset. These parameters include optimizers,
activation function, and learning rates. Furthermore, the number of epochs are
defined before the training starts.

The above-mentioned techniques, segmentation, and wavelet transforms, are used
as data augmentation and fed to the network along with original images for the
training purpose. Apart from this, the basic augmentation is also done which
includes the rotation, translation, scaling, and sheer of the images with
random values.

The dataset is split onto two parts: the training dataset and the validation
dataset. The training dataset is used to train the system by updating the
weights of the layers added while the validation dataset serves as a feedback
to the network after every few iterations and helps the system to improve and
avoid overfitting.

TABLE III: Network Hyper-parameters Parameters | Values
---|---
Max Epochs | 30
Mini Batch Size | 10
Initial Learning Rate | 3e-4
Activation Function | ReLu
Optimizer | SGDM

##  5 Results

For the purpose of validation the results Receiver Operating Characteristic
(ROC) curve is plotted for all the networks under study, it is actually a
probability curve which is used as a performance measurement in a
classification problems related to medical field [41]. For each class a
separate ROC curve is plotted which have True Positive Rate (TPR) and False
Positive Rate (FPR) on its x and y axis respectively [53]. Equation 2 and 3
below shows the formula to calculate TPR and FPR from the classification
results.

$TPR=\dfrac{TruePositive}{TruePositive+FalseNegative}$ (2)
$FPR=\dfrac{FalsePositive}{TrueNegative+FalsePositive}$ (3)

Area under the ROC curves (AUC) tells us how the network performed for the
specific class. The greater the AUC, the better the classification .

In order to obtain the results, first the networks were trained using the
proposed architecture on the original dataset using transfer learning with
pre-trained networks. Then the same networks were trained on the images
obtained after doing the segmentation and wavelet transform of the original
images. The results are shown in the form of ROC curves in the Fig. 8,9 and 10
for the three classes which are Benign, Malignant and Normal respectively
where dotted lines represent the curves of networks trained on only original
images while solid lines show the curves for networks trained on segmented and
wavelet transforms along with the original data. AUC of these classes are
stated in the Table IV, V and VI.

Figure 8: ROC curve for Benign class after classification

Figure 9: ROC curve for Malignant class after classification

Figure 10: ROC curve for Normal class after classification TABLE IV: AUCs of Benign class after classification Transfer Learning Network | Benign Class AUC
---|---
Original Data |  | Data after Segmentation
---
and Wavelet transform
VGG16 | 0.800 | 0.943
GoogleNet | 0.685 | 0.957
AlexNet | 0.55 | 0.959
ResNet50 | 0.659 | 0.966
TABLE V: AUCs of Malignant class after classification Transfer Learning Network | Malignant Class AUC
---|---
Original Data |  | Data after Segmentation
---
and Wavelet transform
VGG16 | 0.620 | 0.930
GoogleNet | 0.707 | 0.954
AlexNet | 0.573 | 0.964
ResNet50 | 0.585 | 0.965
TABLE VI: AUCs of Normal class after classification Transfer Learning Network | Malignant Class AUC
---|---
Original Data |  | Data after Segmentation
---
and Wavelet transform
VGG16 | 0.796 | 0964
GoogleNet | 0.712 | 0.965
AlexNet | 0.665 | 0.979
ResNet50 | 0.665 | 0.985

We can see from the above results consisting of ROC and AUC obtained from
different transfer learning models that models trained on dataset consisting
of wavelet transforms and segmented images performed significantly better on
all the networks. Furthermore we also deduced that ResNet50 performs much
better than than the other models under study while using transfer learning.

###  5.1 Comparison with related work

In Table VII we have provided overview of few related works and their
performances on the Mini-MIAS dataset. However due to the variance in metrics
used and the number of classes, a detailed comparison with the methods of
other works is unfeasible. Most of the previous studies have done two class
classification to mark the presence or absence of cancer tissues and have
acquired some promising results. [50].

TABLE VII: Comparison of classification with related work based on AUC. Paper | Method | Dataset | AUC
---|---|---|---
| Rouhi et al.
---
(2015)[52]
SNN |  | Mini-MIAS,
---
DDSM
0.92 (Binary)
| Valarmathie
---
(2016) [42]
Fuzzy + ANN | Mini-MIAS | 0.99 (Binary)
| Rabidas
---
(2016)[43]
DRLBP, DRLTP | Mini-MIAS | 0.98 (Binary)
| Jaffar
---
(2017)[51]
CNN (COM) |  | Mini-MIAS,
---
DDSM
0.92 (Binary)
| Proposed
---
Method
| Wavelet Transform
---
Segmentation
Transfer Learning
Mini-MIAS |  | (3 Class)
---
0.97 Benign
0.97 Malignant
0.99 Normal

In a comparison of the best results through our method with those found in
related work, it is clear that our method is outstanding for all three classes
with balanced values.

##  6 Conclusions

To summarize, this study effectively implemented the CNN on mammograms for the
detection of abnormal masses in breasts which cause cancer. Different CNN
architectures are used to get the state of the art accuracies on the mini-MIAS
dataset. Hence, the study shows the significance of data augmentation before
feeding the images to the network for training.

Segmentation and wavelet transforms help the network in extracting the
important features from the mammogram. The study’s main contribution, that is
the combination of transfer learning along with wavelet transform,
significantly improves the results by using segmentation and wavelet
transforms as pre-processing for the images before the training process
starts. The combination of segmentation and wavelet transform improves the
overall efficiency of CNN by helping in extracting features meanwhile the
unique hybrid system that uses transfer learning alongside wavelet transforms
and segmentation greatly affects the resulting ROC curves positively.
Segmentation gives the information about edges of tumorous regions while
wavelet transform helps in localization in frequency domain which then gives
the information of tumorous regions in the form of contours in images.

The betterment in results is clear considering the AUC-ROC curve achieves is
greater than 0.95 on all three classes which is considerably good considering
three-class classifications have not been measured using AUC-ROC before.

The dataset utilized for this paper was fairly small and yielded favorable
results, therefore the system can be tested on bigger datasets comprising of a
more huge number of elements to further test its accuracy. One estimated
hurdle could be the application of transfer learning to bigger datasets
however that poses a suitable foundation for future work.

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|  Ahmed Rasheed earned his Bachelor’s degree in Electrical Engineering,
majoring in Electrical Power, from Air University Islamabad, Pakistan. He is
currently an Electrical Engineering Master’s student majoring in Digital
System and Signal Processing, at School of Electrical Engineering and Computer
Science (SEECS) at the National University of Sciences and Technology (NUST)
Islamabad, Pakistan. He is also serving as a Research Assistant at the
laboratory of Adaptive Signal Processing (ASP) at SEECS, specializing in
handling machine learning, deep learning, time series prediction and
biomedical image processing tasks.
---|---
|  Muhammad Shahzad Younis received the bachelor’s degree from National
University of Sciences and Technology, Islamabad, Pakistan, in 2002, the
master’s degree from the University of Engineering and Technology, Taxila,
Pakistan, in 2005, and the PhD degree from University Technology PETRONAS,
Perak, Malaysia in 2009, respectively. Before joining National University of
Sciences and Technology (NUST), he was Assistant Manager at a research and
development organization named AERO where he worked on different signal
processing and embedded system design applications. He is currently working as
an Assistant professor in the Department of Electrical Engineering in School
of Electrical Engineering and Computer Science (SEECS)-NUST. He has published
more than 25 papers in domestic and international journals and conferences.
His research interests include Statistical Signal Processing, Adaptive
Filters, Convex Optimization Biomedical signal processing, wireless
communication modelling and digital signal processing.
---|---
|  Junaid Qadir (SM’ 14) completed his BS in Electrical Engineering from UET,
Lahore, Pakistan and his PhD from University of New South Wales, Australia in
2008. He is currently an Associate Professor at the Information Technology
University (ITU)-Punjab, Lahore, Pakistan. He is the Director of the IHSAN Lab
at ITU that focuses on deploying ICT for development, and is engaged in
systems and networking research. Prior to joining ITU, he was an Assistant
Professor at the School of Electrical Engineering and Computer Sciences
(SEECS), National University of Sciences and Technology (NUST), Pakistan. At
SEECS, he directed the Cognet Lab at SEECS that focused on cognitive
networking and the application of computational intelligence techniques in
networking. He has been awarded the highest national teaching award in
Pakistan—the higher education commission’s (HEC) best university teacher
award—for the year 2012-2013. He has been nominated for this award twice
(2011, and 2012-2013). His research interests include the application of
algorithmic, machine learning, and optimization techniques in networks. In
particular, he is interested in the broad areas of wireless networks,
cognitive networking, software-defined networks, and cloud computing. He is a
regular reviewer for a number of journals and has served in the program
committee of a number of international conferences. He serves as an Associate
Editor for IEEE Access, IEEE Communication Magazine, and Springer Nature Big
Data Analytics. He was the lead guest editor for the special issue “Artificial
Intelligence Enabled Networking” in IEEE Access and the feature topic
“Wireless Technologies for Development” in IEEE Communications Magazine. He is
a member of ACM, and a senior member of IEEE.
---|---
|  Muhammad Bilal is Associate Professor of Big Data and Artificial
Intelligence (AI) at Big Data Laboratory, University of the West of England
(UWE), Bristol. He holds a PhD in Big Data Analytics from UWE, Bristol. He has
multi-disciplinary research interests that span across Intelligent Systems,
Internet of Things (IoTs), AI Product Design, Digital Health, Visual
Analytics, GIS and Semantic technologies. Dr Bilal has led the development of
various enterprise-grade software projects ranging from financials to
healthcare. He has vast expertise in collaborative research design and
execution. So far, he has completed research and development projects of £3.7
Million at Big Data Lab in collaboration with leading UK businesses. He has
also authored more than 50 research articles at high-impact journals and
international conferences.
---|---


# Gravitational-Wave Lunar Observatory for Cosmology

Karan Jani  Department Physics & Astronomy, Vanderbilt University, 2301
Vanderbilt Place, Nashville, TN, 37235, USA  Abraham Loeb  Department of
Astronomy, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA

###### Abstract

Several large-scale experimental facilities and space-missions are being
suggested to probe the universe across the gravitational-wave (GW) spectrum.
Here we propose Gravitational-wave Lunar Observatory for Cosmology (GLOC) -
the first concept design in the NASA Artemis era for a GW observatory on the
Moon. We find that a lunar-based observatory is ideal for probing GW
frequencies in the range between deci-Hz to 5 Hz, an astrophysically rich
regime that is very challenging for both Earth- and space-based detectors.
GLOC can survey binaries with neutrons stars, stellar and intermediate-mass
black holes to $\gtrsim 70\%$ of the observable volume of our universe without
significant background contamination. The sensitivity at
$\mathcal{O}(1~{}\mathrm{Hz})$ allows a unique window into calibrating Type Ia
supernovae. Its unprecedented sensitivity would trace the Hubble expansion
rate up to redshift $z\sim 3$ and test General Relativity and $\Lambda$CDM
cosmology up to $z\sim 100$.

_Introduction_.— Observations from the first set of successful gravitational-
wave (GW) experiments - LIGO and Virgo - have shown the far reaching impact of
the GW spectrum from $10{\sim}1000$ Hz on fundamental physics, astronomy and
cosmology Abbott  _et al._ (2019a). In the next two decades, GW astronomy aim
to probe the universe at lower frequencies and well beyond the sensitivity
reach of the current detectors. The proposals for Earth-based next generation
observatories include the Einstein Telescope Punturo  _et al._ (2010) and
Cosmic Explorer Reitze  _et al._ (2019), which will constitute on the scale of
tens of km intereferometers with enhanced sensitivity up to ${\sim}5$ Hz. By
early 2030s, space missions such as the Laser Interferometer Space Antenna
(LISA) will open the GW spectrum in the milli-Hz regime Amaro-Seoane  _et al._
(2017), while the global network of Pulsar Timing Arrays Manchester (2013)
would be probing nano-Hz GW frequencies. Other space-based concepts have also
been proposed to deeper probe the milli- to micro-Hz frequencies Harry  _et
al._ (2006); Sesana  _et al._ (2019); Baibhav  _et al._ (2019).

One of the most challenging spectral regimes to measure GWs is from deci-Hz to
1 Hz. This frequency range tends to be too low for Earth-based detectors and
too high for space missions. The universe offers a rich set of astrophysical
sources in this regime Mandel  _et al._ (2018), whose observations would lead
to stringent tests of general relativity and physics beyond the Standard Model
Sedda  _et al._ (2020). A few proposals have been put forward for a space-
based deci-Hz detector with geocentric (SAGE Lacour  _et al._ (2019)) and
heliocentric (DECIGO Kawamura  _et al._ (2019), ALIA Crowder and Cornish
(2005), DeciHz Observatory Sedda  _et al._ (2020)) orbits, which rely on
advanced technologies in the post-LISA era (2040s).

In this _Letter_ , we propose a GW detector on the Moon whose primary goal is
to access the deci-Hz range. With the advent of NASA’s
Artemis111https://www.nasa.gov/specials/artemis/ and Commercial
Crew222https://www.nasa.gov/exploration/commercial/crew/index.html programs,
the time is ripe to consider fundamental physics experiments from the surface
of the Moon. We find that the Moon offers an ideal environment for pursuing
uninterrupted deci-Hz GW astronomy for decades and will strongly complement
with the Earth- and space-based network of telescopes. We suggest the acronym
GLOC - Gravitational-wave Lunar Observatory for Cosmology, for a detector that
would survey $30-80\%$ of the observable volume of our universe for a wide-
range of GW sources. In particular, GLOC will be able to measure the evolution
of the Hubble parameter at high-redshfits without multi-messenger followups.

Figure 1: Concept design for GLOC and predicted sensitivity. Left: Three end
stations on the surface of the Moon forming the full triangular-shape GW
detector. The end stations are separated by 40 km. Each end station will
contain a test mirror and a laser, making GLOC equivalent of three
interferoemters. Right: GW sensitivity (as characteristic strain) of space-
based (LISA), Earth-based (aLIGO, ET, CE) and lunar-based (GLOC-optimal and
conservative) detectors. Also plotted for comparison are sensitivity curves of
space-based deci-Hz concepts DECIGO, ALIA, and DeciHz Observatory (DO). Image
of Moon’s surface was adapted from Lunar Reconnaissance Orbiter
(NASA/GSFC/ASU) and Earthrise from the Apollo archives.

_Gravitational-Wave Setup on the Moon_.— The Moon offers a natural environment
for constructing a large-scale interferometer as a GW detector,  and such a
scenario have been mentioned in the literature Thorne ; Lafave and Wilson
(1993); Potter and Wilson (1990). The atmospheric pressure on the surface of
the Moon during sunrise is comparable to the currently implemented 8 km ultra
high vacuum ($10^{-10}$ torr) at each of the LIGO facilities LIGO Scientific
Collaboration (2015). After sunset, the atmospheric pressure on Moon scales
down to $10^{-12}$ torr Johnson  _et al._ (1972). The presence of vacuum just
above Moon’s solid terrain provides a great benefit in extending the LIGO
interferometer length at minimal cost.

The seismometers left from the Apollo missions suggests that the Moon is much
quieter than Earth (see Larose (2005) and the references within). At low-
frequencies ($0.1{\sim}5$ Hz), the seismic noise on the Moon is three orders
of magnitude lower than on Earth Lognonné  _et al._ (2009); Hanada  _et al._
(2005). Seismic noise is a fundamental limitation for the low-frequency
sensitivity of GW detectors on Earth. In aLIGO, the seismic noise dominates at
frequencies $\lesssim 10$ Hz. The next generation Earth-based experiments are
intended to push the limit to ${\sim}3$ Hz through several ambitious
improvements Abbott  _et al._ (2017) using quantum squeezers Goda  _et al._
(2008), covering mirrors with cryogenics and building underground tunnels Hild
_et al._ (2009, 2011). By including some of these technological upgrades, GLOC
can push the sensitivity to frequencies ${\sim}0.1$ Hz (deci-Hz). These
frequencies are too high to pursue with LISA-like space missions, as they are
fundamentally limited by the laser shot noise.

Unlike a similar setup on Earth, a lunar-based detector is only weakly
affected by environmental factors such as winds or lightening. The detector is
very mildly sensitive to the gravitational pull from the Earth’s ocean waves.
Therefore, we expect a near continuous operation of GLOC (close to 100% duty-
cycle). The Moon-quakes occur at much lower frequencies Hanada  _et al._
(2005) and thus should not impact the GW sensitivity in the GLOC spectrum.
Bombardment by cosmic rays and solar flares can be a source of non-Gaussian
noise. To reduce this noise, the end stations securing the test mass and the
optics can be coated with a magnetic shield to keep the excess charge
grounded. In addition, a cooling source can maintain a steady temperature and
mitigate thermal expansion, as the temperature on the surface of the Moon
changes from $-130^{\circ}$ C to $120^{\circ}$ C in a day Williams  _et al._
(2017). Alternatively, a few km long open lava tubes found on the Moon Chappaz
_et al._ (2017) can provide a natural infrastructure for setting up the
intereferometer.

Figure 2: Cosmological reach of GLOC in comoving coordinates. The concentric
circles represents the percentage fraction of the comoving volume of the
observable universe $(V_{\mathrm{obs}}=1.22\times 10^{4}~{}\mathrm{Gpc}^{3}$)
out to a given cosmological redshift, with the outermost being the CMB Planck
Collaboration (2018). The highlighted slices refer to the horizon redshifts in
GLOC  (optimal) for the corresponding GW sources at their detection threshold
(SNR $\geq 8$). The ones on the right are known examples of binary sources
with black holes (BHs) and neutron stars (NSs), while the left exhibits
potential discoveries of intermediate-mass black holes (IMBHs) and
intermediate mass-ratio inspirals (IMRIs). For reference, the circle in the
center represents the maximum reach of aLIGO at its design sensitivity The
LIGO Scientific Collaboration and the Virgo Collaboration (2013).

An additional advantage (thus far) is that the Moon is not corrupted by any
unpredictable noise from human activities. The site selection for the detector
should avoid terrain favorable for potential launches. In case of a lock-loss
in the interferometer, a lunar-based detector can be brought back online from
a control center on Earth. In the event of a serious hardware failure, parts
of the detector can be replaced and repaired by astronauts. The benefit of
performing on-request maintenance is not available for space-based GW
detectors, making the Moon a better long-term investment. In addition, future
space-missions to access the deci-Hz range are limited in their lifetime
(typically a few years), after which the gravitational perturbation from solar
system objects will disrupt their geometry. In contrast, a lunar-based
detector can operate and be steadily improved for decades. Because the science
sensitivity of the lunar-detector is at low frequencies, the data transmission
rate is only a magnitude more than that expected for the space-mission like
LISA.

_GLOC Concept Design_.—  We adopt a design schematic similar to that in next-
generation Earth-based detectors Cosmic Explorer (CE) and Einstein Telescope
(ET). As shown in the left panel of Fig. 1 (left), the arm-length of the
interferometer is set at 40 km and the L-shaped advanced LIGO (aLIGO) type
intereferoemter is replaced by a triangular geometry. Each end station forming
the triangle will consists of a laser and two test masses. Therefore, the full
GLOC would be equivalent of three independent detectors. The end stations can
be designed as dorm shaped compartments that are temperature controlled and
isolated from the rest of the detector. The curvature of the Moon leads to a
${\sim}450$ m vertical offset for the light path between 40 km separation.  An
ideal site for GLOC would be within a bigger craters ($\gtrsim 20$ km),
providing a flat land for at least two end stations and a higher elevation to
place the third station.

The right panel of Fig. 1 shows the target sensitivity of GLOC versus GW
frequencies and compare it with other proposed detectors Robson  _et al._
(2019); Hild  _et al._ (2011). We consider two projections for GLOC - a
moderately ambitious approach to have detection sensitivity down to
$f_{\mathrm{low}}=0.25$ Hz (GLOC-optimal), and a more conservative approach
that reaches $f_{\mathrm{low}}=1$ Hz (GLOC-conservative). The noise curves for
GLOC can be downloaded from https://doi.org/10.5281/zenodo.3948466.

In both the stated cases of GLOC, we assume that the primary limiting noise in
the mid to higher frequencies will be dictated by the quantum noise. Below
$\mathcal{O}(1)$Hz, the sensitivity of GLOC is governed by the seismic noise
and the suspension thermal noise. We expect the seismic noise in GLOC would be
at least three orders of magnitude lower than an ET-like configuration on
Earth. Due to the Moon’s lower surface gravity, the noise from a suspension
setup similar to that on earth would be reduced by a factor of ${\sim}3$.
Further improvements in the thermal noise can be achieved by implementing
mirror coating being planned for the cryogenic detector Craig  _et al._
(2019). With these improvements, GLOC would achieve the projected conservative
sensitivity. To reach the optimal sensitivity will require an unconventional
suspension setup. A possible mechanism it to let the test masses be in a free
fall with a so-called juggled interferometer Friedrich  _et al._ (2014). Such
a setup is more favorable to implement on the Moon due to the freedom with the
atmospheric vacuum.

_Methods_.— For constructing the sensitivity curve of GLOC, we apply the
quantum noise in the range $2-5000$ Hz from the CE2-ifo available on pygwinc
GWINC (2020). To estimate the behavior of seismic and suspension noise below
${\sim 2}$ Hz, we adapt their power-laws from the ET-D design and scale them
to the improvement stated in the concept design. We compute the horizon
distance of GLOC for various astrophysical binaries, characterized by their
masses ($m_{1},m_{2}$), using the methods described in Ref. Jani  _et al._
(2019). For the space-based detectors, we set a timeline of 4 years. For the
GW signal $h(f)$, we utilize the state of the art GW signal model IMRPhenomPHM
London  _et al._ (2018). This model includes the radiated higher harmonics
beyond the quadruple term that are crucial to probe intermediate-mass black
hole binaries Calderón Bustillo  _et al._ (2018). We set GLOC’s detection
threshold at signal to noise ratio (SNR) of 8. The lower limit of integration
for SNR is started from $f_{\mathrm{low}}=0.25$ Hz and includes a factor to
account for the three detector within GLOC’s triangular geometry. We transform
the luminosity distance into redshift through Planck 2018 Cosmology Planck
Collaboration (2018). To estimate the sky-localization error $\Delta{\Omega}$,
we first compute the frequency bandwidth $\sigma_{f}$ and timing accuracy
$\sigma_{t}$ for a given astrophysical source Fairhurst (2011). The effective
baseline $L$ is calculated through the orbital path of the Moon during the
time spent by the GW signal in GLOC’s band. As the binaries spent most time at
the low-frequency regime of GLOC, we approximate the angular uncertainty as
$\sigma_{\theta}\sim\sigma_{t}(c/L)$ Lau  _et al._ (2020).

_Science Case of GLOC_.— As showcased with fig. 2, the detector would have a
rare advantage of accessing GWs at cosmological distances across five orders
of magnitude in mass - from sub-solar dark matter candidates $({\sim}10^{-1}\
M_{\odot})$ Shandera  _et al._ (2018) to stellar mass binaries
$({\sim}10^{1-2}\ M_{\odot})$ to intermediate-mass black holes (IMBHs,
${\sim}10^{3-4}\ M_{\odot})$ Amaro-Seoane and Freitag (2006). Across this
entire mass-range, GLOC’s sensitivity would outperform that of the upcoming GW
experiments on Earth (CE, ET) and space (LISA) (see fig. 3). Furthermore, the
sensitivity band of GLOC is not expected to have any astrophysical foregrounds
from the white dwarf binaries Robson  _et al._ (2019). Thus, any GWs with
redshifted frequency $f_{\mathrm{det}}=(1+z)f_{\mathrm{src}}\gtrsim 0.2$ Hz
could be identified in the data without contamination.

Figure 3: High SNR detections. The area under each line refers to the
detection radius of equal-mass binaries at SNR $\geq 100$ for different GW
detectors. The thick black (optimal) and grey (conservative) lines refer to
sensitivity of GLOC. The dashed lines refer to space-mission and colored lines
indicate the ground-based detectors discussed in this text.

At the detection threshold (SNR$~{}\geq 8$), we find that GLOC can detect the
mergers of neutron stars (NSs) and stellar black holes (BHs) to over 70% of
the entire observable volume of our universe ($z{\lesssim}100$). This provides
an unprecedented cosmological probe of the early universe, comparable to the
21 cm cosmology proposed from the far-side of the Moon Burns  _et al._ (2012)
and the cosmic microwave background (CMB) experiments. While we do not expect
stellar objects to exist beyond $z{\sim}70$ Loeb and Furlanetto (2013), even
one such detection will violate $\Lambda$CDM cosmology Koushiappas and Loeb
(2017). The measurements of stellar binaries at such high redshifts would also
constrain the earliest stellar population Jani and Loeb (2020).

With the low-frequency limit up to ${\sim}0.2$ Hz, GLOC will start detecting
the inspiral of stellar binaries a few months to days before their merger
Peters (1964). This makes GLOC an early alert system for all coalescing GW
sources in the Earth-based network. Unlike other deci-Hz proposals (such as
DECIGO), GLOC would continue to measure the binary all the way up to merger
and ringdown. This ensures that for stellar binaries, (i) the detection radius
in in GLOC is comparable (or better) to the most futuristic concepts proposed
for space-based detectors, (ii) the overall SNR in GLOC is about an order of
magnitude higher than next-generation Earth-based detectors on Earth, and (ii)
the effective baseline $L$ of GLOC would become comparable to the Moon’s
orbital diameter around the Earth. While this study sets the high-frequency
sensitivity of GLOC $(\gtrsim 100$ Hz) to be comparable to CE, this can be
traded for a lower-cost design as most SNR is accumulated with the low-
frequency improvements.

The typical SNRs in GLOC for stellar binaries from high redshifts ($z\sim
3-10)$ would be ${\sim}100$, thus the timing accuracy, $\sigma_{t}$, could be
as good as ${\sim}0.1$ milli-seconds Fairhurst (2011). This combined with the
motion of the Moon at ${\sim}1$ km/s will reduce the angular uncertainty to
$\sigma_{\theta}\sim 10^{-3}~{}\mathrm{deg}$. As a result, the sky-
localization of stellar binaries from GLOC alone will be $\Delta{\Omega}\sim
10^{-5}~{}\mathrm{deg}^{2}$. In Fig. 4, we show this approximate constraint on
sky-localization for coalescing binaries at different redshifts.

A binary neutron star (BNS) at $z\sim 2$ would be in the GLOC band for an
entire orbital period of the Moon, while a nearby BNS (like GW170817
Collaboration and Collaboration (2017)) would be in-band for almost three
months. The sky-localization alert for BNSs can be sent days in advance,
allowing readiness of high-latency electromagnetic followups with reach up to
high redshifts.  Even in the case of GLOC-conservative, the effective baseline
for a BNS would be a quarter of the Moon’s orbit, leading to tight
constraints.

For a relatively light binary black hole (BBH) like GW151226 Collaboration and
Collaboration (2016), GLOC would start measuring its inspiral a day before the
merger. These could constrain the sources at redshifts ${\sim}2$ to about
$0.1~{}\mathrm{arcmin}^{2}$. Next generation Earth-based detectors CE and ET
have their peak sensitivity for BBHs of these total masses Jani  _et al._
(2019). A combined network between these enhanced detectors on Earth and a
geocentric detector like GLOC can further reduce the sky-location error by two
orders of magnitude (see calculations in Grimm and Harms (2020)). As shown in
Fig. 4 (dotted line), these combined network can constrain mergers of light
BBHs to $1~{}\mathrm{arcsec}^{2}$, namely the angular scale of a single
galaxy. These are the tightest constraints on the source location in GW
astronomy, allowing to identify the potential host galaxy without
electromagnetic counterparts Raccanelli  _et al._ (2016); Scelfo  _et al._
(2018, 2020); Calore  _et al._ (2020). The strongest science case of GLOC is
opening such high redshift dark sirens to independently measure the evolution
of the Hubble parameter as a function of redshift Schutz (1986).

Figure 4: Sky-localization with GLOC. Each colored coded curve represents a
face-on binary at different redshifts. The highlighted masses are in the
source frame of the binary. The shaded region refers to the potential
improvement for a $10+10~{}M_{\odot}$ binary with multiband network on GLOC
(optimal) and next-generation Earth-based detectors.

For the emerging population of BBHs in the pair-instability supernovae (PISN)
mass-gap Woosley  _et al._ (2002), such as GW190521 Abbott  _et al._ (2020a,
b) and GW170502 Udall and others others (2020), GLOC would measure their
inspiral a few hours before their mergers. These are the brightest sources in
GLOC, registering SNR $\sim 100$ at $z\lesssim 20$. For comparison, a deci-Hz
space-mission like DO would have similar reach only up to $z\lesssim 1$, while
ET it can reach $z\lesssim 10$ (see fig. 3). A multiband observation between
GLOC and ET would constrain the sky-location to
${\sim}1~{}\mathrm{arcmin}^{2}$, proving critical for associating flares in
Active Galactic Nuclei (AGN) with such binaries Graham  _et al._ (2020).
Furthermore, the inclination angle and orbital eccentricity will be tighter
constrained by measuring the early inspiral in GLOC, thus reducing the error
in redshift and the corresponding source-frame mass of black holes. The
constraints on the secondary black hole will be particularly important in
probing the lower-end of the PISN mass-gap. While a space-mission like LISA
can measure the early-inspiral of these PISN BBHs years in advance Sesana
(2016), it can only do so for sources typically within a Gpc Jani  _et al._
(2019) and retrospective after a detection from Earth Wong  _et al._ (2018).

The enhanced low-frequency sensitivity permits GLOC to survey binaries with
IMBHs of $10^{2-4}\ M_{\odot}$, practically across the entire universe. Such
cosmological reach is crucial for connecting IMBHs with the Pop-III remnants
Madau and Rees (2001) and the seeds of super-massive black holes Gair  _et
al._ (2009); Smith  _et al._ (2017); Pacucci and Loeb (2020). The theoretical
estimates on their populations are fairly weak, but the upper-limits from LIGO
and Virgo detectors The LIGO Scientific Collaboration and the Virgo
Collaboration (2019) suggests that the mergers of lower-range IMBH binaries
are much rarer ($\lesssim 1~{}\mathrm{Gpc}^{-3}~{}\mathrm{yr}^{-1}$) compared
to stellar binaries. Unlike GLOC, space-missions have just a few years of
lifetime ($4-10$ years for LISA), allowing the detection of only a handful of
such rare sources.

In the case of potential IMBH detection, the advent of GLOC opens a new
possibility of multiband observations across three frequency bands - from
early inspiral at milli-Hz (space), to late-inspiral at deci-Hz (Moon) and
mergers/ringdown at ${\sim}10$ Hz (Earth). Such joint measurements of a source
across three bands of frequency spectrum would provide the strongest tests of
the black hole area theorem, and no-hair theorem Vitale (2016); Cutler  _et
al._ (2019); Sedda  _et al._ (2020). The morphology of gravitational-waves in
multiband of LISA and GLOC could provide a direct probe of the environment
surrounding these black holes Toubiana  _et al._ (2020). Furthermore, the
intermediate-mass-ratio inspirals (IMRIs) Amaro-Seoane (2018), which are
relatively weak sources in both LISA and the next-generation Earth detectors,
can be surveyed in GLOC to $z\sim 10$. For IMRIs within redshift $\lesssim 1$,
GLOC would measure them with SNR $\sim 100$. This would increase the detection
confidence of these sources in LISA and ET, thus improving their overall
parameter estimation.

The long-term detection sensitivity at $\mathcal{O}(1~{}\mathrm{Hz})$ is
unique to GLOC. This frequency regime is crucial for probing the explosion
mechanism of Type Ia supernovae (SNe) Mandel  _et al._ (2018). For the the
single degenerate channel of Type Ia, we expect the GW emission at around 1 Hz
Falta  _et al._ (2011); Seitenzahl  _et al._ (2015). If the progenitors of
Type Ia SNe are mergers of two white dwarfs (double degenerate channel), we
expect GLOC to detect them up to ${\sim}2~{}\mathrm{Gpc}$. These detections
will also tightly bound the unknown masses of the white dwarfs, thus reducing
the error budget in calibrating the standard candles.

At the low end of source masses, GLOC can put the tightest bounds on a
putative population of sub-solar dark matter objects ($0.1-1\,M_{\odot})$
Shandera  _et al._ (2018). There are no known astrophysical phenomena that can
create detectable GWs at such low-masses, however, primordial black holes or
dark matter within neutron star cores offer possible scenarios (see Abbott
_et al._ (2019b) and references within). The low-frequency sensitivity of GLOC
allows us to measure the dark matter density of such exotic objects to 30% of
the entire observable volume of the universe ($z\sim 10$).

_Acknowledgements_.—  We are grateful to Rainer Weiss, Stavros Katsanevas,
Scott Hughes, Mathew Evans, Bob Eisenstein, Kevin Kuns, Brian O’Reilly, Stefan
Hild, Kelly Holley-Bockelmann, Jan Harms, Szabolcs Marka, Philippe Lognonné,
Robin Stebbins, Peter Bender, Robert Fisher and the referees for insightful
comments. K.J’s research was supported by the GRAVITY program at Vanderbilt
University. A. L.’s work was supported in part by the Black Hole Initiative at
Harvard University, which is funded by grants from the John Templeton
Foundation and the Gordon and Betty Moore Foundation.

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