# Detecting and Refining HiRISE Image Patches Obscured by Atmospheric Dust

Kunal Kasodekar<EMAIL_ADDRESS>SCAIArizona State UniversityTempeAZUSA

###### Abstract.

HiRISE (High-Resolution Imaging Science Experiment) is a camera onboard the
Mars Reconnaissance orbiter responsible for photographing vast areas of the
Martian surface in unprecedented detail. It can capture millions of incredible
closeup images in minutes. However, Mars suffers from frequent regional and
local dust storms hampering this data-collection process, and pipeline,
resulting in loss of effort and crucial flight time. Removing these images
manually requires a large amount of manpower. I filter out these images
obstructed by atmospheric dust automatically by using a Dust Image Classifier
fine-tuned on Resnet-50 with an accuracy of 94.05%. To further facilitate the
seamless filtering of Images I design a prediction pipeline that classifies
and stores these dusty patches. I also denoise partially obstructed images
using an Auto Encoder-based denoiser and Pix2Pix GAN with 0.75 and 0.99 SSIM
Index respectively.

HiRISE, Dust Storms, Data Collection, Dust Image Classifier, Resnet-50, Auto-
Encoder, Pix2Pix GAN

††conference: ; ; ††ccs: Applied computing Astronomy††ccs: Computing
methodologies Neural networks

##  1\. Introduction

A high-resolution imaging experiment (HiRISE for short) is a powerful camera
on board the Mars Reconnaissance orbiter utilized for imaging the Martian
surface up to a resolution of 3 cms/pixel. Its main objective is to study the
Martian surface and investigate active surface processes and landscape
evolutions as well as the potential for future human habitation/missions. It
can capture images of the Martian surface that span large areas while also
revealing details as minute as surface rocks. It can acquire large-scale image
datasets ( 28Gb) in less than 6 seconds. Thus managing, filtering and
effectively storing a large set of these images is a necessity.

Mars suffers from periodic regional and local dust storms due to wind currents
scooping up and moving the fine Martian dust to higher altitudes in the
atmosphere. Thus, when HiRISE captures the Martian surface during these dusty
conditions the images are redundant as the surface has been heavily obscured
by dust storms. These storms occur sporadically and can hamper the data
pipeline. Designated close-ups images required would not be successfully
captured and would result in loss of flight time and interruptions to the data
collection processes.

I will make use of the dataset provided by Jet Propulsion Lab, named ”HiRISE
Image Patches Obscured by Atmospheric Dust.” It is of utmost importance to
reduce the processing time of the data pipeline, minimize manual efforts and
optimize flight time. It is essential to discard dusty images at the source or
at the very least effectively bifurcate and store them in a dataset for
further processing. This processing can include denoising mission-critical
data or vital images of rare Martian processes.

In order to achieve the aforementioned goals, I train various classifiers that
can detect such dusty patches and create a pipeline using the best-performing
model that can filter out these images from the dataset or at the source
level. The dataset I am working with contains either fully obstructed or clean
images. Hence, I plan to study the dust obscuring the images and create a new
dataset with artificially imbibed noise that partially covers the image. I
train an Auto-encoder and GAN model to denoise the images and recover the
unobstructed Martian surface.

Figure 1. Dusty Patches Obscured by Atmospheric Dust (left), Clear Images of
the Martian Landscape (Right) [Resolution: 100x100]

##  2\. Related Work

There have been no studies on detecting, classifying or denoising Image
Patches obscured by atmospheric dust. However, various studies have been done
on denoising images and classifiers.

Authors of Deep Mars (Wagstaff et al., 2018) do the multi-label classification
of HiRISE orbital images using fine-tuning on HiRISENet and obtain a test
accuracy of 94.5%. The authors make use of transfer-learning on an Alexnet
convolutional network backbone trained on Earth Images to do multi-label
classification on Mars Images and create HiRISENet. With the help of this
network, they develop a content-based search tool to find images of interest.

Images captured using optical telescopes like the Hubble Space Telescope
inherently contain some perturbations and some additive noise. In order to
maximize information gain and for further analysis, denoising is a mandatory
preprocessing step. In the paper ”Learning to Denoise Astronomical Images with
U-nets” (Vojtekova et al., 2020) the authors propose a U-Net architecture
called AstroNet which can recover around 95.9% of the stars from such
astronomical images.

The authors of ”Generative Adversarial Networks Recover Features in
Astrophysical Images of Galaxies beyond the Deconvolution Limit” (Schawinski
et al., 2017) use GANs to reconstruct such perturbed galaxy images.

##  3\. Dataset

I will make use of the dataset provided by Jet Propulsion Lab, named ”HiRISE
Image Patches Obscured by Atmospheric Dust” (Doran, 2019). The dataset
contains two sets of image patches each from the Experimental Data Record
(i.e., Raw images without any corrections) and Reduced Data Record (i.e.,
corrected, and preprocessed images). Both these sets have 20,000 images each
with training, validation, and test split of 10,000, 5,000 and 5,000 images
each. No two splits have the same image. To estimate the inherent noise in the
labels manually vetted labels are provided. I will not use K-folds cross-
validation due to the inherently large size of the dataset. Both EDR and RDR
subsets have an equal distribution of the classes i.e., the dataset is
balanced. The image patches obtained from HiRISE are grayscale and 100 x 100
in dimensions. Currently, the plan is to experiment initially only with both
RDR and later with EDR images due to its lack of comprehensive image
correction and geometric processing (low-quality image data). The resolution
of the RDR images is 100 x 100. Example images are provided in Figure 1.

##  4\. PROPOSED METHODOLOGY

Initially, my goal was to train a classifier to detect dusty patches from
clean Martian images. For classifying images as ”Dusty” or ”Not Dusty” Patches
I do the following steps:

###  4.1. Preprocessing

The dataset contains a CSV file with image location, corresponding classes,
and appropriate dataset split. I do the following steps:

  * •

I convert this file to a data frame, create image-to-class mappings and read
all the images into python lists. Some initial data analysis and visualization
are done to understand the characteristics of this dataset.

  * •

The images are normalized into the 0–1-pixel range and for the Resnet Model, I
have another set of mean and standard deviation normalized images. These lists
are converted to NumPy arrays and resized to sizes as per the classifiers
used. For our PCA + SVM model, the data is reshaped/Flattened to (Batch Size,
and image features) and for our CNN I resize it to 64 x 64 x 1.

  * •

I also shuffle the data randomly, set seeds for multiple experiments and
convert categorical labels, Dusty, and Not Dusty into a NumPy array with
values 1 and 0 respectively. The Train-Val-Test split is 9817-5214-4969 images

###  4.2. Classification

Initially, I made use of Scikit-Learns PCA on our NumPy array with 10
components. I plot the explained variance ratio and pick our components with
the max explained ratio i.e n=3 using our elbow point. Then I train an SVM
with an RBF kernel and C=10000 on our data to predict our targets (Dusty/Not
Dusty).

Then I created a simple convolutional neural network classifier in TensorFlow.
The input shape is a 64x64x1 tensor followed by subsequent Convolution and
Pooling layers with RELU activation. The final layers flatten the input, and
the output is a 2-neuron dense layer with a SoftMax activation. The model is
trained with Adam optimizer, Learning Rate = 3e-4 (Suggested by Andrej
Karpathy) for 10 epochs with Sparse Categorical Cross entropy loss on 9817
images. The accuracy and loss per epoch are calculated. The predictions for
the test images are analyzed to validate the model.

Finally, I experimented on a Resnet-50 Backbone in TensorFlow trained on Image
Net in an attempt to achieve State of Art Results. I append the frozen Resnet
Backbone with some dense and dropout layers after flattening the output
(Following the pyramid model architecture). The final output layer has two
neurons corresponding to the outputs and a Softmax activation. The images are
resized (Upscaled) to a 224 x 224 image and stacked to convert the grayscale
to RGB. The trainable linear probe is fine-tuned for 10 Epochs with the same
aforementioned hyperparameters. The model predictions are tested on the test
dataset to validate the accuracy.

###  4.3. Image Noising

It is crucial to model the noise/dusty patches recorded by the HiRISE Camera
in order to train a model to denoise the images. The dusty patches first
appear completely random, so this is what I did in my initial experiments is
as follows:

I visually analyze the Dusty images to get an idea of the type of Noise
present. Initially, I experiment with perturbing the images with salt-pepper
noise using OpenCv2. To a predetermined group of pixels in each image, I added
salt and pepper noise at random. Unfortunately, the pictures greatly differ
from the actual dusty spots that were photographed. This is mostly due to two
factors: first, the dusty patches are not limited to 255/0 sort of noisy
pixels, i.e., noise is inherently grayscale; and second, even in the grayscale
noise, there is a significant level of randomness. On the higher end of
quantization, some images are formed entirely of noisy pixels, while others
have light noisy pixels on the lower end. The images are nearly entirely made
up of these noisy pixels, which appear to be a mix of light and dark grayscale
pixels upon closer study.

Further, I experimented with grayscale noisy values at the middle and higher
end of the pixel spectrum (105-205 and 205-255) randomly on N points in the
images. By manually adjusting the N values, grayscale noisy pixels, and
proportion of pixel noise in the higher and lower spectrum I was able to get
some visually similar images.

To get an idea of the actual grayscale noise distribution I plot a histogram
to find the peak values and ranges for these two levels of pixel quantization.
I find that most of the pixels lie in the 70 to 220 value range with peaks at
90, 135 and 190.

After experimenting with various methods to model the noise I finally settled
for the following method:

To model the varying level of grayscale noise in the image, I design a
function that adds noise randomly in between two ranges of quantization,
higher and lower. In the lower level of quantization, I add noisy pixels from
70 to 145 and, for the higher range, I add noisy pixels between 145 to 220.
The number of pixels I randomly add noise to will be decided based on the
arguments of the function.

Figure 2. Actual Clear Images (left), Denoised Cleaned Images using an
Upscaled AutoEncoder Architecture (right) [Resolution: 128x128]

Figure 3. Noisy Images for the Corresponding Denoised Images Above
[Resolution: 128x128]

###  4.4. Denoising Models

I have used an auto-encoder as a denoiser, as it can learn the latent
representation of the images by compressing the inputs to a bottleneck layer
and then reconstructing them. For a denoising auto-encoder, this property can
be used to train the previous autoencoder to denoise noisy images at the input
layer to their clean counterparts. However, these models are not extremely
effective at denoising as the model tries to minimize the L2 loss across the
image resulting in averaging out the outputs.

One could argue that a naive approach would be to train a CNN model for image-
to-image translation with a Noisy input image and a clean denoised target
image. Here the goal would be to minimize the pixel-to-pixel euclidean (L2)
distance between the predicted and target image rather than having a high-
level goal of producing an indistinguishable target image. Hence when such a
kind of model is used it results in checkerboard effect/artificats at a large
scale as the model tries to minimize its MSE loss across the whole image
rather than focusing on minimizing actual semantic loss such that visual
fidelity is not lost. Hence I make use of a conditional GAN called Pix2Pix
that in addition to a latent vector for image generation, concatenates the
corresponding denoised image. The generator in Pix2Pix makes use of a modified
UNet with single double strided convolutions and skip connection at the
decoder layers where the decoder layers are concatenated with the encoder
layers. The generator along with its GAN loss has another L1(MAE) loss that
minimizes the difference between real and generated target images. Generally,
the discriminator’s goal is to detect whether the image generated by the
generator is real or fake. Here Pix2Pix makes use of PatchGAN (70x70) as the
discriminator wherein it tries to classify every NxN as real or fake (Similar
to a CNN) and then the results are averaged. These properties help the Pix2Pix
model to not only fool the discriminator but also be as close as possible to
the target in the L1 sense, faster in inference due to patching, better output
quality due to skip connections in U-Net and produce an overall semantically
similar image. Any blurry images produced by the generator will be quickly
discarded by the Discriminator.

Initially, I train an auto-encoder with an input size of 100x100x1 pixels, 4
Encoding layers and a bottleneck/z layer of 25x25. I had experimented with
adding more layers to reduce the bottleneck size but pushing for more encoding
layers resulted in higher reconstruction loss. Due to the unconventional input
size of 100x100 max-pooling odd-sized channels further in the network resulted
in loss of features and spatial information. The clean, non-dusty images are
preprocessed as above into 100x100x1 images and the Auto-Encoder is trained
with the same input and output (Clean Not Dusty Images), with the same
optimizer as above and for 100 epochs and a batch size of 64. Once trained the
Auto-Encoder is retrained with input as Noised images generated above and
clean images as the output. I experiment with various levels of noise and plot
the results.

In order to alleviate some of these problems, I also upscale and downscale the
images to 64x64 and 128x128 respectively and thus correspondingly change the
model architecture by adding more layers and experimenting with a smaller
bottleneck size of 8x8 (64), and 16x16 (256). After experimenting with all
these architectures the upscaled deep architecture with a smaller bottleneck
size of 16x16 (256) was working as the best denoiser (Architecture in Figure
4). A larger bottleneck size of z = 32x32 and shallow layers results in a more
blurry reconstruction and the latent space doesn’t capture the semantic
understanding of the whole image. A highly compressed z=64 is not able to
effectively reconstruct the denoised image from the latent space and suffers
from blurring.

Finally, I train a Pix2Pix model by using the Tensorflow tutorial (Lamberta,
2022) (K, 2022) and the Original Paper (Isola et al., 2016) as reference
implementations. Initially, the conditional pair, (input, translated image)
are prepared. For preprocessing the images from the training set are upscaled
to a size of 256x256 as required by the network, normalized to a range of
[-1,1] and augmented via multiple random image transformations. The model is
compiled with a weighted sum of adversarial loss (BCE) and MAE loss with Adam
optimizer, learning rate = 2e-4, and beta=0.5. The model is trained and
evaluated after 10 Epochs.

Figure 4. Upscaled AutoEncoder Architecture

##  5\. Results

###  5.1. Classifier

For predicting Dusty vs Not Dusty Images the SVM + PCA classifier achieves an
accuracy of 61.96% on the test data. I picked 3 Principal components using PCA
as an input to SVM by finding the elbow point in the explained variance ratio
graph. Averaging across multiple seeds has similar results. The Metrics and
Confusion Matrix (Figure 5) for the model (test data) are below:

SVM+PCA | precision | recall | f1-score | support
---|---|---|---|---
non-dusty | 0.60 | 0.68 | 0.64 | 2554
dusty | 0.65 | 0.56 | 0.60 | 2660
accuracy |  |  | 0.62 | 5214
macro avg | 0.62 | 0.62 | 0.62 | 5214
weighted avg | 0.62 | 0.62 | 0.62 | 5214

Model | Accuracy%(Test)
---|---
SVM + PCA | 61.96
CNN | 89.74
Resnet50 (Finetuned) | 94.05

CNN achieves an accuracy of 93.16% on the train data, 91.49% on the validation
data and 89.74% on the test data with an F1 Score of 0.87(test). The Resnet50
model achieves the highest accuracy of 92.87% on the train data, 95.03% on the
validation data and 94.05% on the test data. The metrics for the Fine-Tuned
Resnet50 model (test data) are as follows:

Resnet50 | precision | recall | f1-score | support
---|---|---|---|---
non-dusty | 0.95 | 0.93 | 0.94 | 2554
dusty | 0.93 | 0.95 | 0.94 | 2660
accuracy |  |  | 0.94 | 5214
macro avg | 0.94 | 0.94 | 0.94 | 5214
weighted avg | 0.94 | 0.94 | 0.94 | 5214

Figure 5. Confusion Matrix

I also experimented with changing the number of layers in the linear probe, by
increasing the dense layers and adding more dropout layers but the accuracy
did not improve. I tried unfreezing the whole network and retraining it with a
low learning rate of 1e-6 however the accuracy drastically reduced. For the
CNN model, I try to use a LeNet-5-esque architecture with a receptive field of
18. The best model (Resnet50 in our case) is saved and used to create a
pipeline that accepts an image folder, does preprocessing and automatically
predicts and stores Dusty Images into an NPZ file. This filtering pipeline can
be used to automatically filter out such dusty patches without manual
intervention. We save all classified images into their respective arrays to an
NPZ file for later use. Future uses can be denoising dusty Images and further
analysis of classified images.

###  5.2. Denoiser

The initial Denoiser an Auto-Encoder with input size 100x100x1 with z=25x25
(625) trained to replicate input images achieves a loss of 0.6163 after 100
iterations. Once trained on the noisy data to clean the images the encoder
achieves a loss of 0.6228. Based on the visual analysis the decoded cleaned
images are somewhat clean of noise and slightly identical to the corresponding
clean target images. However, there is massive blurring and loss of features
in these denoised outputs (Figure 6).

I further experiment with other Auto-Encoder Architectures designed based on
our input image. I created two sets of images upscaled 128x128x1 images and
downscaled 64x64x1 images and correspondingly design Auto-Encoders for these
sizes. I add and reduce layers and change the size of the bottleneck layer.

Based on the validation loss and visual analysis upscaled images using an
auto-encoder with a large bottleneck size and more convolutional layers are
effective denoisers. The results can be inspected visually in Figures 2 and 3.

Then I train a Pix2Pix model and observe that the log-loss of the
Discriminator is on average 0.350 and lower than log(2) i.e the discriminator
performs better than the generator. The generator loss averages around 3.920.
The denoised outputs from the Pix2Pix cgan lack artificats and look sharper
and without loss of visual fidelity. Compared to the AutoEncoder the denoised
images are visually cleaner. The results can be inspected visually in Figures
7 and 8.

Increasing the intensity of noisy pixels beyond 60-70% of the image results in
the auto-encoder model failing in denoising and producing blurry, pixelated
outputs in many cases. However, the Pix2Pix model is able to effectively
denoise the image even in this extreme case. On increasing the noise beyond
80% of the pixels even the Pix2Pix model fails to restore the original clear
image from the noisy image. The reason for this can be attributed to the fact
that the input image is almost made of completely random noisy pixels and
without sufficient clear pixels to determine the martian landscape features
the model fails to generalize and denoise the image.

###  5.3. Quality of Denoising

After the denoising process, I obtained a somewhat clean denoised image. All
my experiment results have been explained through human visual assessment,
which relies on visual fidelity, perception and inspection. To use more
concrete metrics to support my results. I calculate the following metrics on
the test data:

  * •

MAE: It is the L1 pixel-to-pixel loss between the denoised and clear
(Expected) image.

  * •

PSNR: In the case of images it defines the ratio of the original data signal
to the noisy error in our denoising process.

  * •

SSIM: SSIM Index measures the similarity between two images and tells us how
far an image is w.r.t to the expected/original image aligned with the human
perceptual system.

  * •

Multiscale-SSIM: It’s the generalization of the SSIM Index at multiple image
scales/resolutions.

It can see from the tables below that the Pix2Pix model performs better than
the Upscaled AutoEncoder in terms of SSIM Score and MAE. It obtains a 0.99
SSIM score compared to 0.92 from the AutoEncoder thus indicating a higher
visual similarity. Both the models fair almost equally for the other two
metrics however, surprisingly AutoEncoder performs slightly better for these
two metrics. Due to RAM Limitations, we compute the Metrics for the Pix2Pix
image on only half the test images. The metrics for the denoising models can
be viewed in the table below:

Figure 6. Denoised Images with Base Encoder Architecture (Checkerboard Effect
Observed) [Resolution: 100x100] Figure 7. Results from Pix2Pix. Images are
highly perturbed with Random Noise[Resolution: 256x256] Figure 8. More Results
from Pix2Pix

Model | Metric | Value
---|---|---
| MAE (L1) | 0.022845278
Upscaled | SSIM (Mean) | 0.7573066
AutoEncoder | PSNR (Mean) | 31.497534
| Multiscale SSIM (Mean) | 0.9211203
| MAE (L1) | 0.023823868
Pix2Pix | SSIM (Mean) | 0.9998866
| PSNR (Mean) | 30.1209
| Multiscale SSIM (Mean) | 0.88276297

##  6\. Discussion

Based on our observations from the aforementioned results I find that SVM+PCA
with 3 principal components doesn’t work well and has a low F1 Score and
accuracy. CNN works well as expected but doesn’t achieve state-of-the-art
results. It doesn’t generalize well to the test set. Our Fine-Tuned Resnet50
classifier gives the best results, as expected. Resnet-50 is trained on Image
Net and as a backbone will be a strong feature extractor. Freezing all the
layers and replacing the final layer with a sequential model comprising a
stack of linear layers creates an efficient classifier architecture. Fine-
tuning this model results in creating a well-suited accurate classifier. Just
creating and training is model is not enough. It is important to deploy this
model for actual use, i.e classifying and storing dusty and clean patches in
our use case. To achieve this goal a predictor pipeline is created that
accepts an image folder loads it into an np array predicts the images and
stores the results and images in an NPZ file.

For our denoiser, the initial denoiser architecture works well with its
ability to denoise medium perturbed images. By upscaling the image and
applying an Auto-Encoder with more layers and an optimal z value I get better
results even in case of large perturbations. However, in extreme cases the
decoded images are pixelated and lack textures due to the loss function’s
tendency to minimize pixel-to-pixel loss by averaging out the output witnessed
by the low SSIM value and low MAE.

Finally, to overcome the issues from the previous denoising approaches I use
the Pix2Pix model, an image-to-image translation model to denoise the image.
The model effectively denoises images, without the repercussions faced by many
of the other generative models witnessed by the extremely high SSIM value. It
generates images, without artifacts that are sharp and with very few visual
defects. As the Pix2Pix images are already highly upscaled naturally their
MSSIM values will be slightly on the lower side. Although high levels of noise
above the range of 80% noisy pixels can affect its output this is possible to
be circumvented by probably using more data, customizing the model
architecture and training for more epochs.

##  7\. CONCLUSION

I can successfully classify Dusty Patches obstructed by Atmospheric Dust from
Non-Dusty clean Martian Patches. After experimenting with multiple models, I
can conclude that my Fine-tuned Resnet50 with an accuracy of 94.05% works the
best. Further, I have developed a data pipeline to clean dusty images or
filter them out and save them for further post-processing. One of these post-
processing steps is denoising, and I was able to create a deep Auto-Encoder-
based Denoiser that can denoise upscaled input images with decent resolution.
I also train the Pix2Pix model to denoise highly perturbed images effectively
with better denoising capabilities than all my other denoising models. In the
case of highly perturbed images with too many noisy pixels, all the denoising
models fail due to a lack of image features to work with for denoising.

In the Future, I want to create and deploy an end-to-end classifier and
denoising pipeline on the cloud and make it available for public use. I want
to further experiment with EDR images and with designing a model that can
learn and mimic the noise from the Dusty Patches and apply it to our clean
images to generate training data. Finally, I want to create a denoiser that
can filter out Dusty Patches at even 99% noise.

###### Acknowledgements.

This project, report and the research behind it would not have been possible
without the support, insights and feedback from my supervisor Professor Hannah
Kerner. The curriculum of CSE572 offered by her is extensive and covers all
the essential concepts for Data Mining Research. Further, I would like to
thank Liukis, Maria, Steven, et al., for open-sourcing the dataset, ”HiRISE
Image Patches Obscured by Atmospheric Dust”, and for providing me with the
opportunity to work on interesting and relevant real-world classification
problems. Without this dataset, this project would not have been possible.

## Code

Github Link: https://github.com/gremlin97/Detecting-and-refining-HiRISE-Image-
Patches-Obscured-by-Atmospheric-Dust

## SKILLS ACQUIRED

I am thankful to Prof. Kerner for guiding me through this project, from
proposal reviews to report revisions. Working on this project, I was able to
acquire a wide array of skills, specifically those related to deep learning
for Computer Vision and planetary remote sensing. I learned about the
operation of the Mars Reconnaissance Orbiter (MRO), the HiRISE device onboard
it, and the preprocessing considerations (EDR vs RDR) for planetary remote
sensing devices. Furthermore, specifically for Computer Vision, I learned how
to load, preprocess, and analyze planetary images for feature extraction, and
then classify them using state-of-the-art image classifiers to detect
noisy/denoised images. Additionally, I learned how to curate datasets with
noisy images to train denoising models such as Auto Encoders and Pix2Pix GAN
for denoising the images through the aforementioned models trained from
scratch. I also learned how to properly report the results through
classification reports and case-specific metrics such as MAE, PSNR, SSIM, and
MultiScale-SSIM. Learning to train and work with the Pix2Pix GAN was a
challenging and interesting aspect of this study. Finally, the most important
lesson I learned was to identify relevant problems to apply data mining to and
to find probable stakeholders for the same.

## CONTRIBUTIONS

This project is a solo endeavour conducted under the supervision of Prof.
Hannah Kerner. I have performed all the experiments, written the code for
them, and the report as well.

Dust storms frequently disrupt data collection, and subsequent manual removal
of noisy images is inefficient. In order to automate this process, I created a
Dust Image Classifier based on ResNet-50, which achieves 94.05% accuracy. A
prediction pipeline identifies and stores dusty patches, while an Auto
Encoder-based denoiser and Pix2Pix GAN enhance partially obstructed images
with SSIM Indexes of 0.75 and 0.99, respectively.

By acquiring the aforementioned skills, I have successfully classified patches
from HiRISE MRO as Dusty or Non-Dusty using a fine-tuned ResNet-50 with an
accuracy of 94.05%. Furthermore, I developed a cascaded pipeline to segregate
dusty patches and save them to disk for further filtration. For the filtered
dusty patches, I created an AutoEncoder that upscales and denoises the dusty
patches successfully. I also experimented with a Pix2Pix GAN for filtration,
and in the case of low to mid noise, the model successfully denoises the dusty
patches. However, in the case of highly perturbed images, the denoising is not
successful.

## References

  * (1)
  * Doran (2019) Gary Doran. 2019\.  _HiRISE Image Patches Obscured by Atmospheric Dust_.  https://doi.org/10.5281/zenodo.3495068
  * Isola et al. (2016) Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A. Efros. 2016.  Image-to-Image Translation with Conditional Adversarial Networks.  https://doi.org/10.48550/ARXIV.1611.07004
  * K (2022) Bharath K. 2022\.  Pix2pix generative Adversarial Networks.  https://blog.paperspace.com/pix2pix-gan/
  * Lamberta (2022) Billy Lamberta. 2022\.  pix2pix: Image-to-image translation with a conditional GAN | TensorFlow Core.  https://www.tensorflow.org/tutorials/generative/pix2pix.  [Online; accessed 2022-12-09].
  * Schawinski et al. (2017) Kevin Schawinski, Ce Zhang, Hantian Zhang, Lucas Fowler, and Gokula Krishnan Santhanam. 2017\.  Generative adversarial networks recover features in astrophysical images of galaxies beyond the deconvolution limit.  _Monthly Notices of the Royal Astronomical Society: Letters_ 467, 1 (01 2017), L110–L114.  https://doi.org/10.1093/mnrasl/slx008 arXiv:https://academic.oup.com/mnrasl/article-pdf/467/1/L110/10730451/slx008.pdf
  * Vojtekova et al. (2020) Antonia Vojtekova, Maggie Lieu, Ivan Valtchanov, Bruno Altieri, Lyndsay Old, Qifeng Chen, and Filip Hroch. 2020.  Learning to denoise astronomical images with U-nets.  _Monthly Notices of the Royal Astronomical Society_ 503, 3 (11 2020), 3204–3215.  https://doi.org/10.1093/mnras/staa3567 arXiv:https://academic.oup.com/mnras/article-pdf/503/3/3204/36847224/staa3567.pdf
  * Wagstaff et al. (2018) Kiri Wagstaff, You Lu, Alice Stanboli, Kevin Grimes, Thamme Gowda, and Jordan Padams. 2018\.  Deep Mars: CNN Classification of Mars Imagery for the PDS Imaging Atlas.  _Proceedings of the AAAI Conference on Artificial Intelligence_ 32, 1 (Apr. 2018).  https://doi.org/10.1609/aaai.v32i1.11404


# Consistency of higher derivative couplings to matter fields in scalar-tensor
gravity

Tact Ikeda<EMAIL_ADDRESS>Department of Physics, Rikkyo University,
Toshima, Tokyo 171-8501, Japan  Kazufumi Takahashi
<EMAIL_ADDRESS>Center for Gravitational Physics and
Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto
University, 606-8502, Kyoto, Japan  Tsutomu Kobayashi<EMAIL_ADDRESS>Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan

###### Abstract

Recently, a generalization of invertible disformal transformations containing
higher-order derivatives of a scalar field has been proposed in the context of
scalar-tensor theories of gravity. By applying this generalized disformal
transformation to the Horndeski theory, one can obtain the so-called
generalized disformal Horndeski theories which are more general healthy
scalar-tensor theories than ever. However, it is unclear whether or not the
generalized disformal Horndeski theories can be coupled consistently to matter
fields because introducing a matter field could break the degeneracy
conditions of higher-order scalar-tensor theories and hence yield the unwanted
Ostrogradsky ghost. We investigate this issue and explore the conditions under
which a minimal coupling to a matter field is consistent in the generalized
disformal Horndeski theories without relying on any particular gauge such as
the unitary gauge. We find that all the higher derivative terms in the
generalized disformal transformation are prohibited to avoid the appearance of
the Ostrogradsky ghost, leading to the conclusion that only the theories that
are related to the Horndeski theory through a conventional disformal
transformation remain ghost-free in the presence of minimally coupled matter
fields.

††preprint: RUP-23-2, YITP-23-11

##  I Introduction

The mechanism of the accelerated expansion of the late Universe is yet
unknown, which motivates the active study of modified gravity as an
alternative to dark energy. A modification of gravity at high energies is also
strongly motivated because general relativity is considered to be a low-energy
effective theory and it is thus interesting to explore physics beyond general
relativity in a strong gravity regime. Furthermore, modified gravity models
are useful for comparison with general relativity in the context of testing
gravity (see, e.g., Refs. [1, 2, 3] for reviews). Modified gravity is
described, at least effectively, by theories equipped with new gravitational
degree(s) of freedom (DOFs) on top of the massless graviton DOFs, and as such
scalar-tensor theories are often studied where a scalar DOF is taken into
account. Once one goes beyond general relativity, one may naturally consider
higher derivative terms in the gravitational Lagrangian, but higher-derivative
theories are plagued by the Ostrogradsky instability in general [4, 5, 6].
Therefore, in generalizing theories of gravity, one must be careful not to
induce such an instability arising from higher derivatives. Even if the
Lagrangian itself contains higher derivatives, this instability can be avoided
as long as the field equations are intrinsically of second order. The
Horndeski theory [7, 8, 9] is the most general scalar-tensor theory having
second-order Euler-Lagrange equations, and due to this merit it has been
studied extensively over the recent years. However, the Horndeski theory is
not the most general scalar-tensor theory that is free from the Ostrogradsky
instability. The point here is that, if the system is degenerate, higher-order
field equations contain less number of DOFs than anticipated from the order of
derivatives [10, 11, 12, 13, 14]. Degenerate higher-order scalar-tensor
(DHOST) theories [15, 16, 17] exploit this loophole and extend the Horndeski
theory to a large family of higher-derivative scalar-tensor theories having a
single scalar and two tensor DOFs (see Refs. [18, 19] for reviews).

Given that two theories related via invertible field redefinition have the
same number of dynamical DOFs [20, 21], an invertible metric redefinition is a
convenient and useful way of generating nontrivial class of healthy scalar-
tensor theories from existing ones that are manifestly ghost-free. In
particular, derivative-dependent transformations yield higher-derivative
theories that are nevertheless free from the Ostrogradsky ghost. For example,
by applying to the Horndeski theory a disformal transformation, which is a
general metric redefinition involving the first derivative of the scalar DOF
[22], we can obtain a certain subset of DHOST theories [23]. Indeed, the first
example of scalar-tensor theories beyond Horndeski was obtained in that way
[24]. It is important to note that, among subclasses of DHOST theories that
are systematically constructed by imposing the degeneracy conditions, only the
subclass generated via invertible disformal transformation from the Horndeski
theory is physically interesting because cosmological solutions can be stable
(and tensor perturbations remain dynamical) only in that subclass [25,
26].111A noninvertible subclass of disformal transformations can also be used
to generate a certain subclass of DHOST theories, but this subclass does not
accommodate stable cosmological solutions or otherwise the tensor
perturbations are nondynamical [27, 28]. Such disformally generated DHOST
theories include, e.g., the so-called “beyond Horndeski” or GLPV theories [29,
30] as specific cases.

Recently, a generalization of invertible disformal transformations involving
second (and higher) derivatives of the scalar DOF was proposed [31]. By
applying this novel class of generalized disformal transformations to the
Horndeski theory, we can derive yet more general higher-order scalar-tensor
theories than ever constructed, which we call generalized disformal Horndeski
(GDH) theories [32]. Since two theories thus related via invertible disformal
transformation are equivalent, GDH theories are also free from the
Ostrogradsky instability (even though the field equations are apparently of
higher order). However, this statement should be taken with care. The
equivalence holds only in vacuum, and the Horndeski theory with minimally
coupled matter and GDH theories with minimally coupled matter are not
equivalent. Therefore, the inclusion of (minimally coupled) matter fields in
GDH theories could result in extra ghost DOFs. The possible appearance of
extra DOFs can be understood by moving back to the “Horndeski frame” where the
gravity sector is described by the Horndeski theory and the matter sector is
coupled to a generalized disformal metric involving second derivatives of the
scalar DOF. In the Horndeski frame, the matter fields are thus coupled with
second derivatives of the scalar DOF, possibly breaking the degeneracy
conditions. This issue has been pointed out already in the context of DHOST
theories [33, 34]. Motivated by this concern, the consistency of matter
couplings in GDH theories has been investigated in Refs. [32, 35, 36].
However, in those previous studies, the unitary gauge is taken in which the
scalar DOF is homogeneous on constant-time hypersurfaces. The conditions that
remove extra ghost DOFs derived in Refs. [32, 35, 36] should therefore be
weaker than those that would be derived without assuming any particular gauge.
In other words, upon imposing the degeneracy conditions validated only in the
unitary gauge, there would be the Ostrogradsky ghost away from the unitary
gauge. Such a gauge-dependent Ostrogradsky ghost would be a nonpropagating
“shadowy mode,” and hence harmless [37, 38]. Nevertheless, since such theories
are rather tricky and the shadowy mode requires a careful treatment, we are
interested mostly in theories without the shadowy mode. Also, by removing the
Ostrogradsky ghost in any gauge, one may safely consider, e.g., static stars
and black holes dressed with a static scalar profile. Along this line of
thought, in this paper, we explore, without assuming any particular gauge, the
conditions under which matter couplings to GDH theories are consistent and
there is no Ostrogradsky instability.

This paper is organized as follows. In the next section, we briefly review the
generalized disformal transformation [31] and introduce the action for GDH
theories by applying a generalized disformal transformation to the Horndeski
theory. In Sec. III, we study the consistency of matter couplings in the GDH
theory. We review the previous results obtained in the unitary gauge [32, 35,
36], and then investigate the conditions for the matter couplings to be
consistent away from the unitary gauge. We finally draw our conclusions in
Sec. IV.

##  II Generalized disformal transformations

###  II.1 Invertible disformal transformations with higher-order derivatives

Let us consider a generalized disformal transformation defined as [31]

$\displaystyle\bar{g}_{\mu\nu}=F_{0}g_{\mu\nu}+F_{1}\phi_{\mu}\phi_{\nu}+2F_{2}\phi_{(\mu}X_{\nu)}+F_{3}X_{\mu}X_{\nu},$
(1)

where $\phi_{\mu}\coloneqq\nabla_{\mu}\phi$ and
$X_{\mu}\coloneqq\nabla_{\mu}X$ with $X\coloneqq\phi^{\mu}\phi_{\mu}$. Here,
$F_{i}$ ($i=0,1,2,3$) are functions of $(\phi,X,Y,Z)$, where
$Y\coloneqq\phi^{\mu}X_{\mu}$ and $Z\coloneqq X^{\mu}X_{\mu}$. Restricting the
form of the functions to be $F_{0}=F_{0}(\phi,X)$, $F_{1}=F_{1}(\phi,X)$ and
$F_{2}=F_{3}=0$, we obtain the conventional disformal transformation [22]

$\displaystyle\bar{g}_{\mu\nu}=F_{0}(\phi,X)g_{\mu\nu}+F_{1}(\phi,X)\phi_{\mu}\phi_{\nu}.$
(2)

In this sense, the transformation (1) involves the conventional one (2).

Following Ref. [31], we summarize the conditions under which the
transformation (1) is invertible. The essential ingredient of the invertible
generalized disformal transformation is the requirement that a set of
generalized disformal transformations forms a group under the following two
operations:

$\displaystyle\left(\bar{g}\cdot\hat{g}\right)_{\mu\nu}\coloneqq\bar{g}_{\mu\alpha}g^{\alpha\beta}\hat{g}_{\beta\nu},\quad\text{(Matrix
product)}$ (3)

and

$\displaystyle\left(\bar{g}\circ\hat{g}\right)_{\mu\nu}[g,\phi]\coloneqq\bar{g}_{\mu\nu}[\hat{g},\phi].\quad\text{(Functional
composition)}$ (4)

We are then allowed to construct the inverse of the transformed metric and the
inverse transformation. Note that, in contrast to the case of conventional
disformal transformation (2), the closedness under the functional composition
for the generalized disformal transformation (1) is nontrivial as the
transformation law involves the derivative of the metric. For the set of
generalized disformal transformations to form a group, it is sufficient that
the following conditions are satisfied [31]:

$\displaystyle F_{0}\neq 0,\quad\mathcal{F}\neq 0,\quad\bar{X}_{X}\neq
0,\quad\bar{X}_{Y}=\bar{X}_{Z}=0,\quad\left|\frac{\partial(\bar{Y},\bar{Z})}{\partial(Y,Z)}\right|\neq
0,$ (5)

where

$\displaystyle\bar{X}\coloneqq\bar{g}^{\mu\nu}\phi_{\mu}\phi_{\nu},\quad\bar{Y}\coloneqq\bar{g}^{\mu\nu}\phi_{\mu}\bar{X}_{\nu},\quad\bar{Z}\coloneqq\bar{g}^{\mu\nu}\bar{X}_{\mu}\bar{X}_{\nu},$
(6)

and

$\displaystyle\mathcal{F}(\phi,X,Y,Z)$ $\displaystyle\coloneqq
F_{0}^{2}+F_{0}(XF_{1}+2YF_{2}+ZF_{3})+(Y^{2}-XZ)(F_{2}^{2}-F_{1}F_{3}).$ (7)

Note that, among the conditions in Eq. (5), the one that guarantees the
closedness under the functional composition is $\bar{X}_{Y}=\bar{X}_{Z}=0$.
Suppose that these conditions are satisfied. The inverse metric
$\bar{g}^{\mu\nu}$ of $\bar{g}_{\mu\nu}$ is then given by

$\displaystyle\bar{g}^{\mu\nu}=f_{0}g^{\mu\nu}+f_{1}\phi^{\mu}\phi^{\nu}+2f_{2}\phi^{(\mu}X^{\nu)}+f_{3}X^{\mu}X^{\nu},$
(8)

where

$\displaystyle f_{0}$ $\displaystyle\coloneqq\frac{1}{F_{0}},\quad
f_{1}\coloneqq-\frac{F_{0}F_{1}-Z\left(F_{2}^{2}-F_{1}F_{3}\right)}{F_{0}\mathcal{F}},\quad
f_{2}\coloneqq-\frac{F_{0}F_{2}+Y\left(F_{2}^{2}-F_{1}F_{3}\right)}{F_{0}\mathcal{F}},\quad
f_{3}\coloneqq-\frac{F_{0}F_{3}-X\left(F_{2}^{2}-F_{1}F_{3}\right)}{F_{0}\mathcal{F}}.$
(9)

The following formula is also useful for reconstructing the barred metric
$\bar{g}_{\mu\nu}$ [Eq. (1)] when its inverse is given in the form of Eq. (8):

$\displaystyle F_{0}$ $\displaystyle\coloneqq\frac{1}{f_{0}},\quad
F_{1}\coloneqq-\frac{f_{0}f_{1}-Z\left(f_{2}^{2}-f_{1}f_{3}\right)}{f_{0}\mathcal{H}},\quad
F_{2}\coloneqq-\frac{f_{0}f_{2}+Y\left(f_{2}^{2}-f_{1}f_{3}\right)}{f_{0}\mathcal{H}},\quad
F_{3}\coloneqq-\frac{f_{0}f_{3}-X\left(f_{2}^{2}-f_{1}f_{3}\right)}{f_{0}\mathcal{H}}.$
(10)

Here, we have defined

$\displaystyle\mathcal{H}(\phi,X,Y,Z)$ $\displaystyle\coloneqq
f_{0}^{2}+f_{0}(Xf_{1}+2Yf_{2}+Zf_{3})+(Y^{2}-XZ)(f_{2}^{2}-f_{1}f_{3}),$ (11)

which is obtained simply by replacing $F_{i}$ in Eq. (7) by $f_{i}$. Having
constructed the inverse metric explicitly, let us next look at the inverse
transformation. The inverse transformation of (1) is expressed as

$\displaystyle g_{\mu\nu}[\bar{g},\phi]$
$\displaystyle=\frac{1}{F_{0}}\bar{g}_{\mu\nu}-\frac{\bar{X}_{X}^{2}F_{1}-2\bar{X}_{\phi}\bar{X}_{X}F_{2}+\bar{X}_{\phi}^{2}F_{3}}{\bar{X}_{X}^{2}F_{0}}\phi_{\mu}\phi_{\nu}-2\frac{\bar{X}_{X}F_{2}-\bar{X}_{\phi}F_{3}}{\bar{X}_{X}^{2}F_{0}}\phi_{(\mu}\bar{X}_{\nu)}-\frac{F_{3}}{\bar{X}_{X}^{2}F_{0}}\bar{X}_{\mu}\bar{X}_{\nu},$
(12)

where $F_{i}$’s in the right-hand side are given as functions of
$(\phi,\bar{X},\bar{Y},\bar{Z})$. Thanks to the group structure of the set of
generalized disformal transformations, one can thus obtain the inverse metric
and inverse transformation.

###  II.2 Generalized disformal Horndeski theories

We now define a scalar-tensor theory obtained by applying a generalized
disformal transformation to the metric in the Horndeski theory. The Horndeski
theory (in vacuum) is described by the action [7, 8, 9]

$\displaystyle S_{\mathrm{Hor}}[g_{\mu\nu},\phi]\coloneqq\int{\rm
d}^{4}x\sqrt{-g}\,\mathcal{L}_{\mathrm{Hor}}[g_{\mu\nu},\phi],$ (13)

where

$\displaystyle\mathcal{L}_{\mathrm{Hor}}$ $\displaystyle\coloneqq
G_{2}(\phi,X)+G_{3}(\phi,X)\Box\phi+G_{4}(\phi,X)R-2G_{4,X}(\phi,X)[(\Box\phi)^{2}-\phi^{\mu\nu}\phi_{\mu\nu}]$
$\displaystyle\quad+G_{5}(\phi,X)G^{\mu\nu}\phi_{\mu\nu}+\frac{1}{3}G_{5,X}(\phi,X)[(\Box\phi)^{3}-3\Box\phi\phi^{\mu\nu}\phi_{\mu\nu}+2\phi_{\mu\nu}\phi^{\mu\rho}\phi^{\nu}_{\rho}],$
(14)

with $\phi_{\mu\nu}\coloneqq\nabla_{\mu}\nabla_{\nu}\phi$. Performing a
generalized disformal transformation satisfying the invertibility conditions
(5), we obtain a new action for a scalar-tensor theory [32]

$\displaystyle S_{\mathrm{GDH}}[g_{\mu\nu},\phi]\coloneqq
S_{\mathrm{Hor}}[\bar{g}_{\mu\nu},\phi]=\int{\rm
d}^{4}x\sqrt{-g}\,\mathcal{J}\mathcal{L}_{\mathrm{Hor}}[\bar{g}_{\mu\nu},\phi],$
(15)

where we have defined

$\displaystyle\mathcal{J}\coloneqq\frac{\sqrt{-\bar{g}}}{\sqrt{-g}}=F_{0}\mathcal{F}^{1/2}=f_{0}^{-1}\mathcal{H}^{-1/2}.$
(16)

This theory was dubbed the generalized disformal Horndeski (GDH) theory [32].
Since the generalized disformal transformation is just a field redefinition,
$S_{\mathrm{GDH}}[g_{\mu\nu},\phi]$ and
$S_{\mathrm{Hor}}[\bar{g}_{\mu\nu},\phi]$ are mathematically equivalent as
long as the transformation satisfies the invertibility conditions (5). This in
particular means that there are one scalar and two tensor degrees of freedom
in the (vacuum) GDH theory as in the (vacuum) Horndeski theory, even though
the field equations in the former theory contain higher derivatives in
general. However, in the presence of matter fields, things become subtle and
the relation between the two theories must be examined carefully.

To see this point more closely, let us add matter field(s) (collectively
denoted by $\Psi$) minimally coupled to the GDH theory,

$\displaystyle
S_{\mathrm{GDH}}[g_{\mu\nu},\phi]+S_{\mathrm{m}}[g_{\mu\nu},\Psi]=S_{\mathrm{Hor}}[\bar{g}_{\mu\nu},\phi]+S_{\mathrm{m}}[g_{\mu\nu},\Psi].$
(17)

We see that the matter fields are coupled with the generalized disformal
metric (12) in the Horndeski frame (the right-hand side) in which the
gravitational part of the action is written manifestly in the Horndeski form
in terms of the metric $\bar{g}_{\mu\nu}$. Since the generalized disformal
metric (12) contains higher-order derivatives of $\phi$, there is no guarantee
that the coupling to matter is consistent, i.e., no unwanted degree of freedom
appears as an Ostrogradsky ghost through this coupling. If such an additional
dangerous degree of freedom were to appear, then the GDH theory in the
presence of (minimally coupled) matter would be inconsistent, albeit healthy
in vacuum.222We expect that the mass of the Ostrogradsky mode would be
proportional to some negative power of the energy density of the matter field.
Therefore, from the EFT point of view, the ghost would be irrelevant if we
push its mass above the cutoff scale. In the next section, we will study this
point in detail.

##  III Consistency of matter coupling

###  III.1 Unitary gauge

In this subsection, we briefly review the consistency of matter coupling in
the GDH theory under the unitary gauge, where the scalar field is spatially
uniform. Though it is not always justified, one is allowed to take the unitary
gauge at least in the context of cosmology where the scalar field is supposed
to have a timelike gradient. The case of bosonic matter fields was discussed
in Refs. [32, 35, 36]. As we saw in Sec. II.2, in the Horndeski frame, the
matter fields are coupled to the generalized disformal metric (with respect to
the metric that describes the gravity sector), and hence the matter action
involves higher-order derivatives of $\phi$. This indicates that the matter
action yields the time derivative of the lapse function under the unitary
gauge, which generically makes the (otherwise nondynamical) lapse function
dynamical. Thus, the GDH theory would give rise to the Ostrogradsky ghost in
general in the presence of matter fields. Fortunately, one can remove the
Ostrogradsky ghost by restricting the generalized disformal metric to the
following form [32, 35, 36]:

$\displaystyle\bar{g}_{\mu\nu}=\tilde{F}_{0}g_{\mu\nu}+\tilde{F}_{1}\phi_{\mu}\phi_{\nu}+2\tilde{F}_{2}\phi_{(\mu}\mathcal{X}_{\nu)}+\tilde{F}_{3}{\cal
X}_{\mu}\mathcal{X}_{\nu},\qquad\mathcal{X}_{\mu}\coloneqq\left(\delta_{\mu}^{\alpha}-\frac{\phi_{\mu}\phi^{\alpha}}{X}\right)\partial_{\alpha}X,$
(18)

where ${\cal X}_{\mu}$ is the derivative of $X$ projected onto a
constant-$\phi$ hypersurface and $\tilde{F}_{i}=\tilde{F}_{i}(\phi,X,{\cal
Z})$, with

$\displaystyle{\cal Z}\coloneqq{\cal X}^{\mu}{\cal
X}_{\mu}=Z-\frac{Y^{2}}{X}.$ (19)

The point is that the object ${\cal X}_{\mu}$ does not contain the time
derivative of the lapse function under the unitary gauge where $\phi=\phi(t)$.
Note that Eq. (18) is embedded in the original generalized disformal
transformation (1) as

$\displaystyle F_{0}=\tilde{F}_{0},\qquad
F_{1}=\tilde{F}_{1}-\frac{2Y}{X}\tilde{F}_{2}+\frac{Y^{2}}{X^{2}}\tilde{F}_{3},\qquad
F_{2}=\tilde{F}_{2}-\frac{Y}{X}\tilde{F}_{3},\qquad F_{3}=\tilde{F}_{3}.$ (20)

On the other hand, the case of fermionic matter fields needs a separate
treatment as the matter action is written in terms of the tetrad rather than
the metric itself. The authors of Ref. [36] developed the transformation law
for the tetrad to show that the consistency of fermionic matter coupling
requires an additional condition $\tilde{F}_{3}=0$ [36].

Here, it should be emphasized that all these results were obtained in the
unitary gauge. Away from the unitary gauge, there would be an apparent
Ostrogradsky mode. It could be true that such an apparent Ostrogradsky ghost
is harmless because it would be a nonpropagating “shadowy mode” that satisfies
a three-dimensional elliptic differential equation on a spacelike
hypersurface, leading to a configuration completely determined by boundary
conditions. (See Refs. [37, 38, 39] for a more detailed discussion in the
context of U-DHOST theories.) Having said that, we need a careful treatment
for the shadowy mode, and hence theories without the shadowy mode are still of
primary interest to us. By removing the Ostrogradsky mode in any gauge, one
may safely consider, for example, static stars and black holes dressed with a
static scalar field. In what follows, we investigate the consistency of matter
coupling away from the unitary gauge and derive degeneracy conditions without
either Ostrogradsky or shadowy mode.

###  III.2 Away from the unitary gauge

Let us now explore the consistency of the matter coupling away from the
unitary gauge. We start with the Horndeski-frame Lagrangian

$\displaystyle\mathcal{L}[g_{\mu\nu},\phi]=\mathcal{L}_{\mathrm{Hor}}[g_{\mu\nu},\phi]+\mathcal{L}_{\mathrm{m}}[\bar{g}_{\mu\nu},\psi],$
(21)

where $\bar{g}_{\mu\nu}$ is defined in Eq. (1). Note that we have interchanged
the roles of $g_{\mu\nu}$ and $\bar{g}_{\mu\nu}$ as compared to those in Eq.
(17). In any case, so long as the (generalized) disformal transformation is
invertible, the barred metric is some disformal transformation of the unbarred
metric and vice versa, and hence this is just a matter of convention. Note,
however, that the Lagrangian (17) makes sense as a theory of gravity
nonminimally coupled to matter even if the disformal transformation is
noninvertible, though one cannot move to the equivalent description in the GDH
(or Jordan) frame in this case. Therefore, in this subsection, we study a
system described by the Lagrangian (17) without imposing the invertibility
conditions for the disformal transformation from the outset.333Precisely
speaking, we assume the first two conditions in Eq. (5) to guarantee the
existence of the barred inverse metric $\bar{g}^{\mu\nu}$, but do not
necessarily impose the last three conditions. The authors of Ref. [35] used a
similar approach to specify the degeneracy conditions under the unitary gauge
without imposing the invertibility conditions from the outset. For simplicity,
we assume that the matter sector is described by a massless scalar field
$\psi$,

$\displaystyle S_{\mathrm{m}}[\bar{g}_{\mu\nu},\psi]$
$\displaystyle\coloneqq-\frac{1}{2}\int{\rm
d}^{4}x\sqrt{-\bar{g}}\,\bar{g}^{\mu\nu}\psi_{\mu}\psi_{\nu}=-\frac{1}{2}\int{\rm
d}^{4}x\sqrt{-g}\,\mathcal{J}\bar{g}^{\mu\nu}\psi_{\mu}\psi_{\nu},$ (22)

where $\psi_{\mu}\coloneqq\nabla_{\mu}\psi$ and $\bar{g}^{\mu\nu}$ is the
inverse metric associated with $\bar{g}_{\mu\nu}$ defined in Eq. (8).

We expect that the system described by the Lagrangian (17) has four physical
degrees of freedom, where three come from the gravity sector ($g_{\mu\nu}$ and
$\phi$) and one from the matter scalar field $\psi$. In order to avoid an
unwanted fifth (would-be Ostrogradsky) degree of freedom, terms with the
highest time derivatives must possess a degenerate structure. In order to
study the kinetic structure of the Lagrangian (17) in detail, let us introduce
the Arnowitt-Deser-Misner (ADM) variables as

$\displaystyle g_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}=-N^{2}{\rm
d}t^{2}+h_{ij}({\rm d}x^{i}+N^{i}{\rm d}t)({\rm d}x^{j}+N^{j}{\rm d}t),$ (23)

where $N$ is the lapse function, $N^{i}$ is the shift vector, and $h_{ij}$ is
the induced metric. Note that we do not choose the unitary gauge, and hence
the timelike unit normal vector $n_{\mu}=-N\delta^{0}_{\mu}$ associated with a
constant-$t$ hypersurface is not proportional to $\phi_{\mu}$. The extrinsic
curvature is written in terms of the ADM variables as

$\displaystyle K_{ij}=\frac{1}{2N}\left(\dot{h}_{ij}-{\rm D}_{i}N_{j}-{\rm
D}_{j}N_{i}\right),$ (24)

where a dot denotes the time derivative and ${\rm D}_{i}$ denotes the
covariant derivative associated with $h_{ij}$. We also define the variables
associated with the first and second time derivatives of $\phi$ and the first
time derivative of $\psi$ as follows:

$\displaystyle A_{*}\coloneqq n^{\mu}\nabla_{\mu}\phi,\qquad X_{*}\coloneqq
n^{\mu}\nabla_{\mu}X,\qquad\psi_{*}\coloneqq n^{\mu}\nabla_{\mu}\psi.$ (25)

The kinetic structure of the Lagrangian (17) can be captured by the Hessian
matrix $\mathbb{H}$ of the Lagrangian (21) with respect to $K_{ij}$, $X_{*}$,
and $\psi_{*}$. Written explicitly, one has

$\displaystyle\mathbb{H}=\begin{pmatrix}\mathcal{K}^{ij,kl}&0&0\\\
0&\mathcal{A}&\mathcal{M}\\\ 0&\mathcal{M}&\mathcal{P}\end{pmatrix},$ (26)

with

$\displaystyle\mathcal{K}^{ij,kl}\coloneqq\frac{\partial^{2}\mathcal{L}}{\partial
K_{ij}\partial
K_{kl}},\qquad\mathcal{A}\coloneqq\frac{\partial^{2}\mathcal{L}}{\partial
X_{*}^{2}},\qquad\mathcal{M}\coloneqq\frac{\partial^{2}\mathcal{L}}{\partial
X_{*}\partial\psi_{*}},\qquad\mathcal{P}\coloneqq\frac{\partial^{2}\mathcal{L}}{\partial\psi_{*}^{2}}.$
(27)

Note that there is no kinetic mixing between the gravitational and matter
sectors: The gravitational (Horndeski) sector concerns only
$\mathcal{K}^{ij,kl}$, while the matter sector concerns only $\mathcal{A}$,
$\mathcal{M}$, and $\mathcal{P}$. In order to kill the unwanted degree of
freedom revived due to the matter coupling, we require that the $2\times 2$
lower-right submatrix of $\mathbb{H}$ is degenerate, i.e.,

$\displaystyle\mathcal{D}\coloneqq\mathcal{A}\mathcal{P}-\mathcal{M}^{2}=0.$
(28)

The quantity $\mathcal{D}$ can be rewritten in the form of a polynomial in
$\\{A_{*},X_{*},\psi_{*},\mathcal{Q}_{1},\mathcal{Q}_{2},\mathcal{Q}_{3}\\}$
as

$\displaystyle\mathcal{D}=\sum_{i,j,k,l,m,n\geq
0}d_{ijklmn}(\phi,X,Y,Z)\,A_{*}^{i}\,X_{*}^{j}\,\psi_{*}^{k}\,\mathcal{Q}_{1}^{l}\,\mathcal{Q}_{2}^{m}\,\mathcal{Q}_{3}^{n},$
(29)

with

$\displaystyle\mathcal{Q}_{1}\coloneqq
g^{\mu\nu}\psi_{\mu}\psi_{\nu},\quad\mathcal{Q}_{2}\coloneqq
g^{\mu\nu}\psi_{\mu}\phi_{\nu},\quad\mathcal{Q}_{3}\coloneqq
g^{\mu\nu}\psi_{\mu}X_{\nu}.$ (30)

Note that the coefficients $d_{ijklmn}$ depend only on the functions $f_{i}$
characterizing the generalized disformal transformation. In order for
$\mathcal{D}$ to vanish for any configuration of $\phi$ and $\psi$, we shall
fix $f_{i}$’s so that all $d_{ijklmn}$’s vanish.444Under the unitary gauge, we
have $A_{*}=(-X)^{1/2}$, $X_{*}=-Y(-X)^{-1/2}$, and
$\psi_{*}=-(-X)^{-1/2}\mathcal{Q}_{2}$. Hence, we obtain a weaker condition
that the coefficients in front of
$\mathcal{Q}_{1}^{l}\mathcal{Q}_{2}^{p}\mathcal{Q}_{3}^{n}$ vanish, i.e.,
$\sum_{i,j,k\geq
0}(-1)^{j+k}(-X)^{(i-j-k)/2}\,Y^{j}\,\delta_{k+m,p}\,d_{ijklmn}(\phi,X,Y,Z)=0$
for all $l,n,p\,(\geq 0)$. Since the full expression of $\mathcal{D}$ is
extremely involved, we proceed step by step: Among nonvanishing $d_{ijklmn}$,
we first focus on the simplest one(s) to read off condition(s) that $f_{i}$’s
should satisfy. We then substitute the condition(s) back into $\mathcal{D}$,
which simplifies some of $d_{ijklmn}$’s. With simplified $d_{ijklmn}$’s, we
follow the same steps, selecting the simplest one(s) to find additional
condition(s) on $f_{i}$’s. Repeating this procedure, we finally obtain a set
of conditions on $f_{i}$’s under which all $d_{ijklmn}$’s vanish, i.e.,
$\mathcal{D}=0$. In what follows, we apply this strategy to fix the functional
form of $f_{i}$’s. It should be noted that we assume $F_{0}\neq 0$ and
$\mathcal{J}\neq 0$ throughout the following discussion because otherwise one
cannot define the barred inverse metric $\bar{g}^{\mu\nu}$.

The condition that the coefficient of $\psi_{*}^{2}$ vanishes yields

$\displaystyle f_{3}=0.$ (31)

Likewise, from the coefficients of $A_{*}^{4}\mathcal{Q}_{3}^{2}$ and
$X_{*}^{4}\mathcal{Q}_{2}^{2}$, we find that $f_{2}$ must be of the form

$\displaystyle\mathcal{J}f_{2}=\alpha_{2}(\phi,X),$ (32)

where $\alpha_{2}$ is a function of $\phi$ and $X$ which is arbitrary at this
step. From the coefficients of $\mathcal{Q}_{1}$ and
$A_{*}^{2}\mathcal{Q}_{1}$, we see that $f_{0}$ must take the form

$\displaystyle\mathcal{J}f_{0}=\alpha_{0}(\phi,X)+\beta_{0}(\phi,X)Y,$ (33)

where $\alpha_{0}$ and $\beta_{0}$ are arbitrary functions of $\phi$ and $X$.
Then, from the coefficient of $A_{*}^{2}\psi_{*}^{2}$, we find that
$\alpha_{2}$ must be related to $\beta_{0}$ by

$\displaystyle\alpha_{2}=-\beta_{0}.$ (34)

The coefficient of $A_{*}X_{*}^{3}\mathcal{Q}_{2}^{2}$ leads us to the
following relation:

$\displaystyle\beta_{0}(\mathcal{J}f_{1})_{,ZZ}=0.$ (35)

We now have the two branches of solutions, $\beta_{0}=0$ and
$(\mathcal{J}f_{1})_{,ZZ}=0$.

Let us first consider the branch $\beta_{0}=0$. In this case, from the
coefficients of $\mathcal{Q}_{2}^{2}$, $A_{*}^{2}\mathcal{Q}_{2}^{2}$, and
$A_{*}^{4}\mathcal{Q}_{2}^{2}$, we see that $f_{1}$ must take the form

$\displaystyle\mathcal{J}f_{1}=\alpha_{1}(\phi,X),$ (36)

where $\alpha_{1}$ is an arbitrary function of $\phi$ and $X$. Combining the
conditions obtained so far, we obtain the following relations:

$\displaystyle f_{0}=\frac{\alpha_{0}}{\mathcal{J}},\qquad
f_{1}=\frac{\alpha_{1}}{\mathcal{J}},\qquad f_{2}=f_{3}=0.$ (37)

On top of these, we have $\mathcal{J}^{-1}=f_{0}\mathcal{H}^{1/2}$ with
$\mathcal{H}$ defined in Eq. (11), which yields

$\displaystyle\mathcal{J}=\sqrt{\alpha_{0}^{3}(\alpha_{0}+\alpha_{1}X)}.$ (38)

We now know $f_{i}$’s as functions of $(\phi,X)$. Written explicitly, we have

$\displaystyle
f_{0}=\frac{\alpha_{0}}{\sqrt{\alpha_{0}^{3}(\alpha_{0}+\alpha_{1}X)}},\qquad
f_{1}=\frac{\alpha_{1}}{\sqrt{\alpha_{0}^{3}(\alpha_{0}+\alpha_{1}X)}},\qquad
f_{2}=f_{3}=0.$ (39)

By use of Eq. (10), the barred metric can be reconstructed as

$\displaystyle\bar{g}_{\mu\nu}=\sqrt{\alpha_{0}(\alpha_{0}+\alpha_{1}X)}\,g_{\mu\nu}-\frac{\alpha_{0}\alpha_{1}}{\sqrt{\alpha_{0}(\alpha_{0}+\alpha_{1}X)}}\phi_{\mu}\phi_{\nu}.$
(40)

Note that both coefficients are now functions of $(\phi,X)$, and hence this is
nothing but a conventional disformal transformation.

Let us now study the other branch of solutions for Eq. (35), i.e.,
$(\mathcal{J}f_{1})_{,ZZ}=0$. After straightforward manipulations, one can
show that all $d_{ijklmn}$’s vanish if and only if

$\displaystyle f_{0}=\frac{\alpha_{0}+\beta_{0}Y}{\mathcal{J}},\qquad
f_{1}=\frac{\alpha_{0}\alpha_{1}+\beta_{0}^{2}Z}{\mathcal{J}(\alpha_{0}+\beta_{0}Y)},\qquad
f_{2}=-\frac{\beta_{0}}{\mathcal{J}},\qquad
f_{3}=0,\qquad\mathcal{J}^{2}=\alpha_{0}(\alpha_{0}+\alpha_{1}X)(\alpha_{0}+\beta_{0}Y)^{2},$
(41)

with $\alpha_{0}(\neq 0)$, $\alpha_{1}$, and $\beta_{0}$ being functions of
$(\phi,X)$. By use of Eq. (10), the coefficient functions of the barred metric
can be reconstructed from Eq. (41) as

$\displaystyle F_{0}=\sqrt{\alpha_{0}(\alpha_{0}+\alpha_{1}X)},\qquad
F_{1}=-\frac{\alpha_{0}\alpha_{1}}{F_{0}},\qquad
F_{2}=\frac{\alpha_{0}\beta_{0}}{F_{0}},\qquad
F_{3}=\frac{X\beta_{0}^{2}}{F_{0}},$ (42)

or written explicitly,

$\displaystyle\bar{g}_{\mu\nu}=\sqrt{\alpha_{0}(\alpha_{0}+\alpha_{1}X)}\left[g_{\mu\nu}-\frac{\alpha_{1}}{\alpha_{0}+\alpha_{1}X}\phi_{\mu}\phi_{\nu}+\frac{2\beta_{0}}{\alpha_{0}+\alpha_{1}X}\phi_{(\mu}X_{\nu)}+\frac{X\beta_{0}^{2}}{\alpha_{0}(\alpha_{0}+\alpha_{1}X)}X_{\mu}X_{\nu}\right].$
(43)

Interestingly, one can check that the generalized disformal transformation
(43) satisfies the degeneracy condition even in the presence of a k-essence
matter scalar field whose Lagrangian is written as a general function of
$\psi$ and $\bar{g}^{\mu\nu}\psi_{\mu}\psi_{\nu}$. Note, however, that the
above result does not satisfy a part of the invertibility conditions
(specifically, $\bar{X}_{Y}=\bar{X}_{Z}=0$) in general. Indeed, for the above
choice of $f_{i}$’s, we have

$\displaystyle\bar{X}=X\left(f_{0}+Xf_{1}+2Yf_{2}+\frac{Y^{2}}{X}f_{3}\right)=\frac{X\left[\alpha_{0}^{2}+X\alpha_{0}\alpha_{1}-\beta_{0}^{2}(Y^{2}-XZ)\right]}{\mathcal{J}(\alpha_{0}+\beta_{0}Y)},$
(44)

which has a nontrivial dependence on $Y$ and $Z$ unless $\beta_{0}=0$. If
$\beta_{0}=0$, the generalized disformal transformation (43) reduces to Eq.
(40), i.e., the conventional one.

So far, we have found that there exists a nontrivial family of generalized
disformal metrics described by Eq. (43) that allows for consistent coupling of
a k-essence scalar field without either Ostrogradsky or shadowy mode. As
mentioned above, this family does not satisfy a part of the invertibility
conditions in general, meaning that one cannot move to the equivalent
description in the Jordan frame. The only exception is the case $\beta_{0}=0$,
where Eq. (43) reduces to the conventional disformal metric (2). Nevertheless,
even if $\beta_{0}\neq 0$, the theory makes sense as gravity nonminimally
coupled to matter, and hence we could keep it in our consideration. However,
as we shall show in the Appendix, this family does not allow for consistent
coupling of fermionic matter fields unless $\beta_{0}=0$. Therefore, all the
higher-derivative terms in the generalized disformal metric are prohibited
when we require that both bosonic and fermionic matter couplings do not
introduce an Ostrogradsky or shadowy mode, even if we do not impose the
invertibility conditions. Our analysis shows that, if one considers the
generalized disformal transformation with nontrivial higher-derivative terms,
one has to live with either Ostrogradsky or shadowy mode. As clarified in
Refs. [37, 38], the shadowy mode itself is harmless as it satisfies a three-
dimensional elliptic differential equation on a spacelike hypersurface and
hence does not propagate. On the other hand, if an Ostrogradsky mode is
revived due to matter coupling, the theory can be trusted only up to an energy
scale well below the mass of the Ostrogradsky ghost.

##  IV Conclusions

In this work, we have investigated the degeneracy conditions of generalized
disformal Horndeski (GDH) theories in the presence of a minimally coupled
matter field, which is represented by a canonical scalar field. We have
started with the Horndeski-frame Lagrangian (21) where the gravitational
action is given by the Horndeski one while the matter field is coupled to the
generalized disformal metric (1). We have rewritten the total Lagrangian in
terms of the $3+1$ language and constructed the Hessian matrix so that we can
investigate the kinetic structure of the theory. The degeneracy conditions are
required for the matter coupling to be consistent, giving the conditions that
the determinant of the Hessian matrix vanishes. The degeneracy conditions have
been written in the form (29) as a polynomial in
$\\{A_{*},X_{*},\psi_{*},\mathcal{Q}_{1},\mathcal{Q}_{2},\mathcal{Q}_{3}\\}$,
which are independent functions of spacetime constructed out of the
derivatives of the gravitational scalar field $\phi$ and the matter scalar
field $\psi$ [see Eqs. (25) and (30)]. In order for the degeneracy conditions
to be satisfied for arbitrary configurations of $\phi$ and $\psi$, all the
coefficients $d_{ijklmn}$ of the polynomial must vanish. We have thus arrived
at the following conclusions: (i) if one sticks to the invertible
transformations satisfying the conditions (5) so that the equivalent GDH
theory in the Jordan frame exists, only the conventional disformal metric can
be consistently coupled to the kinetic term of the matter scalar field; (ii)
however, if one gives up the invertibility conditions and just considers a
scalar field coupled nonminimally to the gravitational scalar degree of
freedom $\phi$ through the generalized disformal metric, a nontrivial coupling
(containing second derivatives of $\phi$) given by Eq. (41) [or, equivalently,
Eqs. (42) and (43)] is allowed.

We note that our analysis in the present paper is based on the Horndeski-frame
Lagrangian, which itself makes sense even if the metric to which the matter
fields are minimally coupled is not associated with invertible disformal
transformations. In this regard, it would be intriguing to take into account
more general higher-order derivatives that are not covered by our
transformation law (1) (e.g., $\square\phi$) to study what types of higher
derivative couplings to matter fields can survive. It would also be
interesting to investigate the consistency of matter coupling in a class of
scalar-tensor theories with a nondynamical scalar field, i.e., the cuscuton
[40] or its extension [41]. These issues will be left for future work.

###### Acknowledgements.

We thank the authors of Ref. [35] for their helpful correspondence. The work
of TI was supported by the Rikkyo University Special Fund for Research. The
work of KT was supported by JSPS KAKENHI Grant No. JP21J00695. The work of TK
was supported by JSPS KAKENHI Grant No. JP20K03936 and MEXT-JSPS Grant-in-Aid
for Transformative Research Areas (A) “Extreme Universe”, No. JP21H05182 and
No. JP21H05189. *

##  Appendix A Consistency of fermionic matter coupling

In Sec. III.2, we showed that the generalized disformal transformation (1),
with which we define the GDH theory, is restricted to be of the conventional
form (2) in order for a matter scalar field can be consistently coupled
without reviving the Ostrogradsky ghost, provided that the invertibility
conditions (5) are satisfied. On the other hand, our analysis was based on the
Horndeski-frame Lagrangian (21), which itself makes sense as a theory of
gravity nonminimally coupled to matter field(s) even if the disformal
transformation is noninvertible. Interestingly, if we relax the invertibility
conditions, there is a nontrivial family of generalized disformal
transformations given by Eq. (43) that allows for consistent coupling of a
matter scalar field, where the deviation of Eq. (43) from the conventional
disformal transformation is characterized by the function
$\beta_{0}=\beta_{0}(\phi,X)$. However, it remains unclear whether the
transformation (43) with $\beta_{0}\neq 0$ accommodates consistent coupling of
fermionic matter fields. In this Appendix, following the discussion in Ref.
[36], we argue that further imposing the consistency of spinorial matter
coupling leads to $\beta_{0}=0$, i.e., we are again left with the conventional
disformal transformation.

For this purpose, one needs to study the transformation law for the tetrad
under the generalized disformal transformation, since the action of fermions
in curved spacetime is written in terms of the tetrad. The authors of Ref.
[36] developed the tetrad transformation law for the class of generalized
disformal transformations defined by Eq. (18), which we repeat here for
convenience:

$\displaystyle\bar{g}_{\mu\nu}=\tilde{F}_{0}g_{\mu\nu}+\tilde{F}_{1}\phi_{\mu}\phi_{\nu}+2\tilde{F}_{2}\phi_{(\mu}{\cal
X}_{\nu)}+\tilde{F}_{3}{\cal X}_{\mu}{\cal X}_{\nu},\qquad{\cal
X}_{\mu}\coloneqq\left(\delta_{\mu}^{\alpha}-\frac{\phi_{\mu}\phi^{\alpha}}{X}\right)\partial_{\alpha}X,$
(45)

where ${\cal X}_{\mu}$ is the derivative of $X$ projected onto a
constant-$\phi$ hypersurface and $\tilde{F}_{i}=\tilde{F}_{i}(\phi,X,{\cal
Z})$, with ${\cal Z}\coloneqq{\cal X}^{\mu}{\cal X}_{\mu}$. The reason why
they focused on this particular type of generalized disformal transformation
is that it trivially accommodates consistent bosonic matter coupling under the
unitary gauge [32, 35]. The analysis in Ref. [36] shows that $\tilde{F}_{3}=0$
is necessary to avoid the revival of the Ostrogradsky ghost for fermionic
matter coupling.

One can recast the generalized disformal metric (43) into the form

$\displaystyle\bar{g}_{\mu\nu}=\sqrt{\alpha_{0}(\alpha_{0}+\alpha_{1}X)}\bigg{[}$
$\displaystyle
g_{\mu\nu}+\frac{\beta_{0}^{2}Y^{2}+2\alpha_{0}\beta_{0}Y-\alpha_{0}\alpha_{1}X}{X\alpha_{0}(\alpha_{0}+\alpha_{1}X)}\phi_{\mu}\phi_{\nu}$
$\displaystyle+\frac{2\beta_{0}(\alpha_{0}+\beta_{0}Y)}{\alpha_{0}(\alpha_{0}+\alpha_{1}X)}\phi_{(\mu}{\cal
X}_{\nu)}+\frac{X\beta_{0}^{2}}{\alpha_{0}(\alpha_{0}+\alpha_{1}X)}{\cal
X}_{\mu}{\cal X}_{\nu}\bigg{]},$ (46)

but this is not of the form (45) because some of the coefficient functions
depend on $Y$, which cannot be written in terms of $(\phi,X,{\cal Z})$.
Nevertheless, the discussion in Ref. [36] itself applies even if the
coefficient functions $\tilde{F}_{i}$ in Eq. (45) had $Y$-dependence, as we
shall see below.

In what follows, let us promote the coefficient functions $\tilde{F}_{i}$ in
Eq. (45) as functions of $(\phi,X,Y,Z)$. The transformation law for the tetrad
$e^{a}_{\mu}$ associated with the generalized disformal transformation can be
written in the form [36]

$\displaystyle\bar{e}^{a}_{\mu}=\left(E_{0}\delta^{\alpha}_{\mu}+E_{1}\phi_{\mu}\phi^{\alpha}+E_{2}\phi_{\mu}{\cal
X}^{\alpha}+E_{3}{\cal X}_{\mu}{\cal X}^{\alpha}\right)e^{a}_{\alpha},$ (47)

with

$\displaystyle E_{0}=\sqrt{\tilde{F}_{0}},\qquad
E_{1}=\frac{\sqrt{X/\bar{X}}-\sqrt{\tilde{F}_{0}}}{X},\qquad
E_{2}=\frac{\tilde{F}_{2}}{\sqrt{\tilde{F}_{0}+{\cal Z}\tilde{F}_{3}}},\qquad
E_{3}=\frac{\sqrt{\tilde{F}_{0}+{\cal
Z}\tilde{F}_{3}}-\sqrt{\tilde{F}_{0}}}{\cal Z}.$ (48)

Indeed, it is straightforward to verify that
$\bar{g}_{\mu\nu}=\eta_{ab}\bar{e}^{a}_{\mu}\bar{e}^{b}_{\nu}$. Note that one
could add a term ${\cal X}_{\mu}\phi^{\alpha}$ inside the parentheses in Eq.
(47), but it can always be absorbed into a local Lorentz transformation [36].
Having introduced the tetrad transformation law, let us consider the
generalized disformal transformation of the action for a fermionic matter
field represented by a free massless Dirac spinor $\lambda$, i.e.,

$\displaystyle S_{\rm m}[e^{a}_{\mu},\lambda]=\int{\rm
d}^{4}x\,e\left(-\frac{1}{2}\lambda^{\dagger}{\rm
i}\gamma^{\hat{0}}e^{\mu}_{a}\gamma^{a}\nabla_{\mu}\lambda+{\rm c.c.}\right),$
(49)

where $e\coloneqq\det e^{a}_{\mu}$, ${\rm c.c.}$ denotes the complex
conjugate, and $\gamma^{a}$ denotes the gamma matrices in the Minkowski
spacetime such that
$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\eta^{ab}\mathbbm{1}$, with
$\mathbbm{1}$ being the identity matrix in the spinor indices. Note that we
put hats on local Lorentz indices
($a,b,\cdots=\\{\hat{0},\hat{1},\hat{2},\hat{3}\\}$). The covariant derivative
acting on the Dirac field is defined by

$\displaystyle\nabla_{\mu}\lambda\coloneqq\left(\mathbbm{1}\partial_{\mu}+\frac{1}{4}\omega_{\mu}{}^{ab}\gamma_{ab}\right)\lambda.$
(50)

Here, $\gamma_{ab}\coloneqq(\gamma_{a}\gamma_{b}-\gamma_{b}\gamma_{a})/2$ and
the (torsion-free) spin connection $\omega_{\mu}{}^{ab}$ is defined by

$\displaystyle\omega_{\mu}{}^{a}{}_{b}=-e_{b}^{\nu}\left(\partial_{\mu}e^{a}_{\nu}-\Gamma^{\alpha}_{\mu\nu}e^{a}_{\alpha}\right),$
(51)

where $\Gamma^{\lambda}_{\mu\nu}$ is the Christoffel symbol associated with
the metric. We now consider the generalized disformal transformation of the
spinor action (49). Since we are only interested in the degeneracy structure
of the action (49), let us focus on terms that involve time derivatives [36]:

$\displaystyle S_{\rm m}[e^{a}_{\mu},\lambda]\supset\int{\rm
d}^{4}x\sqrt{h}\left(\frac{\rm i}{2}\lambda^{\dagger}\dot{\lambda}-\frac{\rm
i}{2}\dot{\lambda}^{\dagger}\lambda+\frac{\rm
i}{4}\lambda^{\dagger}{}^{(3)}\\!e_{\hat{i}}^{k}{}^{(3)}\\!\dot{e}_{\hat{j}k}\gamma^{\hat{i}\hat{j}}\lambda\right),$
(52)

where ${}^{(3)}\\!e^{\hat{i}}_{k}$ denotes the triad such that
$h_{kl}=\delta_{\hat{i}\hat{j}}{}^{(3)}\\!e^{\hat{i}}_{k}{}^{(3)}\\!e^{\hat{j}}_{l}$
and $h\coloneqq\det h_{kl}=(\det{}^{(3)}\\!e^{\hat{i}}_{k})^{2}$. Replacing
the tetrad by the barred one, we obtain [36]

$\displaystyle S_{\rm m}[\bar{e}^{a}_{\mu},\lambda]\supset\int{\rm
d}^{4}x\sqrt{h}\,E_{0}^{2}(E_{0}+{\cal Z}E_{3})\bigg{[}$
$\displaystyle\frac{\rm i}{2}\lambda^{\dagger}\dot{\lambda}-\frac{\rm
i}{2}\dot{\lambda}^{\dagger}\lambda+\frac{\rm
i}{4}\lambda^{\dagger}{}^{(3)}\\!e_{\hat{i}}^{k}{}^{(3)}\\!\dot{e}_{\hat{j}k}\gamma^{\hat{i}\hat{j}}\lambda$
$\displaystyle-\frac{\rm i}{4}\frac{{\cal Z}E_{3}^{2}}{E_{0}(E_{0}+{\cal
Z}E_{3})}{\cal X}_{m}\dot{\cal
X}_{l}\lambda^{\dagger}{}^{(3)}\\!e_{\hat{i}}^{k}{}^{(3)}\\!e_{\hat{j}}^{l}\gamma^{\hat{i}\hat{j}}\lambda\bigg{]}.$
(53)

As detailed in Ref. [36], under the unitary gauge, the last term inside the
square brackets leads to nondegenerate higher-order derivatives in the
equations of motion for the lapse function and the spinor field. Therefore,
one needs to impose $E_{3}=0$, i.e., $\tilde{F}_{3}=0$ in order not to avoid
the Ostrogradsky ghost. Note that this condition obtained under the unitary
gauge should be a necessary condition for the fermionic matter coupling to be
consistent in an arbitrary coordinate system. Since the transformation (46)
has $\tilde{F}_{3}\propto\beta_{0}^{2}$, the condition $\tilde{F}_{3}=0$
requires $\beta_{0}=0$.

## References

  * [1] K. Koyama, _Cosmological Tests of Modified Gravity_ , _Rept. Prog. Phys._ 79 (2016) 046902 [1504.04623].
  * [2] P. G. Ferreira, _Cosmological Tests of Gravity_ , _Ann. Rev. Astron. Astrophys._ 57 (2019) 335 [1902.10503].
  * [3] S. Arai et al., _Cosmological gravity probes: connecting recent theoretical developments to forthcoming observations_ , 2212.09094.
  * [4] R. P. Woodard, _Ostrogradsky’s theorem on Hamiltonian instability_ , _Scholarpedia_ 10 (2015) 32243 [1506.02210].
  * [5] H. Motohashi and T. Suyama, _Quantum Ostrogradsky theorem_ , _JHEP_ 09 (2020) 032 [2001.02483].
  * [6] K. Aoki and H. Motohashi, _Ghost from constraints: a generalization of Ostrogradsky theorem_ , _JCAP_ 08 (2020) 026 [2001.06756].
  * [7] G. W. Horndeski, _Second-order scalar-tensor field equations in a four-dimensional space_ , _Int. J. Theor. Phys._ 10 (1974) 363.
  * [8] C. Deffayet, X. Gao, D. A. Steer and G. Zahariade, _From k-essence to generalised Galileons_ , _Phys. Rev. D_ 84 (2011) 064039 [1103.3260].
  * [9] T. Kobayashi, M. Yamaguchi and J. Yokoyama, _Generalized G-inflation: Inflation with the most general second-order field equations_ , _Prog. Theor. Phys._ 126 (2011) 511 [1105.5723].
  * [10] H. Motohashi and T. Suyama, _Third order equations of motion and the Ostrogradsky instability_ , _Phys. Rev. D_ 91 (2015) 085009 [1411.3721].
  * [11] H. Motohashi, K. Noui, T. Suyama, M. Yamaguchi and D. Langlois, _Healthy degenerate theories with higher derivatives_ , _JCAP_ 07 (2016) 033 [1603.09355].
  * [12] R. Klein and D. Roest, _Exorcising the Ostrogradsky ghost in coupled systems_ , _JHEP_ 07 (2016) 130 [1604.01719].
  * [13] H. Motohashi, T. Suyama and M. Yamaguchi, _Ghost-free theory with third-order time derivatives_ , _J. Phys. Soc. Jap._ 87 (2018) 063401 [1711.08125].
  * [14] H. Motohashi, T. Suyama and M. Yamaguchi, _Ghost-free theories with arbitrary higher-order time derivatives_ , _JHEP_ 06 (2018) 133 [1804.07990].
  * [15] D. Langlois and K. Noui, _Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability_ , _JCAP_ 02 (2016) 034 [1510.06930].
  * [16] M. Crisostomi, K. Koyama and G. Tasinato, _Extended Scalar-Tensor Theories of Gravity_ , _JCAP_ 04 (2016) 044 [1602.03119].
  * [17] J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui and G. Tasinato, _Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order_ , _JHEP_ 12 (2016) 100 [1608.08135].
  * [18] D. Langlois, _Dark energy and modified gravity in degenerate higher-order scalar–tensor (DHOST) theories: A review_ , _Int. J. Mod. Phys. D_ 28 (2019) 1942006 [1811.06271].
  * [19] T. Kobayashi, _Horndeski theory and beyond: a review_ , _Rept. Prog. Phys._ 82 (2019) 086901 [1901.07183].
  * [20] G. Domènech, S. Mukohyama, R. Namba, A. Naruko, R. Saitou and Y. Watanabe, _Derivative-dependent metric transformation and physical degrees of freedom_ , _Phys. Rev. D_ 92 (2015) 084027 [1507.05390].
  * [21] K. Takahashi, H. Motohashi, T. Suyama and T. Kobayashi, _General invertible transformation and physical degrees of freedom_ , _Phys. Rev. D_ 95 (2017) 084053 [1702.01849].
  * [22] J. D. Bekenstein, _The Relation between physical and gravitational geometry_ , _Phys. Rev. D_ 48 (1993) 3641 [gr-qc/9211017].
  * [23] J. Ben Achour, D. Langlois and K. Noui, _Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations_ , _Phys. Rev. D_ 93 (2016) 124005 [1602.08398].
  * [24] M. Zumalacárregui and J. García-Bellido, _Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian_ , _Phys. Rev. D_ 89 (2014) 064046 [1308.4685].
  * [25] D. Langlois, M. Mancarella, K. Noui and F. Vernizzi, _Effective Description of Higher-Order Scalar-Tensor Theories_ , _JCAP_ 05 (2017) 033 [1703.03797].
  * [26] C. de Rham and A. Matas, _Ostrogradsky in Theories with Multiple Fields_ , _JCAP_ 06 (2016) 041 [1604.08638].
  * [27] K. Takahashi and T. Kobayashi, _Extended mimetic gravity: Hamiltonian analysis and gradient instabilities_ , _JCAP_ 11 (2017) 038 [1708.02951].
  * [28] D. Langlois, M. Mancarella, K. Noui and F. Vernizzi, _Mimetic gravity as DHOST theories_ , _JCAP_ 02 (2019) 036 [1802.03394].
  * [29] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, _Healthy theories beyond Horndeski_ , _Phys. Rev. Lett._ 114 (2015) 211101 [1404.6495].
  * [30] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, _Exploring gravitational theories beyond Horndeski_ , _JCAP_ 02 (2015) 018 [1408.1952].
  * [31] K. Takahashi, H. Motohashi and M. Minamitsuji, _Invertible disformal transformations with higher derivatives_ , _Phys. Rev. D_ 105 (2022) 024015 [2111.11634].
  * [32] K. Takahashi, M. Minamitsuji and H. Motohashi, _Generalized disformal Horndeski theories: cosmological perturbations and consistent matter coupling_ , _PTEP_ 2023 (2023) 013E01 [2209.02176].
  * [33] C. Deffayet and S. Garcia-Saenz, _Degeneracy, matter coupling, and disformal transformations in scalar-tensor theories_ , _Phys. Rev. D_ 102 (2020) 064037 [2004.11619].
  * [34] S. Garcia-Saenz, _On the Coupling of Generalized Proca Fields to Degenerate Scalar-Tensor Theories_ , _Universe_ 7 (2021) 190 [2106.14960].
  * [35] A. Naruko, R. Saito, N. Tanahashi and D. Yamauchi, _Ostrogradsky mode in scalar-tensor theories with higher-order derivative couplings to matter_ , 2209.02252.
  * [36] K. Takahashi, R. Kimura and H. Motohashi, _Consistency of matter coupling in modified gravity_ , 2212.13391.
  * [37] A. De Felice, D. Langlois, S. Mukohyama, K. Noui and A. Wang, _Generalized instantaneous modes in higher-order scalar-tensor theories_ , _Phys. Rev. D_ 98 (2018) 084024 [1803.06241].
  * [38] A. De Felice, S. Mukohyama and K. Takahashi, _Nonlinear definition of the shadowy mode in higher-order scalar-tensor theories_ , _JCAP_ 12 (2021) 020 [2110.03194].
  * [39] A. De Felice, S. Mukohyama and K. Takahashi, _Avoidance of Strong Coupling in General Relativity Solutions with a Timelike Scalar Profile in a Class of Ghost-Free Scalar-Tensor Theories_ , _Phys. Rev. Lett._ 129 (2022) 031103 [2204.02032].
  * [40] N. Afshordi, D. J. H. Chung and G. Geshnizjani, _Cuscuton: A Causal Field Theory with an Infinite Speed of Sound_ , _Phys. Rev. D_ 75 (2007) 083513 [hep-th/0609150].
  * [41] A. Iyonaga, K. Takahashi and T. Kobayashi, _Extended Cuscuton: Formulation_ , _JCAP_ 12 (2018) 002 [1809.10935].