# Learning Scene Structure Guidance via Cross-Task Knowledge Transfer for
Single Depth Super-Resolution

Baoli Sun1 1International School of Information Science $\&$ Engineering,
Dalian University of Technology, China
2Key Laboratory for Ubiquitous Network and Service Software of Liaoning
Province, China
3 Baidu Research, USA
Xinchen Ye1,2 Corresponding author<EMAIL_ADDRESS>This work was supported
by National Natural Science Foundation of China (NSFC) under Grant 61702078,
61772108, 61976038, 61772106. 1International School of Information Science
$\&$ Engineering, Dalian University of Technology, China
2Key Laboratory for Ubiquitous Network and Service Software of Liaoning
Province, China
3 Baidu Research, USA
Baopu Li3 1International School of Information Science $\&$ Engineering,
Dalian University of Technology, China
2Key Laboratory for Ubiquitous Network and Service Software of Liaoning
Province, China
3 Baidu Research, USA
Haojie Li1,2 1International School of Information Science $\&$ Engineering,
Dalian University of Technology, China
2Key Laboratory for Ubiquitous Network and Service Software of Liaoning
Province, China
3 Baidu Research, USA
Zhihui Wang1,2 1International School of Information Science $\&$ Engineering,
Dalian University of Technology, China
2Key Laboratory for Ubiquitous Network and Service Software of Liaoning
Province, China
3 Baidu Research, USA
Rui Xu1,2 1International School of Information Science $\&$ Engineering,
Dalian University of Technology, China
2Key Laboratory for Ubiquitous Network and Service Software of Liaoning
Province, China
3 Baidu Research, USA

###### Abstract

Existing color-guided depth super-resolution (DSR) approaches require paired
RGB-D data as training samples where the RGB image is used as structural
guidance to recover the degraded depth map due to their geometrical
similarity. However, the paired data may be limited or expensive to be
collected in actual testing environment. Therefore, we explore for the first
time to learn the cross-modality knowledge at training stage, where both RGB
and depth modalities are available, but test on the target dataset, where only
single depth modality exists. Our key idea is to distill the knowledge of
scene structural guidance from RGB modality to the single DSR task without
changing its network architecture. Specifically, we construct an auxiliary
depth estimation (DE) task that takes an RGB image as input to estimate a
depth map, and train both DSR task and DE task collaboratively to boost the
performance of DSR. Upon this, a cross-task interaction module is proposed to
realize bilateral cross-task knowledge transfer. First, we design a cross-task
distillation scheme that encourages DSR and DE networks to learn from each
other in a teacher-student role-exchanging fashion. Then, we advance a
structure prediction (SP) task that provides extra structure regularization to
help both DSR and DE networks learn more informative structure representations
for depth recovery. Extensive experiments demonstrate that our scheme achieves
superior performance in comparison with other DSR methods.

##  1 Introduction

To better understand a scene image, depth information is supplemented to the
RGB images, providing the key clue about the scene and enabling wide
applications in 3D reconstruction [9], autonomous navigation [20], monitoring
[2], and so on. However, acquiring depth information for indoor and outdoor
scenes needs expensive cost and great efforts, especially for high-quality and
high-resolution (HR) depth maps. As such, one of the effective post processing
techniques, Depth Super-Resolution (DSR), is greatly desired to yield HR depth
maps to alleviate this problem. Many efforts have been taken along the
direction of DSR. Usually, fine scene structures are easily lost or severely
destroyed in low-resolution (LR) depth map because of the limited spatial
resolution. An RGB image and its associated depth map are the photometric and
geometrical representations of the same scene, and have a strong structural
similarity. Most existing DSR methods learn structural complementarity from
RGB images to recover the degraded depth maps.

Previous color-guided DSR methods take advantage of RGB-D image pairs via a
two-way fusion architecture, in which an extra branch is required to extract
structural guidance from RGB image. As illustrated in Figure 1 (a), RGB image
and LR depth map are often processed by separate branches and filtered
together through a joint branch to output the HR result [26, 31, 4, 21].
However, due to the simple feature aggregation at a specific layer in the
middle of the network, high-frequency structure information from RGB image is
more likely to be lost in the process of feature extraction. Therefore, as
shown in Figure 1 (b), some novel methods [18, 12, 49] incorporate a new
paradigm of feature aggregation, i.e, multi-scale fusion, to allow the network
to learn rich hierarchical features at different levels. This in turn makes
the network to retain more spatial details for recovering both fine-scale and
large-scale structures.

Figure 1: Color-guided DSR paradigms. (a) Joint filtering, (b) Multi-scale
feature aggregation, (c) Our cross-task interaction mechanism to distill
knowledge from RGB image to DSR task without changing its network
architecture.

Although existing color-guided DSR methods have demonstrated remarkable
progress, several limitations still remain. First, these methods require
paired RGB-D data as training examples to jointly recover the degraded depth
map. However, the paired data may be limited or expensive to be collected in
actual testing environment. For example, RGB image and depth map are captured
by separate depth and RGB sensors with different resolutions and views, thus
needing accurate calibration and rectification between them to obtain the
registered pairs. Actually, most of real-world applications still come with
only a single LR depth map, which raises the above question. Second,
considering the memory consumption and computing burden, the processing on the
HR RGB data also hinders the practical application. Moreover, although RGB
features can be used as structural guidance to resolve the degradation in DSR,
RGB discontinuities do not always coincide with those of depth map (structure
inconsistency), which results in noticeable artifacts such as texture copying
and depth bleeding. Therefore, how to leverage RGB information to help recover
the depth map and simultaneously satisfy the actual testing environment, still
needs to be studied.

Motivated by the above analysis, this paper breaks away from the shackles of
general paradigms and introduces a novel scene structure guidance learning
method for the task of DSR, as shown in Figure 1 (c). We explore for the first
time to learn the cross-modality knowledge at training stage, where both RGB
and depth modalities are available, but test on the target dataset, where only
single depth modality exists. Our key idea is to distill the knowledge of
scene structural guidance from RGB modality to the single DSR task without
changing its network architecture.

Specifically, as illustrated in Figure 2, inspired by the success of multi-
task learning [52, 22, 46], we construct an auxiliary depth estimation (DE)
task that takes RGB image as input to estimate a depth map. Upon this, we
propose a cross-task interaction module to realize bilateral knowledge
transfer between DSR task and DE task. Different from the commonly used
distillation techniques [37, 16, 22], we first design a cross-task
distillation that encourages DSR network (DSRNet) and DE network (DENet) to
learn from each other, i.e., the roles of teacher and student will dynamically
switch between both tasks based on their current performances on depth
recovery in the iterative collaborative training. A multi-space distillation
scheme is introduced to distill knowledge from the perspective of output and
affinity spaces, which can better describe the essential structural
characteristics of depth map. Moreover, to address the problem of RGB-D
structure inconsistency, we construct a structure prediction (SP) task that
provides extra structure regularization to help both DSRNet and DENet learn
more informative structure representations for depth recovery. We come up with
an uncertainty-induced attention fusion module to provide a reasonable input
for the SP network (SPNet), in which the uncertainty maps acquired from both
DSRNet and DENet are used to re-weight their features for strengthening
effective structural knowledge. Extensive experiments demonstrate that our
single DSR method even outperforms the color-guided DSR methods on benchmark
datasets in terms of both accuracy and runtime. The main contributions are
summarized as follows,

$\bullet$ So far as we know, our proposed paradigm of DSR is the first work
that learns with multiple modalities as inputs at training stage, but tests on
only single LR depth modality.

$\bullet$ A cross-task distillation scheme is proposed to encourage DSRNet and
DENet to learn from each other in a collaborative training mode.

$\bullet$ A structure prediction network is advanced to provide structure
regularization for helping DSRNet resolve the problem of structural
inconsistency.

##  2 Related Work

Figure 2: Illustration of our proposed framework, which consists of DSRNet,
DENet, and the middle cross-task interaction module. We supervise the outputs
of DSRNet and DENet with the same groundtruth depth map $D_{hr}$. In testing
phase, DSRNet is the final choice to predict HR depth map from only LR depth
map without the help of color image.

Depth Super-Resolution. Compared to single DSR methods [5, 32, 43, 35], color-
guided DSR methods [18, 12, 40, 49] have been widely proposed to improve the
quality of depth map by the guidance of color image. Li _et al._ [26] proposed
a joint filtering approach that leverages color image as guidance to enhance
the spatial resolution of depth map. Hui _et al._ [18] employed a multi-scale
fusion strategy that fuses the rich hierarchical color features at different
levels to resolve ambiguity in DSR. Wen _et al._ [41] presented a data-driven
filter to infer an initial HR depth map with the guidance of color image, then
proposed a coarse-to-fine network to progressively recover the depth map. Guo
_et al._ [12] proposed a hierarchical feature driven method that constructs an
input pyramid and a guidance pyramid for multi-level residual learning. Wang
_et al._ [40] proposed to upsample the depth map with the help of edge map
learned from the color image.

Monocular Depth Estimation. Due to the strong ability of CNN in feature
extraction, many supervised monocular depth estimation methods [45, 10, 42,
23] continue to improve the performance of depth estimation. Laina _et al._
[24] proposed a fully convolutional architecture to model the mapping between
color image and depth map. In [25], a two-stream CNN is proposed to
simultaneously predict depth and depth gradients for accurate depth
estimation. Wang _et al._ [38] presented a depth estimation network via a
semantic divide-and-conquer strategy, in which a scene is decomposed into
semantic segments and then predicts depth for each segment. In contrast,
unsupervised methods [8, 10, 42, 51] use video or stereo data during training
without the need of groundtruth depth maps. Wong _et al._ [42] learned a
robust representation with a two-branch decoder to estimate the depth map.
Different from the above works, we introduce an auxiliary monocular depth
estimation task for achieving the cross-model knowledge learning at training
stage.

Knowledge Distillation. Knowledge distillation [16] is to transfer knowledge
from high-capacity model to a compact model to improve the performance of the
latter one. It has been wildly applied to many applications, including action
recognition [7], style transfer [28], depth estimation [13] and scene parsing
[44]. For example, the task of image classification takes class probabilities
from teacher network as soft targets to train the student network [37, 16] or
transfers the knowledge through intermediate layers [50, 33]. Recently, deep
mutual learning [52] proposed a two-way distillation which transfers knowledge
between the teacher and student and benefits to both networks. Kundu _et al._
[22] tried to extend cross-model distillation to multiple spatially-structured
prediction tasks by using two regularization strategies to minimize the domain
discrepancy. Yao _et al._ [46] presented a dense cross-layer mutual
distillation mechanism to train the teacher and student collaboratively from
scratch. Inspired by the above knowledge distillation techniques, we propose
to learn the scene structure guidance for the single DSR task via our designed
cross-task distillation. Moreover, another difference to the previous works is
that the role of teacher and student is dynamically changed.

##  3 Method

###  3.1 Network Architecture

Figure 2 shows the overall architecture of our framework, which mainly
consists of three components: depth super-resolution network (DSRNet), depth
estimation network (DENet) and the middle cross-task interaction module. Given
a collection of paired LR-HR depth maps
$\\{D_{lr}^{(k)},D_{hr}^{(k)}\\}_{k=1}^{M}$ with the corresponding HR color
images $\\{I^{(k)}\\}_{k=1}^{M}$ as training data, where $M$ is the number of
training data, our goal is to learn a model, i.e., DSRNet, that can predict
the super-resolved depth maps $\\{D_{sr}^{(k)}\\}_{k=1}^{M}$ from their
corresponding downsampled versions $\\{D_{lr}^{(k)}\\}_{k=1}^{M}$.

Specifically, the structure of DSRNet is designed based on a network unit from
deep back-projection network (DBPN) [14], which can effectively improve the
feature representations through iterative projecting HR representations to LR
spatial domain and then mapping back into HR domain111We direct readers to
refer to [14] for more details about the design of DBPN block. . The shallow
features $F_{sr}^{shallow}$ are first extracted from $D_{lr}$ through a simple
convolutional block (including three convolutional layers), and then sent into
$N$ stacked DBPN blocks to obtain HR features $\\{F_{sr}^{n}\\}_{n=1}^{N}$.
$D_{sr}$ is finally reconstructed from $F_{sr}^{N}$ through another
convolutional block. DENet takes $I$ as input and estimates the depth map
$D_{de}$. The architecture of DENet is similar to DSRNet, but replaces the
DBPN blocks with deeper residual blocks [15] to extract informative features
$\\{F_{de}^{n}\\}_{n=1}^{N}$ from color image.

The cross-task interaction module acts as a bridge to connect DSRNet and
DENet, and realizes bilateral knowledge transfer between them. It consists of
two components, i.e., a cross-task distillation scheme and a structure
prediction network (SPNet), where the former focuses on the interaction
between multi-scale features extracted from both networks while the latter
uses structure maps as supervision to further guide the learning of both
networks.

Note that, at the training stage, DSRNet, DENet and the cross-task interaction
module are jointly learned by using both color image and depth map as input.
In testing phase, DSRNet is the final choice to predict HR depth map from only
LR depth map without the help of color image.

###  3.2 Cross-Task Distillation

Knowledge distillation is generally viewed as a technique of transferring
beneficial information from a top-performing model to the other naive one.
Different from the commonly used distillation techniques, in which the teacher
network is trained beforehand and fixed under the assumption that it always
learns a better representation than the student network, our goal is to train
SRNet and DENet collaboratively and encourage them to benefit from each other.

Inspired by mutual learning methods [52, 22], we propose a cross-task
distillation scheme, in which the roles of teacher and student will exchange
between both tasks based on their current performances on depth recovery in
the iterative collaborative training. Specially, at the current round of
training, we need to determine the teacher in advance according to their
performance at the previous round. We compute the average pixel error between
each recovered depth map and its groundtruth for both networks:

$e_{dsr}=\frac{1}{HW}\sum_{h}^{H}\sum_{w}^{W}|D_{sr}(h,w)-D_{hr}(h,w)|,$ (1)
$e_{de}=\frac{1}{HW}\sum_{h}^{H}\sum_{w}^{W}|D_{de}(h,w)-D_{hr}(h,w)|,$ (2)

where $\\{H,W\\}$ are the size of the output depth map. If $e_{dsr}$ is
smaller than $e_{de}$, DSRNet has a relatively better performance, and becomes
the dominant one to guide the learning of DENet, and vice versa.

Next, in order to distill more meaningful knowledge that can accurately
describe the essential structural characteristics of depth map, we introduce a
multi-space distillation scheme to condense the knowledge from the
perspectives of output and affinity spaces, as shown in Figure 2.

Output Space Distillation. To ensure the transfer of local information from
pixel-wise depth values in a depth map, we apply the side-output layer
(containing two successive convolutions) on
$\\{F_{sr}^{n},F_{de}^{n}\\}_{n=1}^{N}$ from both DSRNet and DENet to generate
the corresponding multi-scale depth outputs
$\\{D_{sr}^{n},D_{de}^{n}\\}_{n=1}^{N}$ respectively. Thus, the distillation
loss of output space is designed to indirectly align the features between
DSRNet and DENet:

$\mathcal{L}_{O}=\frac{1}{N}\sum_{i=1}^{N}||D_{sr}^{i}-D_{de}^{i}||_{1},$ (3)

Affinity Space Distillation. Color image and its associated depth map are
different representations of the same scene and have strong structural
similarity. Pixels with similar appearances in a color image have more chances
of belonging to the same object, and should have close depth values. Inspired
by [39, 6, 47] that consider the nonlocal correlations to strengthen
correlated features between pixels and benefit the depth map recovery, we also
transfer non-local structure knowledge on affinity space, which is implemented
by computing pair-wise similarities between pixels.

Assuming the dimension of feature $F$ is $w\times h\times c$, the reshape
function $\mathbb{R}$ recasts $F$ as $\mathbb{R}(F)$ with the dimension of
$wh\times c$. The affinity matrix $A$ is defined as:

$A(F)=\sigma(\mathbb{R}(F)\otimes\mathbb{R}^{T}(F)),$ (4)

where $\sigma(\cdot)$ is the softmax operation, $\otimes$ is the matrix
multiplication and $T$ is the transpose operator. The distillation loss of
affinity space is defined as the following,

$\mathcal{L}_{A}=\frac{1}{N}\sum_{i=1}^{N}||A(F_{sr}^{i})-A(F_{de}^{i})||_{1},$
(5)

The final distillation loss $\mathcal{L}_{distill}$ is expressed as:

$\mathcal{L}_{distill}=\mathcal{L}_{O}+\gamma\mathcal{L}_{A}.$ (6)

where $\gamma$ is an adjustment parameter. Note that, $\mathcal{L}_{distill}$
should be imposed on the training of the student, but not the teacher, which
are determined by the errors comparison between $e_{dsr}$ and $e_{de}$.

###  3.3 Structure Prediction

Figure 3: The proposed SPNet. We fuse $F_{sr}^{N}$ and $F_{de}^{N}$ by our
proposed uncertainty-induced attention fusion module.

The goal of SPNet is to predict a structure map $S$ from the last feature maps
$F_{sr}^{N}$ and $F_{de}^{N}$ generated by DSRNet and DENet respectively.
Through the supervision with the groundtruth structure map $S_{gt}$222$S_{gt}$
is obtained by computing the difference between adjacent pixels (gradients) in
$D_{HR}$. Following [30], we calculate it by a convolution layer with a fixed
kernel to extract the high-frequency parts from $D_{HR}$., SPNet can provide
extra structure regularization to help both DSRNet and DENet learn more
informative structure representations to alleviate the problem of RGB-D
structure inconsistency. As shown in Figure 3, SPNet consists of a fusion
module and a structure CNN, where the latter is a lightweight network with
five stacked ‘Conv+ReLU’ layers and a last ’Conv’ layer.

Usually, the erroneous recovery of DSR and DE tasks occurs in the regions
around depth boundaries and fine structures in a depth map, which are subject
to higher recovery uncertainty. Therefore, instead of simply concatenating
$F_{sr}^{N}$ and $F_{de}^{N}$ and sending into the structure CNN, we design an
uncertainty-induced attention fusion module to strengthen these informative
structure features by attending the recovery uncertainty to the feature map.
Thus, we first compute the uncertainty maps $U_{sr}$ and $U_{de}$ of both
networks by activating the recovery errors:

$U_{sr}=\sigma(Conv_{1\times 1}(D_{sr}-D_{hr}))$ (7)
$U_{de}=\sigma(Conv_{1\times 1}(D_{de}-D_{hr}))$ (8)

where $Conv_{1\times 1}$ is the $1\times 1$ convolution to adjust the
channels. Then, we use the uncertainty maps to re-weight $F_{sr}^{N}$,
$F_{de}^{N}$ and fuse them through an attention module:

$F_{struc}=[F_{sr}^{N}*(1+U_{sr}),F_{de}^{N}*(1+U_{de})]$ (9)

where $F_{struc}$ is the fused features for structure prediction. [$\cdot$]
denotes concatenation operation and $*$ is element-wise multiplication. Note
that, through back-propagating the network gradients from SPNet in the
backward information flow, the parameters of DSRNet and DENet can be updated.

Algorithm 1 Training Details

1:Training data $D_{lr}$, $D_{hr}$, $I$, $S_{gt}$

2:—————- Step 1 —————-

3:Randomly initialize DSRNet and DENet

4:for i = 1; i $\leq$ 100 do

5: Train DSRNet and DENet with $\mathcal{L}_{DSR}$ and $\mathcal{L}_{DE}$,
respectively

6:—————- Step 2 —————-

7:Randomly initialize SPNet

8:for i = 101; i $\leq$ max epoch do

9: Compute the average error value $e_{dsr}$ and $e_{de}$ according to Eq.(1)
and Eq.(2)

10: if $e_{dsr}\leq e_{de}$ then

11: Fix DSRNet and update DENet with

12:
$\mathcal{L}=\mathcal{L}_{DE}+\rho_{1}\mathcal{L}_{struc}+\rho_{2}\mathcal{L}_{distill}$

13: else

14: Fix DENet and update DSRNet with

15:
$\mathcal{L}=\mathcal{L}_{DSR}+\rho_{1}\mathcal{L}_{struc}+\rho_{2}\mathcal{L}_{distill}$

16:$D_{sr}$

###  3.4 Training Algorithm

The training process of our framework can be divided into two steps, as
presented in Algorithm 1. First, we separately train DSRNet and DENet with the
groundtruth $D_{hr}$. The losses are defined as follows:

$\mathcal{L}_{DSR}=||D_{sr}-D_{hr}||_{1},$ (10)
$\mathcal{L}_{DE}=\lambda\frac{1-SSIM(D_{de},D_{hr})}{2}+(1-\lambda)||D_{de}-D_{hr}||_{1},$
(11)

where $\mathcal{L}_{DSR}$ is a common pixel-wise L1 loss for the task of DSR.
Following [10], $\mathcal{L}_{DE}$ is set as a combination of the
reconstruction loss (L1 loss) and structural similarity (SSIM). $\lambda$ is
an adjustment parameter.

Then, we introduce the cross-task distillation between both networks with the
loss $\mathcal{L}_{distill}$ in Eq. (6). At the same time, we randomly
initialize SPNet, and train it together with DSRNet and DENet. The loss for
SPNet is defined as:

$\mathcal{L}_{struc}=||\mathbb{G}(F_{struc})-S_{gt}||_{1},$ (12)

where $\mathbb{G}(\cdot)$ denotes SPNet, $F_{struc}$ is the fused structure
feature in Eq.(9) and $S_{gt}$ is the ground truth of the structure. If DSRNet
is chosen as the student, the parameters of DENet are fixed at the current
epoch, and DSRNet is updated with the following loss:

$\mathcal{L}=\mathcal{L}_{DSR}+\rho_{1}\mathcal{L}_{struc}+\rho_{2}\mathcal{L}_{distill},$
(13)

where $\rho_{1},\rho_{2}$ are the trade-off parameters. Otherwise, DSRNet is
fixed, and DENet is updated with the loss $\mathcal{L}$ by replacing
$\mathcal{L}_{DSR}$ with $\mathcal{L}_{DE}$.

Table 1: Quantitative DSR results (in MAD) on Middlebury 2005 dataset. ‘DSRNet
w/o CT’ and ‘DSRNet’ denote the results of the proposed method without and
with cross-task interaction scheme, respectively.

| Art | Books | Dolls | Laundry | Moebius | Reindeer
---|---|---|---|---|---|---
$\times$2 |  $\times$4 |  $\times$8 |  $\times$16 |  $\times$2 |  $\times$4 |  $\times$8 |  $\times$16 |  $\times$2 |  $\times$4 |  $\times$8 |  $\times$16 |  $\times$2 |  $\times$4 |  $\times$8 |  $\times$16 |  $\times$2 |  $\times$4 |  $\times$8 |  $\times$16 |  $\times$2 |  $\times$4 |  $\times$8 |  $\times$16
Bicubic | 0.48 | 0.97 | 1.85 | 3.59 | 0.13 | 0.29 | 0.59 | 1.15 | 0.20 | 0.36 | 0.66 | 1.18 | 0.28 | 0.54 | 1.04 | 1.95 | 0.13 | 0.30 | 0.59 | 1.13 | 0.30 | 0.55 | 0.99 | 1.88
DJF [26] | 0.12 | 0.40 | 1.07 | 2.78 | 0.05 | 0.16 | 0.45 | 1.00 | 0.06 | 0.20 | 0.49 | 0.99 | 0.07 | 0.28 | 0.71 | 1.67 | 0.06 | 0.18 | 0.46 | 1.02 | 0.07 | 0.23 | 0.60 | 1.36
MSG [18] |  - | 0.46 | 0.76 | 1.53 |  - | 0.15 | 0.41 | 0.76 |  - | 0.25 | 0.51 | 0.87 |  - | 0.30 | 0.46 | 1.12 |  - | 0.21 | 0.43 | 0.76 |  - | 0.31 | 0.52 | 0.99
DGDIE [11] | 0.20 | 0.48 | 1.20 | 2.44 | 0.14 | 0.30 | 0.58 | 1.02 | 0.16 | 0.34 | 0.63 | 0.93 | 0.15 | 0.35 | 0.86 | 1.56 | 0.14 | 0.28 | 0.58 | 0.98 | 0.16 | 0.35 | 0.73 | 1.29
DEIN [48] | 0.23 | 0.40 | 0.64 | 1.34 | 0.12 | 0.22 | 0.37 | 0.78 | 0.12 | 0.22 | 0.38 | 0.73 | 0.13 | 0.23 | 0.36 | 0.81 | 0.11 | 0.20 | 0.35 | 0.73 | 0.15 | 0.26 | 0.40 | 0.80
CCFN [41] |  - | 0.43 | 0.72 | 1.50 |  - | 0.17 | 0.36 | 0.69 |  - | 0.25 | 0.46 | 0.75 |  - | 0.24 | 0.41 | 0.71 |  - | 0.23 | 0.39 | 0.73 |  - | 0.29 | 0.46 | 0.95
GSRPT [4] | 0.22 | 0.48 | 0.74 | 1.48 | 0.11 | 0.21 | 0.38 | 0.76 | 0.13 | 0.28 | 0.48 | 0.79 | 0.12 | 0.33 | 0.56 | 1.24 | 0.12 | 0.24 | 0.49 | 0.80 | 0.14 | 0.31 | 0.61 | 1.07
DSRN [40] | 0.12 | 0.25 | 0.61 | 1.80 | 0.04 | 0.11 | 0.28 | 0.69 | 0.06 | 0.14 | 0.33 | 0.73 | 0.06 | 0.15 | 0.43 | 1.24 | 0.05 | 0.13 | 0.29 | 0.67 | 0.07 | 0.15 | 0.35 | 0.92
DSRNet w/o CT | 0.16 | 0.31 | 0.59 | 1.55 | 0.10 | 0.15 | 0.31 | 0.73 | 0.12 | 0.21 | 0.39 | 0.69 | 0.12 | 0.21 | 0.40 | 0.82 | 0.11 | 0.16 | 0.32 | 0.74 | 0.13 | 0.22 | 0.38 | 0.87
DSRNet | 0.11 | 0.25 | 0.53 | 1.44 | 0.05 | 0.11 | 0.26 | 0.67 | 0.07 | 0.16 | 0.36 | 0.65 | 0.06 | 0.16 | 0.36 | 0.76 | 0.07 | 0.13 | 0.27 | 0.69 | 0.08 | 0.17 | 0.35 | 0.77

Figure 4: Visual comparison of $\times$8 DSR results on Art and Dolls in
Middlebury: (a) Groundtruth depth maps, (b) LR patches, results from (c)
Bicubic, (d) DJF [26], (e) DGDIE [11], (f) DEIN [48], (g) GSRPT [4], (h) DSRN
[40], (i) Ours, and (j) Groundtruth.

##  4 Experiments

We conduct experiments on four datasets, i.e., Middlebury [17], MPI Sintel
[3], NYUv2 [34] and ToFMark [5]. First, following MSG [18], we use 34 RGB-D
images from Middlebury dataset [17] and 58 RGB-D images from MPI Sintel
dataset [3] as training data. To evaluate our performance, we test on
Middlebury 2005 (6 standard RGB-D images: _Art, Books, Moebius, Dolls, Laundry
and Reindeer_). We also evaluate the generalization performance on ToFMark
dataset (3 real-world depth maps captured by Time of Flight (ToF) sensor).
Another training and testing dataset is NYU v2 dataset captured by Kinect
sensor. Following the widely used data splitting manner, we use the first 1000
RGB-D images for training and the rest 449 RGB-D images for evaluation. For
both experiment settings, we randomly extract 10000+ patches of fixed size of
256$\times$256 from HR depth maps and downsample HR depth maps by bicubic
interpolation to get LR inputs. We augment the training data by $180^{\circ}$
rotation. We choose Mean Absolute Difference (MAD) and Root Mean Square Error
(RMSE) as the evaluation metrics. Lower MAD and RMSE values indicate higher
recovery quality.

During training, we set the number of DBPN and Residual blocks as $N=5$. We
set the trade-off parameters as $\gamma=0.5$, $\lambda=0.2$, $\rho_{1}=0.1$
and $\rho_{2}=0.1$. For optimization, we use the Adam optimization algorithm
with momentum $=0.9$, $\beta_{1}=0.9$, $\beta_{2}=0.99$ and $\epsilon=10^{-8}$
to train our models. The initial learning rate is set to 1e-3 and decreased by
multipling by 0.1 for every 50 epochs. We implemented our method using PyTorch
with one RTX 2080Ti GPU.

Figure 5: Visual comparison of $\times$16 DSR results on NYU v2 dataset. (a)
LR depth maps; Results from (b) DJF [26], (c) DJFR [27], (d) P2P [19], (e)
GbFT [1], (f) Ours, and (g) Groundtruth.

Figure 6: Visualization on _Shark_ in ToFMark dataset. (a) Groundtruth
patches, Results from (b) DGDIE[11], (c)GSRPT[4], (d) DSRN[40], and (e) Ours.

###  4.1 Comparison with State-of-The-Art

Middlebury Dataset. We compare our method with state-of-the-art (SOTA) DSR
methods on Middlebury under four different up-scaling factors ($\times 2$,
$\times 4$, $\times 8$ and $\times 16$). All the compared methods and ours are
deep learning based methods, which are trained and tested under the same
conditions for fair comparison. Table 1 summarizes the quantitative results on
the Middlebury dataset. ‘DSRNet w/o CT’ and ‘DSRNet’ denote the results of the
proposed method without and with cross-task interaction scheme, respectively.
Benefiting from the backbone of DBPN network, the performance of ‘DSRNet w/o
CT’ goes beyond most of the previous methods, but slightly inferior to the
recent SOTA methods, e.g., DSRN. When training together with DENet assisted by
cross-task interaction, the performance of our DSRNet steps further compared
to ‘DSRNet w/o CT’, and is comparable to DSRN, even better than it on at least
half of the cases. Note that, different from these color-guided methods, we
stand on single DSR without the help of color image in testing phase, but
achieves satisfactory results on both accuracy and runtime (evaluated later in
this section). The $\times$8 DSR results are visually shown in Figure 4. For
the structural details, e.g., the stick and the teapot handle in _Art_ , and
the toy’s contour in _Dolls_ , we clearly recover these regions without
introducing the texture copying artifacts thanks to our structure prediction
network (SPNet).

NYU Dataset. We further evaluate our method on NYUv2 dataset compared with
other SOTA methods in Table 2. Our method performs the best on all the cases
with different up-scaling factors. Figure 5 presents visual comparisons under
the $\times$16 DSR case. Observing from the recovered depth maps and enlarged
patches, our method achieves the clearest results and preserves the sharpest
and correct object contours, which demonstrates our effectiveness to learn the
information of scene structure guidance from color image.

ToFMark Dataset. We evaluate the generalization ability of our method on
ToFMark dataset. Following DGDIE [11], we fill the missing points in depth
maps and downsample them by $\times$2 downsampling factors, then send them to
the $\times$2 model trained on Middlebury dataset to acquire the
generalization results. As shown in Table 3, our method obtains the best
objective generalization results for all three test examples. Figure 6 shows
the visual comparison on _Shark_. Our method has higher visual quality and
less blur than others, especially at the boundary regions.

Figure 7: Visual analysis of the fusion process. (a) RGB-D pairs, Output
features from (b) DSRNet and (c) DENet, Re-weighted features (d) and (e)
corresponding to (b) and (c), respectively, (f) Fused features, (h) Predicted
structure map; (i) Groundtruth structure map.

Runtime. In Figure 8, we summarize the overall performance by the tradeoff
between accuracy and runtime. We measure the runtime for $\times$8 DSR to
their full resolution (about 1080$\times$1320) on Middlebury dataset. The
color-guided methods i.e., JGF [29] MSG [18], and DSRN [40], run slower than
ours due to the usage of HR color images in testing phase. MS is the single
DSR version of MSG, but still inferior to ours for both accuracy and speed.
Owing to the cross-task interaction, we achieve satisfactory recovered results
with minimum inference time.

Table 2: Quantitative DSR results (in RMSE) on NYUv2 dataset.

| DJF [26] | DJFR [27] | DGDIE [11] | GbFT [1] | P2P [19] | PAC [36] | DKN [21] | Ours
---|---|---|---|---|---|---|---|---
$\times$4 | 3.54 | 3.38 | 1.56 | 3.35 | 4.12 | 2.39 | 1.62 | 1.49
$\times$8 | 6.20 | 5.86 | 2.99 | 5.73 | 6.48 | 4.59 | 3.26 | 2.73
$\times$16 | 10.21 | 10.11 | 5.24 | 9.01 | 10.17 | 8.09 | 6.51 | 5.11

Table 3: Quantitative DSR results (in MAD) on ToFMark dataset.

| MSG [18] | DEIN[48] | DGDIE [11] | GSRPT [4] | DSRN [40] | Ours
---|---|---|---|---|---|---
Books | 12.26 | 12.78 | 12.31 | 13.21 | 11.15 | 11.03
Shark | 14.11 | 15.11 | 14.06 | 15.03 | 13.26 | 13.08
Devil | 12.45 | 14.25 | 9.66 | 12.27 | 9.54 | 9.33

Figure 8: Runtime and performance analysis.

###  4.2 Ablation Study

In this section, we further verify the key designs of our cross-task
interaction, i.e., cross-task distillation and structure prediction task by
ablation study.

Table 4: The impact of each component in cross-task distillation.

| DSRNet | DENet | $\mathcal{L}_{O}$ | $\mathcal{L}_{A}$ | SPNet | Middlebury | NYUv2
---|---|---|---|---|---|---|---
| ✓ |  |  |  |  | 0.398 | 3.13
|  | ✓ |  |  |  | 0.419 | 3.26
| ✓ | ✓ | ✓ |  |  | 0.377 | 2.88
| ✓ | ✓ |  | ✓ |  | 0.384 | 2.93
| ✓ | ✓ | ✓ | ✓ |  | 0.372 | 2.85
| ✓ | ✓ | ✓ | ✓ | ✓ | 0.356 | 2.75

Figure 9: Visualization of recovery errors variation in DSRNet at different
distillation stage. (a) LR depth and HR RGB patches; (b) Error maps w/o cross-
task distillation; Error maps with cross-task distillation after (c) 140$th$
and (d) 180$th$ training epoches; (e) Error maps with cross-task distillation
and structure prediction.

We report the $\times$8 DSR results on Middlebury and NYUv2 dataset under
different experimental settings, as shown in Table 4. The $1^{st}$ and
$2^{nd}$ rows are the objective results of initial DSRNet and DENet trained
separately without cross-task interaction scheme. DENet performs relatively
inferior to DSRNet, because estimating depth map from only color image is more
difficult than from the degraded LR version. Next, $\mathcal{L}_{S}$ and
$\mathcal{L}_{A}$ distill knowledge from output space and affinity space
respectively. When gradually integrating each loss into the baseline, it
brings a progressive performance improvement (6.5% and 8.9% increase on
Middlebury and NYU respectively), which verifies the effectiveness of cross-
task distillation strategy is remarkable. Besides, we also visualize the error
maps between recovered depth map and groundtruth to validate the process of
our iterative collaborative training in Figure 9. As the epoch goes, the
recovery errors for both DSRNet and DENet are decreased at the boundary
regions. Finally, after introducing the SPNet (the final case in Table 4 ),
the values of both error metrics are further decreased (4.3% error reduction
against the previous case). Figure 9(e) also validates this from visual
performance.

Besides, we conduct ablation study to evaluate the effectiveness of our
uncertainty-induced attention fusion module as presented in Table 5. Compared
to simply concatenating features in channel dimension and sending into SPNet
(SPNet w/o U), our proposed fusion module can offer significant assistance for
SPNet to predict the structure map. We also visualize the feature maps in the
fusion process in Figure 7. The object boundaries, e.g., the regions around
the toy head and the strip behind, are obviously highlighted by the
uncertainty map, which facilitate the final structure perception of SPNet.

Table 5: Ablation study of uncertainty-induced attention fusion module
(abbreviated as U) on $\times$8 DSR cases. The numerical results represent the
MADs computed between $S$ and $S_{gt}$.

| Art | Books | Dolls | Laundry | Moebius | Reindeer
---|---|---|---|---|---|---
SPNet w/o U | 1.403 | 2.292 | 1.730 | 1.846 | 1.379 | 1.565
SPNet w/ U | 1.386 | 2.259 | 1.715 | 1.821 | 1.349 | 1.552

##  5 Conclusion

For the first time, we explored to learn the cross-modal knowledge from both
RGB and depth modalities at training stage, but test on only single depth
modality. A cross-task interaction module is advanced to realize bilateral
knowledge transfer between DSRNet and the auxiliary DENet in a well-designed
collaborative training mode. Experiments show our method’s superior
performance for both accuracy and runtime. In the future work, we may extend
our cross-task interaction to more guided image restoration tasks.

## References

  * [1] Badour Albahar and Jia-Bin Huang.  Guided image-to-image translation with bi-directional feature transformation.  In IEEE ICCV, pages 9015–9024, 2019.
  * [2] Guido Borghi.  Combining deep and depth: Deep learning and face depth maps for driver attention monitoring.  CoRR, abs/1812.05831, 2018.
  * [3] D. J. Butler, J. Wulff, G. B. Stanley, and M. J. Black.  A naturalistic open source movie for optical flow evaluation.  In European Conf. on Computer Vision (ECCV), Part IV, LNCS 7577, pages 611–625, Oct. 2012.
  * [4] Riccardo de Lutio, Stefano D’Aronco, Jan Dirk Wegner, and Konrad Schindler.  Guided super-resolution as pixel-to-pixel transformation.  In IEEE International Conference on Computer Vision, ICCV, pages 8828–8836, 2019.
  * [5] David Ferstl, Christian Reinbacher, Rene Ranftl, Matthias Rüther, and Horst Bischof.  Image guided depth upsampling using anisotropic total generalized variation.  In Proc. ICCV, 2013.
  * [6] Yukang Gan, Xiangyu Xu, Wenxiu Sun, and Liang Lin.  Monocular depth estimation with affinity, vertical pooling, and label enhancement.  In European Conference on Computer Vision (ECCV), 2018.
  * [7] Nuno C. Garcia, Pietro Morerio, and Vittorio Murino.  Modality distillation with multiple stream networks for action recognition.  In Computer Vision ECCV 15th European Conference, volume 11212, pages 106–121, 2018.
  * [8] Ravi Garg, B. G. Vijay Kumar, Gustavo Carneiro, and Ian D. Reid.  Unsupervised CNN for single view depth estimation: Geometry to the rescue.  In Computer Vision ECCV 14th European Conference, volume 9912, pages 740–756, 2016.
  * [9] Andreas Geiger, Julius Ziegler, and Christoph Stiller.  Stereoscan: Dense 3d reconstruction in real-time.  In IEEE Intelligent Vehicles Symposium (IV), pages 963–968, 2011\.
  * [10] Clément Godard, Oisin Mac Aodha, and Gabriel J. Brostow.  Unsupervised monocular depth estimation with left-right consistency.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 6602–6611, 2017.
  * [11] Shuhang Gu, Wangmeng Zuo, Shi Guo, Yunjin Chen, Chongyu Chen, and Lei Zhang.  Learning dynamic guidance for depth image enhancement.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 712–721, 2017.
  * [12] Chunle Guo, Chongyi Li, Jichang Guo, Runmin Cong, Huazhu Fu, and Ping Han.  Hierarchical features driven residual learning for depth map super-resolution.  IEEE Trans. Image Processing, 28(5):2545–2557, 2019.
  * [13] Xiaoyang Guo, Hongsheng Li, Shuai Yi, Jimmy S. J. Ren, and Xiaogang Wang.  Learning monocular depth by distilling cross-domain stereo networks.  In Computer Vision ECCV 15th European Conference, volume 11215, pages 506–523, 2018.
  * [14] Muhammad Haris, Gregory Shakhnarovich, and Norimichi Ukita.  Deep back-projection networks for super-resolution.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 1664–1673, 2018.
  * [15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 770–778, 2016.
  * [16] Geoffrey E. Hinton, Oriol Vinyals, and Jeffrey Dean.  Distilling the knowledge in a neural network.  CoRR, abs/1503.02531, 2015.
  * [17] Heiko Hirschmüller and Daniel Scharstein.  Evaluation of cost functions for stereo matching.  In IEEE Computer Society Conference on Computer Vision and Pattern Recognition CVPR, 2007.
  * [18] Tak-Wai Hui, Chen Change Loy, and Xiaoou Tang.  Depth map super-resolution by deep multi-scale guidance.  In Computer Vision ECCV 14th European Conference, pages 353–369, 2016.
  * [19] Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A. Efros.  Image-to-image translation with conditional adversarial networks.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 5967–5976, 2017.
  * [20] Christian Kerl, Jürgen Sturm, and Daniel Cremers.  Dense visual SLAM for RGB-D cameras.  In IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 2100–2106, 2013.
  * [21] Beomjun Kim, Jean Ponce, and Bumsub Ham.  Deformable kernel networks for guided depth map upsampling.  CoRR, abs/1903.11286, 2019.
  * [22] Jogendra Nath Kundu, Nishank Lakkakula, and Venkatesh Babu Radhakrishnan.  Um-adapt: Unsupervised multi-task adaptation using adversarial cross-task distillation.  In IEEE International Conference on Computer Vision, ICCV, pages 1436–1445, 2019.
  * [23] Jogendra Nath Kundu, Phani Krishna Uppala, Anuj Pahuja, and R. Venkatesh Babu.  Adadepth: Unsupervised content congruent adaptation for depth estimation.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 2656–2665, 2018.
  * [24] Iro Laina, Christian Rupprecht, Vasileios Belagiannis, Federico Tombari, and Nassir Navab.  Deeper depth prediction with fully convolutional residual networks.  In Fourth International Conference on 3D Vision, 3DV 2016, pages 239–248, 2016.
  * [25] Jun Li, Can Yuce, Reinhard Klein, and Angela Yao.  A two-streamed network for estimating fine-scaled depth maps from single RGB images.  Comput. Vis. Image Underst., 186:25–36, 2019.
  * [26] Yijun Li, Jia-Bin Huang, Narendra Ahuja, and Ming-Hsuan Yang.  Deep joint image filtering.  In Computer Vision - ECCV - 14th European Conference, pages 154–169, 2016.
  * [27] Yijun Li, Jia-Bin Huang, Narendra Ahuja, and Ming-Hsuan Yang.  Joint image filtering with deep convolutional networks.  IEEE Trans. Pattern Anal. Mach. Intell., 41(8):1909–1923, 2019\.
  * [28] Yanghao Li, Naiyan Wang, Jiaying Liu, and Xiaodi Hou.  Demystifying neural style transfer.  In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI, Melbourne, pages 2230–2236, 2017\.
  * [29] Ming-Yu Liu, Oncel Tuzel, and Yuichi Taguchi.  Joint geodesic upsampling of depth images.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 169–176, 2013.
  * [30] Cheng Ma, Yongming Rao, Yean Cheng, Ce Chen, Jiwen Lu, and Jie Zhou.  Structure-preserving super resolution with gradient guidance.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 7766–7775, 2020.
  * [31] Jinshan Pan, Jiangxin Dong, Jimmy S. J. Ren, Liang Lin, Jinhui Tang, and Ming-Hsuan Yang.  Spatially variant linear representation models for joint filtering.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 1702–1711, 2019.
  * [32] Gernot Riegler, Matthias Rüther, and Horst Bischof.  Atgv-net: Accurate depth super-resolution.  In Computer Vision ECCV 14th European Conference, pages 268–284, 2016.
  * [33] Adriana Romero, Nicolas Ballas, Samira Ebrahimi Kahou, Antoine Chassang, Carlo Gatta, and Yoshua Bengio.  Fitnets: Hints for thin deep nets.  In 3rd International Conference on Learning Representations, ICLR, 2015.
  * [34] Nathan Silberman, Derek Hoiem, Pushmeet Kohli, and Rob Fergus.  Indoor segmentation and support inference from RGBD images.  In Computer Vision ECCV 12th European Conference on Computer Vision, volume 7576, pages 746–760, 2012.
  * [35] Xibin Song, Yuchao Dai, and Xueying Qin.  Deeply supervised depth map super-resolution as novel view synthesis.  IEEE Trans. Circuits Syst. Video Techn., 29(8):2323–2336, 2019\.
  * [36] Hang Su, Varun Jampani, Deqing Sun, Orazio Gallo, Erik G. Learned-Miller, and Jan Kautz.  Pixel-adaptive convolutional neural networks.  In IEEE CVPR, pages 11166–11175, 2019.
  * [37] Gregor Urban, Krzysztof J. Geras, Samira Ebrahimi Kahou, Özlem Aslan, Shengjie Wang, Abdelrahman Mohamed, Matthai Philipose, Matthew Richardson, and Rich Caruana.  Do deep convolutional nets really need to be deep and convolutional?  In ICLR, 2017.
  * [38] Lijun Wang, Jianming Zhang, Oliver Wang, Zhe Lin, and Huchuan Lu.  Sdc-depth: Semantic divide-and-conquer network for monocular depth estimation.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 538–547, 2020.
  * [39] Xiaolong Wang, Ross B. Girshick, Abhinav Gupta, and Kaiming He.  Non-local neural networks.  In 2018 IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 7794–7803, 2018.
  * [40] Zhihui Wang, Xinchen Ye, Baoli Sun, Jingyu Yang, Rui Xu, and Haojie Li.  Depth upsampling based on deep edge-aware learning.  Pattern Recognition, 103:107274, 2020.
  * [41] Yang Wen, Bin Sheng, Ping Li, Weiyao Lin, and David Dagan Feng.  Deep color guided coarse-to-fine convolutional network cascade for depth image super-resolution.  IEEE Trans. Image Processing, 28(2):994–1006, 2019.
  * [42] Alex Wong and Stefano Soatto.  Bilateral cyclic constraint and adaptive regularization for unsupervised monocular depth prediction.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 5644–5653, 2019.
  * [43] Jun Xie, Rogério Schmidt Feris, Shiaw-Shian Yu, and Ming-Ting Sun.  Joint super resolution and denoising from a single depth image.  IEEE Trans. Multimedia, 17(9):1525–1537, 2015.
  * [44] Dan Xu, Wanli Ouyang, Xiaogang Wang, and Nicu Sebe.  Pad-net: Multi-tasks guided prediction-and-distillation network for simultaneous depth estimation and scene parsing.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 675–684, 2018.
  * [45] Dan Xu, Wei Wang, Hao Tang, Hong Liu, Nicu Sebe, and Elisa Ricci.  Structured attention guided convolutional neural fields for monocular depth estimation.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 3917–3925, 2018.
  * [46] Anbang Yao and Dawei Sun.  Knowledge transfer via dense cross-layer mutual-distillation.  CoRR, abs/2008.07816, 2020.
  * [47] Xinchen Ye, Shude Chen, and Rui Xu.  DPNet: Detail-preserving network for high quality monocular depth estimation.  Pattern Recognition, 109, 2020.
  * [48] Xinchen Ye, Xiangyue Duan, and Haojie Li.  Depth super-resolution with deep edge-inference network and edge-guided depth filling.  In IEEE International Conference on Acoustics, Speech and Signal Processing, pages 1398–1402, 2018.
  * [49] Xinchen Ye, Baoli Sun, Zhihui Wang, Jingyu Yang, Rui Xu, Haojie Li, and Baopu Li.  Pmbanet: Progressive multi-branch aggregation network for scene depth super-resolution.  IEEE Trans. Image Process., 29:7427–7442, 2020.
  * [50] Sergey Zagoruyko and Nikos Komodakis.  Paying more attention to attention: Improving the performance of convolutional neural networks via attention transfer.  In 5th International Conference on Learning Representations, ICLR, 2017.
  * [51] Huangying Zhan, Ravi Garg, Chamara Saroj Weerasekera, Kejie Li, Harsh Agarwal, and Ian D. Reid.  Unsupervised learning of monocular depth estimation and visual odometry with deep feature reconstruction.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 340–349, 2018.
  * [52] Ying Zhang, Tao Xiang, Timothy M. Hospedales, and Huchuan Lu.  Deep mutual learning.  In IEEE Conference on Computer Vision and Pattern Recognition, CVPR, pages 4320–4328, 2018.


# Scale-invariant cosmology in de Sitter gauge theory

Tomi S. Koivisto<EMAIL_ADDRESS>Laboratory of Theoretical Physics,
Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia
National Institute of Chemical Physics and Biophysics, Rävala pst. 10, 10143
Tallinn, Estonia  Luxi Zheng<EMAIL_ADDRESS>Laboratory of Theoretical
Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411
Tartu, Estonia

###### Abstract

The Planck mass and the cosmological constant determine the minimum and the
maximum distances in the physical universe. A relativistic theory that takes
into account a fundamental distance limit $\ell$ on par with the fundamental
speed limit $c$, is based on the de Sitter extension of the Lorentz symmetry.
This article proposes a new de Sitter gauge theory of gravity which allows the
consistent cosmological evolution of the $\ell$. The theory is locally
equivalent to Dirac's scale-invariant version of general relativity, and
suggests a novel non-singular extension of cosmology.

##  I Introduction

In the standard $\Lambda$CDM model of cosmology Aghanim:2018eyx , the
background universe is dS (de Sitter). The dS geometry can be seen as a
4-dimensional hyperboloid of curvature $R_{\Lambda}$ and the radius
$\ell_{\Lambda}=\sqrt{3/\Lambda}=\sqrt{12/R_{\Lambda}}$ embedded in a
5-dimensional Minkowski space. The dS scale introduces a horizon, the maximum
proper distance up to which any signal can reach.

At the other end of scales, the space has a resolution limit given by the
Planck length $\ell_{P}$, since the wavelengths of photons required to probe
smaller distances would have enclosed the photon's energy within its
Schwarzschild radius. There are more refined thought experiments that lead to
the existence of a minimum length, and it is either assumed or predicted in
most of the approaches to quantum gravity Garay:1994en .

An observer-independent scale $\ell$ is naturally incorporated into the
physical theory of the universe by postulating the spacetime symmetry SO(4,1)
instead of the usual ISO(3,1). Analogously to the Galilean group being the
$c\rightarrow\infty$ contraction limit of the Poincaré group ISO(3,1), the
latter is the contraction limit $\ell\rightarrow\infty$ of the dS group
SO(4,1) Dyson:1972sd . An extension of the relativity principle that describes
the kinematics with a finite limiting distance $\ell$ has been formulated as
the projective special Licata:2017dpm , the doubly special
AmelinoCamelia:2002wr  and the dS special Aldrovandi:2006vr  relativity. The
gravitational theory is obtained by localisation of the symmetry
Westman:2014yca .

In this paper we propose a dS gauge theory with a dynamical dS scale
$\ell(x)$. Since the Planck length $\ell_{P}$ is defined as

$\ell_{P}=\sqrt{\frac{\hbar G}{c^{3}}}\,,$ (1)

where from now on we will set the speed of light $c=1$ and the Planck constant
$\hbar=1$ to unity, the dynamical Planck length $\ell=\ell(x)$ could
equivalently be considered as the dynamical (squareroot of the) Newton's
constant $G=G(x)$. A well-known realisation of this aspect of the theory is
scalar-tensor gravity, which promotes the gravitational coupling into a
dynamical scalar field Brans:1961sx .

Indeed, we will arrive at an action which is equivalent to the conformally
coupled scalar-tensor gravity Dirac:1973gk , and wherein the $\ell$ plays the
role of a dilaton field. While the dilaton is usually introduced in the
context of Weyl gauge theory Blagojevic:2002du ; Blagojevic:2013xpa ;
Scholz:2017pfo , the kinematical origin of scale invariance in a dS gauge
theory seems not to have been clarified previously.

It was shown long ago that dS gravity can be reduced to Einstein's gravity
with a cosmological constant MacDowell:1977jt ; Pagels:1983pq , and nowadays
it has been well understood that the implied symmetry-breaking is but a
realisation of Cartan's original geometrical construction Wise:2006sm . A fine
introduction to the dynamical symmetry breaking in Cartan geometry, and the
most general polynomial form of such a theory, were presented in
Westman:2014yca . A physical observer requires the further breaking
Gielen:2012fz  of SO(4,1)$\rightarrow$SO(3,1)$\rightarrow$SO(3), where the
final step could be the geometrical origin of cold dark matter Zlosnik:2018qvg
, the CDM part of the $\Lambda$CDM Aghanim:2018eyx .

This framework provides a robust approach also to the problems with the
$\Lambda$ Pagels:1983pq , whilst paving the way towards reconciliation of
gravity and quantum mechanics by lifting the kinematics of dS special
relativity Aldrovandi:2006vr  to the dynamics of dS general relativity
Aldrovandi:2007bi . The proper formulation of a dS gauge theory as a Cartan
geometry where the homogeneous model spaces are flat and their scale $\ell$ is
a function of the coordinates in the quotient dS spacetime has been already
developed by Jennen and Pereira Jennen:2014mba ; Jennen:2015bxa ;
Jennen:2016pqw .

In this article we propose a completion and generalisation of their theory and
explore its cosmological solutions. We begin in Section II by reviewing the
gauge theory of the dS group in Cartan geometry, now specifically adapted to
cosmology. In Section III we study the cosmological background solutions and
derive a family of exact solutions in the presence of perfect fluid which, as
will be argued, should be non-minimally coupled to the dS distance scale
$\ell$. Despite the apparently non-minimal coupling, the theory is
phenomenologically viable and does not lead to drastic violations of the
equivalence principle. We clarify this in Section IV, where it is shown that
particles move along geodesics of an integrable Weyl geometry. For
completeness we also consider a scalar field coupled to the dS gauge theory,
in order to confirm the consistency of the cosmological set-up beyond the
perfect fluid parameterisation. The conclusions are stated in Section V.

##  II The dS gauge theory

The dS space is considered as the 4-dimensional quotient of the dS group by
the Lorentz group. Consequently, in this Section we will be referring to
various distinct sets of coordinates. For clarity, the following table

coordinates | algebra | metric
---|---|---
$\\{x^{i}\\}_{i=1,2,3}$ | $\mathfrak{so}(3)$ | $\delta_{ij}=\text{diag}(1,1,1)$
$\\{x^{a}\\}_{a=0,1,2,3}$ | $\mathfrak{so}(3,1)$ | $\eta_{ab}=\text{diag}(-1,1,1,1)$
$\\{X^{A}\\}_{A=0,\dots,4}$ | $\mathfrak{so}(4,1)$ | $\eta_{AB}=\text{diag}(-1,1,1,1,1)$
$\\{x^{\mu}\\}_{\mu=0,1,2,3}$ | ${\mathfrak{gl}_{\parallel}(3,1)}$ |  $g_{\mu\nu}$ locally $\eta_{\mu\nu}$

summarises our conventions.

The generators of the dS algebra $\mathfrak{so}(4,1)$ satisfy the commutation
relations

$[\Omega_{AB},\Omega_{CD}]=2\left(\eta_{D[A}\Omega_{B]C}-\eta_{C[A}\Omega_{B]D}\right)\,,$
(2)

with $\eta_{AB}$ as given above. The 10 distinct generators
$\Omega_{AB}=-\Omega_{BA}$ can be interpreted as spacetime rotations in 5
dimensions, whilst our spacetime has the 4-dimensional tangent space with the
coordinates $\\{x^{a}\\}_{a=0,1,2,3}$ and the metric $\eta_{ab}$. Therefore we
consider the 4-dimensional rotations to the generated by $\Omega_{ab}$ that
coincide with the corresponding $\Omega_{AB}$, but define the rotations around
the 5${}^{\text{th}}$ dimension as

$\Pi_{a}=\ell^{-1}\Omega_{4a}\,.$ (3)

The 4 generators $\Pi_{a}$ will be interpreted as (generalised) translations.
The algebra inherited from (2) by the new generators is

$\displaystyle\left[\Omega_{ab},\Omega_{cd}\right]$ $\displaystyle=$
$\displaystyle 2\left(\eta_{d[a}\Omega_{b]c}-\eta_{c[a}\Omega_{b]d}\right)\,,$
(4a) $\displaystyle\left[\Pi_{a},\Omega_{bc}\right]$ $\displaystyle=$
$\displaystyle 2\eta_{a[b}\Pi_{c]}\,,$ (4b)
$\displaystyle\left[\Pi_{a},\Pi_{b}\right]$ $\displaystyle=$
$\displaystyle-\ell^{-2}\Omega_{ab}\,.$ (4c)

The $\ell$ is a dimensionful parameter that quantifies how much boost along
the 5${}^{\text{th}}$ dimension is needed for a unit translation. In the limit
$1/\ell\rightarrow 0$, (4) reduces to the Poincaré algebra
$\mathfrak{iso}(3,1)$ and the $\Pi_{a}$ become ordinary translations.

In general, the form of $\Pi_{a}$ will depend upon the geometry of the
symmetry breaking. To illustrate the embedding of the hyperboloid, let us
consider here the flat slicing

$\displaystyle X^{0}$ $\displaystyle=$
$\displaystyle\ell\sinh{(t/\ell})+e^{t/\ell}\delta_{ij}x^{i}x^{j}/2\ell\,,$
(5a) $\displaystyle X^{i}$ $\displaystyle=$ $\displaystyle e^{t/\ell}x^{i}\,,$
(5b) $\displaystyle X^{4}$ $\displaystyle=$
$\displaystyle\ell\cosh{(t/\ell})-e^{t/\ell}\delta_{ij}x^{i}x^{j}/2\ell\,,$
(5c)

since then the induced metric $g_{ab}$ has the most commonly used cosmological
(isotropic and homogeneous) form

$\eta_{AB}\textrm{d}X^{A}\textrm{d}X^{B}=g_{ab}\textrm{d}x^{a}\textrm{d}x^{b}=-\textrm{d}t^{2}+e^{2t/\ell}\delta_{ij}\textrm{d}x^{i}\textrm{d}x^{j}\,.$

By inverting (5),

$\displaystyle t\equiv x^{0}$ $\displaystyle=$
$\displaystyle\ell\log{(X^{0}+X^{4})}-\ell\log{(\ell)}\,,$ (6a) $\displaystyle
x^{i}$ $\displaystyle=$ $\displaystyle\ell X^{i}/(X^{0}+X^{4})\,.$ (6b)

We find the conformal relations between the basis vectors,

$\displaystyle\frac{\partial}{\partial X^{a}}$ $\displaystyle=$ $\displaystyle
e^{-t/\ell}\partial_{a}\,,$ (7a) $\displaystyle\frac{\partial}{\partial
X^{4}}$ $\displaystyle=$ $\displaystyle
e^{-t/\ell}\left(\partial_{t}-\ell^{-1}x^{i}\partial_{i}\right)\,.$ (7b)

Using the dictionary (5,6,7) we can easily write down the orbital generators,

$\displaystyle\Omega_{i0}$ $\displaystyle=$ $\displaystyle
2x_{[i}\partial_{0]}+(t+\ell\alpha)\partial_{i}\,,$ (8a)
$\displaystyle\Omega_{ij}$ $\displaystyle=$ $\displaystyle
2x_{[i}\partial_{j]}\,,$ (8b) $\displaystyle\Pi_{0}$ $\displaystyle=$
$\displaystyle\partial_{t}-\alpha\ell^{-1}x^{i}\partial_{i}\,,$ (8c)
$\displaystyle\Pi_{i}$ $\displaystyle=$
$\displaystyle\beta\partial_{i}-\ell^{-1}x_{i}\left(\partial_{t}-\ell^{-1}x^{k}\partial_{k}\right)\,,$
(8d)

where we used the short-hands

$\displaystyle\alpha$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(1+\ell^{-2}\delta_{jk}{x^{j}x^{k}}-e^{-2t/\ell}\right)=e^{-t/\ell}X^{0}/\ell\,,$
$\displaystyle\beta$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(1-\ell^{-2}\delta_{jk}{x^{j}x^{k}}+e^{-2t/\ell}\right)=e^{-t/\ell}X^{4}/\ell\,.$

As a cross-check, we verified that (4) is satisfied by (8).

An interesting conformal structure is manifest in the stereographic embedding
Aldrovandi:2006vr  alternative to the flat slicing (5), though it will be
shown elsewhere that the Beltrami geometry Guo:2003qm  is convenient for the
representations111In both pictures the $\Omega_{ab}$ have their usual form,
the “transvection” $\Pi_{a}$ becoming a translation contaminated with, in the
stereographic coordinates the special conformal Pereira:2013zxa , and in the
Beltrami coordinates (as well as in the flat slicing) the ordinary conformal
transformation..

To gauge the $\mathfrak{so}(4,1)$, we now introduce the connection 1-form

$\bm{A}=\frac{1}{2}\bm{A}^{AB}\Omega_{AB}=\left(\frac{1}{2}\bm{A}^{ab}{}_{\mu}\Omega_{ab}+\bm{A}^{a}{}_{\mu}\Pi_{a}\right)\textrm{d}x^{\mu}\,.$
(10)

The connection determines the dS-covariant (exterior) derivative
$\textrm{D}=\textrm{d}+\bm{A}$, which further generates the field strength
2-form $\textrm{D}^{2}=\textrm{d}\bm{A}+\bm{A}\wedge\bm{A}\equiv\bm{F}$ and
the 3-form identity $\textrm{D}^{3}=\textrm{D}\bm{F}=0$. It is crucial to note
that because of the definition (3), we have $\bm{A}^{a}=\ell\bm{A}^{4a}$ and
consequently Jennen:2014mba ; Jennen:2015bxa ; Jennen:2016pqw

$\ell
A^{a4}{}_{\mu,\alpha}=A^{a}{}_{\mu,\alpha}-\log{\ell}_{,\alpha}A^{a}{}_{\mu}\,.$
(11)

As a result, the components of the field strength are slightly modified,

$\displaystyle F^{a}{}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\left(A^{a}{}_{[\nu,\mu]}+A^{a}{}_{b[\mu}A^{b}{}_{\nu]}\right)-2\log{\ell}_{,[\mu}A^{a}{}_{\nu]}\,,\,\,\,$
(12a) $\displaystyle F^{ab}{}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\left(A^{ab}{}_{[\nu,\mu]}+A^{a}{}_{c[\mu}A^{cb}{}_{\nu]}-\ell^{-2}A^{[a}{}_{\mu}A^{b]}{}_{\nu}\right)\,.\,\,\,$
(12b)

Thus, a novel term that depends on the dynamics of the scale field $\ell$, now
appears in the translation gauge field strength.

In addition to the connection 1-form (10), a symmetry-breaking scalar field
$\xi^{a}$ is required. Otherwise one cannot introduce the coframe field
$\bm{\mathrm{e}}^{a}$, which is the 1-form defined by

$\bm{\mathrm{e}}^{a}=\bm{A}^{a}+\textrm{D}\xi^{a}\,.$ (13)

From this object, we further obtain the torsion 2-form
$\bm{T}^{a}=\textrm{D}\bm{\mathrm{e}}^{a}$ and the 3-form identity
$\textrm{D}\bm{T}^{a}=\bm{F}^{a}{}_{b}\wedge\bm{\mathrm{e}}^{b}$. One sees
that the translation gauge field strength coincides with the torsion 2-form
once the Lorentz curvature $\bm{F}^{a}{}_{b}=0$ is taken to vanish, since from
(13) we have that $\bm{T}^{a}=\textrm{D}\bm{A}^{a}+\bm{F}^{a}{}_{b}\xi^{b}$.
Assuming that the coframe field has an inverse, all the standard ingredients
of gravitational geometry can now be constructed. In the language of Ref.
Westman:2014yca , our fundamental fields are $V^{A}$ and $\bm{A}^{AB}$, and
$\ell$ corresponds to the norm of the $V^{A}$ and the $\xi^{a}$ to the rest of
its independent components, such that the definition (13) ensures the co-
covariance of the components of
$\bm{\mathrm{e}}^{a}=\mathrm{e}^{a}{}_{\mu}\textrm{d}x^{\mu}$ and its inverse.
Then we can freely project the tangent space indices to spacetime indices and
vice versa.

In particular, we obtain the the spacetime torsion tensor
$T^{\alpha}{}_{\mu\nu}$, and can then construct an action for a translation
gauge theory in terms of the invariant $T$ known as the torsion scalar,

$T=\frac{1}{4}T^{\alpha}{}_{\mu\nu}T_{\alpha}{}^{\mu\nu}+\frac{1}{2}T^{\alpha}{}_{\mu\nu}T^{\nu\mu}{}_{\alpha}-T^{\nu}{}_{\mu\nu}T^{\alpha\mu}{}_{\alpha}\,.$
(14)

If we denote $\bar{\bm{T}}^{a}$ the torsion 2-form in the limit when the
evolution of dS scale is neglected, (12b) tells that
$\bm{T}^{a}=\bar{\bm{T}}^{a}-\textrm{d}\log{\ell}\wedge\bm{A}^{a}$. Plugging
this into (14) we obtain that

$T=\bar{T}+4\log{\ell}_{,\mu}\bar{T}^{\mu}-6\left(\partial\log{\ell}\right)^{2}\,,$
(15)

where we defined $T_{\mu}=T^{\alpha}{}_{\mu\alpha}$. The equivalent result was
reported in Jennen:2015bxa . However, our action integral over this scalar,

$I_{\text{dS}}=-\frac{1}{2}\int\textrm{d}^{4}x\mathrm{e}\left[\ell^{-2}\bar{T}+4\ell^{-3}\ell_{,\mu}\bar{T}^{\mu}-6\ell^{-4}\left(\partial{\ell}\right)^{2}\right]\,,$
(16)

has now different scalings for each of the terms, due to the dilatonic role of
the dS scale. This action turns out to be equivalent to the conformally
coupled scalar-tensor theory Dirac:1973gk . By recalling that the metric Ricci
scalar ${R}$ is related to the torsion scalar via
${R}=-\bar{T}-2\text{}\partial_{\mu}(\mathrm{e}\bar{T}^{\alpha})$, we can
rewrite (16) in the much more conventional (though pedantically speaking, ill-
defined due to higher derivatives) scalar-tensor form

$I_{\text{dS}}=\frac{1}{2}\int\textrm{d}^{4}x\mathrm{e}\left[\ell^{-2}{R}+6\ell^{-4}\left(\partial{\ell}\right)^{2}\right]\,.$
(17)

It is well-known that this theory is invariant under the Weyl rescalings222A
complete classification of scale invariance(s) in the general metric-affine
geometry was given in Ref. Iosifidis:2018zwo . Scale transformations in
torsional geometry have been considered in e.g. Maluf:1985fj ; Maluf:2011kf ;
Bamba:2013jqa ; Wright:2016ayu ; Lucat:2017wtu ; Barnaveli:2018dxo ;
Formiga:2019frd . Dirac:1973gk

$g_{\mu\nu}\rightarrow f^{2}g_{\mu\nu}\,,\quad\ell\rightarrow f\ell\,.$ (18)

This scale invariance allows to reduce the theory explicitly to general
relativity in the gauge $f=\ell_{P}/\ell$, but the symmetry (18) means that is
an equivalence holds regardless of the gauge choice.

###  II.1 On alternative formulations

It could also be interesting to reconsider the geometrical foundation
BeltranJimenez:2019tjy  of the above formulation. In particular, since the
model spaces are characterised by different scales, it may not be justified to
consider the generators to be independent of the coordinates $x^{\mu}$. In
particular, as we see in the cosmology-motivated example (8), the spacetime
dependence enters into the generators via the $\ell=\ell(x)$. The potential
problem with this could however be avoided by reformulating the theory in a
torsion-free geometry. Then one would to begin, instead of (3), with
generators defined by the opposite scaling

$\displaystyle\hat{\Pi}_{a}$ $\displaystyle=$
$\displaystyle\Omega_{4a}=\hat{\eta}_{ab}\Pi^{b}\,,$ (19a)
$\displaystyle\hat{\Omega}_{ab}$ $\displaystyle=$
$\displaystyle\ell^{2}\Omega_{AB}\delta^{A}_{a}\delta^{B}_{b}=\hat{\eta}_{ac}\Omega^{c}{}_{b}\,.$
(19b)

The same algebra (4) in terms of the newly defined generators has to be then
written in terms of the conformally rescaled metric
$\hat{\eta}_{\mu\nu}=\ell^{2}\eta_{ab}$ as

$\displaystyle\left[\hat{\Omega}_{ab},\hat{\Omega}_{cd}\right]$
$\displaystyle=$ $\displaystyle
2\left(\hat{\eta}_{d[a}\hat{\Omega}_{b]c}-\hat{\eta}_{c[a}\hat{\Omega}_{b]d}\right)\,,$
(20a) $\displaystyle\left[\hat{\Pi}_{a},\hat{\Omega}_{bc}\right]$
$\displaystyle=$ $\displaystyle 2\hat{\eta}_{a[b}\hat{\Pi}_{c]}\,,$ (20b)
$\displaystyle\left[\hat{\Pi}_{a},\hat{\Pi}_{b}\right]$ $\displaystyle=$
$\displaystyle-\ell^{-2}\hat{\Omega}_{ab}\,.$ (20c)

This indeed suggests a relation to the Weyl gauge theory and a rationale for
the emergence of the scale symmetry (18). In this basis, the theory can be
formulated consistently using the stereographic projection where the induced
metric is conformally flat and the rotations are $\ell$-independent. In the
end, the $\textrm{d}\log{\ell}$ term does not appear in the
$\hat{\bm{T}}^{a}$, but a corresponding term is found in the
$\hat{\bm{F}}^{a}{}_{b}$, and one can write down the usual quadratic curvature
action that reduces to the dS general relativity. We will not pursue here the
details of this formulation (presumably resulting in the equivalent (17)). In
fact, the geometry of the canonical version would be both torsion-free and
flat Koivisto:2019jra ; BeltranJimenez:2020sih , but such a formulation would
require the enlarging of the gauge group and be superfluous for the present
purpose.

##  III Cosmological solution

Cosmologies inspired by the Jennen-Pereira model Jennen:2015bxa  were analysed
as a dynamical system by Otalora Otalora:2014aoa . However, the class of
models studied therein includes neither the particular case of (16) nor the
version of Ref. Jennen:2015bxa  (obtained from 15), because, firstly, the sign
of the scalar field kinetic term in Ref. Otalora:2014aoa  was flipped, and
secondly, because the trace-coupling was considered as a function of the
scalar field333On more general scalar-torsion modified gravity, see e.g.
Bamba:2013jqa ; Hohmann:2019gmt ; Emtsova:2019qsl ; Flathmann:2019khc ;
Golovnev:2018wbh ; Raatikainen:2019qey ; Bahamonde:2020cfv ; Hohmann:2020dgy .
However, already linear perturbations Hohmann:2020vcv  indicate
Golovnev:2018wbh ; Raatikainen:2019qey  that generic such models are not
viable. This stems from their Lorentz violation Li:2010cg .. We shall now
explore the cosmology of the dS gauge theory (16), and find that is
qualitatively different from models of scalar-torsion modified gravity.

The line element in the flat Friedmann-Lemaître-Robertson-Walker cosmology is

$\textrm{d}s^{2}=-n^{2}(t)\textrm{d}t^{2}+a^{2}(t)\delta_{ij}\textrm{d}x^{i}\textrm{d}x^{j}\,,$
(21)

where $n$ is the lapse function and $a$ the scale factor. In addition to these
two metric components (of which $n$ can always be trivialised by simply a
redefinition of $t$), we have the dS scale $\ell$ which may now evolve in
time. We'll denote the expansion rates as follows:

scale factor | variable | rate
---|---|---
temporal | $n(t)$ | $N=\dot{n}/n$
spatial | $a(t)$ | $H=\dot{a}/a$
dimensional | $\ell(t)$ | $L=\dot{\ell}/\ell$

Including a matter source $I_{M}$ describing a perfect fluid with the energy
density $\rho_{M}$ and pressure $p_{M}$ coupled to the dS gravity (16), the
cosmological mini-superspace action becomes

$I=I_{\text{dS}}+I_{M}=-\int\textrm{d}t\left[\frac{3a^{3}}{\ell^{2}n}\left(H-L\right)^{2}+na^{3}\rho_{M}\right]\,.$
(22)

The 1${}^{\text{st}}$ and the 2${}^{\text{nd}}$ Friedmann equations (obtained
from the variations of $I$ wrt $n$ and $a$, respectively), can now be written
as

$\displaystyle 3H^{2}$ $\displaystyle=$ $\displaystyle{\left(\ell
n\right)^{2}}\left(\rho_{\ell}+\rho_{M}\right)\,,$ (23a)
$\displaystyle-2\dot{H}-3H^{2}+2NH$ $\displaystyle=$ $\displaystyle\left(\ell
n\right)^{2}\left(p_{\ell}+p_{M}\right)\,,$ (23b)

where

$\displaystyle\left(\ell n\right)^{2}\rho_{\ell}$ $\displaystyle=$
$\displaystyle-3L^{2}+6HL\,,$ $\displaystyle\left(\ell n\right)^{2}p_{\ell}$
$\displaystyle=$ $\displaystyle-2\dot{L}-4HL+L^{2}+2NL\,.$

In vacuum, $\rho_{M}=0$, with the time slicing $n=1$, the solutions are
$\ell/a=\text{constant}$. This already yields the insight into the theory that
only the relative calibration of the two scale factors is fixed in vacuum, and
neither of the scale factors alone.

To properly couple matter sources to dS gravity with an evolving $\ell$, we
should take into account the scaling of the energy density with $\ell$. For
the purposes of background cosmology, the energy density of matter with an
equation of state $w_{M}=p_{M}/\rho_{M}$ is then given by, up to a constant,

$\rho_{M}\sim a^{-3(1+w_{M})}\ell^{-1+3w_{M}}\,.$ (24)

This prescription results in the scaling one would expect from physical
arguments in the cases of radiation or dust in the matter sector or, spatial
curvature or a cosmological constant in the geometric sector. In particular,
the effective action for a point particle, studied in more detail in Section
IV.1, suggests the scaling $\rho_{M}\sim\ell^{-1}a^{-3}$ when $w_{M}=0$, and
the scale invariance of the radiation is compatible with that $\rho_{M}\sim
a^{-4}$, independently of $\ell$, when $w_{M}=1/3$. (Furthermore, the energy
density due to a cosmological term is $\sim\ell^{-4}$ and independent of $a$,
whilst the effective energy of a spatial curvature term $\sim(\ell a)^{-2}$.)

Since, according to (24), the field $\ell$ now couples non-minimally to
matter, its equation of motion acquires a source term and reads

$\displaystyle\dot{H}-\dot{L}$ $\displaystyle+$
$\displaystyle\left(2H-L-N\right)\left(H-L\right)$ (25) $\displaystyle=$
$\displaystyle\left(\frac{1}{6}-\frac{1}{2}w_{M}\right)\left(\ell
n\right)^{2}\rho_{M}\,.$

It should be noted that in general, when $L\neq 0$, the energy densities obey
the modified continuity equations,

$\displaystyle\frac{\textrm{d}}{\textrm{d}t}\left(\ell^{2}{\rho}_{\ell}\right)+3H\ell^{2}\left(\rho_{\ell}+p_{\ell}\right)$
$\displaystyle=$ $\displaystyle-2L\ell^{2}\rho_{M}\,,\,\,$ (26a)
$\displaystyle\dot{\rho}_{M}+3H\left(1+w_{M}\right)\rho_{M}$ $\displaystyle=$
$\displaystyle L\left(1-3w_{M}\right)\rho_{M}\,.\,\,$ (26b)

It is easy to see that the $\ell^{2}=8\pi G$, $L=0$ is a solution to the
Friedmann equations (23). Thus it is clear that the model defined above at
least contains viable solutions that describe the standard cosmological
background evolution. In the case of a possible time-evolution of $\ell$, more
general solutions exist to the system of equations.

To investigate such more general solutions, we begin with the power-law ansatz

$n=1\,,\quad a\sim t^{\alpha}\,,\quad\ell\sim t^{\lambda}\,.$ (27)

By plugging this ansatz into the 1${}^{\text{st}}$ Friedmann equation (23a),
we readily see that the power-laws must have the relation

$\lambda=\frac{1}{1+3w_{M}}\left[3\left(1+w_{M}\right)\alpha-2\right]\,.$ (28)

The solution that gives back the expansion law of general relativity is
$\lambda=0$ which implies $\alpha=2/(3+3w_{M})$, but this is only one amongst
the 1-parameter family of solutions parameterised by $\lambda$. Remarkably,
these solutions satisfy also the 2${}^{\text{nd}}$ Friedmann equation (23b),
and consequently they satisfy identically the equation of motion (25) as well.
Accelerating solutions exist. For a background fluid with $w_{M}>-1/3$, the
universe accelerates as if dominated by a quintessence-like field given that
$\lambda>-1$, and further, the universe super-accelerates if
$\ell<-2/(1+3w_{M})$. In general, the universe expands as if was filled with a
fluid that has the equation of state

$w=\frac{\rho_{\ell}+\rho_{M}}{p_{\ell}+p_{M}}=\frac{w_{M}-\left(\frac{1}{3}+w_{M}\right)\lambda}{1+\left(\frac{1}{3}+w_{M}\right)\lambda}\,.$
(29)

More general cosmological solutions, sourced by perfect fluids with
$\dot{w}_{M}\neq 0$ (which can also effectively describe several distinct
perfect fluid components), could be studied numerically.

It is worthy to point out that the dS coupling prescription (24) is
essentially the unique viable possibility. We will briefly comment upon some
alternative prescriptions, omitting the details of the derivations. The
Jennen-Pereira model Jennen:2015bxa  with the standard coupling prescription
(i.e. $\rho_{M}\sim a^{-3(1+w_{M})}$) is not compatible with cosmological
evolution444More precisely, the Friedmann equations would be consistent only
for stiff fluid matter $w_{M}=1$. This was first pointed out to us by Sergio
Bravo Medina.. If this model is supplemented with the $\ell$-dependent
cosmological constant (the sign has to be negative), cosmological evolution
can be recovered such that in the standard Friedmann equation $G\rightarrow
G/(1+3w_{M})$, and therefore in the radiation dominated era the effective
gravitational coupling would be $G/2$, that appears too drastical modification
to allow viable early universe phenomena such as nucleosynthesis and the
formation of the cosmic microwave background. Yet, one could further adjust
the model by retaining the minimal matter coupling but taking into account the
$\ell$-dependence of the gravitational coupling. In such a prescription a
radiation-dominated era is not only phenomenologically excluded, but
incompatible with the Friedmann equations in the first place.

Thus, it turns out that the dS matter coupling prescription (24) that we
justified by physical principles, could actually have been formally deduced by
requiring the existence of viable cosmological background solutions.

###  III.1 On the relevance of the solution

It is lluminating to show that the family of solutions is indeed equivalent
under the symmetry (18). The invariant combination of the metric and the scale
field is $\hat{g}_{\mu\nu}=(\ell_{P}/\ell)^{2}g_{\mu\nu}$, and correspondingly
we denote $\hat{a}=(\ell_{P}/\ell)a$, and $\hat{t}$ the time coordinate when
the lapse function is $\hat{n}=(\ell_{P}/\ell)$. It is then straightforward to
compute the invariant Hubble rate and its time derivative,

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle(\ell/\ell_{P})\left(H-L\right)\,,$ (30a)
$\displaystyle\frac{\textrm{d}}{\textrm{d}\hat{t}}\hat{H}$ $\displaystyle=$
$\displaystyle(\ell/\ell_{P})^{2}\left[\dot{H}-\dot{L}-(H-L)L\right]\,.$ (30b)

Plugging in the relations (28) and (29) we obtain the result for the expansion
rate that corresponds to the effective equation of state $\hat{w}=w_{M}$. In
terms of the ``hatted'' variables, the Friedmann equations (23) of course
assume their standard form.

The gauge freedom allows a radically different re-interpretation of the
expanding universe. In an extreme case, we can understand all the
observational data in a static universe (obtained by setting $\alpha=0$ in the
above family of solutions), where instead the gravitational coupling as well
as the masses of particles are evolving in time (according to
$\lambda=-2/(1+3w_{M})$). For example, the observed cosmological redshift of
photons is then not due to the stretching of the wavelengths together with the
spatial scale factor $a(t)$, but it is due to the shrinking of the dimensional
scale factor $\lambda(t)$. In this description of the universe, we clearly
have no curvature singularity and therefore the cosmological spacetime appears
to be non-singular and extendable to $t\rightarrow-\infty$. Going backward in
time from the present, the dS scale grows indefinitely, and the big bang
would-be-singularity occurs at the point wherein the dS scale becomes infinite
and the hyperboloid flattens out (this is the contraction limit
$\mathfrak{so}(4,1)\rightarrow\mathfrak{iso}(3,1)$), and continuing this naive
extrapolation to still earlier times, the geometry becomes that of anti-dS
with the radius now shrinking indefinitely as we wind backwards towards
$t\rightarrow-\infty$. In this frame, both the metric and the total curvature
invariants are identically zero, though the torsion scalar (14) is $T=6L^{2}$
and thus diverges at $t=0$. The physical matter quantities remain finite. The
radiation555If dust is present at such a primordial stage, its energy density
momentarily disappears at $t=0$. This might be relevant in regards the initial
conditions for the geometric dark matter discovered in Zlosnik:2018qvg . We
note that at least the naive prescription (24) excludes sources with
$w_{M}>1/3$, since their energy density would diverge. energy density and the
pressure are always constant in this static universe frame, as seen from (24).

The possible relevance of the dS kinematics to a new cosmological paradigm has
been foreseen in some discussions Aldrovandi:2004km ; Araujo:2015oqa . Though
no solutions were presented, and the focus was on the opposite contraction
limit $\ell\rightarrow\infty$, the main insight that the conformal property of
the (apparently singular) transition point could be the key in connecting two
aeons in sir Penrose's conformal cyclic cosmology Araujo:2015oqa , is strongly
corroborated by our exact cosmological solution in the consistent dS gauge
gravity (16). By adopting Willem de Sitter's own, projective view of the dS
geometry McInnes:2003xm ; Ong:2016vwr , the solution might naturally be
enclosed into the eternal return of the aeon that is our unique universe.

In the more mainstream context of string theory, the existence of negative
energy vacua seems to be not only a generic prediction in the landscape of
myriad universes, but a requirement for the consistent definition of an
S-matrix, and it has proven quite a challenge to find ways that may lead to
positive vacuum energies compatible with the one observed universe
Kachru:2003sx ; Dasgupta:2019gcd . Previous attempts at realising a non-
singular anti-dS to dS transition have resorted to rather complicated
mechanisms requiring various new indgredients for their realisation
Biswas:2011qe ; Gupt:2013poa . It is remarkable that we seem to consistently
predict the desired non-singular transition, based on an action that is
locally equivalent to general relativity, but underpinned by the principles of
dS gauge theory. -It should be noted that the reinterpretation of the
cosmological expansion as a variation of mass scales is of course well-known
in the context of Fierz-Jordan-Brans-Dicke theory, and in particular, the
possibility that the big bang singularity is a field coordinate singularity
(i.e. removable by a change of variables) was introduced and clarified by
Wetterich Wetterich:2013jsa ; Wetterich:2013aca ; Wetterich:2020oyy  (in the
context of his theory of Variable Gravity Wetterich:2013jsa , which is not a
mere reformulation of general relativity). The removal of black hole
singularities was also considered, employing more general than conformal
change of field coordinates Domenech:2019syf .

Finally, let us mention that Hohmann et al Hohmann:2018shl  have recently
pointed out that the theory (17) formally contains vacuum solutions with
wormholes, despite their local equivalence with the standard vacuum solutions.
The physicality of such wormholes hinges on global, topological issues. As
Hohmann et al Hohmann:2018shl  explained, the key point is that the solutions
may be related by improper Weyl transformations (18), where the factor $f$ may
vanish or become infinite at some points (in other words, the Jacobian of the
field coordinate transformation is not defined at those points). It is
precisely in this sense that the dS gauge theory (16) is inequivalent to
general relativity, and can thus accommodate a more general variety of
physically distinct solutions.

##  IV Implications to matter

At the level of background cosmology it was sufficient to exploit the perfect
fluid parameterisation (24) for matter sources, but the question may remain
whether the proposed dS coupling prescription is consistent for more
fundamental field theory description of massive matter fields. To address this
question, we consider the action for a point particle and for a scalar field.

###  IV.1 Point particle

Consider the massive point particle action,

$I_{pp}=\int m\textrm{d}s\,,$ (31)

where the line element for a time-like curve $x^{\mu}$ is
$\textrm{d}s=\sqrt{-g_{\mu\nu}\textrm{d}x^{\mu}\textrm{d}x^{\nu}}$ and the
mass $m$ is related to the fundamental scale $\ell$ as $m(x)=m_{0}/\ell(x)$,
$m_{0}$ being the dimensionless constant of proportionality. We consider small
variations $\delta x^{\mu}$ of the curve $x^{\mu}(\tau)$ parameterised by an
arbitrary parameter $\tau$,

$\delta I_{pp}=\int\left[\delta
m\frac{\textrm{d}s}{\textrm{d}\tau}-\frac{m}{2\textrm{d}s}\delta\left(g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\right)\right]\textrm{d}\tau\,.$
(32)

The first term we can write as $\int\delta m{\textrm{d}s}=\int
m_{,\mu}\dot{x}^{\mu}\textrm{d}s$ and for the second term we apply integration
by parts. Adding the two terms we get

$\displaystyle\delta I_{pp}$ $\displaystyle=$
$\displaystyle\int\Big{[}mg_{\mu\nu}\frac{\textrm{d}^{2}x^{\nu}}{\textrm{d}s^{2}}+m\frac{\textrm{d}x^{\alpha}}{2\textrm{d}s}\frac{\textrm{d}x^{\nu}}{\textrm{d}s}\left(2g_{\mu(\nu,\alpha)}-g_{\alpha\nu,\mu}\right)$
$\displaystyle+$
$\displaystyle\frac{\textrm{d}x^{\alpha}}{2\textrm{d}s}\frac{\textrm{d}x^{\nu}}{\textrm{d}s}\left(2g_{\mu(\nu}m_{,\alpha)}-g_{\alpha\nu}m_{,\mu}\right)\Big{]}\delta
x^{\mu}\textrm{d}s=0\,.$

Since this holds for arbitrary variations of the path $\delta x^{\mu}$, we
get, by raising one index and dividing by $m$,

$\displaystyle\frac{\textrm{d}^{2}x^{\alpha}}{\textrm{d}s^{2}}$
$\displaystyle+$
$\displaystyle\frac{1}{2}g^{\alpha\beta}\left(g_{\beta\mu,\nu}+g_{\beta\nu,\mu}-g_{\mu\beta,\mu}\right)\frac{\textrm{d}x^{\mu}}{\textrm{d}s}\frac{\textrm{d}x^{\nu}}{\textrm{d}s}$
$\displaystyle=$ $\displaystyle-$
$\displaystyle\frac{1}{2m}\left(\delta^{\alpha}_{\mu}m_{,\nu}+\delta^{\alpha}_{\nu}m_{,\mu}-g_{\mu\nu}{m}^{,\alpha}\right)\frac{\textrm{d}x^{\mu}}{\textrm{d}s}\frac{\textrm{d}x^{\nu}}{\textrm{d}s}\,.$

Since $\log{m}_{,\alpha}=-\log{\ell}_{,\alpha}$, this can be written as

$\ddot{x}^{\alpha}+\overset{\star}{\Gamma}{}^{\alpha}{}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\mu}=0\,,$
(33)

where the overdot now denotes the derivative wrt the proper time $\tau=s$ and
the connection is

$\overset{\star}{\Gamma}{}^{\alpha}{}_{\mu\nu}=\left\\{{}^{\phantom{i}\alpha}_{\mu\nu}\right\\}-\left(\delta^{\alpha}_{(\mu}\log{\ell}_{,\nu)}-\frac{1}{2}g_{\mu\nu}\log{\ell}^{,\alpha}\right)\,.$
(34)

Thus, we predict that the matter moves along the geodesics of a Weyl
connection, for which the Weyl gauge field $\ell_{\mu}=\ell_{,\mu}$ is pure
gauge and thus its curvature vanishes
$F_{\mu\nu}=2\ell_{[\mu,\nu]}=2\ell_{,[\mu\nu]}=0$. Therefore there is no
second clock effect. For more details and the extension of the integrable Weyl
(sometimes called semi-metric) geometry to generic non-metric geometry, see
BeltranJimenez:2020sih .

In terms of the arbitrary parameter $\tau$, (33) generalises to

$\ddot{x}^{\alpha}+\overset{\star}{\Gamma}{}^{\alpha}{}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\mu}=-\dot{s}^{2}\frac{\textrm{d}^{2}\tau}{\textrm{d}s^{2}}\dot{x}^{\alpha}\,.$
(35)

We see that the reparameterisation $\tau=as+b$ where $a,b$ are constants, does
not change the form of the equation (33). We also note that the projective
transformation of the connection by a one-form $p_{\mu}$ can be compensated by
the reparameterisation that satisfies

$\overset{\star}{\Gamma}{}^{\alpha}{}_{\mu\nu}\rightarrow\overset{\star}{\Gamma}{}^{\alpha}{}_{\mu\nu}+\delta^{\alpha}_{\nu}p_{\mu}\,,\quad
p_{\mu}\dot{x}^{\mu}=\dot{s}^{2}\frac{\textrm{d}^{2}\tau}{\textrm{d}s^{2}}\,.$
(36)

A reparameterisation of the curve thus corresponds to a projective
transformation of the affine geometry. The curve abstracted from its
parameterisation i.e. the projective equivalence class of the geodesic is
called a path.

A more elementary modelling of matter fields would begin with spinor fields,
but in the end the classical approximation relevant to our purposes is given
by the point particle action where the $m\sim\ell^{-1}$ is inherited from the
spinor mass term. Spinor fields and gauge fields can be coupled to dS gravity
elegantly with polynomial Lagrangians Pagels:1983pq ; Westman:2012zk .

###  IV.2 Scalar field

Considering a self-interacting scalar field, our coupling prescription
suggests the Lagrangian

$I_{\phi}=-\int\textrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2\ell^{2}}\left(\partial\phi\right)^{2}+\ell^{-4}V(\phi)\right]\,.$
(37)

In the cosmological setting, this gives the total action
$I=I_{\text{dS}}+I_{\phi}$ as

$I=-\int\textrm{d}ta^{3}\left[\frac{3}{\ell^{2}n}\left(H-L\right)^{2}-\frac{1}{2\ell^{2}}\frac{\dot{\phi}^{2}}{n}+\frac{n}{\ell^{4}}{V(\phi)}\right]\,.$
(38)

The scalar field contribution to the Friedmann equations is then given by

$\displaystyle\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left({\dot{\phi}}/{\ell
n}\right)^{2}+{\ell^{-4}}{V(\phi)}\,,$ (39a) $\displaystyle p_{\phi}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left({\dot{\phi}}/{\ell
n}\right)^{2}-{\ell^{-4}}{V(\phi)}\,.$ (39b)

The Klein-Gordon equation is obtained by the variation wrt the scalar field,

$\ddot{\phi}+\left(3H-2L-N\right)\dot{\phi}+n^{2}\ell^{-6}V^{\prime}(\phi)=0\,,$

This equation, by multiplying with $\dot{\phi}$ and rearranging the terms,
reduces to (26a).

To verify the consistency of the system, we consider also the equation of
motion

$\displaystyle\dot{H}-\dot{L}$ $\displaystyle+$
$\displaystyle\left(2H-L-\frac{\dot{N}}{N}\right)\left(H-L\right)$
$\displaystyle=$
$\displaystyle\frac{1}{6}\left(-\dot{\phi}^{2}+4N^{2}\ell^{-2}V(\phi)\right)$
$\displaystyle=$
$\displaystyle\left(\frac{1}{6}-\frac{1}{2}w_{\phi}\right)\left(\ell
N\right)^{2}\rho_{\phi}\,,$

which is in full agreement with (25). Therefore this equation is, as it
should, degenerate with the Friedmann equations.

We note that even though a massless scalar field $V(\phi)\approx 0$ is a
perfect fluid with the stiff fluid equation of state $w_{\phi}=1$, the
heuristic perfect fluid coupling prescription (24), which would naively
suggest the kinetic term to scale as $\ell^{2}$, is not the correct one to
adopt for a proper field theoretical description. The quadratic kinetic term
$\sim(\partial\phi)^{2}$ inherits the scaling dimension
$\sim[\phi^{2}]\sim[\ell^{-2}]$ from the dimension of the scalar field, and
this is the fundamental rationale that determines the coupling.

##  V Discussion

Motivated by the fact that our universe has fundamental limiting scales both
in the infrared and in the ultraviolet ends of the spectrum, we developed a dS
gauge theory of gravity which incorporates the new kinematic invariant $\ell$,
besides the invariant $c$ of Einstein's relativity. The Cartan-geometric
construction, illustrated upon the dS hyperboloid embedded into a spacetime
with one extra dimension, was based upon nothing but the standard gauge field,
a connection 1-form and a symmetry-breaking scalar field. Gravity was realised
as a gauge theory of translations in the sense that the action is quadratic in
the translation gauge field strength $\bm{F}^{a}$ whilst the homogeneous model
spaces are flat $\bm{F}^{a}{}_{b}=0$ (though, as mentioned in II.1, it would
be possible to formulate a more canonical version of translation gauge
theory).

A key insight was that the theory (16) exhibits the rescaling invariance (18).
Thus, the calibration of the scale $\ell$ is arbitrary, and it can changed
without affecting the physics given the accompanying rescaling of the metric.
Incidentally, the theory thus realises the foundational motivations of both
the dS and the Weyl gauge theories. On one hand, the description of our
universe requires observer-independent scales. On the other hand, absolute
scales are physically meaningless. Thus, the new dS theory may provide a
conceptually improved framework to the century-old problem of introducing
scales into physics. From a formal point of view, the orthogonal symmetry is
considerably neater than the Weyl extension of the Poincaré symmetry. An
obvious direction to pursue in the future is the incorporation of the two
limiting scales, $\ell_{P}$ and $\ell_{\Lambda}$ independently, via the
completion of the dS to the conformal symmetry666It is natural to speculate
that Dirac’s original motivation for (17), the large number hypothesis
Dirac:1973gk ; Ray:2007cc , could be vindicated by exploiting the additional
freedom provided by another scalar field, thus yielding the satisfactory
explanation of various other scales in physics..

Let us comment on our theory also in view of the so-called ``teleparallel''
models of gravity. Modifications of gravity in that context are by now well-
known to violate Lorentz symmetry, resulting in extra degrees of freedom which
typically have strong coupling and other unwanted problems777Such concerns
have been raised earlier in the literature Kopczynski_1982 ; Li:2010cg , and
the current state of art in the problematics of the extra degrees of freedom
is reviewed in Blixt:2020ekl ; Golovnev:2020zpv .. In contrast, the new theory
developed in this article, though formulated in terms of a flat connection
$\bm{A}^{a}{}_{b}$ i.e. in a ``teleparallel'' geometry, is not based on a
violation but on an extension of the Lorentz symmetry. Thus our approach also
seems to suggest a way of generating viable ``teleparallel'' gravity models.
However, since the theory (16) we arrived at has the metric scalar-tensor
equivalent (17), it still remains an open question whether a consistent
``genuinely teleparallel'' modification of gravity is possible.

The aim of this article was to explore the cosmology of the dS gauge theory
with time-evolving distance scale $\ell=\ell(t)$. We derived the family of
exact solutions which is characterised by the effective equation of state (29)
for the gravity-fluid system. Incidentally, it turned out that the coupling
prescription (24) that follows from the physical interpretation of the dS
scale $\ell$, is actually necessary for the existence of realistic
cosmological solutions (including even those which reduce to the standard
solutions in general relativity). We considered the possible reinterpretation
of observations in the frame where the universe does not expand but the dS
scale is evolving in time. This is, to our knowledge (despite the often-made
claims otherwise in the vast literature on ``teleparallel'' cosmology), the
first description of the cosmic geometry de facto in terms torsion, without
the metric curvature playing its usual role. We argued that such a novel
description could allow the consistent extension of cosmology beyond the big
bang, and believe that the theory and its cosmological implications merit
further investigation.

###### Acknowledgements.

TK would like to thank Manuel Hohmann and Hardi Veermäe for insightful
discussions on scale invariance, and Sergio Bravo Medina and David Mota for an
earlier collaboration on a related topic. This work was supported by the
Estonian Research Council grants PRG356 ``Gauge Gravity'' and MOBTT86, and by
the European Regional Development Fund CoE program TK133 ``The Dark Side of
the Universe''.

## References

  * (1) N. Aghanim et al., ``Planck 2018 results. VI. Cosmological parameters,'' Astron. Astrophys., vol. 641, p. A6, 2020.
  * (2) L. J. Garay, ``Quantum gravity and minimum length,'' Int. J. Mod. Phys. A, vol. 10, pp. 145–166, 1995.
  * (3) F. Dyson, ``Missed opportunities,'' Bull. Am. Math. Soc., vol. 78, pp. 635–639, 1972.
  * (4) I. Licata, L. Chiatti, and E. Benedetto, De Sitter Projective Relativity.  SpringerBriefs in Physics, Cham: Springer, 2017.
  * (5) G. Amelino-Camelia, ``Doubly special relativity,'' Nature, vol. 418, pp. 34–35, 2002.
  * (6) R. Aldrovandi, J. P. Beltran Almeida, and J. G. Pereira, ``de Sitter special relativity,'' Class. Quant. Grav., vol. 24, pp. 1385–1404, 2007.
  * (7) H. F. Westman and T. Zlosnik, ``An introduction to the physics of Cartan gravity,'' Annals Phys., vol. 361, pp. 330–376, 2015.
  * (8) C. Brans and R. Dicke, ``Mach's principle and a relativistic theory of gravitation,'' Phys. Rev., vol. 124, pp. 925–935, 1961.
  * (9) P. A. Dirac, ``Long range forces and broken symmetries,'' Proc. Roy. Soc. Lond. A, vol. 333, pp. 403–418, 1973.
  * (10) M. Blagojevic, Gravitation and gauge symmetries.  8 2002.
  * (11) M. Blagojević and F. W. Hehl, eds., Gauge Theories of Gravitation: A Reader with Commentaries.  Singapore: World Scientific, 2013.
  * (12) E. Scholz, ``The unexpected resurgence of Weyl geometry in late 20-th century physics,'' Einstein Stud., vol. 14, pp. 261–360, 2018.
  * (13) S. MacDowell and F. Mansouri, ``Unified Geometric Theory of Gravity and Supergravity,'' Phys. Rev. Lett., vol. 38, p. 739, 1977.  [Erratum: Phys.Rev.Lett. 38, 1376 (1977)].
  * (14) H. R. Pagels, ``Gravitational Gauge Fields and the Cosmological Constant,'' Phys. Rev. D, vol. 29, p. 1690, 1984.
  * (15) D. K. Wise, ``MacDowell-Mansouri gravity and Cartan geometry,'' Class. Quant. Grav., vol. 27, p. 155010, 2010.
  * (16) S. Gielen and D. K. Wise, ``Lifting General Relativity to Observer Space,'' J. Math. Phys., vol. 54, p. 052501, 2013.
  * (17) T. Złośnik, F. Urban, L. Marzola, and T. Koivisto, ``Spacetime and dark matter from spontaneous breaking of Lorentz symmetry,'' Class. Quant. Grav., vol. 35, no. 23, p. 235003, 2018.
  * (18) R. Aldrovandi and J. Pereira, ``de Sitter relativity: A New road to quantum gravity,'' Found. Phys., vol. 39, pp. 1–19, 2009.
  * (19) H. Jennen, ``Cartan geometry of spacetimes with a nonconstant cosmological function $\Lambda$,'' Phys. Rev. D, vol. 90, no. 8, p. 084046, 2014.
  * (20) H. Jennen and J. G. Pereira, ``Dark energy as a kinematic effect,'' Phys. Dark Univ., vol. 11, pp. 49–53, 2016.
  * (21) H. Jennen, Dark energy as a kinematic effect.  PhD thesis, Sao Paulo, IFT, 2016.
  * (22) H.-Y. Guo, C.-G. Huang, Z. Xu, and B. Zhou, ``On Beltrami model of de Sitter space-time,'' Mod. Phys. Lett. A, vol. 19, pp. 1701–1710, 2004.
  * (23) J. G. Pereira, A. C. Sampson, and L. L. Savi, ``de Sitter transitivity, conformal transformations and conservation laws,'' Int. J. Mod. Phys., vol. D23, p. 1450035, 2014.
  * (24) D. Iosifidis and T. Koivisto, ``Scale transformations in metric-affine geometry,'' 10 2018.
  * (25) J. W. Maluf, ``Conformal Invariance and Torsion in General Relativity,'' Gen. Rel. Grav., vol. 19, p. 57, 1987.
  * (26) J. Maluf and F. Faria, ``Conformally invariant teleparallel theories of gravity,'' Phys. Rev. D, vol. 85, p. 027502, 2012.
  * (27) K. Bamba, S. D. Odintsov, and D. Sáez-Gómez, ``Conformal symmetry and accelerating cosmology in teleparallel gravity,'' Phys. Rev. D, vol. 88, p. 084042, 2013.
  * (28) M. Wright, ``Conformal transformations in modified teleparallel theories of gravity revisited,'' Phys. Rev. D, vol. 93, no. 10, p. 103002, 2016.
  * (29) S. Lucat and T. Prokopec, ``Is conformal symmetry really anomalous?,'' 9 2017\.
  * (30) A. Barnaveli, S. Lucat, and T. Prokopec, ``Inflation as a spontaneous symmetry breaking of Weyl symmetry,'' JCAP, vol. 01, p. 022, 2019.
  * (31) B. Formiga, J. ``Conformal teleparallel theories and Weyl geometry,'' Phys. Rev. D, vol. 99, no. 6, p. 064047, 2019.
  * (32) J. B. Jiménez, L. Heisenberg, and T. S. Koivisto, ``The Geometrical Trinity of Gravity,'' Universe, vol. 5, no. 7, p. 173, 2019.
  * (33) T. Koivisto, M. Hohmann, and L. Marzola, ``An Axiomatic Purification of Gravity,'' 9 2019.
  * (34) J. Beltrán Jiménez, L. Heisenberg, and T. Koivisto, ``The coupling of matter and spacetime geometry,'' Class. Quant. Grav., vol. 37, no. 19, p. 195013, 2020.
  * (35) G. Otalora, ``A novel teleparallel dark energy model,'' Int. J. Mod. Phys. D, vol. 25, no. 02, p. 1650025, 2015.
  * (36) M. Hohmann, ``Disformal Transformations in Scalar-Torsion Gravity,'' Universe, vol. 5, p. 167, 2019.
  * (37) E. D. Emtsova and M. Hohmann, ``Post-Newtonian limit of scalar-torsion theories of gravity as analogue to scalar-curvature theories,'' Phys. Rev. D, vol. 101, no. 2, p. 024017, 2020.
  * (38) K. Flathmann and M. Hohmann, ``Post-Newtonian Limit of Generalized Scalar-Torsion Theories of Gravity,'' Phys. Rev. D, vol. 101, no. 2, p. 024005, 2020.
  * (39) A. Golovnev and T. Koivisto, ``Cosmological perturbations in modified teleparallel gravity models,'' JCAP, vol. 11, p. 012, 2018.
  * (40) S. Raatikainen and S. Rasanen, ``Higgs inflation and teleparallel gravity,'' JCAP, vol. 12, p. 021, 2019.
  * (41) S. Bahamonde, K. F. Dialektopoulos, M. Hohmann, and J. Levi Said, ``Post-Newtonian limit of Teleparallel Horndeski gravity,'' Class. Quant. Grav., vol. 38, no. 2, p. 025006, 2020.
  * (42) M. Hohmann and C. Pfeifer, ``Teleparallel axions and cosmology,'' 12 2020.
  * (43) M. Hohmann, ``General cosmological perturbations in teleparallel gravity,'' Eur. Phys. J. Plus, vol. 136, no. 1, p. 65, 2021.
  * (44) B. Li, T. P. Sotiriou, and J. D. Barrow, ``$f(T)$ gravity and local Lorentz invariance,'' Phys. Rev. D, vol. 83, p. 064035, 2011.
  * (45) R. Aldrovandi, J. Almeida, and J. Pereira, ``A Singular conformal universe,'' J. Geom. Phys., vol. 56, pp. 1042–1056, 2006.
  * (46) A. Araujo, H. Jennen, J. Pereira, A. Sampson, and L. Savi, ``On the spacetime connecting two aeons in conformal cyclic cosmology,'' Gen. Rel. Grav., vol. 47, no. 12, p. 151, 2015.
  * (47) B. McInnes, ``De Sitter and Schwarzschild-de Sitter according to Schwarzschild and de Sitter,'' JHEP, vol. 09, p. 009, 2003.
  * (48) Y. C. Ong and D.-h. Yeom, ``Instanton Tunneling for De Sitter Space with Real Projective Spatial Sections,'' JCAP, vol. 04, p. 040, 2017.
  * (49) S. Kachru, R. Kallosh, A. D. Linde, J. M. Maldacena, L. P. McAllister, and S. P. Trivedi, ``Towards inflation in string theory,'' JCAP, vol. 10, p. 013, 2003.
  * (50) K. Dasgupta, M. Emelin, M. M. Faruk, and R. Tatar, ``de Sitter Vacua in the String Landscape,'' 8 2019.
  * (51) T. Biswas, T. Koivisto, and A. Mazumdar, ``Could our Universe have begun with Negative Lambda?,'' 5 2011.
  * (52) B. Gupt and P. Singh, ``Nonsingular AdS-dS transitions in a landscape scenario,'' Phys. Rev. D, vol. 89, no. 6, p. 063520, 2014.
  * (53) C. Wetterich, ``Variable gravity Universe,'' Phys. Rev. D, vol. 89, no. 2, p. 024005, 2014.
  * (54) C. Wetterich, ``Universe without expansion,'' Phys. Dark Univ., vol. 2, pp. 184–187, 2013.
  * (55) C. Wetterich, ``Crossing the Big Bang singularity,'' 4 2020.
  * (56) G. Domènech, A. Naruko, M. Sasaki, and C. Wetterich, ``Could the black hole singularity be a field singularity?,'' Int. J. Mod. Phys. D, vol. 29, no. 03, p. 2050026, 2020.
  * (57) M. Hohmann, C. Pfeifer, M. Raidal, and H. Veermäe, ``Wormholes in conformal gravity,'' JCAP, vol. 10, p. 003, 2018.
  * (58) H. F. Westman and T. G. Zlosnik, ``Cartan gravity, matter fields, and the gauge principle,'' Annals Phys., vol. 334, pp. 157–197, 2013.
  * (59) S. Ray, U. Mukhopadhyay, and P. Pratim Ghosh, ``Large Number Hypothesis: A Review,'' 5 2007.
  * (60) W. Kopczynski, ``Problems with metric-teleparallel theories of gravitation,'' Journal of Physics A: Mathematical and General, vol. 15, pp. 493–506, feb 1982.
  * (61) D. Blixt, M.-J. Guzmán, M. Hohmann, and C. Pfeifer, ``Review of the Hamiltonian analysis in teleparallel gravity,'' 12 2020.
  * (62) A. Golovnev and M.-J. Guzmán, ``Foundational issues in f(T) gravity theory,'' 12 2020.