# Federated Learning for Large-Scale Scene Modeling
with Neural Radiance Fields

Teppei Suzuki  Denso IT Laboratory, Inc.
Tokyo, Japan
<EMAIL_ADDRESS>Teppei Suzuki1 1Denso IT Laboratory, Inc.,
Tokyo<EMAIL_ADDRESS>
###### Abstract

We envision a system to continuously build and maintain a map based on earth-
scale neural radiance fields (NeRF) using data collected from vehicles and
drones in a lifelong learning manner. However, existing large-scale modeling
by NeRF has problems in terms of scalability and maintainability when modeling
earth-scale environments. Therefore, to address these problems, we propose a
federated learning pipeline for large-scale modeling with NeRF. We tailor the
model aggregation pipeline in federated learning for NeRF, thereby allowing
local updates of NeRF. In the aggregation step, the accuracy of the clients’
global pose is critical. Thus, we also propose global pose alignment to align
the noisy global pose of clients before the aggregation step. In experiments,
we show the effectiveness of the proposed pose alignment and the federated
learning pipeline on the large-scale scene dataset, Mill19.

##  I Introduction

Neural radiance fields (NeRF) [1] have emerged to represent scenes for view
synthesis and have been used as maps for many robotics applications, such as
vision-based localization, navigation, and SLAM [2, 3, 4, 5, 6, 7]. Although
these studies have been evaluated in relatively small-scale scenes, we believe
that such NeRF-based robotics applications will be available to the systems
operated in large-scale environments, such as self-driving cars and delivery
drones. In addition, since NeRF can lift arbitrary 2D information to 3D (e.g.,
semantic labels [8], and feature vectors [9]), it will facilitate a higher-
level environmental recognition and understanding in robotics applications.

Nowadays, a large number of vehicles are on the road, and in the near future,
many unmanned aerial vehicles, such as delivery drones, will be in the sky. In
such a world, we envision the system to continuously build and maintain an
earth-scale NeRF-based map using data collected from vehicles and drones in a
life-long learning manner and to leverage the map created and maintained with
such a system.

Large-scale scene modeling with NeRF has been studied in recent years [10,
11]. These methods divide a large-scale environment into many small areas and
model the small areas with multiple NeRF models. Although this pipeline
efficiently models a large-scale scene in a distributed training manner, it
has three problems in realizing our envisioned system: (i) The existing
pipelines need to collect data in a server once, but they need high
communication costs to collect data from clients on a server and need a large
number of memory budgets to save the earth-scale data; (ii) these pipelines
require tremendously large computational resources for training a large number
of models to cover an earth-scale environment; and (iii) they can only update
the model for each pre-divided area for the maintenance of the NeRF-based map,
and if the training data is collected only in a part of the area, the model
may suffer from the forgetting problem [3].

To develop a more scalable and maintainable method, we propose the federated
learning pipeline for large-scale modeling with NeRF, as depicted in Fig. 1.
Federated learning (FL) [12] is a data-decentralizing training protocol that
enables clients, such as mobile devices, to collaboratively train a model
while keeping the training data decentralized. Since FL does not need to
aggregate data collected by clients on the server, it can alleviate the
communication cost problem and the data storage problem. FL also alleviates
the computational resource problem because it can leverage the resources of
clients, thereby enabling their overall FLOPs to outperform those of
supercomputers.

Figure 1: Overview of the proposed federated learning pipeline for neural
radiance fields.

To train NeRF in a federated learning manner, we tailor the model aggregation
pipeline, which allows the global model to be locally updated based on the
client data, avoiding the forgetting problem and improving maintainability.
Specifically, we cache the outputs of each local model as the 3D voxel grid,
and then the global model is updated by aggregating cached representation
based on the clients’ pose in the global coordinate. This pipeline does not
affect the outputs of the global model beyond the region modeled by the local
models. In addition, to align the inaccurate global pose of clients due to
sensor noise, we propose the global pose alignment step before the aggregation
step, which is inspired by the vision-based localization using NeRF [2, 7,
13].

Our contribution is summarized below:

  * •

We propose the federated learning pipeline for large-scale scene modeling with
NeRF. By caching the outputs of each local model as the voxel grid, we realize
the local update of the global model.

  * •

We also propose the global pose alignment. To alleviate the sensor noise in
the clients’ pose in the global coordinate, we align the global pose of each
client before the aggregation step by minimizing the difference between the
images rendered by the global and local models.

  * •

We assess the effectiveness of our training pipeline on the Mill19 dataset
[11], which contains thousands of HD images collected from drone footage over
100,000m2 of terrain near an industrial complex.

##  II Related Work

###  II-A Neural Radiance Fields

3D reconstruction from image collections [14, 15, 16] is an important task in
computer vision fields, and the reconstruction methods have been applied to
many robotics applications, particularly SLAM systems [17, 18, 19]. In
principle, objects or whole scenes are reconstructed as 3D points based on
multiple-view geometry.

In recent years, NeRF [1] has emerged for novel view synthesis. Unlike 3D
point-based reconstruction, NeRF models the target scene by a continuous
function and its image rendering process is differentiable. Taking advantage
of these properties, some vision-based robotics systems have been proposed.
iNeRF [2], a vision-based localization method, estimates the camera pose
through the minimization of the loss between the target image and the rendered
image with the gradient method. Maggio et al. [7] and Lin et al. [13] improved
the robustness of iNeRF by leveraging the advantage of Monte Carlo
localization [20]. Adamkiewicz et al. [5] utilized this localization scheme
for robot navigation. Kruzhkov et al. [21] incorporated NeRF into the SLAM
system. They used NeRF as a mapping module and reduced the memory budget for
the map. Moreover, pure NeRF-based SLAM systems, such as iMap [3] and Nice-
SLAM [4], were studied.

As reviewed above, NeRF is applied to robotics applications that use NeRF as a
map. When considering systems using NeRF as a map in a large environment, such
as self-driving cars and delivery drones, we need to model large-scale scenes.
There are certain studies for large-scale scene modeling by NeRF. Block-NeRF
[10] and Mega-NeRF [11] have modeled urban-scale environments. Both methods
divide the target scene into small areas and then model the small areas with
multiple NeRF models. They can efficiently model a large-scale scene in a
distributed training manner.

However, the existing large-scale modeling methods have some problems in terms
of maintainability and computational costs. When part of the scene is changed,
they need to update all models representing the changed scene, even if the
changed area is a part of the modeled area. In addition, due to the forgetting
problem [3], the model needs to be trained with all data in the corresponding
area if a part of the area is changed. Therefore, maintaining the map created
by existing methods is costly. Moreover, these methods require tremendously
large memory budgets to save data and large computational resources to model
the environment if one builds an earth-scale NeRF-based map. Hence, we
approach these problems by adopting a federated learning scheme.

###  II-B Federated Learning

Federated learning (FL) [12] is a data-decentralizing training process for
machine learning models. It distributes the model to clients such as mobile
devices or organizations, trains the model on clients using their own data,
and aggregates the models on the server. By repeating the above process, FL
collaboratively trains a global model under the orchestration of a central
server. FL has some advantages: preserving privacy since it does not collect
user data on the server and leveraging the computational resources of many
clients (millions or more in our scenario). Because of these advantages, FL
has been studied for training machine learning models in various tasks, such
as image classification [12, 22], image segmentation [23, 24], and self-
supervised learning [25, 26].

Many FL approaches update models using a synchronous protocol, which needs to
wait for all updates on clients before aggregation and delays training. The
existence of lagging devices (i.e., stragglers, stale workers) is inevitable
due to device heterogeneity and network unreliability.

To mitigate this problem, asynchronous federated learning (AFL) has been
proposed [27], which aggregates local models asynchronously. Xie et al. [27]
aggregates models by exponential moving averaging. Chen et al. [28] proposed a
method to aggregate the model with a decay coefficient to balance the previous
and current gradients on clients. Since AFL improves the communication
efficiency of FL, our pipeline also adopts the AFL strategy.

##  III Federated Learning for NeRF

We present the overall pipeline of our method in Fig. 1, which consists of
three steps, training local models, global pose alignment, and an aggregation
step. We consider the AFL setting; namely, the server updates the global model
as soon as it receives the local model.

We first describe the assumptions of this study in Sec. III-A. Then, in Secs.
III-B and III-C, we describe the training pipeline for the local models and
the aggregation step. Finally, we describe the global pose alignment in Sec.
III-D.

###  III-A Assumptions

We assume each client knows the relative camera poses between collected
images. This is a realistic assumption because relative poses can be obtained
from some sensors, SLAM, or SfM. In the experiments, we use the camera poses
given by Mill19 [11] that are estimated by PixSfM [29]. Note that if the
obtained poses are noisy, we would be able to correct them by using the
methods that simultaneously optimize poses and the model (e.g., BARF [30] and
NoPe-NeRF [31]).

We also assume that the global pose of each client is known but noisy. Since
the goal of this study is to train the large-scale NeRF model as a map, there
is no map to localize the position111Once the map (i.e., the global model) is
initialized, we can use it to localize each client in the same manner as that
for iNeRF [2].. In this situation, we need to use sensors such as GPS and IMU
to obtain the global pose, but the pose obtained from the sensors basically
contains noise.

In addition, we assume that images collected by clients are captured under the
same lighting and have no dynamic and/or transient object. This assumption is
unreasonable, particularly for outdoor scenes; we will study more realistic
scenarios in future work.

###  III-B Training Local Models

For simplifying the pipeline, we assume that client models are trained from
scratch because NeRF basically “overfits” training data and it does not care
how the model is initialized. Of course, we can consider the standard
pipeline; namely, clients download the global model and then train it with the
local data, which may make convergence faster but increase communication
costs.

The local model consists of two models: the density model $f_{\sigma}$ and the
color model $f_{c}$. Let
$\mathbf{r}=\\{\mathbf{r}_{i}\in\mathbb{R}^{3}|\mathbf{r}_{i}=\mathbf{o}+s_{i}\mathbf{d},\
s_{i}\in\mathbb{R}\\}_{i}$ be a set of sampled points along the ray passing
through a pixel, where $\mathbf{o}$ and $\mathbf{d}$ are a camera origin and a
ray direction. Following [11], we sample 512 points per ray for rendering
images in our experiments. The density at $\mathbf{r}_{i}$ is computed as
$\sigma_{i}=\phi\circ f_{\sigma}(\mathbf{r}_{i})$, where $\phi$ denotes the
softplus function. For representing RGB, we use the spherical harmonics (SH)
as in PlenOctrees [32], which enables caching outputs of the model on the grid
voxels in the aggregation step described in Sec. III-C. Specifically, $f_{c}$
outputs SH coefficients $\mathbf{k}(\mathbf{r}_{i})=f_{c}(\mathbf{r}_{i})$,
where
$\mathbf{k}(\mathbf{r}_{i})=\\{k_{\ell}^{m}\in\mathbb{R}^{3}\\}_{\ell:0\leq\ell\leq\ell_{\max}}^{m:-\ell\leq
m\leq\ell}$; $\ell$ and $m$ are a degree and an order of the SH function,
respectively. Each $k_{\ell}^{m}$ is a set of three coefficients corresponding
to RGB. Then, the view-dependent color at $r_{i}$ is computed by

$\displaystyle\mathbf{c}_{i}=S\left(\sum_{\ell=0}^{\ell_{\max}}\sum_{m=-\ell}^{\ell}k_{\ell}^{m}Y_{\ell}^{m}(\mathbf{d})\right),$
(1)

where $S(\cdot)$ is the sigmoid function and
$Y_{l}^{m}(\mathbf{d}):\mathbb{S}^{2}\rightarrow\mathbb{R}$ is the SH function
at the viewing angle $\mathbf{d}$. Following [33], we use spherical harmonics
of degree 2, which has 27 harmonic coefficients.

The rendering and training procedure is the same as in the original NeRF [1].
Specifically, the pixel color corresponding to the ray $\mathbf{r}$ is
computed as follows:

$\displaystyle\hat{\mathbf{C}}(\mathbf{r})=\sum_{i}\exp(-\sum_{j=1}^{i-1}\sigma_{j}\delta_{j})(1-\exp(-\sigma_{i}\delta_{i}))c_{i},$
(2)

where $\delta_{i}=t_{i+1}-t_{i}$ is the distance between adjacent samples on
the ray. Then, the NeRF model is trained by minimizing the following loss:

$\displaystyle\mathcal{L}=\underset{\mathbf{r}\in\mathcal{R}}{\mathbb{E}}\left\|\hat{C}(\mathbf{r})-C(\mathbf{r})\right\|^{2}_{2},$
(3)

where $C(\mathbf{r})\in\mathbb{R}^{3}$ denotes the ground-truth pixel color
corresponding to $\mathbf{r}$, and $\mathcal{R}$ is the set of rays in the
training data. Note that we do not consider the coarse-to-fine rendering
pipeline [1] in this study, but it can be made available with a larger memory
budget and more computational resources.

###  III-C Updating Global Model

The aggregation step in federated learning often averages the local models.
However, this averaging strategy cannot work for NeRF due to its properties;
NeRF represents only the scene in the training data and assigns random color
and density to the outside of the scene. In addition, the 3D position
$(x,y,z)$ may indicate a different location for each client because clients’
coordinates may not be aligned due to the sensor noise. For these reasons, the
simple averaging strategy will degrade the rendering quality of the global
model.

To avoid the aforementioned problem, we tailor the aggregation step for NeRF.
We represent the global model as the voxel grid, as in Plenoxels [32]. We
cache the output of the local models on the voxel grid, as in FastNeRF [34],
and then aggregate the cached voxel grid and the global model, as shown in
Fig. 2. Since we only cache the outputs in the area modeled by the local model
and aggregate them, this procedure does not affect the output outside the
area. In addition, the voxel grid representation allows us to align the global
pose of the clients before aggregation (Sec. III-D).

Figure 2: Overview of the proposed aggregation step. We first cache outputs of
the local model on the grid voxels, $\tilde{V}^{i}_{\sigma}$ and
$\tilde{V}^{i}_{c}$, and then add them to the global voxel grid, $V_{\sigma}$
and $V_{c}$.

Let $\tilde{V}_{\sigma}^{n}\in\mathbb{R}^{X\times Y\times Z\times 1}$ and
$\tilde{V}_{c}^{n}\in\mathbb{R}^{X\times Y\times Z\times 27}$ be a cached
density grid voxel and a cached SH coefficient grid voxel of the $n$-th
client, where $X$, $Y$, and $Z$ are the voxel resolution and $27$ in $V_{c}$
corresponds to the number of the coefficients. Also, we define the global
model’s voxel grids as $V_{\sigma}\in\mathbb{R}^{X_{g}\times Y_{g}\times
Z_{g}\times 1}$ and $V_{c}\in\mathbb{R}^{X_{g}\times Y_{g}\times Z_{g}\times
27}$, where $X_{g}$, $Y_{g}$, and $Z_{g}$ are the global model’s voxel
resolution. Note that the global model is defined on the global coordinate and
the local voxel grid is defined on the local coordinate, respectively; hence,
the position on the local coordinate, $P\in\mathbb{R}^{3}$, is mapped to the
global coordinate using the global pose, as in $P_{g}=R_{n}P+t_{n}$, where
$R_{n}\in\mathrm{SO}(3)$ and $t_{n}\in\mathbb{R}^{3}$ denote the relative
rotation and translation between the $n$-th client’s local and global
coordinates, respectively, which are obtained from sensors such as GPS and
IMU. Then, we update the density grid voxel of the global model with
exponential moving averaging in the asynchronous federated learning manner as

$\displaystyle V_{\sigma}[P_{g}]\leftarrow\eta
V_{\sigma}[P_{g}]+(1-\eta)\tilde{V}_{\sigma}^{n}[P],$ (4)

where $V_{\sigma}[P]$ denotes an element of $V_{\sigma}$ at $P$; $\eta$
denotes a mixing value that is set to 0.9 in the experiments. The SH
coefficient grid is also updated in the same manner.

The rendering process of the global model is the same as Plenoxels [33];
namely, the density and SH coefficient are sampled from the voxel grids by
trilinear interpolation and then the RGB color is computed following eq. (1)
with the sampled coefficients. Finally, pixel colors are rendered by eq. (2)
with the sampled density and the color. We define the rendered pixel color
through the global model as $\hat{C}_{g}(\mathbf{r})$.

###  III-D Global Pose Alignment

As we mentioned in Sec. III-A, the global pose of the clients contains noises.
Thus, we need to correct it before aggregation because inconsistency between
the local and global models occurs in eq. (4) if the global pose is incorrect.

To align the pose, we optimize it to minimize the difference between the
reference and target images, which are rendered by the local and global
models, respectively. Let $t^{\ast}\in\mathbb{R}^{3}$ and
$\\{R_{j}^{\ast}\in\mathrm{SO}(3)\\}_{j}$ be a translation vector and rotation
matrices for the target views, which are obtained from the area modeled by
both the global and local models. Then, we align the client’s global pose by
solving the following minimization problem:

$\displaystyle(\hat{t},\ \hat{R})=\underset{\\{t,R\\}}{\arg\min}\sum_{j}$
$\displaystyle\underset{\begin{subarray}{c}\mathbf{r}_{t^{\ast},R_{j}^{\ast}},\\\
\mathbf{r}_{t^{\ast}+t,RR_{j}^{\ast}}\end{subarray}}{\mathbb{E}}\lambda|\hat{C}_{g}(\mathbf{r}_{t^{\ast},R_{j}^{\ast}})-\hat{C}(\mathbf{r}_{t^{\ast}+t,RR_{j}^{\ast}})|$
$\displaystyle+(1-\lambda)$
$\displaystyle|\hat{D}_{g}(\mathbf{r}_{t^{\ast},R_{j}^{\ast}})-\hat{D}(\mathbf{r}_{t^{\ast}+t,RR_{j}^{\ast}})|,$
(5)

where $\mathbf{r}_{t,R}$ denotes the ray depending on the camera pose $(t,R)$,
and $\hat{D}_{g}$ and $\hat{D}$ denote the rendered depth through the global
and local models, respectively, which are computed by replacing color values,
$c_{i}$, in eq. (2) with the sampled point position, $s_{i}$, as in [3];
$\lambda\in[0,1]$ is a weight parameter. After optimization, we align the
global pose of the local model based on $\hat{t}$ and $\hat{R}$. Since the
rendering process can be differentiable with respect to $t$ and $R$, we
optimize them using the gradient method, as in the existing NeRF-based
localization methods [2, 7, 13]. In our experiments, we used the same
optimization pipeline as that in Lin et al. [13] to optimize the pose, which
is a Monte Carlo-based method. Note that, unlike existing methods [2, 7, 13],
we leverage multiple views and rendered depth to make optimization stable. We
assess the effectiveness of the multiple target views and the depth loss in
the experiment section.

##  IV Experiments

###  IV-A Experimental Setup

To simulate our method, we use the Mill19 dataset [11] that includes two
scenes, building and rubble, which have thousands of HD images collected from
drone footage over 100,000m2 of terrain close to an industrial complex. We
first generate the data owned by clients in the following procedure: first, we
randomly select one image from training data and compute the Euclidean
distance between the camera position of the selected image and the camera
positions of the other images, and then we collect $k$-nearest neighbors as
the data owned by a client222In our envisioned scenario, the clients are
basically drones and vehicles, and the collected data should be sequential
frames. Thus, we generate the client data from images that are physically
close to each other.. We repeat this procedure for the number of clients, $N$,
that is set to 100 in the experiments. In this experiment, we randomly set $k$
for each client in a range of 100 to 200. After training $N$ local models, we
update the global model with the pose alignment and evaluate it with the
validation data of Mill19. We set the voxel size for caching the output of the
local model to 0.25 m. Following [11], we use the appearance vector to model
the color of each image and optimize it in test time. The experiments were
conducted using the NVIDIA V100 GPU.

We use InstantNGP [35] as the local model because it converges faster and has
fewer FLOPs than the original NeRF architecture, which is suitable if the
computational resources of the clients are limited. Each local model is
trained for one epoch333One epoch corresponds to the iterations of #pixels /
#batchsize. with a batch size of 8192. We optimize the local models with Adam
[36], whose hyperparameters are the same as those of InstantNGP[35], except
for the learning rate; it is set to 5e-3 only for the hash encoding and 5e-4
for the other parameters.

###  IV-B Effect of Pose Alignment

To evaluate our global pose alignment framework, we randomly select one local
model as the global model. We also randomly select another model as a local
model from the models whose training data partially overlap the area modeled
by the global model. After selecting the models, we compute a center position
of the overlapped area as a translation vector, $t^{\ast}\in\mathbb{R}^{3}$,
and obtain rotation matrices, $\\{R^{\ast}_{j}\in\mathrm{SO}(3)\\}_{j}$, from
camera poses of the training data close to $t^{\ast}$. We render the images at
$\\{(t^{\ast},R^{\ast}_{j})\\}_{j}$ as the target values through the global
model. Finally, we randomly sample a translation vector, $t\in\mathbb{R}^{3}$,
and a rotation matrix, $R\in\mathrm{SO}(3)$, which can be regarded as the
sensor noises, and optimize them by solving eq. (5). Thus, the optimal values
will be $t=(0,0,0)$ and $R=I$, and we report the gap from them. Note that
since we use the appearance vector, as described in IV-A, we also optimize it
in addition to the pose. For optimization, we use the Adam optimizer [36] with
a batch size of 4096. We set an initial learning rate to 5e-4 and decay it to
5e-5.

As an ablation study, we show the rotation errors and translation errors with
various $\lambda$ in eq. (5) and a various number of target views in Tab. I.
We randomly sample noises in a range of [-20 m, 20 m] and [-20∘, 20∘].

|
---|---
Figure 3: The alignment results for various initial errors. The gray dots
denote failure cases that increase the initial errors. TABLE I: Rotation and
translation errors with various $\lambda$ and a various number of target
views. We report the mean and standard deviation over 10 trials. We use four
target views on the left-hand side of the table and set $\lambda$ to 0.75 on
the right-hand side of the table.

$\lambda$ | Rot. (deg) | Trans. (m)
---|---|---
0.0 | 24.3$\pm$15.5 | 32.3$\pm$18.7
0.25 | 19.9$\pm$16.7 | 17.7$\pm$11.2
0.5 | 12.9$\pm$16.3 | 9.62$\pm$11.7
0.75 | 0.53$\pm$0.96 | 0.19$\pm$0.33
1.0 | 0.92$\pm$1.63 | 0.43$\pm$0.78

#views | Rot. (deg) | Trans. (m)
---|---|---
1 | 0.34$\pm$0.24 | 0.97$\pm$0.68
2 | 0.33$\pm$0.24 | 0.49$\pm$1.18
4 | 0.53$\pm$0.96 | 0.19$\pm$0.33
8 | 0.12$\pm$0.09 | 0.02$\pm$0.01

As presented on the left-hand side of Tab. I, RGB information is critical for
the pose alignment, but the depth information helps marginally with error
reduction. The multiple target views also contribute to reducing errors, as
depicted on the right-hand side of Tab. I.

We also show the evaluation results of our alignment framework with various
magnitudes of initial errors in Fig. 3. We randomly sample the pose noises and
then align them. We plot the results for 100 trials in Fig. 3. Note that a few
trials with various translation errors have failed, and the failure rate is
5%. Our method can align the translation error up to 75 m, which is comparable
to a typical GPS error. Also, it can correct the rotation error that is larger
than errors in a typical IMU. A few plots in Fig. 3 indicate relatively large
errors after alignment, but we can decrease the errors by increasing the
number of particles and resampling rounds in the Monte Carlo-based
optimization; in fact, our method can reduce errors even when the initial
errors are larger than that of the failure cases. In our method, the increase
in computation time is acceptable because alignment will be performed offline
and does not require real-time processing.

###  IV-C Comparison with Other Training Protocols

To evaluate the rendered image quality of our method, we compare the proposed
training pipeline with two data-centralized training protocols, baseline and
distributed training. The baseline indicates training one model with all data,
which corresponds to the ordinary NeRF training pipeline. The distributed
training indicates the Mega-NeRF’s training pipeline [11]; namely, the scene
is divided into grids, and then models are trained with data corresponding to
each grid. We divide the scene into 4$\times$4 grids for the distributed
training; that is, 16 models are trained, and the number of data for training
each model is approximately 150, which is the same as the expected number of
data used to train each model of our pipeline. For a fair comparison, we
evaluate these training pipelines with the same model and the same
hyperparameters as ours, except for the model size in the baseline. Since the
baseline models the scene using one model, while other training protocols use
multiple models, we double the grid size of the hash encoding for the baseline
to be fair in terms of the model capacity. Note that we assume that the global
position is correctly aligned by our position alignment algorithm in this
experiment.

We show the evaluation results in Tab. II. Note that since the models can be
trained in parallel, the longest training time out of all local models is
reported as the training time in the distributed learning and ours.

TABLE II: Performance for different training protocols on building and rubble scenes of Mill19. | Building | Rubble | Rendering Speed
---|---|---|---
| PSNR | Training Time (h) | PSNR | Training Time (h) | (Pixel/Second)
Baseline | 22.29 | 32.4 | 24.93 | 28.0 | 80.4K
Distributed Training | 18.93 | 0.75 | 21.24 | 0.67 | 69.2K
Ours | 18.06 | 0.97 | 23.01 | 0.95 | 342.6K
|  |
---|---|---
Figure 4: From left to right, the ground-truth image of the test view, the image rendered by the local model trained with a sufficiently large number of viewpoints around the test view, and the image rendered by the local model trained with a relatively small number of viewpoints. |
---|---
Figure 5: An image rendered by the client model (left) and an image rendered
by the cached voxel grids (right). The lattice pattern composed of high-
frequency components is broken.

The training time of distributed training and ours is much shorter than that
of the baseline because of the distributed training protocol. Since the scale
of Mill19 is relatively small compared to our envisioned scenario, the
training time of the baseline is acceptable. However, if we scale it to an
earth-scale, it is impossible to train a model with the standard training
pipeline. In addition to the reduction in training time, our method can
alleviate bandwidth consumption because the size of the local model is 0.1 GB
while that of the local data is up to 1 GB. The rendering speed for the
distributed training is slower than the others because the distributed
training combines the outputs of the multiple models to render one image.
Rendering by ours is the fastest because it only samples cached outputs from
the voxel grids for rendering, unlike the others that consist of hash encoding
and MLP.

The PSNR for our method is worse than that for the baseline for two reasons:
each client trains a local model with a relatively small number of data, while
the baseline trains the model with a sufficiently large number of data.
Basically, NeRF requires a sufficiently large number of viewpoints to
correctly represent the scene in 3D. However, some clients do not satisfy this
requirement around the test view and the local models trained by such clients
cannot represent the scene correctly, as shown in Fig. 4. Consequently, such
local models degrade the global model performance. This disadvantage is also
found in distributed training; in fact, PSNR for distributed training is worse
than that for the baseline. The other reason is the quantization error in the
caching process. We cache the outputs of the local models on the voxel grid in
the aggregation step, and such a quantization operation approximates the
continuous function by the piece-wise linear function. Thus, there are
approximation errors, which cause the gap between PSNR for distributed
training and ours, especially for the building scene that includes high-
frequency components, as shown in Fig. 5. This error can be reduced by
increasing the cached grid size, but it requires larger memory budgets.
Improving local model training and alleviating the quantization errors are
possible directions for future work.

The PSNR for ours on the rubble scene is better than that for distributed
training. The quantization errors in the rubble scene are smaller than those
in the building scene because images in the rubble scenes are basically
composed of low-frequency components. Therefore, in scenes where the effect of
the errors is small, our training protocol would be better than distributed
training in this experimental setting.

##  V Limitations and Future Work

As we mentioned in Sec. III-A, this work does not consider dynamic and/or
transient objects and lighting changes. It may raise privacy concerns because
if a person in the scene is modeled with the lighting, the model may show
where and when the person was doing what. However, there are some approaches
to modeling the scene by ignoring such objects and lighting effects (e.g.,
NeRF-W [37] and Block-NeRF [10]); by exploiting the advantage of such methods,
the proposed pipeline will leverage the privacy-preserving aspect of the
federated learning. Therefore, we believe that considering the transient
objects and lighting changes in the federated learning pipeline is an
important research direction.

Moreover, as we mentioned in Sec. IV-C, the quality of the rendered images is
worse than the images rendered by the baseline. Fortunately, there are many
studies on training NeRF with limited viewpoints [38, 39, 40]. We believe that
we can improve the performance of the local models by leveraging the training
protocol of such methods; consequently, the global model can be improved.
Another approach is to consider the number of viewpoints for aggregating
models. Specifically, by adjusting the mixing coefficient in the exponential
moving average according to the number of viewpoints, we may be able to ignore
the low-quality outputs of the local model in the aggregation step.

##  VI Conclusion

In this study, we proposed the federated learning pipeline for large-scale
scene modeling with NeRF. We designed the aggregation step for training NeRF
in a federated learning manner: The global model is defined as the voxel grids
and the local models are aggregated by caching the outputs of the local model
on the voxel grids, which allows us to update the global model locally. In
addition, we proposed the global pose alignment step to alleviate the sensor
noise in the global pose of each client. We assessed the proposed method on
the Mill19 dataset containing thousands of images collected from drone footage
over 100,000m2 of terrain near an industrial complex and verified its
effectiveness.

We believe our work opens new avenues for large-scale scene modeling using
NeRF and is an important step towards collaboratively training for NeRF in a
federated learning manner.

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# Kalb-Ramond field induced cosmological bounce in generalized teleparallel
gravity

Krishnanand K. Nair<EMAIL_ADDRESS>Mathew Thomas Arun
<EMAIL_ADDRESS>School of Physics, Indian Institute of Science
Education and Research, Thiruvananthapuram 695551, India

###### Abstract

One of the important open questions in high-energy physics is to understand
the lack of evidence of Kalb-Ramond (KR) field, in particular in the present
day cosmology. In this paper we aim to address this issue by showing that a
bounce scenario in the evolution of the Universe strongly advocates their
elusiveness, even if their energy density was very large to start with. We
consider the Kalb-Ramond field and its effects in the context of generalized
teleparallel gravity in (3+1) dimensions. Teleparallel gravity is a
description of gravitation in which the tetrads are the dynamical degrees of
freedom, and the torsion arising from fields with spin are accommodated
naturally as field strength tensors. In order to describe the coupling
prescription, we address the correct generalization of the Fock-Ivanenko
derivative operator for an n-form tensor field. By varying with respect to the
tetrads, this rank-2 field is shown to source the teleparallel equivalent of
Einstein’s equations. We study the possibility of reproducing two well-known
cosmological bounce scenarios, namely, symmetric bounce and matter bounce in
four-dimensional spacetime with with the Friedmann-Lemaître-Robertson-Walker
metric and observe that the solution requires the KR field energy density to
be localized near the bounce. The crucial result in our work is that this
feature also naturally explains the lack of cosmological evidence of the
rank-2 field in the present day Universe for the matter-bounce scenario. Thus,
among the bouncing cosmologies, latter is favored over the former.

††preprint: APS/123-QED

##  I Introduction

The Kalb-Ramond (KR) field has been understood to be essential to correctly
reproduce the low-energy string effective action [2, 1]. These antisymmetric
tensor fields constitute the field content of all superstring models and must
have significant imprint during the primordial epoch of the evolution of the
Universe. Apart from string inspired models, the KR field arise in higher-
dimensional theories, that aim to unify gravity and electromagnetism. Still,
the KR field is not yet detected in any of the experiments.[3].

In this paper, we aim to address the KR field in a generalized teleparallel
setup and show that a natural explanation for its absence in the present-day
Universe is realized in bouncing cosmology. Apart from the benefit that, here,
gravity is understood as a gauge theory of the translation group [4], along
with a conserved energy-momentum gauge current, this description also
naturally accommodates the effects of fields with a spin quantum number
through tetrads, which form the dynamical variables, instead of the metric.
Generalized teleparallel gravity also naturally accommodates an explanation to
cosmological phenomenon like the late-time acceleration of the Universe [5, 6,
7, 8, 9, 10, 11, 12].

Here, we consider a generalized teleparallel gravity setup in $(3+1)$
dimensions appended by an action of the Kalb-Ramond field. With the
appropriate generalization of the Fock-Ivanenko derivative operator for the KR
field, we compute the equivalent of Einstein’s equations by varying the action
with respect to the tetrads. This gives the equivalent energy-momentum tensor
of the antisymmetric field. With the setup in place we now study the
requirement to achieve bouncing cosmology.

Models with bounces [13, 14, 15] provide an elegant solution to the initial
singularity in the big bang paradigm and, in some instances, could generate a
scale-invariant power-law spectrum [16] as well. Even though there have been
immense efforts carried out in modified gravity theories with higher-order
corrections[17, 18] and in braneworld scenarios[19, 20], it is interesting to
understand these phenomena in the teleparallel equivalent of General
Relativity (TEGR) [4]. In this paper, we explicitly compute the energy
spectrum of the tensor field and the appropriate teleparallel gravity model
for symmetric and matter bounce scenarios. We show that the energy and
pressure densities of the tensor field are indeed localized at $t=0$, which
acts as the source for the bounce. We find that, in the case of symmetric
bounce, a significant fraction of the energy density of KR field remains to
the present day Universe, whereas, in the context of matter bounce, the energy
density of the KR field drastically decreases from $3$ M${}_{Pl}{}^{4}$ at the
bounce, to $\sim 0$ at $t=t_{0}$. Hence, we show that the null results from
searches for the KR field strongly suggests matter bounce for the cosmic
evolution.
The paper is categorized as follows. We start with a brief review of TEGR
formalism in Sec.(II) and introduce Kalb-Ramond fields as a source of torsion.
In Sec.(III), we explain the minimal coupling prescription and develop the
Fock-Ivanenko operator for the Kalb-Ramond field. As an application to
cosmology, in Sec.(IV), we compute the energy density and pressure density of
the KR fields in the generalized teleparallel setup that will lead to correct
expansion coefficients in symmetric and matter bounce scenarios. Finally, in
Sec.(V) we summarise our results.

##  II Teleparallel Equivalent of General Relativity

In Einstein’s General Relativity (GR), the affine connection is taken to be
torsionless and satisfies the metricity condition,

$\nabla_{\mu}g_{\nu\rho}=0$ (1)

where $\nabla_{\mu}$ is the covariant derivative with the Levi-Civita
$\tilde{\Gamma}^{\mu}_{\nu\rho}$ playing the role of affine connection.
However, in teleparallel gravity (TG) , the Levi-Civita affine connection is
replaced by the Weitzenb$\ddot{\text{o}}$ck connection, which is torsionfull
but curvatureless and satisfies the metricity condition Eq.(1).

Although teleparallel gravity is an alternative to General Relativity, they
are conceptually distinct [21]. In TEGR, the spacetime metric is constructed
out of tetrads ($h^{a}{}_{\mu}$), which are the dynamical degrees of freedom,
as,

$g_{\mu\nu}=\eta_{ab}h^{a}{}_{\mu}h^{b}{}_{\mu}\ ,$ (2)

where $\eta_{ab}$ is the Minkowski metric of the tangent space. The tetrads
$h^{a}{}_{\mu}$ could be written in terms of flat-space tetrads
($e^{a}{}_{\mu}$ = $\partial_{\mu}x^{a}$) as [21]

$h^{a}{}_{\mu}=e^{a}{}_{\mu}+\omega^{a}{}_{b\mu}x^{b}+A^{a}{}_{\mu}\ .$ (3)

The flat-space tetrads satisfy the relation
$\eta_{\mu\nu}=\eta_{ab}e^{a}_{\mu}e^{b}_{\nu}$, where $\eta_{\mu\nu}$ is the
metric of Minkowski spacetime and the spin connection ($\omega^{a}{}_{b\mu}$)
is given by
$\omega^{a}{}_{b\mu}=\Lambda^{a}{}_{c}\partial_{\mu}\Lambda_{b}{}^{c}$, where
$\Lambda$ is the Lorentz matrix and the translational connection on the
tangent space is denoted by $A^{a}{}_{\mu}$. Note that, in this paper, we will
be referring Greek indices ($\mu$, $\nu$) to the spacetime manifold and the
Latin indices (a, b) to the local Minkowski tangent space. We also assume the
signature of $\eta_{ab}$ as diag $(-+++)$.
Now, the Weitzenb$\ddot{\text{o}}$ck connection $\Gamma^{\rho}{}_{\mu\nu}$ can
be written as [22]

$\Gamma^{\rho}{}_{\mu\nu}=h_{a}{}^{\rho}\partial_{\mu}h^{a}{}_{\nu}+h_{a}{}^{\rho}\omega^{a}{}_{b\mu}h^{b}_{\nu}\
.$ (4)

Since we are interested in the evolution of the Universe, we stick to a
particular choice of tetrads given in Eq.eq39 corresponding to the flat
Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime for which the spin
connection $\omega^{a}{}_{b\mu}=0$ [23, 24, 25, 26, 27]. Given this solution,
one can easily show that the Weitzenb$\ddot{\text{o}}$ck covariant derivative
of the tetrads vanish identically, thus satisfying the metricity condition,

$\nabla_{\mu}h_{\nu}{}^{A}\equiv\partial_{\mu}h^{A}{}_{\nu}-\Gamma_{\mu\nu}^{\rho}h_{\rho}{}^{A}=0\
,$ (5)

where $\nabla_{\mu}$ represents the covariant derivative constructed with the
Weitzenb$\ddot{\text{o}}$ck connection.
Now, the torsion tensor could be constructed from the
Weitzenb$\ddot{\text{o}}$ck connection as given below,

$T^{\rho}{}_{\mu\nu}=\Gamma^{\rho}{}_{\mu\nu}-\Gamma^{\rho}{}_{\nu\mu}\ .$ (6)

Using Eq.(3) and Eq.(4), it is straightforward to see that the torsion tensor
acts as the field strength of the translation potential $A^{a}{}_{\mu}$, for
spin connection $\omega^{a}{}_{b\mu}=0$ [21]

$T^{a}{}_{\mu\nu}=h^{a}{}_{\rho}T^{\rho}{}_{\mu\nu}=\partial_{\mu}A^{a}{}_{\nu}-\partial_{\nu}A^{a}{}_{\mu}\
.$ (7)

The Weitzenb$\ddot{\text{o}}$ck connection in teleparallel gravity
$\Gamma^{\rho}{}_{\mu\nu}$ and the Levi-Civita connections
$\tilde{\Gamma}^{\rho}{}_{\mu\nu}$ in GR are then mathematically related as

$\Gamma^{\rho}{}_{\mu\nu}-K^{\rho}{}_{\mu\nu}\equiv\tilde{\Gamma}^{\rho}{}_{\mu\nu}\
,$ (8)

where $K^{\rho}{}_{\mu\nu}$ is the contorsion tensor given by

$K^{\rho}{}_{\mu\nu}=\frac{1}{2}\left(T_{\mu}{}^{\rho}{}_{\nu}+T_{\nu}{}^{\rho}{}_{\mu}-T^{\rho}{}_{\mu\nu}\right)\
.$ (9)

Note that, we use overtilde to represent quantities calculated using the Levi-
Civita connection in GR to distinguish it from teleparallel gravity in this
paper. It is straightforward to show that the curvature of the
Weitzenb$\ddot{\text{o}}$ck connection also vanishes.

${R}^{\rho}{}_{\lambda\mu\nu}(\Gamma)=0\ .$ (10)

The dual torsion tensor is defined as

$S^{\rho\mu\nu}=\frac{1}{2}\left[K^{\mu\nu\rho}-g^{\rho\nu}T^{\lambda\mu}{}_{\lambda}+g^{\rho\mu}T^{\lambda\nu}{}_{\lambda}\right]\
.$ (11)

Finally, we define a quadratic function of torsion called the torsion scalar T
given by,

$T=T_{\rho\mu\nu}S^{\rho\mu\nu}=T^{\rho}{}_{\mu\nu}T_{\rho}{}^{\mu\nu}/2+T^{\rho}{}_{\mu\nu}T^{\nu\mu}{}_{\rho}-2T^{\rho}{}_{\mu\rho}T^{\nu\mu}{}_{\nu}\
.$ (12)

The gravitational Lagrangian using the torsion scalar can be written as

$\mathcal{L}_{G}=-\frac{h}{16\pi G}T\ ,$ (13)

where $h=det(h^{a}_{\mu})=\sqrt{-g}$

Using Eq.(8) in the above action and reformulating the above Lagrangian in
terms of Levi-Civita connection, we can obtain the mathematical relation
between the torsion scalar $T$ in teleparallel gravity and the Ricci scalar
$\tilde{R}$ in GR

$T\equiv-\tilde{R}+B\ ,$ (14)

where $\tilde{R}$ is the Ricci scalar and
$B=2\tilde{\nabla}_{\mu}(T^{\nu}{}_{\nu}{}^{\mu})$ is a total divergence term.
Thus this action is equivalent to the Einstein-Hilbert action, which gives
Einstein’s field equations of gravity [28].

##  III Coupling prescription using Fock–Ivanenko derivative operator in the
teleparallel geometry

In Minkowski space, the dynamics of the Kalb-Ramond field is described by the
Lagrangian [29]

$\mathcal{L}_{KR}=-H_{abc}H^{abc}\ ,$ (15)

where

$H_{abc}=\partial_{a}B_{bc}+\partial_{b}B_{ca}+\partial_{c}B_{ab}\ ,$ (16)

is the field strength of the Kalb-Ramond field $B_{ab}$, which is a rank-2
antisymmetric tensor field.
On varying the action with respect $B_{ab}$, we get the field equations

$\partial_{a}H^{abc}=0\ ,$ (17)

along with the Bianchi identity

$\partial_{[a}H_{bcd]}=0\ ,$ (18)

For the Lorentz gauge $\partial_{a}B^{ab}=0$, the field equation Eq.(17)
becomes

$\partial_{c}\partial^{c}B^{ab}=0\ .$ (19)

However in teleparallel gravity, the existence of torsion destroys the gauge
invariance of the theory when the KR field is used as the source of the
equation of motion of the field. If we assume the coupling prescription given
by

$\displaystyle\eta^{ab}\rightarrow g^{\mu\nu}=\eta^{ab}h_{a}^{\mu}h_{b}^{\nu}$
(20)
$\displaystyle\partial_{a}\rightarrow\nabla_{\mu}\equiv\partial_{\mu}-\Gamma_{\mu}\
,$

where $\Gamma$ is the Weitzenb$\ddot{\text{o}}$ck connection, the KR field
strength takes the form,

$\displaystyle H_{\mu\nu\rho}$
$\displaystyle={\nabla}_{\mu}B_{\nu\rho}+{\nabla}_{\rho}B_{\mu\nu}+{\nabla}_{\nu}B_{\rho\mu}$
(21)
$\displaystyle=3\partial_{[\mu}B_{\nu\rho]}+3T^{\sigma}{}_{[\mu\nu}B_{\rho]\sigma}\
.$

The last term in Eq.(21) indicates the nonminimal coupling of torsion with the
KR field in teleparallel geometry and thus Eq.(21) is not invariant under U(1)
gauge transformation.

In order to keep the transformation gauge invariant, in the framework of
teleparallel geometry, one needs to use the minimal coupling prescription
[30],

$\displaystyle\eta^{ab}\rightarrow g^{\mu\nu}=\eta^{ab}h_{a}^{\mu}h_{b}^{\nu}$
(22)
$\displaystyle\partial_{a}\rightarrow\mathcal{D}_{\mu}=\partial_{\mu}-\frac{i}{2}\Omega^{ab}{}_{\mu}J_{ab}\
,$

where $\mathcal{D}_{\mu}$ is the Fock–Ivanenko derivative operator [31], which
acts only on the local Lorentz indices. Here, $\Omega^{ab}{}_{\mu}$ is given
by

$\Omega^{ab}{}_{\mu}=-h_{\rho}^{a}K^{\rho\nu}{}_{\mu}h_{\nu}^{b}\ ,$ (23)

and $J_{ab}$ is the generator in the appropriate representation of the Lorentz
group. For instance, $J_{ab}$ acting on any $n$-form field could be written as

$\displaystyle J_{ab}{}(B^{i_{1}i_{2}...i_{n}})=$ $\displaystyle
i\big{(}\delta^{i_{1}}_{a}\eta_{bc}-\delta^{i_{1}}_{b}\eta_{ac}\big{)}B^{ci_{2}...i_{n}}$
(24)
$\displaystyle+i\big{(}\delta^{i_{2}}_{a}\eta_{bc}-\delta^{i_{2}}_{b}\eta_{ac}\big{)}B^{i_{1}c...i_{n}}$
$\displaystyle+....+i\big{(}\delta^{i_{n}}_{a}\eta_{bc}-\delta^{i_{1}}_{b}\eta_{ac}\big{)}B^{i_{1}i_{2}...c}\
.$

It is also important to note that FIDO in teleparallel gravity is equivalent
to the Levi-Civita covariant derivative in the Einstein GR in the absence of
contorsion, as shown in the Appendix (A). More importantly, with this coupling
prescription, torsion does not violate the gauge symmetry of Kalb-Ramond
theory.

Using Eq.(23) and Eq.(24), we get the Fock–Ivanenko derivative acting on
$B^{ab}$ as

$\displaystyle\mathcal{D}_{\mu}B^{ab}$
$\displaystyle=\partial_{\mu}B^{ab}-\frac{i}{2}\Omega^{cd}{}_{\mu}\big{(}i(\delta^{a}_{c}\eta_{dg}-\delta^{a}_{d}\eta_{cg})\big{)}B^{gb}$
(25)
$\displaystyle-\frac{i}{2}\Omega^{cd}{}_{\mu}\big{(}i(\delta^{b}_{c}\eta_{dg}-\delta^{b}_{d}\eta_{cg})\big{)}B^{ag}$
$\displaystyle=\partial_{\mu}B^{ab}-K^{\rho}{}_{\nu\mu}h_{d}{}^{\nu}h^{[a}{}_{\rho}B^{b]d}\
.$

Any spacetime tensor $B^{\mu\nu}$ can be transformed to a Lorentz tensor
$B^{ab}$ by

$B^{ab}=h^{a}{}_{\mu}h^{b}{}_{\nu}B^{\mu\nu}\ .$ (26)

Now, using Eq.(26) and making use of Eq.(5) in Eq.(25), we have the
teleparallel version of the covariant derivative

$\mathcal{D}_{\mu}B^{ab}=h^{a}{}_{\rho}h^{b}{}_{\sigma}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}B^{\rho\sigma}\
,$ (27)

with

$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}B^{\rho\sigma}={\nabla}_{\mu}B^{\rho\sigma}-K^{\rho}_{\lambda\mu}B^{\lambda\sigma}-K^{\sigma}_{\lambda\mu}B^{\rho\lambda}$
(28)

where ${\nabla}_{\mu}B^{\rho\sigma}$ is the Weitzenb$\ddot{\text{o}}$ck
covariant derivative given as

${\nabla}_{\mu}B^{\rho\sigma}=\partial_{\mu}B^{\rho\sigma}+\Gamma^{\rho}_{\mu\lambda}B^{\lambda\sigma}+\Gamma^{\sigma}_{\mu\lambda}B^{\rho\lambda}\
.$ (29)

Thus the teleparallel version of minimal coupling prescription is given as

$\partial_{a}\rightarrow\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}\equiv\partial_{\mu}+\Gamma_{\mu}-K_{\mu}\
.$ (30)

The Fock–Ivanenko derivative $\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}$
in Eq.(22) turns out to be the Weitzenb$\ddot{\text{o}}$ck connection in
teleparallel gravity minus the contorsion tensor.

With the correct prescription ready, let us now consider the Kalb-Ramond
action in the teleparallel background as follows,

$\mathcal{L}_{m}=-hH_{\mu\nu\rho}H^{\mu\nu\rho}\ ,$ (31)

where $h=\sqrt{-g}$ and $H_{\mu\nu\rho}$ is given as,

$\displaystyle H_{\mu\nu\rho}$
$\displaystyle=\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}B_{\nu\rho}+\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\rho}B_{\mu\nu}+\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\nu}B_{\rho\mu}$
(32)
$\displaystyle=\partial_{\mu}B_{\nu\rho}+\partial_{\rho}B_{\mu\nu}+\partial_{\nu}B_{\rho\mu}\
,$

which is U(1) gauge invariant. The teleparallel version of field equation is
given as

$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}H^{\mu\nu\rho}=0\ .$ (33)

And the teleparallel version of the Bianchi identity can be written as

$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{[\mu}H_{\nu\rho\sigma]}=0\ .$ (34)

Assuming Lorentz gauge
$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}B^{\mu\nu}=0$, and using the
commutation relation

$\big{[}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu},\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\nu}\big{]}B^{\lambda\mu}=-Q^{\lambda}_{\sigma\mu\nu}B^{\sigma\mu}-Q_{\mu\nu}B^{\lambda\mu}\
,$ (35)

where

$Q_{\rho\mu\nu}^{\theta}=\nabla_{\mu}K^{\theta}{}_{\rho\nu}-K^{\theta}{}_{\sigma\nu}K^{\sigma}{}_{\rho\mu}-\nabla_{\nu}K^{\theta}{}_{\rho\mu}+K^{\theta}{}_{\sigma\mu}K^{\sigma}{}_{\rho\nu}\
,$ (36)

we can derive the field equations in teleparallel gravity to be

$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}\stackrel{{\scriptstyle\circ}}{{\nabla}}^{\mu}B^{\nu\lambda}-Q^{\nu\lambda\sigma\mu}B_{\sigma\mu}-2Q_{\mu}{}^{[\nu}B^{\lambda]\mu}=0\
.$ (37)

##  IV Non-Singular Cosmological bounce in the presence of Kalb Ramond field

To study the cosmological bouncing in $F(T)$ gravity, lets consider the flat
homogeneous isotropic FLRW metric,

$ds^{2}=-dt^{2}+a(t)^{2}(dx^{2}+dy^{2}+dz^{2})\ ,$ (38)

where $a(t)$ is the scale factor, which is a function of $t$. Corresponding to
this metric, the tetrads become,

$h^{a}{}_{\mu}=\text{diag}\big{(}1,a(t),a(t),a(t)\big{)}\ .$ (39)

In this geometry, the nonzero components of the Weitzenb$\ddot{\text{o}}$ck
connection Eq.(4), torsion tensor Eq.(6), contorsion tensor Eq.(9) and dual
torsion tensor Eq.(11) can be derived as

$\displaystyle\Gamma^{i}{}_{0i}=H\ ,$ (40) $\displaystyle
T^{i}{}_{i0}=-T^{i}{}_{0i}=-H\ ,$ (41) $\displaystyle K^{0}{}_{ii}=-Ha(t)^{2}\
,$ (42) $\displaystyle K^{i}{}_{0i}=-H\ ,$ (43) $\displaystyle
S^{i}{}_{0i}=-S^{i}{}_{i0}=H\ ,$ (44)

where $H=\frac{a^{\prime}(t)}{a(t)}$ is the Hubble parameter. Thus, we can
compute the torsion scalar using Eq.(12) as

$T=6H^{2}\ .$ (45)

Our objective is to find the functional form of the gravitational Lagrangian
$F(T)$ that can give rise to nonsingular bouncing cosmology in the presence of
Kalb-Ramond fields in the FLRW geometry. To do this, let us consider the
action,

$\displaystyle S=\frac{1}{2\kappa^{2}}\Bigg{[}\int d^{4}x\
h\Big{(}\operatorname{F(T)}+\Lambda\Big{)}\Bigg{]}-\frac{1}{2}\int d^{4}x\ h\
H_{\mu\nu\rho}H^{\mu\nu\rho}\ ,$ (46)

where $F(T)=-T+f(T)$, $\kappa=\sqrt{8\pi G}$ and $\Lambda$ is the cosmological
constant. On varying this action with respect to the tetrads $h^{a}_{\mu}$
[28], we get the following equations of motion,

$\displaystyle M_{\mu}{}^{\nu}\equiv$ $\displaystyle\
2hf_{TT}\partial_{\mu}TS_{\nu}{}^{\mu\lambda}+2f_{T}e^{a}{}_{\nu}\partial_{\mu}(hS_{a}{}^{\mu\lambda})$
(47)
$\displaystyle-2hf_{T}T^{\sigma}{}_{\mu\nu}S_{\sigma}{}^{\lambda\mu}-h\big{(}f+\Lambda\big{)}\delta^{\lambda}{}_{\nu}$
$\displaystyle=$ $\displaystyle
h\kappa^{2}\Big{(}3H_{\mu\rho\sigma}H^{\nu\rho\sigma}-\frac{1}{2}\delta_{\mu}{}^{\nu}H_{\rho\sigma\lambda}H^{\rho\sigma\lambda}\Big{)}\
,$

(a)

(b)

(c)

(d)

Figure 1: (a) Time evolution of the scale factor $a(t)$, (b) the Hubble
parameter $H(t)$, (c) Energy density $\rho_{m}$ of the KR field, (d) Matter
pressure $p_{m}$ of the KR field in symmetric bounce for $\beta=7.46$ $\times
10^{-85}$ GeV2 and $t_{0}=6.7\times 10^{41}$ GeV-1

where $f_{T}=\frac{\partial f}{\partial T}$ and
$f_{TT}=\frac{\partial^{2}f}{\partial T^{2}}$.
Varying the KR action with respect to the field $B_{\mu\nu}$, gives the
equation of motion as

$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}H^{\mu\nu\rho}=0\ .$ (48)

The completely antisymmetric three-form field $H^{\mu\nu\rho}$ is physically
equivalent to its Hodge dual, namely a one-form field in 4 dimensions. One can
think of the one-form field to be following from a scalar potential $\phi$ and
is defined as,

$H^{\mu\nu\lambda}=\varepsilon^{\mu\nu\lambda\rho}\partial_{\rho}\phi\ .$ (49)

This however makes the above EOM second order in $\phi$ and can be written as

$\displaystyle\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}H^{\mu\nu\rho}=\varepsilon^{\mu\nu\lambda\rho}\partial_{\mu}\partial_{\rho}\phi-\varepsilon^{\mu\nu\lambda\rho}\big{(}{\Gamma}^{\sigma}{}_{\mu\rho}-K^{\sigma}{}_{\mu\rho}\big{)}\partial_{\sigma}\phi=0\
.$ (50)

The equation of motion of $\phi$ can now be obtained from the Bianchi identity

$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{[\mu}H_{\nu\rho\sigma]}=0\ .$ (51)

Substituting Eq.(49) in Eq.(51) and using the fact
$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{[\mu}\varepsilon_{\nu\rho\sigma\lambda]}\partial^{\rho}\phi=0$
in four dimensions, we obtain the equation of motion of $\phi$ as

$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\lambda}\partial^{\lambda}\phi=0\
.$ (52)

Using Eq.(49), the equation of motion (47) becomes

$M_{\mu}{}^{\nu}=h\kappa^{2}\Big{(}3\delta_{\mu}{}^{\nu}\partial^{\rho}\phi\partial_{\rho}\phi-6\delta_{\lambda}{}^{\nu}\partial^{\lambda}\phi\partial_{\mu}\phi\Big{)}$
(53)

Since we are interested in how the KR field affects the time evolution of the
Universe, for simplicity, we consider $\phi$ as a function of the cosmic time
$t$, satisfying the initial conditions

$\phi(t_{b})=0\ ,\ \ \ \ \ \ \phi^{\prime}(t_{b})=1$ (54)

where $t_{b}$ is the time when the bounce occurs. Now, the equations of motion
Eq.(52) and Eq.(53) takes the form,

$\displaystyle 3H^{2}-6H^{2}f_{T}+\frac{1}{2}\big{(}f+\Lambda\big{)}$
$\displaystyle=\kappa^{2}\rho_{m}\ ,$ (55) $\displaystyle
3H^{2}+2H^{\prime}+\frac{1}{2}\big{(}f+\Lambda\big{)}-6H^{2}f_{T}-$
$\displaystyle 2H^{\prime}f_{T}-2Hf_{TT}T^{\prime}$
$\displaystyle=-\kappa^{2}p_{m}\ ,$ (56)
$\displaystyle\phi^{\prime\prime}+3H\phi^{\prime}$ $\displaystyle=0\ ,$ (57)

where $\rho_{m}$ and $p_{m}$ are the energy density and the matter pressure of
the Kalb-Ramond field in the Universe, given by

$\rho_{m}=3\phi^{\prime 2}\ ,\ \ \ \ \ \ p_{m}=3\phi^{\prime 2}a^{2}\ .$ (58)

Equations (55) and (IV) can be together written as

$2H^{\prime}-2H^{\prime}f_{T}-2Hf_{TT}T^{\prime}=-3\kappa^{2}\phi^{\prime
2}(a^{2}+1)\ .\\\ $ (59)

The Eq.(57) then gives the solution of the KR field as

$\phi(t)=\int_{1}^{t}e^{\big{(}-\int_{1}^{\zeta}3H(\xi)d\xi\big{)}}c_{1}d\zeta+c_{2}\
,$ (60)

where $c_{1}$ and $c_{2}$ are constants set to satisfy the initial conditions
Eq.(54).

In particular, we will be looking into two cases of nonsingular bouncing
cosmology, namely

  1. A.

Symmetric bounce

  2. B.

Matter bounce

###  IV.1 Symmetric bounce

In symmetric bouncing cosmology, the scale factor is given as [32, 33]

$a(t)=a_{0}\operatorname{exp}\Big{(}\alpha\frac{t^{2}}{t_{*}^{2}}\Big{)}\ ,$
(61)

where $a_{0}=a(0)>0$ is the minimum value attained by the scale factor,
$t_{*}>0$ is an arbitrary time and $\alpha>0$ is a parameter. Fig. (1(a))
shows the behavior of $a(t)$ over time, where we chose the parameter $\beta$ =
$\alpha/t_{*}^{2}$. There is a particular time $t_{0}>0$ when the scale factor
becomes unity i.e $a(t_{0})=1$. We define $t_{0}$ to be the present
cosmological time with the present Hubble parameter $H_{0}\equiv H(t_{0})$.
The expression for $t_{0}$ using Eq.(61) is given as

$t_{0}=\sqrt{\frac{-\operatorname{ln}a_{0}}{\beta}}\ .$ (62)

Since $\beta>0$, Eq.(62) restricts the range $a_{0}$ $\in$ $(0,1)$. The
current time is computed to be $t_{0}=6.7\times 10^{41}$ GeV-1, according to
the Planck Collaboration results 2015 [34]. Given the expression of the scale
factor, it is straightforward to calculate the Hubble parameter and the
torsion scalar as

$H(t)=2\beta t\ ,\ \ \ \ \ \ T(t)=24\beta^{2}t^{2}\ .$ (63)

In Fig.(1(b)), we plot the Hubble parameter over time, where $H(t)$ varies
linearly with time. The Hubble parameter’s positivity determines whether a
Universe is expanding or contracting. The phase when $H<0$ for $t<0$ is the
contracting phase followed by the expansion phase where $H>0$ for $t>0$.
Clearly, the bounce occur at $t=0$ (which is a nonsingular bounce), when
$H=0$. The current observational value of Hubble constant is $H_{0}$ =
$H(t_{0})\approx 10^{-42}$ GeV. Using $H_{0}$ and $t_{0}$ in Eq. (63), we get
the value of $\beta$ to be $7.46$ $\times 10^{-85}$ GeV2. Solving the equation
of motion (57) using the initial conditions Eq.(54), we get the expression of
$\phi$ in the symmetric bounce cosmology,

$\phi(t)=\frac{1}{2}\sqrt{\frac{\pi}{3\beta}}\operatorname{erf}\Big{(}\sqrt{3\beta}t\Big{)}\
,$ (64)

where $\operatorname{erf}(x)$ is the error function. In Fig.(2), we plotted
the time evolution of $\phi(t)$. $\phi(t)$ behaves as a sigmoid function,
varying monotonically, but almost saturates after a certain point. This is
evident from the asymptotic behavior of $\phi(t)$,

$\operatorname{\lim}_{t\to\infty}\phi(t)=\frac{1}{2}\sqrt{\frac{\pi}{3\beta}}\
.$ (65)

Figure 2: Time evolution of the scalar field $\phi(t)$ in symmetric bounce for
$\beta=7.46$ $\times 10^{-85}$ GeV2 and $t_{0}=6.7\times 10^{41}$ GeV-1.

The energy density and pressure of the KR field can be obtained using Eq.(58)
as

$\displaystyle\rho_{m}$ $\displaystyle=3\operatorname{exp}\Big{(}-6\beta
t^{2}\Big{)}\ ,$ (66) $\displaystyle p_{m}$
$\displaystyle=3\operatorname{exp}\big{(}-2\beta
t_{0}^{2}\big{)}\operatorname{exp}\Big{(}-4\beta t^{2}\Big{)}\ .$ (67)

The evolution of energy density and matter pressure with respect to the cosmic
time $t$ is plotted in Fig.(1(c)) and Fig.(1(d)) respectively. Both the plots
show a similar behavior with bell-like profile and localization at $t=0$.
Further,it is evident that the evolution depends on the factor $\beta$, which
determines how fast the Universe expands or contracts. The energy density at
the bounce is obtained to be $3$ M${}_{Pl}{}^{4}$, and at the present time
$t_{0}$, it is $1.34$ M${}_{Pl}{}^{4}$. Similarly the matter pressure $p_{m}$
at $t=0$ and $t=t_{0}$ are $1.53$ M${}_{Pl}{}^{4}$ and $0.9$ M${}_{Pl}{}^{4}$
respectively. Clearly, the localization of energy densities at $t=0$ is
responsible for the bounce, but is large enough to have its effects noticeable
at the present day cosmology. Using Eq.(59), we get the differential equation
of f(T) as

$\displaystyle 2Tf_{TT}+f_{T}=$ $\displaystyle
1+\frac{3}{4\beta}\operatorname{exp}\big{(}-\beta
t_{0}^{2}\big{)}\kappa^{2}\operatorname{exp}\Big{(}\frac{-T}{6\beta}\Big{)}$
(68)
$\displaystyle+\frac{3}{4\beta}\kappa^{2}\operatorname{exp}\Big{(}\frac{-T}{4\beta}\Big{)}\
,$

Solving the above differential equation, we finally derive the exact
functional form of $F(T)$ to be

$\displaystyle F(T)$
$\displaystyle=\frac{3}{2}\kappa^{2}\operatorname{exp}\Big{(}\frac{-T}{4\beta}\Big{)}\Big{[}2+3\operatorname{exp}\big{(}-\beta
t_{0}^{2}\big{)}\operatorname{exp}\Big{(}\frac{T}{12\beta}\Big{)}\Big{]}$ (69)
$\displaystyle+6\sqrt{\pi}\sqrt{\frac{T}{\beta}}\kappa^{2}\operatorname{erf}\Big{(}\frac{1}{2}\sqrt{\frac{T}{\beta}}\Big{)}$
$\displaystyle+3\sqrt{6\pi}\operatorname{exp}\big{(}-\beta
t_{0}^{2}\big{)}\sqrt{\frac{T}{\beta}}\kappa^{2}\operatorname{erf}\Big{(}\sqrt{\frac{T}{6\beta}}\Big{)}+C\
.$

where $C$ is an integration constant. Moreover it is important to note that
the reconstructed Lagrangian Eq.(69) is an even function, and hence is
symmetric with respect to the bounce at $t=0$.

(a)

(b)

(c)

(d)

Figure 3: (a) Time evolution of the scale factor $a(t)$, (b) the Hubble
parameter $H(t)$, (c) Energy density of the KR field, (d) Matter pressure of
the KR field in matter bounce for $\sigma=7\times 10^{-6}M^{2}_{Pl}$ and
$t_{0}=6.7\times 10^{41}$ GeV-1

For the gravitational Lagrangian to be able to recover vacuum solutions, $T$
has to be zero in the absence of matter [35]. This is evident from Eq.(63).
Also, as a consequence of Eq.(55) we assume $\Lambda=6\kappa^{2}$ such that it
satisfies the vacuum solution constraint $f(0)=0$. This fixes the integration
constant C to be

$C=-\Big{(}3+\frac{9}{2\operatorname{exp}\big{(}\beta
t_{0}^{2}\big{)}}\Big{)}\ .$ (70)

Figure.(5(a)) shows the function $F(T)$ vs torsion scalar $T$ and Fig.(5(a))
shows the evolution of $F(T)$ with respect to the cosmic time $t$,
corresponding to the symmetric-bounce scenario in the presence of Kalb-Ramond
field described by Eq.(64).

###  IV.2 Matter bounce

In matter bounce cosmology [36, 37, 33] the scale factor is given as

$a(t)=a_{0}\Big{(}\frac{3}{2}\sigma t^{2}+1\Big{)}^{\frac{1}{3}}\ ,$ (71)

where $a(0)$= $a_{0}$ is a positive quantity, and $0<\sigma<<1$ is the a
positive quantity, which is determined from the loop quantum gravity [33]. The
parameter $\sigma$ also determines how fast the bounce occurs [38]. Fig.(3(a))
shows the time evolution of the scale factor in matter bounce cosmology. The
present cosmological time $t_{0}>0$ can be obtained from Eq.(71) as

$t_{0}=\sqrt{\frac{2}{3\sigma}\Big{(}\frac{1}{a_{0}^{3}}-1\Big{)}}\ .$ (72)

Thus, the range of $a_{0}$ is restricted to $(0,1)$, since $\sigma>0$. We have
taken $\sigma=7\times 10^{-6}M^{2}_{Pl}$, which is determined by the amplitude
of the CMB spectrum [38] and the present Hubble constant could be evaluated
using this $\sigma$ to be $H_{0}\approx 10^{-42}$ GeV. The expressions of the
Hubble parameter and the torsion scalar in matter bounce cosmology takes the
form,

$H(t)=\frac{2\sigma t}{3\sigma t^{2}+2}\ \ \ \ \ \
T(t)=\frac{24\sigma^{2}t^{2}}{(3\sigma t^{2}+2)^{2}}\ .$ (73)

$H(t)$ is plotted in Fig (3(b)), which clearly shows that a nonsingular bounce
occurs at $t=0$, with a contracting and expansion phase for $t<0$ and $t>0$
respectively. The torsion scalar at the cosmic time $t_{0}$ can be obtained by
substituting Eq.(72) in Eq.(73), which is given as

$T_{0}\equiv T(t_{0})=4a_{0}^{3}\left(1-a_{0}^{3}\right)\sigma\ .$ (74)

The corresponding energy density and matter pressure of the KR field is
obtained as

$\displaystyle\rho=$ $\displaystyle\frac{12}{(3\sigma t^{2}+2)^{2}}\ ,$ (75)
$\displaystyle p=$
$\displaystyle\frac{6a_{0}^{2}}{2^{-\frac{1}{3}}\big{(}3\sigma
t^{2}+2\big{)}^{\frac{4}{3}}}\ .$ (76)

These are plotted in Fig.(3(c)) and Fig. (3(d)), which shows that the maximum
of energy density and matter pressure is again at $t=0$. At the bounce, the
energy density is $3$ M${}_{Pl}{}^{4}$, but it drastically decreases to
$6.1\times 10^{-234}$ M${}_{Pl}{}^{4}$ at the present time $t=t_{0}$. This
feature explains the lack of cosmological effects of the KR field in the
present day Universe. Upon solving the KR field equation [Eq.(59)] we get the
expression of the scalar field $\phi$ corresponding to the matter bounce
cosmology as

$\phi(t)=\sqrt{\frac{2}{3\sigma}}\operatorname{tan^{-1}}\Big{(}\sqrt{\frac{3\sigma}{2}}t\Big{)}\
.$ (77)

The time evolution of $\phi(t)$ is plotted in the Fig.(4). It can be observed
that the behaviour of $\phi$ is again similar to what we have seen in the case
of symmetric bounce, where it behaves as a sigmoid function. The asymptotic
behavior of $\phi(t)$ in matter bounce as $t\rightarrow\infty$ is given as

$\operatorname{\lim}_{t\to\infty}\phi(t)=\frac{\pi}{\sqrt{6\sigma}}\ .$ (78)

Figure 4: Time evolution of the scalar field $\phi(t)$ in matter bounce for
$\sigma=7\times 10^{-6}M^{2}_{Pl}$ and $t_{0}=6.7\times 10^{41}$ GeV-1.

Solving the functional form of $F(T)$ using Eq.(77) in Eq.(55) and Eq.(IV), we
get $F(T)$ as a function of $t$,

$\displaystyle F(t)$
$\displaystyle=\frac{24\kappa^{2}}{(2+3t^{2}\sigma)^{2}}-\frac{48t^{2}\sigma^{2}}{(2+3t^{2}\sigma)^{2}}$
(79) $\displaystyle+$
$\displaystyle\frac{6t^{2}\sigma}{(2+3t^{2}\sigma)^{2}}\Bigg{[}8\sigma+\kappa^{2}\Big{(}6+9a_{0}^{2}\big{(}\sqrt{2}(2+3t^{2}\sigma)\big{)}^{\frac{2}{3}}\Big{)}\Bigg{]}$
$\displaystyle+$
$\displaystyle\frac{6\sqrt{6}t\sigma^{\frac{1}{2}}}{(2+3t^{2}\sigma^{2})}\Bigg{[}\kappa^{2}\operatorname{tan^{-1}}\Big{(}\sqrt{\frac{3\sigma}{2}}t\Big{)}\Bigg{]}$
$\displaystyle-$
$\displaystyle\frac{18a_{0}^{2}\kappa^{2}t^{2}\sigma}{(2+3t^{2}\sigma^{2})}\operatorname{{}_{2}F_{1}}\Big{[}\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{3t^{2}\sigma}{2}\Big{]}-6\kappa^{2}\
,$

where ${{}_{2}F_{1}}[a,b;c;d]$ represents the hypergeometric function.

The symmetry between the contraction and the expansion phase in matter bounce
requires $F(t)$ to be an even function of $t$. In Fig.(6(b)), we have plotted
the cosmic time evolution of $F(t)$, which shows its symmetric behavior with
respect to the bouncing point at $t=0$. The inverse relation $t(T)$ can be
obtained by the inversion of $T$ in Eq.(73),

$t(T)=\pm\sqrt{\frac{2}{3}}\sqrt{\frac{2}{T}-\frac{1}{\sigma}-\frac{2}{T}\sqrt{1-\frac{T}{\sigma}}}\
.$ (80)

(a)

(b)

Figure 5: The plot $(a)$ shows $F(T)$ vs $T/T_{0}$ in symmetric bounce
scenario for $\beta=7.46\times 10^{-85}$ GeV2, which corresponds to the
present Universe. Here, $T_{0}=24\beta\operatorname{ln}a_{0}$ is the torsion
scalar at $t=0$ and we have chosen $\kappa=M_{Pl}$. The evolution of $F(T)$
with respect to $t/t_{0}$ is plotted in $(b)$

(a)

(b)

Figure 6: The plot $(a)$ shows function $F(T)$ in terms of $T/\sigma$ in the
matter bounce scenario In (a), $F(T)$ is only valid for $T/\sigma\leq 1$,
equivalently $|t|\leq t_{min}$=$\sqrt{\frac{2}{3\sigma}}$ in (b). In plot (b),
evolution $F(T)$ in terms of the $t/t_{min}$ for $\sigma=7\times
10^{-6}M^{2}_{Pl}$ is plotted

Here, we have retained the solution pair that produces the desired result,
$T=0$ at $t=0$. It is important to note that the solution is invertible only
for $-\sqrt{\frac{2}{3\sigma}}\leq t\leq\sqrt{\frac{2}{3\sigma}}$. This
corresponds to a characteristic time period for each matter bounce Universe
corresponding to the critical parameter $\sigma$. Beyond this region of time,
we assume TEGR is valid [38]. As the solution Eq.(79) is an even function of
t, both $\pm$ solutions in Eq.(80) provides the identical form of $F(T)$, with
$-$ and $+$ solutions, representing the contraction and expansion phases
respectively.

Substituting Eq.(80) in Eq.(79), we get the functional form of F(T) as

$\displaystyle F(T)=$ (81)
$\displaystyle\frac{1}{3h(T)^{2}\sigma^{2}}\Bigg{[}3\kappa^{2}\Big{(}a_{0}^{2}(T-2\sigma
h(T))\Big{(}3\times 2^{2/3}T\Big{(}\frac{\sigma h(T)}{T}\Big{)}^{2/3}$
$\displaystyle-2\sigma
h(T)_{2}F_{1}\Big{(}\frac{1}{3},\frac{1}{2};\frac{3}{2};1-\frac{2h(T)\sigma}{T}\Big{)}\Big{)}$
$\displaystyle-$ $\displaystyle 2\sigma
Th(T)\Big{(}\sqrt{\sigma}\sqrt{\frac{2h(T)}{T}-\frac{1}{\sigma}}\tan^{-1}\Big{(}\sqrt{\sigma}\sqrt{\frac{2h(T)}{T}-\frac{1}{\sigma}}\Big{)}$
$\displaystyle+1\Big{)}\Big{)}\Bigg{]}-6\kappa^{2}\ \ \ $

which is also restricted to $T\leq\sigma$ (equivalently,
$-\frac{2}{3\sigma}\leq t\leq\frac{2}{3\sigma}$). In Eq.(81), we have taken
$h(T)$ as a functions of $T$, given as

$\displaystyle h(T)=1-\sqrt{1-\frac{T}{\sigma}}\ .$ (82)

In Fig (6(a)), we have plotted the function $F(T)$ in terms of $T$. Note that
the solution Eq.(81) satisfies the vacuum solution constraint $F(0)=0$.

##  V Discussions and conclusion

TEGR is a successful gravity description, specifically in the presence of
sources that could twist the geometry to create torsion. One such example that
could source torsion is the antisymmetric rank-2 Kalb-Ramond field. These
antisymmetric tensor fields form an integral part of heterotic string models
[29, 39] as massless closed string modes and of some supersymmetric models
like N=2 and N=8 extended SUGRA. They have also been widely studied [40] in
the context of electromagnetic field coupling to the Einstein-Cartan system.
Though essential, it is noteworthy that there are no experimental evidence for
this field in the present day Universe [3].

Since the presence of torsion breaks the $U(1)$ invariance of the gauge
theory, it is important to introduce a suitable coupling prescription in
teleparallel gravity. To do that successfully, we first define an equivalent
of the covariant derivative called the Fock-Ivanenko derivative operator
(FIDO) in teleparallel geometry. In Sec. (III), we generalize FIDO to operate
on any $n$-form tensor field in ($d+1$)-dimensional space, using the equations
Eq.(22), Eq.(23) and Eq.(24) and in particular on KR field. For completeness,
we show the equivalence of Fock-Ivanenko derivative of the KR field in
teleparallel gravity to the Levi-Civita covariant derivative in Einstien’s
gravity in Appendix (A).

We then compute the equations of motion and show that the dynamics of tetrad
fields in teleparallel geometry are governed by the KR field. To keep the
discussion general, we start with a generic function $F(T)$. And we considered
the effect of this setting in producing two bouncing cosmologies, namely
symmetric bounce and matter bounce. The absence of initial singularity in
cosmological evolution has been a significant advantage of bouncing
cosmologies over the inflationary paradigm. In these scenarios, the big bang
is replaced by a continuous phase of expanding and contracting.

Note that the scale factor for both these scenarios are sourced by the
localized energy density of the KR field as shown in Fig.(1(c)) and
Fig.(3(c)). These plots indicate the nature of energy density with time ‘$t$’.
At the present time $t_{0}=6.7\times 10^{41}$ GeV-1, the symmetric-bounce
scenario predicts the KR field energy density of $\rho_{m}=1.34M_{Pl}^{4}$.
Where as matter bounce predicts a much smaller energy density $\rho_{m}\sim
0$. Thus the lack of cosmological evidence of KR field in the present day
Universe strongly advocate matter bounce scenario over symmetric bounce.

In the symmetric bounce, the generalized teleparallel gravity is an increasing
function of $T$, for $T\in(0,\infty)$ as shown in Fig.(5(a)), but for large
$T$, $F(T)$ behaves linearly. Whereas in matter bounce, $F(T)$ again exhibits
similar behavior, albeit the evolution is valid up to some value $T<\sigma$,
as shown in Fig.(6). The analytical results are given in Table.(1). In both
these scenarios, the scalar field shows a wave profile [Fig.(2) and (4)] and
energy profile [Fig.(1(c)) and (3(c))] of a ‘kink’ which could be interesting
to study further. It is also interesting to wonder how an axion/pseudoscalar
field will behave in the teleparallel setting, given the possible parity
violations.

###### Acknowledgements.

Model |  $a(t)$ |  $\phi(t)$ |  F(T)
---|---|---|---
Symmetric bounce |  $a_{0}\operatorname{exp}\Big{(}\alpha\frac{t^{2}}{t_{*}^{2}}\Big{)}$ |  $\frac{1}{2}\sqrt{\frac{\pi}{3\beta}}\operatorname{erf}\Big{(}\sqrt{3\beta}t\Big{)}$ |  $\frac{3}{2}\kappa^{2}\operatorname{exp}\Big{(}\frac{-T}{4\beta}\Big{)}\Big{[}2+3\operatorname{exp}\big{(}\beta t_{0}^{2}\big{)}\operatorname{exp}\Big{(}\frac{T}{12\beta}\Big{)}\Big{]}+6\sqrt{\pi}\sqrt{\frac{T}{\beta}}\kappa^{2}\operatorname{erf}\Big{(}\frac{1}{2}\sqrt{\frac{T}{\beta}}\Big{)}+3\sqrt{6\pi}\operatorname{exp}\big{(}\beta t_{0}^{2}\big{)}\sqrt{\frac{T}{\beta}}\kappa^{2}\operatorname{erf}\Big{(}\sqrt{\frac{T}{6\beta}}\Big{)}-\Big{(}3+\frac{9}{2\operatorname{exp}\big{(}\beta t_{0}^{2}\big{)}}\Big{)}$
Matter bounce |  $a_{0}\Big{(}\frac{3}{2}\sigma t^{2}+1\Big{)}^{\frac{1}{3}}$ |  $\sqrt{\frac{2}{3\sigma}}\operatorname{tan^{-1}}\Big{(}\sqrt{\frac{3\sigma}{2}}t\Big{)}$ |  $\frac{1}{3h(T)^{2}\sigma^{2}}\Bigg{[}3\kappa^{2}\Big{(}a_{0}^{2}(T-2\sigma h(T))\Big{(}3\times 2^{2/3}T\Big{(}\frac{\sigma h(T)}{T}\Big{)}^{2/3}-2\sigma h(T)_{2}F_{1}\Big{(}\frac{1}{3},\frac{1}{2};\frac{3}{2};1-\frac{2h(T)\sigma}{T}\Big{)}\Big{)}-2\sigma Th(T)\Big{(}\sqrt{\sigma}\sqrt{\frac{2h(T)}{T}-\frac{1}{\sigma}}\times\tan^{-1}\Big{(}\sqrt{\sigma}\sqrt{\frac{2h(T)}{T}-\frac{1}{\sigma}}\Big{)}+1\Big{)}\Big{)}\Bigg{]}-6\kappa^{2}$
Table 1: The KR-field $\phi(t)$ and the reconstructed Lagrangian $F(T)$
corresponding to different setups for bouncing cosmology. The function $h(T)$
used in the matter bounce Lagrangian is defined as Eq.(82)

M.T.A. acknowledges financial support of DST through INSPIRE Faculty Grant No.
[DST/INSPIRE/04/2019/002507].

##  Appendix A Coupling prescription using Fock–Ivanenko derivative operator
in the Riemannian geometry

In the framework of Riemannian geometry, the mathematical equivalent of Fock-
Ivanenko derivative of the KR field $B^{ab}$, using Eq.(22) and Eq.(24), is
given by

$\displaystyle\mathcal{D}_{\mu}B^{ab}$
$\displaystyle=\partial_{\mu}B^{ab}-\frac{i}{2}\Omega^{cd}{}_{\mu}\big{(}i(\delta^{a}_{c}\eta_{dg}-\delta^{a}_{d}\eta_{cg})\big{)}B^{gb}$
(83)
$\displaystyle-\frac{i}{2}\Omega^{cd}{}_{\mu}\big{(}i(\delta^{b}_{c}\eta_{dg}-\delta^{b}_{d}\eta_{cg})\big{)}B^{ag}$
$\displaystyle=\partial_{\mu}B^{ab}+\Omega^{a}{}_{c\mu}B^{cd}+\Omega^{b}{}_{c\mu}B^{ac}\
.$

Now, using Eq.(5) and Eq.(8) in Eq.(23), we can write $\Omega^{ab}{}_{\mu}$ as

$\Omega^{ab}{}_{\mu}=h_{\rho}^{a}\tilde{\nabla}_{\mu}h^{b\rho}\ .$ (84)

This can be equivalently written as

$\displaystyle\partial_{\mu}h^{a}{}_{\nu}+\Omega_{b\mu}^{a}h^{b}{}_{\nu}-\tilde{\Gamma}^{\rho}{}_{\nu\mu}h_{\rho}^{a}=0\
.$ (85)

Using the definition of the Fock-Ivanenko derivative of the tetrads,

$\mathcal{D}_{\mu}h_{\nu}^{a}=\partial_{\mu}h^{a}{}_{\nu}+\Omega_{b\mu}h^{b}{}_{\nu}\
,$ (86)

we can rewrite Eq.(85) as

$\displaystyle\mathcal{D}_{\mu}h^{a}{}_{\nu}=\tilde{\Gamma}^{\rho}{}_{\nu\mu}h^{a}{}_{\rho}\
.$ (87)

Substituting Eq.(84) and Eq.(26) in Eq.(83) and using Eq.(87), we get

$\mathcal{D}_{\mu}B^{ab}=h^{a}{}_{\rho}h^{b}{}_{\sigma}\tilde{\nabla}_{\mu}B^{\rho\sigma}\
,$ (88)

where $\tilde{\nabla}_{\mu}B^{\rho\sigma}$ is the Levi-Civita covariant
derivative of $B^{\rho\sigma}$ .
Thus, the Fock–Ivanenko derivative of the antisymmetric Lorentz tensor
$B^{ab}$ reduces to the usual Levi–Civita covariant derivative of general
relativity. In other words, we can say that the minimal-coupling prescription
in Riemannian geometry can be written as,

$\partial_{a}\rightarrow\mathcal{D}_{\mu}=\partial_{\mu}+\tilde{\Gamma}_{\mu}\equiv\tilde{\nabla}_{\mu}\
.$ (89)

Now let us consider the Kalb-Ramond Lagrangian in the background of Riemannian
geometry.

$\mathcal{L}_{m}=-\sqrt{-g}H_{\mu\nu\rho}H^{\mu\nu\rho}\ ,$ (90)

where the field strength $H_{\mu\nu\rho}$ is given by,

$H_{\mu\nu\rho}=\tilde{\nabla}_{\mu}B_{\nu\rho}+\tilde{\nabla}_{\rho}B_{\mu\nu}+\tilde{\nabla}_{\nu}B_{\rho\mu}\
.$ (91)

The corresponding field equation can be written as

$\tilde{\nabla}_{\mu}H^{\mu\nu\rho}=0\ .$ (92)

Assuming the Lorentz gauge $\tilde{\nabla}_{\mu}B^{\mu\nu}=0$, and using the
commutation relation,

$\big{[}\tilde{\nabla}_{\mu},\tilde{\nabla}_{\nu}\big{]}B^{\lambda\mu}=\tilde{R}^{\lambda}{}_{\sigma\mu\nu}B^{\sigma\mu}+\tilde{R}_{\mu\nu}B^{\lambda\mu}\
,$ (93)

we have the field equations of KR fields in teleparallel geometry,

$\tilde{\nabla}_{\mu}\tilde{\nabla}^{\mu}B^{\nu\lambda}+\tilde{R}^{\nu\lambda\sigma\mu}B_{\sigma\mu}+2\tilde{R}_{\mu}{}^{[\nu}B^{\lambda]\mu}=0\
.$ (94)

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