# Self-Supervised Image Representation Learning: Transcending Masking with
Paired Image Overlay

Yinheng Li Han Ding Shaofei Wang Columbia University Columbia University
Columbia University<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Self-supervised learning has become a popular approach in recent years for its
ability to learn meaningful representations without the need for data
annotation. This paper proposes a novel image augmentation technique,
overlaying images, which has not been widely applied in self-supervised
learning. This method is designed to provide better guidance for the model to
understand underlying information, resulting in more useful representations.
The proposed method is evaluated using contrastive learning, a widely used
self-supervised learning method that has shown solid performance in downstream
tasks. The results demonstrate the effectiveness of the proposed augmentation
technique in improving the performance of self-supervised models.

_Keywords_ Self-supervised Learning, Autoencoder, Data Augmentation, Computer
Vision, Masked Modeling,

##  1 Introduction

Self-supervised learning has seen significant success in recent years in both
natural language processing and computer vision domains, particularly with the
advent of attention mechanisms and large language models such as BERT and GPT.
While transformer-based models are the focus in the NLP community for self-
supervised learning, computer vision research has employed a variety of
approaches. One trend is the use of contrastive learning with data
augmentation, while another trend is to adapt the idea of masked training from
NLP to computer vision.

However, applying masked training to images poses challenges. Unlike text,
images have high dimensions and spatial relationships, making it difficult to
mask a single pixel and resulting in large attention matrices. As a result,
researchers have divided images into patches and treated each patch as the
smallest unit, as seen in works such as [7, 8].

In this paper, we propose a new method for image corruption, which is closer
to the idea of "masking" and aims to better leverage masked training in
computer vision tasks. Our method is to overlay one image onto another, a
technique that has been implemented by [10] and shown to be effective. To the
best of our knowledge, this technique has not been used in self-supervised
learning. We demonstrate the effectiveness of this method in comparison to
existing augmentation techniques through experiments on self-supervised
models.

Figure 1: proposed transformation

##  2 Related Work

Autoencoders are a traditional self-supervised learning algorithm that trains
to reconstruct the input from a learned low-dimension representation. Variants
of autoencoders include denoising autoencoders [11], where noise is added to
the input during training. Our proposed method can be viewed as a special type
of denoising autoencoder, where we use image overlap to create a corrupted
image. Recently, self-supervised learning using autoencoders for computer
vision tasks has also achieved great success [8].

Data augmentation is a widely used technique in computer vision research, with
numerous techniques such as flipping, cropping, and coloring ([12]). Image
overlay (also known as image mixture) is also a type of augmentation method,
but has been less explored compared to others.

##  3 Approach

The proposed method is based on the idea of image overlay, which is a form of
image corruption, similar to image masking. There are several reasons why
image overlay is a suitable method for self-supervised representation learning
in computer vision.

First, image overlay is conceptually similar to image masking. When a patch of
an image is masked, it can be viewed as overlaying an image mask onto the
image, which is a special case of overlaying one image on top of another.

Second, image overlay provides better guidance for self-supervised
representation learning compared to standard masking. In masked image
training, the model is asked to reconstruct the missing part of the image,
which is a difficult task in the domain of vision due to the high dimension of
an image and the uncertainty of the real world. For example, predicting what
is behind a window is a challenging task even for humans, due to the infinite
possibilities. On the other hand, distinguishing between two films overlapped
with one another is a relatively easy task as long as one has certain
knowledge of the world. Therefore, training a model to distinguish and recover
the overlapped image demands the model to pick up knowledge about the object
and the world without worrying about uncertainty.

Third, the method of image overlay is also similar to the idea of constructive
learning, where the model needs to distinguish between positive and negative
pairs. In this method, the model is also trained to distinguish between two
samples, except they are in the same image.

Lastly, this method is straightforward to implement as it does not require
constructing any complex transformations on the image or sampling positive and
negative pairs. The only requirement is to change the transparency of an image
and overlay it onto another image.

Figure 2: training pipeline

###  3.1 Pretraining

A detailed approach is shown in Figure2. Denote input image $x_{i}$, current
batch $X_{b}$, encoder $f_{e}(x)$, decoder $f_{d}(x)$

Step1: random sample image $x_{j}$ from $X_{b}$

Step2: generate an augmented image $\hat{x_{i}}=\alpha x_{j}+(1-\alpha)x_{i}$.
$\alpha$ is a hyperparameter which controls the transparency of the image
overlap. It should be less than 0.5 as the input image $x_{i}$ should
dominates the information in the picture.

Step3: train the model to minimize the loss
$L(x_{i})=MSE(f_{d}(f_{e}(\hat{x_{i}})),x_{i})$. MSE is short for mean squared
error.

Once the model is trained, we use $f_{e}$ as the encoder to extract meaningful
representation from the image. The performance will be evaluated in downstream
tasks such as image classification.

###  3.2 Autoencoder

We use a Resnet50 architecture ([13]) as the backbone for the autoencoder,
following the implementation in [14]. The encoder is composed of Resnet50 and
the decoder is a reversed version of Resnet50.

###  3.3 Downstream Evaluation

To evaluate the performance of the pretrained model, the encoder is fine-tuned
on a downstream task. In our experiment, we use image classification as the
downstream task. A linear head is added to the encoder for classification
purposes. During downstream evaluation and testing, no additional augmentation
is applied to the input.

##  4 Object256 Experiment

We conduct our experiments on the Object256 dataset ([15]), which consists of
29,780 images covering 256 objects. The dataset is split into 23,824 images
for the training set and 5,956 images for the testing set. In the pretraining
phase, we train the autoencoder with image overlay $f^{\alpha}$, where
$\alpha\in{0,0.1,0.2,0.3,0.4}$ and $f^{\alpha}=f_{d}(f_{e}(\hat{X}))$.
Additional configuration details can be found in Table a. Our data loader’s
input transformation includes (1) image overlay with $\alpha$ ($\alpha=0$
means no overlay) (2) random resized crop (to ensure the input image is of
size $244\times 244$). To compare our augmentation method with other commonly
used methods, we also trained two sets of models with random mask and
AutoAugment (auto-policy) transformation: $f^{randommask}$ and
$f^{autopolicy}$. The random mask transformation randomly sets $m\%$ (where
$m\in{\\{1,2,3,4}\\}$) of pixels in an image to 0 (blank), which is equivalent
to adding white noise to the input, as shown in Figure 3. AutoAugment ([16])
is a set of augmentations that were optimized on the Imagenet dataset.

Figure 3: an example of random mask transformation Figure 4: This is the
reconstruction result using $f^{\alpha=0.2}$. We only show one result because
results from different $\alpha$’s are visually similar to this picture.

Figure 4 shows the reconstruction results on 64 random samples from the
testing set. All $\alpha$ configurations show very similar reconstruction
results, indicating that $f^{\alpha}$ is sufficiently trained.

We then use the pretrained encoder $f^{\alpha}_{e}$ and fine-tune it on the
downstream image classification task using the obj256 training set for fine-
tuning and the test set for evaluation. In the fine-tuning phase, each encoder
is trained for 20 epochs without any additional transformation. Configuration
details can be found in Table b. The best accuracy on the testing set over 20
epochs is reported. We also train a Resnet model without pretraining for 20
epochs as a baseline for comparison.

config | value
---|---
epoch | 50
learning rate | 0.05
momentum | 0.9
optimizer | SGD
batch size | 256
scheduler | cosine
weight decay | 1e-4
(a) model configuration-pretrain

config | value
---|---
epoch | 20
learning rate | 0.01
momentum | 0.9
optimizer | SGD
batch size | 256
scheduler | cosine
weight decay | 1e-4
(b) model configuration-finetune

The first experiment is to compare the model’s performance on downstream
classification task with different $\alpha$s. According to table 1, the
accuracy across different $\alpha$ is around 37%. This accuracy looks very low
given a resnet50 model can easily achieve 80% on imagenet with one thousand
class. However, we looked up other people’s experiment and confirmed that this
is a reasonable result given our architecture. More results can be found from
this website [[17]]. According to their experiment, the top performer without
transferred learning is only 39.06% . The reason that all models have a low
accuracy on this dataset is that this is a small dataset with a large number
of classes.

The best model in table3 under finetuning is a pre-trained encoder with
$\alpha=0.3$. We also did linear probing where we freeze all layers except the
last layer for classification and trained for 5 epochs. But we found all
models have very similar performance during linear probing.

Lastly, we compared the performance of using image overlap, using random
masking, supervised learning without pretraining and auto-policy augmented
pretraining. Results are shown in table 4. We found the model with image
overlap augmentation still performs the best among all the other models.

model | fine-tune-accuracy | linear-probing-accuracy
---|---|---
$\alpha=0$ | 37.1% | 7.7%
$\alpha=0.1$ | 37.0% | 7.6%
$\alpha=0.2$ | 35.5% | 7.7%
$\alpha=0.3$ | 38.1% | 7.6%
$\alpha=0.4$ | 37.0% | 7.6%
Table 1: accuracy with different alpha model | image classification accuracy
---|---
best-image-overlap | 38.1%
best-image-masking | 36.7%
supervised-learning | 32.5%
auto-policy augmentation | 33.2%
Table 2: accuracy with different models

##  5 Discussion and Conclusion

In this paper, we proposed a new method of image "masking" using image overlay
for self-supervised learning in computer vision. We showed that this method is
simple and easy to implement, and it outperforms other commonly used self-
supervised methods on the Object256 dataset. However, it is important to note
that the Object256 dataset is relatively small and the classification accuracy
is generally low across all models. Therefore, more extensive experimentation
on larger datasets such as Imagenet is needed to fully evaluate the
effectiveness of our proposed method.

Additionally, we only used a Resnet architecture in our experiments due to
computational constraints. Therefore, another direction for future work is to
test our method using transformer-based neural networks.

Overall, our proposed method of image overlay for self-supervised learning in
computer vision provides a promising approach to learn meaningful
representations without the need for data annotation. It is simple to
implement, and we hope that future research will further explore its
potential.

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# Anisotropic LRS-BI Universe with $f(Q)$ gravity theory

Pranjal Sarmah 0000-0002-0008-7228<EMAIL_ADDRESS>Department of Physics,
Dibrugarh University, Dibrugarh 786004, India Avik De 0000-0001-6475-3085
<EMAIL_ADDRESS>Department of Mathematical and Actuarial Sciences,
Universiti Tunku Abdul Rahman, Jalan Sungai Long, 43000 Cheras, Malaysia
Umananda Dev Goswami 0000-0003-0012-7549<EMAIL_ADDRESS>Department of
Physics, Dibrugarh University, Dibrugarh 786004, India

###### Abstract

The possible anisotropic nature in the early phases of the Universe is one of
the interesting aspects of study in cosmology. We investigate the evolution of
the Universe in terms of few cosmological parameters considering an
anisotropic locally rotationally symmetric (LRS) Bianchi type-I spacetime
(LRS-BI) under the $f(Q)$ gravity of symmetric teleparallel theory equivalent
to the GR (STEGR). In this study we consider two $f(Q)$ gravity models, viz.,
$f(Q)=-\,(Q+2\Lambda)$, a simple model with the cosmological constant
$\Lambda$ and the power law model, $f(Q)=-\,\alpha Q^{n}$ with two constant
parameters $\alpha$ and $n$. Considering a proportionality relation between
the directional Hubble parameters with a proportionality constant parameter
$\lambda$ we find a significant contribution of the anisotropic factor in the
evolution of the early Universe for both models. The power law model shows the
more dominating effect of anisotropy in comparison to the simple model
depending on model parameters, especially on the parameter $n$. The power law
model also shows some possible effects of anisotropy on the Universe’s
evolution in the near future. In addition, both models confirm that the
anisotropy does not obtain any appreciable signature in the current stage of
the Universe. From our analysis we also set rough constraints on our model
parameters as $0.5\leq\lambda\leq 1.25$, $0.75\leq\alpha\leq 1.5$ and
$0.95\leq n\leq 1.05$.

##  I Introduction

The cosmological principle suggests that the perspective of an observer on the
cosmos is independent of both his position and the direction in which he is
looking. This very useful assumption implies that the present-day Universe is
largely isotropic and homogenous, and can be modelled perfectly by a
Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime metric. However, this may
not have been the case in the very early stage of the Universe or in the
distant future. Recent results from the Wilkinson Microwave Anisotropy Probe
(WMAP) [1, 2, 3] show that the classic isotropic and homogeneous model of the
Universe requires some tweaking to account for the observational evidence. The
remarkable aspect of the inflationary paradigm [4] that has been supported by
observation is that it isotropizes the early Universe into the FLRW geometry
we observe today. One must still allow for the possibility of spatial
inhomogeneity and anisotropy, and then examine its development into the
observed quantity of homogeneity and isotropy, for a complete version.
However, the observational evidences of smallness of Cosmic Microwave
Background (CMB) quadrupole [5], low anomalies in the large scale angular
distribution of CMB power spectrum [6] etc. suggest some sort of symmetries in
the Universe. Again, studies show that a plane symmetric large scale geometry
of the Universe with the eccentricity of the order $10^{-2}$ can bring the
quadrupole amplitude match to the observational data without affecting the
higher order multipoles of the temperature anisotropy in the CMB angular power
spectrum [7]. Further, the polarization analysis of electromagnetic radiation
propagating over large cosmological distances also indicates the symmetry axis
in the large scale geometry of the Universe [8]. To address the symmetric
nature of the Universe, one option is the locally symmetric Kantowski-Sachs
spacetime, but one may also start with the Bianchi type cosmological models,
which together make up a sizable and nearly exhaustive class of relativistic
cosmology models that are homogeneous but not necessarily isotropic.
Furthermore, the isotropic ones, which may be seen as a specific sub-case [9],
can be properly understood by examining a model that is almost like FLRW but
has fewer symmetries. The simplest Bianchi model which is very suitable for
addressing the Universe with such a symmetry axis is the locally rotationally
symmetric (LRS) Bianchi type-I (BI) model. In LRS-BI model, the metric has the
spatial section with planner symmetry and an axis of symmetry [10]. Thus it
has two equivalent longitudinal directions and a transverse direction. The
metric of it in Cartesian coordinates is

$ds^{2}=-dt^{2}+A^{2}(t)dx^{2}+B^{2}(t)(dy^{2}+dz^{2}).$ (1)

Here, the longitudinal directions are $y$ and $z$ and the transverse direction
is along the $x$-axis. The metric describes an ellipsoidal expansion of space
with the cosmic time which supports the CMB observations and hence it is
widely used [10]. Also the above metric has a unique feature in the sense that
starting from the most general linearly perturbed spatially flat FLRW metric,
under synchronous gauge and the homogeneity bound, it is possible to obtain
such an anisotropic metric (1) [11]. Furthermore, Bianchi cosmologies have
gained popularity in observational cosmology in recent decades, as WMAP data
[1] suggest that the standard cosmological model with a positive cosmological
constant resembles the Bianchi morphology [12, 13, 14, 7, 15]. Furthermore,
according to these findings, the Universe should have been able to develop a
somewhat anisotropic spatial geometry in spite of the inflation, which runs
counter to the predictions of standard inflationary theories [16, 17, 18, 19,
20, 21, 22]. Very recently, a wide range of Bianchi cosmology with the
observational data have been studied (see details in [23, 24, 25, 26, 27,
28]). Investigations on various aspects of anisotropic cosmology in classical
GR by using the LRS-BI metric have been done extensively so far. In Ref. [10]
the connection between deceleration parameter and cosmic shear, and jerk
parameter and ellipsoidal Universe has been studied in GR theory by using the
LRS-BI metric. Another work containing the study of the ellipsoidal Universe
in Braneworld cosmology has been found in Ref. [29]. L. Campanellie et al. in
2011 had tested the isotropy of the Universe with type Ia supernovae data in
the LRS-BI framework [30]. The work on LRS-BI model to examine the anisotropy
in the Universe by using the concept of dynamically anisotropic dark energy
and constant deceleration parameter for perfect fluid is found in Ref. [8]. A
wide range of studies have also been carried out in Modified Theories of
Gravity (MTGs) like $f(R)$, $f(R,T)$ etc. by using the LRS-BI model of the
Universe. X. Liu et al. in 2017 had studied the cosmological dynamics of an
uniform magnetic field in the LRS-BI Universe for the viable $f(R)$ models
[31]. The study of anisotropic cosmological model in $f(R,T)$ gravity with
variable deceleration parameter for the LRS-BI metric has been found in Ref.
[32]. Moreover, the study of de-Sitter and bounce solution in $f(R,T)$
cosmology for LRS-BI Universe has been carried out in the Ref. [33]. Recently
the works on anisotropy cosmology by using the LRS-BI metric have been also
performed in the $f(Q)$-gravity theory. Various cosmological profiles, like
energy density, equation of state, skewness parameters have been studied in
$f(Q)$-gravity by using LRS-BI metric in Ref. [34]. Another work on the
anisotropic Universe in $f(Q)$-gravity with the consideration of Hybrid
expansion law (HEL) for the average scale factors has been accomplished in
Ref. [35].

In addition, the accelerated expansion of the Universe was confirmed in 1998
by the Supernovae Cosmology Project, which utilised IA Supernovae data [36].
This gives rise to a variety of explanations for the acceleration. In general
relativity (GR), the existence of a yet undetected unknown kind of energy in
the Universe, known as dark energy (DE), with exotic properties such as
negative pressure results in a negative equation of state (EoS) parameter. To
circumvent the mysterious DE, as an alternative to GR, modified gravity
theories were investigated by altering the Einstein-Hilbert action while
maintaining the geometry. $f(R)$-gravity theories are the simplest and most
successful in this regard. Other types of theories, such as the teleparallel
theory equivalent to the GR (TEGR) and the symmetric teleparallel theory
equivalent to the GR (STEGR), have been addressed in the past by modifying the
underlying geometry without disrupting the Lagrangian. In these theories, flat
space is considered, and the extremely special (symmetric and metric-
compatible) Levi-Civita connection used in GR is replaced by an affine
connection with either non-vanishing torsion (in TEGR) or non-metricity (in
STEGR) as the guiding force of gravity, and its extension $f(Q)$ theory ($Q$
is the non-metricity scalar) was proposed in [37].

Recently, the $f(Q)$-gravity becomes the point of wide attention, see [38, 39,
40, 41, 42, 43, 34, 44, 45, 46, 47, 48, 49, 50, 51, 48, 52, 53, 54, 55, 56,
57, 58] and the references therein for the recent theoretical studies and
cosmological and astrophysical applications. However, the majority of the
aforementioned cosmological studies have focused on exploring the present
interests of the Universe by considering the spatially flat isotropic and
homogeneous FLRW metric in Cartesian coordinates. To also note that, the
particular line element, so chosen, is automatically a coincident gauge, which
reduces the covariant derivative to merely a partial derivative and simplifies
the calculation considerably. In our present work we are aiming to study the
evolution of the Universe in $f(Q)$-gravity starting from an anisotropic but
homogeneous background metric which offers a broader outlook in this new
gravity theory and its cosmological application.

The present article is organized as follows: In the next section the basic
formalism of $f(Q)$-gravity has been discussed. The equations of motion for
the LRS-BI model defined by the metric (1) in $f(Q)$-gravity are derived in
section III. In section IV the expressions of DE EoS and other cosmological
parameters are developed for the LRS-BI model. For two different $f(Q)$ models
we analyse the cosmological parameters using the numerical calculations and
corresponding graphical representations in section V. Finally, in section VI
summary and conclusion of the work are drawn with the future prospects of the
study.

##  II Basic Formalism of $f(Q)$ Gravity

In $f(Q)$ gravity theory, the spacetime is constructed by using the symmetric
teleparallelism and non-metricity condition, that is
$R^{\rho}{}_{\sigma\mu\nu}=0$ and $Q_{\lambda\mu\nu}\mathrel{\mathop{\mathchar
58\relax}}=\nabla_{\lambda}g_{\mu\nu}\neq 0$. The associated connection
coefficient is given by

$\Gamma^{\lambda}{}_{\mu\nu}=\mathring{\Gamma}^{\lambda}{}_{\mu\nu}+L^{\lambda}{}_{\mu\nu},$
(2)

where $\mathring{\Gamma}^{\lambda}{}_{\mu\nu}$ is the Levi-Civita connection
and $L^{\lambda}{}_{\mu\nu}$ is the disformation tensor. The disformation
tensor is expressed as

$L^{\lambda}{}_{\mu\nu}=\frac{1}{2}(Q^{\lambda}{}_{\mu\nu}-Q_{\mu}{}^{\lambda}{}_{\nu}-Q_{\nu}{}^{\lambda}{}_{\mu})\,.$

In addition, we define the superpotential tensor

$P^{\lambda}{}_{\mu\nu}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{4}\left(-2L^{\lambda}{}_{\mu\nu}+Q^{\lambda}g_{\mu\nu}-\tilde{Q}^{\lambda}g_{\mu\nu}-\frac{1}{2}\delta^{\lambda}_{\mu}Q_{\nu}-\frac{1}{2}\delta^{\lambda}_{\nu}Q_{\mu}\right),$
(3)

and using it the non-metricity scalar is defined as

$Q=-Q_{\lambda\mu\nu}P^{\lambda\mu\nu}.$ (4)

The action of $f(Q)$ gravity is given by

$S=\int\left[\frac{1}{2\,\kappa}\,f(Q)+\mathcal{L}_{m}\right]\sqrt{-g}\,d^{4}x,$

where $\kappa=8\pi G_{N}$, $G_{N}$ being the usual Newtonian gravitational
constant, $g$ is the determinant of the metric tensor and $\mathcal{L}_{m}$ is
the matter Lagrangian. By varying the action with respect to the metric, we
obtain the field equations of $f(Q)$ gravity as

$\frac{2}{\sqrt{-g}}\nabla_{\lambda}(\sqrt{-g}\,f_{Q}\,P^{\lambda}{}_{\mu\nu})+\frac{1}{2}\,f(Q)\,g_{\mu\nu}+f_{Q}(P_{\nu\rho\sigma}\,Q_{\mu}{}^{\rho\sigma}-2P_{\rho\sigma\mu}\,Q^{\rho\sigma}{}_{\nu})=-\kappa\,T_{\mu\nu},$
(5)

where $f_{Q}$ represents the derivative of $f(Q)$ with respect to non-
metricity scalar $Q$ and $T^{m}_{\mu\nu}$ is the energy-momentum tensor of the
matter field.

On the other hand, using the connection coefficient (2), we can have the
following relations between the curvature tensors corresponding to $\Gamma$
and $\mathring{\Gamma}$:

$R^{\rho}{}_{\sigma\mu\nu}=\mathring{R}^{\rho}{}_{\sigma\mu\nu}+\mathring{\nabla}_{\mu}L^{\rho}{}_{\nu\sigma}-\mathring{\nabla}_{\nu}L^{\rho}{}_{\mu\sigma}+L^{\rho}{}_{\mu\lambda}L^{\lambda}{}_{\nu\sigma}-L^{\rho}{}_{\nu\lambda}L^{\lambda}{}_{\mu\sigma}$
(6)

and so

$\displaystyle R_{\sigma\nu}$
$\displaystyle=\mathring{R}_{\sigma\nu}+\frac{1}{2}\mathring{\nabla}_{\nu}Q_{\sigma}+\mathring{\nabla}_{\rho}L^{\rho}{}_{\nu\sigma}-\frac{1}{2}Q_{\lambda}L^{\lambda}{}_{\nu\sigma}-L^{\rho}{}_{\nu\lambda}L^{\lambda}{}_{\rho\sigma},$
$\displaystyle R$
$\displaystyle=\mathring{R}+\mathring{\nabla}_{\lambda}Q^{\lambda}-\mathring{\nabla}_{\lambda}\tilde{Q}^{\lambda}-\frac{1}{4}Q_{\lambda}Q^{\lambda}+\frac{1}{2}Q_{\lambda}\tilde{Q}^{\lambda}-L_{\rho\nu\lambda}L^{\lambda\rho\nu}\,.$

Therefore, by using the symmetric teleparallelism condition, we can rewrite
the field equations in (5) as

$f_{Q}\mathring{G}_{\mu\nu}+\frac{1}{2}\,g_{\mu\nu}(f(Q)-Qf_{Q})+2f_{QQ}\mathring{\nabla}_{\lambda}QP^{\lambda}{}_{\mu\nu}=-\kappa\,T_{\mu\nu},$
(7)

where

$\mathring{G}_{\mu\nu}=\mathring{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathring{R}$

is the Einstein tensor. Being a metric-affine theory, the connection acts as a
dynamic variable in $f(Q)$ theory, however it has been shown in Refs. [59, 60]
that in the coincident gauge choice adopted in the present setting, the affine
connection’s field equation is trivially satisfied in a model-independent
manner. Hence, in this article our sole attention is devoted to the metric
field equation (7).

##  III Equations of motion in the LRS-BI model

For the current scenario, the LRS-BI metric in Cartesian coordinates given by
equation (1) is also a coincident gauge. We have the directional Hubble
parameters $H_{x}=\dot{A}/{A}$, $H_{y}=\dot{B}/{B}$ and the average Hubble
parameter

$\displaystyle(H_{x}+2H_{y})/3=H=\dot{a}/{a},$ (8)

where $a$ is considered to be the average scale factor in the discussed
anisotropic Universe. The anisotropic evolution of the Universe is
characterised by the shear scalar $\sigma$ defined by

$\sigma^{2}=\frac{1}{3}(H_{x}-H_{y})^{2}=3(H-H_{y})^{2}.$ (9)

We can calculate the non-metricity scalar $Q$ in this spacetime metric as

$Q=2H_{y}^{2}+4H_{x}H_{y}.$ (10)

From the field equations (7), we derive the following equations of motion:

$\kappa\rho=\frac{f(Q)}{2}-2f_{Q}\left[2H_{x}H_{y}+H_{y}^{2}\right],$ (11)
$\kappa
p_{x}=-\frac{f(Q)}{2}+2f_{Q}\left[3HH_{y}+\dot{H}_{y}\right]+2H_{y}\dot{Q}f_{QQ},$
(12) $\kappa
p_{y}=-\frac{f(Q)}{2}+f_{Q}\left[3H(3H-H_{y})+\dot{H}_{x}+\dot{H}_{y}\right]+\left(H_{x}+H_{y}\right)\dot{Q}f_{QQ},$
(13)

where $\rho$ is the density of the matter field, and $p_{x}$ and $p_{y}$ are
the pressures of the field along $x$ and $y$ directions respectively. Under a
situation of isotropic pressure, $p_{x}=p_{y}=p$ we have

$f_{Q}(H-H_{y})=k\,a^{-3},$ (14)

for a constant $k$. Therefore, using equation (9), alternately equation (10)
can be expressed as

$Q=6H^{2}-\frac{6k^{2}}{f^{2}_{Q}\,a^{6}}=6H^{2}-2\sigma^{2}.$ (15)

This is a handy expression to calculate the non-metricity scalar once we have
the expressions of the average Hubble parameter and the shear scalar of the
considered anisotropic Universe.

##  IV Dark energy EoS and other cosmological parameters

The field equations (7) can be equivalently written in the following effective
form:

$\mathring{G}_{\mu\nu}=-\frac{\kappa}{f_{Q}}T^{eff}_{\mu\nu}=-\frac{\kappa}{f_{Q}}T_{\mu\nu}+T^{DE}_{\mu\nu},$
(16)

where $-\kappa/f_{Q}$ is the effective gravitational constant and for its
positivity in our construction, we assume $f_{Q}<0$. The DE component emerged
from the modification of STEGR and is given by

$T^{DE}_{\mu\nu}=-\frac{1}{f_{Q}}\left[\frac{1}{2}\,g_{\mu\nu}(f(Q)-Qf_{Q})+2f_{QQ}\mathring{\nabla}_{\lambda}QP^{\lambda}{}_{\mu\nu}\right].$
(17)

Moreover, the Friedmann like equations in the STEGR determine the cosmology of
an anisotropic Universe given by the metric (1) and can be calculated from
equations (11) and (12) as (for simplicity we assume $\kappa=1$)

$\displaystyle\rho^{eff}$ $\displaystyle=$
$\displaystyle\rho-\frac{f(Q)}{2}+3H^{2}f_{Q}-\frac{3\,k^{2}}{f_{Q}\,a^{6}}$
(18) $\displaystyle=$
$\displaystyle\rho-\frac{f(Q)}{2}+3H^{2}f_{Q}-\sigma^{2}f_{Q},$

and

$p^{eff}=p+\frac{f(Q)}{2}-3H^{2}f_{Q}-2H\dot{Q}f_{QQ}+\frac{2k\dot{Q}f_{QQ}}{f_{Q}\,a^{3}}.$
(19)

The additional terms arising in these two equations are due to the non-
metricity of spacetime which behave like a fictitious DE fluid. Thus these DE
fluid components evolving due to the non-metricity can be given by

$\rho^{DE}=-\frac{f(Q)}{2}+3H^{2}f_{Q}-\frac{3k^{2}}{f_{Q}\,a^{6}},$ (20)
$p^{DE}=\frac{f(Q)}{2}-3H^{2}f_{Q}-2H\dot{Q}f_{QQ}+\frac{2k\dot{Q}f_{QQ}}{f_{Q}a^{3}}.$
(21)

On a comparison with the spatially flat FLRW case [52], we could identify two
novel terms $-3k^{2}/f_{Q}a^{6}$ and $2k\dot{Q}f_{QQ}/f_{Q}a^{3}$ appearing in
the energy and pressure equations, respectively. These terms are arised from
the anisotropy of the considered spacetime and these can be noted as the
$\rho_{\sigma}$ and $p_{\sigma}$, respectively.

Now the effective EoS can be written by using equations (18) and (19) as
follows:

$\omega^{eff}\mathrel{\mathop{\mathchar
58\relax}}=\frac{p^{eff}}{\rho^{eff}}=\frac{p+\frac{f(Q)}{2}-3H^{2}f_{Q}-2H\dot{Q}f_{QQ}+\frac{2k\dot{Q}f_{QQ}}{f_{Q}a^{3}}}{\rho-\frac{f(Q)}{2}+3H^{2}f_{Q}-\sigma^{2}f_{Q}}$
(22)

Further, the DE EoS can also be written by using equations (20) and (21) as
given by

$\omega^{DE}\mathrel{\mathop{\mathchar
58\relax}}=\frac{p^{DE}}{\rho^{DE}}=-\frac{\frac{f(Q)}{2}-3H^{2}f_{Q}-2H\dot{Q}f_{QQ}+\frac{2k\dot{Q}f_{QQ}}{f_{Q}a^{3}}}{\frac{f(Q)}{2}-3H^{2}f_{Q}+\sigma^{2}f_{Q}}.$
(23)

The effective EoS and DE EoS will help us to understand the different stages
of the evolution of the Universe. Further, we have derived the expression of
Hubble parameter in $f(Q)$ gravity for LRS-BI model by using the temporal
component of field equation i.e. equation (11) along with equations (10) and
(15). The expression of Hubble parameter takes the form:

$H=\sqrt{\frac{1}{6f_{Q}}\big{[}\frac{f(Q)}{2}-k\rho\big{]}+\frac{\sigma^{2}}{3}}.$
(24)

With the help of this Hubble parameter expression, we can also derive other
cosmological parameters like deceleration parameters, luminosity distance,
distance modulus in $f(Q)$ gravity for the model.

Now, the present scenario gives us a system of three equations (11,12,13) with
five unknowns $H_{x},H_{y},\rho,p_{x},p_{y}$ and the model $f(Q)$. So even for
model specific analysis, we need two additional assumptions to close the
system. We take the first condition in form of the barotropic EOS,
$p=\omega\rho$ and the second as the simplest physically relatable condition
$\sigma^{2}\propto\theta^{2}$. This latter condition is well-known in
anisotropy-based cosmology literature [61, 62, 63, 64, 65, 66, 67] and also
very recently being utilised in probing $f(Q)$ theory in early universe [11],
in the same article it is also showed that in radiation era, a quadratic model
$f(Q)=f_{0}Q^{2}$ reproduces the above condition. Under this proportionality,
we have the relation

$H_{x}=\lambda H_{y},$ (25)

where $\lambda$ is a constant. Therefore, the average Hubble parameter becomes

$\displaystyle H=\frac{(2+\lambda)}{3}H_{y}.$ (26)

Accordingly, equations (10, 11, 12 and 13) take the forms as follows:

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{18(1+2\lambda)}{(2+\lambda)^{2}}H^{2},$ (27)
$\displaystyle k\rho$ $\displaystyle=$
$\displaystyle\frac{f(Q)}{2}-\frac{18(1+2\lambda)}{(2+\lambda)^{2}}H^{2}f_{Q},$
(28) $\displaystyle kp_{x}$ $\displaystyle=$
$\displaystyle-\frac{f(Q)}{2}+\frac{6}{2+\lambda}f_{Q}\big{[}3H^{2}+\dot{H}\big{]}+\frac{216(1+2\lambda)}{(2+\lambda)^{3}}H^{2}\dot{H}f_{QQ},$
(29) $\displaystyle kp_{y}$ $\displaystyle=$
$\displaystyle-\frac{f(Q)}{2}+3f_{Q}\left(\frac{1+\lambda}{2+\lambda}\right)\left[3H^{2}+\dot{H}\right]+\frac{108(1+\lambda)(1+2\lambda)}{(2+\lambda)^{3}}H^{2}\dot{H}f_{QQ}.$
(30)

For equation (25), the condition derived for the pressure isotorpy i.e.
equation (14) reduce to

$\frac{k}{a^{3}}=\big{(}\frac{\lambda-1}{\lambda+2}\big{)}f_{Q}H.$ (31)

Similarly, the effective EoS and the DE EoS given by equations (22) and (23)
respectively can be written as

$\displaystyle\omega^{eff}$
$\displaystyle=\frac{p+\frac{f(Q)}{2}-3H^{2}f_{Q}-(\frac{6}{2+\lambda})H\dot{Q}f_{QQ}}{\rho-\frac{f(Q)}{2}+3H^{2}f_{Q}-\sigma^{2}f_{Q}},$
(32) $\displaystyle\omega^{DE}$
$\displaystyle=-\frac{\frac{f(Q)}{2}-3H^{2}f_{Q}-(\frac{6}{2+\lambda})H\dot{Q}f_{QQ}}{\frac{f(Q)}{2}-3H^{2}f_{Q}+\sigma^{2}f_{Q}}.$
(33)

Further, the value of $\dot{Q}$ can be written in the following form:

$\dot{Q}=36H(1+2\lambda)\Big{[}\frac{p(2+\lambda)+\frac{f(Q)(2+\lambda)}{2}-{18f_{Q}H^{2}}}{6f_{Q}(2+\lambda)^{2}+216(1+2\lambda)H^{2}f_{QQ}}\Big{]}.$
(34)

The above expression of $\dot{Q}$ leads to the required form of effective EoS
to be used for the graphical analysis purpose for various $f(Q)$ models.

For the relation between the directional Hubble parameters in equation (25),
the relation between directional scale factors can be obtained as

$A=cB^{\lambda}.$ (35)

Here, c is a constant and for simplification we have taken $c=1$. Thus, the
average scale factor takes the form: $a=B^{\frac{\lambda+2}{3}}$ and hence the
total energy density of the Universe can be found as

$\rho=\rho_{0}B^{-(1+\omega)(2+\lambda)},$ (36)

which follows from the energy-momentum conservation in the studied setting as
shown in [34]. However, the scale factor parameter is not an observational
parameter. Therefore it is better to express all the important cosmological
parameters in terms of cosmological redshift. As the direct observational data
of various parameters like Hubble parameter, distance modulus etc. against
cosmological redshift are available, so by evaluating the theoretical
expressions of all the cosmological parameters in terms of cosmological
redshift, one can compare theoretical results with the observational data. So,
if $z$ be the cosmological redshift along the $y$-direction in the LRS-BI
Universe, then the average scale factor $a$ can be derived from the metric (1)
in terms of $z$ as

$a=(AB^{2})^{\frac{1}{3}}=(1+z)^{-\frac{(2+\lambda)}{3}},$ (37)

where $B=(1+z)^{-1}$ and $A=(1+z)^{-\lambda}$. All the cosmological parameters
in the following sections will be expressed in terms of cosmological redshift
$z$.

##  V LRS-BI Universe with two different $f(Q)$ gravity models

In this section, we try to obtain various cosmological parameters for two
different form of $f(Q)$-gravity models to develop basic understanding about
the LRS-BI Universe and also try to find out the role of anisotropy in the
early stage of the Universe, or in the near past as follows:

###  V.1 $f(Q)=-(Q+2\Lambda)$ model

First we take a simple form of $f(Q)$, i.e. $f(Q)=-(Q+2\Lambda)$, where
$\Lambda$ is the cosmological constant. This model is just an extension of the
$f(Q)=-Q$ model, which mimics the classical GR results in $f(Q)$-gravity [34,
58]. Here we have added the additional cosmological constant part to get more
physical and realistic results in conformity with the present day cosmology.
It is to be noted here that this model in isotropic cosmology is nothing but
the $\Lambda$CDM model [58, 68]. Here we are interested to see its role in
LRS-BI Universe. Now, the expressions of effective EoS, DE EoS and Hubble
parameter given by equations (32), (33) and (24) respectively take the forms
for this model as

$\displaystyle\omega^{eff}$
$\displaystyle=\frac{\frac{1}{3}\Omega_{r0}(1+z)^{\frac{4(2+\lambda)}{3}}-\Omega_{\Lambda
0}+\Omega_{\sigma
0}(1+z)^{2(2+\lambda)}}{\Omega_{mo}(1+z)^{(2+\lambda)}+\Omega_{r0}(1+z)^{\frac{4(2+\lambda)}{3}}+\Omega_{\Lambda
0}},$ (38) $\displaystyle\omega^{DE}$ $\displaystyle=-1+\frac{\Omega_{\sigma
0}}{\Omega_{\Lambda 0}}(1+z)^{2(2+\lambda)},$ (39) $\displaystyle H$
$\displaystyle=H_{0}\sqrt{\Omega_{mo}(1+z)^{(2+\lambda)}+\Omega_{r0}(1+z)^{\frac{4(2+\lambda)}{3}}+\Omega_{\Lambda
0}+\Omega_{\sigma 0}(1+z)^{2(2+\lambda)}}.$ (40)

Here $H_{0}$ is the current Hubble parameter,
$\Omega_{m0}=8\pi\rho_{m0}/3H_{0}^{2}$ is the density parameter for the matter
content, $\Omega_{r0}=8\pi\rho_{r0}/3H_{0}^{2}$ is the density parameter for
the radiation content, $\Omega_{\Lambda 0}=\Lambda/3H_{0}^{2}$ is the density
parameter for the vacuum energy and $\Omega_{\sigma
0}=\sigma_{0}^{2}/3H_{0}^{2}$ is the density parameter for the anisotropy of
the present Universe [69]. The term $\rho_{m0}$ and $\rho_{r0}$ are current
values of matter density and radiation density of the Universe respectively.
Again, the term $\sigma_{0}^{2}$ in $\Omega_{\sigma 0}$ is the current value
of shear scalar which is related to the shear scalar $\sigma^{2}$ by the
relation $\sigma^{2}=\sigma_{0}^{2}a^{-6}$ and $\Omega_{\sigma 0}\leq
10^{-15}$ [70, 69].

Similarly, the deceleration parameter ($q$) and luminosity distance ($d_{L}$)
for the given $f(Q)$ model can be written as

$q=1-\frac{\lambda}{2}+(\frac{2+\lambda}{2})\big{[}\frac{\Omega_{r0}(1+z)^{\frac{4(2+\lambda)}{3}}-\Omega_{\Lambda
0}+\Omega_{\sigma
0}(1+z)^{2(2+\lambda)}}{\Omega_{mo}(1+z)^{(2+\lambda)}+\Omega_{r0}(1+z)^{\frac{4(2+\lambda)}{3}}+\Omega_{\Lambda
0}+\Omega_{\sigma 0}(1+z)^{2(2+\lambda)}}\big{]},$ (41)
$d_{L}=\frac{(\lambda+2)(1+z)^{\frac{2+\lambda}{3}}}{3H_{0}}\int_{0}^{\infty}\big{[}\frac{(1+z)^{\frac{\lambda-1}{3}}}{\sqrt{\Omega_{mo}(1+z)^{(2+\lambda)}+\Omega_{r0}(1+z)^{\frac{4(2+\lambda)}{3}}+\Omega_{\Lambda
0}+\Omega_{\sigma 0}(1+z)^{2(2+\lambda)}}}\big{]}dz.$ (42)

With the help of equation (42) we can calculate the values of the plot reduce
to distance modulus for different values of $z$ by using the relation,

$D_{m}=5\log{d_{L}}+25.$ (43)

To see and understand the behaviours of all these cosmological parameters at
different stages of evolution of the Universe, we have to plot these
parameters against the cosmological redshift $z$. For this purpose we have to
choose a reliable range of values of the model parameter i.e. $\lambda$. To
estimate this range of $\lambda$ we have plotted the Hubble parameter for
different values of $\lambda$ along with four sets of observational data, viz.
HKP and SVJO5 data [71], SJVKS10 data [72] and GCH09 data [73] on the Hubble
parameter and also with the $\Lambda$CDM model prediction up to $z=2$ as shown
in Fig. 1. The plot shows consistent results with the observational data for
values of $\lambda$ within the range of 0.5 to 1.25 with $\lambda\neq 1$. It
is to be noted that we have used Planck 2018 results [74] on the cosmological
parameters for all numerical calculations.

Figure 1: Behaviour of Hubble parameter $H(z)$ with respect to $z$ for
different values of the parameter $\lambda$ as predicted by the model
$f(Q)=-(Q+2\Lambda)$. The model predictions are in comparison with the four
sets of Hubble parameter data, viz, HKP and SVJO5 data [71], SJVKS10 data [72]
and GCH09 data [73].

Similarly, have plotted distance modulus $D_{m}$ against cosmological redshift
$z$ in Fig. 2 for the same set of $\lambda$ values as considered in the Hubble
parameter plot, along with the SCP Union 2.1 observational data [75] and the
corresponding $\Lambda$CDM model prediction. This plot also shows the
consistent agreement with the observational data.

Figure 2: Behaviour of distance Modulus $D_{m}$ with respect to cosmological
redshift $z$ for different values of the parameter $\lambda$ as predicted by
the model $f(Q)=-(Q+2\Lambda)$. The model predictions are in comparison with
the SCP Union 2.1 observational data [75].

Based on the above two plots we have constrained the values of the model
parameter $\lambda$ within the range 0.5 to 1.25. And hence using the above
considered set of values of $\lambda$, we have plotted the effective EoS
parameter $\omega^{eff}$, DE EoS parameter $\omega^{DE}$ and deceleration
parameter $q$ against the cosmological redshift $z$ in Fig. 3.

Figure 3: Behaviour of effective EoS parameter $\omega^{eff}$ (left), DE EoS
parameter $\omega^{DE}$ (middle) and deceleration parameter $q$ (right)
against the cosmological redshift $z$ for different values of parameter
$\lambda$ as predicted by the model $f(Q)=-(Q+2\Lambda)$ in comparison with
that of $\Lambda$CDM model.

We have seen that both $\omega^{eff}$ and $q$ deviate significantly from the
$\Lambda$CDM case at higher values of $z$ for all the three anisotropic cases.
However the deviation decreases to some level as the $\lambda$ tends to unity
as expected. In the case of $\omega^{DE}$ the deviation from the standard
cosmology is more vivid, prominently abrupt and takes place at smaller values
of $z$ (depending on the value of $\lambda$) in comparison to the cases of
$\omega^{eff}$ and $q$. In this case the model agrees with $\Lambda$CDM upto
$z\sim 10^{2}$, but for $z>10^{2}$ the model shows sharp increase of the value
of $\omega^{DE}$ towards the positive side. This sharp and abrupt behaviour of
$\omega^{DE}$ is due to the contribution of the anisotropic factor in the
early Universe as clear from equation (39). For higher $z$ values this factor
becomes substantial than the $\Lambda$CDM part. Thus this model shows that the
anisotropy was prominent at the early stage of the Universe as the $\lambda$
deviated from one.

###  V.2 $f(Q)=-\alpha Q^{n}$ model

This is the second model we consider in this study, which is a power law model
with a constant multiplier $\alpha$ and with the exponent $n$. The major
advantage of the power law model is that it is mathematically simple yet
powerful [76]. Also the power law model can explain the late time cosmic
acceleration and is also consistent with BBN constrains [77]. The study of
isotropic cosmology in $f(Q)$ theory while considering power law model is
found in Refs. [78, 58, 79]. For this model we can express $f(Q)$ and its
derivatives in terms of cosmological density parameters as follows

$f(Q)=-\alpha
Q^{n}=-\alpha\left[\frac{3^{\frac{1}{n}}H_{0}^{\frac{2}{n}}[\Omega_{mo}(1+z)^{(2+\lambda)}+\Omega_{r0}(1+z)^{\frac{4(2+\lambda)}{3}}+\Omega_{\Lambda
0}]}{(n-\frac{1}{2})^{\frac{1}{n}}\alpha^{\frac{1}{n}}}\right]^{n},$ (44)
$f_{Q}=-\alpha
n\left[\frac{3^{\frac{1}{n}}H_{0}^{\frac{2}{n}}[\Omega_{mo}(1+z)^{(2+\lambda)}+\Omega_{r0}(1+z)^{\frac{4(2+\lambda)}{3}}+\Omega_{\Lambda
0}]}{(n-\frac{1}{2})^{\frac{1}{n}}\alpha^{\frac{1}{n}}}\right]^{n-1},$ (45)
$f_{QQ}=-\alpha
n(n-1)\left[\frac{3^{\frac{1}{n}}H_{0}^{\frac{2}{n}}[\Omega_{mo}(1+z)^{(2+\lambda)}+\Omega_{r0}(1+z)^{\frac{4(2+\lambda)}{3}}+\Omega_{\Lambda
0}]}{(n-\frac{1}{2})^{\frac{1}{n}}\alpha^{\frac{1}{n}}}\right]^{n-2}.$ (46)

In above equations we have used the expression of Hubble parameter for this
model, which takes the form:

$H=\sqrt{\frac{3^{\frac{1}{n}}H_{0}^{\frac{2}{n}}[\Omega_{mo}(1+z)^{(2+\lambda)}+\Omega_{r0}(1+z)^{\frac{4(2+\lambda)}{3}}+\Omega_{\Lambda
0}]}{6(n-\frac{1}{2})^{\frac{1}{n}}\alpha^{\frac{1}{n}}}+H_{0}^{2}\Omega_{\sigma
0}(1+z)^{2(2+\lambda)}}.$ (47)

Now, we are in a position to derive and calculate the effective EoS parameter
($\omega^{eff}$) and DE EoS parameter ($\omega^{DE}$) from equations (22) and
(23) respectively for this model by using above expressions for $f(Q),f_{Q}$
and $f_{QQ}$. Moreover, we can derive the deceleration parameter $q$ and
luminosity distance $d_{L}$ for this model by using the following expressions:

$q=-\big{(}1+\frac{\dot{H}}{H^{2}}\big{)},$ (48)
$d_{L}=\frac{(\lambda+2)(1+z)^{\frac{2+\lambda}{3}}}{3}\int_{0}^{\infty}\big{[}\frac{(1+z)^{\frac{\lambda-1}{3}}}{H(z)}\big{]}dz.$
(49)

Finally, using equation (49) in equation (43), we can derive the expression of
the distance modulus for this power law model.

It is clear that all the cosmological parameters for this power law model
depend on three model parameters, viz. $\lambda$, $\alpha$ and $n$. Similar to
the case of previous model, to constrain these model parameters we have
plotted the Hubble parameter and the distance modulus against cosmological
redshift $z$ for different sets of values of these parameters along with the
HKP, SVJO5, SJVKS10 and GCH09 observational data and the $\Lambda$CDM plot in
Figs. 4 and 5 respectively. From these plots we constrained the parameters
$\alpha$ and $n$ respectively as $0.75\leq\alpha\leq 1.5$ and $0.95\leq n\leq
1.05$ with $n\neq 1$.

Figure 4: Hubble parameter $H(z)$ against cosmological redshift $z$ for
different sets of values of model parameters $\lambda$, $\alpha$ and $n$ as
predicted by the power law model. The model predictions are in comparison with
the four sets of Hubble parameter data, viz, HKP and SVJO5 data [71], SJVKS10
data [72] and GCH09 data [73].

Figure 5:  Distance modulus $D_{m}(z)$ against cosmological redshift $z$ for
different sets of values of model parameters $\lambda$, $\alpha$ and $n$ as
predicted by the power law model. The model predictions are in comparison with
the SCP Union 2.1 observational data [75].

(2019)

We have selected three specific sets of model parameters from their
constrained ranges as mentioned above to plot the effective EoS parameter
$\omega^{eff}$ , DE EoS parameter $\omega^{DE}$ and deceleration parameter $q$
against cosmological redshift $z$ on the basis of the agreements of these
cosmological parameters with the corresponding parameters in the $\Lambda$CDM
Universe. It is found that behaviour of these plots are similar for the same
value of $n$ with different values of other two parameters, but the behaviour
changes significantly for different values of $n$. Therefore we have added
three plots for three different values of $n$ within its constrained range for
$\omega^{eff}$, $\omega^{DE}$ and $q$ in Figs. 6, 7 and 8 respectively.

Figure 6: Effective EoS parameter $\omega^{eff}$ against cosmological redshift
$z$ for different sets of values of model parameters as predicted by the power
law model in comparison with that of $\Lambda$CDM model.

Figure 7: DE EoS parameter $\omega^{DE}$ against cosmological redshift $z$ for
different sets of values of model parameters as predicted by the power law
model in comparison with that of $\Lambda$CDM model.

Figure 8: Deceleration parameter $q(z)$ against cosmological redshift $z$ for
different values of model parameters as predicted by the power law model in
comparison with that of $\Lambda$CDM model.

From the above graphical analysis of effective EoS parameter $\omega^{eff}$
with respect to the cosmological redshift $z$ (Fig. 6) we have observed that
this parameter deviates from the $\Lambda$CDM case for higher $z$ values. The
pattern and prominence of this deviation depends on the set of model
parameters used, specially on the value of the parameter $n$. It is seen that
for higher values of $n$ the deviation becomes prominent even from
comparatively smaller values $z$. However, for all sets of constrained model
parameters this deviation becomes significant for the very large values of $z$
or in the very early Universe. On the other hand the present value of
$\omega^{eff}$ for all sets of model parameters agrees almost well with its
$\Lambda$CDM value. From the plots of the deceleration parameter $q$ against
$z$ (Fig. 8), one can see that this parameter deviate notably from the
$\Lambda$CDM result both in the early as well in the near future of the
Universe. For this parameter such deviations are significantly higher for
smaller values of the parameter $n<1$. For this model the behaviour of
$\omega^{DE}$ (Fig. 7) is peculiar and interesting throughout the evolution of
the Universe. Except for the case with $n=0.95$, other cases with $n>1$ within
the constraint range, $\omega^{DE}$ remains positive for the values of
$z<\;\sim 10^{4}$. This is also the situation to be in near feature except the
case of $\lambda=1.1$. For the values of $z>\;\sim 10^{4}$, $\omega^{DE}$
falls back to large negative values, indicating large negative pressure and
hence rapid expansion of the Universe. In the case with $n=0.95$,
$\omega^{DE}$ always remains positive for $z>0$. Here, in the near the future
part $\omega^{DE}$ remains negative and appears different from the expectation
of the $\Lambda$CDM model, however it agrees in the current phase ($z=0$).
Thus according to the power law model it can be concluded that although in the
current scenario the role of anisotropy is not so prominent, but in the early
as well as in the future Universe it may have or may play some vital role in
the evolution of the Universe.

##  VI Discussion and conclusions

In this study, we have considered a symmetric teleparallel theory equivalent
to GR (STEGR), and its extension the modified $f(Q)$ gravity theory and probed
it in a Universe with small anisotropy (in terms of the LRS-BI spacetime) to
examine the evolution of various cosmological parameters along with the
effective equation of state and hence tried to study the evolution of the
Universe. Here, we have started from deriving the non metricity scalar and
then deriving the field equations for the LRS-BI metric. Since LRS-BI is an
anisotropic model, thus we have also derived the shear scalar for our
calculations. From the field equations, we have finally derived the effective
EoS, DE EoS and other cosmological parameters. To close the system in such
anisotropic scenario, we have considered the physically relatable condition
$\sigma^{2}\propto\theta^{2}$ which in turn yielded the relation
$H_{x}=\lambda H_{y}$, $\lambda$ being a constant. We have finally derived all
the field equations and other cosmological parameters in accordance with this
assumption using the average scale factor in terms of redshift $z$.

We have considered two $f(Q)$ models in this study. In the first model, we
have taken the form of $f(Q)=-(Q+2\Lambda)$ in which $\Lambda$ is the
cosmological constant. All important cosmological parameters have been derived
using this model and found that all these expressions contained the parameter
$\lambda$. To constraint the value of $\lambda$ we have used the observational
data. For this purpose, at first we have plotted the expression of the Hubble
parameter obtained for the model with the HKP and SVJO5 [71], SJVKS10 [72],
and GCH09 [73] data in comparison with $\Lambda$CDM model for up to $z=2$. It
is found that the parameter $\lambda$ shows a good agreement with data within
the range 0.5 to 1.25 of its values with the required restriction $\lambda\neq
1$ for the considered anisotropy. We have further plotted the distance modulus
for the obtained range of $\lambda$ values along with SCP Union2.1 data [75]
and found that the plots excellently agree with the observational data as well
as the $\Lambda$CDM model. Based on these two plots, thus we have found that
$0.5\leq\lambda\leq 1.25$ with $\lambda\neq 1$ is a reliable range for the
parameter $\lambda$ in this $f(Q)$ model. The effective EoS parameter
$\omega^{eff}$, DE EoS parameter $\omega^{DE}$ and deceleration parameter $q$
are plotted using the values of the parameter $\lambda$ within the constrained
range with respect to redshift up to $10^{6}$. We found that although the
model agrees with the current scenario and the near past ($z\leq 10^{2}$), it
deviates significantly from the $\Lambda$CDM model in the early Universe.
These deviations are due to the contribution of $\Omega_{\sigma}$ parameter in
the expressions. Thus, from our analysis of this considered model of $f(Q)$,
we have found that anisotropy may have some contributions in the early
Universe. It is to be noted that for $\lambda=1$, i.e. in the isotropic case,
the expressions reduce to that of the $\Lambda$CDM model.

The second model considered in this study is in the form of power law as
$f(Q)=-\alpha Q^{n}$ in which $\alpha$ and $n$ are the two model parameters.
We have derived all expressions of cosmological parameters for this model as
we did in our previous model and found that all the cosmological parameters
depend on three model parameters $\lambda$, $\alpha$ and $n$ in this case. We
have employed the same technique to constrain these model parameters as in the
previous case, i.e. using the observational data of Hubble parameter and
distance modulus, and plot them with the results from our expressions of
Hubble parameter and distance modulus along with the $\Lambda$CDM model
prediction. It is found that the effective ranges of parameters $\lambda$,
$\alpha$ and $n$ should be $0.5\leq\lambda\leq 1.25$ with $\lambda\neq 1$,
$0.75\leq\alpha\leq 1.5$ and $0.95\leq n\leq 1.05$ with $n\neq 1$
respectively. With these values of model parameters we have plotted the
effective EoS parameter $\omega^{eff}$, DE EoS parameter $\omega^{DE}$ and
deceleration parameter $q$ with respect to the cosmological redshift $z$ and
found that behaviours of these cosmological parameters are similar for same
value of $n$ with different values of other two parameters, but their
behaviours change significantly for different values of $n$, especially at
higher $z$ values. We observed that these cosmological parameters deviate from
their $\Lambda$CDM model predictions at higher values of redshift. Thus, the
power law model also confirms the role of anisotropy in the early Universe.
Further, we have found that these deviations are more prominent for higher
values of $n$ in the case of effective EoS as in this case even in the lower
$z$ values deviations are substantial. $\omega^{DE}$ shows prominent
deviations from the $\Lambda$CDM prediction almost for all values of $n$ with
all values of other parameters nearly along the whole range of $z$ of study
with some peculiar behaviours at very early phases of the Universe depending
on the value of $n$. In the case of the deceleration parameter $q$, we have
found that it diverges too early for $n=0.95$ as compared to the other two
values of $n$. Again, for $n=1.05$, the plot shows downward trend at very high
values of $z$. It needs to be mentioned that all these plots reduce to
$\Lambda$CDM ones, when all the three model parameters i.e. $\alpha$,
$\lambda$ and $n$ become equal to unity. Thus this model gives a scenario of
our Universe where anisotropy has the dominating effect in its early phases
and some possible effect in the near future.

Therefore from our work we have tested the $f(Q)$ theory in LRS-BI metric to
understand that the anisotropic nature of the Universe by considering a
specific relation between directional Hubble parameters and found the
existence of anisotropy in the early Universe for two different models of
$f(Q)$ even though the signature of anisotropy is absent in the current stage
of the Universe. These findings could be confirmed by the cosmological data of
the early Universe, which may be available in the future from the advanced
telescopes such as Thirty Meter Telescope [80], Extremely Large Telescope
[81], CTA [82] etc. In this context more rigorous analysis of the anaisotropic
models of the Universe based on the STEGR would be interesting and necessary
in near future.

## Acknowledgments

A.D. is supported in part by the FRGS research grant (Grant No.
FRGS/1/2021/STG06/UTAR/02/1). U.D.G. is thankful to the Inter-University
Centre for Astronomy and Astrophysics (IUCAA), Pune, India for the Visiting
Associateship of the institute.

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