# Sequential Deep Learning Architectures for Anomaly Detection in Virtual
Network Function Chains ††thanks:

Chungjun Lee1, Jibum Hong2, DongNyeong Heo1, Heeyoul Choi1  1Department of
Information Communication Engineering Handong Global University, Pohang, South
Korea
{chunglee3224, sjglsks<EMAIL_ADDRESS>
2Department of Computer Science and Engineering, POSTECH, Pohang, South Korea
<EMAIL_ADDRESS>
###### Abstract

Software-defined networking (SDN) and network function virtualization (NFV)
have enabled the efficient provision of network service. However, they also
raised new tasks to monitor and ensure the status of virtualized service, and
anomaly detection is one of such tasks. There have been many data-driven
approaches to implement anomaly detection system (ADS) for virtual network
functions in service function chains (SFCs). In this paper, we aim to develop
more advanced deep learning models for ADS. Previous approaches used learning
algorithms such as random forest (RF), gradient boosting machine (GBM), or
deep neural networks (DNNs). However, these models have not utilized
sequential dependencies in the data. Furthermore, they are limited as they can
only apply to the SFC setting from which they were trained. Therefore, we
propose several sequential deep learning models to learn time-series patterns
and sequential patterns of the virtual network functions (VNFs) in the chain
with variable lengths. As a result, the suggested models improve detection
performance and apply to SFCs with varying numbers of VNFs.

###### Index Terms:

network function virtualization, anomaly detection, deep learning

##  I Introduction

Softwarization of computer networks has enabled efficient ways to deploy and
manage services with flexibility and reconfigurability. For example, hardware-
based network functions can now be replaced with virtual network functions to
release constraints related to hardware. In addition, cloud computing with
software-based networks enables service providers to utilize computing
resources for service management in data centers more efficiently [1].

Meanwhile, the number of network devices and linkages has grown over the years
[2], and the softwarization of computer networks also has increased the number
of services provided through virtual networks. The increased complexity and
size of the network introduced problems of quality degradation and
inconsistency in service availability. This extension in the network size
poses a severe management challenge [1, 2].

As service providers are aware that they need to provide dependable service,
they are interested in a data-driven approach for virtual network management,
combining big data and machine learning (ML) to manage large-scale networks
with complex dependencies efficiently. Among much prior work in this line of
research [3, 4, 5], one of the vital network management techniques is anomaly
detection, in which the quality of service is preserved by quickly responding
to a system that shows out-of-normal behavior.

Related to ML-based anomaly detection for virtual network functions (VNFs),
previous works trained ML models with data collected from an operating system
(OS) or hyper-visor of virtual machines (VMs) [1, 6, 7]. For example, in [1],
they collected a dataset and applied methods such as decision tree-based
gradient boosting machine (GBM), XGBoost, and deep neural networks (DNNs) to
detect anomalous cases from the normal ones.

However, these methods have not utilized sequential information over the VNF
sequence, which should be helpful for anomaly detection as VNF instances are
arranged in sequence over which network traffic passes [8]. In addition, if
services require different compositions of SFCs, including type, order, and
the number of VNF instances, existing approaches had to train separate models,
especially when the number of VNF instances is different because of the
discrepancy in input dimension. Furthermore, they have not employed temporal
information over time-series of monitoring data, which should be helpful if
occurrences of abnormality follow temporal patterns as in Figure 1.

To learn sequential information over the VNF sequence, we apply uni and bi-
directional recurrent neural networks (Uni- and Bi-RNNs) and Transformer. As
the sequential architectures can take a variable-length input sequence, our
proposed models are compatible with data from SFCs with different numbers of
VNFs. Such compatibility allows the capacity of joint training (or training
simultaneously) over data from different SFCs. Moreover, to learn time-series
information of monitoring data, we use RNNs to take the input of the
monitoring data at a given time-step and a series of inputs from previous
time-steps.

In order to increase the benefit of learning time-series information from
monitoring data, we add another component to the proposed models to utilize
the predicted result from the preceding time step. This is the same mechanism
as in natural language processing (NLP) models like language model and neural
machine translation [9, 10]. These modified models use prediction of the
preceding time-step as additional input to the current time step in RNNs,
which takes advantage of temporal patterns while maintaining the benefits of
the initially proposed models.

The proposed models improved detection performance over baseline models in
three anomaly detection datasets in our experiment result. Further, the
results show that joint training gives a similar level of performance to that
of the models trained on each dataset individually. Also, in our experiment
setting, the modified models, which use the preceding prediction result,
provide additional improvement for both individual and joint training cases.

The rest of the paper is organized as follows: Section 2 describes the
background for the proposed model. In section 3, we explain the motive and the
proposed deep learning models. In section 4, the experiment process and
results are presented, followed by a conclusion in section 5.

Figure 1: Time-series plot of anomaly detection data. The figure shows a time-
series plot of selected features from an anomaly detection dataset. The plot
shows a sequential pattern related to the occurrence of abnormality, where the
x and y axes indicate the time index and feature values at each time step,
respectively. Cyan and red are for normality and abnormality, respectively.

##  II Background

###  II-A Anomaly Detection Methods

Anomaly detection is a binary classification task where the model determines
whether a given system is in anomalous status. There are two types of
anomalies, including network anomaly and system anomaly which are related to
network traffic pattern and system resource usage. This paper focuses on both
types of anomalies, which lead to degradation of service quality marked by
service level violation (SLA).

In ML-based anomaly detection in the virtual network management domain, prior
literature simulated testbed environments that generate normal and abnormal
patterns of monitoring data from each component where abnormal data is
generated by anomaly injection techniques, which causes a surge effect in OS
resource usage or network traffic [6, 7, 1]. One previous research simulated
IP multimedia service (IMS) environment and anomaly injection techniques to
collect and label data, and it utilized various ML algorithms such as random
forest (RF) to show strong detection performance [6]. Another prior literature
simulated a network environment that utilizes network function virtualization
(NFV), where VNFs were deployed and connected through SFC in web service and
login authentication scenarios, and an array of ML algorithms were trained and
evaluated over the collected dataset [1]. In this paper, we follow the
experiment setting of [1] for anomaly detection in SFC based on monitoring
data from the VNFs.

There is much recent work that uses an unsupervised learning model. In
contrast to supervised learning approaches, they do not require labeling data
because they are trained on only normal patterns of monitoring data [11, 12,
13]. Such models can measure the deviance of observed monitoring data at test
time and raise an alert when the deviance is significant. One of the
approaches computes a set of centroids using the BIRCH algorithm to define the
boundaries for normal and abnormal data [11]. Other approaches utilize various
algorithms for modeling time-series forecasting such as autoregressive
integrated moving average (ARIMA) or recurrent neural networks (RNNs) to learn
patterns of monitored data at normal situations for anomaly detection [12,
13]. Unsupervised learning method has benefits compared to supervised learning
in that it is applicable even if labeling data is not available, while
unsupervised learning models lacked performance compared to supervised
learning models [14].

###  II-B Sequential Deep Learning Models

Sequential deep learning models include Uni(or Bi)-RNNs and Transformer. RNNs
have a recurrent connection to capture sequential patterns in data, and Bi-
RNNs have stacked forms of RNNs to read the sequence in both directions [15,
16]. Transformers are a neural network architecture that can compute the
representation of sequence data based on self-attention without explicit
recurrent connections [17]. Transformer networks showed state-of-the-art
performance in modeling sequential patterns, especially in NLP. More detailed
descriptions of sequential deep learning models can be found in [15, 17].

Figure 2: SFCs with varying numbers of VNFs. The figure describes two
different SFCs deployed in different traffic environments. Our proposed models
have compatibility with monitoring data from different numbers of VNFs. Some
of the shapes in this figure are adopted from [3].

##  III Proposed Method

###  III-A Problem definition

In the previous approach [1], the problem can be defined as learning a
parametric function $\hat{y}_{t}=f(X_{t};\theta)$ which produces a binary
output $\hat{y}_{t}$ where $t$ and $\theta$ are time index and parameter of
the model, respectively. Also, $X_{t}\in\mathbb{R}^{V\times D_{input}}$ is the
input, where $V$ is number of VNFs in the SFC, and $D_{input}$ is the number
of selected metrics (or features) from a VNF instance such as CPU or memory
utilization rates.

In our approach, we consider both the input at current time index and previous
series of inputs. Therefore, input is
$X_{t-l+1}^{t}=[X_{t-l+1},X_{t-l+2},...,X_{t}]$, where $l$ is the sequence
length. Additionally, each $X_{i}$ is considered as a set of monitored data
from each VNF instance, $X_{i}=[x_{i,1},x_{i,2},...x_{i,V}]$ where
$x_{i,v}\in\mathbb{R}^{D_{input}}$ with $t\in[l,...,T]$. Here, $T$ and $i$
denote respectively the number of samples and arbitrary time index. Note that
$X_{i}$ is a sequence of several VNFs and the models should be compatible with
different sizes.

###  III-B Proposed Neural Network model

The proposed model is composed of feature mapping, encoder, readout, and
classifier layers. The first three layers enable learning sequential patterns
over VNF sequence with variable lengths where input $X_{i}$ of arbitrary
sequence length is encoded into fixed-length vector $\tilde{z}_{i}$.
Therefore, proposed architectures improve performance and are compatible with
monitoring data collected from different SFCs, as shown in Figure 2.

First, the feature mapping layer is fully connected layer without activation
function, and it maps input vector $x_{i,v}$ to
$\tilde{x}_{i,v}\in\mathbb{R}^{D_{z}}$, where $D_{z}$ is the dimension of
feature mapping. Second, the encoder layer is sequential deep learning
architecture like Uni-RNN, Bi-RNN and Transformer. It takes input sequence
$\tilde{X}_{i}=[\tilde{x}_{i,1},\tilde{x}_{i,2},...,\tilde{x}_{i,V}]$ and
outputs $Z_{i}=[z_{i,1},z_{i,2},...,z_{i,V}]$. Third, the readout layer is one
of functions like _max_ , _mean_ , or _self-attention_. _Max_ and _mean_ are
conventional pooling methods to combine multiple and varying numbers of input
vectors. The _self-attention_ mechanism summarizes the set of vectors by a
weighted sum, where the weights can be computed based on the sequence with
parameters, $\theta_{att}$ [10, 18]. Output of the readout layer is
$\tilde{z}_{i}=readout([z_{i,1},z_{i,2},...,z_{i,V}])$, where
$\tilde{z}_{i}\in\mathbb{R}^{D_{z}}$. Note the first three layers apply to
monitoring data from SFC with different numbers of VNFs. The forward equation
of the first three layers are summarized as,

$\displaystyle\tilde{X}_{i}$ $\displaystyle=$
$\displaystyle\text{\emph{feature\\_mapping}}(X_{i};\theta_{map}),$ (1)
$\displaystyle Z_{i}$ $\displaystyle=$
$\displaystyle\text{\emph{encoder}}(\tilde{X}_{i};\theta_{enc}),$ (2)
$\displaystyle\tilde{z}_{i}$ $\displaystyle=$
$\displaystyle\text{\emph{readout}}(Z_{i};\theta_{att}).$ (3)

The classifier layer is composed of RNN and DNN. This layer captures
sequential patterns existing over time-steps. In this layer, RNN takes input
sequence, $[\hat{z}_{t-l+1},\hat{z}_{t-l+2},...,\hat{z}_{t}]$, and outputs
$[h_{t-l+1},h_{t-l+2},...,h_{t}]$. Then DNN takes RNN hidden state, $h_{t}$ as
input to produce the detection result, $\hat{y_{t}}=DNN(h_{t})$. In summary,
the forward equations of the classifier layer and the objective function $J$
for the binary classification are given as,

$\displaystyle h_{t-l+1}^{t}$ $\displaystyle=$
$\displaystyle\text{\emph{RNN}}(\tilde{z}_{t-l+1}^{t};\theta_{RNN}),$ (4)
$\displaystyle\hat{y}_{t}$ $\displaystyle=$
$\displaystyle\text{\emph{DNN}}(h_{t};\theta_{DNN}),$ (5) $\displaystyle
J(y_{t},\hat{y_{t}})$ $\displaystyle=$ $\displaystyle-
y_{t}log(\hat{y_{t}})-(1-y_{t})log(1-\hat{y}_{t}).$ (6)

Here, the trainable parameter includes
$\theta_{map},\theta_{enc},\theta_{att},\theta_{RNN}$, and $\theta_{DNN}$.
They are optimized using gradient-descent method to minimize the objective
function $J$ which is the binary cross-entropy (BCE) [19].

Additionally, we use prediction result from the previous time-step for further
improvement. As we observed that incidents of abnormality are highly
correlated to the event in the preceding data point, inputting prediction
results of preceding time-step should be beneficial to prediction at the
current time-step. To use this information, we modify the model to use
previous prediction result, $\hat{y}_{i-1}$ together with current input
$X_{i}$ by concatenation. Therefore, the input is changed to
$X_{i}=[(\hat{y}_{i-1}\oplus x_{i,1}),(\hat{y}_{i-1}\oplus
x_{i,2}),...,(\hat{y}_{i-1}\oplus x_{i,V})]$. The overview of our proposed
model is shown in Figure 3.

Figure 3: Overview of the proposed model. The feature mapping, encoder, and
readout layers process input sequence into fixed dimensional vector,
$\tilde{z}_{t}$. At each time-step, input data, $X_{t}$ is concatenated with
prediction result from previous time-step, $\hat{y}_{t-1}$., so that
$[X_{t-l+1},X_{t-l},...,X_{t}]$ is forwarded to
$[\tilde{z}_{t-l+1},\tilde{z}_{t-l},...,\tilde{z}_{t}]$ and to the classifier
layer. Note the feature mapping layer is omitted for brevity.

##  IV Experiment

This section provides a brief description of the experiment setting, data
acquisition and presents results with analysis. We designed experiments to
check the benefits of two contributions in the proposed models: improvement of
detection performance and compatibility.

###  IV-A Virtual Network Setup

We set up two virtual network scenarios from which we collect anomaly
detection data. In the web hosting service scenario, the client sent requests
to the webserver, going through 5 VNFs: FW, IDS, FM, DPI, and LB, and we refer
to the dataset collected in this setting as web service data (WSD). In the
login authentication scenario, the server was connected to the database
system. The client sent requests to the server through 4 VNFs: FW, FM, DPI,
and IDS, which leads to login authentication data (LAD). Depending on how
strict the SLA was, two sets were gathered: LAD1 and LAD2 (See the next
section). The topology of the testbed [20] was closely related to the multi-
access edge computing (MEC) scenario, and OpenStack (rocky release) was used
for constructing virtual network and SFC. Open source VNFs were considered,
such as firewall (FW, iptables [21]), intrusion detection system (IDS,
Suricata [22]), flow monitor (FM, ntopng [23]), deep packet inspection (DPI,
nDPI [24]), and load balancer (LB, HAProxy [25]).

###  IV-B Data Acquisition Components

To gather data, our ADS had components for monitoring, fault injection, metric
selection, and data labeling. A more detailed description of each component of
ADS can be found in [1].

####  IV-B1 Monitoring

From VMs which composed virtual network, we monitored metrics related to CPU,
memory, disk I/O, and network traffic. The monitoring function was composed of
a monitoring agent, service, and dashboard. Collectd [26], InfluxDB [27], and
Grafana [28] were used for the implementation of each component.

####  IV-B2 Fault Injection

Since anomalies do not exist frequently in a given virtual network setting,
various software and hardware faults were injected into the system on which
VNFs were deployed. The first fault injection method was a generation of
abnormal states to VMs: CPU utilization, memory usage, disk I/O, network
latency, and network packet loss through a tool ‘stress-ng’ [29]. The second
method was a generation of heavy workload by sending tremendous network
traffic using D-ITG traffic generator [30].

####  IV-B3 Metric Selection and Data Labeling

We selected input metrics from monitoring as shown in Table I. The labeling of
the data was based on SLA-related metrics as in [31]. As the response time and
availability were measured, the monitoring data collected when response time
(less than 250ms) or availability (over 99.95% success rate of requests) was
not satisfied are labeled as an anomaly. The LAD2 dataset applies a more
strict SLA than that for WSD and LAD1 datasets (less than 200ms for response
time and 99.99 % success rate of requests). Statistics of the datasets are
summarized in Table II.

Table I: Selected metrics for anomaly detection, adopted from [1] Metrics | Descriptions
---|---
time | Measurement time
instance | VNF instance name
cpu_idle | CPU - idle time
cpu_interrupt | CPU - interrupt time
cpu_nice | CPU - nice status time
cpu_softirq | CPU - softirq time
cpu_steal | CPU - stolen time
cpu_system | CPU - used by kernel mode
cpu_user | CPU - used by user mode
cpu_wait | CPU - I/O wait time
mem_free | Memory - free space
mem_buffered | Memory - buffered space
mem_cached | Memory - cached space
mem_used | Memory - used space
disk_free | Disk - free space
reserved | Disk - reserved space
disk_used | Disk - used space
io_read | I/O read bytes
io_write | I/O write bytes
io_time | I/O - spent time
network_rx_bytes | Received traffic bandwidth
network_tx_bytes | Transmitted traffic bandwidth
network_rx_packets | The number of received packets
network_tx_packets | The number of transmitted packets
network_latency | Hop latency between VNFs
Table II: Statistics of anomaly detection datasets. The numbers of anomalies for training, validation, and test sets are in the parentheses. Dataset | WSD | LAD1 | LAD2
---|---|---|---
VNF instances | 5 | 4 | 4
Total samples | 68,731 | 121,053 | 121,024
Anomalies | 26,354 | 19,913 | 44,513
Training set | 44,665 (16,933) | 78,675 (12,910) | 78,656 (28,734)
Validation set | 6,872 (2,690) | 12,103 (1,997) | 12,100 (4,465)
Testing set | 17,178 (6,731) | 30,259 (5,006) | 30,252 (11,314)
Table III: Performances of initial models which do not employ prediction result of the preceding time-step.  Training setting | Model | WSD | LAD1 | LAD2
---|---|---|---|---
Encoder | Readout | F1-measures | F1-measures | F1-measures
Individual training | GBM [1] | 96.19 | 95.20 | 93.27
XGBoost [1] | 95.81 | 95.30 | 93.81
DNN [1] | 93.14 | 90.27 | 90.77
RNN |  Max / Mean / Self-attention | 97.52 / 97.46 / 97.65 | 97.64 / 97.77 / 97.75 | 95.79 / 96.01 / 95.77
Bi-RNN | 97.59 / 97.72 / 97.59 | 97.95 / 97.95 / 97.77 | 95.74 / 95.97 / 95.81
Transformer | 97.69 / 98.11 / 98.04 | 97.69 / 97.87 / 97.83 | 95.79 / 95.88 / 96.00
WSD & LAD2 | RNN |  Max / Mean / Self-attention | 97.49 / 97.52 / 97.74 |  | 94.91 / 95.04 / 95.19
Bi-RNN | 97.43 / 97.50 / 97.68 | NA | 95.09 / 95.11 / 95.01
Transformer | 97.91 / 98.17 / 98.04 |  | 95.64 / 95.30 / 95.45
Table IV: Performances of the modified models which utilize the output of the previous time-step. Training setting | Model | WSD | LAD1 | LAD2
---|---|---|---|---
Encoder | Readout | F1-measures | F1-measures | F1-measures
Individual training | RNN |  Max / Mean / Self-attention | 100.00 / 100.00 / 100.00 | 99.98 / 99.99 / 99.98 | 99.97 / 99.94 / 99.96
Bi-RNN | 99.99 / 99.99 / 99.99 | 99.99 / 99.99 / 100.00 | 99.95 / 99.97 / 99.97
Transformer | 99.99 / 100.00 / 100.00 | 99.96 / 99.98 / 99.98 | 99.97 / 99.95 / 99.98
WSD & LAD2 | RNN |  Max / Mean / Self-attention | 99.99 / 99.98 / 99.99 |  | 99.92 / 99.91 / 99.90
Bi-RNN | 99.99 / 99.99 / 99.98 | NA | 99.94 / 99.92 / 99.89
Transformer | 100.00 / 100.00 / 100.00 |  | 99.94 / 99.91 / 99.92

###  IV-C Experiment Result

We trained and tested our models on three different datasets: WSD, LAD1, and
LAD2.

####  IV-C1 Initial Models

First, we test our models without the detection result of the previous time-
step, $\hat{y}_{t-1}$. The detection performance of the proposed models are
presented in Table III. In the table, The first group of models is baselines
where GBM and XGBoost are best performing ML models from prior literature [1].
For the second group of models, we experimented with possible combinations of
an encoder (RNN, Bi-RNN, and Transformer) and a readout method (Max, Mean, and
self-attention). For example, F1-measures for Bi-RNN + Max and Transformer +
Mean on LAD1 are 97.95 and 97.87, respectively.

The training setting, ‘individual training,’ shows that one model is trained
and evaluated in each dataset. On the other hand, ‘WSD & LAD2’ indicate that
one model was jointly trained on both datasets and tested on each of them.
Note that LAD1 and LAD2 share the same environment from which the dataset was
generated, but the difference is labeling criteria where LAD2 applied more
strict and suitable criteria for the given setting [1]. Therefore, we carried
out a joint training experiment only with ‘WSD & LAD2’ datasets.

As shown in the table, the proposed models improve F1-measure over the
baselines. In addition, the models have an additional advantage in that they
can use monitoring data from variable length SFCs as input, enabling joint
training. In given datasets, jointly trained models show a similar level of
performance as those trained exclusively on the dataset. Such a result
demonstrates the potential of the proposed models that they can be used for
different SFC setups without requiring two separate detection models.

(a) Detection result of Transformer + self-attention, which does not utilize
the detection results in the previous time-step.

(b) Detection result of Transformer + self-attention, which utilizes the
detection results in the previous time-step, $\hat{y}_{t-1}$

.

Figure 4: Performance comparison between models with DNN classifier and RNN
classifier. The plots highlight detection performance on abnormal points.
Among abnormal data points, correctly classified points are marked with cyan,
and misclassified data points are marked red. Black points are normal data
points. While plots show that both the initial and modified models show high
detection accuracy, the latter performs better.

####  IV-C2 Improvement in the Modified Models

In addition to the initial models, modified models which utilize the
prediction result from the preceding time-step $\hat{y}_{t-1}$ are tested and
the results are summarized in Table IV. The table shows that they result in
significant improvements in detection performance, almost reaching perfect
detection performance. The datasets used in the experiment were especially
suitable to show the benefit of this approach because normal and abnormal data
occur consecutively in intervals. We visualize the detection performance of
two models in Figure 4. Their difference is the usage of $\hat{y}_{t-1}$ to
predict the current time-step. Moreover, similar to individual training, the
models which utilize $\hat{y}_{t-1}$ work well in joint training settings with
different sequence lengths in SFCs as in Table IV.

##  V Conclusion

Based on the sequential deep learning architectures like Uni-RNN, Bi-RNN, and
Transformer, we proposed anomaly detection models composed of several neural
network layers: feature mapping, encoder, readout, and classifier. The
proposed models can learn sequential patterns over the sequence of VNFs and
time steps in anomaly detection data, and they are compatible with the SFCs
that have a varying number of VNF instances. The experiment results show
improvement of F1-measure compared to the baselines. Furthermore, the results
of joint training ensure that our proposed models could learn from different
scenarios simultaneously. The future direction can be to study further the
performance of the proposed models for different network traffic scenarios
such as that of service for mobile connections.

## Acknowledgement

This research was supported by the Institute for Information & communications
Technology Promotion(IITP) grant funded by the Korea government(MSIT) (No.
2018-0-00749, Development of virtual network management technology based on
artificial intelligence).

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,

# Asymptotically safe cosmology with non-canonical scalar field

Rituparna Mandal<EMAIL_ADDRESS>School of Physics, University of
Hyderabad, Central University P.O, Hyderabad 500046, Telangana, India  Soma
Sanyal<EMAIL_ADDRESS>School of Physics, University of Hyderabad,
Central University P.O, Hyderabad 500046, Telangana, India

###### Abstract

We investigate the quantum modified cosmological dynamical equations in a
Friedmann-Robertson-Walker universe filled with a barotropic fluid and a
general non-canonical scalar field characterized by a Lagrangian similar to
k-essence model but with a potential term. Quantum corrections are
incorporated by considering the running of gravitational and potential
couplings, employing the functional renormalization group approach. Covariant
conservation of the non-canonical scalar field and the background barotropic
fluid is considered separately, imposing a constraint resulting from the
Bianchi identity. This constraint determines the evolution of the cut-off
scale with the scale factor and also reveals cosmic fixed points, depending on
whether the flow ceases or continues to evolve. We explore how the general
non-canonical scalar field parameter affects the different types of cosmic
fixed points and how it differs from the canonical case. Furthermore, we
establish a bound on the ratio of the RG parameters involving the non-
canonical parameter for which the universe may exhibit accelerated expansion
for mixed fixed points. This bound indicates the non-canonical scalar field
includes larger sets of RG fixed point which may give rise to an accelerated
universe.

##  I Introduction

A new era of cosmological study began with the observational discovery of the
late-time cosmic acceleration reported in 1998, based on the type Ia
supernovae (SN Ia) Riess:1998 . This has significantly transformed our
understanding of the universe over the past years. The source responsible for
this late-time acceleration, composed of gravitationally repulsive material,
remains unclear and has been dubbed as the “dark energy,” constituting
approximately 68$\%$ of the total energy density at present. The proposal of
the cosmological constant, with a constant energy density, serves as a simple
candidate for generating the late-time cosmic acceleration. This proposal is
consistent with various observations, including the power spectrum of the CMB,
the redshift of distant supernovae, and the distribution of the large-scale
structure. However, the exceedingly small value of the cosmological constant,
compared to the theoretical expectations without definitive reasons, has
prompted theorists to explore a diverse array of alternative explanations.
These alternatives include considering specific matter forms such as
quintessence, k-essence, Chaplygin gas, Phantom in the matter sector, or
modifications to General Relativity (GR) in the gravity sector.
In the matter sector, scalar field models play a significant role in
cosmological models for understanding the accelerating universe, such as
inflation in the early epoch Liddle:2000  and late-time acceleration by dark
energy Tsujikawa:2006 ; Copeland:1998 . If the scalar field potential varies
with energy scale or quantum effects are included through the coupling
constant, the important question we want to ask is how these effects could
impact the dark energy model of the scalar field. A useful and efficient way
of including quantum effects is the Wilsonian Renormalization Group (RG)
approach Wilson:1971 , which relies on coarse-graining procedures and produces
the scale-dependent effective average action that can interpolate between
short and long-distance scales. This effective average action includes all
quantum fluctuations above a certain momentum scale, while quantum
fluctuations below this scale are not included. Hence, this effective action
describes the Renormalization Group trajectories, containing a number of
coupling constants that run along with the cut-off scale, called the RG flow
Reuter:1996cp ; Souma:1999at ; Lauscher:2001ya ; Niedermaier:2006wt ;
Litim:2004 . In this approach, one can conceive of gravity free from
divergence and predictive at a high energy scale, unlike the standard field-
theoretical perturbative approach where gravity is non-renormalizable. The
idea that gravity is non-perturbatively renormalizable and divergenceless at
high energy, as the RG flow converges to a non-trivial UV fixed point, making
it an “Asymptotically Safe” quantum theory of gravity, was first proposed by
S. Weinberg Weinberg:1976  in the late 1970s.
In the past decade, the scenario of asymptotically safe gravity, dictated by
asymptotic safe fixed points at the UV scale, has led to the exciting idea
that it may control the beginning of the universe Bonanno:2001xi . This idea
has been explored in the literature extensively, including those involving
ideal fluid Bonanno:2001xi ; Koch:2011 , even within the context of modified
gravity theories like $f(R)$ theory. The Gaussian fixed point or infrared
fixed point may provide the possibility of an accelerated universe, as
observed today Weinberg:2010 ; Contillo:2011 . The functional RG technique in
asymptotically safe gravity, where the cosmological constant and Newton’s
constant vary with the momentum scale, has been used to study deviations from
standard cosmology Bonanno:2001xi ; Shapiro:2005 ; Mandal:2020 . Moreover,
quantum modifications to the cosmological evolution of the universe with a
barotropic background fluid and the canonical, minimally coupled scalar field
can yield the accelerated universe Hindmarsh:2011 ; Linder:2011 . In our case,
we aim to investigate late-time acceleration by considering quantum
gravitational effects within the framework of the exact renormalization group
flow in a non-canonical scalar field model.
The motivation for introducing a non-canonical scalar field comes purely from
phenomenological considerations, such as from the Born-Infeld action of string
theory Sen:2002 ; Sachs:2003 , initially introduced in the context of
inflation Mukhanov:1999 ; Mukhanov:1999a ; Unnikrishnan:2012 . Later, it was
extended for dark energy Chiba:2000 ; Steinhardt:2000 ; Steinhardt:2001 ;
Chiba:2002 ; Scherrer:2004 ; Chimento:2004  models due to its intriguing
characteristics as a modified matter model. Usually, the most general form for
the Lagrangian of the non-canonical scalar field, containing non-canonical
kinetic term and potential is in the form
$\mathcal{L}(\phi,X)=f(\phi)F(X)-V(\phi)$ where
$X=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$ represents the kinetic
energy, and $\phi$ is the scalar field. The Lagrangian without the need for a
potential term ($V(\phi)=0$) can drive inflationary evolution, and it is
referred to as k-essence Mukhanov:1999 . The Lagrangian, given by
$\mathcal{L}(\phi,X)=F(X)-V(\phi)$ (with $f(\phi)=\text{constant}$), which we
can view as a generalization of quintessence, plays a crucial role in studying
dark energy models Wands:2013 ; Bose:2009 . It can also be employed to unify
dark energy and dark matter models, as discussed in Cota:2011 ; Sahni:2017 ;
Sahni:2021 . This model is referred to as the general non-canonical scalar
field model, first studied in Fang:2007 . In this type of model, we have
incorporated renormalization group corrections through Newton’s constant and
potential to investigate late-time effects in asymptotically safe gravity. We
seek cosmic fixed points using a coupled dynamical equations system approach,
examining how this cosmic fixed point could correspond to RG fixed points in
the RG flow. Our focus is primarily on finding the cosmic fixed points that
may lead to an accelerated universe in the infrared limit (corresponding to
dark energy dominated universe) or the UV limit (inflating universe),
depending on whether the cosmic fixed points correspond to the Gaussian RG
fixed point or the UV fixed point. Our motivation lies in understanding how
the general non-canonical scalar field parameters affect the cosmic fixed
points and how this differs from the canonical case Hindmarsh:2011 ;
Linder:2011 .
In this work, we shall employ the Einstein-Hilbert truncation Reuter:2001ag
in the gravity sector and incorporate corrections at the level of the Einstein
equation, ensuring that the variation of the action with respect to the metric
leaves the form of the Einstein field equations unaltered. The RG flow comes
in the Einstein equations through the non-canonical scalar field model
potential $V_{k}(\phi)$ and Newton’s constant $G_{k}$. As here we have taken
the covariant conservation of the background barotropic fluid and the non-
canonical scalar field, it ensures that the Bianchi identity will provide us a
constraint involving the RG parameters and the dimensionless cosmological
variables (relative energy density of a component with respect to total energy
density). This constraint is the same as the canonical scalar field case in
Hindmarsh:2011  because the kinetic term in Lagrangian does not flow with the
cut-off scale in the non-canonical scalar field similar to the canonical
scalar field. Though the Bianchi identity poses the same constraint, we will
see that the non-canonical parameter significantly changes the cosmic fixed
point.
In here, we incorporate the above ideas to formulate the cosmological
evolution equations for dimensionless cosmological variables, which evolve
with the scale factor for a universe that consists of a background barotropic
fluid and the non-canonical scalar field. We find several cosmological fixed
points from these cosmological dynamical equations using the formalism first
described in Copeland:1998 , which may give an accelerated universe depending
upon the diverse range of possible values for the RG parameters within the
specific renormalization group theory. Moreover, we establish a bound on the
RG parameters involving the non-canonical parameter, which could confirm that
the non-canonical scalar field model includes larger sets of RG fixed points
that may give us an accelerated universe.
The organization of this paper is as follows. In Sec. II, we set up the
Friedmann equations, the continuity equations, and the constraint equations
between the RG parameters and dimensionless cosmological variables for a FLRW
universe which consists of a barotropic fluid and the non-canonical scalar
field taking into account quantum gravity corrections trough the scale-
dependent Newton’s constant $G_{k}$ and the potential of the non-canonical
scalar field $V_{k}(\phi)$. In Sec. III, we use these to write the quantum
corrected cosmological evolution equations and find the different cosmic fixed
points which may give an accelerated universe. We conclude with a discussion
of our results.

##  II RG improved cosmology

Initially, we start by considering a spatially flat Friedmann-Lemaître-
Robertson-Walker (FLRW) universe filled with background matter and a general
non-canonical scalar field that interacts with gravity through the kinetic
energy term. The Lagrangian for this non-canonical scalar field model is given
by

$\displaystyle\mathcal{L}(\phi,X)=F(X)-V_{k}(\phi)$ (1)

where $X=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$ is the kinetic
energy term. We have taken Einstein-Hilbert truncation in the gravity sector
where the quantum corrections are incorporated through the couplings including
the gravitational couplings and the non-canonical scalar field potential
amplitude, which run along with the momentum scale $k$.
The general form of the field equations for the above-mentioned general action
is given by

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi
G_{k}}{3}\left(T_{\mu\nu}^{\phi}+T_{\mu\nu}^{bm}\right)$ (2)

where $T_{\mu\nu}^{\phi}$ describe the energy-momentum tensor for the non-
canonical scalar field and $T_{\mu\nu}^{bm}$ is the background matter energy-
momentum tensor in the form of a perfect fluid. Here, we have used the
subscript $k$ on gravitational constant $G_{k}$ and potential $V_{k}$ to
remind ourselves that these quantities flow with the momentum scale $k$. Note
that the RG improvement has been made at the level of the field equations and
is a kind of “shortcut” that captures the effects of quantum gravity to the
leading order.
For the non-canonical scalar field Lagrangian mentioned in (1), the energy-
momentum tensor for this case is

$\displaystyle
T_{\mu\nu}^{\phi}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}\right)}{\delta
g^{\mu\nu}}=g_{\mu\nu}\left[F(X)-V_{k}(\phi)\right]+F_{,X}\partial_{\mu}\phi\partial_{\nu}\phi$
(3)

where “${,X}$” denotes the partial derivative with respect to $X$. The energy-
momentum tensor of a non-canonical scalar field is equivalent to a perfect
fluid energy-momentum tensor

$\displaystyle T_{\mu\nu}=(p+\rho)u_{\mu}u_{\nu}+pg_{\mu\nu}$ (4)

with velocity $u_{\mu}=\frac{\partial_{\mu}\phi}{\sqrt{2X}}$, pressure
$p_{\phi}(k)=\mathcal{L}=F(X)-V_{k}(\phi)$ and energy density

$\displaystyle\rho_{\phi}(k)=2XF_{,X}-p_{\phi}(k)=2XF_{,X}-F(X)+V_{k}(\phi).$
(5)

In the FLRW universe, in the presence of a background matter field with energy
density $\rho_{bm}$ and pressure $p_{bm}$, we obtain the modified Friedmann
equations from (2) as

$\displaystyle H^{2}$ $\displaystyle=\frac{8\pi
G_{k}}{3}\left(2XF_{,X}-F(X)+V_{k}(\phi)+\rho_{bm}\right)\,$ (6)
$\displaystyle\dot{H}$ $\displaystyle=-4\pi
G_{k}\left(2XF_{,X}+\rho_{bm}(1+\omega_{bm})\right)\,$ (7)

where $H=\frac{\dot{a}}{a}$ is the Hubble constant.
Another important point is to note that the left-hand side of the Einstein
field equation is covariantly conserved, so by virtue of consistency, the
right-hand side must also be covariantly conserved. This is called the Bianchi
identity. Incorporating the Bianchi identity which implies $\left[8\pi
G_{k}\left(T^{\phi\mu\nu}+T^{bm\mu\nu}\right)\right]_{;\nu}=0$, we obtain

$\displaystyle\frac{\partial\rho_{tot}(k)}{\partial
t}=-3H\rho_{tot}\left[1+\frac{p_{tot}(k)}{\rho_{tot}(k)}+\frac{1}{3}\frac{\partial\ln(G_{k}\rho_{tot})}{\partial\ln
k}\frac{\mathrm{d}\ln k}{\mathrm{d}N}\right]$ (8)

where $\rho_{tot}(k)=\rho_{\phi}(k)+\rho_{bm}$ and $N=\ln a$. We work here
with the assumption that the energy-momentum tensor for the non-canonical
scalar field and barotropic fluid is covariantly conserved separately,
indicating no interaction between these two components. The covariant
conservation of energy-momentum tensor for the non-canonical scalar field can
be written as

$\displaystyle\frac{\partial\rho_{\phi}(k)}{\partial
t}+3H\left(\rho_{\phi}(k)+p_{\phi}(k)\right)=0~{},$ (9)

which in turn gives

$\displaystyle\frac{\partial}{\partial
N}\left[2XF_{,X}-F(X)+V(\phi)\right]+6F_{,X}=0~{}.$ (10)

Further, the covariant conservation of the energy-momentum tensor of the
background non-canonical scalar field yields

$\displaystyle\dot{\rho}_{bm}+3H(p_{bm}+\rho_{bm})=0\,.$ (11)

Since we have separately considered the energy-momentum tensor for the non-
canonical scalar field and the barotropic fluid to be covariantly conserved,
this leads to the consistency equation from the Bianchi identity as:

$\displaystyle\frac{\mathrm{d}\ln
k}{\mathrm{d}N}\frac{\partial\ln(G_{k}\rho_{tot})}{\partial\ln k}=0$
$\displaystyle\Rightarrow$ $\displaystyle\frac{\mathrm{d}\ln
k}{\mathrm{d}N}\left[\frac{\mathrm{d}\ln G_{k}}{\mathrm{d}\ln
k}+\frac{\kappa^{2}V_{k}}{3H^{2}}\frac{\partial\ln V_{k}}{\partial\ln
k}\right]=0.$ (12)

Now, we introduce the dimensionless RG flow parameters arising due to the
scale dependence through the gravitational constant and the non-canonical
scalar field potential

$\displaystyle\eta_{RG}=\frac{\partial\ln G_{k}}{\partial\ln
k},~{}~{}\nu_{RG}=\frac{\partial\ln V_{k}}{\partial\ln k}~{}.$ (13)

We find the consistency equation (12) in terms of RG flow parameters as

$\displaystyle\frac{\mathrm{d}\ln
k}{\mathrm{d}N}\left[\eta_{RG}+\frac{\kappa^{2}V_{k}(\phi)}{3H^{2}}\nu_{RG}\right]=0~{}.$
(14)

This condition gives rise to two possibilities; one is $\frac{\mathrm{d}\ln
k}{\mathrm{d}N}=0$ for all time, signifying the evolution of $\ln k$ with $N$.
This comes to a halt at some freeze-in scale. Another possibility can be
expressed as

$\displaystyle\eta_{RG}+\frac{\kappa^{2}V_{k}(\phi)}{3H^{2}}\nu_{RG}=0~{}.$
(15)

These constraints on the RG flow parameters play an important role in
understanding the cosmic dynamics or time evolution of the system when the
quantum modifications are included through the running of the gravitational
constant and the non-canonical scalar field potential.

##  III Renormalization group improved system of dynamical equations

In this section, we express the renormalization group improved system of
dynamical equations to study the cosmic dynamics in the presence of a non-
canonical scalar field and a background fluid. For this purpose, we introduce
two dimensionless variables:

$\displaystyle
x=\frac{\kappa\sqrt{2XF_{,X}-F}}{\sqrt{3}H}~{},~{}~{}~{}~{}y=\frac{\kappa\sqrt{V_{k}(\phi)}}{\sqrt{3}H}~{}$
(16)

where $\kappa^{2}=8\pi G_{k}$. Here, $x^{2}$ represents the relative energy
density of the kinetic part of the scalar field $\rho_{ki}=2XF_{,X}-F$ defined
in (5), and $y^{2}$ represents the relative energy density of the potential
part defined as $\rho_{V}=V_{k}(\phi)$ with respect to the total energy
density $\rho_{tot}(k)=\rho_{\phi}+\rho_{bm}=\frac{3H^{2}}{\kappa^{2}}$
respectively. We define the equation of state for the kinetic part of the
scalar field as:

$\displaystyle\omega_{ki}=\frac{F}{2XF_{,X}-F}~{}.$ (17)

Moreover, the equation of state of the non-canonical scalar field can be
obtained in terms of the dimensionless variable $x$, $y$ and $\omega_{ki}$ as:

$\displaystyle\omega_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}=\frac{\omega_{ki}x^{2}-y^{2}}{x^{2}+y^{2}}~{}.$
(18)

Note that the $\omega_{\phi}$ has implicit scale dependence through
dimensionless variables $x$ and $y$. Now, differentiating the two
dimensionless variables, namely, $x$ and $y$ with $N=\ln
a~{}(\mathrm{d}N=H\mathrm{d}t)$ , we obtain

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}N}$
$\displaystyle=\frac{\kappa}{2H}\frac{\left(2XF_{,XX}+F_{,X}\right)}{\sqrt{3\left(2XF_{,X}-F\right)}}\frac{\mathrm{d}X}{\mathrm{d}N}-x\frac{\dot{H}}{H^{2}}+\frac{x}{2}\frac{\partial\ln
G_{k}}{\partial\ln k}\frac{\mathrm{d}\ln k}{\mathrm{d}N}$ (19)
$\displaystyle\frac{\mathrm{d}y}{\mathrm{d}N}$
$\displaystyle=\frac{y}{2}\left[\frac{1}{V_{k}(\phi)}\frac{\partial
V_{k}(\phi)}{\partial\phi}\frac{\dot{\phi}}{H}+\frac{\partial\ln
V_{k}}{\partial\ln k}\frac{\mathrm{d}\ln
k}{\mathrm{d}N}\right]-y\frac{\dot{H}}{H^{2}}+\frac{y}{2}\frac{\partial\ln
G_{k}}{\partial\ln k}\frac{\mathrm{d}\ln k}{\mathrm{d}N}~{}.$ (20)

To substitute the factor $\frac{\dot{H}}{H^{2}}$ in the above evolution
equations, one can obtain the factor $\frac{\dot{H}}{H^{2}}$ from (7) by
dividing $H^{2}$ in terms of the dimensionless variables, $x$ and $y$ as:

$\displaystyle\frac{\dot{H}}{H^{2}}=-\frac{3}{2}\left[(1+\omega_{bm})(1-y^{2})+x^{2}(\omega_{ki}-\omega_{bm})\right]~{}.$
(21)

To write the above equation, we have written the first Friedmann equation in
terms of the dimensionless variable

$\displaystyle x^{2}+y^{2}+\Omega_{bm}=1$ (22)

where $\Omega_{bm}=\frac{\kappa^{2}\rho_{m}}{3H^{2}}$ is the relative energy
density of matter with respect to the total energy density.
Similar to the canonical scalar field Copeland:1998 , we define a variable
that denotes the slope of the non-canonical scalar field potential, following
Wands:2013 :

$\displaystyle\sigma=-\frac{1}{\kappa\sqrt{3|\rho_{ki}|}}\frac{\partial
V_{k}(\phi)}{\partial\phi}\frac{\sqrt{2X}}{V_{k}(\phi)}~{}.$ (23)

This variable is related as $\sigma=\sqrt{\frac{2}{3}}\lambda$ with the
canonical scalar field variable $\lambda$ defined in Copeland:1998 ,
specifically when $\omega_{ki}=1$.
To express the first term of the evolution equations (19, 20) in terms of the
dimensionless variables $x$, $y$, $\sigma$, and the equation of state of the
non-canonical scalar field, we derive two relations by rearranging Eq. (23)
and utilizing the continuity equation (10):

$\displaystyle\frac{\kappa
V_{k,\phi}\sqrt{2X}}{2\sqrt{3V_{k}(\phi)}H^{2}}=-\frac{3}{2}\sigma xy$ (24)
$\displaystyle\frac{\mathrm{d}X}{\mathrm{d}N}=-$
$\displaystyle\frac{3F}{\left(2XF_{,XX}+F_{,X}\right)\omega_{ki}}\left(\omega_{ki}+1-\frac{\sigma
y^{2}}{x}\right)$ (25)

so that the evolution equations (19) and (20) become

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}N}$
$\displaystyle=\frac{3}{2}\left[\sigma
y^{2}-x(1+\omega_{ki})\right]+\frac{3}{2}x\left[(1+\omega_{bm})(1-y^{2})+x^{2}(\omega_{ki}-\omega_{bm})\right]+\frac{x}{2}\eta_{RG}\frac{\mathrm{d}\ln
k}{\mathrm{d}N}$ (26) $\displaystyle\frac{\mathrm{d}y}{\mathrm{d}N}$
$\displaystyle=-\frac{3}{2}\sigma
xy+\frac{3}{2}y\left[(1+\omega_{bm})(1-y^{2})+x^{2}(\omega_{ki}-\omega_{bm})\right]+\frac{y(\eta_{RG}+\nu_{RG})}{2}\frac{\mathrm{d}\ln
k}{\mathrm{d}N}~{}.$ (27)

In this work, we assume that the renormalization group scale, to which the
gravitational constant and non-canonical scalar field potential flow, is a
function of cosmological time, denoted as $k=k(t)$, or alternatively expressed
as $k=k(N)$. Here, we will discuss the evolution of the RG scale with time. To
facilitate this, we rewrite Eq. (15) in terms of dimensionless variable $y$ as

$\displaystyle\eta_{RG}+y^{2}\nu_{RG}=0~{}.$ (28)

For calculating the evolution of the RG scale with time, that is,
$\frac{\mathrm{d}\ln k}{\mathrm{d}N}$ in terms of $x$, $y$ and $\sigma$, we
differentiate Eq. (28) with respect to $N$ and obtain

$\displaystyle\frac{\mathrm{d}\ln
k}{\mathrm{d}N}=\frac{1}{\alpha_{RG}}\left[\frac{3}{2}\frac{\epsilon_{RG}}{\nu_{RG}}\sigma
x-\frac{3}{2}x^{2}(1+\omega_{ki})-\frac{3}{2}(1+\omega_{bm})\Omega_{bm}\right]$
(29)

where
$\alpha_{RG}=\frac{1}{2}\left[\eta_{RG}+\nu_{RG}-\frac{\partial}{\partial\ln
k}\ln\left(-\frac{\eta_{RG}}{\nu_{RG}}\right)\right]$. The full dynamical
equations for $x$ and $y$ can be obtained by substituting (29) into the right-
hand sides of Eqs. (26, 27). This evolution equation of $k$ with cosmic time
is crucial because we must choose $k(N)$ in a manner that satisfies (29) to
maintain the consistency of the evolution equation. It’s noteworthy that the
non-canonical parameter plays a role in the evolution of the cut-off scale
with cosmic time in Eq. (29), while the consistency condition (28) remains
identical for both the canonical and non-canonical cases.
Another important point is to emphasize how to relate the RG scale to
cosmological time or the Hubble parameter $H$. For this, one can show that
$\frac{\partial\ln H}{\partial\ln k}=0$ when $\frac{\mathrm{d}\ln
k}{\mathrm{d}N}\neq 0$, indicating that $H$ and $k$ are independent phase
space variables. For this reason, the variation of the Hubble parameter with
the scale factor changes exactly like the classical case:

$\displaystyle\frac{\mathrm{d}\ln
H}{\mathrm{d}N}=-\frac{3(1+\omega_{ki})}{2}x^{2}-\frac{3}{2}(1+\omega_{bm})\Omega_{bm}~{}.$
(30)

However, the right-hand side of the above equation implicitly depends on the
RG parameters. Comparing (30) with (29), we indeed obtain:

$\displaystyle\frac{\mathrm{d}\ln
H}{\mathrm{d}N}=\alpha_{RG}\frac{\mathrm{d}\ln
k}{\mathrm{d}N}-\frac{3}{2}\frac{\epsilon_{RG}}{\nu_{RG}}\sigma x$ (31)

which exactly matches with the canonical case. This relation for the non-
canonical scalar field gives the condition that $\alpha_{RG}=1$ together with
$\epsilon_{RG}=0$, $\frac{1}{\nu_{RG}}=0$, $x=0$ or $\sigma=0$ under which $k$
can be mapped with $H$. We will also see in the next section for which of
these cosmic fixed points the RG scale can be chosen with the Hubble
parameter.
Now, as we have not considered any form of $F(X)$ and $V_{k}(\phi)$, the
evolution equations for $\omega_{ki}$ and $\sigma$ can be written as follows,

$\displaystyle\frac{\mathrm{d}\omega_{ki}}{\mathrm{d}N}$
$\displaystyle=3\frac{2\Sigma\omega_{ki}+\omega_{ki}-1}{2\Sigma+1}\left(\omega_{ki}+1-\frac{\sigma
y^{2}}{x}\right)$ (32) $\displaystyle\frac{\mathrm{d}\sigma}{\mathrm{d}N}$
$\displaystyle=-3\sigma^{2}x(\Gamma-1)+\frac{3\sigma\left(2\Sigma(\omega_{ki}+1)+\omega_{ki}-1\right)}{2(2\Sigma+1)(\omega_{ki}+1)}\left(\omega_{ki}+1-\frac{\sigma
y^{2}}{x}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\sigma\left(\epsilon_{RG}-\nu_{RG}-\frac{\eta_{RG}}{2}\right)\frac{\mathrm{d}\ln
k}{\mathrm{d}N}~{}$ (33)

where we have defined two new auxiliary variables in terms of the second
derivative of $F(X)$ and potential $V_{k}(\phi)$ in such a way

$\displaystyle\Sigma\equiv\frac{XF_{,XX}}{F,X}~{},\qquad\Gamma\equiv\frac{V_{k}V_{k,\phi\phi}}{V_{k,\phi}^{2}}~{}.$
(34)

The above defined second order derivative variables will have evolution
equations in terms of dimensionless variables $x$, $y$ and the new third order
derivative variables. This procedure will repeat until we truncate this
succession of equations by fixing the functions $F(X)$ and potential
$V_{k}(\phi)$ of non-canonical Lagrangian terms.
For these, the first assumption is to consider the function that is taken to
be

$\displaystyle F(X)=AX^{n}$ (35)

where $A$ and $n$ are constants. For this choice, the equation of state for
the kinetic part is $\omega_{ki}=\frac{1}{2n-1}$, and the evolution of the
equation of state of the non-canonical scalar field $\omega_{ki}$ simply goes
to zero, indicating $\frac{\mathrm{d}\omega_{ki}}{\mathrm{d}N}=0$. The
evolution equation for the slope of the non-canonical scalar field $\sigma$
can be obtained as

$\displaystyle\frac{\mathrm{d}\sigma}{\mathrm{d}N}=-3\sigma^{2}x\left(\Gamma-1\right)+\frac{3\sigma(1-\omega_{ki})}{2(1+\omega_{ki})}\left(\omega_{ki}+1-\frac{\sigma
y^{2}}{x}\right)+\sigma\left(\epsilon_{RG}-\nu_{RG}-\frac{\eta_{RG}}{2}\right)\frac{\mathrm{d}\ln
k}{\mathrm{d}N}~{}.$ (36)

To truncate the succession of the equations of the non-canonical scalar
potential, the second assumption is that $\Gamma$ is some constant or exactly
equal to one ($\Gamma=1$). If $\Gamma\neq 1$, but some other constant, the
form of the potential can be taken as

$\displaystyle
V(\phi)=V_{0}\left(\phi-\phi_{0}\right)^{\frac{1}{1-\Gamma}}\,.$ (37)

For $\Gamma=1$, the potential form can be taken as an exponential potential

$\displaystyle V(\phi)=V_{0}\exp(-\kappa\lambda\phi)\,.$ (38)

An essential point to consider is that, for a power-law potential where the
Lagrangian is given by $\mathcal{L}=AX^{n}+V_{0}(\phi-\phi_{0})^{\delta}$ with
$\delta=\frac{1}{(1-\Gamma)}$, the exponents of the kinetic and potential
terms must satisfy the following relation from dimensional analysis Wands:2013
:

$\displaystyle\delta=\frac{n}{n+2}~{}.$ (39)

With this two choice of assumptions, we will see which form of potential is
responsible for giving the stable attractor solution for the non-canonical
scalar field dominated universe.

###  III.1 Cosmological Fixed Points

In this subsection, we shall discuss the cosmic fixed points derived from the
renormalization group improved dynamic equations involving the non-canonical
scalar field and the background barotropic fluid. We are particularly
interested in those cosmic fixed points which must be insensitive to initial
conditions for describing the accelerated universe. We will also investigate
whether the cosmic fixed points correspond to the RG fixed points at the
infrared or UV scale.
As evident from (12), two distinct types of cosmic fixed points emerge: one
where the RG scale $\ln k$ becomes independent of $N$, and another where the
RG scale maintains its scaling relationship with $N$ or cosmic time.
In the first case, where $\frac{\mathrm{d}\ln k}{\mathrm{d}N}=0$, the
cosmological fixed points are the same as those calculated from the classical
set of equations mentioned in Wands:2013 . However, it is important to note
that although the classical fixed point occurs when the flow of the RG scale
with cosmic time stops at a certain energy scale, quantum modifications are
still implicitly included in this system through the variables $x$, $y$, and
$\sigma$.
A second type of cosmological fixed point can be calculated by taking the RG
scale, which continues to evolve with the cosmological time as

$\displaystyle\frac{\mathrm{d}\ln k}{\mathrm{d}N}=\text{constant}\neq 0~{}.$
(40)

Here, the dynamical equations of the dimensionless variables, namely $x$, $y$,
and $\sigma$, explicitly depend on the cut-off scale $k$ through the RG
parameters defined in (13). Hence, to achieve a true cosmic fixed point, the
RG parameters must cease to evolve with $k$ and become constant. This is only
possible for RG fixed points, which may include Gaussian and non-Gaussian
fixed points for the underlying quantum theory, where the coefficient of the
right-hand side of Eq. (29) simplifies to
$\alpha_{RG}=\frac{1}{2}\left(\eta_{RG}+\nu_{RG}\right)$.
To proceed further, with the above-mentioned condition, we need to set (26,
27), and (36) to zero to obtain the cosmic fixed points, corresponding to $x$,
$y$, and $\sigma$ being constants. As the variables $x^{2}$ and $y^{2}$ are
constrained by the equation in (22), there can be three types of cosmic fixed
points. Namely, when $x^{2}+y^{2}=1$, the total contribution of energy density
comes from the non-canonical scalar field kinetic part and the potential
energy density part. Another cosmic fixed point occurs when $x^{2}+y^{2}$
equals zero, signifying no contribution from the non-canonical scalar field.
The last type of cosmic fixed point is when $x^{2}+y^{2}$ falls between zero
and one, giving rise to scaling solutions where the energy density of the non-
canonical scalar field scales at the same rate as the background fluid.
In the search for these cosmic fixed points, the trivial fixed point occurs
when both $x_{*}=0$ and $y_{*}=0$ simultaneously, indicating no contribution
from the non-canonical scalar field. In this scenario, the universe is
background matter-dominated.
Now, if we seek the cosmic fixed point where the sole contribution comes from
the non-canonical scalar field, characterized by $x^{2}+y^{2}=1$, three
distinct situations arise. The first one corresponds to $x_{*}=0$ while
$y_{*}\neq 0$. In this case, the RG modified dynamical equations for $x$ and
$y$ take the form as

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}N}$
$\displaystyle=\frac{3}{2}\sigma y^{2}$ (41)
$\displaystyle\frac{\mathrm{d}y}{\mathrm{d}N}$
$\displaystyle=\frac{3}{2}y(1+\omega_{bm})(1-y^{2})+\frac{y}{2}(\eta_{RG}+\nu_{RG})\frac{\mathrm{d}\ln
k}{\mathrm{d}N}~{}.$ (42)

The above equations show that fixing $\sigma_{*}=0$ from the evolution
equation of $x$ is necessary to attain a cosmic fixed point, given that
$y_{*}\neq 0$. Additionally, to achieve $\frac{\mathrm{d}y}{\mathrm{d}N}=0$,
we must fix $y_{*}=1$ or $\omega_{bm}=-1$, along with the condition
$\eta_{RG}=-\nu_{RG}$.
An important point to note is that the consistency equation (15) always holds
$y^{2}=-\frac{\eta_{RG}}{\nu_{RG}}$, restricting only one fixed point as
$y_{*}=1$, as the condition $\eta_{RG}=-\nu_{RG}$ must be satisfied
consistently. This is in contrast to the classical case mentioned in
Wands:2013 , where another fixed point exists for this scenario with $y$ being
arbitrary and $\omega_{bm}=-1$. At this potential-dominated fixed point where
$x_{*}=0$ and $y_{*}=1$, the equation of state for the non-canonical scalar
field becomes $\omega_{\phi}=-1$, potentially leading to an accelerated
universe. To confirm this result, it is necessary to express the effective
equation of state, denoted as $\omega_{eff}$, in the form

$\omega_{eff}=\frac{p_{tot}}{\rho_{tot}}=\frac{p_{\phi}+p_{bm}}{\rho_{\phi}+\rho_{bm}}=\omega_{\phi}\Omega_{\phi}+\omega_{bm}\Omega_{bm}$
(43)

where $\Omega_{\phi}=x^{2}+y^{2}$ and the equation of state of the background
fluid is denoted as $\omega_{bm}=\frac{p_{bm}}{\rho_{bm}}$. In this case, with
$\omega_{eff}=-1$, it becomes intriguing from a cosmological perspective as it
opens up the possibility to describe dark energy or inflation phenomena
depending on the different possible values of RG parameters for the specific
renormalization group theory.
In the second case, where $y_{*}=0$ and $x_{*}\neq 0$, the total energy
density solely comes from the kinetic part of the non-canonical scalar field.
The dynamical equations for $x$ and $y$ evolve as

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}N}$
$\displaystyle=\frac{3x}{2}\left(\omega_{ki}-\omega_{bm}\right)\left(x^{2}-1\right)+\frac{x}{2}\eta_{RG}\frac{\mathrm{d}\ln
k}{\mathrm{d}N}$ (44) $\displaystyle\frac{\mathrm{d}y}{\mathrm{d}N}$
$\displaystyle=0~{}.$ (45)

Now, $\frac{\mathrm{d}x}{\mathrm{d}N}$ will be zero for two cases: either with
$x_{*}=1$ or $\omega_{ki}=\omega_{bm}$ along with the condition
$\eta_{RG}\rightarrow 0$. The condition $\eta_{RG}\rightarrow 0$ is crucial,
as for a kinetic-dominated universe, these cosmic fixed points coincide with
the RG fixed point that arises when gravity is classical, where no
renormalization group improvement is taken into account. However, it is
important to remember that although the fixed point matches the classical
case, the RG modification is included in the theory. These two kinetic-
dominated fixed points are achievable near infrared, where the gravitational
coupling hits the Gaussian RG fixed point and becomes constant. The first
cosmic fixed point occurs when $x_{*}=1$, $y_{*}=0$, along with
$\eta_{RG}\rightarrow 0$, and it is completely dominated by the kinetic energy
density of the non-canonical scalar field. Here, the effective equation of
state is $\omega_{eff}=\omega_{ki}$, calculated from (43), since the equation
of state of the non-canonical scalar field is $\omega_{\phi}=\omega_{ki}$. It
is important to note that this fixed point does not result in an accelerated
universe, as for an accelerated universe, $\omega_{eff}<-\frac{1}{3}$, which,
in turn, implies $n<-1$ in (35). In this case, the cut-off scale evolves with
time according to the following expression:

$\displaystyle\frac{\mathrm{d}\ln
k}{\mathrm{d}N}=\frac{3}{\nu_{RG}}\left[\frac{\epsilon_{RG}}{\nu_{RG}}\sigma_{*}-(1+\omega_{ki})\right]$
(46)

where $\sigma_{*}$ can be fixed setting
$\frac{\mathrm{d}\sigma}{\mathrm{d}N}=0$ in Eq. (36). Solving for $\sigma_{*}$
from $\frac{\mathrm{d}\sigma}{\mathrm{d}N}=0$, one can obtain either
$\sigma_{*}=0$, or $\sigma_{*}$ can be fixed through the equation:

$\displaystyle-3\sigma_{*}\left(\Gamma-1\right)+\frac{3}{2}\left(1-\omega_{ki}\right)+3\left(\frac{\epsilon_{RG}}{\nu_{RG}}-1\right)\left[\frac{\epsilon_{RG}}{\nu_{RG}}\sigma_{*}-\left(1+\omega_{ki}\right)\right]=0~{}.$
(47)

Now, solving for $\sigma_{*}$ from this equation, its value can be calculated
depending on the potential form. For example, for an exponential potential
with $\Gamma=1$, $\sigma_{*}$ can be expressed in terms of the non-canonical
scalar field parameter and RG parameters as:
$\sigma_{*}=\frac{2\left(\frac{\epsilon_{RG}}{\nu_{RG}}-1\right)(1+\omega_{ki})-(1-\omega_{ki})}{2\left(\frac{\epsilon_{RG}}{\nu_{RG}}-1\right)\frac{\epsilon_{RG}}{\nu_{RG}}}$.
Similarly, $\sigma_{*}$ can be determined for a power-law potential where
$\Gamma$ is a constant, expressed in terms of $\omega_{ki}$ and RG parameters.
Consequently, we can conclude that the kinetic-dominated fixed point can be
achieved for both exponential and power-law potentials with different values
of $\sigma_{*}$.
In the second cosmic fixed point, where $x$ is arbitrary with
$\omega_{\phi}=\omega_{bm}$ and $y_{*}=0$, along with $\eta_{RG}\rightarrow
0$, the equation of state for background matter is exactly equal to the
equation of state for the non-canonical scalar field. This type of solution
has been referred to as a kinetically driven scaling solution, as mentioned in
Wands:2013 . Note that for this fixed point, the arbitrariness of $x$ can be
constrained by the first Friedmann equation $x_{*}^{2}+\Omega_{bm}=1$.

case |  $x^{2}$ |  $y^{2}$ |  $\sigma$ |  $\Omega_{bm}$ |  $\omega_{eff}$ |  $\frac{\mathrm{d}\ln k}{\mathrm{d}N}$ |  existence |  type
---|---|---|---|---|---|---|---|---
a |  0 |  1 |  0 |  0 |  -1 |  |  $\frac{\eta_{RG}}{\nu_{RG}}=-1$ |  Potential
b1 |  1 |  0 |  $\sigma_{*}$ |  0 |  $\omega_{ki}$ |  $\frac{3}{\nu_{RG}}\left[\frac{\epsilon_{RG}}{\nu_{RG}}\sigma_{*}-(1+\omega_{ki})\right]$ |  $\eta_{RG}\rightarrow 0$ |  Kinetic
b2 |  arbitrary |  0 |  0 |  $1-x_{*}^{2}$ |  $\omega_{bm}$ |  $-\frac{3}{\nu_{RG}}\left(1+\omega_{bm}\right)$ |  $\eta_{RG}\rightarrow 0$ |  Kinetic Scaling
c1 |  $1+\frac{\eta_{RG}}{\nu_{RG}}$ |  $-\frac{\eta_{RG}}{\nu_{RG}}$ |  0 |  0 |  $\omega_{ki}+\frac{\eta_{RG}}{\nu_{RG}}(1+\omega_{ki})$ |  $-\frac{3\left(1+\omega_{ki}\right)}{\nu_{RG}}$ |  $\Gamma(0)=0$ |  Mixed
c2 |  $1+\frac{\eta_{RG}}{\nu_{RG}}$ |  $-\frac{\eta_{RG}}{\nu_{RG}}$ |  $\sigma_{*}$ |  0 |  $\omega_{ki}+\frac{\eta_{RG}}{\nu_{RG}}(1+\omega_{ki})$ |  $\frac{3}{\nu_{RG}}\left[\frac{\sigma_{*}}{x_{*}}-\left(1+\omega_{ki}\right)\right]$ |  $\epsilon_{RG}=\nu_{RG}$ |  Mixed
d1 |  0 |  $-\frac{\eta_{RG}}{\nu_{RG}}$ |  0 |  $1+\frac{\eta_{RG}}{\nu_{RG}}$ |  $\omega_{bm}+\frac{\eta_{RG}}{\nu_{RG}}(1+\omega_{bm})$ |  $-\frac{3\left(1+\omega_{bm}\right)}{\nu_{RG}}$ |  $\omega_{bm}\neq 0$ |  Scaling
d2 |  $\frac{\left(1+\omega_{bm}\right)}{\left(\omega_{bm}-\omega_{ki}\right)}\left(\frac{\eta_{RG}}{\nu_{RG}}\right)$ |  $-\frac{\eta_{RG}}{\nu_{RG}}$ |  $\sigma_{*}$ |  $1-\left(\frac{\eta_{RG}}{\nu_{RG}}\right)\frac{\left(1+\omega_{ki}\right)}{\left(\omega_{bm}-\omega_{ki}\right)}$ |  $\omega_{bm}$ |  0 |  $\epsilon_{RG}=\nu_{RG}$ |  Scaling
Table 1: Renormalization group improved cosmological fixed points for Einstein
gravity with non-canonical scalar field and background barotropic fluid.

Another important type of cosmic fixed point is the mixed fixed point, where
the total energy density comes from both the kinetic and potential energy
density, satisfying $\Omega_{\phi}=x^{2}+y^{2}=1$, but with no barotropic
fluid components ($\Omega_{bm}=0$). For this kind of cosmic fixed point, $y$
can always be determined from the consistency condition (28) as
$y_{*}=\pm\left(-\frac{\eta_{RG}}{\nu_{RG}}\right)^{\frac{1}{2}}$, which in
turn, fixes $x_{*}=\left(1+\frac{\eta_{RG}}{\nu_{RG}}\right)^{\frac{1}{2}}$.
Here, for these value of $x_{*}$ and $y_{*}$, the evolution equations of $x$
and $y$ reduce to

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}N}$
$\displaystyle=\frac{3}{2}\sigma
x^{2}y^{2}\nu_{RG}\left(1-\frac{\epsilon_{RG}}{\nu_{RG}}\right)$ (48)
$\displaystyle\frac{\mathrm{d}y}{\mathrm{d}N}$
$\displaystyle=-\frac{3}{2}\sigma xy(1-\frac{\epsilon_{RG}}{\nu_{RG}})~{}.$
(49)

To obtain a cosmic fixed point, we need to set
$\frac{\mathrm{d}x}{\mathrm{d}N}=0$ and $\frac{\mathrm{d}y}{\mathrm{d}N}=0$,
which yield two fixed points. One is for $\sigma_{*}=0$, and the other is for
the condition $\epsilon_{RG}=\nu_{RG}$. At one fixed point where
$\sigma_{*}=0$, the RG scale parameter evolves as:

$\displaystyle\frac{\mathrm{d}\ln
k}{\mathrm{d}N}=-\frac{3\left(1+\omega_{ki}\right)}{\nu_{RG}}~{}.$ (50)

For $\omega_{ki}=1$, the evolution of the RG scale matches with the result of
the canonical scalar field mentioned in Hindmarsh:2011 . For another fixed
point where $\epsilon_{RG}=\nu_{RG}$, $\sigma_{*}$ can be derived for the
above-mentioned $x_{*}$ and $y_{*}$ from the equation:

$\displaystyle-3\sigma_{*}x_{*}(\Gamma-1)+\frac{3(1-\omega_{ki})}{2(1+\omega_{ki})}\left(1+\omega_{ki}-\sigma_{*}\frac{y_{*}^{2}}{x_{*}}\right)-\frac{\eta_{RG}}{2}\frac{\mathrm{d}\ln
k}{\mathrm{d}N}=0$ (51)

where the evolution of the cut-off scale can be written for this fixed point
as

$\displaystyle\frac{\mathrm{d}\ln
k}{\mathrm{d}N}=\frac{1}{\alpha_{RG}}\left[\frac{3}{2}\sigma_{*}x_{*}-\frac{3}{2}x_{*}^{2}(1+\omega_{ki})\right]~{}.$
(52)

Similar to the kinetically-driven fixed point, the value of $\sigma_{*}$ can
be derived from the above equations, depending on the potential profile. This
dependency is influenced by both the exponential potential ($\Gamma=1$) and
the power-law potential ($\Gamma=\text{constant}$).
For these two fixed points, the important result comes from the effective
equation of state, which can be written from (43) using (18) as

$\displaystyle\omega_{eff}=\omega_{ki}+\frac{\eta_{RG}}{\nu_{RG}}(1+\omega_{ki})~{}.$
(53)

These cosmic fixed points may offer a dark energy solution with a variety of
equations of state for different values of $\omega_{ki}$ and various
renormalization group theories, where the RG parameters could assume different
possible values. Moreover, it is very interesting to note that the condition
which gives rise to cosmic acceleration must satisfy
$\omega_{eff}<-\frac{1}{3}$, which in turn gives a bound on
$\frac{\eta_{RG}}{\nu_{RG}}$ in terms of the non-canonical scalar field
parameter $\omega_{ki}$

$\displaystyle\frac{\eta_{RG}}{\nu_{RG}}<-\frac{\left(1+3\omega_{ki}\right)}{3\left(1+\omega_{ki}\right)}~{}.$
(54)

For a canonical scalar field, the bound on the RG parameter is
$\frac{\eta_{RG}}{\nu_{RG}}<-\frac{2}{3}$ for which the expansion of the
universe is accelerating. Here, this bound is particularly important because
it gives a general bound of RG parameters in terms of non-canonical scalar
field kinetic term parameter $n$ or $\omega_{ki}$ of Eq. (35).
Another interesting point to note is that at the asymptotically safe matter
fixed point Percacci:2003 ; Narain:2010 ; Rahmede:2010  where all the
couplings of the scalar field vanish at the UV scale, the RG parameters assume
values such as $\eta_{RG}\rightarrow-2$ and $\nu_{RG}\rightarrow 4$. This
scenario, however, does not result in an accelerated universe, as
$\omega_{eff}=0$ for the canonical case where $\omega_{ki}=1$. Surprisingly,
for the non-canonical scalar field, it is observed that for the same RG
parameters, an accelerating universe can be achieved at the UV scale for
$\omega_{ki}<\frac{1}{3}$. This, in turn, imposes a bound on the exponent of
the non-canonical kinetic term in Eq. (35) as $n>2$.
Next, we come to scaling fixed points where the non-canonical scalar field
energy density follows the energy density of background matter. This may be
radiation or matter. As usual, the consistency condition (28) makes sure that
$y_{*}=\left(-\frac{\eta_{RG}}{\nu_{RG}}\right)^{\frac{1}{2}}$. From Eq. (22),
we can fix $x_{*}=0$ and
$\Omega_{bm}=1-y_{*}^{2}=1+\frac{\eta_{RG}}{\nu_{RG}}$. For this particular
scaling fixed point, the evolution equations for $x$ and $y$ take the form

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}N}$
$\displaystyle=\frac{3}{2}\sigma y^{2}$ (55)
$\displaystyle\frac{\mathrm{d}y}{\mathrm{d}N}$ $\displaystyle=0~{}.$ (56)

From the above equations, we can easily see that $\sigma_{*}=0$ for getting
$\frac{\mathrm{d}x}{\mathrm{d}N}=0$. Here, the effective equation of state
$\omega_{eff}$ can be derived from Eq. (53) as

$\displaystyle\omega_{eff}=\omega_{bm}+\frac{\eta_{RG}}{\nu_{RG}}\left(1+\omega_{bm}\right)~{}.$
(57)

From here, we can see that unlike the mixed fixed point, the effective
equation of state does not depend on $\omega_{ki}$. Instead, it depends on the
equation of state of the background matter. However, in this case, the
condition on $\frac{\eta_{RG}}{\nu_{RG}}$ for the accelerating universe can be
set as Eq. (54), the only change being that $\omega_{ki}$ is replaced by
$\omega_{bm}$. Here, at this fixed point, the RG scale evolution is fixed as

$\displaystyle\frac{\mathrm{d}\ln
k}{\mathrm{d}N}=-\frac{3\left(1+\omega_{bm}\right)}{\nu_{RG}}~{}.$ (58)

It is important to note here that at the asymptotic matter fixed point in the
UV scale where $\eta_{RG}=-2$ and $\nu_{RG}=4$ and $\alpha_{RG}=1$, we obtain
the scaling relation $k\propto H$ from Eq. (31) as $x_{*}=0$ for this fixed
point. This scaling cosmic fixed point is exactly the same as the canonical
case and can be mapped with the asymptotically safe matter fixed point in the
UV scale Hindmarsh:2011 .
Another general scaling fixed point can be written by setting
$\omega_{\phi}=\omega_{bm}$ in Eq. (18) which in turn gives

$\displaystyle
x_{*}^{2}=\frac{\left(1+\omega_{bm}\right)}{\left(\omega_{bm}-\omega_{ki}\right)}\left(\frac{\eta_{RG}}{\nu_{RG}}\right)~{}.$
(59)

Obviously, to get the $x_{*}$, we have used
$y_{*}^{2}=-\left(\frac{\eta_{RG}}{\nu_{RG}}\right)$ from consistency
condition. The value of $x_{*}$ and $y_{*}$ will definitely set the fixed
point value of the density parameter of $\Omega_{bm}$ in terms of RG
parameter, non-canonical scalar field parameter and the background equation of
state as

$\displaystyle\Omega_{bm}=1-\left(\frac{\eta_{RG}}{\nu_{RG}}\right)\frac{\left(1+\omega_{ki}\right)}{\left(\omega_{bm}-\omega_{ki}\right)}~{}.$
(60)

At that fixed point, we obtain the evolution equations of $x$ and $y$ as

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}N}$
$\displaystyle=\frac{3y^{2}}{2\left(1-y^{2}\right)}\left[\sigma\left(1-x^{2}\frac{\epsilon_{RG}}{\nu_{RG}}-y^{2}\right)-\frac{\left(1+\omega_{bm}\right)}{x}\left(1-x^{2}-y^{2}\right)\right]$
(61) $\displaystyle\frac{\mathrm{d}y}{\mathrm{d}N}$
$\displaystyle=-\frac{3}{2}\sigma xy(1-\frac{\epsilon_{RG}}{\nu_{RG}})~{}.$
(62)

The evolution equations ensure that for the existence of this scaling fixed
point, two conditions needed to be fixed as $\epsilon_{RG}=\nu_{RG}$ and
$\sigma_{*}=\frac{\left(1+\omega_{bm}\right)}{x_{*}}$ which will give
$\frac{\mathrm{d}x}{\mathrm{d}N}=0$ and $\frac{\mathrm{d}y}{\mathrm{d}N}=0$.
Moreover, we have obtained the RG scale evolution for the scaling fixed point,
putting first the fixed point value of $\sigma_{*}$ at Eq. (29)

$\displaystyle\frac{\mathrm{d}\ln
k}{\mathrm{d}N}=-\frac{2}{\eta_{RG}+\nu_{RG}}\left[\frac{3}{2}\frac{\epsilon_{RG}}{\nu_{RG}}\left(1+\omega_{bm}\right)-\frac{3}{2}\left(1+\omega_{bm}\right)\right]~{}.$
(63)

Now for $\epsilon_{RG}=\nu_{RG}$, the RG scale freezes at some scale $k_{ki}$,
which confirms that this scaling fixed point only exists when
$\frac{\mathrm{d}\ln k}{\mathrm{d}N}=0$. Setting
$\frac{\mathrm{d}\sigma}{\mathrm{d}N}=0$ from Eq. (36) and substituting the
value of $\sigma_{*}$ into this equation, we obtain $\Gamma$ as a constant
given by the following expression:

$\displaystyle\Gamma-1=\frac{\left(1-\omega_{ki}\right)}{\left(1+\omega_{ki}\right)}+\frac{\left(1-\omega_{ki}\right)\left(1-\omega_{bm}\right)}{\left(1-\omega_{ki}\right)\left(\omega_{bm}-\omega_{ki}\right)}~{}.$
(64)

For the canonical case where $\omega_{ki}=1$, we end up with exponential
potential as $\Gamma=1$ but for the non-canonical case where $\omega_{ki}\neq
1$, we obtain a power law potential profile as $\Gamma$ becomes constant
involving the non-canonical parameter $\omega_{ki}$ and the equation of state
of background fluid $\omega_{bm}$.

##  IV Conclusions

In this paper, we have explored the dynamical equations in the presence of a
non-canonical scalar field and a background matter, taking into account
quantum corrections using the functional renormalization group. The important
point we want to address here is how the non-canonical scalar field parameter
influences the dynamical equations; this in turn affects the cosmic evolution
of the universe when quantum modifications are included. Here, we have
particularly studied cosmic evolution by considering the running of the
gravitational constant and the non-canonical scalar field potential. We have
considered the covariant conservation of energy-momentum tensor by both the
non-canonical scalar field and the background matter separately. This in turn
gives another constraint (28) in order to preserve the Bianchi identity. This
constraint relation between the RG flow parameters and the dimensionless
cosmological variables has particular importance because it restricts the
arbitrary behaviour of the RG flow in Einstein-Hilbert truncation.
In this work, we study the dynamical equations showing the difference from the
canonical case and how this evolution depends on the non-canonical scale
parameter $\omega_{ki}$, and the RG flow parameters, namely, $\eta_{RG}$,
$\nu_{RG}$ and $\epsilon_{RG}$. The equation of state for the kinetic energy
of non-canonical scalar field $\omega_{ki}$ represents how this evolution
deviates from the canonical case where $\omega_{ki}=1$; which signifies
$F(X)=X$. We have identified various fixed points, namely, the potential-
dominated fixed points, the kinetic energy dominated fixed points, the mixed
fixed points, and the scaling fixed points depending on the energy
contributions from the non-canonical scalar field and the background matter
fields. The cosmic fixed points have been studied for a Lagrangian in the form
of $\mathcal{L}(\phi,X)=F(X)-V_{k}(\phi)$ with a kinetic term given by
$F(x)=AX^{n}$ and with a power law or exponential potential. In this study,
the cosmological fixed points are attained only when the RG parameters become
constant or $k$-independent. This condition exclusively involves RG fixed
points, resulting in a constant evolution of the RG scale.
The cosmic fixed points exclusively depend on the non-canonical scalar field
parameter $\omega_{ki}$ though the constraint (28) involving the RG parameters
and dimensionless cosmological quantities. The cosmic fixed points which we
identified from dynamical equations may give an accelerated universe depending
on the value of the effective equation of state $\omega_{eff}$ and the RG
parameters. For the mixed fixed points, we observed the possibility of
achieving a dark energy-dominated accelerated universe, depending on the non-
canonical equation of state of kinetic energy $\omega_{ki}$ for different RG
theories. An important result of our work is that we have obtained a bound on
the ratio of the RG parameters $\frac{\eta_{RG}}{\nu_{RG}}$ involving this
non-canonical parameter to attain an accelerated ratio of RG parameters
universe for the mixed fixed points. This bound plays a significant role as
the non-canonical scalar field includes a larger set of RG fixed points, which
may lead to an accelerated universe. For example, at the asymptotically safe
matter RG fixed point in the UV scale, where $\eta_{RG}\rightarrow-2$ and
$\nu_{RG}\rightarrow 4$, this universe does not transition to an accelerated
state for this fixed point in the canonical case where $\omega_{ki}=1$.
However, surprisingly, for the non-canonical case, for the same values of RG
parameters, it results in an accelerated universe at an early epoch for
$\omega_{ki}<\frac{1}{3}$. This, in turn, imposes a bound on the exponent of
the kinetic term as $n>2$. Moreover, the potential-dominated cosmic fixed
point could potentially pave the way for an accelerated universe. This
universe might be analogous to a dark energy-dominated or inflating universe,
depending upon the diverse range of possible values for the RG parameters
within the specific renormalization group theory. In the kinetic-dominated
cosmic fixed point, a noteworthy distinction emerges: unlike the canonical
case where only one cosmic fixed point emerges in the near infrared region,
here, two cosmic fixed points coexist. These fixed points are manifested when
the gravitational coupling reaches the Gaussian RG fixed points and becomes
constant. However, this particular fixed point does not inherently lead to a
dark energy-dominated or inflating universe unless the exponent of the non-
canonical kinetic term is less than $-1$, which would be considered highly
unlikely. Another essential cosmic fixed point is the scaling fixed point,
where the energy density of the non-canonical scalar field scales with the
energy density of background matter. For the first scaling point denoted by d1
in table (1), we have seen that it exactly matches with the canonical case and
the cut off scale $k$ is directly proportional to the Hubble parameter $H$. In
the second scaling fixed point (d2), we illustrate how these fixed points vary
depending on the non-canonical scalar parameter when quantum corrections are
included. In this scenario, the potential profile can be precisely determined
based on whether the scalar field is canonical or non-canonical.

## Acknowledgment

RM would like to thank IOE-UOH for financial support through the IOE-PDRF
scheme.

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