# Spatial Reuse in IEEE 802.11bn Coordinated Multi-AP WLANs: A Throughput
Analysis

David Nunez♭, Francesc Wilhelmi⋆, Lorenzo Galati-Giordano⋆,
Giovanni Geraci♯ ♭, and Boris Bellalta♭ Corresponding author:
<EMAIL_ADDRESS>♭ _Department of Information and Communications
Technologies, Universitat Pompeu Fabra, Barcelona, Spain_ ⋆ _Radio Systems
Research, Nokia Bell Labs, Stuttgart, Germany_ ♯ _Telefónica Research,
Barcelona, Spain_

###### Abstract

IEEE 802.11 networks continuously adapt to meet the stringent requirements of
emerging applications like cloud gaming, eXtended Reality (XR), and video
streaming services, which require high throughput, low latency, and high
reliability. To address these challenges, Coordinated Spatial Reuse (C-SR) can
potentially contribute to optimizing spectrum resource utilization. This
mechanism is expected to enable a higher number of simultaneous transmissions,
thereby boosting spectral efficiency in dense environments and increasing the
overall network performance. In this paper, we focus on the performance
analysis of C-SR in Wi-Fi 8 networks. In particular, we consider an
implementation of C-SR where channel access and inter-Access Point (AP)
communication are performed over-the-air using the Distributed Coordination
Function (DCF). For such a purpose, we leverage the well-known Bianchi’s
throughput model and extend it to support multi-AP transmissions via C-SR.
Numerical results in a WLAN network that consists of four APs show C-SR
throughput gains ranging from 54% to 280% depending on the inter-AP distance
and the position of the stations in the area.

###### Index Terms:

Coordinated Spatial Reuse, IEEE 802.11bn, Multi-Access Point Coordination, Wi-
Fi 8.

##  I Introduction

Next-generation applications such as eXtended Reality (XR), high-quality
holographic video, and zero-delay file exchange to support cooperative and
remote working require further improvements on wireless networks [1]. For
example, the throughput requirements of Virtual Reality (VR) applications must
be above 400 Mbps to properly deliver a 4K VR 360º video experience [2] and
even higher (above 1 Gbps) to provide an ideal (extreme) VR experience [3].
Technologies operating on it like Wi-Fi will be very important for those new
applications because they consume a lot of spectrum resources [4]. Moreover,
achieving those requirements is especially challenging in current dense
scenarios with multiple Wi-Fi networks sharing the same channel, where the
heightened contention and collision levels not only limit the achievable
throughput but also impact the latency and reliability required by such
stringent applications [5, 6, 7].

To address the challenges imposed by next-generation applications in high-
density Wireless Local Area Network (WLAN) scenarios, the future IEEE 802.11bn
amendment [8, 9] is likely to introduce Multi-Access Point Coordination
(MAPC). This framework is envisioned to enhance the overall latency and
reliability performance, through alleviating channel access contentions in
dense Wi-Fi deployments.

MAPC enables the introduction of different methods that perform coordinated
resource allocation, such as Coordinated Time Division Multiple Access
(C-TDMA), where Access Points (APs) agree on splitting time resources, or
Coordinated Orthogonal Frequency-Division Multiple Access (C-OFDMA), where the
agreement is on the allocated Resource Units (RUs) [8]. In certain scenarios,
higher performance can also be achieved by enabling simultaneous transmissions
through Coordinated Spatial Reuse (C-SR) and more complex techniques like
Coordinated Beamforming (C-BF) or Joint Transmission (J-TX). Among these
techniques, C-SR stands out as a particularly promising scheme since it can
reduce latency and, in most scenarios, increase throughput, without requiring
complex and resource-consuming measurement acquisition and coordination
signaling [10, 11, 12]. For this reason, C-SR stands out as one strong
candidate to be included in the next 802.11bn amendment.

Figure 1: Left side: Example of the proposed MAPC operation in a small OBSS
deployment. Right side: coordination frames and coordinated transmission,
where AP1 and AP3 are selected due to their spatial reuse compatibility.

The performance of MAPC techniques is still not fully understood. The
definition of the MAPC framework started with IEEE 802.11be [13], but was then
postponed until IEEE 802.11bn. Since then, multiple flavors of MAPC have been
proposed, some advocating for decentralized solutions [14] and others for
central controllers [15]. Another relevant point of discussion is related to
the way coordination signaling should be performed, and whether over wireless
or wired connections [14, 16]. Close in spirit to our work, [17] introduced a
framework where compatible multi-AP groups are scheduled for performing
simultaneous transmissions within a TDMA-slotted Transmission Opportunity
(TXOP). The results in [17] evidenced that algorithms based on per-AP
selection are preferable to those based on per-group selection.

When it comes to C-SR, its performance has been evaluated in a few IEEE 802.11
standard contributions [14, 18, 19] via simulation results, showing the
potential gains compared with legacy access, C-TDMA and C-OFDMA. In [20], the
authors evaluated the throughput of networks implementing C-TDMA and C-TDMA
with SR (C-TDMA/SR), showing that C-TDMA/SR leads to throughput gains between
50% and 140%, depending on the scenario, when compared to C-TDMA. As for the
analytical characterization of C-SR, a model based on Markov chains was
introduced in [21] to compute the C-SR’s throughput and its spatial efficiency
in comparison to the Distributed Coordination Function (DCF). However, the
model in [21] did not consider backoff collisions and coordination overheads,
which may affect the achievable performance gains. We aim to overcome such
limitations.

Our contributions can be summarized as follows:

  * •

We introduce a C-SR mechanism relying on the existence of spatial reuse groups
of AP-Station (STA) pairs, i.e., groups of devices that can transmit
simultaneously. Spatial reuse groups are periodically updated (e.g., every few
hundreds of msecs) based on the exchanged Received Signal Strength Indicator
(RSSI) reports between APs. Then, once the spatial reuse groups are updated,
the data exchange phase starts, where TXOP sharing is used by the channel
access winner to signal group transmissions.

  * •

We leverage Bianchi’s model to analyze the C-SR mechanism and evaluate its
potential performance under full-buffer conditions. The presented analysis
accommodates different group creation and selection algorithms, thus enabling
to model and compare different implementations.

  * •

Using the proposed analytical model, we evaluate the throughput of C-SR and
compare it against DCF. Our results illustrate how the achievable throughput
is affected by the devices’ location. Moreover, they demonstrate the gains
achieved by the proposed group formation approach, especially in scenarios
with a large number of STAs per AP.

##  II Proposed C-SR Mechanism Model

###  II-A C-SR Operation

We make the following modeling assumptions for the sake of tractability:

  * •

We consider a group of APs within the same coverage area, i.e., Overlapping
Basic Service Set (OBSS), that contend to access the channel using the default
Carrier-Sense Multiple Access with Collision Avoidance (CSMA/CA) mechanism.

  * •

The C-SR groups, i.e., the groups of AP-STA pairs that can benefit from SR
concurrent transmissions, are computed based on the RSSI values exchanged
between the coordinated APs. For low mobility scenarios, the RSSI varies on a
slow time scale and its acquisition incurs limited overhead.

  * •

The C-SR operation is implemented atop DCF in a TXOP-sharing fashion, i.e.,
when a coordinated AP gains access to the channel (i.e., the backoff counter
reaches zero) it initiates a multi-AP transmission along with compatible
transmitters in the OBSS. The AP that wins the contention (leader AP) then
selects one of its STAs as a target receiver, and triggers the group (if
exists) that includes other AP-STA pairs compatible with the target receiver.

The implementation of the overall coordination approach is illustrated in Fig.
1, where it can be seen how a set of coordinated APs compete to gain access to
the channel using DCF. To capture the coordination overheads, we assume a
coordination phase of duration $T_{\rm MAPC}$. This phase includes channel
reservation and information about the set of devices and settings to be used
in the incoming coordinated TXOP, such as bandwidth, Modulation and Coding
Scheme (MCS), TXOP duration, etc.111We assume that the coordination phase is
feasible and aligned with the MAPC mechanism being discussed and likely
included as part of the IEEE 802.11bn amendment. However, details about its
implementation are out of the scope of this paper. After the $T_{\rm MAPC}$
period, AP1 and AP3 start transmitting data in Aggregate MAC Protocol Data
Units (A-MPDUs) to their corresponding STAs, while AP2 sets its Network
Allocation Vector (NAV), waiting its turn for decrementing its backoff.
Subsequently, Block ACKs (BACK) are sent to AP1 and AP3 by their corresponding
STAs using OFDMA.

###  II-B Multi-AP Group Formation

An SR group consists of several APs and STAs pairs that are compatible, i.e.,
pairs of devices that can transmit simultaneously without causing the Signal-
to-Interference-plus-Noise Ratio (SINR) at the served STAs to fall below a
threshold $\gamma_{\textrm{CE}}$ (see Section IV for further details). The
groups are defined based on the RSSI perceived by each STA from the
neighboring APs.222Owing to time division duplexing and channel reciprocity,
downlink RSSI can be estimated from measurements on uplink frames (data or ACK
frames).

Consider $\mathcal{M}$ as the collection of all AP-STA pairs and $\mathcal{C}$
as the ensemble of all valid combinations attainable (via exhaustive search)
from the pairs within $\mathcal{M}$. Then, we define
$\mathcal{M}_{i}\subseteq\mathcal{M}$ as the subset of the
$M_{i}=|\mathcal{M}_{i}|$ active AP-STA pairs in the ${i}$-th combination,
$i\in\mathcal{C}$. Moreover, the quality of each combination $i$ is assessed
by considering the total number of packets that can be transmitted by the
$M_{i}$ APs in combination $i$, i.e.,
$\rho_{i}=M_{i}\sum_{\begin{subarray}{c}\forall j\in\mathcal{M}_{i}\\\
\end{subarray}}\varrho_{i,j}$, where $\varrho_{i,j}$ is the estimated number
of packets correctly transmitted by AP-STA pair $j\in\mathcal{M}_{i}$,
computed based on the MCS index employed by AP-STA pair $j$.

Using the computed combinations of feasible simultaneous transmissions, a
subset $\mathcal{G}\subseteq\mathcal{C}$ is selected. Furthermore, to
guarantee that all the AP-STA pairs are granted with a minimum number of
transmission opportunities, we establish a constraint whereby a given AP-STA
pair $j$ can only be selected exactly $R$ times in $\mathcal{G}$, i.e.,
$\sum_{i\in\mathcal{G}}\mathbbm{1}_{[j\in\mathcal{M}_{i}]}=R$. For the sake of
simplicity and without loss of generality, we set $R=1$ for the remainder of
this paper, which means that an AP-STA pair $j$ will only appear in a single
group $i\in\mathcal{G}$.

To illustrate how the groups $\mathcal{G}$ are selected, we consider the
scenario in Fig. 2(a), where four BSSs (each formed by one AP and one STA) use
the same channel. We assume that the RSSI measurements from all the AP-STA
pairs in the scenario are available to every AP. Table I shows an excerpt of
the combinations of AP-STA pairs and their associated score $\rho_{i}$. To
choose among all possible groups, we sort them in descending order starting by
the ones with the highest values of $\rho_{i}$. Moreover, the groups wherein
one or more AP-STA pairs cannot decode frames properly due to the high
interference generated are discarded, even if their associated $\rho_{i}$
value is high. Following this mechanism, in the example shown in Table I, we
would select combinations $\mathcal{G}=\\{1,5,6\\}$.

Table I: Combinations obtained for Deployment 1. | AP1 | AP2 | AP3 | AP4 | $\rho_{i}$
---|---|---|---|---|---
1 | STA1 | - | - | STA4 | 1480
2 | - | STA2 | - | STA4 | 1218
… |  |  |  |  |
5 | - | STA2 | - | - | 271
6 | - | - | STA3 | - | 271

###  II-C Group Transmission Probability

Every time an AP accesses the channel, i.e., its back-off counter reaches
zero, it randomly selects one of its STAs for transmission and, if the TXOP
sharing is feasible based on the defined groups, it triggers the corresponding
multi-AP group. Therefore, the probability that a group transmits is
proportional to the number of AP-STA pairs that it contains. Thus, for group
$i$, the transmission probability can be computed as:

$\displaystyle\phi_{i}=\sum_{\forall j\in\mathcal{M}_{i}}{\alpha_{j}}\text{,}$
(1)

where $\alpha_{j}$ is the probability that AP-STA pair $j\in\mathcal{M}_{i}$,
and therefore group $i$, is selected for transmission (i.e., STA $j$ is
selected for transmissions since its corresponding AP wins the channel
contention). Following the full-buffer traffic assumption, and assuming that
all the APs use the same access parameters and that STAs associated to each AP
are selected with the same probability, we have that
$\alpha_{j}=\leavevmode\nobreak\ \frac{1}{KS_{j}}$, where $K$ is the total
number of APs in the network, and $S_{j}$ the number of STAs associated to AP
$j$.

For instance, the transmission probability of the groups selected in the
example of Section II-B, i.e., $\mathcal{G}=\\{1,5,6\\}$, are calculated based
on the probabilities that the involved STAs are served at a given TXOP.
Therefore, the transmission probability of group 1,
$\phi_{1}=\frac{1}{4}+\frac{1}{4}=1/2$ (notice that AP1 and AP4 only have one
associated STA). Additionally, $\phi_{5}=\phi_{6}$ = 1/4, because AP2 and AP3
transmit alone in their corresponding groups and have only one associated STA.

##  III Throughput Analysis

To investigate the attainable full-buffer throughput for C-SR, we extend
Bianchi’s DCF model [22] by introducing the concept of multiple AP
transmissions enabled by C-SR within a single TXOP. Bianchi’s DCF model
applies to overlapping sets of devices and assumes a slotted time system,
where the minimum slot unit corresponds to an empty slot (T${}_{e}=9\mu$s).
The empty slot is also used as the unit for decreasing the backoff counter by
one slot while contending for the channel. Apart from the empty slot, we find
successful and collision slots. In the original Bianchi’s model, a successful
slot contains the transmission of only one device. However, in the context of
C-SR, it can contain multiple simultaneous transmissions from the AP-STA pairs
of a C-SR group. A collision slot, on the other hand, occurs when two or more
devices finish their backoff countdown at the same time.

To compute the throughput of a given AP-STA pair $j$, we begin by calculating
the probability of a device accessing the channel in a random slot time, given
by:

$\displaystyle\tau=\frac{1}{\mathbb{E}[B]+1}\text{,}$ (2)

where $\mathbb{E}[B]$ is the expected backoff in number of slots. Considering
an infinite number of retransmissions per packet and utilizing the Binary
Exponential Backoff (BEB) [23], we can compute $\mathbb{E}[B]$ as:

$\displaystyle\mathbb{E}[B]=\frac{\mathrm{CW_{\rm{min}}}+1}{2}\left(\frac{1-p-p(2p)^{m}}{1-2p}\right)-\frac{1}{2}\textrm{,}$
(3)

where $\mathrm{CW_{\rm{min}}}$ denotes the minimum contention window, $m$ is
the number of backoff stages, and $p=1-(1-\tau)^{K-1}$ represents the
conditional collision probability. Since the conditional collision probability
$p$ depends on $\tau$, and vice versa, both parameters need to be iteratively
computed until convergence.

The average duration of a slot, $\operatorname*{\mathbb{E}}[T]$, can be
expressed as:

$\displaystyle\operatorname*{\mathbb{E}}[T]=p_{\rm{e}}T_{\rm{e}}+p_{\rm{s}}\sum_{\forall
i\in\mathcal{G}}{\phi_{i}T_{{\rm{s}},{i}}+p_{\rm{c}}T_{\rm{c}}}\text{,}$ (4)

where, $p_{\rm{e}}=(1-\tau)^{K}$ and $T_{\rm{e}}$ are the probability and the
duration of an empty slot given the coexistence of $K$ transmitters,
respectively, $p_{\rm{s}}=K\tau(1-\tau)^{K-1}$ represents the probability of a
successful transmission slot, $T_{{\rm{s}},i}$ represents the duration of a
transmission when group $i$ is selected, and $T_{\rm{c}}$ is the duration of a
collision slot. Besides, the probability that a slot includes a collision can
be computed as $p_{\rm{c}}=1-p_{\rm{e}}-p_{\rm{s}}$.

A successful transmission comprising group $i$ has a duration of
$T_{{\rm{s}},i}$, as it considers the maximum duration among all AP-STA pairs
that transmit simultaneously in group $i$, i.e.,
$\max_{j\in\mathcal{M}_{i}}\\{T_{i,j}\\}$, and each of them depends on its own
selected transmission rate, $R_{i,j}$. Thus the duration for AP-STA $j$ can be
computed as

$\displaystyle T_{i,j}=$ $\displaystyle T_{\rm MAPC}+2T_{\rm SIFS}+T_{\rm
DATA}(R_{i,j})$ $\displaystyle+T_{\rm BACK}+T_{\rm DIFS}+T_{\rm{e}}\text{,}$
(5)

where all the signaling frames are transmitted using the basic rate.

Finally, the aggregate throughput of the WLAN is

$\displaystyle\Gamma=\frac{p_{\rm{s}}L\bigg{(}\sum_{i\in\mathcal{G}}\phi_{i}\big{(}\sum_{j\in\mathcal{M}_{i}}N_{i,j}\big{)}\bigg{)}}{\mathbb{E}[T]},$
(6)

where $L$ denotes the packet size, and $N_{i,j}$ is the A-MPDU size in number
of packets that the STA in $j$ receives from its corresponding AP for a
transmission in group $i$. The term
$\sum_{i\in\mathcal{G}}\phi_{i}\big{(}\sum_{j\in\mathcal{M}_{i}}N_{i,j}\big{)}$
represents the average number of packets transmitted in a successful slot, and
it characterizes the throughput gain of C-SR compared to DCF. Notice that (6)
holds for the particular case of DCF (without C-SR) if considering that each
group contains a single AP-STA pair.

Similarly, we can compute the throughput of AP-STA pair $j$ by simply
considering the number of transmitted packets by the target AP-STA pair in (6)
instead of the entire group, i.e.,

$\displaystyle\Gamma_{j}=\frac{p_{\rm{s}}L\bigg{(}\sum_{i\in\mathcal{G}}\phi_{i}\big{(}\mathbbm{1}_{[j\in\mathcal{M}_{i}]}N_{i,j}\big{)}\bigg{)}}{\mathbb{E}[T]}.$
(7)

##  IV Performance Evaluation

###  IV-A Scenario and System Model

We consider an OBSS with $K=4$ full-buffered APs deployed throughout a squared
scenario (see deployments in Fig. 2), with each AP separated by a minimum
distance of $d_{\rm{AP-AP}}$ meters from the others. The APs contend to access
the same channel using DCF (they are assumed to be within the communication
range of the others), so packet collisions may occur during that process. Each
AP $k$ has $S_{k}$ stations associated, which serves in a downlink manner. The
STAs are randomly placed $d_{\rm{AP-STA}}$ meters away from their
corresponding AP. For the sake of simplicity, we consider the same number of
STAs associated with each AP.

The transmission rate employed in a given AP-STA transmission depends on the
MCS employed (we consider 802.11ax MCS values), which is selected based on the
estimated RSSI at the receiver. As a result, stations close (far) from their
AP are served using a higher (lower) MCS. The maximum number of aggregated
packets in each A-MPDU depends on the MCS. We consider that all the APs use
the same fixed transmit power. For an SR transmission to be successful, it is
required that the SINR at the receiver surpasses a predefined capture effect
threshold $\gamma_{\textrm{CE}}$ [24]. Notice, as well, that this condition
strongly depends on the set of concurrent AP-STA pairs selected for
transmission in a TXOP.

The path loss effects $P_{\rm{L}}$(dB) are modeled using the TGax model for
Enterprise Scenarios [25]:

$P_{\rm{L}}=40.05+20\log_{10}\left(\frac{\min(d,B_{\rm{p}})f_{\rm{c}}}{2.4}\right)+\mathbbm{1}_{d>B_{\rm
p}}P^{\prime}+7W_{\rm{n}}\text{,}$ (8)

where $d$ is the distance between the transmitter and the receiver in meters
(but no less than 1), $f_{\rm{c}}$ is the carrier frequency in GHz,
$W_{\rm{n}}$ is the number of walls and $P^{\prime}$ is given by
$P^{\prime}=35\log_{10}(d/B_{\rm{p}})$.

We validate the results obtained with our analytical model using a simulator
developed in Matlab reproducing IEEE 802.11 channel access in detail. Table II
collects the parameters used in the evaluation. Note that we consider the same
$T_{{\rm{s}},i}$ for all groups’ transmissions, denoted by $T_{{\rm{s}}}$,
which results in different $N_{i,j}$ values for each STA and group.

###  IV-B Example Deployments

Figure 2: Deployment 1 and Deployment 2 are shown in (a) and (d),
respectively. APs have been placed to a distance $d_{\rm{AP-AP}}=10$ meters,
and 1 STA is associated to each of them. The aggregate throughput of DCF and
C-SR in each deployment is shown in (b) and (e). Moreover, the throughput per
STA in Deployment 1 and Deployment 2 are shown in (c) and (f), respectively.

The two deployments presented in this section have been selected to illustrate
how the performance of C-SR varies from one scenario to another, depending on
the devices’ location. Fig. 2(a) and Fig. 2(d) show the topology of Deployment
1 and Deployment 2, respectively. In both cases, we have $K=4$ APs, the
distance between access points is $d_{\rm{AP-AP}}=10$ meters, and a single STA
per AP is randomly deployed $d_{\rm{AP-STA}}\in[1,5]$ meters away from it.
Besides, Fig. 2(b) and Fig. 2(e) show the aggregate throughput (i.e.,
$\Gamma$) achieved under the DCF and C-SR schemes. Additionally, the bars
drawn in Fig. 2(c) and Fig. 2(f) show the individual throughput experienced by
each station in both deployments, when DCF and C-SR are applied. Black
asterisks represent their corresponding throughput values obtained by
simulation, showing a perfect match between analysis and simulations.

Deployment 1 is an example of a non-favorable deployment for C-SR, where only
STA1 and STA4 can be served simultaneously, while STA2 and STA3 require to be
served using alone transmissions. As indicated in Table I, the selected groups
in this case are $\mathcal{G}=\\{1,5,6\\}$ (see Section II-B). The aggregate
throughput in Fig. 2(b) shows a gain of 54% for C-SR with respect to DCF. This
result is due to the higher number of gained TXOPs, and consequently, more
transmissions by AP1 and AP4 because their group transmission probability is
twice as high as the one of the APs that transmit alone, i.e., $\phi_{1}=1/2$
and $\phi_{5}=\phi_{6}=1/4$.

On the other hand, the STAs in Deployment 2 in Fig. 2(d) are better located to
form compatible C-SR groups. In this case, AP1, AP3, and AP4 are selected to
perform C-SR transmissions to their corresponding STAs, leaving only AP2
alone, hence boosting the network’s performance by around 138% with respect to
DCF (see Fig. 2(e)). The per-STA throughput in Fig. 2(f) shows a remarkable
gain exceeding 180% for C-SR with respect to DCF for all STAs except for STA2,
whose throughput is similar to the one under DCF since it is alone in its
group.

###  IV-C Multiple Random Deployments

We also evaluate $10000$ random deployment realizations for $d_{\rm{AP-
AP}}\in\leavevmode\nobreak\ \\{5,10,20\\}$ meters, obtaining the throughput of
each STA in all realizations. In each case, 10 STAs are deployed for each AP
(for a total of 40 STAs) and placed $d_{\rm{AP-STA}}\in[1,5]$ meters from
their corresponding AP. Fig. 3 illustrates the Cumulative Distribution
Function (CDF) for DCF (continuous-dark line) and C-SR (dashed-green, dotted-
red, and light-blue lines, for 5, 10, and 20 meters, respectively). We observe
that C-SR outperforms DCF in all cases, achieving gains exceeding 92%, 187%,
and 280% in median for $d_{\rm{AP-AP}}=\leavevmode\nobreak\ 5$, $10$ and $20$
meters, respectively. As expected, when $d_{\rm{AP-AP}}$ decreases, the
interference generated by concurring AP transmissions makes it unfeasible to
exploit spatial reuse opportunities, with C-SR performing similarly to DCF. On
the contrary, C-SR transmissions become more profitable when $d_{\rm{AP-AP}}$
increases, thus for $d_{\rm{AP-AP}}=20$ meters, in 20% of the scenarios, all
groups contain all four APs.

Figure 3: CDF of the STA throughput of DCF and C-SR, for $d_{\rm{AP-AP}}=\leavevmode\nobreak\ \\{5,10,20\\}$ meters and 10 STAs per AP. Table II: Simulation Parameters. Parameter | Description | Value
---|---|---
$d_{\rm{AP-AP}}$ | Minimum distance between APs [meters] | {5, 10, 20}
$d_{\rm{AP-STA}}$ | Distance between AP and STAs [meters] | [1, 5]
$B_{\rm{p}}$ | Break-point distance [meters] | 3
$W_{\rm{n}}$ | Number of walls | 1
$BW$ | Bandwidth [MHz] | 80
$f_{\rm{c}}$ | Carrier frequency [GHz] | 6
$N_{\rm{SS}}$ | Number of spatial streams | 2
$T_{\rm{s}}$ | TXOP duration [ms] | 5
$T_{\rm{c}}$ | Duration of a collision slot [$\mu$s] | 137
$T_{\rm MAPC}$ | Coordination overheads [$\mu$s] | 286
$T_{\rm BACK}$ | Block ACK [$\mu$s] | 100
$T_{\rm{e}}$ | Duration of an empty slot [$\mu$s] | 9
$\gamma_{\textrm{CE}}$ | Capture effect threshold [dB] | 15
EIRP | Effective Isotropic Radiated Power [dBm] | 24
$L$ | Length of single data frame [bytes] | 1500

##  V Conclusions

In this paper, we introduced a multi-AP coordination mechanism based on the
formation of spatial reuse groups of AP-STA pairs, i.e., groups of devices
that can transmit simultaneously. We leveraged Bianchi’s model to analyze the
C-SR mechanism and evaluate its potential performance under full-buffer
conditions. Our study compared the throughput of C-SR against that of DCF and
studied the trade-off between group size and interference depending on the
locations of Wi-Fi devices. Our results highlighted the effectiveness of C-SR
group formation and its potential for enhancing overall network performance,
demonstrating a significant throughput improvement of up to $280\%$ in certain
cases. This approach could be further extended by including adaptive power
control.

While we only presented a set of illustrative results due to space
constraints, our analysis enables the evaluation of different group creation
algorithms and TXOP sharing strategies. It also allows to study the effect of
tuning key 802.11 parameters such as the allowed transmission rates, channel
contention settings, and setting different transmission durations (e.g., per
group).

##  VI Acknowledgements

D. Nunez and B. Bellalta were are supported by grant Wi-XR
PID2021-123995NB-I00 (MCIU/AEI/FEDER,UE), and by MCIN/AEI under the MdM
Program (CEX2021-001195-M). G. Geraci was in part supported by the Spanish
Research Agency through grants PID2021-123999OB-I00, CEX2021-001195-M, and
CNS2023-145384, by the UPF-Fractus Chair, and by the Spanish Ministry of
Economic Affairs and Digital Transformation and the European Union
NextGenerationEU.

## References

  * [1] D. Cavalcanti _et al._ , “Wireless TSN–definitions, use cases & standards roadmap,” _Avnu Alliance_ , pp. 1–16, 2020.
  * [2] S. Mangiante _et al._ , “VR is on the Edge: How to Deliver 360° Videos in Mobile Networks,” in _VR/AR Network ’17: Proceedings of the Workshop on Virtual Reality and Augmented Reality Network_ , 08 2017, pp. 30–35.
  * [3] Wireless Broadband Alliance, “WBA Annual Industry Report 2023,” https://wballiance.com/resource/wba-annual-industry-report-2023/, October 2022, accessed on 11/05/2023.
  * [4] E. Oughton _et al._ , “Reviewing wireless broadband technologies in the peak smartphone era: 6G versus Wi-Fi 7 and 8,” _Telecommunications Policy_ , vol. 48, no. 6, p. 102766, 2024.
  * [5] K. Meng _et al._ , “IEEE 802.11 Real Time Applications TIG report,” November 2018, doc.: IEEE 802.11-18/2009r6.
  * [6] A. Aral _et al._ , “Addressing application latency requirements through edge scheduling,” _Journal of Grid Computing_ , vol. 17, no. 4, pp. 677–698, 2019.
  * [7] M. Carrascosa-Zamacois _et al._ , “Wi-Fi multi-link operation: An experimental study of latency and throughput,” _IEEE/ACM Transactions on Networking_ , vol. 32, no. 1, pp. 308–322, 2024.
  * [8] L. Galati-Giordano _et al._ , “What Will Wi-Fi 8 Be? A Primer on IEEE 802.11bn Ultra High Reliability,” _arXiv preprint arXiv:2303.10442_ , 2023\.
  * [9] “IEEE 802.11-22/0078r3: 802.11 UHR Draft Proposed PAR,” https://mentor.ieee.org/802.11/dcn/23/11-23-0078-03-0uhr-uhr-draft-proposed-par.docx, January 2023, accessed on January 2024.
  * [10] K. Aio _et al._ , “Overhead Analysis of Coordinated Spatial Reuse,” May 2023, doc.: IEEE 802.11-23/0616r0.
  * [11] S. Helwa _et al._ , “Thoughts on Coordinated Spatial Reuse,” January 2024, doc.: IEEE 802.11-24/0120r0.
  * [12] L. Lanante _et al._ , “Efficient Coordinated Spatial Reuse,” January 2024, doc.: IEEE 802.11-24/0095r0.
  * [13] A. Garcia-Rodriguez _et al._ , “IEEE 802.11be: Wi-Fi 7 Strikes Back,” _IEEE Communications Magazine_ , vol. 59, no. 4, pp. 102–108, 2021.
  * [14] K. Aio _et al._ , “Coordinated Spatial Reuse Performance Analysis,” September 2019, doc.: IEEE 802.11-19/1534r5.
  * [15] P. Nayak _et al._ , “Resource Management for Multi-AP Coordination,” April 2023, doc.: IEEE 802.11-23/0665r0.
  * [16] T. Zeng _et al._ , “Multi-AP coordination over Fibre,” June 2023, doc.: IEEE 802.11-23/1186r1.
  * [17] D. Nunez _et al._ , “Multi-AP Coordinated Spatial Reuse for Wi-Fi 8: Group Creation and Scheduling,” _arXiv preprint arXiv:2305.04846_ , 2023\.
  * [18] D. Akhmetov _et al._ , “Multi-AP coordination for spatial reuse,” 2020, doc.: IEEE 802.11-20/0107r1.
  * [19] Y. Seok _et al._ , “Coordinated Spatial Reuse (C-SR) Protocol,” April 2020, doc.: IEEE 802.11-20/0576r1.
  * [20] D. Nunez _et al._ , “TXOP sharing with coordinated spatial reuse in multi-AP cooperative IEEE 802.11be WLANs,” in _2022 IEEE 19th Annual Consumer Communications Networking Conference (CCNC)_ , 2022, pp. 864–870.
  * [21] F. Wilhelmi _et al._ , “Throughput analysis of IEEE 802.11bn coordinated spatial reuse,” _arXiv preprint arXiv:2309.09169_ , 2023.
  * [22] G. Bianchi, “Performance analysis of the IEEE 802. 11 distributed coordination function,” _IEEE Journal on Selected Areas in Communications_ , vol. 18, no. 3, pp. 535–547, 2000.
  * [23] B. Bellalta and K. Kosek-Szott, “AP-initiated multi-user transmissions in IEEE 802.11ax WLANs,” _Ad Hoc Networks_ , vol. 85, pp. 145–159, 2019.
  * [24] J. Lee _et al._ , “An experimental study on the capture effect in 802.11a networks,” in _Proc. ACM WiNTECH_ , 2007, pp. 19–26.
  * [25] S. Merlin _et al._ , “TGax Simulation Scenarios,” Nov. 2015, doc.: IEEE 802.11-14/0980r16.


# Exploring Modifications to FLRW Cosmology with General Entropy and
Thermodynamics: A new Approach

<EMAIL_ADDRESS>and M. Monshizadeh2 1Department
of Physics, Faculty of Science, Bu-Ali Sina University, Hamedan 65178, Iran
2Physics Department, Faculty of Science, Islamic Azad University of Hamedan,
Hamedan 65181, Iran

###### Abstract

This paper investigates modifications to the FLRW cosmology resulting from
considering general entropy for the cosmological apparent horizon. Although
the Bekenstein-Hawking entropy has been successful in the study of
thermodynamics of black holes and cosmology, according to the laws of the
thermodynamics, considering this entropy, it imposes a $w=-1$ limit on the
equation of state of the cosmic fluid. To address this limitation, Nojiri and
colleagues in 2022, introduced generalized entropies with four-parameters that
may tend to familiar entropies and can solve some problems of cosmology. In
this paper, we explore how correcting the entropy affects the Energy-momentum
tensor of the cosmic flux from the outset. Our calculations show that in any
way, the modified Bekenstein-Hawking entropy can be still considered as a
general entropy after a modification energy-momentum density tensor with a
correction function $f(\rho)$. The function $f(\rho)$ can be easily
reconstructed, by considering thermodynamics of the apparent horizon. We also
study the first and second laws of thermodynamics in this context, and show
how the problem of limited cosmic flux can be resolved based on the model’s
parameters. At last, In order to study the kinematics of the universe,
cosmography parameters are also calculated and we see that the correction
function $f(\rho)$ has a non-negligible role in cosmography parameters. This
paper provides valuable insights into using general entropies on the apparent
horizon of universe.

FLRW cosmology; Apparent horizon; Laws of thermodynamics; Bekenstein-Hawking
entropy; General entropy

##  I Introduction

Gravity and thermodynamics are two fundamental areas of physics that however
have been long studied independently. After the exploring of black hole
mechanics, particularly Bekenstein-Hawking entropy ($S_{BH}$) Bekenstein
(1973); Hawking (1975), there has been a growing interest in exploring the
link between these two areas Jacobson (1995); Hayward (1998); Padmanabhan
(2005, 2010). After it was proved that Einstein’s equation can be interpreted
as the first law of thermodynamics Padmanabhan (2014), and the development of
FLRW cosmology based on Einstein’s field equation, the analogy between
thermodynamics and cosmology has been also established. The apparent horizon
is the specific type of cosmological horizon in which light can not escape to
infinity and is causally connected to an observer located at a particular
point in spacetime. It has been shown that the apparent horizon can be treated
as a thermodynamic system with temperature ($T$) and entropy ($S$). This idea
was first proposed by Jacobson in 1995 Jacobson (1995), who showed that the
Einstein field equation could be derived from the laws of thermodynamics. The
temperature is given by the surface gravity of the horizon, while the entropy
is given by the area of the horizon in Plank units. Studying the
thermodynamics of the FLRW apparent horizon can shed light on some features
and problems on this area of research Sanchez and Quevedo (2023).

From the context of black hole entropy, some other different entropy functions
have been proposed, such as the entropy of TsallisTsallis (1988), Rényi Rényi
(1960), Barrow Barrow (2020), Sharma-MittalSayahian Jahromi et al. (2018),
KaniadakisKaniadakis (2005); Drepanou et al. (2022), and the entropy of loop
quantum gravity Majhi (2017); Liu (2022). All of them are written as a
monotonic increasing function of $S_{BH}$, obey the third law of thermodynamic
($S\rightarrow 0$ as $S_{BH}\rightarrow 0$), and for suitable choice of
parameter they tend to $S_{BH}$.

Recently some different generalized entropy with three, four, and six
parameters have been proposed so that some of them tend to be known entropies
by choosing suitable parameters. They showed that these entropies can solve
some problems in some areas of cosmology (e.g. black hole, bounce cosmology,
…) Nojiri et al. (2022a, b); Odintsov and Paul (2023a, b). Specially in Nojiri
et al. (2022c) authors showed that after generalizing the entropy from
$S_{BH}$, the formal Friedmann equations should be modified. Also, they
studied generalized entropy and found the following conjecture “the minimum
number of parameters required in a generalized entropy function that can
generalize all the known entropies mentioned above, is equal to four” Odintsov
and Paul (2023b).

The modification of Friedmann equations strengthened my motivation to look
deeper into this issue. Doesn’t the generalization of entropy lead to the
change in the energy-momentum density tensor of the cosmic flux? In this case,
doesn’t the continuity equation also modified? With these changes, will the
laws of thermodynamics still be maintained?

Also, in studying the kinematics of the universe, cosmologists examine some
cosmography parameters, such as Hubble, deceleration, jerk, snap, etc,
parameters Visser (2005); Ivashtenko (2019); Rezaei et al. (2020); Mehrabi and
Rezaei (2021). How do general entropies affect cosmographic parameters?

I will address these questions in this essay. In Sec. II, I will provide a
brief overview of the method presented by Nojiri et.al. Nojiri et al. (2022c)
in the discussion of modifying the Friedman equations as a result of applying
generalized entropy. Then in Sec. III, I suggested that the same changes
created in the previous section can be achieved by modifying the energy-
momentum density of perfect cosmic fluid by a correction function. Of course,
the difference here compared to before is that the continuity equation should
be changed. In Sec. IV, with the help of the first law of thermodynamics and a
four-parameter kind of general entropy, I will try to reconstruct the
correction function applied in the energy-momentum tensor. The second law of
thermodynamics is discussed in Sec. V, and the effect of applying general
entropies on cosmographic parameters is investigated in Sec. VI. Finally, the
work will be summarized with some conclusions.

In this article, I’m using the natural unit in which ‘$c=G=\hbar=1$’, over-dot
will symbolize $d/dt$, and over prime will present the derivative of the
function with respect to its arguments.

##  II Modification of FLRW cosmology by considering a general entropy

In this section, I give a brief review of method presented in Nojiri et al.
(2022c).

In an isotropic FLRW metric,

$ds^{2}=h_{ab}dx^{a}dx^{b}+R^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}),$ (1)

where $x^{0}=t,~{}x^{1}=r,~{}R=a(t)r$ and
$h_{ab}=diag\Big{(}-1,~{}a(t)^{2}/(1-kr^{2})\Big{)}$ with it’s
determinant,‘$h$’, the Bekenstein-Hawking entropy, radius, surface gravity,
and temperature of the apparent horizon are Sanchez and Quevedo (2023); Cai
and Kim (2005); Binétruy and Helou (2015); Hayward (1998)

$\displaystyle S_{BH}$ $\displaystyle=$ $\displaystyle\frac{A}{4}=\pi
R_{h}^{2}=\frac{3}{8\rho}$ (2) $\displaystyle R_{h}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{(H^{2}+\frac{k}{a^{2}})}$ (3) $\displaystyle\kappa$
$\displaystyle=$ $\displaystyle\frac{1}{2\sqrt{-h}}\frac{\partial}{\partial
x^{a}}(\sqrt{-h}h^{ab}\frac{\partial R_{h}}{\partial x^{b}})$ (4)
$\displaystyle T_{h}$ $\displaystyle=$
$\displaystyle\frac{|\kappa|}{2\pi}=\frac{1}{2\pi
R_{h}}|1-\frac{\dot{R_{h}}}{2HR_{h}}|.$ (5)

Following to recent paper of Nojiri et.al. Nojiri et al. (2022c), a four-
parameter general entropy was proposed as

$S_{G}(\alpha_{+},\alpha_{-},\gamma,\beta)=\frac{1}{\gamma}[(1+\frac{\alpha_{+}S_{BH}}{\beta})^{\beta}-(1+\frac{\alpha_{-}S_{BH}}{\beta})^{-\beta}],$
(6)

Using first law of thermodynamics, $dE=-du=-\delta Q-pdV$, where $\delta
Q=TdS$ is heat transfer from the apparent horizon and $p$ is the pressure of
fluid in the volume $V$, it was calculated

$\displaystyle E=\frac{4}{3}\pi R^{3}_{h}\rho,~{}~{}~{}~{}~{}V=\frac{4}{3}\pi
R^{3}_{h},$ (7)
$\displaystyle\frac{dE}{dt}+p\frac{dV}{dt}=-T\frac{dS}{dt}=\frac{4\pi}{3}\dot{\rho}R^{3}_{h}[1+3\frac{\dot{R_{h}}}{R_{h}}\frac{p+\rho}{\dot{\rho}}]$
$\displaystyle\frac{-1}{2\pi
R_{h}}|1+\frac{R^{2}_{h}}{2}(\dot{H}-\frac{k}{a^{2}})|\dot{S}=\frac{4\pi}{3}\dot{\rho}[1-R^{2}_{h}(\dot{H}-\frac{k}{a^{2}})].$
(8)

In these equations, the formal continuity equation $\dot{\rho}=-3H(\rho+p)$ is
used. By considering the general entropy as a function of $S_{G}(S_{BH})$, we
find

$\dot{S}=-2\pi R^{4}_{h}H(\dot{H}-\frac{k}{a^{2}})S^{\prime}({S_{BH}}).$ (9)

Inserting (9) in (8), leads

$\Big{[}1+\frac{\frac{3R_{h}^{2}}{2}(\dot{H}-\frac{k}{a^{2}})}{1-R_{h}^{2}(\dot{H}-\frac{k}{a^{2}})}\Big{]}S^{\prime}(\dot{H}-\frac{k}{a^{2}})=-4\pi(\rho+p),$
(10)

that it has the capability to produce a modified version of the Friedmann
equation. Although by considering the four parameters of general entropy (6),
it is very hard to solve the differential equation (10), by considering a
Tsallis limit of general entropy
($\alpha_{-}=0~{}\&~{}\alpha_{+}\rightarrow\infty$),

$S_{T}=S_{0}(\frac{S_{BH}}{S_{0}})^{\delta}$ (11)

where $S_{0}$ is a constant and $\delta$ is the exponent of Tsallis entropy,
the differential equation (10) was solved and obtained Nojiri et al. (2022c)

$H^{2}+\frac{k}{a^{2}}\sim\pi\Biggl{\\{}\frac{8}{3}\frac{2-\delta}{\delta}S_{0}^{\delta-1}(\rho+\Lambda)\Biggr{\\}}^{\frac{1}{2-\delta}},$
(12)

where $\Lambda$ is a constant of integration which may be interpreted as the
cosmological constant. By rewriting this equation as

$H^{2}+\frac{k}{a^{2}}\sim\frac{8\pi}{3}\rho f(\rho)$ (13)

where

$f(\rho)=\frac{3}{8\rho}\Biggl{\\{}\frac{8\pi}{3}\frac{2-\delta}{\delta}S_{0}^{\delta-1}(\rho+\Lambda)\Biggr{\\}}^{\frac{1}{2-\delta}}$
(14)

By considering relation (13), I express a conjecture in the following form:

conjecture: “By generalizing entropy in an FLRW cosmology, the energy-momentum
density tensor of the cosmic fluid is modified in such a way that a common
correction function multiplies on both the energy density and pressure terms.
This ensures that the perfectness property and the EoS parameter of the cosmic
fluid remain unchanged.”

In the following, I consider this conjecture and try to make a reconstruction
$f(\rho)$ by considering the general entropy.

##  III modification of energy-momentum density tensor corresponding to
general entropy

In the previous section, a conjecture was established that said by considering
the general entropy it might change the energy-momentum density tensor with a
multiplying correction function.

By considering the following modified energy-momentum tensor of a perfect
fluid

$T^{\mu}_{\nu}=diag(-\rho f(\rho),pf(\rho),pf(\rho),pf(\rho)),$ (15)

in which the equation of state parameter (EoS) is equal to $w=p/\rho$. The
conservation law $T^{\mu\nu}_{;\nu}=0$, leads

$\displaystyle T^{\mu\nu}_{;\nu}=0$ $\displaystyle\Rightarrow
a^{3}\frac{d(pf(\rho))}{dt}=\frac{d}{dt}[a^{3}(\rho+p)f(\rho)]\Rightarrow
a^{3}[\dot{p}f(\rho)+f^{\prime}(\rho)\dot{\rho}p]=$ (16) $\displaystyle
3a^{2}\dot{a}[\rho+p]f(\rho)+a^{3}(\dot{\rho}+\dot{p})f(\rho)+a^{3}(\rho+p)f^{\prime}(\rho)\dot{\rho}=0\Rightarrow$
$\displaystyle\dot{\rho}[1+\rho\frac{f^{\prime}(\rho)}{f(\rho)}]+3H(\rho+p)=0$

For example for Eq. (14) which is obtained by considering the Tsallis entropy,
the following modified continuity equation is obtained

$\dot{\rho}+3H(\rho+p)(2-\delta)=0,$ (17)

while in the previous section, the correction term was absent.

By taking the time derivative of Eq. (13) and using (16), the second modified
Friedmann equation leads to

$\dot{H}-\frac{k}{a^{2}}=-4\pi(\rho+p)f(\rho).$ (18)

##  IV Reconstruction $f(\rho)$ by considering the first law of thermodynamics

Using the modified Friedmann equations (13),(18) and the first law of
thermodynamics, $dE=-\delta Q+WdV$, where $W=-1/2T^{ab}h_{ab}$ is the work
density and $\delta Q=TdS$, and also using

$E=\frac{4}{3}\pi R^{3}_{h}\rho f(\rho),~{}~{}V=\frac{4}{3}\pi
R^{3}_{h},~{}~{}W=-\frac{1}{2}(p-\rho)f(\rho),$ (19)

we obtain the following relations

$\displaystyle\frac{dE}{dt}-W\frac{dV}{dt}=\frac{4\pi}{3}\dot{\rho}R^{3}_{h}f(\rho)[1+\frac{\rho
f^{\prime}(\rho)}{f(\rho)}+\frac{3}{2}\frac{\dot{R_{h}}}{R_{h}}\frac{p+\rho}{\dot{\rho}}]$
$\displaystyle-T\frac{dS}{dt}=\frac{4\pi}{3}\dot{\rho}R^{3}_{h}f(\rho)[1+\frac{R^{2}_{h}}{2}(\dot{H}-\frac{k}{a^{2}})][1+\frac{\rho
f^{\prime}(\rho)}{f(\rho)}]\Rightarrow$ $\displaystyle\frac{-1}{2\pi
R_{h}}|1+\frac{R^{2}_{h}}{2}(\dot{H}-\frac{k}{a^{2}})|\dot{S}=\frac{4\pi}{3}\dot{\rho}R^{3}_{h}f(\rho)[1+\frac{R^{2}_{h}}{2}(\dot{H}-\frac{k}{a^{2}})][1+\frac{\rho
f^{\prime}(\rho)}{f(\rho)}].$ (20)

At last by using

$\dot{S}=8\pi^{2}R^{4}_{h}H(\rho+p)f(\rho)S^{\prime}(\tilde{S}_{BH}),~{}~{}~{}~{}\dot{\rho}=\frac{-3H(\rho+p)}{1+\frac{\rho
f^{\prime}(\rho)}{f(\rho)}},$ (21)

where $\tilde{S}_{BH}=\frac{3}{8\rho f(\rho)}$ is the Bekenstein-Hawking
entropy in this new point of view, we find

$S^{\prime}(\tilde{S}_{BH})=1.$ (22)

In the case of standard FLRW cosmology with $f(\rho)=1$, it gives
$S_{G}=S_{BH}+C$ which is expected if the integration constant $C=0$. In order
to reconstruct the correction function $f(\rho)$ from this point of view, we
must compare the correspondence of these two prescriptions. Let’s have

$\displaystyle\tilde{S}_{BH}=S_{G}(S_{BH}),$ (23)

where $S_{G}$ is the generalized entropy (6). Therefore we obtain

$\frac{3}{8\rho
f(\rho)}=\frac{1}{\gamma}[(1+\frac{3\alpha_{+}}{8\beta\rho})^{\beta}-(1+\frac{3\alpha_{-}}{8\beta\rho})^{-\beta}],$
(24)

and simply find that

$f(\rho)=\frac{3\gamma}{8\rho}[(1+\frac{3\alpha_{+}}{8\beta\rho})^{\beta}-(1+\frac{3\alpha_{-}}{8\beta\rho})^{-\beta}]^{-1}.$
(25)

With this approach, we can easily find the correcting function corresponding
to any entropy function that is appropriate in the work and enter it into the
tensor of the energy-momentum density of the cosmic fluid. The continuation of
the process will be the same as standard cosmological calculations. Again, it
is necessary to mention that in this method, the condition of the perfect
fluid is still valid and also the EoS of the cosmic fluid does not change.

In table 1, I give some correcting functions $f(\rho)$ corresponding to any
entropy function $S_{G}$.

| Entropy | $S_{G}$ | $f(\rho)$
---|---|---|---
1 | Tsallis | $S_{0}(\dfrac{S_{BH}}{S_{0}})^{\delta}$ | $(\dfrac{8}{3}S_{0}\rho)^{\delta-1}$
2 | Rényi | $\dfrac{1}{\alpha}\ln(1+\alpha S_{BH})$ | $\dfrac{3\alpha}{8\rho}[\ln(1+\dfrac{3\alpha}{8\rho})]^{-1}$
3 | Sharma-Mittal | $\dfrac{1}{R}[(1+\delta S_{BH})^{R/\delta}-1]$ | $\dfrac{3R}{8\rho}[(1+\dfrac{3\delta}{8\rho})^{R/\delta}-1]^{-1}$
4 | Kaniadakis | $\dfrac{1}{K}\sinh(KS_{BH})$ | $\dfrac{3K}{8\rho}[\sinh(\dfrac{3K}{8\rho})]^{-1}$
5 | Loop quantum gravity | $\dfrac{1}{1-q}\\{\exp[(1-q)\Lambda(\gamma_{0})S_{BH}]-1\\}$ | $\dfrac{3(1-q)}{8\rho}\\{\exp[\dfrac{3}{8\rho}(1-q)\Lambda(\gamma_{0})]-1\\}^{-1}$

Table 1: Reconstruction of $f(\rho)$ for various entropy functions

.

As another result of this discussion, by comparing the two forms of the first
law of thermodynamics in sections II and IV, we obtain

$-W=p\Rightarrow\frac{1}{2}(p-\rho)f(\rho)=p\Rightarrow~{}w=(1-\frac{2}{f(\rho)})^{-1}.$
(26)

Therefore, when we consider the general entropy and consequently the
modification of stress tensor, we see that even though the standard FLRW
cosmology assumes $S_{G}=S_{BH}$ and $f(\rho)=1$, which implies that only the
equation of state (EoS) of the cosmic fluid should be considered under the
first law of thermodynamics with $w=-1$, the generalized entropy prescription
may allow for a wider range of matter with a dynamic EoS. Therefore, the first
law of thermodynamics permits a wider range of matter in the generalized
entropy prescription compared to the standard FLRW cosmology. Although
equation (26) shows that the EoS parameter experiences a singularity at a
certain point, the singularity’s location can be altered by selecting the
appropriate parameters. However, it is important to note that this equation of
state function was derived solely to satisfy the first law of thermodynamics.
Therefore, it only indicates the range in which the first law is satisfied and
does not constrain any particular value.

A noteworthy point in this regard is the power and ease of the new method for
modifying the Friedmann equations compared to the previous method discussed in
Section II. Furthermore, in the future, by choosing other general entropies
such as non-singular entropy or entropy with more parameters, this
modification process can be easily carried out. Also In this method, all kinds
of entropies can be regarded by modified Bekensten-Hawking entropy
‘$\tilde{S}_{BH}=3/(8\rho f(\rho))$’ with its corresponding $f(\rho)$.

##  V second law of thermodynamics

In The second law of thermodynamics, the entropy must be a convex extensive
function of the state variables of the system, so we must have
$\dot{S}\geqslant 0,~{}\&~{}\ddot{S}\leqslant 0$ Sanchez and Quevedo (2023).
Therefore from $S_{G}=\tilde{S}_{BH}$ and modified continuity equation we
obtain

$\displaystyle\dot{S}$ $\displaystyle=$ $\displaystyle 3HS_{G}(1+w)\geqslant
0$ (27) $\displaystyle\ddot{S}$ $\displaystyle=$
$\displaystyle\dot{S}[3H(1+w)+\frac{\dot{H}}{H}+\frac{\dot{w}}{1+w}]\leqslant
0.$ (28)

Therefore Eq. (27) implies that $w\geqslant-1$ and Eq. (28) gives

$3H(1+w)+\frac{\dot{H}}{H}+\frac{\dot{w}}{1+w}\leqslant 0,$ (29)

in Eq. (29), the first term is positive, but two remaining terms may be
negative. However in the universe under a phantom phase, the second term is
also positive and this definitely requires that

$\dot{w}\leqslant-3H(1+w)^{2}-\frac{|\dot{H}|}{H}(1+w),$ (30)

and In a non phantom universe, where $\dot{H}<0$, Condition (29) may be
established even when $\dot{w}$ becomes positive. This case implies

$\dot{H}\leqslant-3H^{2}(1+w)-\frac{H|\dot{w}|}{1+w}.$ (31)

Briefly, by using the general entropy, it is possible to satisfy both the
first and second laws of thermodynamics over a broad range of EoS parameters,
provided that an appropriate set of parameters is chosen.

##  VI the effect of $f(\rho)$ on cosmography parameters

Cosmography or kinematics parameters have been established as applicable tools
to study the kinematical state of our universe. These parameters have been
defined as follows Ivashtenko (2019); Visser (2005)

$\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{\dot{a}}{a}$
$\displaystyle q$ $\displaystyle=$
$\displaystyle-\frac{1}{H^{2}}\frac{\ddot{a}}{a}=-1-\frac{\dot{H}}{H^{2}}$
$\displaystyle j$ $\displaystyle=$
$\displaystyle\frac{1}{H^{3}}\frac{a^{(3)}}{a}=1+3\frac{\dot{H}}{H^{2}}+\frac{\ddot{H}}{H^{3}}$
$\displaystyle s$ $\displaystyle=$
$\displaystyle\frac{1}{H^{4}}\frac{a^{(4)}}{a}=1+6\frac{\dot{H}}{H^{2}}+4\frac{\ddot{H}}{H^{3}}+\frac{H^{(3)}}{H^{4}}+3\frac{\dot{H}^{2}}{H^{4}}$
(32) . . $\displaystyle=$ $\displaystyle\frac{1}{H^{n}}\frac{a^{(n)}}{a}$

where $H,~{}q,~{}j,~{}s$ are Hubble, deceleration, jerk, snap, parameters,
respectively and, superscript $(n)$ denotes the n’th derivative of a function
with respect to cosmic time. Although, in some works, parameters with higher
time derivatives, may be needed Rezaei et al. (2020). According to the
$H^{(n)}$, the above relations can be rewritten in the following form

$\displaystyle\dot{H}$ $\displaystyle=$ $\displaystyle-H^{2}(1+q)$
$\displaystyle\ddot{H}$ $\displaystyle=$ $\displaystyle H^{3}(j+3q+2)$
$\displaystyle H^{(3)}$ $\displaystyle=$ $\displaystyle
H^{4}(s-4j-3q(q+4)-6).$ (33)

Now, by considering modified Friedmann equations(13,16,18), we obtain

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi}{3}\rho
f(\rho)-\frac{k}{a^{2}}$ $\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle-4\pi\rho(1+w)f(\rho)+\frac{k}{a^{2}}$ $\displaystyle\ddot{H}$
$\displaystyle=$ $\displaystyle
12\pi\rho(1+w)^{2}Hf(\rho)-4\pi\rho\dot{w}f(\rho)-\frac{2Hk}{a^{2}}$
$\displaystyle H^{(3)}$ $\displaystyle=$ $\displaystyle 36\pi H(1+w)\rho
f(\rho)[\dot{w}-H(1+w)^{2}]-48\pi^{2}\rho^{2}(1+w)^{3}f(\rho)^{2}-4\pi\rho
f(\rho)\ddot{w}$ (34)
$\displaystyle+\frac{4k}{a^{2}}[H^{2}+3\pi\rho(1+w)^{2}f(\rho)+2\pi\rho(1+w)f(\rho)]-\frac{2k^{2}}{a^{4}}.$

By these relations, one can see that the correction function $f(\rho)$ affects
all cosmography parameters.

##  VII conclusion

This paper investigated the possible modifications to FLRW cosmology by
considering a 4-parameter general entropy model for the universe’s apparent
horizon. Based on the work of Nojiri and colleagues in 2022, we hypothesized
that the general entropies alter the cosmic flux and the energy-momentum
density tensor from the outset, leading to modifications of the Friedman
equations that can be corrected with a function ‘$f(\rho)$’. We also
demonstrated that the continuity equation is modified and that the general
entropy behaves like a pseudo-Bekenstein-Hawking entropy, which allows for the
simple calculation of the function $f(\rho)$ based on the parameters of the
general entropy. In a table, I presented a list of functions $f(\rho)$ for
each known entropies (Tsallis, Rényi, Sharma-Mital, Kaniadakis, and loop
quantum gravity). Also, from the first and second laws of thermodynamics, I
showed that the cosmic flux can contain a wider range of cosmic materials,
thereby removing the limitation created by the Bekenstein-Hawking entropy in
standard cosmology (i.e., restricting the cosmic fluid to $w=-1$). In order to
study the kinematics of the universe, cosmography parameters were also
calculated and we saw that the correction function $f(\rho)$ has an important
role in kinematics of the universe.

Overall, this paper provides important insights into the use of general
entropies in cosmology and highlights the need to consider alternative entropy
models for a deeper understanding of the universe. This new insight could
potentially have significant implications for the evolution of universe in the
future research.

## References

  * Bekenstein (1973) J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
  * Hawking (1975) S. W. Hawking, Commun. Math. Phys. 43, 199 (1975), [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * Jacobson (1995) T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995), eprint gr-qc/9504004.
  * Hayward (1998) S. A. Hayward, Class. Quant. Grav. 15, 3147 (1998), eprint gr-qc/9710089.
  * Padmanabhan (2005) T. Padmanabhan, Phys. Rept. 406, 49 (2005), eprint gr-qc/0311036.
  * Padmanabhan (2010) T. Padmanabhan, Rept. Prog. Phys. 73, 046901 (2010), eprint 0911.5004.
  * Padmanabhan (2014) T. Padmanabhan, Gen. Rel. Grav. 46, 1673 (2014), eprint 1312.3253.
  * Sanchez and Quevedo (2023) L. M. Sanchez and H. Quevedo, Phys. Lett. B 839, 137778 (2023), eprint 2208.05729.
  * Tsallis (1988) C. Tsallis, J. Statist. Phys. 52, 479 (1988).
  * Rényi (1960) A. Rényi, Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, University of California Press, Berkeley and Los Angeles I, 547 (1960).
  * Barrow (2020) J. D. Barrow, Phys. Lett. B 808, 135643 (2020), eprint 2004.09444.
  * Sayahian Jahromi et al. (2018) A. Sayahian Jahromi, S. A. Moosavi, H. Moradpour, J. P. Morais Graça, I. P. Lobo, I. G. Salako, and A. Jawad, Phys. Lett. B 780, 21 (2018), eprint 1802.07722.
  * Kaniadakis (2005) G. Kaniadakis, Phys. Rev. E 72, 036108 (2005), eprint cond-mat/0507311.
  * Drepanou et al. (2022) N. Drepanou, A. Lymperis, E. N. Saridakis, and K. Yesmakhanova, Eur. Phys. J. C 82, 449 (2022), eprint 2109.09181.
  * Majhi (2017) A. Majhi, Phys. Lett. B 775, 32 (2017), eprint 1703.09355.
  * Liu (2022) Y. Liu, EPL 138, 39001 (2022), eprint 2112.15077.
  * Nojiri et al. (2022a) S. Nojiri, S. D. Odintsov, and V. Faraoni, Phys. Rev. D 105, 044042 (2022a), eprint 2201.02424.
  * Nojiri et al. (2022b) S. Nojiri, S. D. Odintsov, and T. Paul, Phys. Lett. B 831, 137189 (2022b), eprint 2205.08876.
  * Odintsov and Paul (2023a) S. D. Odintsov and T. Paul, Phys. Dark Univ. 39, 101159 (2023a), eprint 2212.05531.
  * Odintsov and Paul (2023b) S. D. Odintsov and T. Paul (2023b), eprint 2301.01013.
  * Nojiri et al. (2022c) S. Nojiri, S. D. Odintsov, and T. Paul, Phys. Lett. B 835, 137553 (2022c), eprint 2211.02822.
  * Visser (2005) M. Visser, Gen. Rel. Grav. 37, 1541 (2005), eprint gr-qc/0411131.
  * Ivashtenko (2019) O. Ivashtenko, East Eur. J. Phys. 2019, 5 (2019).
  * Rezaei et al. (2020) M. Rezaei, S. Pour-Ojaghi, and M. Malekjani, Astrophys. J. 900, 70 (2020), eprint 2008.03092.
  * Mehrabi and Rezaei (2021) A. Mehrabi and M. Rezaei, Astrophys. J. 923, 274 (2021), eprint 2110.14950.
  * Cai and Kim (2005) R.-G. Cai and S. P. Kim, JHEP 02, 050 (2005), eprint hep-th/0501055.
  * Binétruy and Helou (2015) P. Binétruy and A. Helou, Class. Quant. Grav. 32, 205006 (2015), eprint 1406.1658.