# The Ricci flow and isoperimetric inequalities on surfaces

###### Abstract

We revisit the connection between the Ricci flow and isoperimetric
inequalities on surfaces which are diffeomorphic to the $2$-sphere. We prove
that the Cheeger isoperimetric constant is non-decreasing under Ricci flow on
topological $2$-spheres. A topological $2$-sphere with non-trivial curvature
is exhibited which is a counterexample to the hypothesis that the Cheeger
constant is a strictly increasing function of the Ricci flow.

Hollis Williams1,∗

1 Division of Physical Sciences and Engineering, King Abdullah University of
Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia

Contact email<EMAIL_ADDRESS>

Keywords: geometric analysis, Ricci flow, isoperimetric inequalities

##  1 Introduction

There is a rich history of the interaction between the mean curvature flow and
geometric inequalities which relate quantities such as area and volume. As an
example, the theory of the one-dimensional mean curvature flow (usually known
as the curve shortening flow) was developed in part using the isoperimetric
inequality. One can also go in the other direction and use the flow theory to
derive inequalities (the theory of the curve shortening flow can be used to
derive an isoperimetric inequality which generalises inequalities of
Alexandrov, Huber and Bol) [1, 2]. Another flow which might be useful in this
context is the Ricci flow. Given a smooth closed manifold $M$ equipped with a
smooth Riemannian metric $g$, the Ricci flow is the following PDE:

$\frac{\partial g}{\partial t}=-2\text{Ric}(g),$

where $\text{Ric}(g)$ is the Ricci curvature of $g$. The Ricci flow was first
used in the context of isoperimetric estimates for surfaces by Hamilton, who
showed the following important result [3, 4]:

Proposition 1: Let $(M,g(t))$ be a solution of the Ricci flow on a topological
$2$-sphere. At any time $t$ such that $h(M,g(t))<16/\sqrt{A}$, one has

$\frac{d}{dt}h(M,g(t))\geq 0.$

If $g_{t}$ is a metric which evolves on the two-sphere $S^{2}$ under Ricci
flow, then $\overline{C}_{S}$ is monotonically non-decreasing for all times
$t$, where $\overline{C}_{S}$ is defined as

$\overline{C}_{S}=\inf_{\gamma}\>C_{S}(\gamma).$

The infimum is taken over all smooth embedded connected curves $\gamma$ and
$C_{S}(\gamma)$ is the isoperimetric ratio $L/\overline{L}$, for $L$ the
length of the curve $\gamma$ and $\overline{\gamma}$ the curve of smallest
length $\overline{L}$ on the round sphere.

The infimum is taken over all smooth embedded connected curves $\gamma$ and
$C_{S}(\gamma)$ is the isoperimetric ratio $L/\overline{L}$, for $L$ the
length of the curve $\gamma$ and $\overline{\gamma}$ the curve of smallest
length $\overline{L}$ on the round sphere.

There are fewer results in higher dimension, but in dimension $3$, Agol, Storm
and Thurston showed that if one evolves the doubled version of a hyperbolic
metric on an acylindrical $3$-manifold with minimal surface boundary under
Ricci flow, then the ratio of area to volume with sectional curvature at least
$-1$ and minimal surface boundary is minimized in the case of a totally
geodesic hypersurface, which could be viewed as an isoperimetric-type
inequality [5]. Another famous example of an isoperimetric-type inequality
from mathematical relativity is the Penrose inequality, which relates the
total mass of an initial data set to the surface area of its outermost
apparent horizon. There is some work which suggests that it might be possible
to use the Ricci flow to prove the time-symmetric version of the Penrose
inequality, but it is unlikely to be useful for proving the general case [6].

In this article, we revisit the Ricci flow as a tool for establishing
isoperimetric inequalities on surfaces. In Section 2, we consider the Cheeger
isoperimetric constant and show that it is non-decreasing under Ricci flow on
topological 2-spheres. In Section 3, we give an example of a topological
$2$-sphere whose Cheeger constant need not be a strictly increasing function
of the Ricci flow, which establishes that a monotonically non-decreasing
Cheeger constant is the optimal possible result. We end in Section 4 with
conclusions and suggestions for future work. We note that the results which we
establish here likely follow from the sharp estimates obtained by Andrews and
Bryan using isoperimetric comparison for normalized Ricci flow on the two-
sphere, so that the main benefit of our approach is the straightforward nature
of the proofs [7].

##  2 The Cheeger Constant under Ricci Flow on Topological Spheres

A natural question to ask in the context of isoperimetric estimates is whether
the Cheeger isoperimetric constant is always a strictly increasing function of
Ricci flow. It is generally not true that the Cheeger constant is a strictly
increasing function under Ricci flow in dimension $3$. To see this, consider
the counterexample of a dumbbell which develops a nondegenerate neckpinch
singularity. Neckpinch singularities do not occur in dimension $2$, so one can
still ask whether the Cheeger constant is a strictly increasing function of
Ricci flow if we restrict only to surfaces. This was posed as an open question
of Manning during his work on the volume entropy of a compact Riemannian
surface with negative curvature under Ricci flow [8]. Papasoglu studied the
growth rate of the Cheeger constant of a non-compact surface of bounded genus
[9].

As mentioned in Proposition 1, Hamilton had previously shown that the
isoperimetric ratio is monotonically non-decreasing on the two-sphere under
Ricci flow [4]. We now define the isoperimetric constant which we will use in
the article, which is known as the Cheeger constant $h(M)$ for a compact
Riemmanian manifold $M$:

$h(M)=\inf_{\gamma}\frac{S(\gamma)}{\min(V(A),V(B))},$

where $S(\gamma)$ is the $(n-1)$-dimensional volume of a submanifold $\gamma$,
$V(A)$ is the volume of an $n$-dimensional submanifold, and the infimum is
taken over all curves $\gamma$ which divide the $n$-manifold $M$ into two
disjoint submanifolds $A$ and $B$. We will prove that $h$ is monotonically
non-decreasing under Ricci flow on topological $2$-spheres. Before proving
this, we will state a few more preliminaries. Since we are restricting to
closed orientable Riemannian surfaces, if $\gamma$ separates $M$ into two open
manifolds $M+$ and $M_{-}$, then the Cheeger constant for the curve $\gamma$
is equal to

$h(\gamma)=L_{g}(\gamma)\bigg{(}\frac{1}{A_{g}(M_{+})}+\frac{1}{A_{g}(M_{-})}\bigg{)}=L_{g}(\gamma)\frac{A_{g}(M)}{A_{g}(M_{+})\cdot
A_{g}(M_{-})},$

where $L_{g}$ is the length and $A_{g}$ is the area measured with respect to
the metric $g$. Furthermore, if $M$ is a closed Riemannian surface, there
exists a smooth embedded closed curve $\gamma$ such that [10]

$h(\gamma)=h(M).$

Finally, an argument of [4] implies that if $(M,g)$ is a Riemannian surface
which is diffeomorphic to $S^{2}$ such that $h(M,g)<16/\sqrt{A}$, then there
exists a smooth embedded loop $\beta$ such that

$h(\beta)=h(M,g).$

We will use these facts to prove that the Cheeger constant is non-decreasing
on topological spheres under Ricci flow.

Proposition 2: Let $(M,g(t))$ be a solution of the Ricci flow on a topological
$2$-sphere. At any time $t$ such that $h(M,g(t))<4\pi/L$, one has

$\frac{d}{dt}h(M,g(t))\geq 0.$

[Proof]: We begin by noting a result of Papasoglu which states that the
Cheeger constant of a topological $2$-sphere $M$ (more specifically, a
Riemannian manifold or simplicial complex which is homeomorphic to the
$2$-sphere) is bounded by its area $A$ [9]:

$h(M)\leq\frac{16}{\sqrt{A}}.$

In the case of a $2$-sphere, this bound is $8/\sqrt{\pi}r$, where $r$ is the
radius of the sphere. We next show that if $(M,g(t))$ is a solution of the
Ricci flow on a topological $2$-sphere, the Cheeger constants $h(\rho,t)$ for
parallel loops $\gamma_{\rho}$ measured with respect to $g(t)$ satisfy a heat-
type equation given by

$\frac{\partial}{\partial t}(\log
h)=\frac{\partial^{2}}{\partial\rho^{2}}(\log
h)+\frac{\Gamma}{L}\frac{\partial}{\partial\rho}(\log h)+\bigg{(}\frac{4\pi-
hL}{A}\bigg{)}\bigg{(}\frac{A_{+}}{A_{-}}+\frac{A_{-}}{A_{+}}\bigg{)},$

where

$\Gamma=\int_{\gamma_{\rho}}k\>ds.$

To prove this, we note that

$\log h=\log L-\log A_{+}-\log A_{-}+\log A.$

By known results from the theory of Ricci flow on topological spheres, we have
[11]

$\frac{\partial}{\partial t}\log L=\frac{\partial^{2}}{\partial\rho^{2}}(\log
L)+\frac{\Gamma}{L}\frac{\partial}{\partial\rho}(\log L),$

$\frac{\partial}{\partial t}\log
A_{\pm}=\frac{\partial^{2}}{\partial\rho^{2}}(\log
A_{\pm})+\frac{L^{2}}{A^{2}_{\pm}}-\frac{4\pi}{A_{\pm}}+\frac{\Gamma}{L}\frac{\partial}{\partial\rho}(\log
A_{\pm}),$

$\frac{d}{dt}\log A=-\frac{8\pi}{A}.$

Algebraic manipulations show that

$\frac{4\pi}{A_{+}}+\frac{4\pi}{A_{-}}-\frac{8\pi}{A}=\frac{4\pi}{A}\bigg{(}\frac{A_{+}}{A_{-}}+\frac{A_{-}}{A_{+}}\bigg{)},$

$\frac{L^{2}}{A^{2}_{+}}+\frac{L^{2}}{A^{2}_{-}}=\frac{L^{2}}{A_{+}A_{-}}\bigg{(}\frac{A_{+}}{A_{-}}+\frac{A_{-}}{A_{+}}\bigg{)}=\frac{hL}{A}\bigg{(}\frac{A_{+}}{A_{-}}+\frac{A_{-}}{A_{+}}\bigg{)}.$

If $h<4\pi/L$, then $\log h$ is a supersolution of a heat-type equation.

To prove Proposition 2, it remains to show that

$\frac{d}{dt}h(M,g(t))\Bigr{\rvert}_{t=t_{0}}\geq-\epsilon,$

at all times $t_{0}$ such that $h(M,g(t))<4\pi/L$. Since there exists a loop
such that

$h(\beta)=h(M,g),$

if this time $t_{0}$ exists, there must be a loop $\gamma_{0}$ whose Cheeger
constant measured with respect to the metric $g(t_{0})$ satisfies

$h(0,t_{0})=h(M,g(t)).$

$\gamma_{0}$ minimises $h$ at time $t_{0}$, which implies that

$\bigg{(}\frac{\partial}{\partial\rho}\log h\bigg{)}(0,t_{0})=0,$

$\bigg{(}\frac{\partial^{2}}{\partial\rho^{2}}\log h\bigg{)}(0,t_{0})\geq 0.$

Since $h\leq 4\pi/L$, the above result for the heat-type equation implies that

$\bigg{(}\frac{\partial}{\partial t}\log h\bigg{)}(0,t_{0})\geq 0.$

This can only happen if for every $\epsilon>0$, there exists a $\delta>0$ such
that for all times $t$ between $t_{0}-\delta$ and $t_{0}$, one has

$\log h(0,t)\leq\log h(0,t_{0})+\epsilon(t_{0}-t)=\log
h(M,g(t_{0}))+\epsilon(t_{0}-t).$

It follows that for all times $t$ between $t_{0}-\delta$ and $t_{0}$, we have

$\log h(M,g(t))+\epsilon t\leq\log h(M,g(t_{0}))+\epsilon t_{0}.$

Remark 3: It follows that $h(M,g(t))$ is non-decreasing under Ricci flow on
topological $2$-spheres by application of the maximum principle, since
$h(M,g(t))$ is a continuous function of $t$ with an upper bound $h(M,g(t))\leq
4\pi/L$.

##  3 Counterexamples to Strictly Increasing Cheeger Constant

We have shown with a heat-type equation that the Cheeger constant is non-
decreasing under Ricci flow, which does not rule out cases where it
momentarily remains the same under Ricci flow. A similar equation can also be
used to show that the isoperimetric ratio is non-decreasing under Ricci flow
[11]. One might ask if it is possible to prove the stronger result that the
constant can only increase under Ricci flow. Some simple counterexamples
obviously rule this out in the general case.

Remark 4: The flat $2$-torus has vanishing curvature, so the Cheeger constant
remains the same under Ricci flow.

Remark 5: Any steady gradient Ricci soliton (these include Witten’s black hole
and the Bryant soliton).

Figure 1: Pair of pants with spherical caps. The caps have positive curvature,
whereas the rest of the surface has negative curvature.

A much more interesting counterexample would be for the case of a compact
surface with variable negative curvature [8]. The methods of this article
cannot be used to construct a counterexample in this case, as we have dealt
only with topological spheres. We can, however, construct a counterexample
amongst topological spheres which have a mixture of both positive and negative
curvature.

Remark 6: Pair of pants with spherical caps. This surface has been studied
extensively in the context of ergodic geodesic flows, since it is an example
of a sphere with such a flow [12, 13]. Note that the spherical caps have
positive curvature, whereas the bulk of the surface has negative curvature
$-1$. Specifically, it is assumed that the caps are rotationally symmetric and
that they have exactly opposite curvature to the body of the pair of pants.
Shorten the length of each trunk so that the loop $\gamma$ used in the
definition of the Cheeger constant circles across the middle of the surface
and splits it into two surfaces $A_{+}$ and $A_{-}$, the former with mostly
positive curvature such that it shrinks under Ricci flow and the latter with
mostly negative curvature such that it expands.

If the lengths of the trunks are chosen so that $A_{+}=A_{-}$, we will have

$\frac{\partial^{2}}{\partial\rho^{2}}(\log
A_{+})=-\frac{\partial^{2}}{\partial\rho^{2}}(\log A_{-}),$

$\frac{\Gamma}{L}\frac{\partial}{\partial\rho}(\log
A_{+})=-\frac{\Gamma}{L}\frac{\partial}{\partial\rho}(\log A_{-}).$

The cuts can also be chosen so that at a certain time $t$ the length of the
loop is $L=2\sqrt{\pi}\sqrt{A_{\pm}}$, which implies that at this time

$\frac{\partial}{\partial t}(\log A_{+})=\frac{\partial}{\partial t}(\log
A_{-})=0.$

Since

$\frac{d}{dt}\log A=-\frac{8\pi}{A},$

we may at this time re-scale the total area $A$ of the pair of pants so that

$\frac{\partial^{2}}{\partial\rho^{2}}(\log
L)+\frac{\Gamma}{L}\frac{\partial}{\partial\rho}(\log L)=\frac{d}{dt}\log A.$

Although re-scaling makes the negative curvature piece less negative and the
positive curvature piece less positive, we are free to re-scale because the
change in both occurs at the same rate. Since there is a time $t$ where

$\frac{\partial}{\partial t}(\log h)=0,$

there is also a time such that

$\frac{\partial h}{\partial t}=0,$

which implies that the Cheeger constant of the surface is non-decreasing, but
not strictly increasing as a function of the Ricci flow.

##  4 Conclusion

In summary, we showed that the Cheeger isoperimetric constant is non-
decreasing under Ricci flow on topological $2$-spheres. It is still an open
question as to whether the Cheeger constant is strictly increasing for Ricci
flow on compact surfaces of negative curvature only. It was shown by Katok
that in this setting the Cheeger constant is at most
$\int_{M}e^{u_{0}}dA_{\infty}\sqrt{-2\pi\chi(M)}$, where $\chi(M)$ denotes the
Euler characteristic and $u_{t}$ denotes a function from $M$ to $\mathbb{R}$.
This is lower than $h_{\infty}=\sqrt{-2\pi\chi(M)}$ (the value for constant
curvature) and it is argued that $h_{0}$ is approached in the limit when $L$
is a closed curve of constant geodesic curvature [14].

## Acknowledgments

The author would like to thank Alexander Baumgartner for useful discussions.

## References

[1] P. Topping, Mean Curvature Flow and Geometric Inequalities, J. reine
angew. Math. 503, 47–61 (1998)

[2] P. Topping, The isoperimetric inequality on a surface, Manuscripta Math.
100, 23-33 (1999).

[3] R. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71, 1988.

[4] R. Hamilton, An isoperimetric estimate for the Ricci flow on the two-
sphere, Mod. Meth. in Comp. Ana. (Annals of Mathematics Studies, 137).
Princeton University Press, pp. 191-200 (1995).

[5] I. Agol, P. Storm and W. Thurston, Lower bounds on volumes of hyperbolic
Haken 3-manifolds, J. Amer. Math. Soc. 20, 1053-1077 (2007).

[6] J. Samuel and S.R. Chowdhury, Energy, entropy and the Ricci flow, Class.
Quantum Grav. 25, 035012 (2008).

[7] B. Andrews and P. Bryan, Curvature bounds by isoperimetric comparison for
normalized Ricci flow on the two-sphere, Calc. Var. 39, 419-428 (2010).

[8] A. Manning, The volume entropy of a surface decreases along the Ricci
flow, Ergod. Theor. Dyn. Sys 24, 171-176 (2004).

[9] P. Papasoglu, Cheeger constants of surfaces and isoperimetric
inequalities, Trans. Am. Math. Soc. 361, 5139-5162 (2009).

[10] J. Hass and F. Morgan, Geodesics and soap bubbles in surfaces, Math.
Zeit. 223, 185-196 (1996).

[11] B. Chow and D. Knopf, The Ricci flow: an introduction (American
Mathematical Society, 2004).

[12] V. Donnay, Geodesic flow on the two-sphere Part I: positive measure
entropy, Ergod. Th. Dynam. Sys. 8, 531-553 (1988).

[13] R. Osserman, Lectures at Geodesic Flow Workshop. Cal. Tech. (1985).

[14] A. Katok, Four applications of conformal equivalence to geometry and
dynamics, Ergod. Th. Dynam. Sys. 8, 139-152 (1988).


# Enhancing Object Detection Robustness: A Synthetic and Natural Perturbation
Approach

Nilantha Premakumara*, Brian Jalaian†*, Niranjan Suri*, and Hooman Samani‡
*Institute for Human & Machine Cognition (IHMC), Pensacola, Florida, USA
†University of West Florida, Pensacola, Florida, USA
‡University of Hertfordshire, Hatfield, UK
{npremakumara, bjalaian<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Robustness against real-world distribution shifts is crucial for the
successful deployment of object detection models in practical applications. In
this paper, we address the problem of assessing and enhancing the robustness
of object detection models against natural perturbations, such as varying
lighting conditions, blur, and brightness. We analyze four state-of-the-art
deep neural network models, Detr-ResNet-101, Detr-ResNet-50, YOLOv4, and
YOLOv4-tiny, using the COCO 2017 dataset and ExDark dataset. By simulating
synthetic perturbations with the AugLy package, we systematically explore the
optimal level of synthetic perturbation required to improve the models’
robustness through data augmentation techniques. Our comprehensive ablation
study meticulously evaluates the impact of synthetic perturbations on object
detection models’ performance against real-world distribution shifts,
establishing a tangible connection between synthetic augmentation and real-
world robustness. Our findings not only substantiate the effectiveness of
synthetic perturbations in improving model robustness, but also provide
valuable insights for researchers and practitioners in developing more robust
and reliable object detection models tailored for real-world applications.

_K_ eywords object detection $\cdot$ synthetic perturbation $\cdot$ natural
perturbation $\cdot$ deep neural network model $\cdot$ data augmentation
$\cdot$ robustness $\cdot$ real-world distribution shifts $\cdot$ ablation
study

##  1 Introduction

Object detection, a fundamental and critical problem in computer vision, aims
to identify and spatially localize objects within images or videos [1]. It is
integral to various computer vision tasks, including object tracking [2],
activity recognition [3], image captioning [4], image segmentation [5], and
visual question answering [6]. Object detection poses a significant challenge
due to high intra-class and low inter-class variance [7].

In recent years, deep learning has achieved unprecedented advances in various
domains, such as generative modeling [8], computer vision [9], and natural
language processing [10]. As deep learning techniques are incorporated into
safety-critical application areas like autonomous vehicles, medical
diagnostics, and robotics, ensuring the reliability and trustworthiness of
these systems is paramount.

Although object detection models have demonstrated remarkable performance,
real-world applications do not guarantee high-quality input images. Therefore,
evaluating the robustness of these models against distribution shifts is
essential before deployment. Numerous robustness interventions have been
proposed to improve neural network robustness, including non-conventional
architectures[11], adding new images to training data [12], alternative losses
[13], and different optimizers [14]. These interventions target specific
distribution shifts, such as noises [15] or synthetic corruptions [16].

Our work presents a novel approach to assessing the robustness of object
detection models by comparing their performance against synthetic and real
natural perturbations. We evaluate four state-of-the-art deep neural network
models, Detr-ResNet-101, Detr-ResNet-50, YOLOv4, and YOLOv4-tiny, using the
COCO 2017 dataset and the AugLy augmentation package, which provides synthetic
approximations of natural perturbations. Subsequently, we conduct a
comprehensive ablation study to perform transfer learning by re-training these
models on synthetic natural perturbations and evaluating their robustness
against real perturbations using the ExDark dataset.

The primary contributions of this study offer a novel perspective on enhancing
the robustness of object detection models against real-world distribution
shifts. We meticulously unravel the intricate relationship between synthetic
and real-world perturbations through the following innovative approaches:

  * •

We delve into a systematic exploration to experimentally identify the optimal
level of synthetic perturbation that effectively enhances the selected models’
robustness, shedding light on the potential benefits of synthetic data
augmentation for real-world deployment.

  * •

Our comprehensive ablation study meticulously evaluates the impact of
synthetic perturbations on object detection models’ performance against real-
world distribution shifts, establishing a tangible connection between
synthetic augmentation and real-world robustness.

Our findings not only substantiate the effectiveness of synthetic
perturbations in improving model robustness, but also provide valuable
insights for researchers and practitioners in developing more robust and
reliable object detection models tailored for real-world applications.

##  2 Related Work

The robustness of CNN-based object detection techniques in the presence of
noise is a critical area of research in computer vision, especially for
surveillance applications where image quality can be challenging. Several
studies have investigated the impact of noise on object detection algorithms,
exploring the effects of synthetic rain, fog, and other degradations on image
quality. [17] found that noisy images significantly affect classification
tasks, while [18] demonstrated that synthetic rain impacts object detection
algorithms similarly to real rain.

[19] proposed a novel approach of adding realistic and varying snow and fog to
existing image datasets to investigate their potential for scene
reconstruction. [20] evaluated various classifiers trained on ImageNet and
found a median 16% decline in classification accuracy when exposed to natural
perturbations. They also discovered that natural perturbations cause
classification and localization errors, resulting in reduced detection mAP for
Faster R-CNN and R-FCN models. [21] assessed the effectiveness of popular deep
learning techniques for object detection and recognition in thermal
surveillance scenarios using a custom dataset collected in different weather
conditions at night.

[22] suggested semantic adversarial editing as a technique to synthesize
believable corruptions and highlighted the challenging data points on which
their target model should be robust. They showed that their proposed technique
can generate a wide range of corruptions and improve model robustness against
natural corruptions. The performance of deep neural network-based methods is
directly influenced by the availability and quality of datasets. Several
challenging large-scale datasets have been introduced, consisting of images or
videos captured under adverse conditions. These datasets include ExDARK,
UNIRI-TID, RESIDE, UFDD, and See in the Dark [23]. Each dataset focuses on
different aspects of challenging conditions, such as low light, various
weather conditions, and occlusions. Table 1 summarizes the main advantages and
limitations of the mentioned approaches.

Table 1: A summary of advantages and limitations of methods tackling object detection in challenging conditions. Literature |  Methods |  Advantages |  Limitations
---|---|---|---
[24] |  Image are transformed and then fed into the RFB-Net. |  Context information fusion allows detection of object in low-light. |  Relies on prior information about type of object, shape etc for detecting them in night-time.
[25] |  Fusion of pre-trained models using Glue layer and information distillation. |  Domain joining with help of glue layer reduces in computation and provides more information for models to learn from different domains. |  Relies on prior domain knowledge.
[26] |  Generative adversarial network with Faster R-CNN. |  Networks learns both day and night-time features. |  Relies on prior information of converting night-time images to day time.
[27] |  Fully convolutional siamese networks with modified binary segmentation task. |  Pre-frame binary segementation mask is used for low-level object representation instead of relying on feature extractor backbone. |  Relies on prior information while generating binary segmentation mask. Fails when faced with motion blur and non-object pattern.
[28] |  Two stream convolutional neural network with attention mechanism. |  Fusion of rgb and thermal image to generate features and noise reduction with convolutional block attention modele. |  Thermal images are not efficient for every envirment use.

Recent studies have demonstrated that data augmentation can enhance model
resilience [29],[16]. Various augmentations can aid resilience, including
different types of noise [30], deep artificial image transformations [12],
natural transformations [31], and combinations of simple image transformations
such as Python Imaging Library operations [32],[16]. In this research, we used
AugLy for image perturbation. AugLy is a state-of-the-art open-source Python
library designed to help AI researchers apply data augmentations to assess and
improve the robustness of their machine learning models. It integrates
multiple modalities, including image, video, and text, which is essential in
many AI research domains [33]. AugLy provides over 100 data augmentations that
mimic what real users do to photos and videos on platforms like Facebook and
Instagram. For our experiment, we used three image augmentations: blur, pixel
degradation, and brightness.

In summary, the related work underscores the importance of developing robust
deep learning models capable of handling natural data variance and
corruptions. Researchers have explored various methods, such as generative
models, data augmentation, and synthetic corruption benchmarks, to achieve
this goal. The availability of challenging datasets and the application of
data augmentation techniques, such as those provided by AugLy, play a crucial
role in assessing and enhancing model robustness under adverse conditions. Our
research builds upon these findings and aims to further investigate the impact
of synthetic natural perturbations on model performance and the potential
benefits of transfer learning and retraining on synthetic perturbations to
improve robustness against real perturbations.

##  3 Methodology

###  3.1 Models, Dataset, and Perturbations

We evaluated model robustness using four pretrained state-of-art neural
network models, a test dataset, and synthetic perturbations. The main
components of our methodology are as follows:

  * •

Models: We used four pretrained neural network models: Detr-ResNet-101, Detr-
ResNet-50, Yolov4, and Yolov4-tiny.

  * •

Dataset: The COCO 2017 dataset was employed for training purposes and for
creating synthetic perturbations. The evaluation of the models’ robustness and
performance was performed using the ExDARK dataset [34], containing images
with real-world perturbations, as part of our ablation study.

  * •

Synthetic Perturbations: We introduced three types of robustness
specifications (blur, brightness, and pixel degradation) using the AugLy
package. Perturbation parameters were adjusted within specific ranges to
simulate natural perturbations. To simulate model robustness with natural
distribution shifts on the image data, we adopted the perturbation strategies
on the dataset. We used perturbations with different levels for each category.
Figure 1 depicts how each model’s confidence score corresponds to the
perturbation level. Example results show that all the model’s detection
confidence scores decrease when increasing perturbation level compared to the
original one.

Figure 1: Illustration of how each model is robust against perturbation level
(a) blur, (b) brightness (light mood), (c) brightness (dark mood), (d)
degrading pixels.

###  3.2 Ablation Study Design

Ablation studies are experimental approaches used to assess the importance of
individual components in complex systems or models, such as deep learning
models, by removing or altering these components [35]. This methodology has
been widely adopted in computer vision, particularly for object detection
tasks [36], and in other domains such as natural language processing and
neuroscience [37].

In this paper, we conduct an ablation study using all four models to evaluate
the effectiveness of synthetic perturbations, specifically focusing on the
brightness modality, in enhancing the robustness of object detection models
against real-world distribution shifts. We test with varied numbers of
synthetic perturbation images to determine how the size of the synthetic
perturbation training set influences detection performance on natural
perturbations. Our ablation study serves two primary purposes:

  1. 1.

Quantify the impact of synthetic perturbations: We apply varying levels of
synthetic brightness perturbations to identify the optimal level that improves
the models’ robustness. This process helps us understand the relationship
between the perturbation magnitude and the performance of the object detection
models.

  2. 2.

Analyze the transferability of improvements: We investigate whether the
models’ robustness improvements due to synthetic brightness perturbations
translate to enhanced performance against real-world distribution shifts,
thereby determining if synthetic perturbations serve as reliable proxies for
natural perturbations in real-world applications.

Our findings from the ablation study provide valuable insights into developing
more robust object detection models and inform future research on the role of
synthetic perturbations in enhancing model resilience.

##  4 Results

###  4.1 Model Performance with Synthetic Perturbations

We first evaluated the models’ performance against synthetic natural
perturbations created using the AugLy augmentation package. Table 2 shows the
mean average precision (mAP) scores of the four object detection models when
tested with these synthetic perturbations. Most models were susceptible to
strong brightness, while all models were more robust to darkness than other
perturbations.

Table 2: The mean average precision (mAP) scores of various models computed
with the original dataset and synthetic corruptions

Model |  mAP Original | Blur(radius=5) |  Brightness (factor=3) |  Brightness (factor=0.2) |  Degrading pixels (radius=5)
---|---|---|---|---|---
mAP | mAP$\Delta$ | mAP | mAP$\Delta$ | mAP | mAP$\Delta$ | mAP | mAP$\Delta$
Detr-ResNet-101 | 73.56 | $67.26$ | $6.3$ | $64.39$ | $9.17$ | $69.02$ | $4.54$ | $64.53$ | $9.03$
Detr-ResNet-50 | 62.37 | $51.26$ | $11.11$ | $47.89$ | $14.48$ | $59.21$ | $3.16$ | $48.52$ | $13.85$
Yolov4 | 71.49 | $63.37$ | $8.12$ | $62.44$ | $8.12$ | $68.21$ | $3.28$ | $61.85$ | $9.64$
Yolov4-tiny | 64.52 | $53.47$ | $11.05$ | $51.04$ | $11.05$ | $60.83$ | $3.69$ | $50.25$ | $14.27$

###  4.2 Comparison of Model Robustness

To provide a comprehensive comparison of the robustness of the four object
detection models (Detr-ResNet-101, Detr-ResNet-50, YOLOv4, and YOLOv4-tiny),
we analyzed their performance under different perturbation conditions. As
shown in Figure 2, Detr-ResNet-101 and Detr-ResNet-50 exhibit similar
performance patterns under various perturbations, with Detr-ResNet-101
performing slightly better than Detr-ResNet-50. YOLOv4 outperforms YOLOv4-tiny
in all scenarios.

Overall, the Detr-ResNet-101 model demonstrated the best performance in
handling synthetic perturbations and real-world perturbations when trained
with an augmented synthetically perturbed dataset. However, it should be noted
that the choice of the best model for specific scenarios may depend on various
factors, including the computational power available, the acceptable trade-off
between accuracy and speed, and the specific perturbation types encountered in
the target domain.

Figure 2: Illustration of performance of multiple object detection models
against AugLy blur and brightness perturbation (a) Detr-ResNet-101 model and
(b) Detr-ResNet-50 model

###  4.3 Impact of Perturbation Levels

To investigate the models’ performance against varying levels of synthetic
perturbation, we conducted experiments by changing the radius and factor
values of each synthetic perturbation. This experiment aimed to identify the
optimal level of synthetic perturbation that should be used to improve the
model’s robustness.

In Figure 3, we observed that the models’ performance worsened when the blur
radius value was greater than $5$, the brightness (light mood) factor value
was greater than $2$, the brightness (dark mood) factor value was less than
$0.2$, and the degrading pixels factor value was greater than $0.2$. Among the
models, Detr-ResNet-101 demonstrated higher robustness compared to the other
three models.

Figure 3: Illustration of model’s performance against different synthetic
perturbation levels.(a),(b),(c),(d) are represent different synthetic blur
level, brightness (light mood) level, synthetic brightness (dark mood) level
and synthetic degrading pixels level respectively

###  4.4 Ablation Study: Synthetic to Real Perturbation Robustness

In the ablation study, we examined the performance of the four object
detection models (Detr-ResNet-101, Detr-ResNet-50, YOLOv4, and YOLOv4-tiny)
against real perturbations from the ExDark dataset, which contains challenging
lighting conditions from an out-of-domain setting. The models were re-trained
with synthetic perturbations before testing. We conducted further tests with
varied numbers of synthetic perturbation images to determine how the size of
the synthetic perturbation training set influences detection performance on
real perturbations. To test the outcomes, we added synthetic poor brightness
perturbations to a subset of the COCO-2017 training dataset at random
percentages of 0%, 20%, 50%, and 70% using the AugLy package.

Figure 4 shows the values of the mAP and loss functions for the models with
subset ratios of 100:00, 80:20, 50:50, and 30:70, respectively. Our results
demonstrate that the mAP has an increasing trend as we increase the percentage
of the augmented synthetically perturbed training dataset and re-train the
models. The common observation across all cases is that while there is a
significant gap between the mAP for the original models, the gap narrows as
the percentage of the augmented synthetically perturbed training dataset
increases. This suggests that adding synthetic perturbations to the training
data improves the models’ robustness to real-world perturbations. However, in
this study, we focused on the results with brightness adjustments only, as
presented in Table 3.

Figure 4: mAP and loss function graphs for different models and synthetic
ratio combinations. (a-d) Detr-ResNet-101, (e-h) Detr-ResNet-50, (i-l) YOLOv4,
and (m-p) YOLOv4-tiny. Columns represent original to synthetic ratios of
100:0, 80:20, 50:50, and 30:70, respectively. Table 3: The mAP scores of
various models computed with the original dataset and synthetic corruptions

Model |  Original:Synthetic ratio on traing dataset | Model’s performance (mAP)
---|---|---
COCO-testing | ExDARK
Detr-ResNet-101 | 100:00 | $73.56$ | $51.85$
| 80:20 | $73.24$ | $61.43$
| 50:50 | $75.23$ | $70.56$
| 30:70 | $77.92$ | $76.47$
Detr-ResNet-50 | 100:00 | $62.37$ | $48.28$
| 80:20 | $64.76$ | $58.59$
| 50:50 | $63.47$ | $62.58$
| 30:70 | $67.58$ | $66.89$
Yolov4 | 100:00 | $71.49$ | $49.48$
| 80:20 | $68.78$ | $60.45$
| 50:50 | $70.37$ | $61.57$
| 30:70 | $72.45$ | $62.45$
Yolov4-tiny | 100:00 | $64.52$ | $32.45$
| 80:20 | $65.43$ | $38.58$
| 50:50 | $67.84$ | $42.59$
| 30:70 | $69.51$ | $56.73$

##  5 Limitations and Future Work

While our study provides valuable insights into the performance of object
detection models under various synthetic perturbations and their robustness
against real-world perturbations, there are several limitations that should be
acknowledged. First, our analysis was limited to the ExDark dataset, which
primarily focuses on poor lighting conditions. To generalize our findings, it
is crucial to create and test the models on datasets containing different
modalities of natural perturbations, such as occlusions, weather variations,
and sensor noise, among others.

Second, our experiments only considered four pre-trained object detection
models. To provide a more comprehensive understanding of model robustness,
future work could include a wider range of models and architectures, as well
as customized models specifically designed to handle perturbations.

Another limitation is that our study focused mainly on brightness adjustments
in the ablation study. Expanding the scope of the ablation study to include
other types of synthetic perturbations could provide a more complete
understanding of model performance and generalization.

Lastly, the choice of augmentation techniques and their parameter settings may
also impact the results. Future work could explore alternative data
augmentation methods and fine-tune the parameters to find the optimal balance
between model robustness and performance.

In summary, future work should aim to:

  1. 1.

Develop and test object detection models on diverse datasets containing
various natural perturbation modalities.

  2. 2.

Investigate the performance and robustness of a broader range of object
detection models and architectures.

  3. 3.

Expand the ablation study to encompass different types of synthetic
perturbations.

  4. 4.

Explore alternative data augmentation techniques and optimize their parameters
to enhance model robustness and performance.

##  6 Conclusion

In this study, we investigated the robustness of four state-of-the-art object
detection models, namely Detr-ResNet-101, Detr-ResNet-50, YOLOv4, and
YOLOv4-tiny, to natural perturbations by simulating synthetic perturbations
using the AugLy package. We conducted a series of experiments to evaluate the
performance of the models against synthetic and real-world perturbations,
assess the impact of the size of the synthetic perturbation training set on
detection performance under real perturbations, and determine the optimal
level of synthetic perturbation for improving model robustness. Ablation
studies were also performed to further analyze the models’ performance.

Our findings indicate that the performance of object detection models can be
significantly improved by augmenting the training dataset with synthetically
perturbed images, particularly in scenarios involving challenging lighting
conditions, such as those found in the ExDark dataset. Among the models
tested, Detr-ResNet-101 exhibited the best robustness to perturbations.
However, the choice of the most suitable model may depend on various factors,
including computational power, the trade-off between accuracy and speed, and
the specific perturbation types encountered in the target domain.

These results provide valuable insights for researchers and practitioners
seeking to enhance the robustness of object detection models to natural
perturbations and offer a foundation for further research into methods for
improving model performance under a wide range of challenging conditions. By
addressing the limitations outlined in the previous section and expanding the
scope of investigation, future work can contribute to the development of more
robust object detection models capable of handling diverse and complex real-
world scenarios.

## References

  * [1] P F Felzenszwalb, R B Girshick, D McAllester, and D Ramanan.  Object detection with discriminatively trained part-based models.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 32:1627–1645, 9 2010.
  * [2] Kai Kang, Hongsheng Li, Junjie Yan, Xingyu Zeng, Bin Yang, Tong Xiao, Cong Zhang, Zhe Wang, Ruohui Wang, Xiaogang Wang, and Wanli Ouyang.  T-cnn: Tubelets with convolutional neural networks for object detection from videos.  IEEE Transactions on Circuits and Systems for Video Technology, 28:2896–2907, 10 2018.
  * [3] N. Vaswani, A. Roy Chowdhury, and R. Chellappa.  Activity recognition using the dynamics of the configuration of interacting objects.  In 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings., volume 2, pages II–633, 2003.
  * [4] Qi Wu, Chunhua Shen, Peng Wang, Anthony Dick, and Anton van den Hengel.  Image captioning and visual question answering based on attributes and external knowledge.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 40:1367–1381, 6 2018.
  * [5] Bharath Hariharan, Pablo Arbeláez, Ross Girshick, and Jitendra Malik.  Hypercolumns for object segmentation and fine-grained localization.  11 2014.
  * [6] Chris Alberti, Jeffrey Ling, Michael Collins, and David Reitter.  Fusion of detected objects in text for visual question answering.  pages 2131–2140. Association for Computational Linguistics, 2019.
  * [7] Mark Everingham, Luc Van Gool, Christopher K. I. Williams, John Winn, and Andrew Zisserman.  The pascal visual object classes (voc) challenge.  International Journal of Computer Vision, 88:303–338, 6 2010.
  * [8] Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros.  Unpaired image-to-image translation using cycle-consistent adversarial networks.  pages 2223–2232, 2017.
  * [9] Sara Sabour and Nicholas Frosst.  Dynamic routing between capsules.  pages 3856–3866, 2017.
  * [10] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.  Bert: Pre-training of deep bidirectional transformers for language understanding.  2018\.
  * [11] Cristina Vasconcelos, Hugo Larochelle, Vincent Dumoulin, Nicolas Le Roux, and Ross Goroshin.  An effective anti-aliasing approach for residual networks.  11 2020.
  * [12] Hongyi Zhang, Moustapha Cisse, Yann N. Dauphin, and David Lopez-Paz.  mixup: Beyond empirical risk minimization.  10 2017.
  * [13] Simon Kornblith, Ting Chen, Honglak Lee, and Mohammad Norouzi.  Why do better loss functions lead to less transferable features?  10 2020.
  * [14] Luke Metz, Niru Maheswaranathan, Jonathon Shlens, Jascha Sohl-Dickstein, and Ekin D. Cubuk.  Using learned optimizers to make models robust to input noise.  6 2019.
  * [15] Yiren Zhou, Sibo Song, and Ngai-Man Cheung.  On classification of distorted images with deep convolutional neural networks.  1 2017.
  * [16] Dan Hendrycks, Norman Mu, Ekin D. Cubuk, Barret Zoph, Justin Gilmer, and Balaji Lakshminarayanan.  Augmix: A simple data processing method to improve robustness and uncertainty.  12 2019.
  * [17] Samuel Dodge and Lina Karam.  Understanding how image quality affects deep neural networks.  IEEE International Conference on Quality of Multimedia Experience, QoMEX 2016, 6 2016.
  * [18] Georg Volk, Stefan Muller, Alexander Von Bernuth, Dennis Hospach, and Oliver Bringmann.  Towards robust cnn-based object detection through augmentation with synthetic rain variations.  2019\.
  * [19] Alexander von Bernuth, Georg Volk, and Oliver Bringmann.  Simulating photo-realistic snow and fog on existing images for enhanced cnn training and evaluation.  2019\.
  * [20] Vaishaal Shankar, Achal Dave, Rebecca Roelofs, Deva Ramanan, Benjamin Recht, and Ludwig Schmidt.  A systematic framework for natural perturbations from videos.  7 2019.
  * [21] Mate Kristo, Marina Ivasic-Kos, and Miran Pobar.  Thermal object detection in difficult weather conditions using yolo.  IEEE Access, 8:125459–125476, 2020.
  * [22] Rakshith Shetty, Mario Fritz, and Bernt Schiele.  Towards automated testing and robustification by semantic adversarial data generation.  European Conference on Computer Vision (ECCV), 2020.
  * [23] Muhammad Ahmed, Khurram Azeem Hashmi, Alain Pagani, Marcus Liwicki, Didier Stricker, and Muhammad Zeshan Afzal.  Survey and performance analysis of deep learning based object detection in challenging environments, 8 2021.
  * [24] Yuxuan Xiao, Aiwen Jiang, Jihua Ye, and Ming-Wen Wang.  Making of night vision: Object detection under low-illumination.  IEEE Access, 8:123075–123086, 2020.
  * [25] Yukihiro Sasagawa and Hajime Nagahara.  Yolo in the dark - domain adaptation method for merging multiple models -.  08 2020.
  * [26] Kun Wang and Mao Zhen Liu.  Object recognition at night scene based on dcgan and faster r-cnn.  IEEE Access, 8:193168–193182, 2020.
  * [27] Qiang Wang, Li Zhang, Luca Bertinetto, Weiming Hu, and Philip H. S. Torr.  Fast online object tracking and segmentation: A unifying approach, 2019\.
  * [28] Zhengzheng Tu, Yan Ma, Zhun Li, Chenglong Li, Jieming Xu, and Yongtao Liu.  Rgbt salient object detection: A large-scale dataset and benchmark, 2022\.
  * [29] Geirhos Robert, Rubisch Patricia, Michaelis Claudio, Bethge Matthias, Wichmann Felix A., and Brendel Wieland.  Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness.  2019\.
  * [30] Raphael Gontijo Lopes, Dong Yin, Ben Poole, Justin Gilmer, and Ekin D. Cubuk.  Improving robustness without sacrificing accuracy with patch gaussian augmentation.  arXiv:1906.02611, 6 2019.
  * [31] Alfred Laugros, Alfred Laugros@atos Net, Alice Caplier, and Matthieu Ospici.  Using synthetic corruptions to measure robustness to natural distribution shifts.  2021\.
  * [32] Ekin D. Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc V. Le.  Autoaugment: Learning augmentation strategies from data.  https://arxiv.org/abs/1805.09501v3, 5 2018.
  * [33] Papakipos Zoe and Bitton Joanna.  Augly: Data augmentations for robustness.  1 2022.
  * [34] Yuen Peng Loh and Chee Seng Chan.  Getting to know low-light images with the exclusively dark dataset.  Computer Vision and Image Understanding, 178:30–42, 2019.
  * [35] Sina Sheikholeslami, Moritz Meister, Tianze Wang, Amir H Payberah, Vladimir Vlassov, and Jim Dowling.  Autoablation: Automated parallel ablation studies for deep learning.  In Proceedings of the 1st Workshop on Machine Learning and Systems, page 1–7, 2021.
  * [36] Ross Girshick, Jeff Donahue, Trevor Darrell, and Jitendra Malik.  Rich feature hierarchies for accurate object detection and semantic segmentation.  In Proceedings of the IEEE conference on computer vision and pattern recognition, page 580–587, 2014.
  * [37] Richard Meyes, Melanie Lu, Constantin Waubert de Puiseau, and Tobias Meisen.  Ablation studies in artificial neural networks.  arXiv preprint arXiv:1901.08644, 2019.


]

# On the distribution of dark matter in galaxies: quantum treatments

Carlos R. Argüelles<EMAIL_ADDRESS>Remo Ruffini  Dipartimento
di Fisica and ICRA, Sapienza Università di Roma, P.le Aldo Moro 5, I–00185
Rome, Italy  ICRANet, P.zza della Repubblica 10, I–65122 Pescara, Italy  Ivan
Siutsou  Dipartimento di Fisica and ICRA, Sapienza Università di Roma, P.le
Aldo Moro 5, I–00185 Rome, Italy  ICRANet, P.zza della Repubblica 10, I–65122
Pescara, Italy  Bernardo Fraga  Dipartimento di Fisica and ICRA, Sapienza
Università di Roma, P.le Aldo Moro 5, I–00185 Rome, Italy  Université de Nice
Sophia di Antipolis, Nice, France

([)

###### Abstract

The problem of modeling the distribution of dark matter in galaxies in terms
of equilibrium configurations of collisionless self-gravitating quantum
particles is considered. We first summarize the pioneering model of a
Newtonian self-gravitating Fermi gas in thermodynamic equilibrium developed by
Ruffini and Stella (1983), which is shown to be the generalization of the King
model for fermions. We further review the extension of the former model
developed by Gao, Merafina and Ruffini (1990), done for any degree of fermion
degeneracy at the center ($\theta_{0}$), within general relativity. Finally,
we present here for the first time the solutions of the density profiles and
rotation curves corresponding to the Gao et. al. model, which have a definite
mass $M_{h}$ and circular velocity $v_{h}$, at the halo radius $r_{h}$ of the
configurations, typical of spiral galaxies. This treatment allow us to
determine a novel core-halo morphology for the dark matter profiles, as well
as a novel particle mass bound associated with them.

Dark matter - Self-gravitating systems: fermions - Galaxies: density profiles
and rotation curves

###### pacs:

68.37.Ef, 82.20.-w, 68.43.-h

##  I INTRODUCTION

One of the most quoted papers in the study of dark matter distribution in
galactic halos is certainly the work of Tremaine & Gunn PRL79Tremaine , there
the authors established a lower limit on the mass of neutrinos composing
galactic halos, by considering an isothermal classical distribution of self-
gravitating particles, and imposing quantum constraints on the phase space
density in the core of the galaxies. This treatment presenting a peculiar
mixture of quantum and classical considerations has always attracted the
attention and the suspicion of many astrophysicists and theoretical
physicists. It was in particular the opinion of one of us (R. Ruffini), that a
self-consistent treatment of quantum constraints in a self-consistent quantum
description of the microphysical system was needed. A long effort so started.

##  II HISTORICAL REVIEW: TWO PIONEERING WORKS

One of the preliminary works in order to prove the analogy and differences
between a classical and quantum self-gravitating system was advanced within a
Newtonian approach in Ruffini & Stella (1983) AA83Ruffini . There, the problem
of a semi-degenerate system of fermions under gravitational interaction was
approached, and compared and contrasted with the classical King model.

They proposed a distribution function built for non-relativistic particles,
which reads:

$\displaystyle f(v)$ $\displaystyle=$
$\displaystyle\frac{1-\exp{[-j^{2}(v_{e}^{2}-v^{2})]}}{\exp{[j^{2}(v^{2}-\bar{\mu})]}+1},\,v\leq
v_{e}$ $\displaystyle=$ $\displaystyle 0,\qquad\qquad\qquad\qquad\quad
v>v_{e},$

where $v_{e}$ is the escape velocity. In the limit $v_{e}\rightarrow\infty$,
the usual Fermi distribution is obtained. The other two constant parameters
are $j^{2}=m/(2kT)$ and $\bar{\mu}=2\mu/m$. The relevance of this $f(v)$ is
that it is an extension of the King model to the case of a Fermi gas.
Moreover, if the degeneracy parameter $\theta=j^{2}\bar{\mu}$ is defined, it
can be easily seen that when $\theta\rightarrow-\infty$, the non-degenerate
limit is reached and the distribution function used by King is recovered.
Instead, when $j^{2}\rightarrow\infty$ and $\bar{\mu}\rightarrow v_{e}^{2}$
the degenerate limit is obtained, and the escape velocity is associated with
the Fermi energy.

The energy integral,

$E=v^{2}/2+V(r)\,,$ (1)

together with the Jeans theorem for spherical systems, allowed them to simply
relate the escape velocity with the gravitational potential by $v_{e}^{2}=-2V$
(being $V=0$ at the surface of the configuration). They finally solved the
Poisson equation for $W=-2j^{2}V$, being the mass density given by
$\rho\propto\int f(v)v^{2}dv$ which is related with $W$ via the $j$ parameter.

For simplicity, in the attempt to understand the physical interpretation of
the parameters, an only value for the central degeneracy parameter was assumed
for the sake of example,

$\theta(0)\equiv\theta_{0}=0\,,$ (2)

and no other values for $\theta_{0}$ were explored at the time. The different
normalized mass density solutions were given for different values of
$W(0)\equiv W_{0}$ as shown in Fig. 1.

Figure 1: Normalized density profiles for different values of $W_{0}$ and
fixed $\theta_{0}=0$. The dotted curve corresponds to the analogous King
profiles while the dashed curve represents the degenerate limit (taken from
AA83Ruffini  with permission).

Under these special conditions the analogy between the self-gravitating system
of fermions and the King model was proven, as well as a first attempt to
justify the Tremaine & Gunn limit.

It soon became clear that these solutions, although interesting in reproducing
classical results of the King profiles for a self-gravitating fermion gas,
were really extremely restrictive, and not representative of the general
solutions for a relativistic self-gravitating system of massive fermions.
These restrictions correspond to three different constraints: 1)
$\theta_{0}=0$; 2) the application of a cut-off in the phase space which
implies the elimination of an important family of solutions; and 3) the use of
a non relativistic Newtonian approach.

A fundamental step was made by Gao, Merafina and Ruffini (1990) AA90Gao , to
include special relativity effects in the phase space of the distribution
function, as well as general relativity. Thus, they considered the
relativistic Fermi-Dirac distribution function for the ‘inos’ footnote
without any cut-off in their momentum space, i.e.
$f(\epsilon)=(\exp[(\epsilon-\mu)/(kT)]+1)^{-1}$, where
$\epsilon(p)=\sqrt{c^{2}p^{2}+m^{2}c^{4}}-mc^{2}$ is the particle kinetic
energy and $\mu$ the chemical potential with the particle rest-energy
subtracted off.

They wrote the system of Einstein equations in the spherically symmetric
metric $g_{\mu\nu}={\rm
diag}(e^{\nu},-e^{\lambda},-r^{2},-r^{2}\sin^{2}\theta)$, where $\nu$ and
$\lambda$ depend only on the radial coordinate $r$, together with the
thermodynamic equilibrium conditions of Tolman PR30Tolman , and Klein
RMP49Klein ,

$e^{\nu/2}T=const.\,,\quad e^{\nu/2}(\mu+mc^{2})=const.$

in the following dimensionless way,

$\displaystyle\frac{d\hat{M}}{d\hat{r}}=4\pi\hat{r}^{2}\hat{\rho}$ (3)
$\displaystyle\frac{d\theta}{d\hat{r}}=\frac{\beta_{0}(\theta-\theta_{0})-1}{\beta_{0}}\frac{\hat{M}+4\pi\hat{P}\hat{r}^{3}}{\hat{r}^{2}(1-2\hat{M}/\hat{r})}$
(4)
$\displaystyle\frac{d\nu}{d\hat{r}}=\frac{\hat{M}+4\pi\hat{P}\hat{r}^{3}}{\hat{r}^{2}(1-2\hat{M}/\hat{r})}$
(5) $\displaystyle\beta_{0}=\beta(r)e^{\frac{\nu(r)-\nu_{0}}{2}}\,.$ (6)

There, the following dimensionless quantities were introduced:
$\hat{r}=r/\chi$, $\hat{M}=GM/(c^{2}\chi)$, $\hat{\rho}=G\chi^{2}\rho/c^{2}$
and $\hat{P}=G\chi^{2}P/c^{4}$, where $\chi=2\pi^{3/2}(\hbar/mc)(m_{p}/m)$ is
the dimensional factor which has unit of length and scales as $m^{-2}$; with
$m_{p}=\sqrt{\hbar c/G}$ the Planck mass, and the temperature and degeneracy
parameters, $\beta=kT/(mc^{2})$ and $\theta=\mu/(kT)$, respectively. The mass
density $\rho$ and pressure $P$ are expressed in terms of the standard
infinite integrals in momentum space weighted with the $f(\epsilon)$ already
given, for a relativistic and semi-degenerate Fermi gas (see AA90Gao ).

In that work, they solved the initial condition problem for the variables of
the system, $\theta(r)$, $\beta(r)$, $\nu(r)$, and $M(r)$, by giving at $r=0$
(and indicated by a subscript ‘0’) $M_{0}=0$, while giving arbitrary values
for the temperature and degeneracy parameters $\beta_{0}$ and $\theta_{0}$,
respectively. In Figs. 2–3, different normalized mass density solutions for
different $\theta_{0}<0$ and $\theta_{0}\geq 0$ respectively are shown, for a
fixed non-relativistic central temperature parameter $\beta_{0}$.

Figure 2: Different density profiles for different $\theta_{0}<0$ and fixed
$\beta_{0}$ in dimensionless variables. To note the simple cored plus $r^{-2}$
morphology (taken from AA90Gao  with permission). Figure 3: Different density
profiles for different $\theta_{0}\geq 0$ and fixed $\beta_{0}$ in
dimensionless variables. To note the more complex core plus ‘plateau’ plus
$r^{-2}$ morphology (taken from AA90Gao  with permission).

It is important to note that the system (3–6) has no particle mass $m$
dependence when solved in the dimensionless variables, while instead the
physical magnitudes such as $r$ and $\rho$ have an explicit dependence on $m$
trough the dimensional factor $\chi(m)$. The fact that they were mainly
interested in the general properties of the solutions without going through
the physical magnitudes, no particle mass constraints were put there.

##  III NOVEL PROCEDURE AND DISCUSSION

We have recently returned to the Gao et. al. work, and propose a completely
different way for solving the boundary condition problem for the system (3–6),
in order to fulfill the observationally inferred values of typical dark matter
halos in spiral galaxies as given in AJ08deBlok . Namely, for given initial
conditions $M_{0}=\nu_{0}=0$, arbitrary $\theta_{0}$ (depending on the chosen
central degeneracy), and defining the halo radius $r_{h}$ at the onset of the
flat rotation curve, we solve an eigenvalue problem for the central
temperature parameter $\beta_{0}$, until the observed halo circular velocity
$v_{h}$ is obtained. After this, we solve a second eigenvalue problem for the
particle mass $m$ until the observed halo mass $M_{h}$ is reached at the
radius $r_{h}$.

The quest has been to use all these information in order to put constraints on
the mass of the ‘ino’ in galactic halos by introducing the observational
properties possibly to be utilized in this research.

Interestingly enough, as detailed in PRL13Ruffini , it turns out that only for
an specific range of $\theta_{0}>0$ these two eigenvalue problems can be
solved together, implying as a consequence, a novel reach morphology for the
density profiles as well as a novel particle mass bound associated with it.
The density profiles presents a quantum degenerate core, followed by a low
degenerate plateau until it reach the $r^{-2}$ Boltzmannian regime
corresponding to the flat part in the rotation curve. In Fig. 4 we show a
family of density profiles for different values of $\theta_{0}$ which fulfills
the mentioned halo constraints. We also plotted for comparison the purely
Boltzmannian curve which agrees with the same observed halo magnitudes.

Figure 4: Physical density profiles for specific ino masses $m$ and central
degeneracies $\theta_{0}$ fulfilling the observational constraints
$M_{h}=1.6\times 10^{11}M_{\odot}$ and $v_{h}=168$ km/s at $r_{h}=25$ Kpc (as
taken from AJ08deBlok  and detailed in PRL13Ruffini ). In dot-dashed line the
purely Boltzmannian profile for comparison.

As can be seen from Fig. 4, we obtain from this novel analysis a more
stringent lower mass bound for the ‘ino’ mass $m$, which is $\sim 10$ times
higher than the ones inferred in AA83Ruffini  and PRL79Tremaine . This is,
$m\geq 0.42$ keV$/c^{2}$ for typical spiral galaxies.

It is interesting that the quantum and relativistic treatment of the
configurations considered here are characterized by the presence of central
cored structures unlike the typical cuspy configurations obtained from a
classic non-relativistic approximation, such as the ones of numerical N-body
simulations in ApJ97Navarro . This naturally leads to a possible solution to
the well-known core-cusp discrepancy AJ01deBlok .

###### Acknowledgements.

We appreciate to Asia Pacific Center for Theoretical Physics (APCTP) for its
hospitality during completion of this work.

## References

  * (1) S. Tremaine, and J. E. Gunn, Phys. Rev. Lett. 42, 6 (1979).
  * (2) R. Ruffini, and L. Stella, Astron. Astrophys. 119, 35 (1983).
  * (3) J. G. Gao, M. Merafina, and R. Ruffini, Astron. Astrophys. 235, 1 (1990).
  * (4) R. C. Tolman, Phys. Rev. 35, 904 (1930).
  * (5) O. Klein, Rev. Mod. Phys. 21, 531 (1949).
  * (6) W. J. G. de Blok, et al., Astron. J. 136, 2648 (2008).
  * (7) R. Ruffini, C. R. Argüelles, J. A. Rueda, and I. Siutsou, Phys. Rev. Lett. (to be submitted).
  * (8) J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 490, 493 (1997).
  * (9) E. de Blok, S. McGaugh and R. Vera, astro.ph. 7366D (2001).
  * (10) A. Boyarsky, O. Ruchayskiy and M. Shaposhnikov, Annuarev. Nucl. 59, 191 (2009).
  * (11) ‘Ino’ refers to any beyond standard model neutrino which is suitable within the astrophysical application of the model here presented. For example: the sterile neutrino (see e.g. ARNPS09Boyarsky ).