Gravitational lensing of pulsars as a probe of dark matter halos Francesca von Braun-Bates

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Gravitational lensing of pulsars as a

probe of dark matter halos

Francesca von Braun-Bates

Masters by Research submitted in fulﬁllment of

the requirements for the degree of

Master of Science (Research)

School of Physics

University of Sydney

April 2014

arXiv:2210.06151v1 [astro-ph.CO] 11 Oct 2022

Abstract

A key question in cosmology is the properties of dark matter. A particular open problem

is whether dark matter on small scales is clumpy, forming gravitationally-bound halos

distributed within the Galaxy. The practical diﬃculties inherent in testing this hypoth-

esis stem from the fact that, on astrophysical scales, dark matter is solely observable via

its gravitational interaction with other objects.

This thesis presents a gravitational-lensing-based solution for the mapping and char-

acterisation of low-mass dark matter halos via their signature in millisecond pulsar

observations. This involves numerical calculations in three stages: ﬁrst, determining the

time delay and magniﬁcation surfaces generated in the frame of reference of the halo;

second, obtaining the corresponding pulsar signature in the reference frame of the ob-

server; and last, generalising the method to multiple halos at varying distances. In both

the single-lens and multiple-lens cases, we discuss whether the delay is observationally

detectable.

Dark matter halos act as gravitational lenses which produce a variable ﬂux and induce

additional time delays in (tangent) bundles of photons passing near or through the

halo. The key dependency of the mass estimate is the density proﬁle adopted for the

halo. I utilise a variety of proposed halo mass proﬁles — namely the elliptical model

of Kochanek &al. [28], the axially symmetric Schwarzschild and homogenous disc lenses

(both [48]) and the Navarro–Frenk–White density proﬁle [39,40] — which are applicable

over a broad range of halo masses. The pulsar simulations use the most realistic and

sophisticated of these, the empirically-derived proﬁle of Navarro, Frenk & White. I

justify the adoption of a radially-symmetric density proﬁle by showing that this greatly

simpliﬁes the calculation of the lens convergence. Moreover, I demonstrate that the use

of Hankel transforms is a novel way to increase the eﬃciency of the relativistic time

delay.

The observational signatures of such halos are best identiﬁed using millisecond pulsars.

This remarkable subset of the pulsar population has both the highest rotational frequen-

cies and the most period stability of all known pulsars. Furthermore, the potential for

gravitational wave detection using millisecond pulsars will result in an abundance of new

data from pulsar surveys. I propose that observational techniques do not require major

adjustments when searching for signs of gravitational lensing, thus it is unnecessary to

implement specialist data reduction pipelines, which enable the data from existing and

future surveys to be examined for lensing with relative ease.

This thesis provides a practical method to search for dark matter halos within our Galaxy

and is readily extensible to nearby globular clusters and galaxies, pending the discovery

of millisecond pulsars in these more distant systems.

Statement of Originality

The work in this thesis is entirely my own, with supervision by Prof. Geraint Lewis.

It has not been reproduced in publications. The chapter on dark matter Chapter 2

is a literature review. The mathematics in §3.2-§3.4 can be found elsewhere in the

literature (as cited), apart from §3.4.2,§3.4.3. Otherwise, all of the thesis is my own

work, including the code in Appendix B. The matlab code used is all of my own design

using existing functions (as of R2012a), except for bessel zeros used in the Hankel

convolution code in §3.4.2 and cb*,mtit and polyrev, which are used for plotting.

Geraint Lewis assisted in debugging the code.

I certify that this report contains work carried out by

myself except where otherwise acknowledged. It has

not been submitted to any other institution for the

award of a degree.

Signed: Date: 26/03/2014

Related material

Code repository

https://github.com/vonbraunbates/pulsar-lensing

Presentations Given

2014 cosmology seminar University of Oxford

iii

Contents

Abstract i

Chapter 1. Introduction 1

1.1. Purpose................................................................... 1

1.2. Context................................................................... 2

1.3. Structure and remarks on notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Chapter 2. Dark matter halos 4

2.1. Evidence for dark matter.................................................. 4

2.1.1. Cosmological mass fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2. Mass in galaxies....................................................... 8

2.1.3. Mass in stars.......................................................... 9

2.2. Baryonic dark matter..................................................... 10

2.2.1. Primordial hydrogen................................................... 10

2.2.2. Massive compact halo objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.3. Stellar Remnants...................................................... 13

2.3. Non-baryonic candidates.................................................. 16

2.3.1. Neutrinos and Sterile Neutrinos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2. Axions................................................................ 16

2.3.3. WIMPs and SuperWIMPs............................................. 18

2.3.4. Hidden dark matter................................................... 19

2.4. Conclusions............................................................... 19

Chapter 3. Simulations of gravitational lensing 21

3.1. Lens models............................................................... 22

3.1.1. Schwarzschild lens..................................................... 22

3.1.2. The homogeneous disc model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.3. The Navarro-Frenk-White proﬁle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.4. Elliptical lens.......................................................... 24

3.2. Light propagation in gravitational lens systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3. Fermat’s principle......................................................... 26

3.3.1. The lens equation..................................................... 26

3.3.2. Image locations........................................................ 29

3.3.3. Time delay............................................................ 30

3.4. The lensing potential...................................................... 31

3.4.1. Surface mass density .................................................. 31

3.4.2. Simpliﬁcations due to axisymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.3. Calculating the potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5. The ﬂux theorem.......................................................... 35

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GravitationallensingofpulsarsasaprobeofdarkmatterhalosFrancescavonBraun-BatesMastersbyResearchsubmittedinfulllmentoftherequirementsforthedegreeofMasterofScience(Research)SchoolofPhysicsUniversityofSydneyApril2014AbstractAkeyquestionincosmologyisthepropertiesofdarkmatter.Aparticularopenproblemiswhet...

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Gravitational lensing of pulsars as a probe of dark matter halos Francesca von Braun-Bates

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