Fast radio bursts as astrophysical and
cosmological probes
Karel Geraedts
August 2, 2020
Studentnummer 11432055
Report Bachelor Project Physics and Astronomy
Size 15 EC
Conducted 27/03/2020 to 24/07/2020
Institute Anton Pannekoek Institute
Faculty Faculteit der Natuurwetenschappen, Wiskunde en Informatica
University University of Amsterdam
Supervisor Jason Hessels
Abstract
In 2007 Lorimer et al. [1] discovered a new type of fast transients. They noticed a burst with a very high dispersion measure (DM) which suggested that it orig-inated from outside the Galaxy. Now a couple of hundred of these bursts have been detected and we refer to them as fast radio bursts (FRBs). FRBs typically have a duration of a couple of milliseconds, are bright (fluence ∼ 0.1 − 100 Jy ms) and emitted in radio. Some of these FRBs are one-off events and some have seen to repeat suggesting that the progenitors could be collisions between two compact objects and compact objects interacting with their environment respectively. Recently Bochenek et al. [2] and Kirsten et al. [3] present the detection of FRBs from a magnetar. On the basis of current detections, it is estimated that about a couple of thousand could be seen across the sky per day [4]. Besides understanding FRB’s progenitors and their emission mecha-nism, research has been done into how FRBs can be used to probe cosmology and astrophysics. They can be used to study the interstellar- and intergalac-tic medium, much like pulsars are being used to study the interstellar medium ([5], [6]), and also to probe cosmology like estimating the Hubble constant H0
([7], [8]). In this thesis we discuss how this can be done and review the rele-vant propagation effects needed to understand the research. Finally, we explain how strongly gravitationally lensed FRBs can be used to make estimates of the lense’s properties such as its mass and source position in the observer’s plane. We find that for localized FRBs one could improve the precision of lens mass and source position estimates by considering how many lensed images were seen in the data and the relative magnification between the images, which can be obtained by looking at the signal to noise ratios.
Popular scientific abstract
Snelle transi¨enten zijn licht flitsen die een hele korte duur hebben, van een paar nanoseconden tot een paar seconden, en deze transi¨enten worden over het hele electromagnetische spectrum gezien. In 2007 hebben Lorimer et al. [1] een nieuwe soort transi¨ent gevonden, die we nu ‘Fast Radio Bursts’ (FRBs) noemen. Deze licht flitsen worden gemeten in radio, duren meestal een paar millisecon-den, zijn extreem fel en komen van buiten ons sterrenstelsel, de Melkweg. De eerst gemeten FRB in 2007 had een geschatte afstand tot de bron van ongeveer 3.2 miljard lichtjaren (1 gigaparsec). Sinds deze eerste detectie zijn er al een paar honderd FRBs gedetecteerd en is de verwachting dat men er een paar duizend per dag over de hemel zou kunnen detecteren. Dit maakt ze gelukkig niet heel zeldzaam - de eigenschappen van FRBs maken ze namelijk heel hand-ige objecten om cosmologie te bestuderen. FRBs reizen door veel verschillende media voordat wij ze waarnemen. Ze reizen door de bron zijn lokale omgeving, door het sterrenstelsel waar de bron in zit, vervolgens door het intergalactische media (de media tussen sterrenstelsels) en uiteindelijk reizen ze door de Melk-weg voordat ze bij ons op de telescoop terecht komen. Al deze verschillende media zijn geioniseerd en gemagnetiseerd en terwijl de FRB hier doorheen be-weegt, ervaart het verschillende propegatie effecten. Deze effecten kunnen we meten en gebruiken om onder andere een betere kaart te maken van de heelal en belangrijke vragen te beantwoorden zoals, waar zit al de baryonische massa? Of, hoe snel dijt het heelal uit?
In deze scriptie vertellen we meer over hoe we de uitdijing van het heelal kunnen berekenen met behulp van FRBs. De uitdijing van het heelal wordt gerelateerd aan een constante die we de Hubble constant (H0) noemen. Er zijn verschillende
methoden waarop we H0 kunnen berekenen maar de nauwkeurigheid van deze
methoden is nog niet klein genoeg en het blijft daarom interessant om nieuwe methoden te bedenken en uit te testen. E´en van de nieuwere methoden maakt gebruik van FRB zwaartekracht lensen. Een zwaartekracht lens is een term die we geven aan zware objecten die door hun zwaartekracht licht van een bron achter hun buigt naar de waarnemer’s zichtlijn. Dit effect wordt geillustreerd in Figuur 5b en we noemen de ring die je hier ziet de Einstein ring. Hoewel de ring er altijd is kun je hem niet altijd waarnemen en zie je voor hele verre lensen eerder meerdere afbeeldingen/bogen in plaats van een hele ring.
Het licht van deze afbeeldingen komt meestal niet tegelijkertijd aan en uit het verschil in aankomst tijd kan H0 berekend worden. Om dit te doen hebben
we wel nog wat andere eigenschappen van de lens nodig zoals zijn massa en de positie van de bron relatief tot de lens. De massa kan geschat worden door te kijken naar spectraal lijnen van de lens maar dit geeft niet altijd een hele nauwkeurige schatting. Deze scriptie laat zien hoe simulaties van verschillende massa en bron-positie scenarios een betere H0 schatting mogelijk maken.
Contents
1 Introduction 5
2 Progenitors 5
3 Propagation effects and their uses 7
3.1 The missing Baryon problem . . . 7
3.1.1 Dispersion . . . 7
3.1.2 Scattering . . . 7
3.1.3 Macquart et al. 2020 [9] . . . 8
3.2 Probing the Hubble constant with lensed FRBs . . . 10
3.2.1 Gravitational lensing of FRBs . . . 11
3.2.2 Time delay cosmography . . . 11
3.2.3 From time delay to the Hubble constant . . . 12
3.2.4 Current research . . . 12
3.3 Faraday Rotation . . . 13
3.4 Scintillation . . . 14
4 Analysis & Simulations 15
5 Results 17
6 Discussion 20
1
Introduction
Fast transients are pulses that last for a short period of time, from a couple of nanoseconds to a couple of seconds. Transients have been seen all across the electromagnetic spectrum and are interesting to study because they experience propagation effects that can be measured and used to probe the media through which the light has travelled. In 2007 Lorimer et al. [1] discovered a new type of fast transient which are now called fast radio bursts (FRB). A phase-space diagram showing various types of radio transients can be seen in Figure 1. These FRBs are seen in radio, are very bright (luminosities that are 109− 1012higher
than pulses from pulsars) and they typically last for a couple of milliseconds. The first burst was discovered because it had such an extremely high dispersion measure (DM) (as discussed in greater detail later in the text) which indicated that the source of the burst was about 1 Gpc away. Since their discovery, a couple hundred of FRBs have been found, most of which are one-off events and some have been seen to repeat. Now the estimate is that about 103FRBs could
be seen across the sky per day.
The properties of FRBs are such that they can be used as cosmological and astrophysical probes much like the properties of pulsars are being used to study our Galaxy, but the benefit here is that FRBs are of extra-galactic origin. One of the ways FRBs can be used to probe cosmology is that strongly gravitationally lensed FRBs can be used to estimate the Hubble constant (H0), model
indepen-dently and with more precision than with most other techniques. This thesis explores how analysis of lensed FRB simulations could improve the precision of lens mass and source position estimates - thus contributing to the precision of Hubble constant estimates.
The outline of this thesis is as follows: In Section 2 we briefly discuss possible progenitors of FRBs and describe some recent findings about these progeni-tors; in Section 3 we will discuss how FRBs can be used as cosmological and astrophysical probes and explain the propagation effects needed to understand this; Section 4 will explain how the lens mass and the source position can be estimated from strongly gravitationally lensed FRBs and describe which simu-lations were made to do this; the results will be shown and explained in section 5; Section 6 will discuss the relevance of the results and present ways in which they can be used in further research; in Section 7 provides conclusions.
2
Progenitors
The precise emission mechanism for the production of FRBs as well as the type of objects that emit FRBs are not yet known. Though there are many theo-ries about the possible progenitors, ranging from neutron stars to black holes,
Figure 1: A phase-space diagram showing various types of radio transients. On the y-axis the luminosity (Jy kpc) and on the x-axis the frequency·width (GHz s). The dotted diagonal lines are lines of constant brightness temperature and the light blue lines are lines of constant distance to the source. This plot is often used to illustrate the uniqueness of FRBs.
to white dwarfs and even exotic stars like strange stars. In order to produce an FRB a lot of energy is needed (fluence ∼ a couple of tens of Jy ms), the duration of the mechanism should be only a couple of milliseconds and a small emission region (< 10 km) is needed to be able to produce such a high energy density, and where particles can emit coherently. Also some of these theories involve situations where one expects to see only one bursts like for example at the birth or death of compact stars whereas others would predict a repeating burst signal, such as magnetars interacting with their local environment. Obvi-ously more than one of the theories that have been put forward could be correct depending on the properties of the bursts, like its luminosity and whether the source repeatedly emits FRBs or not. We refer the reader to Platts et al. [10] catalogue of FRB theories for more information.
Recently Bochenek et al. 2020 [2] and CHIME/FRB Collaboration et al. 2020 [11] presented a discovery of an FRB associated with a Galactic magnetar and Kirsten et al. 2020 [3] present a discovery of two more bursts from this same Galactic magnetar source.
3
Propagation effects and their uses
Fast transients experience propagation effects as they travel through the mag-netized and ionized media that make up the interstellar medium (ISM), the intergalactic medium (IGM) and the hosts galaxy’s ISM. Measuring these prop-agation effects can thus lead to a better understanding of the composition of these media but they can also be used to study cosmology. In this section we will discuss relevant propagation effects and how FRBs can be used as astrophysical and cosmological probes.
3.1
The missing Baryon problem
3.1.1 Dispersion
When light travels through a dispersive, i.e. any medium with an inhomoge-neous structure, medium its velocity changes depending on the density and the frequency of the light. Light with a higher frequency travels at a faster velocity than light with a lower frequency. Though at the source, the burst is emitted simultaneously across different frequencies dispersion causes the higher frequen-cies to arrive earlier than the lower frequenfrequen-cies. The quantity associated with the amount of dispersion is called the dispersion measure (DM).
The dispersion measure directly depends on the amount of free electrons in the intervening media and can be related to the density along the path as follows,
DM = Z d
0
ne(l)dl (1)
with ne the free electron density and d the distance to the source. Figure 2 is
a plot of an FRB and it shows this frequency dependent time of arrival. If the time delay is measured, then one can infer the total DM of the bursts;
∆t = e
2
2πmec
(νlo−2− νhi−2) DM ≈ 4.15 (νlo−2− νhi−2) DM ms (2) with methe mass of an electron, νlo−2 and ν
−2
hi the lowest and highest observed
frequencies. It is important to note that dispersion is an effect that can only be measured for transients and that the precision of the measurements depends on the duration of the pulse. Non-transients also experience propagation effects but these can not be measured as easily because temporal effects are hard to distinguish for constant sources.
3.1.2 Scattering
The scattering of light causes FRBs to be temporally broadened. The tempo-ral broadening arises from the density fluctuations within the medium which
Figure 2: A plot of the first discovered FRB by Lorimer et al. 2007 [1]. The emissions at higher frequencies arrive earlier than those at lower frequencies.
impose small light-travel-time differences for rays propagating through the gas. The light-travel-time differences depend on the frequency of the light and the distance to the source. The three smaller plots in Figure 3 shows the effect of scattering on a FRB. As is shown in these plots the time it takes for the signal to decay depends on the frequency with higher frequency decaying faster than lower frequencies. To be more specific:
τ ∝ ν−4 (3)
with ν the frequency and τ the decay time of the tail. Unlike the dispersion measure the time delay depends not on the total amount of electrons but instead on how these are distributed along the line of sight. The scattering measure is used to quantify the total amount of scattering and is linked to the free electron distribution as follows,
SM = Z d
0
Cn2e(l)dl. (4)
The distance to the source is given by d and C2
ne indicates the strength of the
fluctuations in the electron density along the line of sight. These fluctuations depend on the turbulence and geometry of the medium along the line of sight. Stinebring et al. 2018 discuss a model of the ISM in which the bulk of the scattering is dominated from one or several relatively thin ’screens’ of material about half the distance to the source.
3.1.3 Macquart et al. 2020 [9]
About 3/4 of the baryonic matter in the Universe is hard to detect and is ex-pected to live in the highly diffuse media outside galaxies and within galaxy
Figure 3: Main panel: The delay between the burst’s arrival time as a function of frequency. From the delay the DM can be measured with help of eq. 1, ∼ 8.5DMM W,
which implied that the burst was of extragalactic origin. Inset: The flux density as a function of time, notice the exponentially decaying tail due to scattering. Image from Thornton et al. (2013) [12]
clusters. In [9] Macquart et al. demonstrate a new technique for detecting the baryons in these diffuse states using FRBs. The rest of this section will describe how this can achieved and what results Macquart et al. found.
The dispersion measure of an FRB depends on the electron density along the line of sight and this relation is given by Equation 1. This equation can be separated as,
DMF RB = DMM W+ DMIGM+
(DMHost+ DMLocal)
1 + z (5)
with DMM W the Milky Way contribution, DMIGM the contribution of the
intergalactic medium, DMHostthe contribution of the host galaxy and DMLocal
the contribution of the source’s local environment. In order to detect the missing baryons the contribution of the IGM needs to be accurately measured, since it is expected that more than 3/4 of the baryons reside here. If the total DM is measured from the data, the Galactic contribution can be estimated with models such as the NE2001 ([5]) and YMW2016 ([6]), and if the FRB has been localized the host galaxy and local environment contributions can be estimated using other optical data. Now all that is left is the contribution from the IGM, which can be related to the baryon mass fraction in the IGM,
DMIGM= 21cH0Ωb,0 64πGmp Z z 0 fIGM(¯z)(1 + ¯z) E(¯z) d¯z (6)
Figure 4: Properties of 8 CHIME FRBs. This table was taken from [13].
with fIGM the baryon mass fraction in the IGM. For the purpose of this thesis
it is not important to know what these other constants are but just to realize that one can infer fIGM from DMIGM.
A typical FRB has a DM of about 500 cm−3 pc, see Tabel 4 for the DM of 8 CHIME FRBs. DMM W can be estimated by using known models of the
Galactic electron density, like NE2001 and YMW2016. Generally if the FRB was detected at b > 10 deg (i.e. not in the Galactic plane) DMM W ∼ 30 cm−3 pc.
The expected contribution of DMHost and DMLocal depends strongly on the
host galaxy and type of progenitor but are often assumed to be between 50−100 cm−3 pc. In order to get a good estimate of DMIGM it is therefore important
that one can estimate all these other components.
3.2
Probing the Hubble constant with lensed FRBs
The Hubble constant (H0) is a measure of how fast the Universe is expanding at
the current time and it sets the scale, size and the age of the Universe. One of the most important and hardest quantities to measure in astrophysics is the distance to objects outside our Milky Way and H0 is one of the most important factors
when calculating such distances because it links measured redshifts to distances. Therefore many efforts have been made to accurately estimate H0using, e.g. the
cosmic distance ladder (nearby cepheids, the tip of the red giant branch, etc.), from the Planck satellite CMB (cosmic microwave background) measurements, gravitational waves, lensed quasars, etc. However the results obtained from these different techniques are not consistent with cosmological models based on the CMB measurements. Finding more precise ways of measuring H0 is
therefore still important. Gravitationally lensed FRBs have been proposed to estimate H0in a similar way as has been done with lensed quasars except with
much greater precision due to the improved time-delay measurements. In the remainder of this section we will briefly discuss how lensed FRBs can be used
(a) Image of quasar 2237+0305. (b) Artist impression of the Einstein ring.
for this task and illustrate why this is more precise than other estimates. 3.2.1 Gravitational lensing of FRBs
According to general relativity, heavy objects curve space-time around them. Because of this light that was originally travelling in straight lines, now travels in curved lines. When the source, lens and observer are perfectly aligned lensing causes the observer to see a ring around the lens, the so called Einstein ring (Figure 5b). However it is rarely the case that all these 3 objects are aligned perfectly which causes the ring to change into several arcs or dots as is shown in Figure 5a. Moreover, the amount of images can change and depending on the mass distribution of the lens, its velocity and the distance from the source to the lens one would expect to see 1-5 images. The different images do not have the same arrival times, magnifications nor do the images always have to have the same relative position to each other as in Figure 5a. The relative position of these images to each other can actually be used to estimate the mass distribution.
3.2.2 Time delay cosmography
As was mentioned above, the different lensed images do not have the same arrival times and the delay between the arrival times of an image i and an image j can be related to physical properties of the lens system as follows;
∆tij= D∆t c [ (θi− β)2 2 − φ(θi) − (θj− β)2 2 + φ(θj)] (7)
with D∆t the so called time-delay distance given by,
D∆t= (1 + zl)
DolDos
Dls
, (8)
φ the lens’ gravitational potential, β the source position, Dolthe distance from
the observer to the lens, Dosthe distance from the observer to the source, Dls
the distance from the lens to the source and θi and θj the lensed image of the
Figure 6: A sketch of the lens system. The figure was take from [14]
3.2.3 From time delay to the Hubble constant
In astrophysics distances to objects outside our Milky Way are often estimated by measuring the redshift and converting this to a distance by using the follow-ing approximation, z =dH0
c . So for localized lensed FRBs one can estimate H0
by measuring the redshifts using spectral lines of the optical data of the lens and host galaxy. One can convert the measured time delay between 2 images to the time-delay distance (Equation 7) and finally calculate H0 using the
equa-tion menequa-tioned earlier. Precise measurement of H0 therefore relies on precise
measurements of, 1) the redshift to the lens, 2) the redshift to the source, 3) time delay measurements, 4) the source position with respect to the lens, 5) the lens potential (i.e. lens mass distribution) and 6) the image positions. Since FRBs are millisecond duration bursts it is clear that they can improve point (3) compared to lensed quasars whose time delay accuracy is typically ∼ 1 day. In other words lensed quasars have an uncertainty in arrival times of ∼ 107 ms
whereas with FRBs this uncertainty is ∼ ms, which is effectively zero relative to the other uncertainties. To constrain these other observations, good quality images of the lens object will be needed to measure the lens gravitational poten-tial. High resolution is needed to measure the different images on the sky, which for FRBs are typically seperated by 1 arcsecond or less, and precise location of the FRB’s host galaxy is needed.
3.2.4 Current research
Much of the techniques discussed above were obtained from Oguri et al. 2019 [14] where they discuss gravitational lensing of explosive transients. Li et al. 2018 [8] argue that strongly lensed repeating FRBs can be used to probe H0
and their main results can be seen in Figure 7. They find that, assuming H0 = 70 km s−1 Mpc−1, the precision can be dropped to a sub-percent level
(∼ 0.91%). Which is significantly better compared to for instance local distance measurements (∼ 3%) and lensed quasars (∼ 4%). Wucknitz et al. 2020 [7] also discuss the promise of repeating lensed FRBs for conducting cosmography. Instead of using the time delay between 2 lensed images to do this, as discussed
Figure 7: Estimates of the Hubble constant H0using different techniques. The black
curve is an estimate made using 10 strongly gravitationally lensed FRBs and assuming H0= 70 km s−1 Mpc−1. This plot was taken from [8]
above, they propose instead to consider the time differential of these time delays as multiple bursts of a repeating source are measured over months, years. They show that, by using the time differential, their method is independent of the lens mass distribution which would remove a significant challenge of modelling this distribution.
3.3
Faraday Rotation
A magnetized medium rotates the plane of polarization of light that travels through it. The total rotation is proportional to the electron density and the component of the magnetic field in the direction of propagation. The amount of rotation is associated with a quantity called the rotation measure (RM). This quantity can be measured in the data by looking at the angle of polarization between different frequencies,
Θ = RM
ν2 (9)
with RM the Faraday rotation measure. As mentioned before, RM depends on the component of the magenetic field in the direction of propagation B||(l) but
also on the free electron column density ne(l) and can be written as,
RM = −0.81 Z d
0
B||(l)ne(l)dl (10)
Like the dispersion measure RM, can also be subdivided in the same components as in Equation 5. However, note that a magnetic field in the intervening medium can change the total angle of polarization by adding or subtracting from the total contribution along the line of sight, and so making it harder to say something about the total magnetic field along the line of sight. Nonetheless, studying the RMs of FRBs can still be fruitful and research like F. Vazza et al. 2018 [15] and Prochaska et al. 2019 [16] shows this. Vazza et al. show that measuring the RM of 102- 103samples of FRBs can discriminate between extreme scenarios for the origin of cosmic magnetic fields. Prochaska et al. combine the measurements of DM and RM of localized bursts to probe the magnetization and turbulence of galactic haloes.
3.4
Scintillation
The DM of FRBs can be used to estimate the distance to its source. For z < 1 a rough estimate of the redshift can be made by noting that z ≈ DMIGM/1000
cm−3 pc. To convert the redshift to distance we can use the approximation dL ≈ 2z(z + 2.4) Gpc, which is also valid for z < 1. From these two
approxi-mations one can estimate the distance to the FRB source and thus also make an estimate of the luminosity and the brightness temperature. It is found that FRBs have TB ∼ 3.5 · 1035K which indicates that they are probably emitted
coherently at the sources and thus scintillate.
Light travelling through a turbulent and clumpy medium is scattered due to the different refractive indices along the line of path. For non-coherent sources the angular broadening is significantly large so that little interference happens between the different light rays. However for coherent sources, like FRBs, this is not the case and the scattered light rays recombine and constructively or de-structively interfere with each other leading to a complex frequency structure in the observer’s plane, which changes with time. Looking at the second spectrum (i.e. a Fourier transform) of such a pattern yields at which specific frequencies these scintills happen. The frequencies at which the patterns change is called the scintillation bandwidth, and the time for a specific pattern to persist is called the scintillation time. The scintillation bandwidth scales with with frequency as follows,
∆νscint∝ ν4. (11)
The total scintillation of a pulse depends not only on the frequency but also on the intervening medium, the relative motion between the source and the observer and the angular broadening of the burst. The angular broadening of FRBs is small enough to cause noticeable scintillation in regions like CGM where this was not able to be measured before, (H. K. Vedantham (2018)) [17].
4
Analysis & Simulations
We have discussed the progenitors of FRBs, some of the relevant propagation effects they experience, and how FRBs can be used to probe cosmology. In the sections that follow we will elaborate further on gravitational lensing of FRBs and try to give the reader a more intuitive feeling of the lensing system and the important aspects that influence what we observe. We now describe the analyses and simulations we performed.
Recall from Section 3.2 that we expect to see multiple lensed images in the data (1-4) depending on the the source position, the lens mass and its distribution. Figures 8a, 8b and 8c illustrate how this looks like in the observer’s plane for different β and thus different number of images. These images as well as much of the analysis performed makes use of the Lenstronomy python package [18]. In all of these images the lens mass distribution is spherically symmetric and the mass of the lens was 1012M
sun. The red curve in these figures is called the
critical curve and it represents the position on the sky where one would expect to see the Einstein ring if the source was close enough. The green curve is called a caustic curve1and whenever the source position moves across such a caustic curve the number of images increases by 2. Another parameter that is often used in lens modelling to describe lens system is the gamma parameter. gamma is related to the shearness of the lens and Figure 9 illustrates the influence of gamma on the lens system.
The delay in arrival times between an image i and j can be described by Equa-tion 7. It is important to be able to accurately constrain all the parameters here so that D∆tcan be accurately estimated and thus also H0. Assuming that
a lensed FRB as well as the lensing galaxy has been localized we propose a method in which the precision could be improved without other observations but instead a simple elimination of lens scenarios.
For these simulations D∆t is estimated from zl and by assuming some prior
value for H0. Now to calculate ∆ti,j the following values need to be known β,
φ, θiand θj. The lens mass distribution is assumed to be spherically symmetric.
The image positions i and j can be found by solving the lens equation,
β = θ − α(θ) (12)
see Figure 6 for more information about the angles. Since the mass distribution is known now the only parameter that influences φ is the lens mass (mlens). The
1The caustic is a curve or surface to which each of the light rays is tangent, defining a boundary of an envelope of rays as a curve of concentrated light. For gravitationally lensed FRBs this corresponds to points on the observer’s plane for which the magnification is largest. See Oguri et al. [14] for more detail.
(a) 4 lensed images (b) 2 lensed images
(c) 1 lensed image
Figure 8: Plot of the lens system in the observer’s plane, these plots were created using the Lenstronomy package [18]. Notice that the amount of images depends on whether the source position (black star) has passed a caustic or not (green curve). The red curve is the Einstein ring, the diamonds are the lensed image positions and the size of the diamonds scale with the magnification of each image. The units on the axes are in arcseconds
Figure 9: Plot of the lens system in the observer’s plane, these plots were created using the Lenstronomy package [18]. The grey diamonds are the image positions and the size of the diamonds scale with the magnification of the image. The black star is the source position. The red curve and green curve are the critical and caustic curves respectively . The units on the axes are in arcseconds
simulations find the best combinations of (mlens) and β for which the expected
time delay is the same as the delay seen in the data and in the results we argue how this in combination with complementary optical data could be used to improve the mass constraints.
5
Results
In the following, we have also assumed that; 1) the distance to the lens is 1 Gpc, 2) the distance ratio is 0.75 which is the ratio between the distance to the lens from the source and the distance to the source, i.e. Dls/Dosand 3) a gamma
of 2. We used 500 different values for mlensranging from 103 - 1012 Msun and
for each mass find the best value for β from which one would expect the same time delay as measured in the data.
The results, under these assumptions, are shown in Figure 10. On the y-axis the lens mass is plotted and on the x-axis the source position β is plotted. The
Figure 10: Scatter plot of the possible combinations mlensand β for which the time
delay between 2 lensed images is expected to be 100 ms. The color coding indicates how many images one can expect to see in the data.
color coding of the dots indicates how many lensed images one would expect to see in the data. As the distance from the source to the lens gets larger the mass of the lens needs to be bigger in order for light to be deflected enough to end up in the observer’s line of sight. This is also what we see in the plot. Notice that the dots plotted here are all configurations of β and mlensfor which
the expected time delay between 2 images is the same as the time delay seen in the data, within some error, i.e. a time delay between the 2 images of ∼ 100 ms. For localized FRBs other optical data could be available and perhaps from this data an upper and lower limit on the possible masses for the lens can be es-timated. A representation of this is shown in Figure 11. Here the horizontal bar represents the mass range that was found from other optical data and all dots that do not lie in this range have been put in the background so that only the possible configurations are left. In this range we can distinguish between scenarios where one would expect to see 1,2,3 or 4 images in the data. Take the case that 3 images were found, then the range of possible mlensshrinks. See a
representation of this in Figure 12.
Now that mlens is better constrained, this has also given better constraints
to the possible values of β. What one could also do is look at the expected magnifications of the lensed images and compare these to the S/N ratios seen in the data, and in a similar way as shown above exclude some more of the possible scenarios.
Figure 11: Scatter plot of the possible combinations mlensand β for which the time
delay between 2 lensed images is expected to be 100 ms. The color coding indicates how many images one can expect to see in the data. The grey horizontal bar represents the constraints on mlensobtained from optical data of the lens.
Figure 12: Scatter plot of the possible combinations mlensand β for which the time
delay between 2 lensed images is expected to be 100 ms. The color coding indicates how many images one can expect to see in the data. The grey horizontal bar represent the constraints on mlensobtained from optical data and assuming 3 images were observed.
6
Discussion
The technique presented here shows how mlens and β can be more precisely
estimated in case only limited optical data about the lens is available. However, an elaborate error analysis would be needed to quantify the improvement and to indicate how likely one can distinguish between the number of expected images. To explain how, we briefly recap: More precise measurements of H0 depend on
having a precise estimate of the time-delay distance (D∆t). According to
Equa-tion 7 and along with the assumpEqua-tions that were made in the analysis we show that the precision of D∆t depends on the precision of mlens, β and ∆t. In the
analysis and simulations presented here we assumed some prior D∆t and then
solved Equation 7 to find the combinations of mlens and β for which one can
expect the same time delay (∆t), within some error, as seen in the data. Instead of this it would be more relevant to fix ∆t and then find the best combinations of mlens and β which yield a value of D∆t allowed within the already weak
constraints obtained on D∆t by other optical data. Then for those solutions go
through a series of accept/reject trials based on the number of images and the relative magnifications between these images for a given solution. The hope is then that this will help improve the precision of D∆t by eliminating a lot of the
possible solutions.
No lensed FRB has been detected yet and as Li et al. [8] indicate it might take about 30 years before we detect a significantly large group of lensed FRBs (∼ 10), which is needed to accurately measure H0. If these lensed FRBs are
measured a lot of effort will be needed to model the lens mass which can be used to predict the lens potential (φ) as well as the environment along the line of sight (LOS) which could also influence the time delay between the images. Other massive structures, along the LOS, could do this by acting as smaller lenses which could influence the time delay.
Another application of lensed FRBs is that they could be used to confirm that the Hubble constant changes with time and also measure this time dependency. This can be done by simply measuring the Hubble constant at different red-shifts. Though most FRBs have been seen at redshift of < 1 the detection of a lensed repeating FRB at z=4 or higher would be key for measuring H(z) and determining the nature of dark energy.
The all-sky rate of FRBs is roughly 103 sky−1 day−1. Currently a couple of
hundred FRBs have been detected but with the development of new radio tele-scopes and upcoming surveys like CHIME in Canada, UTMOST in Australia and the Square Kilometer Array (SKA) the expectation is that we will be able to detect more than 100 FRBs per day. Though no gravitationally lensed FRBs have been detected yet the probability for an FRB to be lensed is about 10−4. As a result we will have the ability to detect > 10 lensed FRBs per year of
which about 3% are expected to be observable as repeating FRBs.
7
Conclusion
In this thesis we have introduced the reader to a relatively new class of radio transients called FRBs. We review several propagation effects that these bursts experience and how understanding the influence of these effects on the mea-sured data enables new applications that were not possible before. Applications like improving the precision of Hubble constant estimates (Section 3.2.3) and measuring the baryon mass fraction in the IGM (Section 3.1.3).
Gravitationally lensed FRBs can be used to improve the precision of H0
es-timates because the time-delay distance (D∆t) is influenced by the expansion
of the Universe and thus H0. Obtaining accurate estimates of H0 using this
method rely on being able to precisely measure other key observations like, 1) zl, 2) zsthe redshift of the lens and source respectively, 3)∆ti,j the time delay
between 2 images i and j, 4) φ the lens potential which is influenced by the lens mass and its distribution, 5) β the source position and 6) the lensed source positions. If we only consider point 3 then clearly FRBs are significantly better because their time delay can be measured with millisecond precision compared to the the 1 day precision from lensed quasars. But to constrain these other ob-servations we will need to localize a lensed FRB to measure its redshift, gather other optical data to constrain the lens mass and its distribution and be able to resolve the different lensed images for which < 1 arcsecond resolution is needed. In Section 4 we aim to give the reader a more intuitive feeling for how these dif-ferent observations influence the lens system and introduce some basic concepts that one should consider when modelling lens systems. We make the simple assumption of a spherically symmetric lens and show how gamma (lens shear-ness), β and the time delay influence the lens system.
The results in Section 5 showed how lens system modelling can be used to improve estimates of mlensand β by assuming some prior model and then
im-proving this model by distinguishing between situations where 1,2,3 or 4 lensed images are expected and comment that this same technique could be used by also calculating the expected magnification of each of the images and comparing this to the S/N ratios measured in the data.
The aim of this thesis was to investigate whether simulations of gravitation-ally lensed FRBs would improve the precision of mlensand β estimates and to
increase the reader’s understanding of the different aspects of lens system mod-elling. Further research will be needed, including more elaborate simulations
and error analysis, to verify if such a method could indeed improve the preci-sion of mlens and β estimates, which would increase our capability to conduct
cosmology and astrophysics.
Acknowledgements
I would like to thank my supervisor Jason Hessels for the effort and time he put into this project and for including me into the weekly group meetings, were I learned a lot about current research on FRBs and got to meet a great group of people. Also I would like to thank Liam Connor for helping me and for introducing me to the Lenstronomy python package.
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