Applying non-harmonic analysis in a search for periodic sources in Antares data

U

NIVERSITY OF

A

MSTERDAM AND

THE

F

REE

U

NIVERSITY

B

ACHELOR

P

ROJECT

P

HYSICS AND

A

STRONOMY

Applying non-harmonic analysis in a search

for periodic sources in Antares data

Author:

Bowie B

REWSTER

(11288876)

Supervisor:

Prof.dr. Maarten

DE

J

ONG

Second Examiner:

Prof.dr. Paul

DE

J

ONG

1

Abstract

Quickly rotating, extremely dense and emitting beams of radiation aligned with their poles, pulsars are some of the most impressive and rare astronomical objects in our galaxy. Pul-sars generally have frequencies of about 1 Hz but some pulPul-sars have frequencies of less than 1 mHz. Under the assumption that pulsars emit neutrino radiation in the same direc-tion that they emit electromagnetic radiadirec-tion we search for periodic signals in neutrino-data collected by the Antares observatory. Going beyond a Fourier analysis we apply a non-harmonic analysis to the data-set to search for periodic signals corresponding to known pulsar frequencies. Non harmonic analysis consist of 4 algorithms which each look for local minima of a cost-function. These algorithms converge to the local minimum most near the initial frequency provided by an initial Fourier analysis. All 4 algorithms are applied iteratively to subsequently find the most powerful frequencies in the data-set. We show the viability of the method in reproducing frequencies found by a Fourier analysis and converging to values with a higher precision than Fourier analysis.

2

Samenvatting

Wanneer een ster aan het einde van zijn leven explodeert in een supernova kan er een relatief kleine, snel draaiende, bal materie met een hoge dichtheid overblijven. Deze zoge-naamde pulsar heeft ook een sterk magnetisch veld en zend elektro-magnetische straling (voornamelijk in radio gebied) uit. Doordat de pulsar (en dus zijn polen) snel ronddraaien is deze straling, net zoals licht afkomstig van een vuurtoren, soms wel te zien en soms niet te zien. De frequentie waarmee de pulsar ronddraait kan gemeten worden door de straling te meten op aarde. In dit onderzoek wordt aangenomen dat pulsars ook neutrinos, fun-damentele deeltjes met weinig massa, afgeven in dezelfde richting als dat de pulsars licht afgeven. Als dat zo zou zijn zouden de neutrino’s van zo een pulsar dus dezelfde frequentie hebben als de radiostraling die de pulsar afgeeft. Antares is een observatorium dat specifiek kijkt naar neutrino’s. Met behulp van een meting van Antares zoeken wij met een wiskunde analyse naar frequenties in de data van Antares. De wiskundige analyse die gebruikt wordt heet “non-harmonic analysis”. Doordat we laten zien dat de analyse in de buurt komt van al bekende resultaten laten we ook zien dat het resultaat in principe nauwkeuriger bepaald kan worden dan met voorgaande analyses.

Contents

1 Abstract 1 2 Samenvatting 1 3 Introduction 3 4 Background 3 4.1 Pulsars . . . 3 4.2 Neutrinos . . . 3 5 Methodology 4 5.1 Antares . . . 4 5.2 Fourier transform . . . 5 6 Non-harmonic analysis 5 6.1 Steepest descent . . . 6 6.2 Newtons method . . . 7 6.3 Low energy neutrinos . . . 8

3

Introduction

In the 50 years since the discovery of pulsars, quickly rotating neutron stars that are highly magnetized which emit beams of radiation out of their poles, the study of these astronomical objects has captured physicist and astronomers and helped verify theoretical predictions. In 1974 predictions based on Einstein’s theory of general relativity were verified by the obser-vation of a binary system consisting of a pulsar and neutron star. This system was losing angular momentum by emitting gravitational waves, this monumental discovery was done by Joseph Hooton Taylor, Jr. and Russell Hulse whom for their work received the 1993 Nobel prize in physics. Currently more than 2800 pulsars have been cataloged by the Australia Telescope National Facility(ATNF)[2]. Pulsars are probably sites of high-energy neutrinos and γ-rays. For example, Link & Burgio [1] predict high energy muon neutrinos could be produced in pulsars via resonant scattering. Nagataki [3] estimates the flux of neutrinos generated by the decay of pions. In this paper we present a search for periodic sources of neutrinos in data collected by the Antares observatory under the assumption that the neu-trinos produced in the pulsar would be aligned to the electromagnetic radiation produced by the pulsar. Using non-harmonic analysis we find inconclusive evidence whether the fre-quencies found in the Antares data set orignate from pulsars, furthermore we are able to approximately reproduce the values found in a fast fourier transform applied to the Antares data set and note future improvements in this method that would allow for higher accuracy and a greater capacity to resolve the questions posed here.

4

Background

4.1

Pulsars

Pulsars are highly magnetized neutron stars, remnants of supernovae of massive (∼5-10M) stars. Pulsars emit electromagnetic radiation from their poles with their beams of

emission aligned with the magnetic axis of the pulsar. When a supernova which will produce a pulsar occurs, the core of the star collapses in on itself causing the residual extremely dense remnant to rotate with a high angular velocity. The measured period of the pulsars from the ATNF catalogue are shown in figure 1. Pulsars in the ATNF catalogue have an average rotational period of of about 1 second but some pulsars have periods of less than 1 millisecond. The power required to sustain the electromagnetic radiation of the pulsar can come from three different sources. Rotation-Powered pulsars which lose rotational energy to supply their emissions. Accretion-Powered pulsars where the power is provided by the gravitational energy of matter falling into the pulsar. And magnetars, in which a decaying magnetic field provides the power. Magnetars are of particular interest to our research since they can accelerate charged particles to the energies required to produce neutrinos.

4.2

Neutrinos

Postulated in 1930 by Wolfgang Pauli and experimentally verified in 1956, neutrinos are el-ementary particles with a very low rest mass (long thought to be zero). There are 3 types of neutrinos each corresponding to one of 3 charged leptons, the electron, muon and tauon neutrinos. In contrast to the electromagnetic force where the massless photon mediates interactions, the weak force is mediated by the W or Z boson, which are both massive par-ticles. The large mass of the mediating particles causes the weak force to have a relatively short range and apparently weak coupling. As a result neutrinos rarely interact with normal

Figure 1: The occurrence rate of the periods of pulsars in the ATNF database [2]

matter and usually pass through unimpeded and without being detected. The rare times that neutrinos do interact with matter they couple to a W or Z boson when near the an atom. This boson in turn couples to a nucleon in the atom which can produce a variety of particles which can be detected. All the charged particles produced, either directly or indirectly due to a an neutrino interaction which are relativistic , can contribute to a detectable signal.

5

Methodology

5.1

Antares

Antares is the name of the neutrino detector located 2.5 km beneath sea level near Toulon, France[1]. Antares (Astronomy with a Neutrino Telescope and Abyss environmental RE-Search project) is a collection of 12 lines of about 350 meters length. Each line has 75 optical modules and each of these optical modules consists of a large area photo-multiplier and its associated electronics. The analogue signals from the photo-multipliers are digitised off-shore and all digital data are sent to shore where they are processed in real time by a farm of commodity PCs. The data are organised in frames with a length of about 100 ms; data covering the same time period are sent to the same PC. On the rare occasion a neu-trino interacts with the water surrounding the optical modules, Antares detects the Cerenkov radiation produced by the relativistic particles emerging from these interactions. Because neutrinos seldom interact with regular matter, other sources of light should be avoided by the large depth of water. Antares has provided the data for our analysis and for this re-search, specifically we focus on a large data set in which no selections have been applied. For contrast, other research related to Antares uses data that has been filtered before hand. The requirements can be very strict to the extend that low energy(MeV) neutrinos do not survive. In our research we look at a continuous stream of all photons detected for a dura-tion of 10 minutes to search for a periodic signal of low energy neutrinos which could have originated in a pulsar.

Figure 2: The discrete Fourier spectrum that is found when an FFT is applied to the entire Antares data-set. There are 2 sets of artefacts. 1)The spike at 100 Hz is due to the fre-quency of the AC current used to supply the detector. 2)The spikes at regular intervals of approximately 10Hz are integer multiples of 1/frame length.

5.2

Fourier transform

The go-to equipment for spectral analysis, the fast Fourier transform (FFT) takes in a discrete signal xn with N data-points and returns a discrete signal Xk of the frequencies present in

that data set also with N data points. In figure 2, the output of a FFT analysis on a given 10 minute data set from ANTARES is shown. As opposed to its continuous counterpart the Fourier transform, FFT is limited by the finite domain to which it is applied yielding an accuracy that increases linearly with N.

Xk = 1 N N −1 X n=0 xnexp( −2πikn N ) , k = 1, 2, 3, ..., N − 1

If the original periodic signal that the FFT is applied to has a frequency of k/N ∆t, Xk

ac-curately reflects the original signal behind xn. However if this is not the case xn is instead

represented as a combination of several frequencies which ultimately do not reflect the the nature of xn. Ordinarily this can be remedied by increasing N, this method however has an

inverse effect if there is a temporal fluctuation in the original data set, worsening the fre-quencies detected. Because of the inherent uncertainty that comes with applying FFT to a finite data set, when applying FFT there will be a number of frequencies found (with a relatively lower power) surrounding the actual frequency that created xn. To solve this the

Fourier coefficient used in FFT can be estimated by solving a nonlinear equation by applying non-harmonic analysis(NHA).

6

Non-harmonic analysis

In a non-harmonic analysis the Fourier coefficient is determined using a least squares method. This way the frequency f can take on any real number and is not bound to in-teger multiples of k/N ∆t where ∆t is the time between data points. NHA also provides an accurate estimate of the phase φ and the amplitude A of the wave responsible for xn. When

the value of ∆t has less impact on the final result. This way NHA can predict surrounding information from a part of the signal. In figure 3 the iterative process of NHA is outlined. First a FFT is applied to xn to find the frequency that has the highest amplitude in the wave

spec-trum. These initial values A0and f0are the starting point for the next step, steepest descent.

In this steepest descent a cost function is used to iteratively determine f and φ until they converge to fsd and φsd. Then after fsd and φsd have been found, using the same principle

Asd is found. Next is Newtons methods using as a starting point fsd, φsd and Asd, Newtons

method converges f and φ to a high accuracy but over a relatively short domain resulting in fnmand φnm. As the penultimate step fnm and φnmare used in another iteration of steepest

descent to find Asd2. These results are used to determine a new discrete time graph xm

which is subtracted from the original time graph xn, after which the entire process can start

again. One by one the strongest frequencies with their respective phase and amplitude are found and subsequently subtracted from the original data set.

Figure 3: The NHA process, picture taken from [4]

6.1

Steepest descent

NHA minimises the difference between a generalised wave function A cos(2πf t + φ) and the wave with the highest amplitude in xnusing the following cost function

F (A, f, φ) = 1 N N −1 X n=0 [xn− A cos(2πf ∆t + φ)]2

Starting with initial values A0 and f0 derived using the FFT and using φ0 = 0.

fsd,i+1 = fsd,i− µ dF (f, A0, φ0) df φsd,i+1 = φsd,i− µ dF (f0, A0, φ) dφ

Where the index i is used to denote iteratively applying equation 3 and 4 i times. Steepest descent converges to values covering a large domain relative to Newtons method. µ is a

weighting coefficient based on the retardation method. Where µ has a value between 0 and 1 to make the cost function monotonically converge.

To illustrate the range over which steepest descent can optimize a set of values a short test function of 6 waves is used

10

X

k=5

k cos(10k 2πx + k/10)

To create our test data we sample this function 200 times with regular intervals of ∆x = ₂₀₀1 . These values are then inserted into the cost function with A0 = 10to create a function with

2 variables f and Φ. This cost function is shown in figure 4 to illustrate the space over which steepest descent attempts to minimize the cost. The correct values for the wave with the biggest amplitude from function 3 are: φ = 1 and f = 100. It is clear that the phase would converge regardless of its initial value, the frequency, however would only converge if the original frequency was within a range of about 1Hz. This range of convergence is

greater for a higher sampling frequency (or a lower ∆x) but is notably shorter is there is greater number of sample points.

Figure 4: Created with a test function, the cost function as a function of f and φ. Steepest descent and Newton’s method are looking for maxima in this space.

6.2

Newtons method

NHA improves upon the accuracy achieved by steepest descent by applying newtons method. Newtons method converges over a smaller range than steepest descent but with a higher accuracy. Newton’s method is also an iterative process using the following

recurrence formulas. fnm,i+1 = fnm,i− µ J      dF (f,A,φ) df d2_{F (f,A,φ)} dφdf dF (f,A,φ) dφ d2_{F (f,A,φ)} dφ2      φnm,i+1 = φnm,i− µ J      d2_{F (f,A,φ)} df2 dF (f,A,φ) df d2_{F (f,A,φ)} dφdf dF (f,A,φ) dφ      Where J is given by J =      d2_{F (f,A,φ)} df2 d2_{F (f,A,φ)} dφdf d2_{F (f,A,φ)} dφdf d2_{F (f,A,φ)} dφ2     

Note that before calculating φnm,i+1 and fnm,i+1, f ,φ and A are replaced respectively by

fnm,i, φnm,iand Anm,i. After applying newtons method, steepest descent is applied once

more to allow A to converge to Asd2

6.3

Low energy neutrinos

Other research at Antares is focused on high energy neutrinos where E > 1 TeV. High energy neutrinos produce more particles and in turn allow more properties of the individual neutrinos interactions to be determined. In high energy neutrino physics, necessarily the data are filtered to exclude background due to radioactive decays in sea water. As a consequence, the detection threshold is about 100 GeV. The rate of detected neutrinos is then about 5 per day. To lower the detection threshold and to access frequencies of 1 Hz or less, unfiltered data are used in this analysis.

7

Results

To estimate the performance of the NHA process applied to the Antares data set we try to reproduce known frequencies present in the data. The 2 strongest frequencies are the artefacts at f ≈ 11 Hz and f ≈ 19 Hz, see figure 2. Furthermore we look for 2 Pulsars with the strongest flux on earth, namely ljg+15 and dml02 with respective frequencies 29.95 Hz and 11.19 Hz [2]. In this analysis an Antares data set was used with N = 6485442 and ∆t = 100 µs, as a result the frequencies present in the FFT are integer multiples of fk= k × 9.54 × 10−3Hz.

f

F F T

A

F F T

f

sd

A

sd

φ

sd

f

md

φ

md

A

sd2

artefact 1

9.53

3.96

9.54

3.93

-1.63

9.54

-1.63

4.16

artefact 2

19.07

1.51

19.07

7.718

-1.33

19.07

-2.086

6.65

pulsar 1

29.96

0.03

29.96

0.01

0.48

29.96

1.184

-0.02

pulsar 2

11.20

0.02

11.20

0.06

-1.36

11.20

-1.04

0.06

null

47.76

0.01

47.75

0.06

-1.51

47.75

2.19

-0.02

NHA reproduces known frequencies within a significant margin of error, since the waves can take on non integer multiples of k/N ∆t, this is to be expected. The null measurement returns a value of -0.02 (negative values just represent a flipped wave) which shows the margin of error to be significant. In this analysis the frequency would often converge to a value very near the original value ±10−7 Hz. This makes sense due to the large number of sample points in the original data. Since the FFT spectrum is discretized, the initial value the steepest descent method starts on may not be the local minimum corresponding to the peak in the FFT spectrum, this could explain the discrepancy between the found amplitude of the second artefact Asd2= 6.65and the amplitude in the FFT spectrum AF F T = 1.51. In

future research this could be mitigated by initialising multiple searches on frequencies near the frequency of interest thus searching over all the minima in the cost function for a width of 9.54 × 10−4_Hz.

In this analysis the strongest frequencies were not iteratively subtracted from the data set. This is due to a lack of processing power required to perform this operation. A future study

with a more optimised algorithm, more processing power or more available time could apply this technique in its totality to do a more comprehensive analysis. In future research the phase of the pulsars found in an analysis could be compared to the relative phase of the pulsars in the ATNF catalogue, if these were in agreement with each other that would be strong evidence that the frequencies found in the analysis were produced by pulsars. In more expansive research a precise expectation value for the neutrino flux on earth produced via neutrino showers originating from pulsars could be created.

References

[1] Bennett Link and Fiorella Burgio. “Flux predictions of high-energy neutrinos from pulsars”. In: Monthly Notices of the Royal Astronomical Society 371.1 (Aug. 2006), pp. 375–379.ISSN: 0035-8711.DOI: 10.1111/j.1365-2966.2006.10665.x. eprint: https://academic.oup.com/mnras/article-pdf/371/1/375/3032820/mnras0371-0375.pdf.URL: https://doi.org/10.1111/j.1365-2966.2006.10665.x.

[2] R. N. Manchester et al. “The Australia Telescope National Facility Pulsar Catalogue”. In: The Astronomical Journal 129.4 (Apr. 2005), pp. 1993–2006.DOI: 10.1086/428488.

URL: https://doi.org/10.1086%2F428488.

[3] Shigehiro Nagataki. “High-Energy Neutrinos Produced by Interactions of Relativistic Protons in Shocked Pulsar Winds”. In: The Astrophysical Journal 600.2 (Jan. 2004), pp. 883–904.DOI: 10.1086/380095.URL: https://doi.org/10.1086%2F380095.

[4] T. Yoshizawa, S. Hirobayashi, and T. Misawa. “Noise reduction for periodic signals using high-resolution frequency analysis”. In: EURASIP Journal on Audio, Speech, and Music Processing (Sept. 2011).DOI: