UBC Undergraduate Research

Characterization of CHIME0s Complex Gain Using New Transits of CygA and CasA Hertig, Emilie Marie 2019-04

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Characterization of CHIME′s Complex Gain Using NewTransits of CygA and CasAbyEmilie Marie HertigA HONOURS THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFBachelor of ScienceinTHE FACULTY OF SCIENCE(Physics and Astronomy)The University of British Columbia(Vancouver)April 2019c© Emilie Marie Hertig, 2019AbstractThis work aims to improve the calibration of the Canadian Hydrogen IntensityMapping Experiment (CHIME), a new radiotelescope built in Penticton (BC) andinaugurated in 2017. CHIME’s main goal is to observe baryon acoustic oscillationsin order to probe the evolution of the universe at redshifts between 0.8 and 2.5, inthe period where the standard model of cosmology predicts that dark energy startedto dominate over matter and radiation. Accurate expansion measurements duringthis period would provide tighter constraints on the Hubble parameter and the darkenergy equation of state. This would lead to invalidation or further verification ofour current cosmological theories, therefore improving our understanding of thenature of dark energy.It has been observed that the the steel structures of the telescope, as well asthe cables and antennae have temperature-dependent behaviour, which affects thequality of the data. Previous attempts of characterizing this dependency weren’tsuccessful; therefore, this works aims to explore an innovative method based onnew direct sky observations and allowing to determine the influence of externaltemperature on CHIME’s complex gain. Thermal susceptibilities are obtained fromanalyzing observations of bright radiosources, mainly CygA and CasA, and calcu-lating linear fits of the gain fractional variation as a function of external temper-ature. A correction for the nonlinear behaviour of antennae, based on laboratoryexperiments, is included in later stages of the analysis, as well as a detailed inves-tigation and treatment of outliers.This project is a step towards the making of a full thermal model that couldbe included in CHIME’s calibration algorithm in order to significantly improve thequality of cosmological data.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Theoretical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Notions of cosmology . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 ΛCDM model . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Constraining cosmological parameters . . . . . . . . . . . 92.1.3 Baryon Acoustic Oscillations (BAO) as standard rulers . . 112.2 The CHIME telescope . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Interferometry . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Intensity mapping . . . . . . . . . . . . . . . . . . . . . 162.2.3 CHIME’s working principle . . . . . . . . . . . . . . . . . 173 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1 Gain calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 21iii3.2 Data pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.1 Linear fits . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.2 Low Noise Amplifier (LNA) correction . . . . . . . . . . 243.3.3 Singular value decomposition . . . . . . . . . . . . . . . 254 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.1 Preliminary observations . . . . . . . . . . . . . . . . . . . . . . 264.2 First susceptibility results . . . . . . . . . . . . . . . . . . . . . . 304.3 LNA correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.4 Singular value decomposition . . . . . . . . . . . . . . . . . . . . 394.5 Analysis of outliers . . . . . . . . . . . . . . . . . . . . . . . . . 455 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57ivList of FiguresFigure 2.1 Hubble’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 2.2 BAO in the Cosmic Microwave Background (CMB) . . . . . . 12Figure 2.3 BAO in galactic surveys . . . . . . . . . . . . . . . . . . . . . 13Figure 2.4 Constraints from BAO . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.5 Intensity mapping resolution . . . . . . . . . . . . . . . . . . 17Figure 2.6 Photograph of the CHIME telescope . . . . . . . . . . . . . . 18Figure 2.7 CHIME antenna . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 2.8 Analog signal chain . . . . . . . . . . . . . . . . . . . . . . . 19Figure 4.1 Individual channel gain amplitudes 1 . . . . . . . . . . . . . 27Figure 4.2 Individual channel gain amplitudes 2 . . . . . . . . . . . . . 27Figure 4.3 Waterfall plot with extreme gain values . . . . . . . . . . . . 28Figure 4.4 Waterfall plot of gain amplitudes 1 . . . . . . . . . . . . . . . 29Figure 4.5 Waterfall plot of gain amplitudes 2 . . . . . . . . . . . . . . . 29Figure 4.6 Linear fits of gain variation VS temperature . . . . . . . . . . 31Figure 4.7 Linear fits for individual channels . . . . . . . . . . . . . . . 32Figure 4.8 Thermal susceptibility from CygA and CasA data . . . . . . . 33Figure 4.9 Thermal susceptibility from CygA, CasA and TauA data . . . 34Figure 4.10 LNA temperatures . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 4.11 Linear fits before and after LNA correction . . . . . . . . . . . 36Figure 4.12 Root Mean Square (RMS) of residuals before and after LNAcorrection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 4.13 Comparison of susceptibilities before and after LNA correction 38Figure 4.14 Comparison of susceptibilities for 36 and 70 CasA transits . . 39vFigure 4.15 RMS of data for 70 CasA transits . . . . . . . . . . . . . . . . 40Figure 4.16 Outliers and linear fits 1 . . . . . . . . . . . . . . . . . . . . 41Figure 4.17 Outliers and linear fits 1 . . . . . . . . . . . . . . . . . . . . 42Figure 4.18 Ratio of first to second singular values . . . . . . . . . . . . . 43Figure 4.19 Right singular vector . . . . . . . . . . . . . . . . . . . . . . 44Figure 4.20 Right singular vector excluding outliers . . . . . . . . . . . . 44Figure 4.21 RMS across channels . . . . . . . . . . . . . . . . . . . . . . 46Figure 4.22 Linear fits excluding outliers . . . . . . . . . . . . . . . . . . 47Figure 4.23 Linear fits excluding outliers 2 . . . . . . . . . . . . . . . . . 48Figure 4.24 RMS ignoring outliers . . . . . . . . . . . . . . . . . . . . . . 50Figure 4.25 Rain statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 4.26 Final susceptibilities . . . . . . . . . . . . . . . . . . . . . . 52Figure 4.27 Dataset dependency of susceptibility . . . . . . . . . . . . . . 53viGlossaryCHIME Canadian Hydrogen Intensity Mapping ExperimentDRAO Dominion Radio Astrophysical ObservatoryCMB Cosmic Microwave BackgroundFRB Fast Radio BurstsBAO Baryon Acoustic OscillationsSDSS Sloan Digital Sky SurveyWMAP Wilkinson Microwave Anisotropy ProbeLNA Low Noise AmplifierRFI Radio-Frequency InterferenceSVD Singular Value DecompositionFLA Filter AmplifierRMS Root Mean SquareviiAcknowledgmentsI would like to extend my heartfelt thanks to my supervisor Dr. Gary Hinshaw forgiving me the opportunity to be a part of his team and contribute to this fascinatingproject, as well as for his advice and guidance throughout the process of research.A special thanks goes to Mateus Fandino for his precious help, his clear explana-tions, his kindness and patience. Many thanks also to Dr. Seth Siegel for storingand pre-processing the gain data. Finally, I am very grateful to the whole CHIMECollaboration for the enjoyable research experience and the friendly atmosphere inthe experimental cosmology lab.viiiChapter 1Introduction1.1 MotivationSince its first observation in 1998, the accelerated expansion of the universe hasremained an unsolved mystery and a flourishing domain of investigation for mod-ern cosmologists. While an intuitive view of the Big Bang model suggests thatthe initial expansion should be slowing down under the effect of gravity, observa-tions yielded by a type Ia supernovae survey conducted in 1998 [15] and numeroussubsequent experiments show the exact opposite: an unknown force seems to bepushing galaxies away from each other at an increasing rate. The mysterious phe-nomenon, due to its puzzling nature and its apparent lack of directly observableproperties, has been given the name of ”dark energy”; understanding its physicalorigin has become one of the greatest challenges of today’s cosmology.The most widely accepted cosmological theory at present corresponds to theΛCDM model, where Λ stands for a cosmological constant representing dark en-ergy, and CDM refers to ”cold dark matter”. The adjective ”cold”, in this context,means that dark matter is seen as non-baryonic particules moving at sub-relativisticvelocities. In this model, the initial stage of the universe was characterized by ex-tremely high density and temperature; the emergence of expanding space and time,commonly known as ”Big Bang”, was triggered 13.79 billion years ago [14] byquantum fluctuations of yet unknown nature. Within the first few fractions of asecond, a very brief stage of extreme exponential expansion known as inflation is1believed to have taken place. Such a sudden and violent event (supposedly increas-ing the size of the universe by a factor of 1026 in the span of 1032 second [19])can seem difficult to conceive, but this model actually solves some fine-tuning is-sues arising from Big Bang cosmology and provides an explanation for CosmicMicrowave Background (CMB) anisotropies [16]. Inflation was followed by a longperiod of decelerating expansion during which the universe was successively dom-inated by radiation and matter. The present acceleration phase suggests that thedominating component is now dark energy, the proportion of which has been esti-mated to 68.89% by the latest Planck results [14]. Representing it by a cosmologi-cal constant Λ indicates that the dark energy density is assumed to remain constantin time. This means that dark energy doesn’t ”dilute” with expansion like mat-ter and radiation; while this would justify why it only becomes dominant in laterstages of the universe’s evolution, it still doesn’t explain what dark energy actuallyis. Many hypotheses have been made, none of them has been confirmed yet. One ofthe candidates is the vacuum energy predicted by quantum field theory [9], whichwould be consistent with a cosmological constant; however, the predicted energydensity exceeds the measured cosmological constant by no less than 120 orders ofmagnitude. Several modern cosmologists are also questioning the fundamental as-sumptions of ΛCDM and proposing alternative theories such as a time-dependantdark energy density [20], modified gravity [17] or scale invariance of vacuum [11].Most of the observational evidence for the ΛCDM model is contained in theCMB, which constitutes the relic of a much hotter and denser state of the universe.The near-perfect correlation between the CMB spectrum and the predicted black-body at 2.7255 K definitely tipped opinions in favor of Big Bang theory; since then,ΛCDM has been generally accepted as today’s leading cosmological model. In itssimplest form, ΛCDM only requires the knowledge of 6 cosmological parametersin order to reconstitute the past evolution of the universe and predict its future.These parameters have to be constrained by experiments, most of them consistingin measuring the relation between distance and redshift of various remote celestialobjects. Such measurements are the main goal of the Canadian Hydrogen Inten-sity Mapping Experiment (CHIME) telescope. Numerous surveys with a similarpurpose have already been conducted, mostly focusing on the very early universe(e.g. Wilkinson Microwave Anisotropy Probe (WMAP) [10] and Planck [14]) or the2closer neighbourhood of our galaxy (e.g. Sloan Digital Sky Survey (SDSS) [18]).CHIME’s wide frequency range and use of intensity mapping will allow for obser-vations at redshifts 0.8 < z < 2.5, an intermediate range of major importance as itis believed to enclose the moment at which dark energy became dominant. Mea-surements of the Hubble parameter (see Chapter 2) and of the dark energy equationof state deduced from CHIME’s 3D map of this zone could be the key to a betterunderstanding of the nature of dark energy, by determining if it really is a cosmo-logical constant. These results might allow to invalidate or further verify ΛCDM,and maybe even to discover a relation between the current accelerating expansionand the initial inflation stage.Additionally, CHIME is already used for Fast Radio Bursts (FRB) detection [6](for which its wide field of view is a significant advantage) and pulsar monitoring[12]. Two FRBs have already been observed [4], and many more are expected to bedetected in the near future. While this is a very encouraging result, my work willbe focused on the cosmological purpose of the instrument, and I will therefore notgo into more detail on the topic of FRBs.With foreground signals 103 to 105 times brighter than the actual cosmologi-cal sources [5], CHIME requires a very precise calibration. Previously conductedlaboratory experiments as well as studies of the CHIME Pathfinder have indicatedthat the telescope’s components present a temperature-dependent response whichaffects the quality of the data. This paper will focus on the effect of external tem-perature on the amplitude of the complex gain, corresponding to the direction-independant part of the total gain as generated by amplifiers and cables. Thermalexpansion of cables and steel structures produces variations of the complex gainwhich need to be analyzed and taken into account in the calibration algorithm.Previous attempts of characterizing CHIME’s thermal susceptibility consisted ingenerating a controlled artificial signal and correlating the telescope’s responseto the temperature at the time of measurement. This Radio-Frequency Interfer-ence (RFI) producing method had the disadvantage of perturbing other instrumentsat the Dominion Radio Astrophysical Observatory (DRAO), and had to stop beingused before it could yield satisfactory results. This work therefore aims to explorean alternative method in which the artificial signal is replaced by direct sky data.The new archive of stable observations of CygA, CasA and more recently also3TauA will constitute the core of the present analysis; these bright and well-knownradiosources will be used to deduce CHIME’s complex gain for each frequencybin and channel over a few months’ period. Variations of this complex gain willthen be analyzed in the hopes of determining CHIME’s thermal susceptibility andimproving its calibration algorithm.1.2 OverviewThis paper will begin with a theoretical section summarizing the ΛCDM model’sfundamental concepts and equations. Techniques used to constrain cosmologicalparameters will be introduced, with a focus on Baryon Acoustic Oscillations (BAO)measurements. CHIME’s design and working principle will then be described.The chapter on experimental methods will cover the various steps of gain ex-traction and data pre-processing, before describing the different stages of the de-velopment of the thermal model. Details will be provided on the fitting techniquesand statistical concepts used to obtain the results.All the important results will then be presented and commented, in a logicalorder following the various stages of investigation. The significance of these resultswill be evaluated and opportunities for further investigation will be discussed.4Chapter 2Theoretical concepts2.1 Notions of cosmologyThis section will provide some conceptual and mathematical details about theΛCDM model and CHIME’s scientific purpose.2.1.1 ΛCDM modelThe fundamental framework for modern cosmology is given by Einstein’s generalrelativity; the theory describes a four-dimensional spacetime whose curvature isinfluenced by matter-energy distribution, and where the trajectories of all particles(including photons) correspond to geodesics. The separation between two events inspacetime is determined by a metric; at cosmological scales, where the universe canbe assumed to be isotropic and homogeneous, the Robertson-Friedmann-Walkermetric is commonly used. Its mathematical expression is as follows [16]:ds2 =−c2dt2+a(t)2[dr2+Sκ(r)2dΩ2] (2.1)In this equation, expressed in spherical coordinates, Sκ(r) is a curvature-dependentfactor defined in Equation 2.2 and dΩ2 = dθ 2+ sin(θ)2dφ 2. R0 represents the ra-dius of curvature, and the curvature scalar κ equals 1 for positive curvature, 0 forflat space and -1 for negative curvature.5Sκ(r) =R0 sin(r/R0) (κ = 1)r (κ = 0)R0 sin(r/R0) (κ =−1)(2.2)The a(t) factor appearing in Equation 2.1 is called ”scale factor”, and is offundamental importance as it characterizes the expansion of the universe. It is de-fined to be equal to 1 at present time. The coordinates (r,θ ,φ) in Equation 2.1 arereferred to as ”comoving coordinates”; two points having constant comoving co-ordinates implies the absence of proper motion relative to each other, even thoughthe separation between the two objects increases due to the expansion of spacetime.At large scales, expansion can be assumed to be isotropic and the proper distancebetween two objects can be written dp(t) = a(t)r with r the comoving distance. Inthis case, Equation 2.3 holds.d˙p(t) = a˙(t)r =a˙(t)a(t)dp(t) (2.3)Figure 2.1: Edwin Hubble’s original diagram showing the linear relation be-tween distance and recession speed of neighboring galaxies. The dis-tances were underestimated at the time, leading to an excessive value forH0. Source: https://astro.unl.edu/naap/distance/graphics/hubble orig.png6The proportionality constant H(t) = a˙(t)a(t) is called ”Hubble parameter”, refer-ring to Edwin Hubble’s first observation of the expansion of the universe in 1929(Figure 2.1). Equation 2.3, evaluated at present time t = t0, is known as Hubble’slaw.While Hubble’s law expresses the fact that the universe is expanding (i.e. a˙(t)>0), it doesn’t tell us anything about the sign of a¨(t). Evidence of accelerating ex-pansion was only discovered decades later, in 1998 [15], when a survey conductedby the High-Z Supernova Search Team yielded larger luminosity distances to var-ious Type Ia Supernovae than would be observed in the absence of a positive ac-celeration. Since then, this result has been confirmed by multiple experiments andit is now generally accepted that a¨(t)> 0, despite the lack of satisfactory physicalexplanation for this phenomenon.The evolution of a(t) as a function of cosmic time can be computed usingFriedmann’s equation. Derived from Einstein’s field equations, this equality relatesthe Hubble parameter to the energy distribution and the curvature of the universe.Its mathematical form is as follows, with G Newton’s gravitational constant, ε thetotal energy density (summed over all components of the universe), κ the curvaturescalar and Λ the cosmological constant:(a˙a)2=8piG3c2ε− κc2R20a2+Λ3(2.4)The other equation involving a derivative of a(t) is the acceleration equation,where P represents the pressure:a¨a=−4piG3c2(ε+3P)+Λ3(2.5)Two important observations can be drawn from Equation 2.4 and Equation 2.5.First, the only way to produce accelerating expansion is to include a componentwith negative pressure in Equation 2.5. More precisely, we can define the equationof state as the relation between energy density and pressure for a given substance:P = wε (2.6)Then a component producing positive acceleration must have w < −13 . The7second observation is that, neglecting Λ in Equation 2.4, there exists a criticalenergy density for which spacetime is flat (κ = 0). This value is determined byEquation 2.7.εcrit(t) =3c28piGH(t)2 (2.7)The time evolution of the energy density for a component with equation ofstate w is ruled by the fluid equation (Equation 2.8), derived from the first law ofthermodynamics.ε˙+3a˙a(ε+P) = 0 (2.8)Substituting Equation 2.6 into Equation 2.8 and solving the resulting differen-tial equation for ε , the following general expression can be obtained:ε(a) = ε0a−3(1+w) (2.9)From Equation 2.7, we can define a density parameterΩ= εεcrit for each compo-nent of the universe; written with a subscript 0, Ω0 refers to the density parameterat present time. The latest Planck results [14] yield a value very close to 0 for thecurvature of our universe; therefore, ΛCDM assumes spacetime to be flat, whichmeans the total energy density is equal to the critical density (1−Ωm−Ωr−ΩΛ =0). The Friedmann equation can now be rewritten as a function of the density pa-rameters at present time and equations of state of baryonic matter (w= 0), radiation(w = 1/3) and dark energy (according to ΛCDM, w =−1):(HH0)2=Ωr,0a−4+Ωm,0a−3+ΩΛ (2.10)Equation 2.10 can be expressed as a function of an observable quantity, redshift(z), thanks to the following relation:z =λobs−λemλem=1a−1 (2.11)Therefore, the scale factor at the time of emission is related to the redshift ofthe observed object by the simple formula a−1 = 1+ z. Friedmann’s equation now8takes the form: (H(z)H0)2=Ωr,0(1+ z)4+Ωm,0(1+ z)3+ΩΛ (2.12)From Equation 2.12, it appears that measurements of H(z) can be used to yieldconstraints on cosmological parameters. Using the values obtained by previousexperiments such as WMAP [10] or Planck [14] for H0 and the density parameters,the theoretical evolution of H(z) can be computed and compared to measurements;any observed deviations could for example invalidate the cosmological constantassumption and yield new constraints on the dark energy equation of state. This isthe main objective of the CHIME mission: by measuring the distance-redshift rela-tion for standard rulers at 0.8 < z < 2.5, the radiotelescope will be able to retracethe expansion history of the universe during this crucial period and distinguishpossible disagreements with the predictions of ΛCDM. This new study of the darkenergy equation of state might enable us to confirm or rule out some of our currenthypotheses, and could therefore be a step towards a better understanding of thephysical nature of one of the universe’s biggest mysteries.2.1.2 Constraining cosmological parametersMeasuring the distance-redshift relation mentioned in Section 2.1.1 is not as easyas it seems; indeed, it is important to realize that the proper distance, defined as thelength of the spatial geodesic between two events in spacetime, is not an observablequantity. The mathematical expression of the proper distance at present time isgiven by Equation 2.13, where te is the emission time:dp(t0) = c∫ t0tedta(t)(2.13)From this definition, it is clear that the time evolution of the proper distanceto an object directly depends on the time evolution of the scale factor a(t), whichis determined by the parameters of the model used to describe the universe. Thisexplains why distance measurements are so widely employed to constrain cosmo-logical parameters and confirm or invalidate possible cosmological models.The two observable quantities related to proper distance are luminosity distance9and angular diameter distance. To determine luminosity distance, one needs to ob-serve the energy flux F received from a source of known luminosity L. Luminositydistance is then given by Equation 2.14.dL =√L4piF(2.14)Luminosity distance is related to proper distance by a simple equation involv-ing another observable quantity, reshift (z):dL = (1+ z)dp (2.15)Experiments aiming to measure luminosity distances in order to constrain cos-mological parameters are conducted by observing bright sources of known lu-minosity, called ”standard candles”. The first discovered standard candles wereCepheid variables: a relation was established between the period of variation andthe luminosity of these very bright stars, which allows astronomers to deduce theintrinsec luminosity by analyzing the variation pattern. The obtained value is thencompared to the received flux and Equation 2.14 is used to deduce the luminositydistance. Due to their supergiant nature and intense brightness, Cepheids can yieldmeasurements of intergalactic distances; however, their range is limited to about30 Mpc [16]. Other methods are required in order to explore the universe at highredshifts.Another famous category of standard candles includes type Ia supernovae.These stellar explosions occur when a white dwarf orbiting around a companionstar attracts enough matter to exceed the Chandrasekhar mass and collapse underthe effect of its self-gravity. It has been discovered that the light curve of such asupernova is related to its peak brightness. Therefore, by observing the time takenby the supernova to reach its peak and to fade away, it is possible to deduce themaximal luminosity and to use Equation 2.14 again in order to determine the lumi-nosity distance. As supernovae are extremely violent events, their brightness canexceed that of an entire galaxy, which makes them visible from very far away; SNIa observations have been conducted up to z ≈ 1. At such redshifts, the observeddistances present a significant deviation when compared to those computed for adecelerating or steady expansion. Therefore, type Ia supernovae measurements re-10sulted in the first experimental evidence for dark energy and are also used to placeconstraints on the Hubble parameter and on the energy densities of matter and ofthe cosmological constant.While SN Ia surveys have been very successful over the years, our growingunderstanding of the early universe and of the mechanisms of gravitational insta-bility and structure formation have led us to explore a new class of objects, knownas ”standard rulers”. Standard rulers correspond to great structures of known, con-stant comoving size. Measuring the apparent diameter of such objects allows todetermine the angular diameter distance dA, related to proper distance by Equa-tion 2.16:dA =dp1+ z(2.16)In Section 2.1.3, we will see how CHIME uses a specific type of standard rulers,Baryon Acoustic Oscillations (BAO), in order to retrace the expansion of the uni-verse at relatively high redshifts and constrain the dark energy equation of state.Finally, it is important to note that constraints on cosmological parameters canalso be deduced from observations of the very early universe, especially the CMB.The state of the universe at the time of last scattering contains a mine of informationthat can be used to analyze the underlying physics and test our current models.The most accurate constraints are obtained by combining CMB and SN Ia or BAOmeasurements.2.1.3 BAO as standard rulersDuring the first few fractions of a second of the universe’s history, when space-time had just been formed and matter hadn’t been created yet, quantum fluctua-tions occured and were exponentially amplified by inflation. These perturbationsremained after inflation, as potential wells to which matter (both dark and bary-onic) was gravitationnally attracted, producing slight density anisotropies in thevery early cosmos; these initially infinitesimal anomalies would later become thesource of the great structures (galaxy clusters and superclusters) observed in thepresent-day universe. Before recombination, photons and hot, ionized baryonicmatter constantly interacted through Thomson scattering. As this photon-baryon11plasma was attracted to the aforementioned potential wells, pressure would buildup until a state of maximum compression was reached. The plasma would then”bounce back” from this contracted state, and the spherical density perturbationswould propagate outwards as acoustic waves. Once the temperature of the universehad decreased enough for hydrogen atoms to form and matter to become electri-cally neutral, photons decoupled from baryons; the absence of radiation pressurecaused these waves to stop propagating and leave their mark as concentric hot(overdense) and cold (underdense) rings on the CMB (Figure 2.2). Since then,these rings (which are actually the projection of three-dimensional spherical shellson the two-dimensional last scattering surface) have kept a constant comoving size,and their maximal radius, called ”sound horizon”, has been calibrated from CMBmeasurements, yielding a value of 146.8±1.6 Mpc [2].Figure 2.2: Zoom on hot and cold spots and rings observed in the CMB. TheWMAP data shows temperature anisotropies, while the Planck resultsfocus on polarization. Inverse temperature and polarization patterns areobserved in both cases. These anisotropies constitute evidence of theexistence of BAO and have allowed to calibrate the comoving size of thesound horizon. Image credit: CHIME Collaboration.As more and more dark and baryonic matter was attracted to the primordial12overdense shells, gravitational instability progressively led to the formation ofstars, galaxies, clusters and superclusters. Billions of years after the emission ofthe CMB photons, the imprint of BAO is still detectable in the great structures of theuniverse, as a preferred scale for galaxy clustering resulting in a peak in the correla-tion function of galaxy distributions. This feature was first observed by Eisensteinet al. in 2005 [8] and is shown in Figure 2.3.Figure 2.3: Correlation function of the galaxy distribution obtained from theSloan Digital Sky Survey (SDSS). The peak at d ≈ 100h−1 Mpc illus-trates the higher correlation between galaxies separated by a comovingdistance d. With h = H0/100 and H0 the Hubble constant (67.66 kms−1 Mpc−1), the observed preferred scale is in good agreement withthe sound horizon determined from CMB measurements. Image credit:Eisenstein et al. [8].BAO structures having a constant comoving radius means that they can be usedas a reference distance. As they can only be recovered from sky maps using corre-lation functions, they belong to the category of statistical standard rulers.Once a BAO structure has been detected at a certain redshift z, observations13across and along the line of sight allow the computation of the angular diameterdistance dA(z) and the Hubble parameter H(z) respectively. With s the diameterof the sound horizon, ∆θ the measured angular diameter across the sky and ∆z theredshift difference between both extremities of the structure, the relevant formulae[2] are given by Equation 2.17 and Equation 2.18.dA(z) =s∆θ(1+ z)(2.17)H(z) =c∆zs(2.18)Therefore, using BAO as standard rulers has the significant advantage of pro-viding measurements of two important quantities at once, and offers the possibilityof cross-checking the obtained results using Equation 2.19.dA(z) ∝∫ z0dz′H(z′)(2.19)Consequently, for the same quality of data, BAO measurements can providetighter constraints on the dark energy equation of state than type Ia supernovaeobservations, for which part of the physical mechanism is very complex and stillnot fully understood [13], and which only provide measurements of dL(z). Deter-mining dA(z) and H(z) simultaneously is more powerful than only knowing one ofthese quantities. The simulation in Figure 2.4 illustrates how combining BAO datawith CMB observations by Planck can significantly improve constraints on the darkenergy equation of state, and yield similar results to much more expensive surveysproposed for the next few years (EUCLID and BigBOSS).14-1.4 -1.2 -1.0 -0.8 -0.6-1.5-1.0-0.50.00.51.01.5w0waFigure 2.4: Comparison of simulated constraints from Planck + DEFT StageII (blue), PLanck + DEFT Stage II + EUCLID/BigBOSS (green) andPLanck + DEFT Stage II + CHIME (black). Combining CMB and BAOresults in the tightest constraints on the dark energy parameters w0 andwa, with the model assuming w = w0 +wa z1+z . ΛCDM corresponds tow0 =−1 and wa = 0. Image credit: CHIME Collaboration.2.2 The CHIME telescope2.2.1 InterferometryCHIME is a ground-based transit interferometer, which has no moving parts butuses the Earth rotation in order to map half of the sky every day. One of the mainprinciples behind the obtention of these maps is interferometry, which consists incombining the signals received by separate antennae in a way that preserves thephase data, and analyzing the interference pattern in order to reconstitute the fullsky information. Increasing the number of antennae results in a more accuratereconstruction of the signal.15In the case of CHIME, four adjacent half-cylinders oriented North-South eachreflect the incoming light towards their respective focal line. Along each focal lineare placed 256 dual-polarization feed antennae. The resolution requirements forintensity mapping (see Section 2.2.2) are met in the East-West direction by thefocusing effect of the cylindrical dishes, and in the North-South direction by theinstantenous correlation of the signals received by each pair of feeds.2.2.2 Intensity mappingThe technique of intensity mapping consists of detecting the 21 cm spin-flip emis-sion line of neutral hydrogen. The transition from triplet to singlet state in theground state of the hydrogen atom is extremely rare, but the resulting emission re-mains detectable at cosmological scales due to the abundance of neutral hydrogenin regions such as damped Lyman alpha systems, mostly situated inside galaxies(often quasars). The intensity of the emission is related to the density of matter; de-tecting this signal can therefore allow to reconstitute the power spectrum of matterdistribution in the zone of interest.The significant advantage of intensity mapping is that it does not require resolv-ing individual galaxies, as denser regions simply appear as brightness temperaturefluctuations on the map. Consequently, the use of this technique allows CHIME tohave a much lower resolution than instruments dedicated to galactic surveys suchas SDSS, from which the first evidence of BAO was obtained. An comparative ex-ample is shown in Figure 2.5. This greatly increases the mapping speed, and is thereason why CHIME’s maps will be able to encompass a wide redshift range and amuch larger volume of sky than previous surveys.16Figure 2.5: Left panel: map of a slice of sky up to z = 0.25 obtained by theSloan Digital Sky Survey. Each dot is an individual galaxy. Observ-ing every single galaxy for a significant amount of time results in a lowmapping speed. Right panel: the same map with a resolution similar tothat of CHIME. No individual galaxies are visible, but density fluctua-tions appear as intensity variations and the BAO scale is resolved. Imagecredit: [3] and CHIME Collaboration.2.2.3 CHIME’s working principleAs mentioned in Section 2.2.1, CHIME consists of four adjacent parabolic cylindersmade of steel, whose role is to reflect and focus incoming light towards the North-South oriented focal lines. Figure 2.6 shows a photograph of the external structureof the telescope. After being reflected by the cylinders, light is captured by the1024 dual-polarization antennae distributed between the four focal lines. Thesecloverleaf shaped antennae [7] are sensitive to both linear polarizations, East-Westand North-South, between 400 and 800 MHz. This range was chosen in order tomatch the frequency of the redshifted 21 cm emission line between z = 0.8 andz = 2.5. One of CHIME’s feeds is pictured in Figure 2.7. Directly after beingcaptured by an antenna, the signal goes through a Low Noise Amplifier (LNA),and is transported by a 50 m coaxial cable to the Filter Amplifier (FLA). Thesedifferent steps before digitization correspond to the analog signal chain, which isschematized in Figure 2.8.17Figure 2.6: Photograph of the CHIME instrument taken at the DRAO. Themetallic structures of the four cylinders and focal lines are visible, aswell as the containers enclosing the X-engine. The smaller dish anten-nae in the background belong to a different instrument. Image credit:CHIME Collaboration [1].Figure 2.7: Photograph of one of CHIME’s cloverleaf shaped dual-polarization antennae. Image credit: CHIME Collaboration.18Figure 2.8: Diagram of CHIME’s analog signal chain. (1) Reflector. (2) An-tenna. (3) LNA. (4) 50 m coaxial cable. (5) Outer bulkhead. (6) Shieldedroom. (7) FLA. (8) A/D converter input. Image credit: CHIME Collabo-ration.The analog chain is the source of CHIME’s thermal susceptibility. Thermalexpansion of the steel structures of the reflectors and the 50 m cable are the mostdominant factors, but the temperature-dependant response of the LNAs has alsobeen confirmed and analyzed in the lab.After being digitized, the signal arrives to the F-engine which performs a fastFourier transform in real time, with an input data rate of 13 terabits/s, in order toconvert data from position space to frequency space, divided in 1024 frequencybins. Finally, the signal is sent to the X-engine, which computes the visibilitymatrix (see Chapter 3). From the X-engine, data is transferred to three differentbackends, each characterized by different sampling rates and analysis methods:the FRB search engine, the pulsar timing monitor and the cosmology backend.19Table 2.1: Summary of CHIME’s main characteristics [1].No. of cylinders 4Dimensions of a cylinder 20 m × 100 mNo. of feeds 256/cylinderNo. of channels 2048 (4 cyl., 256 feeds each, 2 pol.)Redshift range 0.8-2.5Frequency range 400-800 MHzNo. of frequency bins 1024Frequency resolution 0.39 MHzAngular resolution 15’-25’Spatial resolution 15 MpcInstantaneous FOV 200 sq. deg.Table 2.1 summarizes the main characteristics of the instrument. The angularresolution has been adjusted to correspond to 1/10 of the BAO scale at the redshiftsof interest. The large instantaneous field of view combined with the Earth rotationallows the telescope to map half of the celestial sphere each day.20Chapter 3Methods3.1 Gain calibrationThe signals detected by CHIME arrive in the form of electromagnetic waves, com-posed of oscillating electric and magnetic fields. A general way of mathematicallyexpressing such an electric field is by using a complex number:E = E0eiφ (3.1)If one feed, identified by the subscript i, receives a signal described by Equa-tion 3.1, another feed j separated from the first one by a vector ~bi j will receivethe signal E0ei(φ+2pinˆ·~bi jλ). nˆ corresponds to the unit vector in the direction of theincoming radiation, and λ is the wavelength of the signal.Cross-correlating these two inputs, each multiplied by the corresponding com-plex gain gi introduced by the system, yields the following expression:EiE∗j = giE0eiφ(g jE0ei(φ+2pinˆ·~bi jλ))∗= gig∗j |E0|2e−2piinˆ·~bi jλ (3.2)Replacing~bi j by~r j−~ri, Equation 3.2 becomes:EiE∗j = |E0|2gieinˆ·~riλ 2pi(g jeinˆ·~r jλ 2pi)∗(3.3)21Equation 3.3 shows that a matrix containing elements Vi j = EiE∗j for all possi-ble pairs of feeds can be written as the outer product |E0|2~v~v†, with~vi = gieinˆ·~riλ 2pi .This implies that this matrix can be expressed as V = BDB†, with B a unitary ma-trix and D a diagonal matrix containing the eigenvalues of V . In the ideal casedescribed above, only one eigenvalue is nonzero (|E0|2) and the correspondingeigenvector is equal to ~v. Therefore, if the complex gain of the system is known,the power of the received signal can be calculated by performing an eigendecom-position of the matrix Vi j. Inversely, if the intensity of the observed source isknown, the complex gains can be deduced from the eigenvector corresponding tothe nonzero eigenvalue. This property is very useful for the calibration procedureas it allows to retrieve and analyze the system gains based on the observation ofbright radiosources.In the case of CHIME, the situation is slightly more complicated than the simpleand ideal example derived here. The output of the correlator, called ”visibility ma-trix”, is obtained by cross-correlating signals for each pair of feeds and integratingover every possible direction nˆ; it also incorporates a noise factor ni j. Mathemati-cally, the visibility matrix elements are given by Equation 3.4.Vi j =∫|E0(nˆ)|2gi(λ )g j(λ )∗e2piinˆ·~riλ e−2piinˆ·~r jλ d2nˆ+ni j (3.4)The previously mentioned eigendecomposition procedure can be generalized toEquation 3.4, therefore CHIME’s complex gains are obtained from the eigenvaluesand eigenvectors of the visibility matrix.3.2 Data pre-processingOnce the complex gains have been deduced from the correlator outputs, a few stepsof pre-processing have to be performed before the data is ready to be analyzed.The first of these steps is the removal of digital gains. Indeed, the system gainscalculated from the eigendecomposition of Equation 3.4 include the complex gain(the component of interest here, produced by the steel structures and cables), and anadditional component generated by the digitization of the data. These digital gainsobviously do not depend on external conditions such as temperature, therefore theirinfluence on the analyzed data must be removed. The digital gain values are stored22separately from the system gains in the CHIME database, and a function built in theCHIME-specific Python environment allows for their removal.The second pre-processing step is the filtering of the data. Transits of the ob-served radiosources can be contamined by external perturbations such as a simul-taneous solar transit or Radio-Frequency Interference (RFI); contamined datasetshave to be excluded from the analysis. This step is performed by a ”flagging”algorithm, which takes into account a complex combination of external parame-ters such as weather conditions, solar transits and interference sources, and sub-sequently evaluates the validity of a given dataset. The full development of theflagging algorithm is not the object of this paper, therefore it will not be presentedin more detail; it is however important to note that flagged data is systematicallyfiltered out before any analysis is performed. This is, once again, made possible bythe CHIME-specific Python environment used throughout this project.3.3 Data analysis3.3.1 Linear fitsThe main objective of this work is to determine CHIME’s thermal susceptibility; inorder to do this, a relation has to be found between the telescope’s complex gainand the external temperature at the time of observation. While the complex gainis, as its name indicates, a complex number, this paper will focus on its amplitude.Analyzing the gain phase constitutes a perspective for future investigation.Expressing the complex gain for channel i and frequency f as g(i, f )= g0(i, f )eiφ(i, f ),the amplitude then corresponds to g0(i, f ). The absolute value of the gain itself isof little interest in this context; however, relative fluctuations between differenttransits are of crucial importance. Therefore, we can define the fractional gainvariation for a transit at time t:g f rac(i, f , t) =g0(i, f , t)− g¯0(i, f )g¯0(i, f )(3.5)In this equation, g¯0(i, f ) represents the gain amplitude averaged over time forchannel i and frequency f .23The fundamental hypothesis of this work is that g f rac should follow a lineartrend as a function of external temperature. This assumption is supported by thefact that the expansion of the 50 m cable, which is considered to be the main sourceof CHIME’s thermal susceptibility, obeys a linear law. Previous experiments per-formed on the CHIME Pathfinder instrument also indicated a satisfactory linearmodel.The frequency-dependent thermal susceptibility bp( f ) for one polarization pwill therefore be given by Equation 3.6:g¯ f rac( f , t) = a+bp( f )T (t) (3.6)g¯ f rac( f , t) corresponds to the median or mean (both will be investigated) ofthe fractional gain variation over all channels coupled to the same polarization. Itwould also be possible to fit individually for each channel, and find the medianor mean of the susceptibilities afterwards; however, results detailed in Chapter 4show that averaging before fitting is the best choice as it suppresses some isolatedoutliers which would significantly impact individual channel susceptibilities.3.3.2 LNA correctionLab experiments performed by the CHIME Collaboration prior to this project haveshown that the LNAs do not have a linear response to temperature variations. There-fore, the simple model presented in Section 3.3.1 does not apply to the componentof the complex gain produced by the antennae. In order to determine the signifi-cance of the LNA contribution to the total gain, it is possible to predict the expectedLNA response to a given temperature. This operation is performed by a Pythoncode developed as a result of the aforementioned laboratory experiments.Temperature sensors have been placed directly at 59 of CHIME’s LNAs; the datacollected by these thermometers allows to calculate the median LNA temperatureat the time of a CygA or CasA transit, and deduce the expected LNA gain response.The total complex gain is then divided by the obtained value, and fitted to theexternal temperature measured by the on-site meteorological station as explainedin Section 3.3.1.In order to evaluate the quality of the fits, the Root Mean Square (RMS) of24residuals given by Equation 3.7 will be calculated. N is the number of transits con-sidered, f (T ) represents the linear fit (Equation 3.6) and g¯ f rac(T ) is the fractionalgain variation, averaged over channels, measured at temperature T .rms =(∑N0 ( f (T )− g¯ f rac(T ))2N) 12(3.7)Equation 3.7 shows that a lower RMS indicates a more accurate fit; therefore,the LNA contribution can be considered significant if the RMS of residuals is lowerafter the correction than before. The RMS of the data itself will also be calculatedover time, frequencies or channels depending on the context (see Chapter 4).3.3.3 Singular value decompositionIn order to verify that temperature really is the main explanatory factor for thecomplex gain variations, a linear algebra technique known as Singular Value De-composition (SVD) can be employed. This method consists of building a matrixin which each column contains complex gain values for all channels, for a givendate and frequency. Therefore, there will be 1024 matrices (one per frequency bin)with dimensions m×n with m the number of channels (2048) and n the size of thedataset.Such a matrix can be written as a product of two unitary matrices (A and B)and a rectangular diagonal matrix (D).M = ADB† (3.8)The values on the diagonal of D are known as singular values. In an idealcase where one single factor dominates the behaviour of the complex gain, thematrix M can be written as an outer product of two vectors multiplied by the onlynonzero singular value. In reality, we are not expecting all the other singular valuesto be zero; however, a first singular value significantly bigger than the other onesis a good indication that the one-parameter model considered so far is a decentapproximation. This method will therefore be used to test the assumption thattemperature variations are the main reason behind the observed gain fluctuations.25Chapter 4Results4.1 Preliminary observationsBefore performing any fitting or advanced analysis, some general, qualitative ob-servations about the raw gain data can be mentioned. Figure 4.1 and Figure 4.2show the gain amplitude values obtained as a function of frequency for 4 differentchannels, for the same CygA transit (08/10/18). Three main characteristics directlyappear.First, a global increasing pattern is visible on all 4 panels, meaning that thecomplex gain amplitude tends to be greater for higher frequencies.Second, an oscillatory pattern with a characteristic frequency of 30 MHz ap-pears, and is more defined in Figure 4.2 than in Figure 4.1. This oscillation can beexplained by an interference effect produced by multiple reflections between thecylindrical structure and the antennae. At certain frequencies, this interference isconstructive and results in an amplified received intensity, thus increasing the mea-sured gain; this occurs when the distance between the cylinder and the focal line isan integer multiple of the wavelength of the signal. In between these peaks, lowergain values are consequences of destructive interference. Calculations carried outprior to this project predicted a period of about 30 MHz for this interference phe-nomenon, which is in agreement with the observed pattern. Thermal expansion ofthe cylinder will modify the distance to the focal line; therefore, we can expect thewhole interference figure to be shifted as a consequence of temperature variations.26The third interesting feature is the presence of some isolated outliers whichare significantly higher or lower than the rest of the data. Despite being clearlyoff the trend, these points were not filtered out by the flagging algorithm. As thenext few sections will show, such events can have a non-negligible impact on themeasured susceptibilities and will therefore have to be understood and possiblyeven predicted.Figure 4.1: Gain amplitude for the CygA transit on 08/10/18. The left panelshows the data obtained for channel 1, and the right panel for chan-nel 500. Solar transits and RFI-contaminated frequencies have been re-moved, which explains the blank spots. A global rising pattern com-bined with a 30 MHz oscillation can be observed.Figure 4.2: Gain amplitude for the CygA transit on 08/10/18. The left panelshows the data obtained for channel 1000, and the right panel for chan-nel 1500. Similar trends as in Figure 4.1 are visible.27The waterfall plot shown in Figure 4.3 displays the gain amplitude obtainedfrom the same CygA transit, as a function of frequency and channel. The whitestripes represent missing data (frequencies that have been filtered out by the flag-ging algorithm). While channels seem to behave rather uniformly, it is difficult todistinguish any details; this is due to the presence of a few isolated extremal gainvalues, about one order of magnitude greater than the average. Being so rare, thesevalues are not properly visible on the plot; however, their existence is indicatedby the extended range of the colorbar. Understanding what produces such extremevalues and why they are not filtered out will be an important step in the later stagesof analysis.Figure 4.3: Waterfall plot of the gain amplitude obtained for the CygA tran-sit on 08/10/18, over all frequencies and all channels. The white linescorrespond to filtered out data. The wide global amplitude range is dueto the presence of a few extreme outliers, but most of the amplitudes arecontained in a range of 0 to 400.Figure 4.4 shows two slices of Figure 4.3, for two different sets of 200 channelseach. The behaviour of different channels appears to be sufficiently unifom overthis range to justify the averaging process explained in Chapter 3. Indeed, it onlymakes sense to average the fractional gain variation over all channels if they allfollow a similar trend, which appears to be the case in both of these plots. Similar28results have been observed for most other channels (except the very few containingextremal gain amplitudes).Figure 4.4: Gain amplitude for the CygA transit on 08/10/18. The left panelshows the data obtained for channels 1 to 200, and the right panel forchannels 601 to 800. These two ranges don’t contain any extreme out-liers, therefore more detailed patterns than in Figure 4.3 can be distin-guished. Comparing the two panels shows that the gain amplitude tendsto behave rather uniformly over channels.Figure 4.5: Gain amplitude for CygA transits displayed over channels 1 to200. The left panel shows the data obtained on 09/24/18, and the rightpanel on 10/24/18. Here again, the uniform behaviour of channels isnoticeable. Slight differences can be observed between the two dates,but the general pattern is similar and the 30 MHz oscillation is visible.Analogous comments can be made about Figure 4.5. This time, the group of29200 channels was kept the same, but two different CygA transits are represented.Both plots still display a uniform tendency for all considered channels. The visi-ble differences between the two graphs are an illustration of the fact that the gainamplitude varies depending on the considered transit, a phenomenon that is pri-marily explained by the instrument’s sensitivity to external temperature and will bestudied in detail in the next few sections.4.2 First susceptibility resultsIn order to qualitatively test the hypothesis that the gain amplitude varies linearlywith temperature, Figure 4.6 displays the correlation between median gain frac-tional variation and external temperature, for 107 transits of CygA and 4 differentfrequencies. It appears that the data follows the expected linear trend, which con-firms the fundamental hypothesis of this project (at least in first approximation). Alarge majority of the points are moderately scattered around an obvious linear pat-tern. A few isolated values are significantly off the expected trend; they are greatlyoutnumbered by the other points and are, in this case, not sufficient to visibly per-turb the fits.Performing similar fits for individual channels can result in a much less distinctlinear behaviour, as shown in Figure 4.7. In the left panel, the data is way tooscattered to follow any clear pattern, and the linear fit is completely off. A slighttendency to align can be seen in the left half of the graph, but the group of points inthe bottom right corner prevents us from drawing any conclusion from this figure.The right panel, however, shows that while a channel sometimes does not displaythe expected pattern for a given frequency, it can follow a linear trend (similar tothat observed in Figure 4.6 for the median of all channels) at another frequency.In general, this comparison indicates that it is preferable to average the fractionalgain variation over all channels before performing the fits, as the fits for individualchannels can be more easily perturbed by the scattering of data or the presence ofextremal values.30Figure 4.6: Linear fits of the median gain fractional variation as a functionof temperature for different frequencies (pol. E-W). Top left: 721.875MHz. Top right: 682.815 MHz. Bottom left: 643.75 MHz. Bottomright: 604.6875 MHz. The represented dataset includes 107 CygA tran-sits between 05/31/18 and 10/30/18. The data appears to follow a clearlinear trend despite moderate scattering and some isolated outliers.31Figure 4.7: Linear fits of the gain fractional variation as a function of tem-perature for channel 1001 and two different frequencies. Left panel:721.875 MHz. Right panel: 643.75 MHz. The fit in the left panel isclearly off as the points do not follow a linear trend. However, the samechannel exhibits a mostly linear behaviour at a different frequency.The slopes of all the linear fits of the fractional gain variation as a function oftemperature can be represented in a single plot, as frequency-dependent thermalsusceptibility values. Such a graph is displayed in Figure 4.8, containing the sus-ceptibility values obtained from the aforementioned 107 CygA transits and fromanother dataset including 74 CasA transits. It is immediately clear that the 30 MHzoscillation that affects the gain amplitudes, also has an influence on the thermalsusceptibility. The oscillatory pattern is mostly visible between 600 and 750 MHzfor both polarizations. As expected, the susceptibilities obtained from CygA andCasA data are very close; the complex gain being defined as direction-independent,noticing a strong dependency on the observed source would be abnormal.32Figure 4.8: Thermal susceptibility obtained as a function of frequency, fromthe 107 CygA transits and from 74 CasA transits between 07/25/18 and11/01/18. Top panel: pol. E-W. Bottom panel: pol. N-S. A clear os-cillatory pattern is visible, especially between 600 and 750 MHz, witha characteristic frequency of 30 MHz. The susceptibility values exhibitlittle to no source dependency.A plot very similar to Figure 4.8 was obtained with two additional datasets:the first one containing 34 CygA transits between 09/21/18 and 11/05/18, and thesecond one containing transits of TauA, another bright radiosource that could po-tentially be used for calibration. While the new CygA results follow the same trendas the two first datasets and are only slightly shifted upwards, the TauA data has avery strange behaviour. This dataset was introduced in order to confirm the absenceof dependency on the observed source; however, the results actually indicate theexact opposite. The TauA susceptibilities are much lower than the rest and do notexhibit a distinct pattern. Furthermore, a significant fraction of the data is missing,meaning that it has been filtered out during pre-processing. While it would be in-teresting to understand the reason behind this odd behaviour, the scarceness of dataand the encouraging results obtained for the two other sources pushed us to mo-mentarily discard TauA. The TauA datasets will be further analyzed in Section 4.5.33Figure 4.9: Thermal susceptibility obtained as a function of frequency. Redpoints: 34 CygA transits between 09/21/18 and 11/05/18. Green points:46 TauA transits between 09/20/18 and 11/24/18. While the two CygAand the CasA datasets are in good agreement, apart for a slight shift up-wards, the TauA susceptibilities are completely off the expected trend.4.3 LNA correctionAs explained in Chapter 3, we have attempted to account for the nonlinear be-haviour of LNAs in the hopes of reducing scattering and improving the quality ofthe fits. The LNA temperatures measured by the 59 on-site sensors are displayed inFigure 4.10.34Figure 4.10: Temperature measurements as a function of time for the sensorsplaced at 59 of the LNAs. The grey curves represent the individualmeasurements, the blue curve corresponds to the median of the 59 val-ues.Here again, the first step of the analysis was to check the coherence of thelinear model and to observe the influence of the correction on the slopes of thefits. Figure 4.11 shows the fits obtained before and after correction for a CasAdataset including 36 transits. The plots show a linear trend accompanied by quitesignificant scattering, both before and after correction; this scattering has a moreimportant effect on these fits than on the ones obtained in the previous section dueto the lower number of data points available. The correction influences the slopes,which means that it will produce a shift in the susceptibilities. However, it doesnot seem to significantly improve the scattering problem.35Figure 4.11: Linear fits of the median gain fractional variation as a func-tion of temperature for different frequencies (pol. E-W), before andafter removing the LNA gain component. Top left: 721.875 MHz.Top right: 682.815 MHz. Bottom left: 643.75 MHz. Bottom right:604.6875 MHz. The represented dataset includes 36 CasA transitsbetween 11/17/18 (date of first LNA temperature measurement) and01/03/19. Linear trends are visible both before and after correction,and the correction visibly impacts the slope. Important scattering ispresent in both cases.As suspected from the qualitative observation of Figure 4.11, the values of theRMS of residuals before and after the LNA correction are almost identical. Thisindicates that the correction does not significantly impact the quality of the fitsand the accuracy of the model. As expected, most of the temperature-dependentresponse comes from the cables and steel structures; the antennae themselves donot play a major role.36Figure 4.12: RMS of residuals before and after LNA correction, for pol. E-W(top panel) and pol. N-S (bottom panel). An oscillatory pattern appearswith a characteristic frequency of about 30 MHz. The correction doesnot significantly lower the RMS values.Figure 4.13 represents the susceptibility values deduced from the 36 CasA tran-sits, before and after the correction. While the corrected values are shifted down-wards (removing the LNA component slightly lowers the global susceptibility), theyfollow the same oscillatory trend as the original data points.37Figure 4.13: Thermal susceptibility obtained as a function of frequency fromthe 36 CasA transits, before and after LNA correction. Top panel: pol.E-W. Bottom panel: pol. N-S. The typical 30 MHz oscillation is stillvisible. The original and corrected data follow a very similar pattern,however the correction introduces a downwards shift in the suscepti-bility values.Finally, Figure 4.14 illustrates the strong dependency of the susceptibility onthe considered dataset. The blue points are obtained from 70 CasA transits; theorange points are deduced from a 36 transits sample of this same dataset, whichcorresponds to the time period over which LNA temperature measurements havebeen available. Both datasets are represented without LNA correction. Such adependency on the considered time period is a surprising feature; just like thecomplex gain is supposed to be source-independent, it is also assumed to be time-independent at constant temperature (if two different transits of the same sourcewere observed with the same external temperature, they should yield the samecomplex gain). Nevertheless, Figure 4.14 seems to prove the opposite. This ob-servation has led us to wonder if temperature really is the only dominating factorexplaining gain variations; Section 4.4 and Section 4.5 seek to answer this ques-tion.38Figure 4.14: Thermal susceptibility obtained as a function of frequencyfrom the full 70 CasA transits dataset (Dataset 1, from 09/21/18 to01/03/19), compared to the ones deduced from the 36 transits sample(Dataset 2), without LNA correction. Top panel: pol. E-W. Bottompanel: pol. N-S. A strong dependency on the considered time periodis visible, as the susceptibility values are significantly different despitethe fact that the two time ranges overlap.4.4 Singular value decompositionAs seen in the previous section, the RMS of residuals has an oscillatory behaviour,with peaks corresponding to the most imprecise fits and flat regions correspondingto zones in which the linear model accurately represents the data. The same obser-vations can be made about the RMS of the data and the RMS of residuals for the 70CasA transits without LNA correction.39Figure 4.15: RMS of data and RMS of residuals for the 70 CasA transits (pol.E-W) without LNA correction. The values highlighted with red starscorrespond to frequencies for which the RMS of residuals is high, in-dicating scattered data or an imprecise fit. The green stars indicatefrequencies with lower RMS, where the linear fits are satisfactory. TheRMS of residuals is globally lower than that of the data, which indi-cates that using the linear model already improves the quality of thedata.In order to better understand what causes some fits to be more accurate thanothers, we plotted the median fractional gain variation as a function of externaltemperature, with the corresponding linear fits, for 6 ”good” frequencies (greenstars) and 6 ”bad” frequencies (red stars). Figure 4.16 and Figure 4.17 show thatthe peak RMS frequencies (on the left) systematically contain between 4 and 6outliers that are way off the linear trend, while the rest of the points tend to befairly well aligned. The good frequencies do not contain such clear outliers.40Figure 4.16: Linear fits of the median fractional gain variation (pol. E-W)obtained from the 70 CasA datasets, for peak RMS (left side) and lowRMS frequencies (right side). From left to right and top to bottom:791.016 MHz, 782.031 MHz, 787.891 MHz, 776.172 MHz, 730.078MHz, 721.094 MHz. Frequencies with high RMS tend to contain iso-lated outliers that do not appear in the plots for the ”good” frequencies.41Figure 4.17: Same as Figure 4.16, for different frequencies. From left toright and top to bottom: 726.953 MHz, 714.844 MHz, 669.141 MHz,660.156 MHz, 666.016 MHz, 653.906 MHz.While the results of the LNA correction and the apparent dependency on theconsidered dataset seemed to indicate that temperature was not the only factor ofimportance, the results displayed in the two previous figures show that the linearthermal model is actually rather correct, except for a few frequencies for which asmall number of isolated points are sufficient to perturb the fits. This might meanthat there are a few particular days on which the transit data was contaminated by anexternal source, which could explain the recurring 4-6 clear outliers. The frequencydependent nature of these outliers, with a 30 MHz characteristic frequency, seemsto indicate a relation between these points and the reflectors.To make sure that temperature really is the main parameter, a singular value42decomposition was performed on the CasA data. Figure 4.18 shows the ratio be-tween the first and second singular values, for the 6 peak RMS frequencies (orange)and the 6 low RMS frequencies (blue). In both cases, this ratio varies between30 and 65; this result demonstrates the fact that the first singular value is largelydominant compared to the others, meaning that a one-parameter model is a goodapproximation of the situation.Figure 4.18: Ratio of first to second singular values for the 6 good frequen-cies and the 6 bad frequencies from Figure 4.16 and Figure 4.17. Thefirst singular value clearly dominates in all cases.If the dominating parameter is indeed temperature, the values contained in theright singular vector~v should be proportional to the considered temperature values.This hypothesis was tested in Figure 4.19, where the right singular vector multi-plied by the first singular value s1 was plotted against temperature (right side). Alinear trend is distinguishable, however the fit is thrown off by some clear outliersin the bottom of the plot. In the left panel, the values contained in s1~v are multipliedby the inverse slope a of the fit on the right and compared to the measured externaltemperatures. The two curves exhibit some similar tendencies but do not coincide.43Figure 4.19: Right panel: linear fit of right singular vector values VS externaltemperature. The slope of the obtained fit is used in the left panel,in order to compare the temperature evolution to the singular vectorvalues multiplied by a constant. Due to 7 points visible on the rightside, the calculated fit does not follow the actual linear trend of thedata and the agreement between the two curves in the left panel isquestionable.However, if the 7 clear outliers are removed, the linear trend in the right side ofFigure 4.20 is evident and the agreement between as1~v and the measured temper-atures is greatly improved. This result illustrates the fact that the thermal model isonly perturbed by a restricted number of points.Figure 4.20: Same as Figure 4.19, without the 7 outliers (in blue). The linearfit now describes the remaining data (orange) in a satisfactory way, andthe right singular vector is clearly related to temperature.The next step of the analysis will be to determine to which transits the out-liers in Figure 4.19 correspond. If they are the same as the transits producing thesystematic outliers in Figure 4.16 and Figure 4.17, this would be a very encourag-44ing result: it would mean that, except for a restricted group of abnormal transits,CHIME’s thermal response is well-described by a linear model. If this hypothesis isconfirmed, the perturbed transits will have to be further investigated (rain and windstatistics, as well as potential interference problems, might have to be considered).Ideally, this would allow us not only to suppress these contaminated transits in or-der to obtain a more accurate model, but also to predict when such anomalies aresusceptible of happening again.4.5 Analysis of outliersA more profound analysis allowed us to identify two distinct types of outliers.Studying the mean of the fractional gain variation instead of the median over chan-nels exhibited another family of anomalous points, now referred to as ”type 1 out-liers”, which did not appear in the previous investigations. In these particular tran-sits, channels do not behave in a uniform way. It is therefore suspected that theseoutliers result from the influence of an external source on some isolated channels;the primary hypothesis, which will be strengthened by later results, is that raincauses water to enter some of the antennae and trigger an abnormal response. Thephenomenon significantly perturbing a restricted number of feeds explains whytype 1 outliers were undetectable when considering the median of channels, butare visible when working with the mean (see Figure 4.22 and Figure 4.23). Astype 1 outliers are the only transits for which channels exhibit a non-homogeneousbehaviour, they can be detected by taking the RMS over channels of the fractionalgain variation. As shown in Figure 4.21 for CasA, the RMS values for type 1outliers are clearly superior to those of normal days. The difference is especiallysignificant between 400 and 500 MHz. This allowed to determine a criterion todetect type 1 outliers: the RMS over channels is superior to 0.1 in this frequencyrange. This method successfully identified outliers of this type in TauA datasets,as shown in Figure 4.22 and Figure 4.23.45Figure 4.21: RMS of the fractional gain variation across channels, as a func-tion of frequency, for the 34 CasA transits. Each curve corresponds toone transit. Type 1 outliers (blue) clearly have a higher RMS than gooddays (orange), especially between 400 and 500 MHz. This comes fromthe fact that only some of the antennae exhibit an anomalous responseon these days, resulting in a higher variation across channels. Note thattype 2 outliers are not detected by this method.The second family of outliers is the one already observed in Section 4.4, whichalso appears when considering the median fractional gain variation and exhibits astrong frequency dependance. These anomalies, referred to as type 2 outliers, arenot channel-dependent, so they cannot be detected by the previously mentionedmethod. However, these outliers being sensitive to frequency allows to define anew criterion: this time, RMS values are computed across frequencies instead ofchannels, around 0.1 of the median, and the threshold is fixed at 0.014. This cri-terion, defined by observing CasA data, effectively picked out TauA outliers asshown in pink on Figure 4.22 and Figure 4.23. These two figures clearly illus-trate the two types of outliers, with type 1 points always appearing off and type2 points only causing problems in ”peak” RMS frequencies. The linear fit of themean fractional gain variation as a function of external temperature, when ignoringboth types of outliers, accurately describes the remaining data.46Figure 4.22: Linear fits of the mean fractional gain variation VS temperature,excluding type 1 (blue) and type 2 outliers (pink), for 97 TauA transitsbetween 09/20/18 to 01/23/19 and different frequencies. The remain-ing data clearly follows the linear trend, and the frequency-dependentnature of type 2 outliers is visible. The left column shows peak RMSfrequencies, the right one shows trough frequencies. From left to rightand top to bottom: 669.1 MHz, 651.2 MHz, 671.1 MHz, 657.0 MHz,703.5 MHz, 685.5 MHz.47Figure 4.23: Same as Figure 4.22 for different frequencies. From left to rightand top to bottom: 705.5 MHz, 691.4 MHz, 772.7 MHz, 719.9 MHz,774.6 MHz, 726.2 MHz.Having computed these new, improved fits after detecting and filtering outanomalous transits allows to calculate the new RMS of residuals and the RMS ofthe mean fractional gain variation across days. These two quantities are plotted in48Figure 4.24 for all three sources. The red and green stars correspond to the ”peakRMS” and ”low RMS” frequencies represented in the two previous figures; afterfiltering out outliers, peaks in the RMS of residuals have disappeared and no signif-icant frequency dependance is observed. The target value of 0.003 is reached forpart of the frequency range. This means that the linear model, combined with theappropriate treatment of outliers, improves the quality of the data sufficiently tomeet the accuracy criteria of the instrument. This remark is also supported by thefact that the RMS of residuals is lower than that of the data without thermal model,especially in the 550-800 MHz range.Now that the thermal model has been confirmed to be satisfactory when outliersare ignored, it is important to find the reasons behind these anomalous transits inorder to be able to predict them. Figure 4.25 displays rain quantities cumulatedfor a six hour period at the DRAO, over the whole observation time. These statis-tics are superposed to the times of TauA transits, with type 1 and 2 outliers beingrepresented in blue and pink respectively. It appears that type 1 outliers exhibit acorrelation with the peaks in the graph: all type 1 points closely coincide with arain event. This supports the primary hypothesis that these anomalies stem fromwater entering some of the antennae. A few of the minor peaks do not coincidewith any type 1 outlier; this does not go against the hypothesis, as it is possiblefor rain to occur without causing significant amounts of water to perturb the re-sponse of some of the feeds. Type 2 outliers are more difficult to explain and nodefinite cause for their existence has been found yet. It is interesting to note thatall the TauA type 2 points correspond to a single group of consecutive days, rightafter New Year. A similar constatation has been made with CasA type 2 outliers,which coincide with the TauA values. This series of dates is preceded by severalsignificant peaks in the rain statistics, therefore it is possible that these outliers arealso related to water in some way. However, this does not explain the inherentlydifferent characteristics of these transits compared to type 1 outliers, with the ho-mogeneous channel behaviour and the frequency dependance seemingly indicatinga relation with reflector effects.49Figure 4.24: RMS of the data and RMS of residuals as a function of frequency,ignoring outliers, for the mean fractional gain variation of the 3 sources(E-W polarization). From top to bottom: CasA, CygA, TauA. The RMSof residuals is systematically lower than that of the data and does notexhibit a strong frequency dependance anymore. The target of 0.003is attained between 600 and 700 MHz.50Figure 4.25: Comparison of TauA outlier days with rain accumulation over a6 hour period. A correlation seems to exist between type 1 outliers andpeak rain events. The series of type 2 outliers also follows an importantrain accumulation, but the relation is not certain yet.The improved linear fits resulting from the appropriate detection and treatmentof outliers can be used to compute the final thermal susceptibility values for allthree sources, over the whole frequency range. The results are displayed in Fig-ure 4.26. Note that this figure shows a general increasing trend and positive values,while previous susceptibility results exhibited the opposite tendency: this is due tothe fact that Figure 4.26 was computed using inverse gain values. To first order,the slopes of the fits are therefore opposite of those found for the gains themselves.This plots contains similar features as previous susceptibility plots: 30 MHz rip-ples and a slight dependency on the considered source. It appears that the qualityof TauA datasets was greatly improved by the removal of outliers, as the TauA datanow closely matches the CasA values. This is a significant progress compared toFigure 4.9.51Figure 4.26: Susceptibilities obtained from linear fits of the mean fractionalgain variation, filtering out outliers, for the 70 CasA transits, the 34CygA transits and the 97 TauA transits. Despite being shifted withrespect to each other, the three sources produce similar results andthe behaviour of TauA is now coherent with the two other datasets.Scattering is significant and the clearest trend is visible between 600and 700 MHz.Finally, the dependency of susceptibilities on the considered portion of thedataset was tested by performing the linear fits using only half of the transits, in fourdifferent configurations. The first half and second half were used separately, thenthe even and odd days were also extracted and employed for the fits. The resultsshown in Figure 4.27, despite some differences, all exhibit the same general trendand the order of the susceptibility values is in agreement between the four panels.This is a progress compared to the results obtained before excluding outliers, whichexhibited a very strong dependency on the considered time period. This improve-ment can be explained by the fact that removing outliers related to irregular eventssuch as rain allowed to obtain a more reliable one-parameter model. An ideal one-parameter model would be completely independent on the chosen dates. In ourcase, the uncertainty introduced by considering different time periods is compara-ble to the difference between the three radiosources, and to the amplitude of the30 MHz ripples. Within these limitations, the thermal model can be considered tobe consistent with the direction-independent and dataset-independent nature of thecomplex gain.52Figure 4.27: Susceptibilities obtained from only taking into account one halfof each dataset, in different configurations. A dependency on the con-sidered dataset is still present, but a lot less significant than before ex-cluding outliers. The general increasing trend and the 30 MHz ripplesare visible in each panel.53Chapter 5DiscussionThe results presented in this paper are illustrative of the validity of a linear thermalmodel, and also serve to understand the limitations of the employed methods. RMSof residuals values obtained after excluding outliers show how the thermal modelcan improve the data, and the target accuracy for the model being reached for asignificant part of the frequency range is a very encouraging result. The final sus-ceptibility plots demonstrate that the general increasing tendency and the order ofthe values are agreeing for all sources, independently of which part of the datasetis taken into account. It should therefore be possible to fit a smooth function to thisgeneral trend, and already significantly improve the calibration of the instrument.However, uncertainties are still present in all these measurements. Considering di-verse sources or time periods results in differences that limitate the model. The 30MHz oscillations linked to the expansion of the reflector introduce ripples whichwould also decrease the accuracy of a smooth fit through the susceptibility values.Therefore, taking into account these ripples by trying to model the expansion ofthe cylinders and the resulting shift in the interference pattern is an idea for furtherinvestigation. This could be facilitated by comparing current results with CHIMEPathfinder data in order to determine if a similar oscillation with lower amplitudewas also occuring in the CHIME prototype. Other perspectives for future study in-clude conducting an analysis similar to this work using the phase of the complexgain instead of its amplitude, and performing tests of the thermal model by apply-ing it to simulated or actual sky maps and evaluating the resulting improvements.54Chapter 6ConclusionThe objective of this study was to build a thermal model that would characterizethe dependency of CHIME’s complex gain on external temperature and be used toimprove its gain calibration algorithm.The results, obtained by analyzing transits of bright radiosources (CygA, CasAand later TauA), have shown that the complex gain amplitude variations can besatisfactorily described by a first order model combined with the appropriate datapre-processing.After obtaining encouraging preliminary results despite significant scattering,different potential solutions were explored in order to reduce the impact of outliersas well as the dependency of the obtained susceptibilities on the considered sourceand dataset. The nonlinear response of LNAs was proposed as a first hypothesisand corrected for, but this modification only shifted the susceptibility values anddid not improve the quality of the fits. External factors other than temperature werethen considered; the SVD decomposition of the gain variation values demonstratedthat a one-parameter model was a valid description of most of the data, but that anisolated group of outliers probably related to other factors such as rain needed tobe explained, predicted and filtered out. Further investigation of outliers allowed todistinguish, detect and exclude two categories of anomalous transits, which greatlyimproved the quality of the linear fits and final susceptibility values and allowed toattain the target accuracy.In the light of these results, we can consider the goals of this analysis to have55been met. Future work will be needed in order to include this model in the calibra-tion algorithm and apply it to actual sky data, which will in time allow to performinnovative and possibly groundbreaking cosmological measurements, hopefully re-sulting in a better understanding of dark energy.56Bibliography[1] CHIME. https://chime-experiment.ca/. Accessed on 2019-04-19.[2] B. A. Bassett and R. Hlozek. Baryon acoustic oscillations. InP. Ruiz-Lapuente, editor, Dark Energy. Cambridge University Press, 2010.ISBN 9780521518888.[3] M. R. Blanton et al. The galaxy luminosity function and luminosity densityat redshift z=0.1. The Astrophysical Journal, 592:819–838, 2003.[4] P. J. Boyle. First detection of fast radio bursts between 400 and 800 MHz byCHIME/FRB. http://www.astronomerstelegram.org/?read=11901, August2018. Accessed on 2019-04-15.[5] CHIME Collaboration. Calibrating CHIME, a new radio interferometer toprobe dark energy. The Astrophysical Journal, submitted, 2018.[6] CHIME/FRB Collaboration. The CHIME fast radio burst project: systemoverview. The Astrophysical Journal, submitted, 2018.[7] M. Deng and D. Campbell-Wilson. The cloverleaf antenna: A compactwide-bandwidth dual-polarization feed for CHIME. 2014 16th InternationalSymposium on Antenna Technology and Applied Electromagnetics(ANTEM), Victoria, BC, pages 1–2, 2014.[8] D. J. Eisenstein et al. Detection of the baryon acoustic peak in thelarge-scale correlation function of SDSS luminous red galaxies. TheAstrophysical Journal, 633, 2005.[9] A. Gomez-Valent. Vacuum energy in quantum field theory and cosmology.PhD thesis, University of Barcelona, 2017.[10] G. Hinshaw et al. Nine-year Wilkinson Microwave Anisotropy Probe(WMAP) observations: Cosmological parameter results. The AstrophysicalJournal Supplement Series, 208, 2013.57[11] A. Maeder. An alternative to the ΛCDM model: the case of scale invariance.The Astrophysical Journal, 834, 2016.[12] C. Ng. Pulsar science with the CHIME telescope. Proceedings of IAUSymposium No. 337, submitted, 2017.[13] S. Perlmutter, M. S. Turner, and M. White. Constraining dark energy withSNe Ia and large-scale structure. Physical Review Letters, 83:670–673,1999.[14] Planck Collaboration. Planck 2018 results. VI. cosmological parameters.Astronomy & Astrophysics, 2018.[15] A. G. Riess. Observational evidence from supernovae for an acceleratinguniverse and a cosmological constant. Astronomical Journal, 116:1009–1038, 1998.[16] B. Ryden. Introduction to Cosmology. Cambridge University Press, 2edition, 2017. ISBN 9781107154834.[17] F. Sbisa`. Modified Theories of Gravity. PhD thesis, University of Milan andUniversity of Portsmouth, 2014.[18] SDSS Collaboration. The fourteenth data release of the Sloan Digital SkySurvey. The Astrophysical Journal Supplements, forthcoming.[19] J. A. Vazquez, L. E. Padilla, and T. Matos. Inflationary cosmology: fromtheory to observations. Lecture notes in cosmology, 2018.[20] D. Wang and X. Meng. Observational constraints and diagnostics fortime-dependent dark energy models. Physics Letters B, forthcoming.58

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