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Towards precision measurements of the Hubble constant with the Canadian Hydrogen Intensity Mapping Experiment Pinsonneault-Marotte, Tristan 2018

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Towards precision measurements of the Hubble constant with theCanadian Hydrogen Intensity Mapping ExperimentbyTristan Pinsonneault-MarotteB.Sc. Honours Physics, McGill University, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Astronomy)The University of British Columbia(Vancouver)April 2018c© Tristan Pinsonneault-Marotte, 2018Committee PageThe following individuals certify that they have read, and recommend to the Faculty of Graduate andPostdoctoral Studies for acceptance, the thesis entitled Towards precision measurements of theHubble constant with the Canadian Hydrogen Intensity Mapping Experiment, submitted byTristan Pinsonneault-Marotte in partial fulfillment of the requirements for the degree of Masterof Science in Physics.• Gary Hinshaw, Physics and Astronomy (Supervisor)• Kiyoshi Masui, Physics and Astronomy (Examining Committee Member)iiAbstractThe Canadian Hydrogen Intensity Mapping Experiment (CHIME) is a transit interferometer locatedat the Dominion Radio Astrophysical Observatory in Penticton, BC. It is designed to map large-scale structure in the universe by observing 21 cm emission from the hyperfine transition of neutralhydrogen between redshifts 0.8 and 2.5. CHIME will perform the largest volume survey of theuniverse yet attempted and will characterize the BAO scale and expansion history of the universewith unprecedented precision in this redshift range. CHIME achieved first light in the fall of 2017and instrument commissioning is underway. In this work I present sensitivity forecasts and deriveconstraints on cosmological parameters given CHIME’s nominal survey. The broad redshift range ofthe observations will enable tight constraints to be placed on the Hubble constant H0 , independentof CMB or local recession velocity measurements. Precision measurements of this epoch will shednew light on the tension between direct measurements of the Hubble constant vs. those inferredfrom high-redshift observations, notably the CMB anisotropy. CHIME measurements together witha prior on the baryon density from measurements of deuterium abundance are enough to placeconstraints on H0 at the 0.5% level assuming a flat ΛCDM model, with uncertainty increasing to∼ 1% if curvature is allowed to vary, or up to ∼ 3% for a dark energy equation of state with w 6=−1. Including priors from CMB measurements, in the scenario where the datasets are consistent,narrows these uncertainties further, most significantly in the model where w is a free parameter.iiiLay SummaryThe Canadian Hydrogen Intensity Mapping Experiment (CHIME) is a radio telescope located inPenticton, BC. It is designed to survey a large volume of the universe to map the distribution ofmatter at the largest scales, distances broad enough to encompass entire groups of galaxies. Bytracking the evolution of these structures over a significant fraction of the lifetime of the universe,CHIME will measure the expansion of space. The rate of expansion today has been measured usingtwo very different methods, and their results disagree. CHIME will make a novel, independentmeasurement of this same quantity, and inform whether the current tension can be attributed toerrors in the measurements or if the cosmological model needs to be modified. I have producedforecasts for the precision that CHIME will be able to achieve. I find that expected measurementerrors are small enough to allow CHIME to distinguish between the conflicting observations.ivPrefaceThis thesis is original, unpublished work by the author, Tristan Pinsonneault-Marotte, conductedas part of the CHIME collaboration, under the supervision and guidance of Gary Hinshaw. Someelements of the analysis and hardware contributions presented here were based on existing work,as cited in the text. In particular, some of the code used for the analysis was branched off existingwork from Richard Shaw, available at the neutrino-forecast repository on Github. Thecontributions I made to the FLA power control modules were building on an effort led by Kwintenvan Gassen, a student researcher also part of the CHIME team at UBC.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Forecasts for contraints on H0 with CHIME . . . . . . . . . . . . . . . . . . . . . . . . . 41 The Hubble constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Measuring H0 with BAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Baryonic acoustic oscillations . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Imprint of the BAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Measuring the BAO standard ruler with CHIME . . . . . . . . . . . . . . . . . . . 113.1 BAO in the matter power spectrum . . . . . . . . . . . . . . . . . . . . . . 113.2 Fitting the power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Fisher forecast of errors on DA and H . . . . . . . . . . . . . . . . . . . . . . . . 164.1 Fisher information matrix overview . . . . . . . . . . . . . . . . . . . . . 164.2 S/N for CHIME power spectrum measurements . . . . . . . . . . . . . . . 174.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Parameter sensitivity forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.1 Determining rs from primordial deuterium abundance . . . . . . . . . . . 215.2 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Constraining H0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4 Constraining w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25viCommissioning CHIME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 FLA power control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.1 Hardware installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2 Software development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Receiver software development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Noise integration in Pathfinder sky maps . . . . . . . . . . . . . . . . . . . . . . . 304 Telescope assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37viiList of TablesTable 1 Errors on DA/rs and Hrs from the Fisher matrix. ρH,DA is the correlation coeffi-cient of the errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Table 2 Marginalised 68% confidence intervals quoted as a relative 1σ deviation on themean for the parameter H0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23viiiList of FiguresFigure 1 The CHIME cylinders, Dominion Radio Astrophysical Observatory, Penticton,BC. (Photo taken by Andre Renard) . . . . . . . . . . . . . . . . . . . . . . . 2Figure 2 Stacks of hot and cold spots in the WMAP CMB anisotropy map. The BAOfeature as an acoustic ring located at the sound horizon rs. (Figure credit: GaryHinshaw) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 3 The matter power spectrum (top) at redshift z = 2.5, computed with CAMB[10], and the corresponding correlation function (bottom). The vertical dashedline indicates the scale of the BAO feature, which manifests itself as a bumpin the correlation function. In Fourier space, this bump produces ringing – thewiggles that modulate the smooth shape of the power spectrum. . . . . . . . . 13Figure 4 One hundred realisations of the power spectrum were computed with CAMBusing different sets of cosmological parameters taken from an Monte-CarloMarkov chain from Section 5. These are plotted in the left panel to show that thedifferent parameters produce harmonic series that are slightly offset. In the rightpanel, the same curves are shown, but with the k axis rescaled for every one bythe sound horizon calculated for the corresponding parameters, realigning thewiggles. Note that the curves are the ratio to a power spectrum computed withlow baryon density to emphasize the wiggles. . . . . . . . . . . . . . . . . . . 14Figure 5 Forecasted CHIME HI power spectrum measurement (left column) and signalto noise ratio (right column) for highest (top row) and lowest (bottom row)redshift bands used for this analysis. The dashed lines indicate the scale chosenas the cutoff beyond which non-linear evolution becomes important, based onEquation 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 6 MCMC chains as parameter constraints for CHIME+D/H with a model withfixed curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 7 MCMC chains as parameter constraints for CHIME+D/H with a model wherew 6=−1 is free. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 8 MCMC chains as parameter constraints for CHIME+CMB with a model wherew 6=−1 is free. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25ixFigure 9 68% confidence regions for w and H0 based on the forecasted CHIME sensitiv-ity. In one case a prior on Ωbh2 from deuterium abundance is included and inthe other it is priors on both Ωbh2 and Ωmh2 from measurements of the CMB. . 26Figure 10 Installed FLA bulkhead. The FLA themselves are housed in the metallic boxeswith SMA ports protruding from the white support rack. The power controlboards are the green circuit boards mounted horizontally between rows. (Phototaken by Mohamed Shaaban.) . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 11 Estimate of noise level in an average of up to 16 Pathfinder ringmaps. TheRMS is computed across pixels from a patch of sky that was chosen to be freeof artifacts, after differencing adjacent frequency bins to remove sky signal. Thecolors indicate difference frequencies, given in the legend in MHz. . . . . . . . 32Figure 12 A CHIME feed with amplifiers being installed in a cassette, showing the ‘cari-bou’ support. (Photo taken by Mark Halpern.) . . . . . . . . . . . . . . . . . . 34xAcknowledgmentsI want to thank the entire UBC CHIME team for their guidance and support. It has been a privilegeto participate in the development of one of the most exciting experiments in cosmology today,especially surrounded by an enthusiastic group of intelligent people from whom I have learned agreat deal. In particular, I’ll thank Gary and Kiyo for their advice over the course of preparing thisthesis, and my office mates Carolin, Deborah, Mateus, and Looney who have been companions andmentors in grad school.xiIntroductionWe live in an age of precision cosmology, wherein theories so broad in scope as to describe the evo-lution of spacetime itself over billions of years can rigourously be put to the test. The simplest andmost successful description of the Universe so far, the “cosmological standard model”, is known asΛCDM (for the cosmological constant Λ, a static model of dark energy, and Cold Dark Matter thatare its defining features). Although the fundamental assumptions of this model remain unchallengedby observation, the nature of its major constituents – dark energy and dark matter – are not under-stood. Particularly pressing is the case of dark energy, which is found to be at least very close to acosmological constant. However, its observed energy density is separated by orders of magnitudefrom the vacuum energy one expects from quantum field theory, which itself has achieved greatexperimental confirmation in particle physics. Even more confounding would be to find that darkenergy is not constant but evolves dynamically. Setting observational constraints on the dark energyis one of the main goals of the Canadian Hydrogen Intensity Mapping Experiment (CHIME), a radiointerferometer that will perform the largest volume survey yet attempted of the large scale structureof matter in the universe by measuring 21 cm emission over redshifts z= 0.8 to 2.5 (800-400 MHz),at a time when the universe was transitioning into the current period of accelerated expansion drivenby dark energy. CHIME is currently in the final stages of commissioning and achieved first light atthe end of summer 2017.The telescope is a compact transit interferometer composed of four cylindrical reflectors (seeFigure 1) with dimensions 100×20 m, each instrumented with 256 dual-polarization antennae. Thestructure has no moving parts. Its cylindrical geometry only focuses incoming light in one direction(east-west), creating a field of view shaped like a narrow band spanning from one horizon to theother. This band scans the sky as it drifts overhead due to the rotation of the Earth, exposing theinstrument to the entire northern sky every day.The 2048 inputs are digitized and correlated in an “FX” scheme before being saved to disk. Theanalog signal is alias sampled and channelized into 1024 frequency bins between 400-800 MHz (the“F” part), after which every pair of inputs is multiplied and integrated (“X”), typically into a timebin of 10 s. The correlated inputs are referred to as visibilities, and the full set forms a 2048×2048hermitian matrix, for every frequency, for every time frame. This produces a raw data rate ofaround 135 TB per day. Although some of this information is redundant (which may be useful forcalibration purposes), the large data volume allows for much greater mapping speeds than can beachieved with single-dish or few-baseline telescopes. The interferometric phases can be adjusted in1Figure 1: The CHIME cylinders, Dominion Radio Astrophysical Observatory, Penticton, BC.(Photo taken by Andre Renard)the analysis stage to resolve angular structure within the broad primary beam, up to the diffractionlimit of the longest baseline, effectively pointing the telescope in any direction after the fact.The large scale structure in the distribution of matter in the universe is dominated by dark matter,which is five times more abundant than the matter we can observe directly. Since the dark mattercan only be detected via its gravitational effects, observations of other astronomical phenomenaare used as tracers of its density. Previous surveys of large scale structure have used individualgalaxies to map the underlying web of dark matter, but this is a slow process since spectra must beacquired on every one to determine the redshift. Additionally, the high angular resolution necessaryto observe single galaxies is not a requirement for mapping the large scale structure, the features ofwhich are at much greater angular scales. The signal CHIME aims to measure is 21 cm emissionfrom neutral hydrogen as a tracer of matter density, at the comparatively low resolution afforded byradio observations. Thus CHIME will not resolve individual galaxies, but measure an aggregate ofall the 21 cm radiation within its large field of view, a technique refferred to as intensity mapping.This allows for a very fast mapping speed, as the entire sky can be imaged every night, rather thanhaving to point at individual locations. The effectiveness of neutral hydrogen as a tracer of largescale structure has recently been confirmed by correlating intensity mapping observations from theGreen Bank Telescope with galaxy surveys over a patch of sky at redshift z = 0.8, as reported byMasui et al. [11]. The two techniques are found to agree, bolstering the case for intensity mappingas a new way forward in the field.Intensity mapping produces a 3-d map of the density of neutral hydrogen, with the third dimen-sion given by frequency, or redshift. As light travels to us from a distant source, the expansion of2the universe dilutes its energy, effectively stretching the wavelength as a Doppler shift. The amountby which the observed 21 cm emission has been redshifted is a measure of the time and distanceto the source. CHIME’s frequency band of 400-800 MHz corresponds to redshifts of 2.5-0.8 onthe rest frequency of 1420 MHz. This spans a period between ∼ 2− 6 billion years after the BigBang. Such a large survey volume, covering a significant fraction of the lifetime of the universe,will allow CHIME to track the evolution of the large scale structure at high signal to noise. Precisemeasurements of the expansion history at intermediate redshifts have the potential to shed light ondark energy and the accelerated expansion of the universe. The primary goal of the experiment isto set experimental constraints on the dark energy equation of state, but this is by no means theonly cosmological question that can be investigated using the rich dataset. Of particular interestare the constraints these observations may place on the Hubble constant, given the apparent tensionbetween measurements at high and low redshift. CHIME will also make observations outside ofcosmology, notably for pulsar timing and Fast Radio Bursts.In this work I will present forecasts for constraints on the Hubble constant H0 based on CHIMEsensitivity estimates. I will also describe work I performed over the course of the master’s towardsthe commissioning of the instrument.3Forecasts for contraints on H0 withCHIME1 The Hubble constantThe Hubble constant, H0, is a measurement of the expansion rate of the universe today. It is namedafter Edwin Hubble, who first observed that galaxies are systematically moving away from us ata velocity that is proportional to their distance, suggesting that the space between them is itselfexpanding, and kickstarting the field of observational cosmology. This was in 1929 [8], and sincethen cosmology has undergone rapid development, with significant discoveries in theory and ob-servation building up our current understanding of the universe on the largest scales. Originatingfrom a Big Bang singularity, the universe plausibly underwent a period of rapid inflation followedby a more gradual cooling leading up to the formation of the structures we observe today: stars,galaxies, and on. Spacetime has a globally non-euclidean geometry and evolves dynamically ininteraction with the energy content of the universe, as described by general relativity. Observationshave found precise agreement with the theory for a universe made up mostly of cold dark matter(CDM) and a cosmological-constant-like dark energy (Λ), with matter accounting for only ∼ 5%of the energy density today. This model is referred to as ΛCDM. The “dark” constituents of theuniverse are so named because apart from their gravitational effects they have never been directlyobserved, and currently have no counterparts in the standard model of particle physics which hasbeen successful in describing all other domains within the reach of observation (up to a handful ofnotable exceptions).Even putting aside the puzzle surrounding the nature of the dominant constituents of the uni-verse, increasingly precise observations have begun to at least suggest inconsistencies in the stan-dard ΛCDM model. One of these tensions arises from disagreement between independent measure-ments of H0. There have been two prominent approaches to measuring the Hubble constant. Thefirst attempts to measure the expansion rate directly by observing a large number of nearby galaxiesand fitting the relationship between their recession velocities and distances. They are carried awayby the expansion of space according to Hubble’s law: v=H0d. This method relies on supernovae asstandard candles to estimate the distance to individual galaxies, calibrated using a series of increas-ingly remote sources for which the luminosity can be estimated, known as the “distance ladder”. It4must also separate or suppress the component of the measured velocity that is due to local orbitalmotions of the galaxies, and not cosmological expansion. Recent results from Riess et al. [14] us-ing this technique give H0 = 73.48± 1.66km/s/Mpc (the strange choice of units is adapted to thequantities found in Hubble’s law).A second approach is to infer the value of the Hubble constant from measurements of the cos-mic microwave background (CMB), at the limit of the observable universe, its light having beenproduced only a few hundred thousand years after the Big bang. The patterns of anisotropy inthe observed CMB encode information about the relative abundances of the different energy con-stituents of the universe and the initial conditions of fluctuations in their otherwise homogeneousdistribution throughout space. Excellent agreement with the data is found for a minimal six param-eter ΛCDM model assuming a globally flat geometry. The value of the Hubble constant inferred inthis way is H0 = 69.7±2.4 km/s/Mpc for WMAP [7], and H0 = 67.51±0.64 km/s/Mpc for Planck[13]. Note that the difference between these two can be attributed to the Planck data extending tohigher angular resolution. If H0 is evaluated using only the low resolution data common to bothexperiments, they are in good agreement. So the high-resolution end of the Planck data pulls H0lower. The CMB value is in tension with the local distance ladder result quoted above – Riesset al. [14] note a difference of 3.7σ with the Planck result. However, the value of H0 thus derivedis largely degenerate with curvature if the latter is allowed to be non-zero, which is why priorsfrom local distance ladder measurements are necessary to obtain constraints on the energy densityin curvature. Together, these datasets are consistent with a flat universe, with Ωk = 0.04± 0.040[13]. Allowing dark energy to deviate from the cosmological constant form also has the potentialto alter the expansion history and modify the inferred value of H0. So the apparent tension betweenCMB-only and local measurements of H0 is encountered for a minimal model of the universe –flat with a cosmological constant – and adding additional freedom by allowing for non-negligiblecurvature or evolving dark energy may eventually be necessary. Alternatively, systematic errorsin either measurement could be responsible for the disagreement. In either case, to make furtherprogress precision measurements at an epoch much more recent than the CMB, when curvature anddark energy are both potentially important, will be useful in resolving the question.Surveys of large scale structure at high-redshift searching for the imprint of baryonic acous-tic oscillations (BAO, described in the following section) are a promising way forward. The BAOsignature is present at all epochs since last scattering, opening the door to observations at periodsbetween the production of the CMB and today, spanning essentially the full age of the universe. Al-ready, the BAO signal has been detected by galaxy surveys at low redshifts (z. 1) – most notably bythe Sloan Digital Sky Survey [2] – and this data has been used to tighten constraints on cosmologicalparameters from the CMB. See Addison et al. [1] for a thorough review of these efforts. The lackof spectral lines and slow mapping speeds prevent galaxy surveys from pushing to further redshift,but novel experiments like CHIME are aiming to sidestep this limitation with the use of hydrogenintensity mapping. CHIME will be sensitive to a redshift range of 0.8− 2.5, corresponding to aperiod ∼ 2−6 billion years after the Big Bang. This is when the ΛCDM universe is expected to be5on the verge of transitioning from an energy distribution dominated by matter to one dominated bydark energy. If curvature is non-zero, this may also be a window in time when it could have brieflymade a significant and detectable contribution to the energy density.2 Measuring H0 with BAO2.1 Baryonic acoustic oscillationsExtrapolating the presently observed expansion of the universe back in time suggests it was oncein a hot and dense state. At some point, the temperature of radiation would have been sufficientto ionize hydrogen, coupling photons and matter through Compton scattering. This represents aboundary beyond which the universe becomes opaque to the propagation of light. The discovery andsubsequent observations of the cosmic microwave background (CMB) have confirmed this pictureand characterized it to exquisite precision. The CMB is an image of the universe at this instant whenit cooled sufficiently to allow hydrogen to form again and photons to travel unimpeded and reachus today. Although the temperature of the CMB is remarkably uniform over the entire sky (to apart in 104), the small anisotropies that are observed encode a wealth of information about the earlyuniverse. Acoustic Oscillations in the CMBAverage hot spot Average cold spotThe most distinctive and important signature in the CMB is the acoustic sound front that surrounds every hot & cold spot in the map.Figure 2: Stacks of hot and cold spots in the WMAP CMB anisotropy m p. The BAO featureas an acoustic ring located at the sound horizon rs. (Figure credit: Gary Hinshaw)Temperature fluctuations are related to fluctuations in the matter density through the tight photon-baryon coupling (where baryon is taken to mean all non-dark matter, as is conventional). In this6regime, the photon-baryon plasma can support density waves (sound waves), with the restoringforce provided by radiation pressure. These are the baryonic acoustic oscillations (BAO). Initialfluctuations in the matter density produce impulses that then propagate away at the sound speed ofthe plasma, cs. When radiation decouples from matter, the restoring force vanishes and the wave-front gradually comes to a halt. The distance the wave travelled in the time before this decouplingis the sound horizon at recombinationrs =∫ t∗0dtcsa(t)=∫ ∞z∗dzcsH(z), (1)where the factors a(t) and H(z) account for the expansion over this period, as detailed in the fol-lowing section. This scale is detected statistically in the anisotropies of the CMB, as illustrated inFigure 2. It is also present in the matter density fluctuations that eventually grew into the large scalestructure of the late-stage universe. The characteristic scale of this feature is the same as given inEquation 1, apart from a relatively small correction to the bounds of the integral to account for theinertia of the density front that cause it to travel further after radiation decouples and the CMB isproduced. This residual velocity is wiped out by gravitational drag, so the corrected scale is oftenreferred to as the sound horizon at the end of the drag epoch. In what follows, I will use rs and“sound horizon” to refer to the latter quantity.2.2 Imprint of the BAOThe BAO left an imprint in the density field of matter at a characteristic scale corresponding tothe radius they travelled before being frozen in place – the sound horizon at last scattering, rs. Asthe universe continues to cool, the initial matter over and under-densities grow due to gravitationalcollapse to form stars, galaxies, and clusters – altering the initial distribution at increasingly largescales over time. The BAO scale is a sufficiently large distance to have survived as a coherentfeature up to this point in cosmic history, largely unaffected by local growth of structure. Thepassive evolution of the BAO make it an ideal “standard ruler” to track the global expansion ofspace over cosmological timescales.The BAO standard ruler is present along all three spatial dimensions, with no preference givento any choice of coordinates, but observationally, it can be detected in two orthogonal directions:in the angular distribution of matter on the celestial sphere (i.e. a spherical slice with the Earth atthe centre), or radially, as density fluctuations along the line of sight, at depths determined by theredshifts of the measured signal. These directions will be referred to as perpendicular and parallel tothe line of sight. They constrain two physical parameters of the cosmological model1 independently.The first is angular diameter distance, DA, defined as the distance separating an observer froman object of known diameter l that they would infer given its apparent angular size ∆θ . In euclidean1The overview of fundamental cosmological concepts that will follow is based on similar discussions found in thetextbooks by Dodelson [5] and Weinberg [21].7space, the small angle approximation takes the familiar form∆θ =lDA. (2)In an expanding or curved space, this distance is in general different, since the trajectory of lighttravelling from the object is not a straight line. The specific geometry of the space is described by ametric, a prescription for measuring distances given some choice of coordinate basis, expressed as aset of coefficients gµν : ds2 = gµν dxµ dxν . A foundational principle in cosmology is the assumptionof homogeneity and isotropy – that there are no preferred locations or directions in the universe.This requirement greatly reduces the number of allowed metrics to describe spacetime, the mostgeneral expression being the Friedmann-Robertson-Walker (FRW) metricds2 =−dt2+a2(t)[ dr21−Kr2 + r2 dΩ2], (3)where a(t) is a scale factor, parametrizing the expansion of space over time, and K is the curvature,which can be negative, zero, or positive. For the special case of a static (a ≡ 1) and flat (K ≡ 0)space, this expression reduces to the Minkowski metric. It is a convention to define the scale factortoday to be unitya(t = t0)≡ 1 , (4)in which case K > 0 can be interpreted as the radius of curvature of the universe today.In coordinate space, (t,r,θ ,φ), light travels in straight lines, so it is straightforward to writedown the angular diameter in the small angle approximation∆θ =l/a(t)r, (5)where the physical extent l has been expressed in coordinate space by removing the effect of thescale factor a(t) at the time light left the object, and r is the coordinate distance to the observer. Thisdefines angular diameter distance (for an object of known physical diameter) asDA =l∆θ= a(t)r . (6)Information about the expansion history of the universe is thus encoded in DA, through its depen-dence on the scale factor.Coordinate distance r may not appear to be a physically meaningful quantity, but it is simplyrelated to another useful distance measure in cosmology: comoving distance Dc. This is defined as8the distance travelled by light in time t with the effect of the scale factor removedDc(t) =∫ t0dt ′a(t ′)=∫ r0dr√1−Kr2 =arcsin(r), K > 0r, K = 0arcsinh(r), K < 0(7)using the fact that the spacetime interval for light vanishes to turn the time integral into an integralover distance. So in flat space we haveDA = Dca(t) =Dc1+ z, (8)where I’ve taken the opportunity to define the redshift as 1+ z = a−1.So far, I’ve defined distances in a geometrical setting, but haven’t considered how they changewith the evolving universe. The dynamics of the standard cosmological model, in particular thehistory of a(t), are dictated by the Einstein equations of general relativity, which take a simplifiedform for the FRW metric. One of them relates the evolution of the scale factor to the (uniform)energy density ρa˙2+K =8piGρa23. (9)This equation defines a critical density for which the curvature vanishes todayK = 0∣∣∣t=t0⇔ ρcr = 3(a˙/a)28piG∣∣∣∣∣t=t0=3H208piG, (10)where H0 is the Hubble parameter, H = a˙/a, evaluated today, at t = t0. Using this definition, separatethe different contributions to the energy density in terms of their ratio to the critical density todayρ = ρcr(ΩΛ+Ωma−3+Ωra−4) ; Ωi =ρiρcr∣∣∣t=t0. (11)ΩΛ is the energy density of the cosmological constant Λ which by definition doesn’t evolve overtime, Ωm is the density of matter, diluting as a−3, and Ωr is radiation, evolving like a−4. One wayto understand the power of −4 diluting the radiation density is to think of photons with numberdensity decreasing as a−3, but also redshifting as their wavelength is stretched by the expansion ofspace, thus reducing their energy by an additional factor of a.With these definitions, Equation 9 becomes(HH0)2=ΩΛ+Ωma−3+Ωra−4− KH20a−2 , (12)evaluated at t = t0,1 =ΩΛ+Ωm+Ωr− KH20, (13)9The deviation from flatness can be interpreted as an energy density in curvatureΩk =− KH20= 1− (ΩΛ+Ωm+Ωr) , (14)which leaves a final expression for the Friedmann equation governing the evolution of the scalefactor with time (HH0)2=ΩΛ+Ωka−2+Ωma−3+Ωra−4 . (15)As a differential form, this equation isdt =daaH0√ΩΛ+Ωka−2+Ωma−3+Ωra−4=−adzH0√ΩΛ+Ωk(1+ z)2+Ωm(1+ z)3+Ωr(1+ z)4.(16)With this form, the integral for the comoving distance to a given redshift can be evaluated, such thatthe angular diameter distance isDA =1H0(1+ z)∫ z0dz′√ΩΛ+Ωk(1+ z′)2+Ωm(1+ z′)3+Ωr(1+ z′)4. (17)This expression depends on the parameters of the cosmological model H0 and {Ωi}. Observationalconstraints on these can therefore be obtained from measurements of the apparent angular diameter∆θ and redshift of an object of known size, the BAO feature in this case.The scale of the BAO tracks the expansion of space, so its physical size is changing over time,going like a(t)rs, where rs is it size in comoving coordinates (measured today, at a = 1). The soundhorizon is not measured by surveys of large scale structure such as CHIME, it must be taken as anadditional parameter or prior information. So the angular measurement on its own is a combinedmeasurement of DA and rs, for a given redshiftDA(z)rs=1(1+ z)∆θ(z). (18)Equations 17 and 18 are the model and data for measurements of the BAO scale in the transversedirection.In the direction parallel to the line of sight, the coordinate axis of the observations is redshift (orfrequency), so physical distances must be mapped into separations in redshift. Imagine two photonsemitted a different distance from the observer along the line of sight at times t1 and t2 such that theyarrive simultaneously. One will have travelled further than the other by the time it is detected, soit will have been redshifted by an additional factor that encodes the separation between their pointsof origin. Since the comoving distance is defined as the travel distance of light, this separationcan be written in comoving coordinates as a difference of the distances to each source, leading to10an expression for the separation in redshift. Let’s take the distance between them to be the soundhorizonrs = r2− r1 =∫ t2t1dta(t)=∫ z+∆zzdz′H(z′)≈ ∆zH(z), ∆z 1 . (19)In the parallel direction, the BAO feature measured as a separation in redshift is thus a measurementof H(z)rs. The model for the Hubble parameter as a function of z is given by Equation 15 asH(z)rs = H0rs√ΩΛ+Ωka−2+Ωma−3+Ωra−4 . (20)These two quantities DA(z)/rs and H(z)rs are the cosmological observables that can be probedby the BAO standard ruler observed in the transverse and parallel directions. Their evolution overredshift constrain the parameters {H0,rs,Ωm,ΩΛ,Ωr} (Ωk is not independent).3 Measuring the BAO standard ruler with CHIME3.1 BAO in the matter power spectrumCHIME aims to measure the imprint of BAO on the large scale distribution of matter by surveying asignificant fraction of the volume of the observable universe. The result of this survey will be a three-dimensional map of the density in neutral hydrogen within this volume. It is the statistics of thisdistribution that are of interest for cosmology, not the localization of any specific hydrogen cloud.In the standard model of cosmology, over and under-densities of matter are seeded by quantumfluctuations in the very early universe that result in a gaussian random field for the matter densityρ(x) over space. Fractional deviations from the mean density ρ¯δm =ρ− ρ¯ρ¯, (21)are thus a zero-mean gaussian random field, which can be characterized entirely by an isotropictwo-point correlation functionξ (x,x′) = ξ (∣∣x−x′∣∣) = 〈δm(x)δm(x′)〉 , (22)where the angle brackets denote an ensemble average over all of space. As time progresses, theseover-densities grow under the influence of gravity, the dynamics of which can be described byperturbation theory to linear order as long as the over-densities remain small. Assessing the timeand scale at which non-linearities become important will be addressed later in this work. As long asperturbations remain linear, Fourier modes evolve independently from one another, so it is commonto work in Fourier space, with wavenumber k conjugate to position x. The Fourier transformeddensity perturbations δm(k) are described by the power spectrum P(k)〈δm(k)δm(k′)∗〉≡ (2pi)3P(k)δ 3(k−k′), (23)11whereP(k) =∫d3x e ix·kξ (x,0) (24)is the Fourier transform of the correlation function. Note that P(k) = P(k) since the correlationfunction is isotropic.The most important data product CHIME will generate for cosmology is a measurement of thepower spectrum of matter density perturbations P(k⊥,k‖) (based on the neutral hydrogen tracer).The 3-d vector k in Fourier space is divided into components k⊥ and k‖ in the transverse and radialdirections relative to the line of sight. We make this distinction to emphasize the nature of themeasurements along those two directions. The power spectrum for k⊥ is derived from angularseparations on the sky, averaged azimuthally, whereas for k‖ it corresponds to separations along theredshift (or frequency) direction. See Shaw et al. [19] for details on the planned power spectrumestimation program. The BAO feature as a series of harmonic peaks and troughs modulating thesmooth broadband shape of the power spectrum – harmonic ringing associated with the peak in thecorrelation function at the primordial sound horizon. These “wiggles” are the Fourier-space analogto the BAO standard ruler, and can be used to infer cosmological distances as a function of redshift,and eventually constrain the Hubble constant H0 and energy densities Ωi. See Figure 3 in whicha power spectrum/correlation function pair are plotted to demonstrate the correspondence betweenthe peak at the BAO scale and the wiggles in the power spectrum.Recall, in the transverse direction, the characteristic scale is encoded in the angular diameterdistance to the sound horizon rs, measured as an angular size ∆θDA(z)/rs =crsH0(1+ z)∫ z0dz′√1+Ωm[(1+ z′)3−1]=1(1+ z)∆θ, (25)and in the radial direction it is measured as a size in redshift ∆zH(z)rs = H0rs√1+Ωm[(1+ z)3−1]≈ ∆z, ∆z 1 , (26)(where a FRW geometry is assumed and energy density in radiation or curvature neglected forbrevity).The overall shape and amplitude of the observed power spectrum will differ from the underlyingtrue spectrum due to effects such as redshift space distorsions, non-linear growth of structure, andbias from the 21 cm emission tracer, but the scale of the BAO feature is expected to be unaffected.This can be appreciated most clearly in real space, where it is difficult to imagine an astrophysicaleffect that could shift the centroid of the peak in the correlation function systematically over theentire survey volume. A careful treatment of this question can be found in Eisenstein et al. [6].3.2 Fitting the power spectrumGiven an observation of the power spectrum Pobs(k⊥,k‖;z) for a number of redshift bins, I’d liketo derive DA(z)/rs and H(z)rs, from which I can constrain cosmological parameters as outlined in120.0 0.1 0.2 0.3 0.4 0.5k (Mpc−1)P(k)0 50 100 150 200s (Mpc)s2ξ(s)Figure 3: The matter power spectrum (top) at redshift z = 2.5, computed with CAMB [10],and the corresponding correlation function (bottom). The vertical dashed line indicates thescale of the BAO feature, which manifests itself as a bump in the correlation function. InFourier space, this bump produces ringing – the wiggles that modulate the smooth shapeof the power spectrum.Section 2. The harmonic series of peaks and troughs that modulate the power spectrum correspondto the Fourier transform of an isolated feature in the correlation function localized at rs. The oscil-lations in the power spectrum will therefore have a phase evolving like krs, i.e. the wavelength ofthe wiggles encodes the BAO scale. Varying the parameter rs from its expected value will vary theseparation in the BAO crests. This effect is illustrated in Figure 4, where many realisations of thepower spectrum were computed using CAMB [10] for slightly different values of the cosmologicalparameters. Each spectrum has a slightly offset series of wiggles, but when the k axis is rescaled bythe value of rs calculated for the corresponding parameters, the wiggles line up again.The quantities DA(z) and H(z) are degenerate with rs at any given redshift, for power spectrameasured along the transverse and parallel directions respectively. In this way, the history of ex-pansion can be compared to the theoretical expectation by the relative rescaling of the P(k) wigglesas a function of redshift. Deviations from the fiducial model can be measured relative to D˜A(z) andH˜(z), where the tilde denote the fiducial values of the parameters we would like to fit for. Definethe rescaled k˜ for deviations from the fiducial parameters ask˜⊥ =DAD˜Ak⊥; k˜‖ =H˜Hk‖ . (27)130.00 0.05 0.10 0.15 0.20k0.50.60.70.80.91.01.10.00 0.05 0.10 0.15 0.20k rs/(rs)00.50.60.70.80.91.01.1Figure 4: One hundred realisations of the power spectrum were computed with CAMB usingdifferent sets of cosmological parameters taken from an Monte-Carlo Markov chain fromSection 5. These are plotted in the left panel to show that the different parameters produceharmonic series that are slightly offset. In the right panel, the same curves are shown, butwith the k axis rescaled for every one by the sound horizon calculated for the correspondingparameters, realigning the wiggles. Note that the curves are the ratio to a power spectrumcomputed with low baryon density to emphasize the wiggles.Since the shape of the power spectrum is well-constrained by theory and measurements of the CMB,it can be fit to the observations to extract these scaling factors at every redshift bin. The methodI will adopt is to fit a fiducial power spectrum Pf id [k˜⊥(DA), k˜‖(H);z] computed from the currentstandard cosmological model (using the numerical solver CAMB [10]) that depends on DA and Has parameters (through the k˜).It is desirable to extract these quantities from the data in a way that is model-independent, toproduce a simple processed dataset that can later be compared against model predictions. Modelindependence should be a significant advantage of working with a standard ruler, so it may appearsurprising to employ a fiducial model and carry out the fitting process in Fourier space, for whicha cosmology needs to be assumed. Localizing the peak in the correlation function might be a morestraightforward approach to measuring the BAO, since the characteristic scale is encoded there insuch an obvious way. However, even this approach requires some model-specific assumptions. Tofind the peak in the correlation function along the redshift direction, it is necessary to average overa large enough redshift range to resolve the BAO scale, and achieve high signal to noise. But thelocation of the peak is changing with redshift – the very effect we are attempting to measure – whichwill cause smearing of the peak and degrade the measurement. Accounting for this effect is done14by transforming the data from redshift into “real” space (i.e. in coordinates x), an operation thatrequires a cosmology to be assumed. See Shaw et al. [19] for details.In the end, the same limitation affects both methods, but working in Fourier space has otheradvantages. The effects of the evolution of structure over time are easier to separate from the signalin k-space, most importantly the onset of non-linearity, which first occurs on small scales (high k)and propagates to the larger scales over time. Since the BAO signal is damped with increasing k(for scales smaller than the diffusion length over the time until last scattering, thermal motions washout the acoustic oscillation – Silk [20]), information at the tail-end of the spectrum will quickly beerased by non-linear evolution. To avoid contaminating the measurement with these polluted data,it is straightforward in Fourier space to restrict the range of k to values well-below the thresholdof non-linearity. Procedures to undo some of this non-linear evolution and recover the BAO up tohigher k have been developed, but for the purpose of forecasting, I will take a conservative approachoverall and retain the simple hard cut. Where to set the upper limit on k will vary: lower redshiftswill need a tighter cut-off value.To determine kmax, I will adopt the criteria specified in Seo and Eisenstein [15], that the RMSof density perturbations averaged over a ball of diameter equal to the corresponding cutoff scale beone-half:σR(z) = 0.5; R =pi2kmax, (28)defined asσ2R(z)≡〈(34piR3∫|x|<Rd3x δm(x,z))2〉∝∫ ∞0dk k2P(k) | f (kR)|2 , (29)where f (kR) is the Fourier conjugate to a 3-d tophat of radius R (see Weinberg 8.1.45). TakingR→ ∞, f becomes a δ -function and σ∞ = 0, since over a large enough volume density fluctuationsaverage out. If σR approaches unity, then there exist over or under-densities on the scale of R forwhich δm = ∆ρ/ρ¯ ∼ 1, and linear perturbation theory breaks down. As matter collapses, theseover and under-densities grow, such that this cutoff scale becomes larger. σR therefore tracks theevolution of the non-linear scale for density fluctuations.Another consideration in fitting the power spectrum is that its broadband shape may deviatefrom the fiducial model. The measured power spectrum is of neutral hydrogen, which is expectedto be a biased tracer of the overall matter density. The harmonics of the BAO should be robust toany bias – it is difficult to imagine any physical process that would systematically shift its location– but other features of the spectrum may look significantly different. An accurate measurementof the BAO should therefore be insensitive to anything but the wiggles. This can be achievedby parametrizing the broadband shape of the model power spectrum to isolate the information itcarries. These nuisance parameters can then be marginalized over to remove their influence on themeasurement. Seo et al. [17] explored a number of polynomial schemes for fitting the broadbandshape and found that a fourth-order additive polynomial and a second-order multiplicative term aresufficient to remove the shape without introducing an excess of nuisance terms to the fit. The model15power spectrum that will be employed isPmod(k‖,k⊥) = Pmod(k) = B(k)Pf id(k˜)+A(k) (30)A(k) = a0+a1k+a2k2+a3k3+a4k4 (31)B(k) = b0+b1k+b2k2 , (32)where Pf id is the power spectrum calculated for a fiducial cosmology and used as a template to fitthe BAO feature. The power spectrum is modelled as isotropic, with no individual dependence onthe components k‖,k⊥, since no anisotropy is expected in the data that cannot be accounted for inthe analysis. The dependence on DA and H is encoded in the k˜ as indicated in Equation 27.4 Fisher forecast of errors on DA and H4.1 Fisher information matrix overviewA useful tool for forecasting errors on an upcoming measurement is the Fisher Information Matrix(FIM). For observations Pobs(k) over some domain in k and a model Pmod(k;αi) with parametersαi, the log-likelihood of the model given the data islnL (α) =−12∑i j(Pobs(ki)−Pmod(ki))Σ−1i j (Pobs(k j)−Pmod(k j)) , (33)where Σi j is the covariance between data-points i and j.The FIM is defined as the curvature of the log-likelihood in parameter-spaceFαβ =−〈∂ 2 lnL∂α∂β〉=∑i j∂Pmod(ki)∂αΣ−1i j∂Pmod(k j)∂β=(∂P∂α)TΣ−1(∂P∂β), (34)where in the last expression the subscripts have been suppressed and the sum is represented as amatrix multiplication. Note that data does not appear in this expression, only the data covariancematrix Σ carries information about the measurement procedure. If the model is linear in the param-eters – Pmod = f Tα for some basis f – the likelihood is a gaussian centred on the true values αˆ(denoted now as a vector)lnL (α) =−12(α− αˆ)T f Σ−1 f T (α− αˆ)=−12(α− αˆ)TΠ−1(α− αˆ) ,(35)and the inverse covariance of the parameters corresponds exactly to the FIM:Fαβ = (Π−1)αβ .For a model that is an arbitrary function of the parameters, P(α), the FIM corresponds to the16first term in a Taylor expansion around the true valueslnL (α)≈−12(α− αˆ)T(∂P∂α∣∣∣αˆ)TΣ−1(∂P∂α∣∣∣αˆ)(α− αˆ) . (36)It can further be shown that the inverse of the FIM sets a lower bound on the covariance of theparameters (the Cramr-Rao bound): Παβ ≥ (F−1)αβ . The FIM can thus be employed as a forecastfor the errors on the model parameters, with an estimate of the data covariance based on the designof the instrument, and assuming the parameter estimation is performed in a near-optimal way.4.2 S/N for CHIME power spectrum measurementsTo proceed with error estimation using the Fisher matrix formalism of the previous section, deriva-tives of the model power spectrum must be evaluated. Those for the polynomial coefficients arestraightforward, but the dependence on the parameters of interest DA(z) and H(z) must be arrivedat through the components of k˜∂P∂α=∂P∂ k˜∂ k˜∂α=∂P∂ k˜∂∂α√√√√(DAD˜A)2k2⊥+(H˜H)2k2‖ (37)=1DAk2⊥k∂P∂k , α = DA− 1Hk2‖k∂P∂k , α = H ,(38)where in the last step, I have evaluated the expression at the fiducial values D˜A = DA, H˜ = H.The full set of derivatives that appear in the Fisher matrix for the power spectrum model given inEquation 30 are∂P∂DA=1DAk2⊥k∂P∂k∂P∂an= kn∂P∂H=− 1Hk2‖k∂P∂k∂P∂bn= knP(k) ,with all values from the fiducial model, and n spanning the set of polynomial coefficients.The remaining component of the Fisher matrix calculation is the covariance for the observedpower spectrum Pobs(k⊥,k‖;z). It can be estimated from models of the instrumental response tothe expected signal. Seo et al. have carried out this forecasting calculation for a generic transitinterferometer hydrogen intensity survey that can be evaluated for the CHIME specifications. Signalto noise, S/N, is calculated for every discrete bin in three-dimensional k-space within which thepower spectrum is to be estimated. The volume of such an element isdk‖ d2k⊥ = 2pik⊥ dk⊥ dk‖ . (39)17For a survey volume Vsur, bandwidth ∆ f , and integration time tint , they find a signal to noise ratio ofSN=√2pik⊥ dk⊥ dk‖Vsur2(2pi)3PHI(k⊥,k‖)Wˆ 2PHI(k⊥,k‖)+[gT¯sky+T¯agT¯sig√tint∆ f]2VR+Nshot. (40)PHI(k⊥,k‖)Wˆ 2 is the power spectrum for neutral hydrogen – the signal we wish to detect – weightedby a window function to account for the instrumental response at the corresponding k⊥,k‖. Theprefactor multiplies the signal to noise by the square root of the number of modes averaged inthat k-space bin; the first term in the denominator is sample variance, the second is receiver andforeground noise (VR is the pixel volume; this expression is derived in the paper), and the last termis galactic shot noise. Since CHIME does not resolve individual galaxies, Nshot is set to zero in whatfollows. Finally, the integration time is set to 5 years, the nominal operation time for CHIME.Richard Shaw has implemented this calculation for the CHIME instrument in a public codehosted at https://github.com/jrs65/neutrino-forecast. I have branched off the code to compute thesignal to noise in the observed power spectrum and also return the fiducial power spectrum itself andits derivative. The window function Wˆ (k⊥,k‖)was defined as a sinc function in the parallel directionand a triangle function in the perpendicular direction. The sinc function is the Fourier transform ofa top-hat, which is the transfer function used along the frequency direction to divide it into bins.Instrumental response along the k⊥ direction is determined by the sensitivity to different angularscales achievable using the interferometric array. For a compact, redundant interferometer likeCHIME, this window function turns out to be a triangle, bounded by the angular scale correspondingto the longest baseline of the array. Details are provided in Seo and Hirata [16]. These choices forthe window function are a simplification. The real instrumental response will necessarily be morecomplicated, but refinement of this aspect of the analysis will be left for future work.To accurately model the signal to noise, redshift space distorsions (RSD) are included in theneutral hydrogen power spectrum PHI(k⊥,k‖), introducing anisotropy between the k⊥ and k‖ direc-tions. RSD are modelled by the widely used approximation introduced by Kaiser [9], modulatingthe power spectrum with a factor(b+µ2 f )2; µ =k‖k, (41)where f is a function that tracks the growth of perturbations over cosmic time and b = δHI/δm is abias term to account for the fact that the overdensity in neutral hydrogen may not be exactly equal tothe overdensity in matter (see Dodelson [5] chapter 9.4). For the purpose of forecasting, I set b = 1.After estimating the signal to noise, this effect is removed from the power spectrum, restoring itsisotropy so it can be used as the template Pf id(k) for the fitting function. This reflects the expectedprocedure for analysis of measured data.The (k⊥,k‖) grid for which forecast errors are generated is set by the cut-off kmax determinedat every redshift from the expected onset of non-linear evolution (as prescribed in Section 3.2), anda choice for the number of uniformly spaced bins across the range. The length of one wiggle inthe power spectrum is about 0.05 Mpc-1, so a resolution of 0.01 Mpc-1 should be amply sufficient18z kmax/h DA/rs σDA Hrs σH ρH,DA2.16 0.568 11.66 0.069 (0.59 %) 0.1095 0.00074 (0.68 %) -50.24 %1.58 0.426 11.92 0.043 (0.36 %) 0.0832 0.00037 (0.45 %) -58.38 %1.18 0.328 11.66 0.039 (0.33 %) 0.0672 0.00031 (0.46 %) -55.80 %0.89 0.255 10.96 0.059 (0.53 %) 0.0567 0.00044 (0.77 %) -41.50 %Table 1: Errors on DA/rs and Hrs from the Fisher matrix. ρH,DA is the correlation coefficientof the errors.to Nyquist sample the BAO feature. The sampling of k‖ will determine the maximum number ofredshift bins that can partition the full frequency band while still spanning large enough scales toachieve that resolution. This is analogous to the familiar case of a time-domain discrete Fouriertransform for which the frequency resolution is given by the inverse of the total time spanned bythe sample: ∆ f = 1/t. The bounds of the redshift bins and the respective values of kmax used in thiswork can be found in Table 1The signal to noise estimated in this way is displayed in Figure 5 for two redshift bins locatedin different regions of the CHIME band. In these figures, the cut above kmax has not been enforced.Comparison of the two panels shows how the sensitivity of the instrument across k⊥ changes withredshift, mainly due to the decreasing angular resolution with increasing redshift, which correspondsto increasing wavelength. Thus the high k region of the spectrum can be probed with high signalto noise at lower redshifts, which is just when it becomes increasingly polluted with non-linearities.For the high redshift bin, 64% of the signal to noise is located within the non-linear cut-off, butthis drops to only about 3.5% in the low redshift case. These competing trends are inherent to anyhydrogen intensity survey, and provide a strong motivation to investigate methods for recovering theinformation at these higher k. Some of the non-linear evolution can be modelled and corrected forusing the method of Lagrangian displacements, based on the Zel’dovich approximation, as describedin Eisenstein et al. [6]. Integrating this method into the present sensitivity forecasts will be left forfuture work.4.3 ResultsThe Fisher matrix for errors on DA(z) and H(z) can now be evaluated given the signal to noiseforecast derived in the previous section. The fiducial cosmology I use here is standard ΛCDM, withvalues for the cosmological parameters taken from the TT,TE,EE+lowP+lensing+ext likelihood fitsof Planck 2015 [13]. The power spectrum for this fiducial model is computed using the softwarepackage Cora [18]. Table 1 lists the derived errors and the fiducial values of the distance measuresfor every redshift. Note the large correlation coefficient between DA and H. This is to be expectedgiven the nature of the measurement, for which most data points represent some combination of k⊥and k‖, so the two parameters are not determined independently.190.2 0.4 0.6 0.8 1.00.20.40.60.81.0k(Mpc−1)z=2. 160.81.21.62.02.42.83.23.64.0log10P(Mpc3)0.2 0.4 0.6 0.8 1.00.20.40.60.81.0z=2. 16061218243036424854S/N0.2 0.4 0.6 0.8 1.0k (Mpc−1)0.20.40.60.81.0k(Mpc−1)z=0. 890.81.21.62.02.42.83.23.64.0log10P(Mpc3)0.2 0.4 0.6 0.8 1.0k (Mpc−1)0.20.40.60.81.0z=0. 89061218243036424854S/NFigure 5: Forecasted CHIME HI power spectrum measurement (left column) and signal tonoise ratio (right column) for highest (top row) and lowest (bottom row) redshift bandsused for this analysis. The dashed lines indicate the scale chosen as the cutoff beyondwhich non-linear evolution becomes important, based on Equation 285 Parameter sensitivity forecastsIn the previous sections, the necessary machinery has been established in order to proceed to pa-rameter sensitivity forecasts. Equations 17 and 20 provide the model predictions for the observ-ables DA(z) and H(z), with parameters {H0,rs,Ωm,Ωk,Ωr}. Note that ΩΛ is not included since itis determined by the other three (Equation 14). The energy in radiation is negligible at late times(Ωr ∼ 10−4, with at most (1+ z)4 ∼ 102) compared to the other constituents, so it will have little in-cidence on the fits. The temperature of the CMB has been measured to extremely high precision andas a result uncertainties on Ωrh2 are negligible. I will simply take them to be zero in the likelihood,fixing Ωr. H0 and rs are manifestly degenerate – some prior knowledge of rs will be necessary20to set any constraints on the Hubble constant. The most precise estimates of the sound horizonare from measurements of the CMB, which is what is typically used to break this degeneracy inBAO observations. However, in the context of apparent tension between CMB and distance laddermeasurements of H0, it will be interesting to estimate the precision of constraints from CHIMEBAO alone, independent of the CMB and in general without assuming flatness. Another handle onthe sound horizon can be obtained from measurements of the primordial deuterium abundance andmodelling of the early universe nucleosynthesis.5.1 Determining rs from primordial deuterium abundanceThe predictions from the theory of Big Bang nucleosynthesis for the abundances of light elementsin the universe is one of the pillars of modern cosmology. Starting with minimal assumptions abouta universe that started with a Big Bang – homogeneity and isotropy, that physics was described bythe same laws at early times as it is today – the principles of thermodynamics and nuclear physicsmake definite predictions for the relative abundances of helium and deuterium that condense outof the primordial plasma as it cools. Observations of the helium fraction in low metallicity starshave confirmed these predictions to remarkable accuracy [12]. The relic abundance of deuterium isdetermined by the efficiency of the reaction that converts it to helium: D + p↔ 3He + γ , which inturn depends on the physical density of baryons Ωbh2. A higher density would have allowed moredeuterium to be produced before expansion brought the reaction to a halt. This makes the deuteriumabundance relative to hydrogen, D/H, a sensitive probe of Ωbh2. Following Addison et al. [1] I willuse the value derived from observations of the absorption spectra of high-redshift quasars by Cookeet al. [4], in their most recent publication:Ωbh2 = 0.02235±0.00037 . (42)A prior onΩbh2 is sufficient to break the H0–rs degeneracy because the sound horizon of BAO isdetermined largely by Ωb and Ωm. Evaluating the integral in Equation 1 cannot be done analyticallyin general, but fitting formulae have been derived that are accurate in restricted regions of parameterspace. Aubourg et al. [3] provide a formula for the sound horizon at the drag epoch, numericallycalibrated from the linear perturbation code CAMB [10]rs ≈ 55.154exp[−72(ων +0.0006)2](Ωmh2)0.25351(Ωbh2)0.12807Mpc≈ 55.154(Ωmh2)0.25351(Ωbh2)0.12807Mpc, (43)where ων ≡ 0.0107(∑mν/1.0eV)≤ 0.006 is a correction for non-zero neutrino mass. Investigatingthe effects of neutrino masses or additional relativistic species is outside the scope of this work, so Iwill fix ων = 0. Given the limit set by Planck, ∑mν ≤ 0.6eV , this results in < 0.5% variation in thevalue of rs. Aubourg et al. [3] quote an accuracy of 0.021% compared to the numerical solutions.215.2 Parameter estimationAdopting this scaling relation for rs removes it as a free parameter from the model for DA(z) andH(z) in favour of Ωbh2. The full set of fitting parameters I will consider for the ΛCDM modelis {H0,Ωb,Ωm,Ωk}. Uniform priors on all of these will implicitly be assumed with the exceptionof the combination Ωbh2, for which the gaussian standard deviation is given in Equation 42. Thelikelihood of the data will be approximated as e−χ2 , with the errors estimated from the sensitivityforecasts of Section 4. The posterior probability density for the parameters is given by Bayes’theoremP(H0,Ωb,Ωm,Ωk) ∝ e−χ2P(Ωbh2) . (44)Evaluating the posterior is most conveniently performed with the Monte Carlo Markov Chainmethod, a widely used stochastic algorithm for sampling the probability distribution. It producesa cloud of points in parameter space with a number density proportional to the distribution. Thismethod is advantageous for exploring large-dimensional spaces since its computational expensescales like the number of points in the chain for any number of parameters, whereas evaluatingthe likelihood on an explicit grid grows to the power of the number of dimensions. Additionally,analyzing the statistics of the posterior distribution is very straightforward since the Markov chainproduces a representative sample from which it is trivial to compute the moments.5.3 Constraining H0Different models and datasets will be considered to evaluate the performance of the CHIME surveyfor estimating H0 and distinguishing between variations on the cosmological model. The onlyextension I will consider to ΛCDM is a deviation from cosmological-constant dark energy. Theevolution of the dark energy is determined by its equation of state,pde = wρde , (45)relating density and pressure, parametrized by the constant w. Λ has w = −1. For comparison,w = 0 for pressureless matter and w = 1/3 for radiation. For values of w 6= −1, dark energy nolonger has constant density throughout the expansion of space, i.e. Equation 15 must be modifiedto include some power of the scale factor multiplying ΩΛ.Constraints using priors from measurements of the CMB will be considered separately to getan idea of the precision achieved by combining these complimentary observations, assuming theyare found to be consistent. In what follows I will use ‘CHIME’ to refer to the sensitivity forecastsestimated in this work, ‘D/H’ to indicate a prior on Ωbh2 from Cooke et al. [4], ‘CMB’ to indicatepriors on (Ωbh2,Ωmh2) from the Planck 2015 parameter fits [13], and ‘w’ when deviations from thecosmological constant are allowed. The resulting marginalized 68% confidence intervals on H0 forall of these combinations are summarised in Table 2.22Table 2: Marginalised 68% confidence intervals quoted as a relative 1σ deviation on the meanfor the parameter H0.CHIME+D/H CHIME+CMB CHIME+D/H+Ωk CHIME+D/H+w CHIME+CMB+w0.44% 0.39% 1.02% 2.66% 0.83%Figure 6: MCMC chains as parameter constraints for CHIME+D/H with a model with fixedcurvature.CHIME+D/H This is the minimal ΛCDM model favoured by observations so far. Curvature isfixed at the negligible value obtained in Planck 2015, Ωk = 0.04± 0.040 [13]. A corner plot ofthe MCMC posterior distributions for the parameters is shown in Figure 6. The contraints on Ωbh2appear to reproduce the input prior. This shows that CHIME BAO measurements are not partic-ularly sensitive to the baryon density. The constraining power is confined to the H0–Ωm plane.Marginalising over Ωm results in a 0.44% standard deviation, competitive with the CMB-derivedmeasurements and many times more precise than distance ladder estimates.CHIME+D/H+Ωk Next, we can relax the assumption of an essentially flat universe to see how thisaffects the CHIME constraint. In this fit, Ωk is a free parameter, with a uniform prior. The resultingposterior on Ωk spans roughly ±0.1, well beyond any plausible curvature based on all observationsto date. Nevertheless, the CHIME sensitivities produce constraints of 1.02% on H0. This reflectsthe fact that over such a broad range of redshift, measurements of the Hubble constant from BAO23are not degenerate with curvature, reinforcing their complimentary nature to CMB observations.CHIME+D/H+w Now we go beyond ΛCDM, allowing values of w 6=−1. This adds considerablymore freedom to the model, resulting in errors of ∼ 2.7% on H0. The chains extend about ±0.2around w =−1, as can be seen in the corner plot in Figure 7.Figure 7: MCMC chains as parameter constraints for CHIME+D/H with a model where w 6=−1 is free.CHIME+CMB Observations of the CMB constrain the baryon and matter densities very precisely.In this fit, gaussian priors on Ωbh2 and Ωmh2 are included. The 68% confidence limits on theparameters quoted in Planck 2015 are used as standard deviations for these priors. This should bea good approximation since there is little covariance between these parameters in the CMB fits, ascan be seen in Figure 6 of Planck 2015 [13]. The CMB priors improve the derived constraint on H0slightly, mainly as a result of including any prior on Ωm at all, since the measurement of Ωbh2 fromCooke et al. [4] is competitive with the CMB precision.CHIME+CMB+w The inclusion of the same CMB priors in the free w fit has a much more signif-icant effect, as can be seen by comparing the parameter constraints shown in Figures 8 and 7. BAOfrom CHIME together with CMB can measure H0 to ∼ 1% accuracy even with w 6=−1.24Figure 8: MCMC chains as parameter constraints for CHIME+CMB with a model where w 6=−1 is free.5.4 Constraining wAlthough it is not the main focus of this work, measuring w with high precision is the primarymotivation for CHIME. It is therefore worth taking a moment to note the sensitivity forecasts ob-tained on w. Figure 9 shows contours for the posterior in the H0–w parameter plane. Two caseswere considered: setting a prior on the baryon density from deuterium abundance or priors on thebaryon and matter densities from CMB observations. The CMB prior is much more constrainingthan deuterium since it includes Ωmh2. This is because w determines the power of the scale factorwith which the weight of dark energy evolves when calculating the Hubble parameter, Equation 20,where the matter density appears in a similar role. The measurement of Ωmh2 from the CMB breaksthis degeneracy and allows for constraints at the 2.5% level on w, whereas CHIME+D/H produceerrors around 6.8%..250.96 0.98 1.00 1.02 1.04H0/(H0)true1.101.051.000.950.90wCHIME+D/HCHIME+CMBFigure 9: 68% confidence regions for w and H0 based on the forecasted CHIME sensitivity. Inone case a prior on Ωbh2 from deuterium abundance is included and in the other it is priorson both Ωbh2 and Ωmh2 from measurements of the CMB.26Commissioning CHIMEIn the introduction, I gave a brief description of the Canadian Hydrogen Intensity Mapping Experi-ment (CHIME), the telescope which has been at the center of my master’s work. Over the past twoyears, CHIME has been undergoing the final stages of commissioning, leading up to a ceremonialfirst light in the fall of 2017 followed shortly by the beginning of data acquisition in spring 2018.This represents the culmination of years of effort from a group of researchers scattered mostlybetween the main partner institutions at McGill, the University of Toronto, and the University ofBritish Columbia. In this chapter, I will focus on my individual (not single-handed!) contributionsto the project as a master’s student. Given the scale and complexity of the instrument, these havebeen miscellaneous and provided the opportunity to practice wielding a wide range of tools andideas. As such, I will not try and provide a narrative here, and the following contributions will bepresented in no particular order.1 FLA power controlThe last stage of the analog chain that spans from the antenna feeds to the digitizer at the input ofthe correlator is a filter amplifier (FLA). In addition to amplifying the signal to the appropriate levelfor optimal digitization, this component applies a 400-800 MHz spectral bandpass filter. Effectivefiltering is crucial to the fidelity of the digitization because the latter operates by alias sampling. Thesignal is sampled at 800 MHz – only half the Nyquist rate – which causes aliasing of the 400-800MHz band into 0-400 MHz. Since this frequency range is cleared by the FLA, there is nothing toconfuse the aliased signal, and no information is lost. The FLA also provides a DC level on thetransmission line to power the upstream amplifiers.CHIME has 2048 feeds, and thus 2048 signal chains. With such a large number of components,there are bound to be failures and these can be disruptive to the operation of the instrument. Anoscillating amplifier can produce loud radio frequency interference (RFI) that will contaminate thechannels near it for example. To mitigate this issue, it is desirable to have control over the powergoing to individual FLA, so that they can be shut down without disrupting the rest of the system.It is also useful to have the ability to query and record the FLA power state over time, to moni-tor the system and inform later data analysis. These considerations led to the development of anFLA power control module, including hardware and software components. Kwinten van Gassen, astudent researcher at UBC, led the development of these tools and I contributed to their physical in-27stallation and subsequent feature updates necessary to integrate them into the full CHIME softwareframework.1.1 Hardware installationThe receiving and frequency channelization component of the CHIME correlator is housed in twoRF shielded rooms, themselves embedded in repurposed shipping containers – one collects signalsfrom two cylinders. The coaxial cables carrying the signal interface with the RF room via a brassbulkhead through which the FLA ports protrude. The installation procedure involved drilling holesin this bulkhead to mount the FLA support rack and to allow the FLA connectors through, cleaningthe brass surface to ensure good electrical contact and prevent RFI leakage, inserting the FLA inthe bulkhead and connecting them to the power distribution system while recording their serialnumbers and location, and bolting them into place with the appropriate torque to ensure no gaps inthe conducting surface. These steps were repeated for all 2048 amplifiers.The power distribution is divided between control boards that can provide power to 16 FLA.Each output can be toggled by a photoelectric relay, and groups of 8 relays are set by sending a byteto the corresponding individually addressed controller chip over the Inter-integrated circuit (I2C)protocol. Power control boards are grouped into buses that share a data line, with 16 addresses(8 boards) on each line. Instructions are sent over the buses by an Arduino-based module. Eachmodule controls four buses and has a persistent state, i.e. it will restore the power state of the arrayof FLA if it is power-cycled. In all, there are 4 modules with 4 buses (16 branches) each with 16addresses (256 branches) that control 8 FLA (2048 branches). The modules are powered by thesame source as the FLA themselves, so that when the underlying power supply is powered on or offthe state of the FLA is restored automatically.I helped to install and cable up the modules and power supply in the receiver room. One powersupply provides constant voltage to 1024 FLA, with a total output of around 2 kW at roughly 300A. This current is carried by eight heavy-duty copper cables strung across the ceiling. The Arduinomodules interface over USB with a ‘housekeeping’ PC that runs the control software which I willdescribe in the following section. The photograph in Figure 10 shows the inner FLA bulkhead.1.2 Software developmentThe Arduino-based modules that communicate over I2C with the power control boards are managedthrough a software package written in Python. The software wraps an I2C client into abstract objectsrepresenting the bulkheads, power boards, and individual FLA locations. It was developed initiallyby Kwinten van Gassen, with a command line interface for user interaction. Later, it was decidedthat CHIME would adopt the Representational state transfer (REST) standard interface for systemmonitoring and control, so it was necessary to write a new interface to the FLA control software.REST operates over the HTTP protocol so this involved building a web server that could be run onthe housekeeping PC and listen for commands from users over HTTP. I also put together a commandline interface to the REST server for convenience.28Figure 10: Installed FLA bulkhead. The FLA themselves are housed in the metallic boxeswith SMA ports protruding from the white support rack. The power control boards arethe green circuit boards mounted horizontally between rows. (Photo taken by MohamedShaaban.)292 Receiver software developmentThe full data rate output by the CHIME correlator is almost 135 TB per day, in a stream of2048× 2048 visibility matrices for 1024 frequencies generated every 10 s. Capturing and pro-cessing this data to write it to disk for later analysis is not a trivial task, especially since it needsto be performed in real time to maintain a continuous acquisition over a period of months. Thereceiver software pipeline that handles the data flow from the correlator is divided into multiplestages, distributed between different machines but all under a common software framework built inC++ named kotekan.The ‘X’-engine of the correlator receives the stream of frequency channelized data from the‘F’-engine for every one of the 2048 inputs. It is made up of 256 nodes, each with 4 GPUs, suchthat every GPU receives the data from all inputs for a single frequency. The inputs are multipliedand integrated to produce the visibility matrix and then passed on to the kotekan pipeline. Thefirst stages of the pipeline occur on the GPU nodes themselves, but the 256 data streams need to becollected before they can be written out and archived. This happens on two dedicated receiver nodes,networked to the GPU array over 40 GbE links. The data files written out by the receiver nodesfinally need to be compressed and copied over to an archive node where they are made available foranalysis (and distributed to redundant archives).The kotekan pipeline is completely modular, composed of a string of unit tasks that performsome operation on the data. Some examples of pipeline tasks are to collect the streams from thearray of GPU nodes, accumulate high cadence data streams down to lower rates, perform real-time calibration, or select subsets of data products to save separately. The pipeline can branch andproduce multiple datasets in parallel. For the initial CHIME data acquisition period, the ultimatecompression scheme was not ready to be implemented but it is infeasible to write out the full datarate to disk (referred to as N2), so the pipeline was configured to select a subset of products andfrequencies with higher sensitivity to save in the meantime. But having access to N2 data is veryvaluable at this stage of the experiment for the purpose of characterizing the instrument, and themodularity of the pipeline allows saving a second data product composed of N2 visibilities for onlya handful of frequencies.I have been heavily involved in the development of the receiver pipeline for the period followingfirst light and leading up to the initial data acquisition run. In particular, I implemented a numberof kotekan tasks required for the first run and overhauled the existing pre-archiver compressionprogram to meet the performance requirements of the full CHIME data rate. I also helped withsoftware deployment on the receiver system.3 Noise integration in Pathfinder sky mapsThe CHIME Pathfinder is a smaller scale telescope with the same design as CHIME, also locatedat the Dominion Radio Astrophysical Observatory in Penticton, BC. It has been operating as a testbed for CHIME systems and analysis since 2015. As of the fall of 2016, initial data acquisition30and calibration efforts had progressed sufficiently to generate stable sky maps on a daily basis. Thisproject aimed to test the noise properties of these maps from day to day.The dataset that was considered spanned 16 sidereal days, with some interruptions. Each daycorresponds to a full scan of the visible sky from the latitude at Penticton, so the signal for everyone is nominally identical. The visibility timestreams generated by the telescope measure the skydrifting through the beam as the Earth rotates. The time axis of the data therefore maps to an axisover right ascension (RA). Along the declination direction, the individual visibilities have a reso-lution that corresponds to the broad primary beam, covering an area of sky spanning from horizonto horizon. The structure along this direction can be resolved interferometrically by decoding theadditional information stored in the array of visibilities. This is done by summing the visibilitiesweighted by the set of phases that exactly compensate for the delay between the signal received byeach feed in the pair from a given declination. The signal from that direction adds constructivelywhereas the other directions on average cancel out. In this way, the visibility timestreams can bephased up to be sensitive to the sky at a specific declination, mapping out a ring on the celestialsphere. Doing this for a number of declinations produces a corresponding number of these ringsthat can be stitched together to form a ‘ringmap’. These are formed on grids of pixels along theRA and declination directions, for every sidereal day. CHIME Pathfinder observes in the same bandas CHIME, with 1024 frequency bins, but for this analysis only four pairs of adjacent frequencieswere considered, chosen to span the full band and avoid its RFI contaminated regions. The pairs arelocated around (769.5, 644.1, 518.8, 443.8) MHz.The goal of this project was to average a successively larger number of daily maps togetherand try and estimate the instrumental noise persisting in this integrated map at every step. If theinstrumental noise in the maps is randomly distributed and uncorrelated from day to day, we expectfrom the central limit theorem that the RMS of this noise should drop off as the inverse of the squareroot of the number of daily maps that went into the average.To estimate the noise level in a given map, the following procedure was used:1. Select a patch of sky that excludes any bright sources and shows no visual anomalies, i.e. thatappears to be mostly background.2. Take a difference of the pairs of adjacent frequency bins to attempt to strip out the sky signaland leave only noise.3. Form all possible subsets of days and take an average of the maps in each.4. Compute the pixel-by-pixel RMS across all average maps of n days (of which there would be(Nn)if there are N days in total).5. Record the average of the pixel-by-pixel RMS across the entire map/patch as the estimatednoise level for n days.The results of the procedure outlined above are shown in Figure 11. The dashed line shows theexpected 1/√n dependence, and the RMS results for increasing numbers of days are normalized to310.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.0RMS / σ0EE769.5644.1518.8443.80.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.20.30.40.50.60.70.80.91.0ES0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0days^(-1/2)0.10.20.30.40.50.60.70.80.91.0RMS / σ0SE0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0days^(-1/2)0.30.40.50.60.70.80.91.0SSFigure 11: Estimate of noise level in an average of up to 16 Pathfinder ringmaps. The RMS iscomputed across pixels from a patch of sky that was chosen to be free of artifacts, afterdifferencing adjacent frequency bins to remove sky signal. The colors indicate differencefrequencies, given in the legend in MHz.the average of the RMS accross individual days (n = 1). Two of the curves are close to the idealcase in all panels, indicating that at those frequencies the map patches were dominated by randomnoise. The green curve (518.8 MHz) does not average down nearly as quickly. Closer inspectionof the maps at that frequency revealed that artifacts attributed to cross-talk between inputs werestill present after processing. These features are correlated from day to day and make a plausibleexplanation for the high residual noise. More surprising is the black curve (443.8 MHz), whichappears to be suppressed faster than 1/√n. One way to obtain such behaviour would be for at leastsome days in the set to have features that are anti-correlated between them. Even if this affectedonly a few days from the set, it could be enough to alter the curve significantly, and would have alarger effect on its left side, where they would be present in a greater number of the combinationsof days that are considered. Investigating the maps that were part of the set, such a feature was infact found in a pair of days. This was hypothesized to be leftover sky signal from the frequencydifferencing. It happened that the difference ended up negative in one case and positive in the other,so that individually these maps increase the noise in the average, but if they are both present it issuppressed.324 Telescope assemblyA significant portion of the work for this master’s was spent assembling various components ofthe telescope over the course of regular visits to the construction site. These contributions to theexperiment may also be among the most important, since they were direct and necessary stepstowards the physical realisation of the instrument, a prerequisite to any subsequent analysis efforts.I will describe a few of these tasks to give a taste for this aspect of the project, starting with theassembly of the analog signal chain.Feeds and focal line electronics The antennae are located along the focal lines of the cylindricalreflectors, numbering 256 per cylinder. They are attached to aluminum “cassettes” – boxes sus-pended on rails below the focal line walkway – which house the first stage amplifiers. Each of thecassettes holds fours feeds and eight amplifiers with trailing cables that join up in a connector plate.Every cassette (there are 256) was first assembled inside before being transported to the telescopefocal line. The procedure for installing a cassette involved two people standing on the walkway whowould attach ropes to its corners and swing the assembled cassette below the focal line and onto therails. One would then lay flat on the walkway to adjust the cassette’s position before securing it tothe edge of the adjacent ones.Each of the amplifiers then needed to be connected to the appropriate 50 m coaxial cable thatcarries the signal down to the receiver huts. To record all of these connections and ensure they weremade in the right order, each component has a barcode serial number that would be scanned as theywere connected.Cable pulling The 50 m coaxial cables are routed from the receiver huts to the focal line throughducts that run up along walkway support legs (these are made from the same stuff as ventilationducts). There are four ducts per cylinder, each holding 64 cables. These were pulled up in bundles ofeight, using a specially designed spool that can unwind eight cables simultaneously. The procedureinvolved two people – one would be on the walkway hauling the cables while the other stayed at thebottom pulling them off the spool and shoving them into the duct. These cables are stiff and heavy soconsiderable force was required. However, care had to be taken to unwind them smoothly withoutintroducing any kinks or torsion that would impact the performance of the coaxial transmissionline. The bundles would initially be guided up the duct using a rope that would stay within theducts throughout the process. Squeezing in the bundles without losing the rope could occasionallybe challenging. Once the cables were in, they still needed to be arranged in trays that would supportthe extra lengths of cables with shorter spans on the focal line and connected to the bulkhead thatallows the signals into the receiver huts, recording these connections along the way.Layout database All of the connections between components of the telescope at a given time arerecorded in a MySQL database. This centralized database is a very valuable tool for keeping trackof the state of a complex instrument like CHIME and greatly facilitates the interpretation of the33data it produces. To ensure the accuracy and completeness of the layout database, it had to bepopulated as the components were installed, by recording the barcode serial numbers as mentionedabove. This process was implemented in Python scripts running on a portable tablet computerthat would guide the user through the installation process for a specific subsystem and save a recordof the connections created. The connection data would then be automatically copied to a locationaccessible from UBC and added to the layout database. I helped to keep this system running andadd new features as they became necessary, and was responsible for updating the database with thenew connections.Machining parts on the water-jet cutter A number of small mechanical parts that went into theCHIME assembly were made from bent aluminum sheet metal. For example, Figure 12 shows apart that supports the amplifiers in the cassettes and provides strain relief to the output cables –nicknamed the caribou due to its shape. The water-jet cutter was the ideal tool for prototypingthese parts and producing them in the relatively small numbers needed for CHIME. I operated themachine and made hundreds of ‘caribou’ and a few other components for CHIME.Figure 12: A CHIME feed with amplifiers being installed in a cassette, showing the ‘caribou’support. (Photo taken by Mark Halpern.)Focal line weatherproofing The first stage amplifiers, antenna feeds, and multiple cable connec-tions are located on the focal line of the reflectors, beneath the walkway. Although the electroniccomponents are housed in sealed enclosures and the walkways provide some shelter from the brunt34of the elements, more stringent weatherproofing will improve the stability and lifetime of the in-strument. For instance, water accumulating near the feeds could change their dielectric properties,and regular exposure to the damp can promote corrosion. In addition, an unanticipated concernis to keep wildlife, birds specifically, from nesting near the electronics. It was realised during theoperation of the Pathfinder that these small enclosed spaces, kept warm by the powered amplifiers,make desirable shelter for starlings. Sealing up in total nearly 400 m of elevated walkway is nota small task. I helped with installing flashing, caulk, chicken wire, weather stripping, etc. over anumber of visits to site.Receiver hardware Apart from my work on the FLA installation, I also participated in puttingtogether other subsystems of the receiver and correlator. These tasks usually involved elaboratecable management. For instance, the ‘X’-engine and ‘F’-engine of the correlator are located indifferent RF-shielded huts, roughly 100 m apart, and the signals are transmitted between them overfibre optics. There is one dedicated fibre link for every frequency bin output by the channelizer,providing data to every one of the 1024 GPUs. There are thus 1024 fibres running between the huts.I helped with routing them on the GPU side, bringing the cables to the correct racks where the GPUnodes are mounted.35ConclusionThe past two decades have seen experimental cosmology develop at a rapid pace, with increasinglyprecise observations converging with remarkable consistency towards a rather simple model for theuniverse on the largest scales. However, significant questions remain unanswered – it appears the‘dark sector’ is the dominant constituent of the universe – and tensions in the model parameters maybe beginning to appear. Most notably, measurements of H0 from the CMB and those from directobservation of recession velocities have developed a disagreement at the 3.7σ level. These questionscall for further observations – especially in areas that have so far gone unexplored – that will provideeither evidence for new physics or validation of the currently accepted theories. CHIME is poisedto help shed light on at least two of these problems by measuring the BAO in a large volume atintermediate redshifts.In this work, I have obtained sensitivity forecasts for the CHIME survey and propagated theresults to produce expected constraints on the Hubble constant H0 and dark energy equation of stateparameter w. Independently of the CMB, CHIME together with measurements of the baryon den-sity from deuterium abundance can constrain H0 at ∼ 0.5% precision. This is comparable to theprecision attained from the CMB and much tighter than the errors on the distance ladder measure-ments. CHIME thus has the potential to distinguish between these two measurements and informthe interpretation given to the existing tension. In the event that CHIME and CMB find consistentvalues for H0, combining them produces a joint constraint that is a mild improvement over CHIMEalone. Constraints on w in contrast are greatly improved by combining CHIME with CMB obser-vations. 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