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|