Instrumentation for Radio Interferometers with Antennas on a Regular Grid by Deepthi Bhavana Gorthi a Dissertation Submitted In
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Instrumentation for Radio Interferometers with Antennas on a Regular Grid by Deepthi Bhavana Gorthi A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Astrophysics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Aaron Parsons, Chair Professor Jessica Lu Professor Uros Seljak Spring 2021 Instrumentation for Radio Interferometers with Antennas on a Regular Grid Copyright 2021 by Deepthi Bhavana Gorthi 1 Abstract Instrumentation for Radio Interferometers with Antennas on a Regular Grid by Deepthi Bhavana Gorthi Doctor of Philosophy in Astrophysics University of California, Berkeley Professor Aaron Parsons, Chair In the past two decades, a rebirth of interest in low-frequency radio astronomy, for 21 cm tomography of the Epoch of Reionization, has given rise to a new class of radio interferom- eters with N 100 antennas. The availability of low-noise receivers that do not require cryogenic cooling has driven down the cost of antennas, making it affordable to build sensi- tivity with numerous small antennas rather than traditional large dish structures. However, the computational- and storage-costs of such radio arrays, determined by the (N 2) scaling of visibility products that need to be computed for calibration and imaging, becomeO propor- tional to the cost of the array itself and drive up the overall cost of the radio telescope. When antennas in the array are built on a regular grid, direct-imaging methods based on spatial Fourier transforms of the array can be exploited to avoid computing the intermediate visibility matrices that drive the unfavorable scaling. However, such methods rely on the availability of calibrated antenna voltages which are themselves difficult to obtain without using visibility matrices. In this thesis, I explore two real-time calibration strategies that can operate on subsets of visibility matrices, which can be computed without compromising on the (N log N) scaling of direct-imaging systems. O For more general radio interferometer layouts, baseline-dependent averaging with fringe stop- ping can be used to decrease the data rate of visibility products. While the computational cost is nearly unchanged, this technique can decrease the data volume of cross-correlation products, making it more tenable to store, process, and calibrate the output of the corre- lator. In this thesis, I describe the entire signal processing pipeline built for the Hydrogen Epoch of Reionization Array (HERA), which is currently being commissioned for detecting and characterizing the power spectrum of neutral hydrogen in the redshift range 5 < z < 28. The HERA correlator implements both fringe stopping and baseline dependent averaging to bring down the data rate from nearly 1 Tbps to 15 Gbps. i To my parents and brother, For tolerating my obsession with space and being my first audience. ii Contents Contents ii List of Figuresv List of Tables xiii 1 Introduction1 1.1 Epoch of Reionization..............................1 1.1.1 Ly-α forest probes............................3 1.1.2 Evidence from CMBR..........................5 1.2 Neutral Hydrogen.................................6 1.2.1 Physics of 21 cm..............................7 1.2.2 Spin Temperature.............................7 1.2.3 Global Signal Model........................... 11 1.2.4 Power Spectrum.............................. 13 1.3 Radio Interferometry............................... 16 1.3.1 Radio Antennas.............................. 17 1.3.2 Two-element Interferometer....................... 18 1.3.3 Visibility Equation............................ 19 1.3.4 Beam Chromaticity............................ 22 1.4 Measuring the 21 cm Power Spectrum...................... 24 1.4.1 Delay Transform............................. 25 1.4.2 Foreground Contamination........................ 26 1.4.3 Experiments and Current Status.................... 31 1.5 Correlators.................................... 33 1.5.1 Design Architectures........................... 34 1.5.2 Hardware Platforms........................... 35 1.6 FFT Correlators................................. 38 1.6.1 Theoretical Motivation.......................... 39 1.6.2 Calibration................................ 41 1.7 Hydrogen Epoch of Reionization Array..................... 43 1.8 Thesis Layout................................... 46 iii 2 Software FFT Correlator 48 2.1 FPGA-based Channel Selector.......................... 50 2.2 File Writing Pipeline............................... 55 2.3 Observing Runs.................................. 58 2.4 Analysis and Results............................... 62 2.4.1 Data from Green Bank.......................... 62 2.4.2 Data from South Africa......................... 66 3 Reduced Redundant-Baseline Calibration 68 3.1 Metrics for Evaluating Calibration Methods.................. 69 3.1.1 Uncertainty in Antenna Gains...................... 72 3.1.2 Scatter in Visibilities of Redundant Baselines............. 73 3.1.3 Simulation................................. 74 3.2 Low-Cadence Calibration............................. 76 3.2.1 Scaling in Gain Variance with Integration Time............ 77 3.2.2 Scaling in Gain Variance with an (N log N) Calibrator....... 77 3.3 Subset Redundant Calibration..........................O 80 3.3.1 Brief Discussion on Using Short Baselines............... 81 3.3.2 Degeneracy Criterion........................... 81 3.3.3 Scaling in Gain Variance with Number of Baselines.......... 82 3.3.4 Covariance in Estimated Gains..................... 84 3.3.5 Scaling in Gain Variance with an (N log N) Calibrator....... 87 3.4 Comparing the two Calibration Schemes....................O 89 3.4.1 Scaling in Gain Variance......................... 90 3.4.2 Variance in Calibrated Redundant Visibilities............. 91 3.4.3 Bias in Estimated Variables....................... 94 2 3.4.4 χr of Estimated Gains and Visibilities................. 96 4 HERA Correlator System 99 4.1 Architecture Overview.............................. 100 4.2 Design Specifications............................... 102 4.2.1 Frequency Resolution........................... 102 4.2.2 Integration time.............................. 103 4.2.3 Fringe Rotation.............................. 104 4.2.4 Baseline Dependent Averaging...................... 104 4.2.5 Phase Switching.............................. 105 4.3 F-engine...................................... 106 4.3.1 SNAP Boards............................... 106 4.3.2 Clocking.................................. 108 4.3.3 FPGA Design............................... 108 4.3.4 Firmware................................. 113 4.3.5 Control Software............................. 113 iv 4.4 Networking.................................... 117 4.5 X-engine...................................... 119 4.5.1 Hashpipe................................. 119 4.5.2 Cross-correlation Pipeline........................ 120 4.5.3 Data Acquisition............................. 125 4.6 Preliminary Results................................ 125 5 Conclusion 129 Bibliography 131 v List of Figures 1.1 Artist's impression of the evolution of the Universe in the standard cosmologi- cal paradigm. The first luminous structures are theoretically expected to have formed at a definite time, and for a finite period called the Epoch of Reionization. Observations of this time span could answer many questions about the evolution of our Universe from a homogeneous isotropic medium at recombination to the hierarchical structures we see today. (Credit: NAOJ)...............2 1.2 Spectra of QSOs located in the redshift range of the epoch of reionization, bor- rowed from Fan et al. (2006). The Gunn-Peterson trough, evident in the spectra at redshifts of z 6 gradually disappears and is replaced by the Ly-α forest at lower redshifts, indicating∼ the presence of large quantities of neutral hydrogen at high redshifts that disappear over time........................4 1.3 Electronic transitions in a hydrogen atom that produce the Wouthuysen-Field effect, or a change in the spin-state of the atom due to Ly-α coupling (solid lines). The dotted transitions are permissible but do not result in a change in spin state. The quantum orbital notation nFLJ is explained in the text......9 1.4 Evolution of the brightness temperature of the sky-averaged 21 cm signal, taken from Pritchard and Loeb (2012, Figure 1)...................... 11 1.5 Results from the Experiment to Detect the Global EoR Signature (EDGES; Bow- man et al. 2018) showing an absorption profile that is 50% higher in amplitude than theoretically possible. The different lines show the∼ best-fit profile obtained using different hardware configurations........................ 13 1.6 Figure illustrating the relationship between spherically averaged coordinates and 2 2 cylindrical coordinates, given by k = k? + kk. Panel (a): A simple mock power spectrum plotted against sphericallyq averaged k-modes. Panel (b): The same power spectrum profile, plotted as a color-map against cylindrically averaged coordinates....................................... 15 1.7 In a simple two element interferometer, the small difference in the arrival time of a signal between the two antennas generates interferometric fringes in the combined response......................................... 19 vi 1.8 Panel (a): Relationship between the aperture plane, the electric field beam pat- tern of the reflector, the synthesized beam (primary beam) of an interferometer (single dish), and the uv-plane of the telescope.