Measuring Hierarchical Structure Formation Using Quasars and Lyman-Alpha Forests
In this study, we examine hierarchical structure at large redshifts utilizing Sloan Digital Sky Survey (SDSS) Lyman-alpha forest data. Lyman-alpha forests are absorption lines in the spectrum of quasars that serve as tracers for clouds of primordial hydrogen. These data serve as a 1-dimensional probe of the matter density field at high redshift. Using a measure sensitive to hierarchical structure formation assembled around a discrete wavelet transform, we were able to detect hierarchical structure formation in the spectrum of every quasar studied. Some systems showed power law behavior in the scale-scale correlation, while others did not have a preferred scale. Future work exploring the entirety of the data set is needed to determine, what if any patterns emerge, as well as the underlying factors involved with potential patterns. Along with a deeper analysis of the data set, we hope to investigate factors such as redshift dependence with extended research.
What's the Big Picture?
Figure 1 (above): An artistic representation of the Big Picture. Light emitted from quasars interacts with hydrogen clouds.
We aim to determine the evolutionary structure of the universe based on Lyman-Alpha absorption data from quasars as proxies for dark matter and its formation. Figure 1 (above) details what is going on out in space regarding our research. A quasar is a young galaxy (all quasars are galaxies, but not all galaxies are quasars). When galaxies are in this young, quasar stage of their evolution, they are especially bright (due to the large amount of matter accelerating in the quasar accretion disk)--and this brightness is great for the kind of research we do because we can see quasars from extremely far away. Light travels from quasars to Earth, and along the way interacts with large scale hydrogen clouds. These clouds are dispersed throughout the universe and are trapped by underlying dark matter. The goal of this project is to understand how these clouds of hydrogen change shape over time, because if we understand how they evolve, we can glean a better understanding of how all the matter (and dark matter) in the universe develops through time.
Hierarchical Struture
Hierarchical structure describes how the structure of the universe changes over time. Structure formation occurs when gravity splits up or combines structures (clouds of hydrogen, in our case). If we are looking at two large hydrogen clouds, learning about their hierarchical structure entails answering the following questions: Where did these clouds come from? Do they both originate from the same "Mother Cloud"? Or is this story more complex? In essence, understanding hierachical structure boils down to being able to construct a family tree (like the one shown in Figure 2 below) for hydrogen clouds (our structures of interest).
Figure 2 (above):An illustration of simple clustering at different levels. This provides an explanation for how hierarchical structure formation occurs.
The Lyman-Alpha Forest
In Figure 3, we show the physics of how the absorption characteristic of the Lyman-alpha forest works. Quasars emit a wide spectrum of light, including light in the right range to excite the ground state (n=1) electron of neutral hydrogen to its first excited state (n=2). As the radiation from the quasar interacts with neutral hydrogen clouds, the quasar's radiation in this wavelength will be absorbed. Thus, the quasar's spectrum will be missing this contribution. Historically this part of the hydrogen spectrum is called the Lyman-alpha spectrum. The electron will quickly fall back to its ground state by emitting a photon at the same wavelength that was absorbed. The emitted photons create a transmitted flux of radiation from the clouds, and this flux data is what we run our analysis on.
Figure 3 (above): A depiction of hydrogen absorption. A light source emits light at a wavelength of 1216 angstroms at a neutral hydrogen atom. When hydrogen interacts with light of this wavelength, it is able to move its electron from n=1 (ground) to n=2, resulting in an absorption line in its spectrum.
Methods
The Wavelet Transform
The wavelet transform is a mathematical tool that helps break up a signal or function into different scale components. In this study, the discrete wavelet transform (DWT) is utilized to determine scale-scale correlations between adjacent hydrogen clouds. A scale-scale correlation is used to determine the likelihood of matter clustering hierarchically. The DWT is computed by breaking the signal vector (which in the case of this research, is the binned vector of absorption fluxes taken from the quasar spectrum) into two vectors referred to as the app coefficient and the detail coefficient. A schematic of computing the wavelet transform is given in Figure 4 (below).
Figure 4 (above): An example calculation of the DWT. The pink boxes on the left represent app coefficients (local averages), and the blue boxes on the right represent detail coefficients (local average differences). The numbers in this figure are merely random numbers used for the sake of example, not actual data points.
Scale-Scale Correlation
We use the scale-scale correlation function first defined in Pando et al. (Pando et al., 1998):
This equation helps to determine the correlation, C, between scales from the wavelet. The δ quantities are taken from the detail coeficient vectors calculated in the wavelet transfrom, with j and l detailing positions in the detail coefficient vector for particular δ's. If there is a high correlation from one scale to the next scale, there is a higher chance that the previous structure is where the next structure derived from.
Results and Discussion
Figure 5 (below) shows a typical result for the scale-scale correlation function on a quasar spectrum from our data set. The horizontal axis is the scale, j at which the correlations are calculated. The red points and error bars are generated from bootstrapped random data, and the black points are the real data from the SDSS Lyman-alpha transmitted flux. Any point that differs from the random data has hierarchical structure (due to normalization in the denominator of our correlation function). All Lyman-alpha forest data we studied showed a clear hierarchical structure above random data. Some data seemed to show power law behavior in this measure, but we could not determine any common trend as of now other than significantly different from random data.
A result of the scale-scale correlation function plotted against scale j. The red error bars pertain to the observed scale-scale correlation of bootstrapped random data, while the black points represent scale-scale correlation of the data set.
Acknowledgements
I would like to extend thanks to Jesús Pando for his supervision of our research group, writing and technical assistance, and continued support and guidance throughout the research process. Additionally, I thank Zeph Kaffey and Grace Longbons for their contributions to the study as fellow research assistants, as well as those who helped revise and edit this paper.
I would like to thank DePaul University's College of Science and Health (CSH) for funding this research through the Undergraduate Summer Research Program, and the CSH Research and Faculty Development Committee for approving this research project. I also acknowledge the source of our data set, the Sloan Digital Sky Survey.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck- Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.
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