Saturday, January 29, 2011

Publication delay

The recent east coast snow storms have played havoc with our power and internet access and this has set me back in progress in the redshift quantization posts.

I'm pushing back releases by one week while I try to finish our cleanup and await a more stable power and internet access. 

Sorry for the inconvenience.

Wednesday, January 26, 2011

Posts of Interest on Other Blogs...

I recently discovered Rob Knop's Galactic Interactions blog on  Dr. Knop has several posts of potential interest to followers of this blog.
Another nice piece from science history that covers topics relevant to this blog (thanks to Physics & Physicists: Dark Energy and Dark Matter - The "Placeholders" of Our Time
for drawing my attention to this article).

    Saturday, January 22, 2011

    Quantized Redshifts. II. The Fourier Series

    The Tools
    The mathematical tool commonly used in reports of redshift quantization is the power spectral density (PSD) (Wikipedia).  You've seen simple versions of the PSD in the form of a graphic equalizer in computer applications such as iTunes.  Sound engineers use a graphic equalizer to adjust audio power in multiple frequency bands (Wikipedia).  While the PSD is an excellent tool for identifying well-defined frequencies, broadband signals which cover many frequencies, require considerably more effort to interpret.  To understand this requires a bit more exploration of just what the PSD is and does.

    Charles Fourier was the father of what we now call the set of transformations that bear his name.  Why did Fourier develop these transformations?  Fourier was working on the problem of heat transfer, examining solutions of the equation (Wikipedia).

    For very simple cases of conduction between two planar surfaces, the solution to the equation seemed intractable.   But Fourier decided to approach the problem in a standard reductionist way - can the big problem that is unsolvable be broken down into a set of simpler problems that can be solved by the existing techniques.  Because the heat transfer equation was a differential equation of 2nd-order, one possible set of simpler solutions might be a weighted sum of sines and cosines.  Fourier explored the idea that any function f(x) in a range of position, x, from -L to +L, might have an alternative representation as a sum of sines and cosines.  In mathematical notation, this is written,
    where n is an integer ranging from one to infinity.  Fourier found that this equation would be true if the coefficients, an, and bn, were given by the integrals:

    The technique provided the means for Fourier to re-write the heat transport equation into a sum of 2nd-order differential equations where the individual terms did have simple solutions.  Then the method would allow him to recombine the simple solutions into the complete solution to the problem.

    Caveat: Care should be exercised as some sources define a0 and the normalization of an and bn different than defined here.  So long as a consistent set of series and coefficients are used, there should be no problem.

    Fourier Series: Frequencies Everywhere!
    The technique would prove to be incredibly powerful, and would open the door to the more generalized methods of orthogonal functions (wikipedia).  These techniques would become very important for solving the more complex equations that would be developed for electromagnetism (Maxwell's Equations) and quantum mechanics (Schrodinger Equation).

    But the Fourier series has another implications important for our story

    All functions of a finite range could be expressed as a sum of sine and cosine waves.  The cosine wave is equivalent to a sine wave shifted in phase by 90 degrees (pi/2 radians).

    Caveat: I say 'All functions', but there are some limitations, called Dirichlet conditions (Wikipedia).  However, it is easy to see that in a practical sense, it means that almost any function that can represent a physical system will have a Fourier Series. 

    Let's look at a function and its Fourier series.

    We'll start with a square wave (Wikipedia).  The square wave can be expressed mathematically as

    and graphically, we'll show two periods of the wave.

    From the 'recipe' above, we can compute the Fourier coefficients for this profile, a & b:

    We plot the amplitude of these coefficients, using different colors for the sine and cosine components.  The dot represents the actual amplitude of the coefficient (vertical axis) for a given integer, n (the horizontal axis).

    In this example, we see that for the cosine terms in blue, only the first coefficient, a0, is non-zero.  All other cosine terms are zero.  The sine terms, in red, are very different - we see that the even frequencies are zero and the odd frequencies are non-zero.

    As a check on our result, and to illustrate that the Fourier series really does work, we can plug these coefficients into the Fourier series and add them up and see how they compare to the original function.  To examine how the result behaves when we include more terms, we'll use multiple colors to distinguish the curves with a different number of terms.  The black curve is the original square wave.  The cyan curve is constructed from only two terms, red includes four terms, blue includes eight terms, and green is 32 terms..

    Click to Enlarge

    Notice that the more terms we include, the more closely the series value matches the original function in black.  There is only one discrepancy, and that is the Gibbs phenomena (wikipedia) visible at the top of the square where it turns down.  This occurs at strong discontinuities in the function.  The more terms we include, that overshoot will get narrower in width, but it is an unfortunate artifact of the discontinuity in the square wave.

    Next Weekend: The PSD connection

    Saturday, January 15, 2011

    Quantized Redshifts. I. Introduction

    In the previous post, I examined the claim that the SDSS galaxy survey visibly exhibits symmetry that places the Earth in a favored position.  The claim did not hold up very well to some very simple visual tests (see Delusions of Geocentric Quantization...).

    This is the first of a series of posts dealing with claims of 'quantized' redshifts of extragalactic objects (galaxies and quasars) using the power spectral density (PSD), a popular tool for searching for frequencies in data.  Redshift quantization is often claimed as a means for dismissing the Big Bang as a cosmological model and is commonly invoked by supporters of Plasma Cosmology and/or Electric Universe as well as various flavors of creationism (CreationWiki). Not all creationists support claims of redshift quantization, most notably Reasons To Believe is an Old-Earth Creationism (OEC) group that accepts much of Big Bang cosmology.  See also RationalWiki & Wikipedia

    The power spectral density (PSD, Wikipedia) is a popular analysis tool for temporal (time series) and spatial datasets.  At one time it was performed in analog circuits with wide frequency bins using band-pass filters for such applications as audio signal analysis.  While once a very computationally-intensive technique for digital signals, the development of the Fast Fourier Transform (FFT) combined with the speed and memory of modern computing hardware, has made the PSD relatively easy to compute.  Unfortunately, many who have access to the PSD make the mistake of assuming that it blindly 'finds frequencies' in datasets.  They don't take the time to understand what the PSD actually does with a dataset.

    It would be nice if there were a comprehensive tutorial with detailed demonstrations of what types of datasets work well with the PSD as well as what types of artifacts are created by poorly chosen sampling and other misuses, but I have yet to find one.  Most researchers working with this tool, myself included, have had to discover these problems for themselves, by generating test datasets of known content and processing them through the PSD.  Then one discovers just how easy non-periodic processes can create peaks in the PSD that can be misinterpreted as real frequencies. 
    The PSD has had popular use in time-series and one-dimensional spatial analysis.  There is a well-defined extension for computing the PSD in 3-D space.  It is tempting to reduce 3-D spatial datasets to 1-D for their simplicity, but if done, one must be careful to transform the coordinates correctly, lest errors and artifacts be introduced.

    For most of those researchers reporting redshift periodicities with the radial technique, you find many of them do most of their work in areas other than cosmology - so they don't take the time to learn the nuances of the technique on the data they are using. They often seem to have a MatLab or IDL toolbox that generates an FFT and they use it without experimenting with what their process does to signals of KNOWN content.

    Now for some quick overview questions.

    Would 'quantized redshifts' be evidence against the Big Bang?

    Possibly, but not definitively.

    If the quantized redshifts were a property of the light as it propagated over cosmological distances, there are scenarios where it would not impact actual cosmological structure. 

    The serious impact for Big Bang cosmology would be if the quantization represented the actual distribution of galaxies in concentric shells around the Milky Way galaxy.  Such structure formation would be very difficult to reconcile with current known physics.

    Are 'periodicities' the same thing as 'quantization'?


    The action of gravity is expected to create large scale structures in Big Bang cosmology, with density enhancements and voids created by gravity attracting material from one region to collect it to form structure.  This can create create regions with preferred size scales just as water waves have preferred scales in bodies of water.  We have actually observed the enhancement predicted by Big Bang cosmology as it has propagated from the cool spots of the cosmic microwave background into the large galaxy collections we observe today.  See Baryon Acoustic Oscillations are NOT 'Redshift Quantization'

    Most of those who try to deny Big Bang cosmology evidence have been attempting to hijack the terminology, trying to claim that the periodicities actually observed are the same as quantization.  In this way, they can cite mainstream cosmology references as evidence for their claims.  From a Christian perspective, such willful distortion of the professional use of the term could be considered equivalent to bearing false witness (Wikipedia: Ten Commandments).

    Do Periodicities imply geocentrism or galacto-centrism?


    Structure formation limits homogeneity (the uniformity or smoothness of the distribution of matter in the cosmos) to particular size scales.  When compared to the large scale, these inhomogeneities are so small that they do not significantly violate the conditions of homogeneity for Big Bang cosmology any more than the existence of mountains is a killer for the 'round earth' theory (see Wikipedia: Flat Earth Society The Flat-out Truth).  See also Testing the Homogeneity of Large-scale Structure with the SDSS Data and The scale of homogeneity of the galaxy distribution in SDSS DR6)

    In cosmology, homogeneity is defined at a cosmic scale value, often designated by the variable, a, defined over all space, which increases with cosmic time, which I will represent with the Greek letter, tau. When we do observations, we see a galaxy at distance, d, we are sampling the cosmos at time tau-d/c (due to the curvature of space, this relationship is not strictly true and as a result, there are several different measures of distance in cosmology - wikipedia). Therefore the density distribution and structure will appear to change with distance due to this sampling effect.

    In addition, one of the beauties of the power spectrum technique used for identifying structure in these datasets is that while it can identify periodicities in a system, it does so without regard to any 'center' for those periodicities.  Note that even the periodicity of the baryon acoustic oscillations identified in the SDSS data (see Baryon Acoustic Oscillations are NOT 'Redshift Quantization') of about 500 million light years is not obviously visible in the SDSS data.  In the SDSS sample from the previous post (Delusions of Geocentric Quantization...), the graphic is about 500 million light years in radius and should contain about two oscillations across its diameter.  Can you visibly identify it?  If so, where is it's center?  Use the methods outlined in the previous post.

    My Goals for this series of posts

    The basic topics I plan to cover in this series of posts (and some of these will require more than one post) are
    • An introduction to Fourier Series and its extension to the Fourier Transform, the FFT, and the PSD
    • Explore the PSD produced with ideal data as input
    • Explore what happens to the PSD when noise is introduced into the data
    • Discuss some common misuses of the PSD
    • Problems created in the PSD by radially sampling a 3-D dataset
    • More experiments with the PSD, demonstrating some specific claims found in creationist literature
    I've scheduled these at about one post per week, with perhaps a short digression in between some posts.  I think I've got these split out in just about reasonable chunks suitable for blog posts, but I will no doubt miss some nuances.  There are a substantial number of graphs and equations that need to be installed via the Blogger interface, which is not really ideal for these types of communications.  Let's see how it goes...

    Next: The Fourier Series and its Applications

    Saturday, January 8, 2011

    Delusions of Geocentric Quantization...

    In a couple of comments sections of this blog (links), Mr. Rick DeLano claims that, despite evidence to the contrary, he SEES periodicities in some of the skymaps produced by such groups as the Sloan Digital Sky Survey (SDSS).  In particular, he mentions skymaps such as those available at the SDSS at links like the one reproduced here.
    Click for larger version
    I suggested Mr. DeLano conduct an exercise with this graphic to test his statement but I find no evidence that he has actually done so.  A LOT of bad science is driven by researchers claiming they 'see' something in a dataset that can't be objectively identified (see Pathological Science, Pareidolia).  Persistence in pursuit of these claims has destroyed more than a few careers.

    Since Mr. DeLano is unwilling or unable to make any actual effort to validate his claim in an objective way, I will examine the claim in detail here, performed the test which I described to him.

    Let's examine the issues in several steps to make sure we have a reasonably complete understanding of the data we are examining.

    What does the SDSS plot represent in its projection from a 3-dimensional space? 

    I have taught several astronomy classes and occasionally found that students unfamiliar with the ways in which 3-D datasets are sometimes projected into a 2-dimensional page genuinely do not understand what they are seeing.

    The SDSS  plot is a 'slice' of the sky 1.25 degrees above and below the celestial equator.  In this case, the two-dimensional plot of galaxies on the sphere of the sky is projected in to the third dimension with the value of the redshift, z, which is a proxy for the distance of the galaxy from the observer.  Once extended into three dimensions, a slice is cut through the sphere, creating a circular plane on which we will project a small amount of data above and below the slice.
    Click for larger version
    In this construction, the Earth is in the center of the sphere is represented by the blue dot in the center of the plot.  The pie-slice shaped regions marked in yellow are areas where data could not be collected because the Milky Way obstructs too many of the more distant objects.  This map represents a very small section of the entire sky visible from Earth, so one needs to exercise caution when extending anything 'seen' in this dataset to the entire sky.

    What is meant by 'quantization' in the rigorous scientific sense? 

    Historically, describing a physical quantity as 'quantized' has meant that it has discrete measured values.   In atomic physics, the energy levels of atoms are described as quantized because they would correspond to a fixed energy in each state.  In the case of a hydrogen atom, the electron energy levels were proportional to 1/n^2, where n is an integer, 1,2,3,4,...  Intermediate values, such as energies corresponding to n=1.2 or 5.7, are never observed.

    For redshifts to be 'quantized', they would have to only occur at certain discrete values.  For example, if redshifts were quantized in steps of z = 0.02, we would expect to only see galaxies with redshifts that were integral multiples of this value.  For a quantized redshift of z=0.02, we would only find galaxies on the green circles surrounding the Earth in the graphic below.
    Click for larger version
    In the plot above, there is not even the suggestion of alignment of galaxies along these curved lines.  Note that Hartnett & Hirano, using power spectral analysis (Galaxy redshift abundance periodicity from Fourier analysis of number counts N(z) using SDSS and 2dF GRS galaxy surveys) reported redshift periodicities at z = 0.0102, 0.0246, and 0.0448.  All of these values, and their integral harmonics, should be visible in this graphic as well-defined walls of galaxies confined between the green circles.  As I will illustrate in the coming posts, many different things can create peaks in power spectra.

    Yet we see many of these 'walls' of galaxies cutting across the green circles, in violation of the claim that the distribution is spherically symmetric around the Earth. 

    Here's some structures I've identified in the SDSS map.  None of them exhibit an Earth-centered symmetry.
    Click for larger version

    What is meant by 'periodicity' in the rigorous scientific sense?

    Substances that support wave-type motions, such as gases and fluids, can support various periodic behaviors, both in time and space.  In fact, Fourier analysis was developed to mathematically handle just these types of physics problems.  The superposition of these wave motions will create density enhancements in otherwise uniform gases and fluids.

    Is there structure in the SDSS survey?

    Absolutely!  Modern cosmological simulations predict a pattern of clumping under gravity (including some energy loss by radiative processes in the plasma, which forms due to the energy release of the collapse).  Here is a snapshot from one of the modern simulations (see more at Simulating the joint evolution of quasars, galaxies and their large-scale distribution) which exhibit some similarity to a collection of soap bubbles, where the bubbles enclose 'empty' voids with membranes and filaments of soap and water.

    It is possible to identify a number of apparent cross-sections of 'bubbles' in the structure.  I mark just a few in the graphic with light-blue ovals, but many more, with overlaps can clearly be identified.  These are like the slices through many of the cosmological simulations
    Click for larger version
    This is a slice through the data incorporating distances inferred from the galaxy's redshift value.

    What happens when you look through the data in directions perpendicular to this, if you were to see these galaxies projected on the sky at night?  Does it retain a similar bubble-like structure?  Here's a sample from the NYU value-added catalog.
    This is how the SDSS galaxy distribution would look if we could see it projected on a section of the sky about 100 degrees x 60 degrees in area.  The animated gif steps through the data at different values of redshift, z.  We see structures, very similar to the filaments and bubbles in the SDSS projection in z,  out to about z = 0.2, suggesting that the structures we observe look the same from at least two very different directions.   Beyond z = 0.2, the galaxies become too sparse to identify any structure.

    Astronomy Picture of the Day also recently posted a release of the 2MASS survey that plotted one million galaxies on the sky.  I leave it as an exercise to the reader to identify structures (walls & bubbles) in this map.  The structures revealed in this map resemble those in the SDSS survey in angle and z plotted above, consistent with the idea that the universe is homogeneous.

    Does the structure in the SDSS surface exhibit a high degree of symmetry around the Milky Way Galaxy?   

    There is a selection effect created by the fact that observers look outward from the Earth radially and this places us in the center of the data, with everything else scattered beyond that.  These plots only go out to z = 0.14 (or about 0.14*(3e5 km/s)/(72 km/s/Mly) = 580 million light years).  You can obtain a more accurate distance using the cosmology calculator at Ned Wright's Cosmology Tutorial site.  The SDSS survey extends far beyond this.  To use this aspect of the geometry to claim the Earth is the center of the Universe is as bizarre as standing on a mountaintop, noticing that your view extended equally in all directions around you, and then declaring YOU are the center of the universe.

    So I've tried to identify the 'concentric/geocentric structures' claimed by Mr. DeLano and others, but no objective tests seem to support the claim.  This suggests that the 'concentric structures' are a form of pareidolia and only exist in the mind of the observer who wishes them to exist.

    As I have demonstrated above, this was a very simple set of tests, which I performed with very simple, and freely available, graphics tools.  Yet Mr. DeLano was unable, or unwilling, to do it himself.  Why?  
    Annotations installed in SDSS graphics using Inkscape.

    Saturday, January 1, 2011

    Initiating Power-up for the 2011 New Year...

    What a 6 weeks...
    • Numerous parties, both hosting and attending
    • A funeral that involved a few hundred miles of 2-way travel, some through lake-effect snow
    • Some (minor) snowstorms at home
    • A small and mildly disruptive home repair/remodel
    • I organized a presentation by Dr. John Mather (wikipedia) for NCAS and had lunch with Dr. Mather & Dr. Robert Park (wikipedia).  The presentation is available online at YouTube.
    • My new computer system is almost completely configured.  I've run a few compute-intensive simulations on it as testing for the redshift quantization project.
    And through all this I managed to make a lot of progress on my To Do list.
    • Project 1: The Quantized Redshift tutorial is largely complete.  Four of the articles, with graphics, are currently staged, with more to come.  I expect to release one per week.
    • Project 2: The GPS project is a little less developed since it has a number of sub-components that I want the capability to address
    • Project 3: Five Lagrange Points don't exist in geocentrism. Pretty far along due to it's simplicity, but it still requires a fair amount of background material.  However, Mr. Delano's arrogance has provided an interesting opportunity and I'll give this some time to develop.
    • Project 4: For the Michelson interferometers operating in space, I've received a reply from the instrument team.  I will start assembling more detailed documentation about what we know from these spacecraft.
    • Project 5 & 6: Hardly touched these two.
    An important note to potential commenters:
    Any comments made to the current series of posts which is addressed in a future article will not be released until the future article is released so I can appropriately link it as a response.  I already have several additional posts addressing currently pending comments and I plan to hold them as well.

    I've been thoroughly enjoying this time away from weekly posting.  I have made an incredible amount of progress on material that I have been wanting to develop into tutorials that would be appropriate for classroom use.  Since I now have a backlog of material on quantized redshifts that still need to be converted to blog posts.  I expect to release them about once per week but that frequency may slow as I catch up to material still under development.

    Teachers should free to e-mail me with suggestions of specific topics on this site that you'd like to see developed into material more suited to classroom use.