A few weeks ago I had the privilege to attend a talk by Simon Singh at the Adler Planetarium. It was a engaging and entertaining talk. If you have the opportunity to see one of his presentations, I recommend it.

At the close of the presentation I found out that Dr. Singh was signing copies of his new book Big Bang: The Origin Of The Universe. I figured what the heck and I picked up a copy. If you have the slightest interest in the history of astronomy and cosmology then I high recommend this book. Dr. Singh has a great writing style and the book is a quick read.

I simply cannot remember which is normalized and which is denormalized. I cannot do coordinate transforms -- I consider myself "spatially impared". And I can never remember the truth table for implies.

"p implies q" or "p only if q" has the following truth table:

     p | q | p -> q
    ---+---+--------
     T | T |   T
     T | F |   F
     F | T |   T
     F | F |   T

where p and q are propositions.

It is logically equivalent to say:

• p implies q
• q is a necessary condition for p
• p is a sufficient condition of q
• in order that p be true it is necessary that q be true
• if p is true then q is true

For example:

    If a person is a father then a person is male.

This statement is of the form p -> q where:

• p: A person is a father
• q: A person is male

It is necessary for a person to be male to be a father. Being a father is a sufficient condition for being male. If a person is not a father, nothing can be said about if they are male. Whereas if a person is not male, they may not be a father. This last statement is the contrapositive of the proposition.

     p | q | -p | -q | p -> q | -q -> -p
    ---+---+----+----+--------+----------
     T | T |  F |  F |   T    |    T
     T | F |  F |  T |   F    |    F
     F | T |  T |  F |   T    |    T
     F | F |  T |  T |   T    |    T

where - represents negation. Since both p -> q and -q -> -p have identical truth tables they are said to be logically equivalent.

While doing some research for a greedy algorithm that I'm working on, I stubled across the McNugget(tm) Number. Who says that mathematicians can't be fun?

Aristotle stated that heavier objects fell faster than lighter ones. Galileo stated the following thought experiment:

Imagine two unequal balls dropped from a height at the same time; according to Aristotle, the heavier ball would drop faster. Now imagine the same experiment with one difference: the two unequal balls are joined by a cable between them. If it was true that the heavy ball moves faster and the light one moves slower, then the light ball will hold back the heavy one. If Aristotle was correct the two balls tied together would not reach the ground as quickly as the heavy ball alone. But if we assume that the cable between the balls has the effect of turning the two balls into a single mass that is heavier than either one by itself, the joined balls should drop faster than either one by itself.

Another way of looking at this is the reverse:

A heavy object is dropped from a height. As it descends it cracks and splits into two objects, each of which is lighter than the original object. Will the two objects suddendly slow to half speed?

One can continue the above thought experiment and question what would occur if each of the two objects divided into two .... If the speed did in fact halve then one could imagine subdividing further to a point at which it appeared that the objects would fall very very slowly.

The Transit of Venus (where Venus will cross the face of the sun) will occur tomorrow, June 8th, 2004 in the early morning. I'm going to head to my local planetarium if I can get my sorry butt out of bed that early.

A little known figure in the history of the transit of Venus is Jeremiah Horrocks. He was a self-trained astronomer whose work Venus in Sole Visa influenced Issac Newton. Take the time to read about this facinating man.

The final set of notes on measuring dark matter and energy (Part 3 of 3):

• Why does everything in the universe look as if it is receeding from us? Are we at some ideal location in the universe? Pick any point on a sphere. As the sphere increases in size (think about blowing up a balloon) any other point on that sphere moves away from it. So we are not at some special point in the universe. Every point moves away from every other. The distance a point is away from another is proportional to the velocity at which it is receeding.
• Light is red shifted for the same reason as above. As space expands it stretches light making it redder (longer wavelenghts).
• Experimental evidence shows that this linearity of distance to velocity (determined via measuring the red shifts) breaks down with large distances. This implies that there is an acceleration.
• How are distances measured in cosmology?
  • Standard Ruler uses the apparent angular separation compared to a known (actual) fixed separation. An exmaple of this technique is measuring how far a car is away from you by measuring the distance between the headlights (given that you know the actual distance between the headlights).
  • Standard Candle uses apparent brightness compared to a known (actual) fixed luminosity. An example of this can be done by measuring the brightness of a car's headlights at some distance away (given that the actual brightness of the headlights are known).

• Various experiements to use the measuring techniques:

  • The universe is spatially flat. This is not to be confused with being flat in space-time which would go against Einstein's theories of relativity (equating curvature of space with mass providing for gravity). Spatially flat means that two parallel light beams will remain parallel (when ignoring the expansion of the universe). Coming back to the fact that space-time is not: since the expansion of the universe is subtracted to provide for the light remaining parallel, this shows that space-time is not flat.
  • Since there is no direct way to measure parallel photons, the elongation of the cosmic microwave background (CMB) (the red shift) can be used to measure the expansion of the universe. That is, since the expansion of space causes a red shift (expanding space elongates the waves) and variance in the spectra from distant galaxies can be easily measured, the CMB can be used as a standard ruler.
  • Supernovae are considered to be standard candles and are therefore used for measuring distances. Because supernovae are formed by drawing in mass from another object until a critcal mass (the Chandrashekar mass) is achieved and the mass explodes, the brightness of this explosion should be known. In practice, supernovae are known to not be standard candles but there are techniques to compensate for the fact that the luminosity does vary. These techniques involve measuring the time over which the explosion occurs and the brightness.
  • The geometry of space-time and the expansion history is determined by the matter-energy content of the universe.

• The power spectrum of the CMB has a very unique signature as seen in the graph. (The x-axis is the inverse of the spot size (measured as an angle) and the y-axis is the brightness.) The "bumps" or peaks in the graph on the right side can be explained by harmonics in the theory of sound.

  The spectrum of sound has a fundamental frequency and harmonic overtones. The parallel in cosmology is that space (the distance over which the sound wave travels) is swapped for time. Very early in the big bang (t ~ 10^-36s) the universe was much more dense (~1000x smaller than it is now) and composed of elementary particles and photons. As the universe expanded, recombination occurred which means that it cooled enough for the particles to form atoms (i.e. hydrogen). In comparison to sound waves, recombination is the maximum displacement for photons.

• The close correlation of the peaks in the power spectrum to that of the theory proves that only the initial conditions (i.e. the big bang) contributed to the signature as there are no other mechanisms that would produce that signature.
• One can use a standard ruler of the fundamental frequency to measure the curvature of space.
• The density of dark matter can be determined directly from the power spectrum as one of the harmonics can be directly attributed to dark matter. The first peak is the fundamental frequency. The second peak is attributed to baryons ("normal" matter). The thrid peak is attributed to dark matter.
• The heights of the peaks in the power spectrum indicates the contribution of each harmonic (component of matter). For example, a small second peak is indicative of baryon density comprable to the photon density. Also, without dark matter (matter that does not interact with light (photons)) the harmonic peaks would be much smaller.
• Dark energy could not have contributed at the time of recombination as it would be observed in the peaks (the relative heights of the peaks).
• Dark energy's density decreases much more slowly than other matter as the universe expands. As the universe expands, the density of dark matter and normal matter decreases but the dark energy is constant (check the "constant" part). We are at a very unique time in which the density of dark matter and dark energy are comprable. This is relavant in the formation of large structures (see previous notes).
• As the density of dark and normal matter decreases (with the expansion of the universe), dark energy is much more prevalent. Dark energy has an effect that is opposite of gravity. This is believed to be the reason behind why the universe is currently accelerating.
• The discrepancy between particle theory and the measurements of dark energy is approximately 120 orders of magnitude. We need a new theory -- enter string theory.
• These higher order theories (such as dark energy and string theory) may be due to a fundamental misunderstanding of the effects of gravity at very large distances.

Lecture notes are available from Wayne Hu.

Reference links:

• http://www.astro.ucla.edu/~wright/CMB-DT.html

The last thing that the MRI technician said to me before leaving the room was: don't move. "Don't move", piece of cake ... or so I thought!

Within moments of listening to the ear-plug muted rumblings of the machine I found myself being lulled into a semi-conscious state. Like a student nodding off in a boring lecture, I began the vicious cycle of stupor then jerking myself awake, stupor ... awake. Oh crap, I thought, if I don't find some way of staying awake I'm going to have to go through all of this again.

Frantically recalling what Meg did in A Wrinkle in Time to save her from control of IT, I began to recite the multiplication tables in my head. I started with nine (I just like nine). 9 x 1 = 9 ... I quickly blew through to 9 x 10 = 90. This ain't gonna cut it -- too easy. 9 x 11 = 99, 9 x 12 = 108, 9 x 13 = ... crap! Grade school multiplication tables only went up to tweleve. I could just add nine to one-oh-eight and cheat my way along. But what if I got to 9 x 100 and I didn't get nine-hundred due to cumulative error? What would my friends think? My mother and her weak heart, it might just kill her to find out! There has to be a better way.

We're going to have to do this the long way:

  13
 x 9
----
  27
+ 9
----
 117

Now why do we do this whole "multiply by the ones and write down the result and then multiply by the tens and write that down shifted over one" nonesense? There has to be a simple answer but no one bothered to tell me!

Rewriting the multiplication in a more linear form we get:

But thirteen can be written as a sum:

Bringing in the distributive law of multiplication:

Simplifies to:

Ah ha! I recalled one of my grade school teachers saying that you can put a zero in that "shifted over" spot.

  13
 x 9
----
  27
+ 90
----
 117

That makes the two statements identical! This is pretty obvious when you think about it. Since we're taught the mechanics and not the theory it's just all taken for granted.

By the time I worked through all of this (as well as some other interesting tidbits involving nine that I will write up later) the technician was rolling me out of the bowels of the great machine. I made it!

(After the fact I did recall that it was IT that was trying to gain control over Meg by forcing her to recite the multiplication tables and was therefore the worst thing that I could have done but it had saved me. Thank you Madeleine L'Engle!)

Some notes on the experimental side of dark matter (Part 2 of 3):

• Supersymmetry: A symmetry relating fermions and bosons in which every "ordinary" particle has a corresponding "superpartner" which differs in spin by half a unit. This theory attempts to find a common explanation for forces.

• One possible explanation for dark matter is WIMPs (weakly interacting massive particles). A possible candidate for WIMPs are supersymmetric particles. These particles have a very low probability of interaction (less than that of the lighter neutrino). Fortunately supersymmetric particles interact with baryons via nuclear recoils (billiard-ball interactions).

• Current detectors can detect ~1 recoil per kg per year. To detect a WIMP a ~1 recoil per ton per year is required.

• There are a number of problems in creating a detector sensitive enough to detect WIMPs. The primary problem is background radiation. This radiation can come from cosmic sources, radon from the ground, naturally occurring isotopes in metals used for constructing the detector, and even potassium 40 found in human sweat. What's most unfortunate is that WIMPs are low energy which corresponds directly with this background radiation.

  Interesting factoid: radon pools if ventilation is not adequate. Radon levels were ~2x greater than recommended in the underground "red button" rooms of France's nuclear agency due to poor air circulation. These are the unknown casualities of the cold war.

• CDMS (Cryogenic Dark Matter Search) is a detector for WIMPs. It is housed in the Soudan underground laboratory.

• DRIFT is an experiment to eliminate background radiation by exploting properties of the WIMP "wind". It is assumed that WIMPs are isotropic throughout the universe. Because the earth and sun are moving through the galaxy, there should be a "wind" of WIMPs. The wind will have a particular signature due to the rotation of the earth about the sun and the earth about its axis (For example during July the earth will be moving in the direction of the motion of the sun through the galaxy whereas in December it will be moving in the opposite direction). This signature caused by the WIMP wind should be different than that seen by any other background radiation.

• MINOS (Main Injector Neutrino Oscillation Search) involves a gun at Fermi that shoots neutrinos at a detector in the Soudan mine. This will be the first attempt to detect neutrinos from a source other than cosmic rays.

Some notes from a lecture on Dark Matter and Structure formation in the Universe (Part 1 of 3):

• Zone of avoidance: the region along the galactic equator that cannot be easily observed due to the absorption of light from the dust in the plane of the Milky Way
• Universe expansion does not play in local effect such as star - planet, star - star, and even local galaxy - galaxy. "Close in" galaxies (such as ours and Andromeda) are actually approaching (display blue shifts). Once the 1 / r^2 of gravity falls off the expansion (red shift) effects are seen.
• Sloan Digital Sky Survey (SDSS) is an effort to map one million galaxies in the universe. The process is done via measuring red shifts.

  Stars can be separated from galaxies by looking for a 4 spike cross. Stars are essentially points that cannot be resolved which causes a star pattern from the lenses. Galaxies will appear as blobs as they can be resolved. (I need to find a definitive reference for this phenomena.)

• NASA's Wilkinson Microwave Anisotropy Probe (WMAP) is measuring the cosmic background radiation.
• By combining the results of the SDSS and WMAP physical evidence is found for the existance of dark energy. Dark energy is gravitationally repulsive. What's interesting about dark energy is that it corresponds to Einstein's cosmological constant. The cosmological constant was added by Einstein to his equations to achieve a static model of the universe. After it was determined that the universe was not static (it's currently expanding) this constant was considered to be Einstein's blunder. It seems that Einstein was right after all.
• Dark matter does not lose energy like standard matter does. Standard matter (baryons) loses energy (by radition) and will condense forming dense matter. Dark matter on the other hand does not lose energy and therefore has a lower bound on its density. Because of this dark matter does not have an effect on the planetary scale as standard matter dominates. As the scale moves to intergalactic distances the density of dark matter is such that it contributes to interactions.
• "Gravity is like the economy: the rich get richer and the poor get poorer." The results of gravity follow that of chaos -- that is, it is highly dependent on initial conditions. Small initial anisotropies in a distribution of matter will quickly become large clusters of matter. A locally dense region will pull more and more matter into it ("the rich get richer") and a locally spare region will have more and more matter pulled from it ("the poor get poorer").
• Using the background microwave radiation (from WMAP) as a baseline, simulations have been performed to show that fibers similar to that seen in the results of SDSS evolve.

  It has also been shown that the effects of dark matter / energy must be added into these simulations in order to maintain the structure of the filaments. The dark matter / energy dominates over the gravitional effects to curtail further collapse of the filaments

Course slides are available.

A useful astronmy reference.

Thank you Andrey Kravtsov for a great talk!