Q: What are gravitons?

duhoc asked the following question in a comment to my post calling for reader questions:

I love cosmology and physics but understand very little about it.  What are gravitons, in simple terms? 

Gravitons can be complicated.  To that end, I want to state that I am no expert on the subject...  But maybe that's just right for a simple answer.  If my answer here doesn't do the job, let me know and I will track down a friend who researches quantum gravity or string theory to do a better job.

Gravity is one of the four fundamental forces, along with electromagnetism, strong force (which holds the nucleus of atoms together) and weak force (which is responsible for radioactive decay).  Every other fundamental force has a particle associated with it that communicates the force.  Electromagnetism has the photon, the strong force has gluons an the weak force has W and Z bosons (I know the least about these).  The only exception here is that gravity does not have a fundamental particle associated with it that has been observed.  However, it is safe to work on a theory that establishes the graviton as its communicating particle.  [Constructing theories by extending observations from similar situations to a new one that is not completely explained has proven effective before - the neutrino was a theoretical particle assumed to exist to account for the missing energy carried away in some forms of radioactivity and it was later proved to be true!]

One of the reasons that it is so difficult to directly detect the graviton is that gravity is the weakest of the fundamental forces (even though it holds the Universe together - that's because there is no negative mass and we don't know why that is either).  The weaker the force, the more effort there is needed to detect it.  For the graviton, it isn't a lack of motivation on scientists' part but a limit on technology... Just the contamination shielding for a detector the size of Jupiter, which would let us observe a gravitation once every 10 years, would be so massive that it would collapse into a back hole.

So, instead of observing the graviton directly, we can observe the effects of gravitons and gain knowledge about the its properties.  One of the most promising ways is by observing gravitational waves (and I know a lot about this!).  You may (or may not) have hear that we expect gravitational waves to travel at the speed of light.  The reason we expect it to is that we expect the graviton to be massless and massless particles cruise through the Universe at the speed of light.  But, if we detect a gravitational wave and it takes significantly longer the the time light would take to travel between the LIGO detectors and her international partners, then that would imply that the graviton has mass.  Think about that for a second...  Gravity exerts forces between masses and if the particle that communicated this force has mass itself, that can really complicate things.  There are theoretical physicists who spend their careers thinking about what if some detail of general relativity that hasn't been thoroughly verified is different than we think; what would be the consequences of that?  They know and they have more complicated theories worked out and the results of a massive graviton has been pondered by them as well.  Again, I am not a theoretical physicist but I do appreciate the complications involved.  However, one property of the Universe that has been shown time and time again is that the simple solution is usually the correct one which is why we say the we expect gravitational waves to travel at the speed of light.

There are some really interesting theories in cosmology that focus on the graviton.  For example, in string theory it is theorized that our Universe lives on a brane inside a higher dimensional bulk (where there are more dimensions than length, width, depth and time).  Here, gravitons are able to travel to other branes (universes).  If this theory is correct, then this could explain the missing mass in the Universe we call dark matter.

I like to include pictures with each of my blog post to help keep things interesting.  But, since the graviton has never been observed I haven't been able to find any strictly scientific images for this post.  However, if you would like a graviton of your very own to cuddle with, the Particle Zoo (which sells stuffed versions of elementary particles) has one you will be attracted to (ha, ha...  get it?):

I'm not endorsing or advertizing this product, I just thought it was interesting.

I hope this answers your question.  If not or if you anyone has more question, feel free to leave a comment below or ask me on Twitter @livingligo.

Q: What Was/Will Be the Detection Range of Initial/Advanced LIGO?

REMINDER: I DO NOT speak for the LIGO Scientific Collaboration or the Virgo Collaboration.  Therefore, these answers are mine alone as are any mistakes.

@vicnice137 asked the following question:

What's the effective range for LIGO? and for adv LIGO?  Will it include Virgo Cluster?  

The answer to this includes calibrations (which are fundamental to any experiment), astronomy, and General Relativity (I promise, no math).

First let's talk about calibration.  Any time you do an experiment, you use an instrument to measure something and before you can make statements with any confidence about your result, you need to know exactly how your instrument responds or works.  For LIGO, that means we need to understand how the light at the output of LIGO (where the interference pattern between the arms and gravitational waves are observed) changes when the mirror moves a known amount.  This is a bit more complicated than it sounds since there are feedback systems in LIGO that serve to keep the mirrors still (mostly to cancel vibrations of the mirror from our environment).  There are measurements that are made at the beginning and the end of each science data run that are used as reference characterizations of these feedback systems and there are measurements of small vibrations with a specific frequency that are purposefully and continually applied to the mirrors to characterize the calibration of LIGO at a given time.  (These calibration vibrations only affect our ability to detect gravitational waves at that specific frequency.)  Together, this information allows us to convert the intensity of the interference pattern LIGO produces into the change in length of LIGO's arms (which is what a passing gravitational wave will induce).

Now that we understand how LIGO responds to passing gravitational waves, we need to establish a reference gravitational wave that we will use to talk about how far out into space we would be able to detect that specific gravitational wave.  This reference source is known in traditional astronomy as a standard candle.  There are four main kinds of gravitational waves:

• binary inspiral mergers (like 2 dense stars, 2 black holes or a dense star/black hole pair that orbit each other rapidly and then merge to become one), 
• continuous gravitational waves (like from deformations on otherwise spherical rotating stars),  
• stochastic gravitational waves (weak gravitational waves from many sources at once; perhaps the relic gravitational waves from the Big Bang), and 
• burst gravitational waves (these are short gravitational waves from previously unknown sources or sources that are not well modeled - like what happens inside a star as it collapses right before the bright burst of light from a supernova).  

Since there is more than one kind of gravitational wave to choose from, we need to establish a priority and LIGO's priority is that:

1. The gravitational wave should be from a source we theoretically know very much about.  That is, a simple system that produces a well characterized gravitational wave.
2. The gravitational wave source should be abundant in the near Universe so that we can reasonably expect to detect it as one of the first direct gravitational wave measurements.

From these priorities the binary inspiral mergers stand out since bursts are inherently unknown, stochastic gravitational waves are inherently weak and the population of sources that produce continuous gravitational wave is not well established.

Now that we have settled on a source, we want to use a pair that is representative of the source but not exceptional in strength (otherwise, we would be overestimating how far out into space we can realistically expect to detect these gravitational waves).  LIGO chose a 1.4 solar mass neutron star paired with another 1.4 solar mass neutron star (FYI: 1.4 solar masses is on the lower limit of the expected mass of a neutron star so LIGO picked a very conservative measure for our detection distance measure).  Since General Relativity tells us how strong a gravitational wave this source will produce with respect to how far away the source is, we can combine this with our calibration (or current sensitivity) of LIGO to establish how far into the Universe we can expect to detect this representative gravitational wave.

This is such an important measure to us, that we constantly measure this value and project it on the wall at each of the control rooms in LIGO (there are 2 LIGOs: the one I work at in Louisiana and the other in Washington state).  By glancing at this and other figures-of-merit, we can quickly assess how the detector is working at that time.  Below is an example figure-of-merit that shows how far away we could detect two 1.4 solar mass neutron stars inspiraling:

The green line represents the detection range for the LIGO here in Livingston, Louisiana (known to us as L1) and the red line is other range of the LIGO detector in Hanford, Washington (known to us as H1).  The horizontal axis shows what time the measurement was taken (going back in time moving from right to left; the far right [Time = 0] was when this plot was produced) and the vertical axis measures the distance into space in Mpc (megaparsecs).  Parsecs are a somewhat odd unit of distance to anyone who doesn't study astronomy but it is equal to about 3.26 light years.

During the last data run with LIGO before the advanced LIGO upgrades began, you can see from the plot above that we were able to detect our neutron star pair out to about 20 Mpc or 65 million light years.  This did indeed encompass the Virgo Cluster of galaxies (which are between 53.5 and 54.1 million light years away).  However, LIGO was not able to maintain this detection distance all the time (you can see on the plot above that starting at about -1.5 hours the L1 range dropped from about 20 Mpc to about 15 Mpc - this was due to increased seismic activity in the region that was beyond our control).  Since the Virgo cluster is located at about 16.5 Mpc, you can see that it was not always within our detection range.

Advanced LIGO will increase our sensitivity, and therefore our detection range, by 10 times.  Once advanced LIGO reaches its design sensitivity, we will be able to detect our standard candle gravitational waves out to about 200-300 Mpc (or 650-978 million light years).  This will enable us to see 1000 times more of the Universe (since volume is proportional to the radius [distance] cubed and 10x10x10 = 1000):

Each dot in this illustration is an entire galaxy, not just a star!

It is also important to consider the limitations of this detection distance since we are limiting ourselves to thinking about only one particular kind of gravitational wave.  For example, I specialize in searching for burst gravitational waves.  These were not chosen as our standard candle because we don't know much about them (we like to call them the gravitational waves that go bump in the night).  But to apply the detection distance measured in our figure-of-merit based on a different class of gravitational waves would be short-sighted since bursts could very well be stronger (gamma ray bursts are some of the most energetic light signals astronomers have observed and, not only do we not know what causes them, many of them are VERY far away).  When we use a catalog of galaxies to target our burst searches (after all, a gravitational wave is much more likely to come from an area on the sky containing a galaxy unless we lucked out and had a nearby gravitational wave from our own galaxy), we include all known galaxies up to about 2.5 times the standard candle detection range (about 50 Mpc, or about 163 million light years).

I hope this answered this questions satisfactorily!  I know this is a bit long, but I wanted to be thorough and explain as much jargon as possible.

If you have questions about this or anything else, feel free to ask in a comment below or send it to me on Twitter @livingligo!

Questions to be answered...

Wow... I can't believe the response that I got to my open call for questions!  I am so happy to see the interest in LIGO (and science in general) and I'm eager to start answering questions.  Here is the list of those that I have so far and who asked them (if you are unfamiliar with Twitter, the screen names listed after the @ sign are people who asked me questions through Twitter; otherwise the screen name is what was used to post a comment to my blog post):

duhoc - I love cosmology and physics but understand very little about it.  What are gravitons, in simple terms?  Answered on 25 August 2011

@AstroGuyz - You know the question on every science bloggers' mind is the Big One; "When will LIGO discover gravity waves?"  Are the prospects for gravitational wave detection good before AdLIGO goes online?  Think we'll nab it before the Higgs? Answered on 4 November 2011

@umbonfo - What about the Citizen Science project Einstein@Home? I'm running it but I don't know which GW data it's analyzing.  Answered on 22 September 2011

@HughScot - What do you hope to discover about gravitational waves that will help mankind in the future?  Answered on 26 January 2012

David Dickinson - Am curious if any "spin-off" discoveries are expected from the discovery of gravity waves... (i.e. exotic objects, new cosmological theories etc) 

@EclipseMaps - What are consequences for theory of gravity/relativity if null results for gravitational waves after extended observations?  Answered on 11 November 2011

@vicnice137 - What's the effective range for LIGO? and for adv LIGO?  Will it include Virgo Cluster?    Answered on 12 August 2011

***

ADDED 12 August 2011:

BDR - On the twitter feed it says LIGO/you look for space-time ripples; is that something that occurs naturally? If there was some kind of intentionally produced ripple effect (for time travel, maybe, or maybe just galaxy/universe disruption), would it be possible to tell that it was created by another being as opposed to a natural occurrence? Would it be possible to disrupt the universe with a big or chronic enough space-time ripple? I must know these things; tia.

***

I promise that I will address each of these questions in the coming weeks (and I will be sure to contact the inquirer when I answer their question so that they don't miss it).  As always, feel free to send me a question anytime and I will be happy to try and answer it on this blog!

Come back tomorrow and I will answer one of these questions (I haven't decided which one yet).  I may not answer them in order since I like to make sure I am giving you the clearest most accurate answer I can which may include consulting with colleagues.  

Today's picture - Advanced LIGO input beam tubes arriving on site at Livingston (yesterday):

These tubes replace ones that were a smaller diameter in Initial LIGO.  We now need bigger tubes to accommodate the new upgrades near the input of the laser into the detector and near the output.

This picture is from the new LIGO - Livingston Facebook page.  If you are interested in knowing more about what is happening, like us and you will see updates in your News Feed.

What Do You Want to Know About?

In all of my posts so far, I have talked about my day-to-day life as a LIGO scientist (which is the point of the blog after all).  That means I tell you about the things I think are interesting (otherwise I wouldn't be writing about them).

This time, I would love you hear from you!  What do you want to know about?  I can tell you more about becoming a scientist, answer questions about my research or general questions about LIGO and gravitational waves - whatever.  Let me know what you want know!

Post what your interested in as a comment below or tell me on Twitter @livingligo.

Below is a picture of me in my office not two seconds ago as seen from my laptop's webcam:

Bye!

Using Astronomy to Teach Physics Workshop and AAPT Summer Meeting

I've just returned from a week long trip to Nebraska for 2 conferences, one in Lincoln and the other in Omaha.

Using Astronomy to Teach Physics Workshop:

The Lincoln trip was for a special workshop on Using Astronomy to Teach Physics (UATP).  The goal of this workshop is to bring educators in physics and astronomy together, share the information on the state-of-the-art science projects in their fields and then breakout into small groups to identify ways to bring this frontier science into the undergraduate physics curriculum.  I put together a professional poster (see my post on the different types of posters for more information on scientific posters) on the educational work done in the Science Education Center (SEC) at LIGO, both locally and nationally as part of the LIGO Scientific Collaboration's Education and Public Outreach (EPO) group [this group has just announced its partnership with the 2012 US Science & Engineering Festival].  (I will put a link to this post as soon as it becomes publicly viewable.) While I was making this poster, I found pictures of all of the SEC staff and myself engaging students and the public in some way.  For my picture, I found one that I adore (which is shocking since I hate nearly all pictures of myself) that shows me talking to school students while giving a tour of the LIGO control room:

I found this workshop particularly interesting for 2 reasons: 1) there were talks on the different frontier astronomy projects to make sure everyone in attendance at least had a working knowledge of what the state-of-the-art is and 2) there were breakout sessions where groups who were interested in similar goals met to discuss actionable ways to incorporate the new astronomy into the undergraduate physics curriculum.  My big project that I am now working on as a direct consequence of this is a document I intend to publish in the American Journal of Physics on connections between the basic physics concepts taught in the undergraduate physics courses and the technology that makes LIGO possible.  For example:

LIGO is looking for gravitational waves that will change the length of its 4 km arms less than 1/1000th the diameter of a proton (that's 0.000000000000000001 meters).  At this length scale, one must consider the effects of quantum mechanics.  So, here's the issue: the thermal vibration of the atoms in the mirrors used in LIGO is going to be much bigger than the "big" gravitational wave cited above.  How can LIGO possibly hope ever detect gravitational waves distinctly from this thermal mirror vibration?  (The 'no math' answer is at the end of this post.)  

After publication of this document, I am thinking about approaching LIGO's EPO group to propose that we create a web site to support this kind of effort.  Basically, I would like to take the document apart and use it to make a skeleton for the web site.  Then, any time a LIGO member has a homework question, activity, etc. that they use in their classroom, they can contribute that content to the site so that anyone who is interested can also use that content.

AAPT Summer Meeting:

After 3 days in Lincoln, I then went to the American Association of Physics Teachers (AAPT) Summer Meeting in Omaha.  There I got to make new friends in the Physics Instructional Resource Association (PIRA) and we did something together just about every night (which, if you know me, is remarkable since I am the type to hole up in my hotel room during down time).  I also co-presented an invited talk on LIGO outreach using demonstrations with my colleague Kathy Holt.  Demonstrations are really at the core of the outreach we so since we do demonstrations with teachers and give them the tools they need to take those demonstrations back to their classrooms, we do demonstrations during public open houses and there is often a demonstration or two when we work with student field trips.  It was great doing this talk with Kathy since the combination of our backgrounds (she was a teacher before working at LIGO) give different and useful perspectives on what we do in the SEC.

Then, besides going to other talks, I also had some academic service obligations.  I believe that I have mentioned that I am on the APS Forum on Education (FEd) Executive Committee.  Even though the AAPT is not affiliated with the APS, they work together closely since they both serve physicists but with different focus.  Because of this, the AAPT Executive Board meets with the APS FEd Exec. Comm. to coordinate efforts and we had a good lunch meeting this year.  I am also on the AAPT Committee on Graduate Education in Physics and this committee also met to make plan for the upcoming AAPT Meetings and to discuss the broader impact activities we are undertaking.

In past blog posts, I have included pictures of the city I happened to travel to as seen from my hotel window (see Long Beach, Milwaukee, Anaheim).  The view out of my window was a horrible view of a parking lot and another hotel.  Fortunately, the view from my colleague's room was much better (she got a room in the main hotel for the conference and I didn't since I waited too long to make my reservations).  So, here is the view of Omaha from the window:

OH!  One more thing...  When I was flying from Omaha (whose airport in actually in Iowa), the north terminal of airport of evacuated for a "suspicious package".  Luckily for me, I was flying out of the south terminal.  Once I got home, I found out that the hours long closing was due to someone's physics classroom apparatus that got TSA's panties in a bunch (it went through security as a carry-on).  I suppose it's good that they were awake enough to think something suspicious but what concerns me is that it is almost a certainty that the same thing was carried on the plane to get to the meeting at all.  I guess it wasn't as threatening then :) 

Answer to the quantum mechanics question posed above (highlight text below to uncover the answer):

The thermal vibration of the mirror's atoms would indeed make it impossible to measure a gravitational wave only if the laser was so well focused as to only shine on the area of a few of the mirror's surface atoms.  Fortunately, that is not the case in LIGO!  The beam spot on the mirror is about 10 cm (4 inches) in diameter.  In that area, there are MANY surface atoms that are vibrating.  By observing this large area, LIGO effectively averages over the vibration of all of the atoms that the light falls on yielding a zero net motion.  Thus, LIGO works just fine!

Academic Genealogy

I think that I have mentioned in other posts in this blog that one of my hobbies is genealogy.  I am planning to write more on my family history someday (it's always fun to find a great-grandfather in the state penitentiary when doing a census search) but I wanted to talk a little today about my academic genealogy. 

When earning an graduate degree in an academic field, one normally had an advisor that mentors the students in their research.  So, you can trace back through time who your advisor's advisor was and so on.  Normally, a modern student has only one advisor unless their research is interdisciplinary or other reasons.  Therefore, unlike a normal family tree where each child has a mother and a father, an academic genealogy doesn't branch as much.

I have done some research in the past on my academic genealogy.  My doctoral advisor was Lee Samuel Finn at Penn State and his advisor was Kip Thorne at Caltech.  On my own, I was able to trace my academic genealogy back about 10 'generations'.

Recently, I found the Mathematics Genealogy Project (MGP).  This work by North Dakota State University tracks the academic genealogy of mathematicians both their 'ancestors' and their 'descendants'.  Lucky for me, physics is closely related to mathematics (after all, Newton did pioneer calculus in order to do his physics) and my immediate academic family is documented in the MGP.  I found my advisor and started moving back through my ancestors.  It was amazing going back in time like this!  My academic genealogy goes back through 7 centuries to the High Middle Ages (essentially the founding of universities in Europe) and spans the fields of physics, astronomy, mathematics, chemistry, biology, medicine, philosophy and theology.  Before my academic great-grandfather all my 'ancestors' were educated in Europe, mostly in Germany, Austria, France and Italy.  And while this is not at all surprising, I am the only female in the tree (if you click on the poster sized image below, it will be a 6 MB JPEG; you can view the < 1 MB PDF here):

Warning: Clicking on this image will load a 6 MB JPEG.  Click here for the < 1 MB PDF.

You can also view my page on the Mathematics Genealogy Project.

Some of my academic ancestors of note (just the historically significant names I am familiar with):

• Nicolas Copernicus - known for heliocentrism (a system where the Sun is the center of the solar system) which was in opposition the accepted geocentrism (a system where the Earth is stationary and the center of the Universe).
• Christiaan Huygens - known as the first theoretical physicist, Huygens is also known for explaining Saturn's rings, wave theory and centrifugal force, among other things.
• Jacob Bernoulli - known for discovering the mathematical constant e (2.71828...) among other mathematical contributions.
• Johann Bernoulli - known for his development of infinitesimal calculus and other mathematical contributions
• Leonhard Euler - mathematician and physicist who made contributions to many sub-fields including mathematical notation (e.g. using the Greek capital sigma as notation for summation), graphing and astronomy.
• Joseph-Louis Lagrange - known for development of the calculus of variations and Lagrangian mechanics among other things.
• Jean Baptiste Joseph Fourier - known for the series approximation for discontinuous functions and related transformation that are both named after him, he also was the first to discover the greenhouse effect.
• Siméon Denis Poisson - known for the Poisson distribution which described the probability of a regular event that has no memory (dependency) on the events that happened before the present, among many other contributions to mathematics and physics.
• Johann Peter Gustav Lejeune Dirichlet - mathematician credited with the modern formal definition of a function.
• Jean-Baptiste le Rond d'Alembert - known for his contributions to fluid mechanics and testing for the convergence of a series, among other things.
• Pierre-Simon Laplace - known for work in celestial mechanics (especially work concerning the stability of the orbits in the solar system), the dependence of the speed of sound on temperature, and was also the first to expound upon an object similar to a black hole, among many other things.

So, what does this illustrious heritage say about me?  NOTHING!  Even if you are mentored by the best, it is ultimately up to you to establish and prove yourself.  However, it is humbling to have any connection not only to history but to the science that I've been using for many years.

If you are interested in learning more about academic genealogies and why they are documented in the article "A Trace of Greatness" from the Times Higher Education (6 May 2010).

Slaying My Own Dragons

I haven't written in a while.  I've been working and traveling but that isn't why I haven't been writing.  I've stayed away because I have recently been dealing with my own personal demons (which surface for me at least on an annual basis)... 

When I first started this blog, I promised a look into my everyday life as a LIGO scientist.  Almost everything that I have shared has been positive and, truly, that is how the majority of my life passes - I am blessed beyond my dreams and I love my life and my work.  However, there are the other days where I feel like nothing I've done has amounted to anything other than keeping me busy.  Deep down, I know that isn't true but I have a horrible way of marginalizing everything I do.  Basically, if I did it then anyone could have or it wasn't meaningful.  This is a well known phenomenon called the Impostor Syndrome.  I've heard about this is various places; I think most recently it was in regard to women in physics but this is a widespread phenomenon in both genders. 

Honestly, I am hesitant to even write about this here.  Physics is a competitive profession.  I feel like a person's worth is usually judged on what you've done lately.  I am always afraid that I haven't accomplished enough to not be forgotten let alone respected.  And with my job being a temporary (I am a postdoctoral scholar - this is much like when a medical doctor goes through residencies after earning their medical degree) and on a yearly contract, not constantly earning respect means that I could lose my job all together.

It isn't something that I discuss with my co-workers; after all these are the people whose respect I am trying to earn and maintain.  I don't even bring it up to my friends because, since I really don't have much of a life outside of work, my friends are also physicists - sometimes even people I feel are my competition.  On my latest trip (to Santa Barbara, CA for the APS Executive Board retreat), I did bring this up in conversation over dinner (I felt more comfortable around these physicists since they are not in the same research circle as myself and I rarely see them).  As soon and I mentioned I'd been dealing with a bit of Impostor Syndrome the immediate response I got was, "We all feel that way."  At that, I didn't know how to respond since I was surprised at how open this person was with me.

So, how do I go about slaying this dragon?  Well, the first stage is messy and usually involves much anxiety and panic about the difference between what I feel I've accomplished and what I should have accomplished.  This then moves into a planning phase where I decide what I am going to do and is followed by a series of email feelers to people I need to collaborate with to perform the work.  By this time, I have usually exhausted myself (at least emotionally) and I wait for responses from collaborators.  If they are prompt, a new determined calm can begin to take root; otherwise, the anxiety increases again.  I start thinking, "Wow, I was right and everyone thinks so little of me that they don't want to work with me!"  (Note to self: next time check your spam filter before you flip out again.)  With a new plan of action intact, I get to start the cycle of the Impostor Syndrome again: "I have all this great work to do but I don't think that I am talented enough to complete it."  But, I plug away at it, complete tasks and rarely acknowledge what I've done.

The one good thing that comes from these episodes of mine is that it jump starts new projects for me.  It also reminds me of how lucky I am to have my husband since he is the only person with whom I share this insecurity.  The poor guy is my sounding board for all of the anxiety I've built up and there really isn't anything he can do for me.

A double rainbow taken from the LIGO Livingston Observatory parking lot on 30 June 2008.

This is a good article on the Impostor Syndrome: Laursen, Lucas, "No, You're Not an Impostor", Science Careers (15 February 2008).

***

Revised Erdös Number: 4

A friend of mine read my last blog post and showed that both of us (as members of the LIGO Scientific Collaboration) have a lower Erdös Number (4) than I noted in that post.  Here are the references establishing this network:

1: Paul Erdős & Mark Kac
    Erdös, P.; Kac, M. "The Gaussian law of errors in the theory of additive number theoretic functions",  Amer. J. Math.  62,  (1940). 738–742.
2: Mark Kac & Theodore A. Jacobson
    Gaveau B.; Jacobson T. ; Kac M.; Schulman L. S. "Relativistic extension of the analogy between quantum mechanics and Brownian motion", Phys. Rev. Lett. 53 (1984), no. 5, 419–422.
3. Theodore A. Jacobson & Bruce Allen
    Allen, Bruce; Jacobson, Theodore "Vector two-point functions in maximally symmetric spaces", Comm. Math. Phys. 103 (1986), no. 4, 669–692. 
4. Bruce Allen & Amber Stuver
    Abbott, B.; et al. "Detector description and performance for the first coincidence observations between LIGO and GEO," Nucl. Instrum. Methods A 517 (2004), 154 – 179.