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Physicists see the cosmos in a coffee cup

April 16, 2009
Courtesy Duke University
and World Science staff

A pro­fes­sor and a grad­u­ate stu­dent have found a “u­ni­ver­sal prin­ci­ple” that they say un­ites the in­ter­play of light and shade on the sur­face of your cof­fee, with the way gra­vity dis­torts dis­tant ga­lax­ies’ light.

They think sci­en­tists will be able to use vi­ola­t­ions of this prin­ci­ple to map un­seen clumps of mys­te­ri­ous “dark mat­ter” in the uni­verse. 

Light re­flected from met­al ring's sides al­so pro­duces the coffee-cup 'cusp curve' ef­fect. (Cour­te­sy Hen­rik Wann Jen­sen, UCSD; home­page image © L. Usi)


Light rays nat­u­rally re­flect off a curve like the in­side sur­face of a cof­fee cup in a curv­ing, ivy leaf pat­tern that comes to a point in the cen­ter and is bright­est along its edge. 

Math­e­mati­cians and phys­i­cists call that shape a “cusp curve,” and they call the bright edge a “caus­tic,” based on an al­ter­na­tive dic­tion­ary def­i­ni­tion mean­ing “burn­ing bright,” ex­plained Ar­lie Pet­ters, a phys­i­cist and math­e­ma­ti­cian at Duke Uni­ver­s­ity in Dur­ham, N.C. “It hap­pens be­cause a lot of light rays can pile up along curves.”

Drawn by the math­e­mat­ic­ally-inclined art­ist Leonardo da Vin­ci in the early 16th cen­tu­ry, caus­tics can be seen else­where in eve­ry­day life, in­clud­ing sun­light re­flecting across a swim­ming pool’s sur­face and chop­py wave-light pat­terns re­flecting off a boat hull.

Caus­tics al­so show up in gravita­t­ional lens­ing, a phe­nom­e­non caused by ga­lax­ies so mas­sive that their gra­vity bends and dis­torts light from more dis­tant ga­lax­ies. “It turns out that their gra­vity is so pow­er­ful that some light rays are al­so go­ing to pile up along curves,” said Pet­ters, a gravita­t­ional lens­ing ex­pert.

“Moth­er Na­ture has to be cre­at­ing these things,” Pet­ters said. “It’s amaz­ing how what we can see in a cof­fee cup ex­tends in­to a math­e­mat­ical the­o­rem with ef­fects in the cos­mos.”

From Earth’s van­tage point, the whole cos­mos looks like a vast in­ter­play of gra­vity and light that can ex­tend far back in­to space and time. “As with any illu­mina­t­ion pat­tern, some ar­eas will be brighter than oth­ers,” Pet­ters said. “And the bright­est parts will be along these caus­tic curves.”

Un­der­stand­ing da­ta from tel­e­scope sur­veys, he added, re­quires un­der­stand­ing the dis­tor­tions in­her­ent in lens­ing. These some­times warp a dis­tant point of light in­to mul­ti­ple and mag­ni­fied cop­ies of them­selves.

Pet­ters and oth­er re­search­ers pre­vi­ously found that, if such a light source seems to be jux­ta­posed with­in the con­fines of a caus­tic arch, two du­pli­cate im­ages will ap­pear to be po­si­tioned ab­nor­mally close to each oth­er and al­so seem equally bright. And be­cause these clones are of seem­ingly equal bright­ness, sub­tract­ing one lu­minos­ity from the oth­er re­sults in a dif­fer­ence of ze­ro.

In an ar­ti­cle ap­pearing in the March 23 Jour­nal of Math­e­mat­i­cal Phys­ics, Pet­ters and grad­u­ate stu­dent Amir Aazami ex­tended the math­e­mat­ics of such rel­a­tively sim­ple ex­am­ples to in­clude what Pet­ters called “high­er or­der caus­tics.” In such situa­t­ions the in­ter­play of light and gra­vity may ex­tend fur­ther in­to space­time and un­dergo var­i­ous forms of “caus­tic meta­mor­pho­sis” in the pro­cess.

Aazami was in­for­mally test­ing out a spe­cial case of their evolv­ing caus­tics the­o­rem called an “el­lyp­tic um­bil­ic” by us­ing a tech­ni­cal com­put­ing soft­ware pro­gram called Math­e­mat­ica when he no­ticed a pat­tern.

“It kept get­ting ze­ro over and over again,” Aazami said, no mat­ter what sce­na­rio he tried the soft­ware on. “So I thought, ‘it’s mak­ing a mis­take.’ And I went back and looked again, and I kept get­ting ze­ro. And I said, ‘this is be­gin­ning to make sense!’ That was the ‘Aha!’ mo­men­t.”

Pet­ters con­clud­ed that his grad­u­ate stu­dent had found a un­iver­sal math­e­mat­ical prin­ci­ple so per­va­sive that it can im­pose bal­ance on the most com­pli­cat­ed gravita­t­ional lens­ing il­lu­sions. For in­stance, if lens­ing pro­duces four light source cop­ies of un­even bright­nesses, the rel­a­tive dim­ness of some is pre­cisely bal­anced by the rel­a­tive lu­minos­ity of oth­ers so they can­cel each oth­er out.

“It’s mi­rac­u­lous that they can­cel out,” Pet­ters said. “This re­lates to very soph­is­t­icated math­e­mat­ics that you would nev­er think could have any­thing to do with na­ture.”

The Duke re­search­ers said that for the sim­plest caus­tics, the the­o­rem has al­ready been cor­rob­o­rat­ed by a few gravita­t­ional lens­ing ob­serva­t­ions. And they ex­pect the high­er or­der caus­tics to be ob­served once the Large Syn­op­tic Sur­vey Tel­e­scope, now be­ing as­sem­bled in Chile, be­gins what Pet­ters called “the most mas­sive sur­vey of the sky known” in a few years.

“We feel very con­fi­dent that these un­iver­sal in­vari­ants will show them­selves in the da­ta to come from the LSST,” he said.

Anoth­er sce­na­rio he pre­dicts are ex­cep­tions to the rule: “For one of the high­er or­der caus­tics, if there are two pairs of lensed im­ages that are close to each oth­er but not equally bright, then the the­o­rem is vi­olated,” he said.

“The rea­son would be some sub­struc­ture in the ga­laxy,” he said, likely dark mat­ter near one of the im­ages that causes it to be demag­ni­fied.

Dark mat­ter is a mys­te­ri­ous sub­stance that as­tro­no­mers can­not di­rectly ob­serve but can “sense” by its gravita­t­ional tug on light. By us­ing the LSST in con­junc­tion with their the­o­rem, as­tro­no­mers “would be able to iden­ti­fy dark mat­ter sub­struc­tures in com­plex ga­lac­tic sys­tems,” Pet­ters pre­dicted.


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A professor and a graduate student have found a “universal principle” that they say unites the interplay of light and shade on the surface of your coffee, with the way gravity distorts distant galaxies’ light. They think scientists will be able to use violations of this principle to map unseen clumps of mysterious “dark matter” in the universe. Light rays naturally reflect off a curve like the inside surface of a coffee cup in a curving, ivy leaf pattern that comes to a point in the center and is brightest along its edge. Mathematicians and physicists call that shape a “cusp curve,” and they call the bright edge a “caustic,” based on an alternative dictionary definition meaning “burning bright,” explained Arlie Petters, a physicist and mathematician at Duke University in Durham, N.C. “It happens because a lot of light rays can pile up along curves.” Drawn by the mathematically-inclined artist Leonardo da Vinci in the early 16th century, caustics can be seen elsewhere in everyday life, including sunlight reflecting across a swimming pool’s surface and choppy wave-light patterns reflecting off a boat hull. Caustics also show up in gravitational lensing, a phenomenon caused by galaxies so massive that their gravity bends and distorts light from more distant galaxies. “It turns out that their gravity is so powerful that some light rays are also going to pile up along curves,” said Petters, a gravitational lensing expert. “Mother Nature has to be creating these things,” Petters said. “It’s amazing how what we can see in a coffee cup extends into a mathematical theorem with effects in the cosmos.” From Earth’s vantage point, the whole cosmos looks like a vast interplay of gravity and light that can extend far back into space and time. “As with any illumination pattern, some areas will be brighter than others,” Petters said. “And the brightest parts will be along these caustic curves.” Understanding data from telescope surveys, he added, requires understanding the distortions inherent in lensing. These sometimes warp a distant point of light into multiple and magnified copies of themselves. Petters and other researchers previously found that, if such a light source seems to be juxtaposed within the confines of a caustic arch, two duplicate images will appear to be positioned abnormally close to each other and also seem equally bright. And because these clones are of seemingly equal brightness, subtracting one luminosity from the other results in a difference of zero. In an article appearing in the March 23 Journal of Mathematical Physics, Petters and graduate student Amir Aazami extended the mathematics of such relatively simple examples to include what Petters called “higher order caustics.” In such situations the interplay of light and gravity may extend further into spacetime and undergo various forms of “caustic metamorphosis” in the process. Aazami was informally testing out a special case of their evolving caustics theorem called an “ellyptic umbilic” by using a technical computing software program called Mathematica when he noticed a pattern. “It kept getting zero over and over again,” Aazami said, no matter what scenario he tried the software on. “So I thought, ‘it’s making a mistake.’ And I went back and looked again, and I kept getting zero. And I said, ‘this is beginning to make sense!’ That was the ‘Ah Ha!’ moment.” Petters concluded that his graduate student had found a universal mathematical principle so pervasive that it can impose balance on the most complicated gravitational lensing illusions. For instance, if lensing produces four light source copies of uneven brightnesses, the relative dimness of some is precisely balanced by the relative luminosity of others so they cancel each other out. “It’s miraculous that they cancel out,” Petters said. “This relates to very sophisticated mathematics that you would never think could have anything to do with nature.” The Duke researchers said that for the simplest caustics, the theorem has already been corroborated by a few actual gravitational lensing observations. And they expect the higher order caustics to be observed once the Large Synoptic Survey Telescope, now being assembled in Chile, begins what Petters called “the most massive survey of the sky known” in a few years. “We feel very confident that these universal invariants will show themselves in the data to come from the LSST,” he said. Another scenario he predicts are exceptions to the rule: “For one of the higher order caustics, if there are two pairs of lensed images that are close to each other but not equally bright, then the theorem is violated,” he said. “The reason would be some substructure in the galaxy,” he said, likely dark matter near one of the images that causes it to be demagnified. Dark matter is a mysterious substance that astronomers cannot directly observe but can “sense” by its gravitational tug on light. By using the LSST in conjunction with their theorem, astronomers “would be able to identify dark matter substructures in complex galactic systems,” Petters predicted.