Relaxing to Three Relaxing to Three DimensionsDimensions

Lisa Randall, Harvard Lisa Randall, Harvard UniversityUniversity

Localization: w/ Sundrum, KarchLocalization: w/ Sundrum, KarchHigher-dimensional cosmology: w/KarchHigher-dimensional cosmology: w/Karch

ContextContext

• In some scenarios (string theory, In some scenarios (string theory, inflation)inflation)

• There are many possible vacuaThere are many possible vacua

• We live in one of themWe live in one of them

• Landscape ideaLandscape idea

• That part of the story is very likely That part of the story is very likely truetrue

AnthropicAnthropic

• But the idea gets combined with But the idea gets combined with • Anthropic (“environmental”) principleAnthropic (“environmental”) principle• The universe we live in is chosenThe universe we live in is chosen• Not by energetic or other considerationsNot by energetic or other considerations• But because that is the only one that But because that is the only one that

will support life?/galaxies?/??will support life?/galaxies?/??• Used primarily to address cosmological Used primarily to address cosmological

constant problemconstant problem

Other ideas?Other ideas?

• Perhaps correct, but worthwhile to Perhaps correct, but worthwhile to explore other aspects of cosmology explore other aspects of cosmology with many possible vacuawith many possible vacua

• Is there any other reason we end up Is there any other reason we end up in the universe we find ourselves in?in the universe we find ourselves in?

• Could there be a dynamical Could there be a dynamical explanation for more observable explanation for more observable aspects of the universe?aspects of the universe?

The Relaxation PrincipleThe Relaxation Principle

• Alternative: The Relaxation PrincipleAlternative: The Relaxation Principle• The universe naturally evolves into the The universe naturally evolves into the

favored vacuumfavored vacuum• Cosmological evolution determines which Cosmological evolution determines which

of many possible vacua are favoredof many possible vacua are favored• Shinji Mukohyama and I applied this ideaShinji Mukohyama and I applied this idea(with partial success) to the cosmological (with partial success) to the cosmological

constant problemconstant problem

New application of this New application of this principleprinciple• In this talk, we will consider a cleaner In this talk, we will consider a cleaner

application:application:• Why are there three dimensions of Why are there three dimensions of

space that we experience?space that we experience?• Why do forces act in three spatial Why do forces act in three spatial

dimensions?dimensions?• Why should three dimensions be Why should three dimensions be

special at all??special at all??

CommentsComments

• We’ll ask some obvious questions We’ll ask some obvious questions about cosmology in a higher-about cosmology in a higher-dimensional universedimensional universe

• Application speculative butApplication speculative but

• New insights/questions about New insights/questions about cosmology, gravitycosmology, gravity

First, comments about First, comments about dimensions of spacedimensions of space• We don’t know why there appear to be four We don’t know why there appear to be four

dimensionsdimensions

• No physical laws mandate 4DNo physical laws mandate 4D

• General Relativity works for any number of General Relativity works for any number of dimensionsdimensions

• A big question for physicists in the last A big question for physicists in the last decade has been what happens to those decade has been what happens to those dimensionsdimensions

• Why are they there? Do they have effects? Why are they there? Do they have effects? Why don’t we see them?Why don’t we see them?

Many Possibilities for Many Possibilities for Multidimensional WorldsMultidimensional Worlds

• Number of DimensionsNumber of Dimensions

• Their SizeTheir Size

• Their ShapeTheir Shape

• Uniform distribution of matter and Uniform distribution of matter and energy?energy?

• Are all extra dimensions the same?Are all extra dimensions the same?

First question to ask: Why are First question to ask: Why are Extra Dimensions Acceptable?Extra Dimensions Acceptable?

• Old answerOld answer: Extra dimensions can be : Extra dimensions can be rolled up to a tiny sizerolled up to a tiny size

• New answersNew answers: : Not necessarily tinyNot necessarily tiny Could be bounded without being rolled upCould be bounded without being rolled up Not even rolled up or bounded—Infinite Not even rolled up or bounded—Infinite

dimensionaldimensional Space-time appears to be four-dimensional Space-time appears to be four-dimensional

locally, but not necessarily everywhere!locally, but not necessarily everywhere!

Compactification: Curled up Compactification: Curled up DimensionsDimensions

• If a dimensions is If a dimensions is wound sufficiently wound sufficiently tightly you won’t see tightly you won’t see itit

• e. g. Curled up to a e. g. Curled up to a very small circle very small circle

• Very intuitive; if Very intuitive; if sufficiently small, it sufficiently small, it doesn’t look like it’s doesn’t look like it’s therethere 1 Dimensional can see 2 or can see 2 or

3D with small 3D with small probeprobe

can see 2 or can see 2 or 3D with small 3D with small probeprobe

Second question: Why are Second question: Why are there four dimensions?there four dimensions?

• Brandenberger-Vafa gave a possible Brandenberger-Vafa gave a possible answer answer

• In context of string theoryIn context of string theory

• With simple (toroidal) rolled-up With simple (toroidal) rolled-up dimensionsdimensions

• Gravity involves a length scale, lGravity involves a length scale, lPlPl

• At shorter distances, we know general At shorter distances, we know general relativity breaks downrelativity breaks down

• Need a more fundamental theory—very Need a more fundamental theory—very likely string theorylikely string theory

• But string theory only makes sense with But string theory only makes sense with ten ten (eleven?)(eleven?) dimensions dimensions

• Makes sense to ask why four dimensions get Makes sense to ask why four dimensions get picked out in string theorypicked out in string theory

String Theory:One Reason

To consider extra D

Brandenberger-Vafa Brandenberger-Vafa Explanation for 4 Dimensions Explanation for 4 Dimensions in String Theoryin String Theory• In general, if dimensions are rolled-up,In general, if dimensions are rolled-up,• There will be wrapped stringsThere will be wrapped strings• UNLESS the strings can meet and annihilateUNLESS the strings can meet and annihilate• However, strings won’t meet if there are more However, strings won’t meet if there are more

than four-dimensions in spacetime (2+2=4)than four-dimensions in spacetime (2+2=4)• But if there are four or fewer dimensions, the But if there are four or fewer dimensions, the

dimensions will grow largerdimensions will grow larger•

• Conclude: 4 (or fewer) dimensions will be largeConclude: 4 (or fewer) dimensions will be large

Nice ideaNice idea

• But…But…

• Depends on initial conditionsDepends on initial conditions

• Depends on poorly understood dynamics—Depends on poorly understood dynamics—everything is happening at the Planck scaleeverything is happening at the Planck scale

• Depends on moduli stabilization (what Depends on moduli stabilization (what ultimately determines size and shape)ultimately determines size and shape)

• Neglects nonstring objects in “string”theoryNeglects nonstring objects in “string”theory—branes—branes

• Extra dimension can be curled upExtra dimension can be curled up

• Can be simple (circle, torus)Can be simple (circle, torus)

• Complicated (Calabi-Yau)Complicated (Calabi-Yau)

• They can be bounded between They can be bounded between branesbranes

• They can be flat but they can also be They can be flat but they can also be warpedwarped

Catalyst for Recent Catalyst for Recent Developments about extra Developments about extra dimensionsdimensions

• Branes: Membrane-Branes: Membrane-Like Objects that Like Objects that can confine can confine particles and forcesparticles and forces

• Differentiate space Differentiate space on and orthogonal on and orthogonal to braneto brane

Extra-dimensions With Extra-dimensions With Branes?Branes?

• Space can be bounded between Space can be bounded between branes; not rolled-upbranes; not rolled-up

• More importantly for experiment and More importantly for experiment and for this talkfor this talk

• Particles can be confined to branesParticles can be confined to branes

• Everything but gravity on a braneEverything but gravity on a brane

Brane-WorldBrane-World

Branes that Distort Space:Branes that Distort Space:New Way to Hide Dimensions New Way to Hide Dimensions

• If we’re confined to branes, only problematic If we’re confined to branes, only problematic aspect of dimensions is gravityaspect of dimensions is gravity

• Forces and particles are confined to branes (3-Forces and particles are confined to branes (3-dimensional?)dimensional?)

• However, energetic branes in an energetic bulk However, energetic branes in an energetic bulk can distort spacecan distort space

• So much so that an infinite extra dimension is So much so that an infinite extra dimension is possiblepossible

• Gravitational field (and graviton) get localized Gravitational field (and graviton) get localized near a brane—near a brane—RS2RS2

Scenario for Warped Scenario for Warped GeometryGeometry

Graviton in this geometryGraviton in this geometry

Find Localized gravity Find Localized gravity

• Lower-dimensional gravity survives in this Lower-dimensional gravity survives in this warped spacewarped space

• dsds22=dr=dr22+e+e-kr-kr(dx(dx dx dx • Warped metric: overall scale factorWarped metric: overall scale factor• Here it exponentially decreasesHere it exponentially decreases• Consequence is that the zero mode in a KK Consequence is that the zero mode in a KK

reduction, ereduction, e-kr-kr, is normalizable, is normalizable• Find you get a four-dimensional gravitonFind you get a four-dimensional graviton• Even though space is fundamentally five-Even though space is fundamentally five-

dimensionaldimensional

Gravitational Field Near Gravitational Field Near BraneBrane

Even More Dramatic: Even More Dramatic: Locality of Four DimensionsLocality of Four Dimensions

• Why should you need to know about Why should you need to know about space far from the brane?space far from the brane?

• w/ Karch, an example based on AdS w/ Karch, an example based on AdS branebrane

• Warp factor turns aroundWarp factor turns around

• We find four-dimensional gravity We find four-dimensional gravity (mediated by massive graviton) near (mediated by massive graviton) near the brane!the brane!

Even more dramatic Even more dramatic possibilitypossibility

• Dimensionality depends on locationDimensionality depends on location

• Could see different dimensions in Could see different dimensions in different placesdifferent places

• Determined by gravity bound state in Determined by gravity bound state in that regionthat region

4D gravity4D gravity

5D gravity

5D gravity

4D gravity

4D gravity

5D gravity

5D gravity

AdS5/AdS4AdS5/AdS4

Again, graviton gets localized near brane

No gravity localization

Unless another brane

Four dimensions might be a Four dimensions might be a local phenomenon local phenomenon

• Why should you need Why should you need to know about space to know about space far from the brane?far from the brane?

• four-dimensional four-dimensional gravity (mediated by gravity (mediated by massive graviton) near massive graviton) near the branethe brane

• But higher-dimensional But higher-dimensional elsewhereelsewhere

• Copernican revolution Copernican revolution continues!continues!

Four dimensions in context of Four dimensions in context of branes and localized gravitybranes and localized gravity

• Make opposite assumptions from BVMake opposite assumptions from BV• Assume the number of large dimensions of Assume the number of large dimensions of

space is fixed (by string theory?)space is fixed (by string theory?)• Ask instead which branes (that is which Ask instead which branes (that is which

dimensionalities of branes) survivedimensionalities of branes) survive• If 3-branes survive, possible candidates for If 3-branes survive, possible candidates for

a four-dimensional universea four-dimensional universe• If 3-branes have the biggest filling fraction, If 3-branes have the biggest filling fraction,

they are the most likely candidatethey are the most likely candidate

IdeaIdea

• Consider a ten-dimensional universeConsider a ten-dimensional universe• And let it evolve according to conventional And let it evolve according to conventional

FRW evolutionFRW evolution• Assume universe starts with an equal Assume universe starts with an equal

number of branes and antibranes (generic number of branes and antibranes (generic initial conditions)initial conditions)

• Of all dimensions (with possible exception Of all dimensions (with possible exception of 8-branes for reasons we will get to)of 8-branes for reasons we will get to)

• Let the energy of the branes determine Let the energy of the branes determine the equation of state that enters the FRW the equation of state that enters the FRW evolutionevolution

What happens?What happens?

• During the universe’s evolution, some During the universe’s evolution, some branes will dilute more than othersbranes will dilute more than others

• After some time, we’ll be left with a After some time, we’ll be left with a universe in which certain types of universe in which certain types of branes will be much more numerousbranes will be much more numerous

• Those are likely to be the ones on Those are likely to be the ones on which we livewhich we live

10 d FRW evolution10 d FRW evolution

•n+1 dimensional FRW metric

•n-dimensional maximally symmetric spatial geometry

•k=-1,0,1

ds2=-dt2+a2(t) d k2

evolutionevolution

• Assume the equation of state is dominated by a Assume the equation of state is dominated by a single component (single w)single component (single w)

• Even if not initially true, will be the stable Even if not initially true, will be the stable attractor of the evolutionattractor of the evolution

• SolveSolve

~a-n(1+w)->t~an/2(w+1)

Values of w?Values of w?

• w=0 pressureless dustw=0 pressureless dust

• w=1/n for radiation with traceless stress w=1/n for radiation with traceless stress tensortensor

• w=-1 for cosmological constantw=-1 for cosmological constant

• For a d-brane with a d+1-dimensional world For a d-brane with a d+1-dimensional world volume in n spatial dimensionsvolume in n spatial dimensions

• w=-d/nw=-d/n

• From this, we concludeFrom this, we conclude

• a~ta~t(n-d)/2(n-d)/2

How do branes evolve in this How do branes evolve in this expanding universe?expanding universe?

• If branes are non-interacting, we know If branes are non-interacting, we know how they scale how they scale

• Even without knowing the FRW evolution Even without knowing the FRW evolution preciselyprecisely

• The volume of the branes goes like aThe volume of the branes goes like add

• Whereas the volume of space goes like aWhereas the volume of space goes like ann

• So the density of d-branes (here d is So the density of d-branes (here d is merely dimension)merely dimension)

• aad-nd-n

• Clearly if this were all that were going on, Clearly if this were all that were going on, the largest branes would dominate the the largest branes would dominate the energy density after enough time has energy density after enough time has elapsedelapsed

• But this is not the caseBut this is not the case

• In a 10-dimensional universe, 9-branes and In a 10-dimensional universe, 9-branes and anti-9-branes overlap everywhere anti-9-branes overlap everywhere

• So they will instantaneously annihilateSo they will instantaneously annihilate

• And cannot dominate the energy densityAnd cannot dominate the energy density

Other branesOther branes

• Here’s where things become interestingHere’s where things become interesting

• And we see that 3-branes are specialAnd we see that 3-branes are special

• Higher-dimensional branes must always Higher-dimensional branes must always intersectintersect

• Their world volumes are such thatTheir world volumes are such that

• 2(d+1)≥10 (=n+1 for what we are 2(d+1)≥10 (=n+1 for what we are interested in)interested in)

• So for us, rather than 2+2=4, use 4+4<10!So for us, rather than 2+2=4, use 4+4<10!

ConsequencesConsequences

• The density of branes with more than 3+1 The density of branes with more than 3+1 dimensions is less than you would thinkdimensions is less than you would think

• We use the same (now standard) argument We use the same (now standard) argument that is used for strings to establish their that is used for strings to establish their dilution ratedilution rate

• Strings that intersect spawn loops Strings that intersect spawn loops

• Those loops radiate energy away through Those loops radiate energy away through gravitational wavesgravitational waves

• We’ll assume similar dynamics for branesWe’ll assume similar dynamics for branes

Cont’dCont’d

• Horizon volume is tHorizon volume is tnn so so

True for any branes that can find each other and interact and trigger decay that’s limited only by causality

id~td-n

Intersecting brane densityIntersecting brane density

• Assume decay processes happen at the Assume decay processes happen at the maximum efficiency allowed by causalitymaximum efficiency allowed by causality

• The network at all times looks the same The network at all times looks the same when viewed on the scale “t”, the horizon when viewed on the scale “t”, the horizon sizesize

• Scaling solutionScaling solution• Length of string is some number times tLength of string is some number times t• Volume of d-brane some number times tVolume of d-brane some number times tdd

Who Wins?Who Wins?

• Suppose 3-branes dominateSuppose 3-branes dominate• w=-1/3 w=-1/3 t~an/2(w+1)

• a~ta~t1/3; 1/3; 7-branes: t7-branes: t-(9-7)-(9-7) ; 3-branes: a ; 3-branes: a-(9-3)-(9-3)

• 7-branes and 3-branes dominate7-branes and 3-branes dominate• Density for both decreases as tDensity for both decreases as t22

• All other branes dilute more quicklyAll other branes dilute more quickly• (Possible concern about 8-branes(Possible concern about 8-branes• But no worse than domain walls for But no worse than domain walls for

cosmic-string dominated scenarios)cosmic-string dominated scenarios)

Where are we?Where are we?

• The only stable evolution has 3-branes and The only stable evolution has 3-branes and 7-branes dominating7-branes dominating

• No other set of branes can consistently No other set of branes can consistently dominate the energy densitydominate the energy density

• Remarkable resultRemarkable result

• 3-branes and 7-branes appear in particle 3-branes and 7-branes appear in particle physics and cosmologyphysics and cosmology

• Used only generic assumptions and Used only generic assumptions and evolutionevolution

• 3-branes and 7-branes are both needed to 3-branes and 7-branes are both needed to have viable gauge theory with matter have viable gauge theory with matter

• Furthermore, 3-branes and 7-branes can Furthermore, 3-branes and 7-branes can generate an inflationary scenariogenerate an inflationary scenario

• Also potentially relevant to AdS/CFT (many Also potentially relevant to AdS/CFT (many 3-branes)3-branes)

• Also perhaps F-TheoryAlso perhaps F-Theory• A very interesting systemA very interesting system

Localized Gravity in Ten Localized Gravity in Ten Dimensions Dimensions

• 4-d gravity in a universe with only 3-4-d gravity in a universe with only 3-branes and 7-branesbranes and 7-branes

• System of 3 intersecting 7-branesSystem of 3 intersecting 7-branes

• Gravity gets localized on the Gravity gets localized on the intersectionintersection

• Find 4d gravityFind 4d gravity

Stable configuration of threeStable configuration of three 7-branes 7-branes

• (0,1,2,3,4,5,6,7,8,9)(0,1,2,3,4,5,6,7,8,9)• (x,x,x,x,x,x,x,x,-,-)(x,x,x,x,x,x,x,x,-,-)• (x,x,x,x,x,x,-,-,x,x,)(x,x,x,x,x,x,-,-,x,x,)• (x,x,x,x,-,-,x,x,x,x)(x,x,x,x,-,-,x,x,x,x)• Generically the 3 branes intersect Generically the 3 branes intersect

over a four-dimensional world volumeover a four-dimensional world volume• Preserves supersymmetry and should Preserves supersymmetry and should

be stablebe stable

ConclusionsConclusions

• Lots of new results about extra dimensionsLots of new results about extra dimensions• Most recent: maybe 3+1-dimensions really Most recent: maybe 3+1-dimensions really

is specialis special• New geometries, cosmologies with New geometries, cosmologies with

localized gravitylocalized gravity• Ripe for further explorationRipe for further exploration• Dynamical relaxation principle very Dynamical relaxation principle very

satisfying-merits further studysatisfying-merits further study