Relaxing to three dimensions
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Relaxing to Three Dimensions. Lisa Randall, Harvard University Localization: w/ Sundrum, Karch Higher-dimensional cosmology: w/Karch. Context. In some scenarios (string theory, inflation) There are many possible vacua We live in one of them Landscape idea

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Relaxing to Three Dimensions

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Relaxing to three dimensions

Relaxing to Three Dimensions

Lisa Randall, Harvard University

Localization: w/ Sundrum, Karch

Higher-dimensional cosmology: w/Karch



  • In some scenarios (string theory, inflation)

  • There are many possible vacua

  • We live in one of them

  • Landscape idea

  • That part of the story is very likely true



  • But the idea gets combined with

  • Anthropic (“environmental”) principle

  • The universe we live in is chosen

  • Not by energetic or other considerations

  • But because that is the only one that will support life?/galaxies?/??

  • Used primarily to address cosmological constant problem

Other ideas

Other ideas?

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

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

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

The relaxation principle

The Relaxation Principle

  • Alternative: The Relaxation Principle

  • The universe naturally evolves into the favored vacuum

  • Cosmological evolution determines which of many possible vacua are favored

  • Shinji Mukohyama and I applied this idea

    (with partial success) to the cosmological constant problem

New application of this principle

New application of this principle

  • In this talk, we will consider a cleaner application:

  • Why are there three dimensions of space that we experience?

  • Why do forces act in three spatial dimensions?

  • Why should three dimensions be special at all??



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

  • Application speculative but

  • New insights/questions about cosmology, gravity

First comments about dimensions of space

First, comments about dimensions of space

  • We don’t know why there appear to be four dimensions

  • No physical laws mandate 4D

  • General Relativity works for any number of dimensions

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

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

Many possibilities for multidimensional worlds

Many Possibilities for Multidimensional Worlds

  • Number of Dimensions

  • Their Size

  • Their Shape

  • Uniform distribution of matter and energy?

  • Are all extra dimensions the same?

First question to ask why are extra dimensions acceptable

First question to ask: Why are Extra Dimensions Acceptable?

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

  • New answers:

    • Not necessarily tiny

    • Could be bounded without being rolled up

    • Not even rolled up or bounded—Infinite dimensional

    • Space-time appears to be four-dimensional locally, but not necessarily everywhere!

Compactification curled up dimensions

Compactification: Curled up Dimensions

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

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

  • Very intuitive; if sufficiently small, it doesn’t look like it’s there

can see 2 or 3D with small probe

1 Dimensional

Second question why are there four dimensions

Second question: Why are there four dimensions?

  • Brandenberger-Vafa gave a possible answer

  • In context of string theory

  • With simple (toroidal) rolled-up dimensions

Relaxing to three dimensions

String Theory:One Reason

To consider extra D

  • Gravity involves a length scale, lPl

  • At shorter distances, we know general relativity breaks down

  • Need a more fundamental theory—very likely string theory

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

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

Brandenberger vafa explanation for 4 dimensions in string theory

Brandenberger-Vafa Explanation for 4 Dimensions in String Theory

  • In general, if dimensions are rolled-up,

  • There will be wrapped strings

  • UNLESS the strings can meet and annihilate

  • However, strings won’t meet if there are more than four-dimensions in spacetime (2+2=4)

  • But if there are four or fewer dimensions, the dimensions will grow larger

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

Nice idea

Nice idea

  • But…

  • Depends on initial conditions

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

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

  • Neglects nonstring objects in “string”theory—branes

Relaxing to three dimensions




  • Extra dimension can be curled up

  • Can be simple (circle, torus)

  • Complicated (Calabi-Yau)

  • They can be bounded between branes

  • They can be flat but they can also be warped

Catalyst for recent developments about extra dimensions

Catalyst for Recent Developments about extra dimensions

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

  • Differentiate space on and orthogonal to brane

Extra dimensions with branes

Extra-dimensions With Branes?

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

  • More importantly for experiment and for this talk

  • Particles can be confined to branes

  • Everything but gravity on a brane

Brane world


Branes that distort space new way to hide dimensions

Branes that Distort Space:New Way to Hide Dimensions

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

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

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

  • So much so that an infinite extra dimension is possible

  • Gravitational field (and graviton) get localized near a brane—RS2

Scenario for warped geometry

Scenario for Warped Geometry

Graviton in this geometry

Graviton in this geometry

Find localized gravity

Find Localized gravity

  • Lower-dimensional gravity survives in this warped space

  • ds2=dr2+e-kr(dxm dxnhmn)

  • Warped metric: overall scale factor

  • Here it exponentially decreases

  • Consequence is that the zero mode in a KK reduction, e-kr, is normalizable

  • Find you get a four-dimensional graviton

  • Even though space is fundamentally five-dimensional

Gravitational field near brane

Gravitational Field Near Brane

Even more dramatic locality of four dimensions

Even More Dramatic: Locality of Four Dimensions

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

  • w/ Karch, an example based on AdS brane

  • Warp factor turns around

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

Even more dramatic possibility

Even more dramatic possibility

  • Dimensionality depends on location

  • Could see different dimensions in different places

  • Determined by gravity bound state in that region

4D gravity

4D gravity

5D gravity

5D gravity

Ads5 ads4


No gravity localization

Unless another brane

Again, graviton gets localized near brane

Four dimensions might be a local phenomenon

Four dimensions might be a local phenomenon

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

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

  • But higher-dimensional elsewhere

  • Copernican revolution continues!

Four dimensions in context of branes and localized gravity

Four dimensions in context of branes and localized gravity

  • Make opposite assumptions from BV

  • Assume the number of large dimensions of space is fixed (by string theory?)

  • Ask instead which branes (that is which dimensionalities of branes) survive

  • If 3-branes survive, possible candidates for a four-dimensional universe

  • If 3-branes have the biggest filling fraction, they are the most likely candidate

Relaxing to three dimensions


  • Consider a ten-dimensional universe

  • And let it evolve according to conventional FRW evolution

  • Assume universe starts with an equal number of branes and antibranes (generic initial conditions)

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

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

What happens

What happens?

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

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

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

10 d frw evolution

10 d FRW evolution

ds2=-dt2+a2(t) d Sk2

  • n+1 dimensional FRW metric

  • n-dimensional maximally symmetric spatial geometry

  • k=-1,0,1



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

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

  • Solve


Values of w

Values of w?

  • w=0 pressureless dust

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

  • w=-1 for cosmological constant

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

  • w=-d/n

  • From this, we conclude

  • a~t(n-d)/2

How do branes evolve in this expanding universe

How do branes evolve in this expanding universe?

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

  • Even without knowing the FRW evolution precisely

  • The volume of the branes goes like ad

  • Whereas the volume of space goes like an

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

  • ad-n

Relaxing to three dimensions

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

  • But this is not the case

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

  • So they will instantaneously annihilate

  • And cannot dominate the energy density

Other branes

Other branes

  • Here’s where things become interesting

  • And we see that 3-branes are special

  • Higher-dimensional branes must always intersect

  • Their world volumes are such that

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

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



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

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

  • Strings that intersect spawn loops

  • Those loops radiate energy away through gravitational waves

  • We’ll assume similar dynamics for branes

Cont d


  • Horizon volume is tn so


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

Intersecting brane density

Intersecting brane density

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

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

  • Scaling solution

  • Length of string is some number times t

  • Volume of d-brane some number times td

Who wins

Who Wins?

  • Suppose 3-branes dominate

  • w=-1/3 t~an/2(w+1)

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

  • 7-branes and 3-branes dominate

  • Density for both decreases as t2

  • All other branes dilute more quickly

  • (Possible concern about 8-branes

  • But no worse than domain walls for cosmic-string dominated scenarios)

Where are we

Where are we?

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

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

  • Remarkable result

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

  • Used only generic assumptions and evolution

Relaxing to three dimensions

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

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

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

  • Also perhaps F-Theory

  • A very interesting system

Localized gravity in ten dimensions

Localized Gravity in Ten Dimensions

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

  • System of 3 intersecting 7-branes

  • Gravity gets localized on the intersection

  • Find 4d gravity

Stable configuration of three 7 branes

Stable configuration of three 7-branes

  • (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)

  • Generically the 3 branes intersect over a four-dimensional world volume

  • Preserves supersymmetry and should be stable



  • Lots of new results about extra dimensions

  • Most recent: maybe 3+1-dimensions really is special

  • New geometries, cosmologies with localized gravity

  • Ripe for further exploration

  • Dynamical relaxation principle very satisfying-merits further study

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