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


Context

Context

  • 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


Anthropic

Anthropic

  • 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??


Comments

Comments

  • 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

dims

dims

Shape

  • 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

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

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

Idea

  • 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


Evolution

evolution

  • 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

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


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!


Consequences

Consequences

  • 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

Cont’d

  • Horizon volume is tn so

rid~td-n

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


Conclusions

Conclusions

  • 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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