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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

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
slide13

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
slide16

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
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
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
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
slide30
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
slide36
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
slide43
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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