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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

- 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

- 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?

- 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

- 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

- 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?

- Number of Dimensions
- Their Size
- Their Shape
- Uniform distribution of matter and energy?
- Are all extra dimensions the same?

- 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!

- 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

- Brandenberger-Vafa gave a possible answer
- In context of string theory
- With simple (toroidal) rolled-up 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

- 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

- 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

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

- Branes: Membrane-Like Objects that can confine particles and forces
- Differentiate space on and orthogonal to brane

- 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

- 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

- 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

- 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!

- 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

No gravity localization

Unless another brane

Again, graviton gets localized near brane

- 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!

- 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

- 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

- 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

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

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

- 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

- 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

- 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

- 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

- 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

- 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

- 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)

- 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

- 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

- 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

- (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