Extra spatial dimensions

Claudia de Rham

Black Hole Session

Perimeter Institute, 5th May 2007

Main questions

1. How well does the Cosmological Paradigm work?

2. What are dimensions, and what are the different ways to think about extra dimensions?

3. Why then do we live in three spatial dimensions?

The Cosmology Paradigm

The early Universeunderwent exponentialexpansion in a tiny fraction of a second

The Cosmology Paradigm

Matter was created at the end of this accelerated expansion, as a transfer of energy, called “reheating”

The Cosmology Paradigm

Current Universe composition

4% Visible Matter22% Dark Matter74% Dark Energy

The expansion of universe is currentlyaccelerating

The Cosmology Paradigm

Current Universe composition

4% Visible Matter22% Dark Matter74% Dark Energy

The expansion of universe is currentlyaccelerating

Other Cosmological puzzles

Other Cosmological puzzles

• What is the Big-Bang?

• What created it?• Did time begin?• Was there anything before?

Other Cosmological puzzles

• What is the Big-Bang?

• The Hierarchy Problem…

• What created it?• Did time begin?• Was there anything before?

What is the Hierarchy Problem?• A hierarchy problem arises when 2 fundamental quantities

with same unit, don’t share the same order of magnitude.

• Naïvely, one expects all numbers coming from a fundamental theory to share the same order of magnitude.

• This is not the case for the scales at which quantum effects for gravity and for the other three forces of nature kick in.

• We expect quantum gravity effect to kick in at ~ 1018 GeV (at the Planck mass), while quantum effects for the other forces kick in at ~ 250 GeV (at the Higgs mass)

• The Planck mass is thus 1016 times heavier that the Higgs mass.

What is the Hierarchy Problem?• A hierarchy problem arises when 2 fundamental quantities

with same unit, don’t share the same order of magnitude.

• Naïvely, one expects all numbers coming from a fundamental theory to share the same order of magnitude.

• This is not the case for the scales at which quantum effects for gravity and for the other three forces of nature kick in.

• We expect quantum gravity effect to kick in at ~ 1018 GeV (at the Planck mass), while quantum effects for the other forces kick in at ~ 250 GeV (at the Higgs mass)

• The Planck mass is thus 1016 times heavier that the Higgs mass.

ie. 10 000 000 000 000 000 times heavier

How important is this Hierarchy?New born baby 3 kgs

5000 kgsAsian Elephant

400 tones Boeing 747

46 000 tones The Titanic

Egyptian Pyramid 1 million tone

200 thousand million tones Mount Everest

Asteroid Gaspra 10 million million tones

1

103

105

107

109

1014

1016

Can extra dimensions have anything to say about these

puzzles?

Can extra-dimensions have anything to say about these

puzzles?

• what are dimensions?• how many dimensions do we live in / do we see?

Can extra dimensions have anything to say about these

puzzles?

• what are dimensions?• how many dimensions do we live in / do we see?

• Trains live in 1 dimension

Can extra dimensions have anything to say about these

puzzles?

• what are dimensions?• how many dimensions do we live in / do we see?

• Trains live in 1 dimension• Boats live in 2 dimensions

Can extra dimensions have anything to say about these

puzzles?

• what are dimensions?• how many dimensions do we live in / do we see?

• Trains live in 1 dimension• Boats live in 2 dimensions• Planes can travel in

3 dimensions

“Our” dimensions• We live in a world with

3 spatial dimensions.• In addition to these spatial

dim, there is time.• There is only one time-like

dim, so we live in a (3+1)-dim world.• We will only consider extra spatial dimensions.• Extra dim of time are not compatible with

quantum mechanics.• Extra dimensions will be represented by taking 1

or 2 dim out.

Kaluza and Klein Idea

• The notion of extra dimensions wasintroduced by

• General Relativity was extended to (4+1)-dim• As an attempt to unify gravity and electromagnetism.

Theodor Kaluza in 1919and Oscar Klein in 1926

Size of this extra dimension

• If the extra dim was very large, we would see it.

• The size of the extra dim should be so small, we wouldn’t be able to resolve it with our own eyes.

Kaluza-Klein compactification

• In KK compactification, our world is spread along a very small extra dimension.

• At low-energy, gravity in (4+1)-dim compactified on a tiny circle would look like our gravity in (3+1)-dim plus electromagnetism (photons).

Large extra dimensions• Alternative point of view:

Extra dim could be “large”, but we would not be able to travel through them.

• We could be confined to a surface which itself would live

in higher dimensions.• All the particles we are made

of could be forced to remain on that surface.

Gravity

• To fully understand the physics on the surface, we need to understand how gravity behaves along these extra dimensions.

Yet, some forces such as gravity would be free to spread along all the dimensions.

Hu DL, Chan B, Bush JWM.  Water-walkingPHYSICS OF FLUIDS 15 (9): S10 SEP 2003

How would such a surface arise?

• Naturally arise from string theory,they represent surfaces on which open strings end.

How would such a surface arise?

• Naturally arise from string theory.

• But independently of string theory, they can spontaneously arise as topological defects.

• What is a topological defect?

Picture from National High Magnetic Field LaboratoryPrecholesteric to Cholesteric Phase transition

What is a topological defect?

• A topological defect arise when evolving from one configuration to another, there is a choice associated with the new configuration.

What is a topological defect?

• A topological defect arise when evolving from one configuration to another, there is a choice associated with the new configuration.

• Example: Crystal structure and magnetic domains

Picture from physics department, Brown University

What is a topological defect?

• In different regions of space, different choices can be made, (ex. magnet alignment).

• The boundaries between these different regions are topological defects and carry energy.

• When the Universe cools down, the same kind of process happens, leading to some topological defects.

• We could be “trapped” on one of these surfaces, “unaware” of other dimensions.

Movie from Oslo SuperConductor Lab

Consequences for us

• We could live on such a “membrane”.

• This would modify the way we perceive gravity.

• Gravity would be diluted, and would “leak” within the extra dimensions.

• For a large volume of the extra dim, this could explain the Hierarchy problem.

Newton’s law?

• But if gravity is modified, what about Newton’s law?

• Newton’s law (and general relativity) work remarkably well within the solar system.

• But how well do they work at distances much smaller or much larger than the size of the solar system?

Gravity at small distances

• Newton’s law has only been tested at scales up to ~ 0.01 mm (in particular by the Eöt-Wash experiment in Seattle.)

Torsion pendulum, Eöt-Wash Group

• At smaller distances, gravity could behave very differently.

• This could be the signature of the presence of extra dimensions…

Gravity at small distances• For instance when the size of the extra-

dimension is finite (ex. bounded by two branes).

• At large distances compared to l, the extra dim is not very much excited.

• At small distances compared to l, the extra dim can be very much excited.

Randall-Sundrum 1999 (RS model)

Gravity would behave very differently a small distances.

l

l

Gravity at large distances

• How well do we understand gravity at very large distances (cosmological scales)?

• We think we understand it, but what we observe at these scales is also very surprising,

Gravity at large distances

• How well do we understand gravity at very large distances (cosmological scales)?

• We think we understand it, but what we observe at these scales is also very surprising,

• The expansion of the Universe seems to be accelerating.

• To explain observations, the Universe should be composed of 70% of Dark Energy

Gravity at large distances

• How well do we understand gravity at very large distances (cosmological scales)?

• We think we understand it, but what we observe at these scales is also very surprising,

• The expansion of the Universe seems to be accelerating.

• To explain observations, the Universe should be composed of 70% of Dark Energy

• maybe gravity is actually not so well understood at these scale…

Gravity at large distances

• Extra dimensions can for instance provide a new way to think about the accelerated expansion of the Universe.

• In a model where the size of the extra dim is not finite.• Gravity can be localized on the brane by another

mechanism,• So at small distances, gravity will just behave as in four

dimensions.• At large distances, gravity will feel the extra dimension,

and would become higher-dimensional (ex. 5d gravity).• Gravity will then be weaker at large distances,• This could explain the accelerated expansion.

Dvali, Gabadadze, Porrati 2000 (DGP model)

Extra dim and the Big-Bang

What is the

Big-Bang ???

Extra dim and the Big-Bang

We could start with attracting branes initially parallel, the branes would then move towards each other and collide.

Our 4d big-bang may simply be a collision of two branes in some higher dimensions.

Turok & Steinhardt 2001

Branes can have charges, like particles, and can attract or repulse each others.

Extra dim for cosmology

• Extra dimensions thus provide new potential ways to think about some of the key problems of particle physics and cosmology:

• The cosmological evolution of the Universe can be described as the motion of a brane within an extra dimension.

• Hierarchy problem,• Accelerated expansion and dark energy • Origin of the Big-Bang…

Extra dim for cosmology

• Extra dimensions thus provide new potential ways to think about some of the key problems of particle physics and cosmology:

• The cosmological evolution of the Universe can be described as the motion of a brane within an extra dimension.

• Hierarchy problem,• Accelerated expansion and dark energy • Origin of the Big-Bang…

But if they existed, why would we live in (3+1)d ???

Dimensionality puzzle

Why do we live in (3+1)d?

• In Kaluza-Klein compactifications,

• In the brane setup,

• Why only three dim are large, and all other dimensions are small?

• Why is our surface 3-dimensional?

Is there any selection mechanism?

Collision of extended objects

• Two point-like particles will

• collide in 1 spatial dimension• but not in 2• or any higher spatial dimensions

Collision of extended objects

• Strings will

• Strings are “extended” objects with 1 intrinsic dimension.

• occupy the entire world in 1 dim,• intersect in 2 spatial dim at all

times,• find each other in 3 spatial dim

• but not in higher dim.

Collision of extended objects

• Walls will

• Walls are “extended” objects with 2 intrinsic dimensions.

• occupy the entire world in 2 dim,• intersect in 3 spatial dim at all

times,

• find each other in 4 dim, etc…

Collision of extended objects

• The same will hold for any extended object.• By analogy, any two extended objects

of intrinsic dimension p, living in a (d+1)-dimensional world,

will collide and can annihilate each other if

1)(d1)p(2

Collision of extended objects

• The same will hold for any extended object.• By analogy, any two extended objects

of intrinsic dimension p, living in a (d+1)-dimensional world,

will collide and can annihilate each other if

1)(d1)p(2 • p=0: points only collide in d=1 dim• p=1: strings only collide in d=3 dim or less• p=2: walls collide in d=5 dim or less• p=3: 3-d surfaces, collide in d=7 dim or less• etc…

Annihilation of strings

• Let’s consider a world with d dim, all of which are identical.

• We consider them to be very small and wrapped with strings (p=1) (and other objects),

• When some dim spontaneously grow large, strings compress them back.

• But strings can annihilate in 3 dim (or less),• So the maximum number of dimensions

that can grow large is 3 !

1)(d1)p(2 3d1pBrandenberger-Vafa 1989

Annihilation of strings• Strings would naturally meet and annihilate each

others only in 3 spatial dimensions or less, • So only 3 of these dimensions would be let free

and allowed to grow while the other will remain naturally small.

Extra dimensions à la

Kaluza-Klein.

Selection of 3d-branes• The same philosophy can be taken starting from

a different scenario. • The Universe could be initially filled with branes

of different dimensions.

Majumbar-Davis 2003

Selection of 3d-branes

• The same philosophy can be taken starting from a different scenario.

• The Universe could be initially filled with branes of different dimensions.

• In String theory, there are (d+1)=10 dimensions• And in some kind of string theories, there only

are either odd or even-dimensional branes.

Selection of 3d-branes• In the theory with odd-dimensional branes, there

are branes (extended objects) with dimensions:

p=1 (strings),p=3 (like our world),p=5 p=7 (higher dim objects) p=9

Selection of 3d-branes• In the theory with odd-dimensional branes, there

are branes (extended objects) with dimensions:

p=1 (strings),p=3 (like our world),p=5 p=7 (higher dim objects) p=9

• Extended objects withcan annihilate each other, here (d+1)=10.

1)(d1)p(2

Selection of 3d-branes• In the theory with odd-dimensional branes, there

are branes (extended objects) with dimensions:

p=1 2(p+1)= 4 < 10 p=3 2(p+1)= 8 < 10 p=5 2(p+1)=12 > 10 p=7 2(p+1)=16 > 10 p=9 2(p+1)=20 > 10

• Extended objects withcan annihilate each other, here (d+1)=10.

1)(d1)p(2

Selection of 3d-branes• In the theory with odd-dimensional branes, there

are branes (extended objects) with dimensions:

p=1 2(p+1)= 4 < 10 p=3 2(p+1)= 8 < 10 p=5 2(p+1)=12 > 10 p=7 2(p+1)=16 > 10 p=9 2(p+1)=20 > 10

• Extended objects withcan annihilate each other, here (d+1)=10.

1)(d1)p(2

Will annihilate and disappear

Will survive

Selection of 3d-branes• In the theory with odd-dimensional branes, there

are branes (extended objects) with dimensions:

p=1 2(p+1)= 4 < 10 p=3 2(p+1)= 8 < 10 p=5 2(p+1)=12 > 10 p=7 2(p+1)=16 > 10 p=9 2(p+1)=20 > 10

• Surfaces with 3 dimensions would be the surfaces of highest dimension that would survive through this mechanism!

Will annihilate and disappear

Will survive

Conclusions

• Higher spatial dimensions would have profound consequences for the physics we see, both in particle physics, cosmology and their interface.

• They would modify our understanding of physics either at small (sub-millimeter) or large (astrophysical) distances.

• If they existed, why our world appears four-dim would be something of a puzzle,

Conclusions

• Higher spatial dimensions would have profound consequences for the physics we see, both in particle physics, cosmology and their interface.

• They would modify our understanding of physics either at small (sub-millimeter) or large (astrophysical) distances.

• If they existed, why our world appears four-dim would be something of a puzzle,

• But their absence would yet be an even bigger mystery!