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Extra spatial dimensions Claudia de Rham Black Hole Session Perimeter Institute, 5 th May 2007.

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1 Extra spatial dimensions Claudia de Rham Black Hole Session Perimeter Institute, 5 th May 2007

2 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?

3 The Cosmology Paradigm The early Universe underwent exponential expansion in a tiny fraction of a second

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

5 The Cosmology Paradigm Current Universe composition 4% Visible Matter 4% Visible Matter 22% Dark Matter 74% Dark Energy The expansion of universe is currently accelerating

6 The Cosmology Paradigm Current Universe composition 4% Visible Matter 4% Visible Matter 22% Dark Matter 74% Dark Energy The expansion of universe is currently accelerating

7 Other Cosmological puzzles

8 What is the Big-Bang?What is the Big-Bang? What created it?What created it? Did time begin?Did time begin? Was there anything before?Was there anything before?

9 Other Cosmological puzzles What is the Big-Bang?What is the Big-Bang? The Hierarchy Problem…The Hierarchy Problem… What created it?What created it? Did time begin?Did time begin? Was there anything before?Was there anything before?

10 What is the Hierarchy Problem? A hierarchy problem arises when 2 fundamental quantities with same unit, don’t share the same order of magnitude.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.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.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 ~ GeV (at the Planck mass), while quantum effects for the other forces kick in at ~ 250 GeV (at the Higgs mass)We expect quantum gravity effect to kick in at ~ 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 times heavier that the Higgs mass.The Planck mass is thus times heavier that the Higgs mass.

11 What is the Hierarchy Problem? A hierarchy problem arises when 2 fundamental quantities with same unit, don’t share the same order of magnitude.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.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.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 ~ GeV (at the Planck mass), while quantum effects for the other forces kick in at ~ 250 GeV (at the Higgs mass)We expect quantum gravity effect to kick in at ~ 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 times heavier that the Higgs mass.The Planck mass is thus times heavier that the Higgs mass. ie times heavier

12 How important is this Hierarchy? New born baby3 kgs 5000 kgsAsian Elephant 400 tones Boeing tones The Titanic Egyptian Pyramid 1 million tone 200 thousand million tones Mount Everest Asteroid Gaspra 10 million million tones

13 Can extra dimensions have anything to say about these puzzles?

14 Can extra-dimensions have anything to say about these puzzles? what are dimensions?what are dimensions? how many dimensions do we live in / do we see?how many dimensions do we live in / do we see?

15 Can extra dimensions have anything to say about these puzzles? what are dimensions?what are dimensions? how many dimensions do we live in / do we see?how many dimensions do we live in / do we see? Trains live in 1 dimensionTrains live in 1 dimension

16 Can extra dimensions have anything to say about these puzzles? what are dimensions?what are dimensions? how many dimensions do we live in / do we see?how many dimensions do we live in / do we see? Trains live in 1 dimensionTrains live in 1 dimension Boats live in 2 dimensionsBoats live in 2 dimensions

17 Can extra dimensions have anything to say about these puzzles? what are dimensions?what are dimensions? how many dimensions do we live in / do we see?how many dimensions do we live in / do we see? Trains live in 1 dimensionTrains live in 1 dimension Boats live in 2 dimensionsBoats live in 2 dimensions Planes can travel in 3 dimensionsPlanes can travel in 3 dimensions

18 “Our” dimensions We live in a world with 3 spatial dimensions.We live in a world with 3 spatial dimensions. In addition to these spatial dim, there is time.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.There is only one time-like dim, so we live in a (3+1)-dim world. We will only consider extra spatial dimensions.We will only consider extra spatial dimensions. Extra dim of time are not compatible with quantum mechanics.Extra dim of time are not compatible with quantum mechanics. Extra dimensions will be represented by taking 1 or 2 dim out.Extra dimensions will be represented by taking 1 or 2 dim out.

19 Kaluza and Klein Idea The notion of extra dimensions was introduced byThe notion of extra dimensions was introduced by General Relativity was extended to (4+1)-dimGeneral Relativity was extended to (4+1)-dim As an attempt to unify gravity and electromagnetism.As an attempt to unify gravity and electromagnetism. Theodor Kaluza in 1919 and Oscar Klein in 1926

20 Size of this extra dimension If the extra dim was very large, we would see it.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.The size of the extra dim should be so small, we wouldn’t be able to resolve it with our own eyes.

21 Kaluza-Klein compactification In KK compactification, our world is spread along a very small extra dimension.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).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).

22 Large extra dimensions Alternative point of view: Extra dim could be “large”, but we would not be able to travel through them.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.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.All the particles we are made of could be forced to remain on that surface.

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24 Gravity To fully understand the physics on the surface, we need to understand how gravity behaves along these extra dimensions.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-walking PHYSICS OF FLUIDS 15 (9): S10 SEP 2003

25 How would such a surface arise? Naturally arise from string theory, they represent surfaces on which open strings end.Naturally arise from string theory, they represent surfaces on which open strings end.

26 How would such a surface arise? Naturally arise from string theory.Naturally arise from string theory. But independently of string theory, they can spontaneously arise as topological defects.But independently of string theory, they can spontaneously arise as topological defects. What is a topological defect?What is a topological defect? Picture from National High Magnetic Field Laboratory Precholesteric to Cholesteric Phase transition

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

28 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.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 domainsExample: Crystal structure and magnetic domains Picture from physics department, Brown University

29 What is a topological defect? In different regions of space, different choices can be made, (ex. magnet alignment).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.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.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.We could be “trapped” on one of these surfaces, “unaware” of other dimensions. Movie from Oslo SuperConductor Lab

30 Consequences for us We could live on such a “membrane”.We could live on such a “membrane”. This would modify the way we perceive gravity.This would modify the way we perceive gravity. Gravity would be diluted, and would “leak” within the extra dimensions.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.For a large volume of the extra dim, this could explain the Hierarchy problem.

31 Newton’s law? But if gravity is modified, what about 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.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?But how well do they work at distances much smaller or much larger than the size of the solar system?

32 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.)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.At smaller distances, gravity could behave very differently. This could be the signature of the presence of extra dimensions…This could be the signature of the presence of extra dimensions…

33 Gravity at small distances For instance when the size of the extra- dimension is finite (ex. bounded by two branes).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 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.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

34 Gravity at large distances How well do we understand gravity at very large distances (cosmological scales)?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,We think we understand it, but what we observe at these scales is also very surprising,

35 Gravity at large distances How well do we understand gravity at very large distances (cosmological scales)?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,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.The expansion of the Universe seems to be accelerating. To explain observations, the Universe should be composed of 70% of Dark EnergyTo explain observations, the Universe should be composed of 70% of Dark Energy

36 Gravity at large distances How well do we understand gravity at very large distances (cosmological scales)?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,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.The expansion of the Universe seems to be accelerating. To explain observations, the Universe should be composed of 70% of Dark EnergyTo explain observations, the Universe should be composed of 70% of Dark Energy maybe gravity is actually not so well understood at these scale…maybe gravity is actually not so well understood at these scale…

37 Gravity at large distances Extra dimensions can for instance provide a new way to think about the accelerated expansion of the Universe.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.In a model where the size of the extra dim is not finite. Gravity can be localized on the brane by another mechanism,Gravity can be localized on the brane by another mechanism, So at small distances, gravity will just behave as in four dimensions.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).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,Gravity will then be weaker at large distances, This could explain the accelerated expansion.This could explain the accelerated expansion. Dvali, Gabadadze, Porrati 2000 (DGP model)

38 Extra dim and the Big-Bang What is the Big-Bang ???

39 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.

40 Extra dim for cosmology Extra dimensions thus provide new potential ways to think about some of the key problems of particle physics and 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.The cosmological evolution of the Universe can be described as the motion of a brane within an extra dimension. Hierarchy problem,Hierarchy problem, Accelerated expansion and dark energyAccelerated expansion and dark energy Origin of the Big-Bang…Origin of the Big-Bang…

41 Extra dim for cosmology Extra dimensions thus provide new potential ways to think about some of the key problems of particle physics and 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.The cosmological evolution of the Universe can be described as the motion of a brane within an extra dimension. Hierarchy problem,Hierarchy problem, Accelerated expansion and dark energyAccelerated expansion and dark energy Origin of the Big-Bang…Origin of the Big-Bang… But if they existed, why would we live in (3+1)d ???

42 Dimensionality puzzle Why do we live in (3+1)d? In Kaluza-Klein compactifications,In Kaluza-Klein compactifications, In the brane setup,In the brane setup, Why only three dim are large, and all other dimensions are small?Why only three dim are large, and all other dimensions are small? Why is our surface 3-dimensional?Why is our surface 3-dimensional? Is there any selection mechanism?

43 Collision of extended objects Two point-like particles willTwo point-like particles will collide in 1 spatial dimensioncollide in 1 spatial dimension but not in 2but not in 2 or any higher spatial dimensionsor any higher spatial dimensions

44 Collision of extended objects Strings willStrings will Strings are “extended” objects with 1 intrinsic dimension.Strings are “extended” objects with 1 intrinsic dimension. occupy the entire world in 1 dim,occupy the entire world in 1 dim, intersect in 2 spatial dim at all times,intersect in 2 spatial dim at all times, find each other in 3 spatial dimfind each other in 3 spatial dim but not in higher dim.but not in higher dim.

45 Collision of extended objects Walls willWalls will Walls are “extended” objects with 2 intrinsic dimensions.Walls are “extended” objects with 2 intrinsic dimensions. occupy the entire world in 2 dim,occupy the entire world in 2 dim, intersect in 3 spatial dim at all times,intersect in 3 spatial dim at all times, find each other in 4 dim, etc…find each other in 4 dim, etc…

46 Collision of extended objects The same will hold for any extended object.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 ifBy analogy, any two extended objects of intrinsic dimension p, living in a (d+1)-dimensional world, will collide and can annihilate each other if

47 Collision of extended objects The same will hold for any extended object.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 ifBy analogy, any two extended objects of intrinsic dimension p, living in a (d+1)-dimensional world, will collide and can annihilate each other if p=0: points only collide in d=1 dimp=0: points only collide in d=1 dim p=1: strings only collide in d=3 dim or lessp=1: strings only collide in d=3 dim or less p=2: walls collide in d=5 dim or lessp=2: walls collide in d=5 dim or less p=3: 3-d surfaces, collide in d=7 dim or lessp=3: 3-d surfaces, collide in d=7 dim or less etc…etc…

48 Annihilation of strings Let’s consider a world with d dim, all of which are identical.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),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.When some dim spontaneously grow large, strings compress them back. But strings can annihilate in 3 dim (or less),But strings can annihilate in 3 dim (or less), So the maximum number of dimensions that can grow large is 3 !So the maximum number of dimensions that can grow large is 3 ! Brandenberger-Vafa 1989

49 Annihilation of strings Strings would naturally meet and annihilate each others only in 3 spatial dimensions or less,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.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.

50 Selection of 3d-branes The same philosophy can be taken starting from a different scenario.The same philosophy can be taken starting from a different scenario. The Universe could be initially filled with branes of different dimensions.The Universe could be initially filled with branes of different dimensions. Majumbar-Davis 2003

51 Selection of 3d-branes The same philosophy can be taken starting from a different scenario.The same philosophy can be taken starting from a different scenario. The Universe could be initially filled with branes of different dimensions.The Universe could be initially filled with branes of different dimensions. In String theory, there are (d+1)=10 dimensionsIn String theory, there are (d+1)=10 dimensions And in some kind of string theories, there only are either odd or even-dimensional branes.And in some kind of string theories, there only are either odd or even-dimensional branes.

52 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=9In 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

53 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=9In 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 with can annihilate each other, here (d+1)=10.Extended objects with can annihilate each other, here (d+1)=10.

54 Selection of 3d-branes In the theory with odd-dimensional branes, there are branes (extended objects) with dimensions: p=12(p+1)= 4 10 p=72(p+1)=16 > 10 p=92(p+1)=20 > 10In the theory with odd-dimensional branes, there are branes (extended objects) with dimensions: p=12(p+1)= 4 10 p=72(p+1)=16 > 10 p=92(p+1)=20 > 10 Extended objects with can annihilate each other, here (d+1)=10.Extended objects with can annihilate each other, here (d+1)=10.

55 Selection of 3d-branes In the theory with odd-dimensional branes, there are branes (extended objects) with dimensions: p=12(p+1)= 4 10 p=72(p+1)=16 > 10 p=92(p+1)=20 > 10In the theory with odd-dimensional branes, there are branes (extended objects) with dimensions: p=12(p+1)= 4 10 p=72(p+1)=16 > 10 p=92(p+1)=20 > 10 Extended objects with can annihilate each other, here (d+1)=10.Extended objects with can annihilate each other, here (d+1)=10. Will annihilate and disappear Will survive

56 Selection of 3d-branes In the theory with odd-dimensional branes, there are branes (extended objects) with dimensions: p=12(p+1)= 4 10 p=72(p+1)=16 > 10 p=92(p+1)=20 > 10In the theory with odd-dimensional branes, there are branes (extended objects) with dimensions: p=12(p+1)= 4 10 p=72(p+1)=16 > 10 p=92(p+1)=20 > 10 Surfaces with 3 dimensions would be the surfaces of highest dimension that would survive through this mechanism!Surfaces with 3 dimensions would be the surfaces of highest dimension that would survive through this mechanism! Will annihilate and disappear Will survive

57 Conclusions Higher spatial dimensions would have profound consequences for the physics we see, both in particle physics, cosmology and their interface.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.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,If they existed, why our world appears four-dim would be something of a puzzle,

58 Conclusions Higher spatial dimensions would have profound consequences for the physics we see, both in particle physics, cosmology and their interface.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.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,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!But their absence would yet be an even bigger mystery!


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