Relaxing to Three Dimensions

1. Relaxing to Three Dimensions Lisa Randall, Harvard University Localization: w/ Sundrum, Karch Higher-dimensional cosmology: w/Karch

2. 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

3. 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

4. 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?

5. 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

6. 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??

7. Comments • We’ll ask some obvious questions about cosmology in a higher-dimensional universe • Application speculative but • New insights/questions about cosmology, gravity

8. 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?

9. Many Possibilities for Multidimensional Worlds • Number of Dimensions • Their Size • Their Shape • Uniform distribution of matter and energy? • Are all extra dimensions the same?

10. 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!

11. 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

12. Second question: Why are there four dimensions? • Brandenberger-Vafa gave a possible answer • In context of string theory • With simple (toroidal) rolled-up dimensions

13. 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

14. 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

15. 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

16. 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

17. Catalyst for Recent Developments about extra dimensions • Branes: Membrane-Like Objects that can confine particles and forces • Differentiate space on and orthogonal to brane

18. 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

19. Brane-World

20. 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

21. Scenario for Warped Geometry

22. Graviton in this geometry

23. 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

24. Gravitational Field Near Brane

25. 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!

26. 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

27. AdS5/AdS4 No gravity localization Unless another brane Again, graviton gets localized near brane

28. 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!

29. 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

30. 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

31. 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

32. 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

33. 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)

34. 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

35. 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

36. 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

37. 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!

38. 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

39. 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

40. 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

41. 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)

42. 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

43. 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

44. 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

45. 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

46. 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