Gravity and Large Extra Dimensions

Cédric Deffayet (APC, Paris) “Brane” worlds :

Usual (4 dimensional ) space-time: a “brane” 4D

n extra-dimensions + non localized fields (e.g. gravitational field)

Localized Degrees of freedom: typically « Bulk » space-time those of the particle physics standard has 4+n dimensions model U(1) × SU(2) × SU(3) String theories « Old » phenomenological approaches: « Universal Extra Dimensions » (UED)

XA

RD

Space-times extra-dimensions T: string tension 2 (D=10:superstrings) T ∝ M string of size RD

16 MGUT ∼ 10 GeV

-1 At energies lower than R D and Mstring : the effective theory is a supersymmetric field theory TeV More recently: non UED discovery of “D-branes” and string dualities

Dirichlet Gauge fields localized on the world volume of the brane 1/ Introduction

1.1/ Kaluza-Klein mechanism 1.2/ Models and motivations

An ingredient of superstring theories:

In the last 10-12 years : brane universe • A new way to do “phenomenology” beyond the standard model of particle physics. • Opens a new window to make string theory closer from experiments • Gives new ways and new ideas to modify gravity

2/ Brane World Phenomenology Space-time extra-dimensions in field theories… or how to hide space-time extra dimensions

Standard model of General Relativity particle physics

Kaluza-Klein mechanism (∼ 1920) One Example:

A scalar field Φ ( x ö ,y ) in a 4 ( x ö ) +1 ( y ) dimensional space-time Obeyes the following equations of motion ö y (∂ ∂ö + ∂ ∂y) Φ =0

Let us now assume the fifth dimension is compact 5 dimensional d’Alembertian

Φ (x ö ,y +2ùR)=Φ (x ö ,y) ⇒ The fifth dimension (y) is compact 4 large dimensions (xö) with radius R

ö ö iky Φ ( x ,y)= Φ k ( x ) exp ( R ) ⇒ k P Inserting this in the scalar field equation of motion:

ö y 2 k (∂ ∂ö + ∂m∂ky ) Φ k =0 With mk = R

4 m4 = Equation of motion for a R scalar field of mass mk 3 A 5D scalar field is seen as a bunch of 4D m3 = R massive scalar fields (called Kaluza-Klein) modes: 2 Experiments made at energies much m2 = R below m1 do only see the zero-mass mode (the “zero mode”): 1 m1 = R The low energy effective theory is 4 dimensional m0 = 0 Gravity and compact extra-dimensions

The same reasoning holds for the graviton: Small perturbation around the reference “cylinder” : the graviton

μ μ avec gμν(x ,y) = ημν + hμν(x ,y)

Metric describing Décomposed in Fourier the 4+1 dimensional seriee: space-time • One massless graviton Flat metric describing • A tower of massive the reference cylinder “Kaluza-Klein” graviton An other way to see the same result :

Two test masses separated by a distance r << R Interact with a (4+n) dimensional Newton’s law: m1m2 1 Short distance modification of V(r) R ∝ r n+1 M 2+n the gravitational interaction (4+n) r

The same masses, a distance r >> R apart, interact with a 4 dimensional Newton’s laws: m m V(r) 1 2 1 ∝ rR n M 2+n (n) (4+n) 4D Planck Mass (Newton’s constant) 2 = 2+n n MP M(4+n)R(n) So the Kaluza-Klein mechanism allows the existence of unseen space-time extra dimension accessible to a given field Φ if Very Large 1/ Those dimensions are “compact” experimental difference 2/ Their size is <<< the smallest distance scale at which(“hierarchy” the theory ) of the field Φ has been tested experimentally

For the Standard model Particle Physics: 1 18 R<(Tev)à 10à m Another very large “hierarchy”ø (“gaugemaximum hierarchy”) energy : accessible in For gravitation: particle physics accelerators Characteristic scale of4 quantum gravity R<1019à m MP 10 GeVMinimum distance at which one has ø tested the gravitational force Electroweak symetry breaking scale M 103 GeV EW ø One motivation for brane-world: the « gauge hierarchy » problem

Standard Planck energy: model of Electroweak breaking Quantum gravity? particle scale physics Grand unification?

4 16 19 10à eV TeV 10 GeV10 GeV énergie No more distance « gauge hierarchy» 2 17 30 33 10à cm 10à cm 10à cm 10à cm

StringString tension?tension? Size of Size of extra- Quantumextra-dimensions? gravity? dimensions ?

In brane world models In a field theory language

Arkani-Hamed, Dimopoulos, Dvali models (1998) 4D Brane N compact dimensions of size R(n) 2 = 2+n n MPlanck M(4+n)R(n) Graviton Kaluza-Klein rd da State an St el d if M(4+n) TeV mo ø 13 R(1) 10 cm ï ø 2 12 R(2) 10à cm mKK 10à GeV ï ø 7 ⇒ ø 7 R 10à cm m 10à GeV ï (3) ø ⇒ KK ø Modification of gravity at Accessible in energetic macroscopic distances ? processes ? BeyondSo far… compact compact extra-dimensionsextra-dimensions

branebrane

“Warped”Compact extra-dimensionsextra-dimension

Arkani-Hamed & Compact Randall & Sundrum (1999) Dimopoulos & Dvali (1998) Extra dimensions(s) “Anti-de-Sitter”space-time

brane But gravity equations of motion are modified such that one has large distance modification of gravity DGP model

Dvali, Gabadadze, Porrati (2000) In a 5D flat space-time infinite extra-dimension String theory, M theory Non stringy approaches to beyond standard Low energy effective model approaches: theories of some a new way to tackle some superstring problems Compact constructions. (e.g. the “gauge hierarchy”) extra-dimensions(ADD & Antoniadis) Strings with a low Arkani-Hamed & “fundamental”Dimopoulos scale: & Dvali An alternative to a new way to do compactification phenonomelogy with superstrings

Warped extra-dimension Toy model to study Linked to the AdS-CFT some brane world correspondanceRandall & Sundrum characteristics and other constructions brane

Compact In both cases gravity is standard extra-dimensionsCompact extra-dimensions of size

R(n) for r >> R(n) or RAdS Arkani-Hamed & Because of the “bulk” geometry Dimopoulos & Dvali

brane DGP model Dvali, Gabadaze, Gravity is modified Porrati, 2000 at large distance! Warped extra-dimensions

Randall & Sundrum “Anti-de-Sitter” space-

time of radius RAdS 2/ Brane world phenomenology

•ADD 2.1/ Particle Physics

•ADD 2.2/ Astrophysics

•DGP 2.3/ Cosmology

• ADD 2.4/ Tests of gravitation •DGP Some examples… Taken in the framework of the “canonical” ADD model (n compact extra-dimensions) and DGP model (large distance modification of gravity) Kaluza-Klein gravitons / Standard model interactions

In standard (4D) gravity, we know from Einstein that gravity couples to everything in the form of a metric (Equivalence principle) 4 Sint = d xg√ LSM

In brane world models, this is also the case,R but the metric is the « projection » on the brane of a higher dimensional metric

4D coordinates n extra coordinates 4 n I (fixed) setting the Sint = d xd yî(y ) √g LSM position of the brane R This results in a coupling between all the Fourier (along the dimensions transverse to the brane) modes (KK modes) of the metrics and the standard model fields InteractionsKaluza-Klein gravitons gravitons de /Kaluza-Klein/ Standard model Modèle interactions Standard

4 Sint = d xg√ MS L Energy momentum tensor of standard model particles 4 R n~ ö÷ d xhö÷ + ... T R ðñ í n~ Kaluza-Klein (Fourier) hö÷ modes of graviton

í Coupling like 1/MPlanck

Production rate: Number of accessible Kaluza-Klein û N2 modes: M ()Energy n ∝ Planck N ∝ 2.1/ Accelerator tests:

•Real graviton emission:

+ ex: e eà í/Z GKK ppö → gG ... → KK •Virtual graviton exchange:

•Precision tests of the standard model •New effective interactions

•Other exotic processes:

•Production of string states (with high spin)? •Black hole production?

(Most of those calculations are done in the framework of an effective field theory) 2.2/ Constraints from astrophysics 2.1/SN1987A: Linked to the duration of some phase of the life of a star

Emission of visible energy neutrinos

Known source Observable: time scale of energy Duration of the neutrino burst (10s) Main unknown: Temperature of the SN core Gravitationnal collapse of the SN Iron core Changed by possible invisible energy emission, here in the form of Kaluza- Klein gravitons M 14TeV,M 1.6TeV... (6) õ (7) õ Other astrophysical constraints:

• Decay of Kaluza-Klein gravitons by gamma ray emission .

Gravitons produced in the primordial universe

Gravitons produced while the formation of a neutron star

• Abnormal heating of a neutron star by the decay of Kaluza-Klein gravitons (Hannestad & Raffelt 2002) M 750TeV,M 35TeV... (6) õ (7) õ 2.3/ Cosmoloy 2.3.1/ « Perturbative » cosmological constraints

• In the primordial Universe, above some temperature T*, the Universe is cooling down faster by Kaluza-Klein graviton emission than by expansion

T* > T(Nucleosynthesis) for M(4+n) ~ TeV. This can change baryogenesis, or “reheating” after inflation.

• “Over closure of the Universe” by Kaluza-Klein graviton production in the early universe

• Kaluza-Klein graviton decay in the form of gamma rays

2.3.2/ One example of « non perturbative » cosmology So far… we mostly discussed models where gravity is modified at small distances

Relevant scale for cosmology: the Hubble radius H-1 associated with the rate at which the Universe expands In standard (4D) cosmology…

• The Universe is described by a FLRW metric: 2 2 2 i j ds = -dt + a (t) γij dx dx

Scale factor of the a(t ) Universe 0 a(t) Small distance modifications of gravity Modification of cosmology at small Hubble radius, I.e. at early times

Large distance modifications of gravity Modification of cosmology at large Hubble radius, I.e. at late times

Why modify gravity at large distance ?

One way to get rid of dark energy ?

Changing the dynamics Dark energy ? of gravity ? One way to do that is the DGP brane world model ! (C.D., C.D., Dvali, Gababadze 2001)

The standard Friedmann (4D) equations (relate the expansion rate of the Universe to its material content)

Becomes

Analogous to standard (4D) Friedmann equations -1 for H << rc i.e. in the primordial universe

In the late Universe, -1 ρ decreases and one has H ∼ constant ∼ rc In this model the graviton is a resonance of Kaluza- Klein gravitons (massive gravitons). In contrast to other brane world models (ADD, RS), there is no massless graviton.

An old result (known as vDVZ discontinuity) is that a massive graviton can be distinguished from a massless graviton, irrespectively of the smallness of the graviton mass!

e.g. light deflection by the Sun Sun

The same would be true in DGP gravity in the linearized theory… but disappears due to the non linearity of the model A lot of interesting developments to deal with this issue Exciting playground to test the idea that the accelerated expansion of the Universe can be due to a large distance modification of gravity….

This model predicts modifications with respect to General Relativity in the solar system

Universal perihelion precession Best prospect to detect this effect: lunar ranging experiments (or BEPICOLUMBO mission ?) Very lively debate on the internal consistency of the model (crucial issue: UV completion ?)

But a lot of new ideas and related models (bigravity and « massive gravity », cascading DGP, « degravitation » , Galileons …) 2.4/ tests of gravity

Models we have presented give also new motivations for experiments testing gravity at small and large distances M(6) > 3.6 TeV

1999 2004 2006

From Adelberger et al., Kapner et al. Conclusions:

• Brane world models: very rich domain with a lot of new ideas in high energy theoretical physics, cosmology, gravitational physics … New superstring “phenomenology” (strings with a low “fundamental” scale).

New theoretical and phenomenological ideas in particular in the gravity sector.

• Possibility to put strong constraints on some constructions.

Maybe extra-dimensions will show up in experiments in the very close future !