Cosmological Signatures of Brane Inflation

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1 March 22, 2008

2 Milestones in the Evolution of the Universe mm.html

3 Information about the Inflationary period The amplitude of the large-scale temperature fluctuations: δ H = 2 5 PR = , is determined by the energy density during inflation. The data suggests V GeV. The spectral index and the running of the spectral index: n 1 = d ln δ2 H (k) = 2η 6ɛ d ln k dn = 16ɛη + 24ɛ 2 + 2ξ 2 d ln k Present value from WMAP (only) data: n scalar = dn scalar /d log k = ,

4 Braneworld Closed Strings NS-NS: Graviton G µν, Dilaton φ, Antisymmetric Tensor B µν and RR higher-spin fields C µ0 µ p couple to the brane world-volume. Open Strings NS and R: Gauge Bosons, Fermions and Tachyons The ends of the open strings cannot leave the branes. The distance between the branes becomes the mass of the particles: M 2 = Y µ Y µ 4π 2 α 2 + L 0 α Figure: The Braneworld model can describe all the particles and interactions of the Standard Model. Gravity is present as well but being represented by a closed string mode is not confined to propagate only inside the branes.

5 Candidates for the Inflaton Closed string modes Radion/dilaton: gravitational strength couplings Open string modes Tachyons: their potential is too steep Brane separation as inflaton: brane interaction has a relatively flat potential In general, the brane interaction has gravitational strength very weak very flat potential But the inflaton has standard model strength coupling to standard model particles can reheat efficiently G. Dvali and H. Tye, hep-ph/

6 Brane Inflation Expansion of the universe driven by the energy of the inflaton field. Inflaton field identified with inter-brane separation. Parallel branes experience no force. V (y) = (1 NS NS ± 1 RR ) V 0 (y) The interaction between branes gives the inflaton its potential. A brane and an anti-brane experience a Coulonb-type attractive force. V (y) = C V 0 y n Figure: There in no interaction between parallel branes. The interaction between a brane and an anti-brane is attractive.

7 Stabilization of the compactification Stabilizing the compactification (GKP, hep-th/ ) produces a warped geometry. D3 branes sink to the bottom of the warped region.

8 KKLT Stabilizing the compactification produces AdS vacua, hep-th/ Use euclidean D3 branes or the gaugino condensation on wrapped D7 branes to generate the superpotential: W = w 0 + Ae iaρ The volume modulus has a non-trivial Kähler potential: K = 3 ln [i (ρ ρ)] The F-term potential has the form: V F = e K [ K ab D a W D a W 3 W 2] V F has a negative niminum. One must add anti-branes to obtain Minkowski space.

9 Uplifting An anti-brane breaks SUSY and contributes a positive term to the potential. Its contribution is affected by the warp factor. V D3 = T 3h 4 A σ 2 An anti-brane prefers to move to the bottom of the warped throat. V = aae aσ 2σ 2 ( ) 1 3 σaae aσ + W 0 + Ae aσ + N T 3hA 4 σ 2

10 Adding mobile branes If we now add brane-anti-brane pairs to this geometry: The anti-branes will sink to the bottom of the throat. They will uplift the potential to a ds minimum realizing Inflation. The mobile branes will be attracted towards the anti-branes, and will move towards the bottom of the throat. Inflation ends when the branes and the anti-branes collide and annihilate. However: The Kahler potential for the mobile branes depends on the volume modulus: K = 3 ln [ i (ρ ρ) + k ( φ, φ )] This introdces a new η-problem.

11 Moduli Stabilization and the η Problem KKLMMT, hep-th/ , argue that the potential for the inflaton takes the form: V = V 0 (σ c ) (σ 1 3 ϕ 2) 2 V 0 (σ c ) ( ) ϕ 2 This results in a value η = 2/3, so the slow-roll condition is not satisfied. A possible solution was suggested by Baumann et.al. arxiv: : It is possible to find a value of the field where the η vanishes.

12 The curse of the inflexion point Take into account the embedding of the D7 in the CY. The superpotential take the form: ( ) f (φ) 1/n W = w + A 0 e aρ f (0) The corresponding F-term potential features an inflexion point. Most of the e-folds come from the region around the inflexion point. Brane-anti-brane interaction also gives an inflexion point: V Coulomb = k ) 2 (σ ϕ 2 3 (φ φ0 ) 4

13 Inflationary Dynamics The effective action has the form: S = d 3 xdt ( ) 1 g 2 G ij µ φ i µ φ j V + R Assuming an FRW 4D metric the equations of motion are: φ k + 3H φ k ka V + G φ a + Γk φ ij i φj = 0 with the Hubble constant given by: H 2 = 1 3 ( ) G ij φ i φ j + V Γ k ij are the connection coefficients obtained from the target space metric G ij.

14 Density Perturbations The power spectrum of a multi-field inflationary model is given by: V ij N N P (k) = 75π 2 MP 2 G φ i φ j N=60 (M. Sasaki and E. D. Stewart, astro-ph/ ). The COBE normalization implies P (k 0 ) at the scale k Mpc. In order to obtain the right amplitude for the density fluctuations, we have to rescale the potential, which amounts to rescaling the string mass. In our model we find: which is about M GUT. M S GeV

15 Multibrane trajectories

16 Signatures of Multiple Branes We now consider the most general perturbations around the background: For the metric: g µν + δg µν = ( (1 + 2A (t, x)) a (t) B,i (t, x) ) a (t) B,i (t, x) a 2 (t) [(1 2ψ (t, x)) δ ij + 2E,ij (t, x)] and the fields φ I as: φ I = φ I (t) + δφ I (t, x)

17 Perturbations of the Scalar field The equation for the scalr field perturbations: δφ K + 3ȧ δφ a K + k2 KI V = 2AG φ I + [ a 2 δφk + G KI 2 V G KI φ I + φj φ J ] φ K [Ȧ + 3 ψ + k 2 Ė k2 a B 2Γ K IJφ I δφ J ΓK IJ φ L φ I φ J δφ L = 0 ] V φ I δφ J and similarly for the metric fluctuations. The equation simplifies if we use the Mukhanov-Sasaki variables.

18 Mukhanov-Sasaki Variables Q I δφ I φ + I H ψ For these varibles the equation of motion becomes: Q K + 3HQ K + k2 a 2 QK + 2Γ K IJφ I Q J + [ G KI 2 V G KI V φ I + φj φ J φ I + ΓK IJ φ L φ I φ L ( ) ( ) 8πG a 3 φ K φ I a 3 G IJ t H ( ) ( ) ] 8πG 8πG Γ ILJ φ K φ I φ L Γ KLM H H φ L φ M G IJ φ I Q J = 0

19 Initial Conditions We take the initial conditions to be given by the Buch-Davies vacuum Qk I (η) = H 2k 3 e iηk In practical terms we evolve the each mode from just inside the horizon until the end of inflation. There is some non-trivial evolution outside the horizon:

20 Adiabatic and Entropy Directions Velocities of the background fields define a vector in field space: ( ) φ = φ 1... φ N One can now decompose the M-S vector Q in two components: adiabatic φ Qσ = G IJφ I Q J + G kl φk φl G H ψ KL φk φl entropy φ δs = «G ij»q i G kl φ k δφ l Gmn φ m φ n s Qp «G ab φ a Q b G cd φ c φ d φ i Q j We can now define the curvature and entropy perturbations φ p

21 Adiabatic and Entropy Spectra The adiabatic and entropy perturbations are defined as: R = S = H G ij φi φj Q σ H G ij φ i φ j δs The corresponding spectra for the two variables are: P R (k) = k3 R (k) 2 2π2 P S (k) = k3 S (k) 2 2π2

22 Results Field trajectories, adiabatic spectrum.

23 More Results Entropy spectra for different brane trajectories. Left: non-coincident branes. Right: coincident branes.

24 Conclusions The Brane World model accomodates both inflation and the Standard Model It also offers a natural mechanism for ending Inflation. The potential features an inflexion point: η vanishes there. Many models fit the present data. Entropy perturbations are too small to be observable.