ASPECTS OF D-BRANE INFLATION IN STRING COSMOLOGY

Size: px

Start display at page:

Download "ASPECTS OF D-BRANE INFLATION IN STRING COSMOLOGY"

Jessie Jackson
6 years ago
Views:

1 Summer Institute Fujiyoshida August 5, 2011 ASPECTS OF D-BRANE INFLATION IN STRING COSMOLOGY Takeshi Kobayashi (RESCEU, Tokyo U.)

2 TODAY S PLAN Cosmic Inflation and String Theory D-Brane Inflation and its Difficulties arxiv: with Shunichiro Kinoshita and Shinji Mukohyama Inflation from Rapid-Rolling D-Branes arxiv: with Shinji Mukohyama arxiv: with Shinji Mukohyama and Brian A. Powell Curvatons in Warped Throats arxiv: with Shinji Mukohyama arxiv: with Masahiro Kawasaki and Fuminobu Takahashi

3 COSMIC INFLATION image: NASA/WMAP Science Team

4 COSMIC INFLATION homogeneous and isotropic flat without unwanted relics tiny inhomogeneities image: NASA/WMAP Science Team

5 TOWARDS MICROSCOPIC REALIZATION Inflaton : a scalar field which acts like vacuum energy V (φ) slow-roll conditions: M 2 p V 2, η M 2 V p 1 2 V V φ

6 FUNDAMENTAL QUESTIONS What is the inflaton? What microphysics governed the inflationary universe? UV-sensitivity of inflationary cosmology How valid is the effective field theory description?

7 FUNDAMENTAL QUESTIONS What is the inflaton? What microphysics governed the inflationary universe? UV-sensitivity of inflationary cosmology How valid is the effective field theory description? String Theory may help!

8 STRING COSMOLOGY Good things for Cosmology allows description of the universe at high energy provides new ideas, new ingredients, and new ways of thinking about the early universe Good things for String Theory may provide tests of string theory through cosmological observables

9 STRING THEORY quantum gravity candidate based on closed / open strings

10 STRING THEORY quantum gravity candidate based on closed / open strings open strings end on Dp-branes, i.e. physical objects with p spatial dimensions

11 STRING THEORY quantum gravity candidate based on closed / open strings open strings end on Dp-branes, i.e. physical objects with p spatial dimensions string theory predicts 10 (or 11) spacetime dimensions the extra 6 (or 7) dimensions need to be compactified

12 COMPACTIFICATION OF TYPE IIB STRING THEORY Klebanov, Strassler 00 Gidding, Kachru, Polchinski 02 Kachru, Kallosh, Linde, Trivedi 03 warped compactification to 4-dim. ds with all moduli fixed via fluxes, brane sources, and nonperturbative effects extra 6-dim. space R-R flux warped throat region NS-NS flux ds 2 = h(y) 2 g (4) µν dx µ dx ν + h(y) 2 g (6) mndy m dy n

13 D-BRANE INFLATION IN A WARPED THROAT A brane-antibrane pair drives inflation. Kachru, Kallosh, Linde, Maldacena, McAllister, Trivedi 03 extra 6-dim. space warped throat D3 region D3

14 D-BRANE INFLATION : A CLOSER LOOK inflaton radial φ position of the D3 D3 D3 φ Coulomb interaction D3-D3 annihilation

15 INFLATON POTENTIAL φ : (normalized) D3 position h 0 : warp factor at the throat tip V (φ) =2h 4 0T 3 1 µ4 φ 4 D3-tension Coulomb interaction + H 2 φ 2 + from moduli stabilization V (φ) φ 2h 4 0T 3 φ

16 INFLATON POTENTIAL φ : (normalized) D3 position V (φ) =2h 4 0T 3 1 µ4 φ 4 D3-tension Coulomb interaction h 0 : warp factor at the throat tip + H 2 φ 2 + from moduli stabilization H 2 V 3M 2 p η = M 2 p V V 2 3 η-problem slow-roll inflation (unless delicate fine-tuning) Baumann et al

17 ALTERNATIVE APPROACH: DBI INFLATION Silverstein, Tong 04 Inflation from D-branes moving with relativistic velocities.

18 ALTERNATIVE APPROACH: DBI INFLATION Silverstein, Tong 04 Inflation from D-branes moving with relativistic velocities. However... Throat too short for a relativistically moving D3-brane to drive sufficient inflation. Baumann, McAllister 06 Lidsey, Huston 07

19 ALTERNATIVE APPROACH: DBI INFLATION Silverstein, Tong 04 Inflation from D-branes moving with relativistic velocities. However... Throat too short for a relativistically moving D3-brane to drive sufficient inflation. Baumann, McAllister 06 Lidsey, Huston 07 Not only for D3-branes, but also for higher-dimensional wrapped branes (D5 and D7). TK, Mukohyama, Kinoshita 07

20 ALTERNATIVE APPROACH: DBI INFLATION Silverstein, Tong 04 Inflation from D-branes moving with relativistic velocities. However... Throat too short for a relativistically moving D3-brane to drive sufficient inflation. Baumann, McAllister 06 Lidsey, Huston 07 Not only for D3-branes, but also for higher-dimensional wrapped branes (D5 and D7). TK, Mukohyama, Kinoshita 07 Relaxation of constraints by multi-field/galileon extensions? (talks by Tsutomu & Shuntaro yesterday)

21 SUMMARY SO FAR FOR D-BRANE INFLATION Slow-Roll Inflation suffers from the η-problem (i.e. Hubble size mass) DBI Inflation (relativistic limit) suffers from geometrical constraints

22 INFLATION FROM RAPID-ROLLING D-BRANES Kofman, Mukohyama 07 TK, Mukohyama 08 TK, Mukohyama, Powell 09 η O(1) An inflationary attractor solution DOES exist even for. This corresponds to D-branes moving rapidly, but non-relativistically. The inflaton is quickly decelerating / accelerating. ( slow-roll inf.)

23 RAPID-ROLL INFLATION L g = 1 2 gµν µ φ ν φ V (φ) M 2 p 2 V V 2 η M 2 p V V Hubble eq. EOM 3M 2 p H 2 = φ V φ +3H φ = V

24 RAPID-ROLL INFLATION L g = 1 2 gµν µ φ ν φ V (φ) M 2 p 2 V V 2 η M 2 p V V conditions 1 η < 3 4 Hubble eq. EOM 3M 2 p H 2 = φ V φ +3H φ = V 3M 2 p H 2 V ch φ V where c = η 2

25 RAPID-ROLL INFLATION L g = 1 2 gµν µ φ ν φ V (φ) M 2 p 2 V V 2 η M 2 p V V slow-roll recovered upon η 0 conditions 1 η < 3 4 Hubble eq. EOM 3M 2 p H 2 = φ V φ +3H φ = V 3M 2 p H 2 V ch φ V where c = η 2

26 INFLATION FROM RAPID-ROLLING D3-BRANE φ L = 1 g 2 gµν µ φ ν φ V 0 1+O(1) φ2 Mp 2 µ4 φ 4 V (φ) h 0 /α 1/2 φ inflationary attractor φ H 6η η φ

27 INFLATION FROM RAPID-ROLLING D3-BRANE φ L = 1 g 2 gµν µ φ ν φ V 0 1+O(1) φ2 Mp 2 µ4 φ 4 V (φ) φ h 0 /α 1/2 inflationary attractor φ V 1/ φ H 1.0 6η η φ φ/m p

28 INFLATION FROM RAPID-ROLLING D3-BRANE φ L = 1 g 2 gµν µ φ ν φ V 0 1+O(1) φ2 Mp 2 µ4 φ 4 V (φ) 1.0 h 0 /α 1/2 inflationary attractor φ H φ φ V 1/2 0 strong warping gives sufficient inflationary period η η φ φ/m p

29 INFLATION FROM RAPID-ROLLING D3-BRANE D3-D3 system in a warped geometry can give rise to rapid-roll inflation no need to cancel the Hubble size mass remedy to the η-problem sufficient inflationary expansion can be obtained in a single throat geometrical constraints circumvented However, the inflaton itself cannot produce scale-invariant density perturbations. perturbations need to be generated by something else

30 Angular directions of warped throats can generate curvature perturbations σ during inflation TK, Mukohyama 10 and work in progress as multi-field inflation light fields modulating rapid-roll inflation can generate scale-invariant perturbations after inflation (as curvatons) TK, Mukohyama 09 D-branes oscillating in warped throats can be curvatons

Angular directions of warped throats can generate curvature perturbations σ during inflation TK, Mukohyama 10 and work in progress as multi-field inflation light fields

31 Angular directions of warped throats can generate curvature perturbations σ during inflation TK, Mukohyama 10 and work in progress as multi-field inflation light fields modulating rapid-roll inflation can generate scale-invariant perturbations after inflation (as curvatons) TK, Mukohyama 09 D-branes oscillating in warped throats can be curvatons

32 CURVATON MECHANISM curvaton : a light field (i.e. m 2 H 2 ) which generates density perturbations after inflation Linde, Mukhanov 97 Enqvist, Sloth 01 Lyth, Wands 01 Moroi, Takahashi m2 σ 2 log ρ ρ φ a 4 H σ δρ σ ρ σ a 3 log a

33 CURVATON MECHANISM curvaton : a light field (i.e. m 2 H 2 ) which generates density perturbations after inflation Linde, Mukhanov 97 Enqvist, Sloth 01 Lyth, Wands 01 Moroi, Takahashi m2 σ 2 log ρ ρ φ a 4 H σ δρ σ δρ σ ρ σ a 3 log a

34 CURVATON MECHANISM curvaton : a light field (i.e. m 2 H 2 ) which generates density perturbations after inflation Linde, Mukhanov 97 Enqvist, Sloth 01 Lyth, Wands 01 Moroi, Takahashi m2 σ 2 log ρ ρ φ a 4 H σ δρ σ δρ σ ρ σ a 3 δρ σ ρ log a

35 CURVATONS IN A WARPED THROAT 6D Bulk Deformed Conifold S 3 S 2 warp S 3 S 2 angular isometry

36 CURVATONS IN A WARPED THROAT 6D Bulk Deformed Conifold S 3 S 2 warp S 3 S 2 angular isometry Standard Model brane (anti-d3)

37 ANGULAR POTENTIAL isometry breaking bulk effects moduli stabilizing non-perturbative effects Standard Model brane (anti-d3) + warping small mass to the angular degrees of freedom of the SM brane = CURVATON σ eventually decays into other open string modes (reheating)

38 S = T 3 ACTION d 4 ξ det G µν 1 ΨiD/ Ψ T3 C 4 d 4 x g (4) ( σ) 2 + ψid/ ψ (m 2 bulk + m 2 np)σ 2 + g sm 1/2 α 3/2 h 3 0 m 2 bulk m2 np m 2 bulk + m2 np σ ψid/ ψ + m 2 bulk = h 2 0 g s Mα : bulk effects m 2 np = hλ 2 0 g s Mα : nonperturbative effects h 0 : warp factor at the tip ds 2 = h 2 g µν (4) dx µ dx ν + h 2 g mndx (6) m dx n

39 λ PARAMETER CONSTRAINTS : nonperturbative effects 6.0 masslessness m σ <H inf BBN g s =0.1 M pl α 1/2 = 300 h 0 = hence N = f NL < 100 H inf f NL M pl 4.5 subdom. σ dom : bulk effects

40 λ PARAMETER CONSTRAINTS : nonperturbative effects 6.0 masslessness m σ <H inf BBN g s =0.1 M pl α 1/2 = 300 D7-brane (Kuperstein embedding) 5.5 hence h 0 = 10 5 N = f NL < 100 H inf f NL M pl 4.5 subdom. σ dom KS throat : bulk effects

41 Here we have considered quadratic curvaton potentials, but density perturbations can be quite different for non-quadratic potentials. Kawasaki, TK, Takahashi 11 related works by Enqvist, Takahashi 08 Kawasaki, Nakayama, Takahashi 08

42 Here we have considered quadratic curvaton potentials, but density perturbations can be quite different for non-quadratic potentials. Kawasaki, TK, Takahashi 11 related works by Enqvist, Takahashi 08 Kawasaki, Nakayama, Takahashi 08

43 Curvatons with Non-Quadratic Potentials log ρ ρ φ : inflaton V H a 4 δρ σ ρ σ : curvaton σ a 3 log a ζ c 1 δρ σ ρ σ

44 Curvatons with Non-Quadratic Potentials log ρ non-uniform onset of oscillation ρ φ : inflaton V H a 4 δρ σ ρ σ : curvaton σ a 3 δh osc log a ζ c 1 δρ σ ρ σ

45 Curvatons with Non-Quadratic Potentials log ρ non-uniform onset of oscillation ρ φ : inflaton V H a 4 δρ σ ρ σ : curvaton σ a 3 δh osc log a ζ c 1 δρ σ ρ σ c 2 δh osc H osc Additional contributions to the density perturbations!

46 Density Perturbations P ζ = N H σ 2π 2 N σ = r 4+3r (1 X(σ osc )) 1 V (σ osc ) V (σ osc ) 3X(σ osc) σ osc V (σ osc ) V (σ ) r ρ σ ρ r X(σ osc ) horizon curvaton decay osc onset of curvaton oscillation 1 σosc V (σ osc ) 2(c 3) V 1 (σ osc ) : effects due to non-uniform onset of oscillation Non-Gaussianity fnl also modified.

47 towards the hilltop: strong enhancement of linear-order density pert. with mild increase of fnl fnl = O(10) even for a dominant curvaton

48 λ PARAMETER CONSTRAINTS : nonperturbative effects 6.0 masslessness m σ <H inf BBN g s =0.1 M pl α 1/2 = 300 D7-brane (Kuperstein embedding) 5.5 hence h 0 = 10 5 N = f NL < 100 H inf f NL M pl 4.5 subdom. σ dom KS throat : bulk effects

49 λ PARAMETER CONSTRAINTS : nonperturbative effects 6.0 masslessness m σ <H inf BBN g s =0.1 M pl α 1/2 = 300 D7-brane (Kuperstein embedding) f NL < 100 hence Non-quadratic curvaton in a h 0 = 10 5 N = H inf f NL M pl warped throat can further σ dom. broaden the parameter window. 4.5 subdom KS throat : bulk effects

50 Additional features can show up on the spectral tilt when the inflationary background is given by rapid-roll inflation. δρ ρ H inf

51 SPECTRAL TILT IN RAPID-ROLL INFLATION The Hubble parameter in rapid-roll inflation does NOT possess a hierarchy amongst its higher-order time derivatives: 1 H d dt ln H 1 H d dt 2 ln H 1 H d dt 3 ln H cf. SLOW-ROLL INFLATION: 1 H d dt ln H 1 H d dt 2 ln H 1 H d dt 3 ln H

52 SPECTRAL TILT IN RAPID-ROLL INFLATION The Hubble parameter in rapid-roll inflation does NOT possess a hierarchy amongst its higher-order time derivatives: 1 H d dt ln H 1 H d dt 2 ln H 1 H d dt 3 ln H The resulting curvature perturbation spectrum can have large running (and its running, and so on). d d ln k ln δρ ρ d 2 δρ d(ln k) 2 ln d 3 δρ, ρ, d(ln k) 3 ln ρ,

53 CONSTRAINTS ON RAPID-ROLL INFLATION P (k) H 2 When, #$#) TK, Mukohyama, Powell 09 P (k) = P (k 0) 1+A 1+A n s (k 0 ) 1= AB 1+A k k 0 B dns/d ln k #$#* #$#+ #$#' #!#$#'!#$#+!#$#* dn s d ln k (k 0)= AB2 (1 + A) 2 Tightly constrained by CMB & LSS data.!#$#)!#$#&!#$#( #$% #$%& ' '$#& '$'! n s =1+d ln P (k)/d ln k

54 CONSTRAINTS ON RAPID-ROLL INFLATION P (k) H 2 When, #$#) TK, Mukohyama, Powell 09 P (k) = P (k 0) 1+A 1+A n s (k 0 ) 1= AB 1+A k k 0 B dns/d ln k #$#* #$#+ #$#' #!#$#'!#$#+!#$#* dn s d ln k (k 0)= AB2 (1 + A) 2 Tightly constrained by CMB & LSS data.!#$#)!#$#&!#$#( #$% #$%& ' '$#& '$'! distinguishing features especially at small scales n s =1+d ln P (k)/d ln k

55 SUMMARY Slow-roll and relativistic limits of D-brane inflation suffer from the η-problem and geometrical constraints, respectively. Rapid-rolling D-branes exhibit stable inflationary attractors, as well as provide sufficient inflationary expansions. Rapid-roll inflationary backgrounds give rise to density perturbation spectra with running spectral index. Angular oscillations of D-branes at throat tips can source the primordial density perturbations through the curvaton mechanism. Various roles in inflationary cosmology can be shared by the many degrees of freedom that show up in string theory.

56 BACKUP SLIDES

57 INFLATON KINETIC TERM ρ throat geometry ds 2 = h(ρ) 2 g (4) µν dx µ dx ν + h(ρ) 2 (dρ 2 + ρ 2 dσ 2 X 5 ) DBI action inflaton dφ = T 3 dρ S DBI = = = T 3 T 3 T 3 d 4 x det(g µν ) d 4 x g (4) h 4 1+h 4 g (4)αβ α ρ β ρ d 4 x g h (4) g(4)αβ α ρ β ρ d 4 x g T (4) 3 h g(4)αβ α φ β φ

58 ALTERNATIVE APPROACH: DBI INFLATION Silverstein, Tong 04 Inflation driven by D-branes moving with relativistic velocities. ρ S DBI = ds 2 = h(ρ) 2 g (4) µν dx µ dx ν + h(ρ) 2 (dρ 2 + ρ 2 dσ 2 X 5 ) T 3 = T 3 d 4 x det(g µν ) d 4 x g (4) h(ρ) 4 1 ρ2 h(ρ) 4 The warping of the throat enforces the D-brane to slow down, regardless of the potential. ρ 2 h(ρ) 4 < 1

59 DBI INFLATION L = T (φ) 1 φ 2 g (4) T (φ) + T (φ) V (φ) T (φ) T 3 h(φ) 4 T (φ) inflation can occur for some, V (φ) a new stringy inflation mechanism a remedy to the η-problem (no need for slow-roll) produces large non-gaussianity can be tested in future experiments

60 NEW DIFFICULTIES: GEOMETRICAL CONSTRAINTS ON FIELD RANGE Baumann, McAllister 06 Lidsey, Huston 07 DBI inflation generates (too) large non-gaussianity. In order to suppress non-gaussianity down to a level consistent with WMAP data, the inflaton need to travel at least some field range. φ On the other hand, the inflaton field range is geometrically restricted by the throat length, which is restricted by the Planck mass. M 2 p = 2V 6 (2π) 7 gsα 2 4 > 2Vol(X 5)( φ) 6 (2π) 7 gsα 2 4 h 4 T3 3

61 NEW DIFFICULTIES: GEOMETRICAL CONSTRAINTS ON FIELD RANGE Baumann, McAllister 06 Lidsey, Huston 07 DBI inflation generates (too) large non-gaussianity. In order to suppress non-gaussianity down to a level consistent with WMAP data, the inflaton need to travel at least some field range. φ On the other hand, the inflaton field range is geometrically restricted by the throat length, which is restricted by the Planck mass. M 2 p = 2V 6 (2π) 7 gsα 2 4 > 2Vol(X 5)( φ) 6 (2π) 7 gsα 2 4 h 4 T3 3 CONTRADICTION! Throat too short for a relativistically moving D3-brane to drive sufficient inflation.

62 WRAPPED BRANES ρ TK, Mukohyama, Kinoshita 07 Becker, Leblond, Shandera 07 dφ T 1/2 3 dρ dφ T 1/2 d 2n ξ 1/2 det(g kl B kl ) 3+2n dρ effective field range increased for wrapped branes Geometrical constraint for DBI inflation can be solved!

63 BACKGROUND CHARGE A long throat is sufficient in any case. Large flux number required. tadpole-cancellation condition N = χ 24 χ : Euler number of a Calabi-Yau fourfold The large flux number required exceeds the largest known Euler number for a CY 4-fold χ = Klemm, Lian, Roan, Yau 1998

64 SLOW-ROLL AND RAPID-ROLL INFLATIONS V (φ) =V 0 1+µ φ2 M 2 p p = φ V 1/2 0 q = φ M p p 0.0 p q q µ =1/200 µ =1/3

65 COSMOLOGICAL CONSTRAINTS ON RAPID-ROLL WMAP ACBAR l(l+1)c l /2!"K ! = 0.001! = 0.05! = l n s

Non-Gaussianity from Curvatons Revisited

RESCEU/DENET Summer School @ Kumamoto July 28, 2011 Non-Gaussianity from Curvatons Revisited Takeshi Kobayashi (RESCEU, Tokyo U.) based on: arxiv:1107.6011 with Masahiro Kawasaki, Fuminobu Takahashi The