Observing the Dimensionality of Our Parent Vacuum

Transcription

1 Observing the Dimensionality of Our Parent Vacuum Surjeet Rajendran, Johns Hopkins with Peter Graham and Roni Harnik arxiv:

2 Inspiration Why is the universe 3 dimensional? What is the overall shape and structure of the universe?

3 Inspiration Why is the universe 3 dimensional? What is the overall shape and structure of the universe? Do we have a landscape of vacua, extra dimensions? Does this explain properties of our universe (e.g. the Cosmological Constant)? Did we have a period of eternal inflation in our past? Is our universe a vast, inhomogeneous multiverse?

4 Inspiration Why is the universe 3 dimensional? What is the overall shape and structure of the universe? Do we have a landscape of vacua, extra dimensions? Does this explain properties of our universe (e.g. the Cosmological Constant)? Did we have a period of eternal inflation in our past? Is our universe a vast, inhomogeneous multiverse? How will we know?

5 Outline 1. Motivation 2. The Anisotropic Universe 3. Observables

6 Lower Dimensions Lower dimensional vacua seem generic Even SM has a landscape of lower dimensional vacua Arkani-Hamed et. al. (2008) More ways to compactify more dimensions We assume our universe came from a lower dimensional vacuum Possibly all dimensions began compact? Dimensions tend to decompactify Brandenberger & Vafa (1989) Giddings & Myers (2004)

7 Lower Dimensions Lower dimensional vacua seem generic Even SM has a landscape of lower dimensional vacua Arkani-Hamed et. al. (2008) More ways to compactify more dimensions We assume our universe came from a lower dimensional vacuum Possibly all dimensions began compact? Dimensions tend to decompactify Brandenberger & Vafa (1989) Giddings & Myers (2004) Creates large initial anisotropy, diluted by slow-roll inflation We will look for residual signs of this special direction

8 Landscape Signals For signals to be observable, inflation must not have lasted too long. Inflation needs tuning. Few e-folds may be generic. Many landscape signals require this e.g. curvature, bubble collisions Aguirre, Johnson & Shomer (2007), Chang, Kleban & Levi (2007)

9 Landscape Signals For signals to be observable, inflation must not have lasted too long. Inflation needs tuning. Few e-folds may be generic. Many landscape signals require this e.g. curvature, bubble collisions Aguirre, Johnson & Shomer (2007), Chang, Kleban & Levi (2007) These also assume other vacua are 3+1 dimensional What if we relax this assumption? These signals could reveal our history of decompactification (see also Blanco-Pillado and Salem, 2010)

10 The Anisotropic Universe

11 Initial Transition If the parent vacuum is 2+1 dimensional Coleman - De Luccia tunneling creates a bubble of 3+1 dimensional space Could be radion tunneling (or change in fluxes, etc.)

12 Initial Transition If the parent vacuum is 2+1 dimensional Coleman - De Luccia tunneling creates a bubble of 3+1 dimensional space Could be radion tunneling (or change in fluxes, etc.) spatial dimensions = R 2 S 1 y x z x

13 Initial Transition If the parent vacuum is 2+1 dimensional Coleman - De Luccia tunneling creates a bubble of 3+1 dimensional space Could be radion tunneling (or change in fluxes, etc.) y spatial dimensions = R 2 S 1 t x z x x creates an infinite, open FRW universe, in 2 dimensions negative curvature only in 2 dimensions, third dimension flat

14 Initial Transition Alternatively, if the parent vacuum is 1+1 dimensional: The single uncompactified dimension is flat Other two may be any compact 2-manifold with geometry S 2, E 2, or H 2 Generic compactifications have large curvature

15 Initial Transition Alternatively, if the parent vacuum is 1+1 dimensional: The single uncompactified dimension is flat Other two may be any compact 2-manifold with geometry S 2, E 2, or H 2 Generic compactifications have large curvature We won t consider the 0+1 dimensional case. In general, expect anisotropic curvature after transition

16 After the Transition We assume after the transition: dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 b(t) 2 dz 2 k = ±1

17 After the Transition We assume after the transition: dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 FRW equations: ȧ 2 a 2 +2ȧ a ḃ b + k a 2 = 8πGρ ä a + b b + ȧ a ḃ b = 8πGp r 2ä a + ȧ2 a 2 + k a 2 = 8πGp z b(t) 2 dz 2 k = ±1

18 After the Transition We assume after the transition: dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 b(t) 2 dz 2 k = ±1 FRW equations: ȧ 2 a 2 +2ȧ a ḃ b + k a 2 = 8πGρ ä a + b b + ȧ a ḃ b = 8πGp r 2ä a + ȧ2 a 2 + k a 2 = 8πGp z z-dimension is flat anisotropic curvature

19 After the Transition We assume after the transition: dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 b(t) 2 dz 2 k = ±1 FRW equations: ȧ 2 a 2 +2ȧ a ḃ b + k a 2 = 8πGρ ä a + b b + ȧ a ḃ b = 8πGp r 2ä a + ȧ2 a 2 + k a 2 = 8πGp z z-dimension is flat anisotropic curvature anisotropic pressure

20 After the Transition We assume after the transition: dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 b(t) 2 dz 2 k = ±1 FRW equations: ȧ 2 a 2 +2ȧ a ḃ b + k a 2 = 8πGρ ä a + b b + ȧ a ḃ b = 8πGp r 2ä a + ȧ2 a 2 + k a 2 = 8πGp z z-dimension is flat anisotropic curvature anisotropic pressure normal FRW eqn

21 After the Transition We assume after the transition: dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 b(t) 2 dz 2 k = ±1 FRW equations: ȧ 2 a 2 +2ȧ a ḃ b + k a 2 = 8πGρ ä a + b b + ȧ a ḃ b = 8πGp r 2ä a + ȧ2 a 2 + k a 2 = 8πGp z z-dimension is flat anisotropic curvature anisotropic pressure normal FRW eqn anisotropic curvature anisotropic expansion: H a ȧ a = H b ḃ b

22 Curvature Dominance Assume immediately after the tunneling ḃ 0 t a(t) t 1+O G Λ t 2 b(t) b 0 1+O G Λ t 2 inflation so b(t) is frozen H b 0 x but H a is large

23 Curvature Dominance Assume immediately after the tunneling ḃ 0 t a(t) t 1+O G Λ t 2 b(t) b 0 1+O G Λ t 2 inflation so b(t) is frozen H b 0 x but H a is large a(t) will expand, diluting curvature until t 2 G Λ when Ω k < Ω Λ slow-roll inflation takes over and drives all dimensions to expand H b H a

24 Evolution of the Anisotropic Universe normal FRW eqn: 2 H a +3H 2 a + k a 2 = 8πGp z our universe is approximately isotropic: H H a H b H Ω k 1 and

25 Evolution of the Anisotropic Universe normal FRW eqn: 2 H a +3H 2 a + k a 2 = 8πGp z our universe is approximately isotropic: H H a H b H Ω k 1 and d dt H +3H a H + k a 2 =8πG (p r p z ) 0

26 Evolution of the Anisotropic Universe normal FRW eqn: 2 H a +3H 2 a + k a 2 = 8πGp z our universe is approximately isotropic: H H a H b H Ω k 1 and d dt H +3H a H + k a 2 =8πG (p r p z ) 0 thermal equilibrium isotropic pressure and during MD pressure is small enough p p H H

27 Evolution of the Anisotropic Universe normal FRW eqn: 2 H a +3H 2 a + k a 2 = 8πGp z our universe is approximately isotropic: H H a H b H Ω k 1 and d dt H +3H a H + k a 2 =8πG (p r p z ) 0 thermal equilibrium isotropic pressure and during MD pressure is small enough p p H H eqn for b(t): 2 H b +3H 2 b k a 2 = 8πGp a(t) expands normally, b(t) expands as if curvature was opposite sign

28 inhomogeneous solutions are: Return of Curvature d dt H +3H a H + k a 2 =0 Ω k = k a 2 H 2 Inflation RD MD H H a H H a H H a = Ω k = 1 3 Ω k = 2 5 Ω k homogeneous solutions sourced at every transition but die off quickly

29 Return of Curvature d dt H +3H a H + k a 2 =0 Ω k = k a 2 H 2 inhomogeneous solutions are: Inflation RD MD H H a H H a H H a = Ω k = 1 3 Ω k = 2 5 Ω k ~ 1 in curvature dominance ~ e -120 after inflation ~ 10-5 at recombination ~ 10-2 today homogeneous solutions sourced at every transition but die off quickly

30 Return of Curvature d dt H +3H a H + k a 2 =0 Ω k = k a 2 H 2 inhomogeneous solutions are: Inflation RD MD H H a H H a H H a = Ω k = 1 3 Ω k = 2 5 Ω k ~ 1 in curvature dominance ~ e -120 after inflation ~ 10-5 at recombination ~ 10-2 today homogeneous solutions sourced at every transition but die off quickly a(t) t Ω k we need the full 5 solutions during MD: b(t) t Ω k 5

31 Observables

32 Measuring Curvature Sound horizon at recombination provides a standard ruler

33 Measuring Curvature Sound horizon at recombination provides a standard ruler Curvature is measured by observing angular size of ruler ~ 1 a high-l observable

34 Measuring Curvature Sound horizon at recombination provides a standard ruler Curvature is measured by observing angular size of ruler ~ 1 a high-l observable What does anisotropic curvature look like?

35 Standard Rulers dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 universe roughly flat before recombination rulers are fixed physical length ds z b(t) 2 dz 2 0 r

36 Standard Rulers dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 universe roughly flat before recombination rulers are fixed physical length ds z b(t) 2 dz 2 0 r

37 Standard Rulers dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 universe roughly flat before recombination rulers are fixed physical length ds z b(t) 2 dz 2 0 Δθ r

38 Standard Rulers dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 universe roughly flat before recombination rulers are fixed physical length ds z b(t) 2 dz 2 0 Δθ r transform to locally flat frame observable angle is tan (θ) = a (t) b (t) dr dz + O 1m 28 Gpc

39 Standard Rulers dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 b(t) 2 dz 2 universe roughly flat before recombination rulers are fixed physical length ds z θ Ω k θ Ω k 0 r θ Ω k transform to locally flat frame observable angle is tan (θ) = a (t) b (t) dr dz + O Should be easier to measure than isotropic curvature 1m 28 Gpc

40 Effect of Geometric Warp

41 Effect of Geometric Warp

42 Effect of Geometric Warp

43 CMB Flux today = CMB Flux z Φ 0 = dn 0 dω 0 da 0 dt 0 de 0 P 0 r

44 CMB Flux today = CMB Flux z Φ 0 = dn 0 dω 0 da 0 dt 0 de 0 P Anisotropic Curvature: 1. Non-sphericity of LSS 2. Bending of photon path 3. Angle dependent redshift Late time effect acts on all multipoles 0 r

45 CMB Flux today = CMB Flux z Φ 0 = dn 0 dω 0 da 0 dt 0 de 0 P Anisotropic Curvature: 1. Non-sphericity of LSS 2. Bending of photon path 3. Angle dependent redshift Late time effect acts on all multipoles 0 r

46 CMB Flux today = CMB Flux z Φ 0 = dn 0 dω 0 da 0 dt 0 de 0 P Anisotropic Curvature: 1. Non-sphericity of LSS 2. Bending of photon path 3. Angle dependent redshift Late time effect acts on all multipoles 0 r

47 CMB Flux today = CMB Flux z Φ 0 = dn 0 dω 0 da 0 dt 0 de 0 P Anisotropic Curvature: 1. Non-sphericity of LSS 2. Bending of photon path 3. Angle dependent redshift 0 r Late time effect acts on all multipoles Φ 0 (E 0, θ 0 )=Φ P (E P, θ P ) dωp dω 0 dap dt P da 0 dt 0 dep de 0

48 CMB Flux today = CMB Flux z Φ 0 = dn 0 dω 0 da 0 dt 0 de 0 P Anisotropic Curvature: 1. Non-sphericity of LSS 2. Bending of photon path 3. Angle dependent redshift 0 r Late time effect acts on all multipoles Φ 0 (E 0, θ 0 )=Φ P (E P, θ P ) dωp dω 0 dap dt P da 0 dt 0 a 2 b dep de 0

49 CMB Flux today = CMB Flux z Φ 0 = dn 0 dω 0 da 0 dt 0 de 0 P Anisotropic Curvature: 1. Non-sphericity of LSS 2. Bending of photon path 3. Angle dependent redshift 0 r Late time effect acts on all multipoles Φ 0 (E 0, θ 0 )=Φ P (E P, θ P ) dωp dω 0 dap dt P da 0 dt 0 dep de 0 Φ 0 (E 0, θ 0 )= exp E 2 0 E 0 T LSS (θ 0 +δθ) (1 + Ω k 0 Y 20 (θ 0 )) 1

50 CMB Flux today = CMB Flux z Φ 0 = dn 0 dω 0 da 0 dt 0 de 0 P Anisotropic Curvature: 1. Non-sphericity of LSS 2. Bending of photon path 3. Angle dependent redshift 0 r Late time effect acts on all multipoles Φ 0 (E 0, θ 0 )=Φ P (E P, θ P ) dωp dω 0 dap dt P da 0 dt 0 dep de 0 Φ 0 (E 0, θ 0 )= exp E 2 0 E 0 T LSS (θ 0 +δθ) (1 + Ω k 0 Y 20 (θ 0 )) 1 #1

51 CMB Flux today = CMB Flux z Φ 0 = dn 0 dω 0 da 0 dt 0 de 0 P Anisotropic Curvature: 1. Non-sphericity of LSS 2. Bending of photon path 3. Angle dependent redshift 0 r Late time effect acts on all multipoles Φ 0 (E 0, θ 0 )=Φ P (E P, θ P ) dωp dω 0 dap dt P da 0 dt 0 dep de 0 Φ 0 (E 0, θ 0 )= exp E 2 0 E 0 T LSS (θ 0 +δθ) (1 + Ω k 0 Y 20 (θ 0 )) 1 #1 #2

52 CMB Flux today = CMB Flux z Φ 0 = dn 0 dω 0 da 0 dt 0 de 0 P Anisotropic Curvature: 1. Non-sphericity of LSS 2. Bending of photon path 3. Angle dependent redshift 0 r Late time effect acts on all multipoles Φ 0 (E 0, θ 0 )=Φ P (E P, θ P ) dωp dω 0 dap dt P da 0 dt 0 dep de 0 Φ 0 (E 0, θ 0 )= exp E 2 0 E 0 T LSS (θ 0 +δθ) (1 + Ω k 0 Y 20 (θ 0 )) 1 #1 #2 #3

53 The Quadrupole Φ 0 (E 0, θ 0 )= exp E 2 0 E 0 T LSS (θ 0 +δθ) (1 + Ω k 0 Y 20 (θ 0 )) 1 T = T + lm a lm Y lm

54 The Quadrupole Φ 0 (E 0, θ 0 )= exp E 2 0 E 0 T LSS (θ 0 +δθ) (1 + Ω k 0 Y 20 (θ 0 )) 1 T = T + lm a lm Y lm only redshift affects the monopole

55 The Quadrupole Φ 0 (E 0, θ 0 )= exp E 2 0 E 0 T LSS (θ 0 +δθ) (1 + Ω k 0 Y 20 (θ 0 )) 1 T = T + lm a lm Y lm only redshift affects the monopole contribution to CMB quadrupole anisotropy: a 20 8 π 15 5 Ω k 0 T tuning likely range ~ 10 4 Ω k ~ cosmic variance low-l multipoles have high cosmic variance local ISW effect may raise quadrupole Francis & Peacock (2009), WMAP7 (2010)

56 Angular Correlations a lm = size of temperature fluctuation in Y lm mode statistical isotropy except a l1 m 1 a l 2 m 2 =0 a lm a lm C l

57 Angular Correlations a lm = size of temperature fluctuation in Y lm mode statistical isotropy except a l1 m 1 a l 2 m 2 =0 a lm a lm C l (1 + Ω k0 # lm ) a lm a l 2,m Ω k0 (C l 2 # lm + C l # lm ) anisotropic curvature

58 Angular Correlations a lm = size of temperature fluctuation in Y lm mode statistical isotropy except a l1 m 1 a l 2 m 2 =0 a lm a lm C l (1 + Ω k0 # lm ) a lm a l 2,m Ω k0 (C l 2 # lm + C l # lm ) anisotropic curvature a good measure of anisotropy: A LM ll = mm a lm a l m ( 1) m C LM l,m,l, m = 0 for isotropic

59 Angular Correlations a lm = size of temperature fluctuation in Y lm mode statistical isotropy except a l1 m 1 a l 2 m 2 =0 a lm a lm C l (1 + Ω k0 # lm ) a lm a l 2,m Ω k0 (C l 2 # lm + C l # lm ) anisotropic curvature a good measure of anisotropy: A LM ll = mm a lm a l m ( 1) m C LM l,m,l, m = 0 for isotropic anisotropic curvature gives: A 20 ll Ω k0 C l l A 20 l,l 2 Ω k0 (l (C l C l 2 )+C l ) l These are our high-l observables - low cosmic variance

60 WMAP Anomaly WMAP sees only two nonzero: A 20 ll and A 20 l,l 2 More precision than isotropic curvature, no degeneracy with scale factor expansion history possibly due to instrumental systematics Planck should improve measurement

61 Is it just another anisotropy?

62 Is it just another anisotropy? Symmetries of Bubble Nucleation => Specific initial geometry dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 b(t) 2 dz 2

63 Is it just another anisotropy? Symmetries of Bubble Nucleation => Specific initial geometry dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 b(t) 2 dz 2 Power in just one linearly independent harmonic e.g. Y20

64 Is it just another anisotropy? Symmetries of Bubble Nucleation => Specific initial geometry dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 b(t) 2 dz 2 Power in just one linearly independent harmonic e.g. Y20 Expect power in all harmonics for generic anisotropy.

65 Is it just another anisotropy? Symmetries of Bubble Nucleation => Specific initial geometry dr ds 2 = dt 2 a(t) kr 2 + r2 dφ 2 b(t) 2 dz 2 Power in just one linearly independent harmonic e.g. Y20 Expect power in all harmonics for generic anisotropy. Symmetries valid in thin wall regime. Thick wall?

66 Signals of Compact Topology Eternal inflation seems to imply space is infinite But we re led to finite, compact topology in at least one dimension

67 Signals of Compact Topology Eternal inflation seems to imply space is infinite But we re led to finite, compact topology in at least one dimension Observe matched circles in the sky Cornish, Spergel & Starkman (1996) current limit = 24 Gpc may get to ~ 28 Gpc diameter of our universe

68 Signals of Compact Topology Eternal inflation seems to imply space is infinite But we re led to finite, compact topology in at least one dimension Observe matched circles in the sky Cornish, Spergel & Starkman (1996) current limit = 24 Gpc may get to ~ 28 Gpc diameter of our universe 2+1 dimensional parent: curvature and topology are in different directions 1+1 dimensional: same directions

69 Other Measurements CMB is a snapshot - only 2 dimensional information 3D info can directly distinguish anisotropy from inhomogeneity 21 cm and galaxy surveys 21 cm can observe curvature to Ωk ~ 10-4

70 Other Measurements CMB is a snapshot - only 2 dimensional information 3D info can directly distinguish anisotropy from inhomogeneity 21 cm and galaxy surveys 21 cm can observe curvature to Ωk ~ 10-4 Quadrupole from anisotropy generates correlated E-mode polarization.

71 Other Measurements CMB is a snapshot - only 2 dimensional information 3D info can directly distinguish anisotropy from inhomogeneity 21 cm and galaxy surveys 21 cm can observe curvature to Ωk ~ 10-4 Quadrupole from anisotropy generates correlated E-mode polarization. Anisotropic curvature also causes differential Hubble expansion H Ω k H Visible directly in Hubble measurements Current limits ~ few % May improve to < 10-2 with e.g. GW sirens Schutz (2001)

72 Conclusions + Future Directions Have high-l, low cosmic variance, observables of dimension changing transitions Due to late time effect of anisotropic curvature Not statistical predictions, though provide evidence for landscape/eternal inflation Can test an observation of curvature for isotropy Anisotropy implies lower dimensional parent vacuum Isotropy is evidence for 3+1 dimensional parent vacuum Interesting to explore dimension changing transitions Other observables, e.g. bubble collisions, gravitational waves? Does the landscape provide a reason for 3+1 dimensions?